Extrema, Increase, Decrease
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- Increasing, Decreasing
- Extrema with Example 1
- Extrema with Example 2
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30

Inflection, Convex, Concave
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- Concavity & Convexity
- Inflection Points
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29

Vertical Asymptotes
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- Vertical Asymptotes
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31
- Exercise 32
- Exercise 33
- Exercise 34

Horizontal Asymptotes
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- Horizontal Asymptotes
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31
- Exercise 32
- Exercise 33
- Exercise 34
- Exercise 35
- Exercise 36
- Exercise 37

Oblique Asymptotes
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- Oblique Asymptotes
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14
- Exercise 15
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21
- Exercise 22
- Exercise 23
- Exercise 24
- Exercise 25
- Exercise 26
- Exercise 27
- Exercise 28
- Exercise 29
- Exercise 30
- Exercise 31
- Exercise 32
- Exercise 33
- Exercise 34
- Exercise 35
- Exercise 36

Curve Plotting
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Global Extrema
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[{"Name":"Extrema, Increase, Decrease","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Increasing, Decreasing","Duration":"14m 28s","ChapterTopicVideoID":1653,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1653.jpeg","UploadDate":"2019-10-29T13:10:42.7070000","DurationForVideoObject":"PT14M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this clip, I\u0027m going to talk about increasing and decreasing functions,"},{"Start":"00:04.770 ","End":"00:10.260","Text":"and I\u0027ll make use of a couple of sketches."},{"Start":"00:10.260 ","End":"00:14.500","Text":"Let\u0027s see. Here I\u0027ll draw 1 function,"},{"Start":"00:14.870 ","End":"00:19.260","Text":"call it f of x, and unrelated to"},{"Start":"00:19.260 ","End":"00:23.980","Text":"this f of x is another function which I\u0027m also going to call f of x."},{"Start":"00:26.690 ","End":"00:32.070","Text":"The other 1 will look something like this,"},{"Start":"00:32.070 ","End":"00:35.050","Text":"and I\u0027ll also call it f of x."},{"Start":"00:36.700 ","End":"00:42.350","Text":"If I take a walk along this function here from left to right,"},{"Start":"00:42.350 ","End":"00:44.150","Text":"and we always travel from left to right,"},{"Start":"00:44.150 ","End":"00:45.800","Text":"that\u0027s the positive direction."},{"Start":"00:45.800 ","End":"00:49.460","Text":"If I\u0027m taking a walk along this function,"},{"Start":"00:49.460 ","End":"00:51.770","Text":"I noticed that I\u0027m going upwards,"},{"Start":"00:51.770 ","End":"00:53.230","Text":"I\u0027m on an incline."},{"Start":"00:53.230 ","End":"00:55.995","Text":"Other words, if I\u0027m going,"},{"Start":"00:55.995 ","End":"01:00.355","Text":"let\u0027s say from this point to this point,"},{"Start":"01:00.355 ","End":"01:04.920","Text":"I notice that as I go from left to right,"},{"Start":"01:05.200 ","End":"01:12.075","Text":"the x increases, let\u0027s say this is x_1 and this is x_2."},{"Start":"01:12.075 ","End":"01:13.635","Text":"But as x increases,"},{"Start":"01:13.635 ","End":"01:16.960","Text":"so does y, which means that I\u0027m going upwards."},{"Start":"01:16.960 ","End":"01:21.285","Text":"If I take the corresponding y,"},{"Start":"01:21.285 ","End":"01:24.910","Text":"y_1 and y_2, y_2 is bigger than y_1."},{"Start":"01:24.910 ","End":"01:28.485","Text":"That\u0027s actually the definition of increasing for a function,"},{"Start":"01:28.485 ","End":"01:30.570","Text":"is that when the x increases,"},{"Start":"01:30.570 ","End":"01:34.460","Text":"so does the y. Graphically what it means is as I go along,"},{"Start":"01:34.460 ","End":"01:37.400","Text":"I\u0027m going in the direction from left to right,"},{"Start":"01:37.400 ","End":"01:39.800","Text":"of course, then I\u0027m going upwards."},{"Start":"01:39.800 ","End":"01:45.020","Text":"The converse applies to a function which is decreasing."},{"Start":"01:45.020 ","End":"01:50.360","Text":"If I\u0027m on traveling along this function and I\u0027m decreasing,"},{"Start":"01:50.360 ","End":"01:53.195","Text":"then as I go from, say,"},{"Start":"01:53.195 ","End":"01:58.715","Text":"this point and I continue up to this point, then once again,"},{"Start":"01:58.715 ","End":"02:07.035","Text":"the xs are increasing in both cases because I always travel from left to right,"},{"Start":"02:07.035 ","End":"02:09.950","Text":"same as in this graph."},{"Start":"02:09.950 ","End":"02:12.620","Text":"I also travel from left to right."},{"Start":"02:12.620 ","End":"02:14.575","Text":"But in this case,"},{"Start":"02:14.575 ","End":"02:19.915","Text":"if I look at the corresponding ys,"},{"Start":"02:19.915 ","End":"02:22.955","Text":"the first y was here,"},{"Start":"02:22.955 ","End":"02:25.460","Text":"and the second y from x_2,"},{"Start":"02:25.460 ","End":"02:27.530","Text":"that\u0027s y_2 is below it."},{"Start":"02:27.530 ","End":"02:30.470","Text":"In other words, in the case of an increasing function,"},{"Start":"02:30.470 ","End":"02:32.150","Text":"as I went from left to right here,"},{"Start":"02:32.150 ","End":"02:34.055","Text":"I went upwards also."},{"Start":"02:34.055 ","End":"02:38.160","Text":"But in this case, as I go along, I\u0027m going downwards."},{"Start":"02:43.970 ","End":"02:49.265","Text":"That\u0027s generally the idea of increasing and decreasing."},{"Start":"02:49.265 ","End":"02:53.720","Text":"Now it\u0027s all very well and easy to say what\u0027s increasing,"},{"Start":"02:53.720 ","End":"02:55.610","Text":"what\u0027s decreasing when you\u0027re given a sketch."},{"Start":"02:55.610 ","End":"02:59.255","Text":"I mean, I look at this and I just see by the shape that it\u0027s"},{"Start":"02:59.255 ","End":"03:03.080","Text":"increasing and I look at this 1 and"},{"Start":"03:03.080 ","End":"03:07.415","Text":"I just see from the general shape that this is decreasing."},{"Start":"03:07.415 ","End":"03:10.025","Text":"But in practice we\u0027re not going to get a sketch."},{"Start":"03:10.025 ","End":"03:13.610","Text":"We\u0027re going to get a definition of a function."},{"Start":"03:13.610 ","End":"03:16.940","Text":"We\u0027re going to get something like f of x equals"},{"Start":"03:16.940 ","End":"03:21.575","Text":"natural log of x plus x plus x squared or some formula."},{"Start":"03:21.575 ","End":"03:25.310","Text":"We\u0027re going to be given the definition of the function and not a picture."},{"Start":"03:25.310 ","End":"03:29.930","Text":"How then will we determine when the function is increasing or decreasing?"},{"Start":"03:29.930 ","End":"03:32.480","Text":"Well, in general, the answer is that we\u0027re going to use"},{"Start":"03:32.480 ","End":"03:35.315","Text":"the derivative to do this and I\u0027ll"},{"Start":"03:35.315 ","End":"03:41.695","Text":"show you now how are we going to tell increasing and decreasing by use of the derivative."},{"Start":"03:41.695 ","End":"03:44.600","Text":"1 last thing before I get to the derivative,"},{"Start":"03:44.600 ","End":"03:49.460","Text":"I just want to go over again the definition of increasing and decreasing."},{"Start":"03:49.460 ","End":"03:55.935","Text":"This is more of theoretical interest than practical is that we define, like I said,"},{"Start":"03:55.935 ","End":"04:03.255","Text":"increasing means that if x_1 is less than x_2,"},{"Start":"04:03.255 ","End":"04:05.700","Text":"means I\u0027m going from left to right,"},{"Start":"04:05.700 ","End":"04:09.810","Text":"then y_2 is bigger than y_1."},{"Start":"04:09.810 ","End":"04:13.330","Text":"In other words, y_1 is less than y_2."},{"Start":"04:13.330 ","End":"04:18.860","Text":"That\u0027s a general property for checking increasing even if we don\u0027t have a derivative,"},{"Start":"04:18.860 ","End":"04:21.145","Text":"but it\u0027s not used in practice that much."},{"Start":"04:21.145 ","End":"04:24.755","Text":"However, it is the definition generally of increasing."},{"Start":"04:24.755 ","End":"04:30.590","Text":"For decreasing, we have a similar definition that a function f is"},{"Start":"04:30.590 ","End":"04:37.080","Text":"decreasing means that if x_1 is less than x_2,"},{"Start":"04:37.080 ","End":"04:46.165","Text":"then y_1 is bigger than y_2."},{"Start":"04:46.165 ","End":"04:48.020","Text":"This is the difference."},{"Start":"04:48.020 ","End":"04:51.590","Text":"Here it\u0027s bigger, but here it\u0027s smaller."},{"Start":"04:51.590 ","End":"04:56.300","Text":"As we go from left to right, we go downwards."},{"Start":"04:56.300 ","End":"04:59.915","Text":"Now let\u0027s say this is sometimes used to check"},{"Start":"04:59.915 ","End":"05:05.270","Text":"theoretically when we don\u0027t have a derivative this property increasing and decreasing,"},{"Start":"05:05.270 ","End":"05:09.270","Text":"but we\u0027ll move now to the derivative."},{"Start":"05:09.580 ","End":"05:13.385","Text":"Let\u0027s look here at this graph,"},{"Start":"05:13.385 ","End":"05:17.790","Text":"and let\u0027s take 1 of these."},{"Start":"05:17.790 ","End":"05:22.410","Text":"I don\u0027t know, say this 1 as our x, not just x_1,"},{"Start":"05:22.410 ","End":"05:28.060","Text":"let\u0027s going to take it as my x. I want to draw a tangent to the graph at this point."},{"Start":"05:28.060 ","End":"05:33.305","Text":"Here let\u0027s say this line is the tangent line at this point."},{"Start":"05:33.305 ","End":"05:35.960","Text":"Then because the function is increasing,"},{"Start":"05:35.960 ","End":"05:38.630","Text":"the tangent is also increasing,"},{"Start":"05:38.630 ","End":"05:40.595","Text":"meaning it has a positive slope."},{"Start":"05:40.595 ","End":"05:42.935","Text":"This is quite clear, if this increases,"},{"Start":"05:42.935 ","End":"05:47.955","Text":"this is got to be upward sloping and not downwards,"},{"Start":"05:47.955 ","End":"05:53.330","Text":"and we know that if the equation of this tangent is a straight line,"},{"Start":"05:53.330 ","End":"05:58.819","Text":"so it will be of the form y equals ax plus b."},{"Start":"05:58.819 ","End":"06:04.300","Text":"Then we know that the slope is a and we also know"},{"Start":"06:04.300 ","End":"06:10.765","Text":"that a is given by the derivative of f at this point x."},{"Start":"06:10.765 ","End":"06:13.465","Text":"What this means is, if the function is increasing,"},{"Start":"06:13.465 ","End":"06:15.130","Text":"the tangent has a positive slope,"},{"Start":"06:15.130 ","End":"06:18.855","Text":"so this is positive or greater than 0,"},{"Start":"06:18.855 ","End":"06:21.320","Text":"and then so because they\u0027re equal,"},{"Start":"06:21.320 ","End":"06:22.975","Text":"f prime of x,"},{"Start":"06:22.975 ","End":"06:32.865","Text":"the derivative is also going to be positive or greater than 0.I won\u0027t sketch it,"},{"Start":"06:32.865 ","End":"06:34.690","Text":"but conversely on this graph here,"},{"Start":"06:34.690 ","End":"06:35.740","Text":"if we draw a tangent,"},{"Start":"06:35.740 ","End":"06:39.925","Text":"it will be going downwards and we\u0027ll have a negative slope."},{"Start":"06:39.925 ","End":"06:44.810","Text":"This is so important that I\u0027d like to write it as a theorem,"},{"Start":"06:44.810 ","End":"06:47.130","Text":"let say a proposition."},{"Start":"06:49.420 ","End":"07:01.860","Text":"That is that if f prime of x is positive at some x,"},{"Start":"07:02.800 ","End":"07:10.530","Text":"then f is increasing."},{"Start":"07:13.230 ","End":"07:19.635","Text":"So there. If f prime of x is positive,"},{"Start":"07:19.635 ","End":"07:22.530","Text":"the function f is increasing,"},{"Start":"07:22.530 ","End":"07:25.110","Text":"and if f prime of x is negative,"},{"Start":"07:25.110 ","End":"07:28.695","Text":"then the function f is decreasing."},{"Start":"07:28.695 ","End":"07:37.340","Text":"Let me highlight this. Yeah. So derivative positive function is increasing."},{"Start":"07:37.620 ","End":"07:44.110","Text":"Derivative negative function is decreasing."},{"Start":"07:44.110 ","End":"07:48.740","Text":"We\u0027ll have to see some example of how to use this."},{"Start":"07:49.760 ","End":"07:53.260","Text":"Here is an example."},{"Start":"07:54.140 ","End":"08:02.505","Text":"We\u0027ll take f of x equals x cubed"},{"Start":"08:02.505 ","End":"08:07.245","Text":"minus 1 over x"},{"Start":"08:07.245 ","End":"08:14.320","Text":"plus 4x plus 10."},{"Start":"08:14.320 ","End":"08:16.270","Text":"I asked the question,"},{"Start":"08:16.270 ","End":"08:21.200","Text":"is f increasing or decreasing?"},{"Start":"08:26.730 ","End":"08:33.630","Text":"So the solution: what I have to do is to"},{"Start":"08:33.630 ","End":"08:36.630","Text":"figure out f prime of x and then I\u0027ll see if it is"},{"Start":"08:36.630 ","End":"08:40.335","Text":"positive the function\u0027s increasing and negative it\u0027s decreasing."},{"Start":"08:40.335 ","End":"08:43.115","Text":"Let\u0027s see, what is f prime of x?"},{"Start":"08:43.115 ","End":"08:49.435","Text":"Well, derivative of x cubed is 3x squared minus 1 over x."},{"Start":"08:49.435 ","End":"08:53.920","Text":"Well, derivative of 1 over x is minus 1 over x squared,"},{"Start":"08:53.920 ","End":"08:56.185","Text":"so with the minus makes it plus."},{"Start":"08:56.185 ","End":"08:59.065","Text":"It\u0027s plus 1 over x squared."},{"Start":"08:59.065 ","End":"09:04.045","Text":"Derivative 4x is 4 and the 10 gives nothing,"},{"Start":"09:04.045 ","End":"09:07.225","Text":"and this is f prime."},{"Start":"09:07.225 ","End":"09:11.290","Text":"Now, if I look at each of the terms,"},{"Start":"09:11.290 ","End":"09:17.290","Text":"this 1 is bigger or equal to 0,"},{"Start":"09:17.290 ","End":"09:20.799","Text":"it could be 0, but it\u0027s generally positive,"},{"Start":"09:20.799 ","End":"09:23.270","Text":"so positive or 0."},{"Start":"09:23.460 ","End":"09:31.630","Text":"This is bigger or equal to 0 as a plus and the next 1,"},{"Start":"09:31.630 ","End":"09:34.465","Text":"1 over x squared, well, x can\u0027t be 0,"},{"Start":"09:34.465 ","End":"09:37.465","Text":"so x squared is positive strictly,"},{"Start":"09:37.465 ","End":"09:41.695","Text":"and so is 1 over x squared though it\u0027s strictly positive I\u0027ll write as plus plus."},{"Start":"09:41.695 ","End":"09:43.960","Text":"4 is certainly bigger than 0."},{"Start":"09:43.960 ","End":"09:47.275","Text":"Positive can\u0027t be 0 also plus plus."},{"Start":"09:47.275 ","End":"09:52.870","Text":"So strictly positive plus positive plus positive of zero is got to be positive."},{"Start":"09:52.870 ","End":"09:56.290","Text":"This is definitely bigger than 0,"},{"Start":"09:56.290 ","End":"09:57.985","Text":"it\u0027s a plus plus."},{"Start":"09:57.985 ","End":"10:01.270","Text":"Since f prime of x is always positive,"},{"Start":"10:01.270 ","End":"10:07.130","Text":"therefore, I conclude that f is increasing."},{"Start":"10:10.170 ","End":"10:14.300","Text":"That\u0027s the answer to that example."},{"Start":"10:14.520 ","End":"10:19.165","Text":"Now, to slightly complicate things,"},{"Start":"10:19.165 ","End":"10:24.399","Text":"a function isn\u0027t always increasing or always decreasing."},{"Start":"10:24.399 ","End":"10:27.130","Text":"A function can change direction in the middle."},{"Start":"10:27.130 ","End":"10:28.630","Text":"Let me give you an example."},{"Start":"10:28.630 ","End":"10:30.670","Text":"I will draw a freehand sketch here."},{"Start":"10:30.670 ","End":"10:38.425","Text":"Take the y-axis, will take the x-axis here."},{"Start":"10:38.425 ","End":"10:42.220","Text":"So there\u0027s why there\u0027s x."},{"Start":"10:42.220 ","End":"10:46.120","Text":"Let\u0027s take something like, oh, I don\u0027t know,"},{"Start":"10:46.120 ","End":"10:52.374","Text":"the function goes up to here and then dips down here,"},{"Start":"10:52.374 ","End":"11:00.070","Text":"and maybe comes up here."},{"Start":"11:00.070 ","End":"11:04.120","Text":"We can see that this function is increasing up to"},{"Start":"11:04.120 ","End":"11:08.845","Text":"a point and then decreasing and then increasing."},{"Start":"11:08.845 ","End":"11:12.459","Text":"Let me mark some x value that say this is 0,"},{"Start":"11:12.459 ","End":"11:14.545","Text":"that says minus 4."},{"Start":"11:14.545 ","End":"11:17.515","Text":"What interests me is also this peak here,"},{"Start":"11:17.515 ","End":"11:20.800","Text":"let\u0027s say that\u0027s minus 1."},{"Start":"11:20.800 ","End":"11:27.235","Text":"Let\u0027s say that this is 1, this is 3."},{"Start":"11:27.235 ","End":"11:29.020","Text":"Let\u0027s make this peak here,"},{"Start":"11:29.020 ","End":"11:32.395","Text":"which also interests me to be 2."},{"Start":"11:32.395 ","End":"11:34.660","Text":"Let me add a bit of color just to emphasize."},{"Start":"11:34.660 ","End":"11:40.930","Text":"The function is increasing here up to here."},{"Start":"11:40.930 ","End":"11:48.579","Text":"Then it starts decreasing and it\u0027s increasing again from here and onwards."},{"Start":"11:48.579 ","End":"11:57.340","Text":"We presume that this graph goes on indefinitely and will also mark where it\u0027s decreasing."},{"Start":"11:57.340 ","End":"12:02.759","Text":"So from here down to here,"},{"Start":"12:02.759 ","End":"12:05.220","Text":"up to here, it\u0027s decreasing."},{"Start":"12:05.220 ","End":"12:07.665","Text":"We have increasing, decreasing, increasing."},{"Start":"12:07.665 ","End":"12:10.245","Text":"If I want to summarize that,"},{"Start":"12:10.245 ","End":"12:14.140","Text":"I can say that this function is"},{"Start":"12:14.820 ","End":"12:23.800","Text":"increasing for when x is less than minus 1,"},{"Start":"12:24.800 ","End":"12:29.890","Text":"or x is bigger than 2."},{"Start":"12:30.560 ","End":"12:34.450","Text":"That it\u0027s decreasing,"},{"Start":"12:38.100 ","End":"12:44.380","Text":"where the red is for x is between minus 1 and 2."},{"Start":"12:44.380 ","End":"12:48.955","Text":"So minus 1 less than x, less than 2."},{"Start":"12:48.955 ","End":"12:56.320","Text":"Usually we\u0027re more interested in the values of x rather than the points on the graph."},{"Start":"12:56.320 ","End":"13:00.910","Text":"What I\u0027m interested in highlighting really is this range of"},{"Start":"13:00.910 ","End":"13:06.415","Text":"where on the x is the function increasing."},{"Start":"13:06.415 ","End":"13:12.115","Text":"I get this range here and also from here onwards."},{"Start":"13:12.115 ","End":"13:17.260","Text":"That\u0027s the x less than minus 1."},{"Start":"13:17.260 ","End":"13:20.605","Text":"That\u0027s the x bigger than 2."},{"Start":"13:20.605 ","End":"13:23.560","Text":"In the middle, use a different color here,"},{"Start":"13:23.560 ","End":"13:30.025","Text":"we\u0027ll take this part as where the function is decreasing between minus 1 and 2."},{"Start":"13:30.025 ","End":"13:32.455","Text":"1 of the questions I want to ask is,"},{"Start":"13:32.455 ","End":"13:35.455","Text":"how did I get this minus 1 and the 2?"},{"Start":"13:35.455 ","End":"13:37.990","Text":"Well, I just took it by looking at the graph and guessing,"},{"Start":"13:37.990 ","End":"13:40.495","Text":"but we\u0027re going to be more precise with this,"},{"Start":"13:40.495 ","End":"13:43.300","Text":"and there\u0027s going to be a name for this kind of point where"},{"Start":"13:43.300 ","End":"13:47.425","Text":"the function goes up and then goes down or down then up."},{"Start":"13:47.425 ","End":"13:53.319","Text":"These points actually they have a name in general,"},{"Start":"13:53.319 ","End":"14:01.390","Text":"they\u0027re called extrema, which is the plural of each 1 is called an extramum,"},{"Start":"14:01.390 ","End":"14:04.960","Text":"that\u0027s just the Latin way of making the plural."},{"Start":"14:04.960 ","End":"14:08.500","Text":"Each extramum will be 1 of 2 kinds."},{"Start":"14:08.500 ","End":"14:14.335","Text":"It will be a minimum or a maximum."},{"Start":"14:14.335 ","End":"14:17.305","Text":"I\u0027m just preceding the next section what\u0027s to come."},{"Start":"14:17.305 ","End":"14:19.510","Text":"This will be a maximum point which is the"},{"Start":"14:19.510 ","End":"14:22.210","Text":"highest in its area, and this will be a minimum."},{"Start":"14:22.210 ","End":"14:25.630","Text":"But more of that in the next clip,"},{"Start":"14:25.630 ","End":"14:28.250","Text":"when I\u0027m done here."}],"ID":1663},{"Watched":false,"Name":"Extrema with Example 1","Duration":"21m 55s","ChapterTopicVideoID":1651,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1651.jpeg","UploadDate":"2019-10-29T12:58:58.2300000","DurationForVideoObject":"PT21M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.695","Text":"In this clip, I\u0027m going to talk about what are called extrema."},{"Start":"00:04.695 ","End":"00:07.320","Text":"That\u0027s the Latin plural of extremum."},{"Start":"00:07.320 ","End":"00:10.530","Text":"An extremum is a maximum or a minimum."},{"Start":"00:10.530 ","End":"00:16.975","Text":"Let\u0027s start with a sketch and draw just a rough and ready sketch."},{"Start":"00:16.975 ","End":"00:20.550","Text":"This is the y-axis,"},{"Start":"00:20.550 ","End":"00:25.385","Text":"the x-axis and the function will look something"},{"Start":"00:25.385 ","End":"00:32.840","Text":"like up to here then down here and then here again."},{"Start":"00:32.840 ","End":"00:40.965","Text":"Let\u0027s say that this corresponds to 1 and this corresponds to 2."},{"Start":"00:40.965 ","End":"00:44.960","Text":"This is y equals f of x,"},{"Start":"00:44.960 ","End":"00:47.495","Text":"where this is x and this is y."},{"Start":"00:47.495 ","End":"00:51.800","Text":"We assume that this function doesn\u0027t just stop here,"},{"Start":"00:51.800 ","End":"00:56.730","Text":"it keeps going in either direction."},{"Start":"00:56.730 ","End":"01:05.195","Text":"What we see is that the function is increasing up to the point where x equals 1,"},{"Start":"01:05.195 ","End":"01:11.360","Text":"then decreases up to the point where x equals 2 and then increases again."},{"Start":"01:11.360 ","End":"01:15.515","Text":"I can write this down as f is"},{"Start":"01:15.515 ","End":"01:22.865","Text":"increasing for x less than 1 or x bigger than 2,"},{"Start":"01:22.865 ","End":"01:29.710","Text":"and f is decreasing for x between 1 and 2."},{"Start":"01:29.710 ","End":"01:33.935","Text":"Shall write as 1 less than x less than 2."},{"Start":"01:33.935 ","End":"01:43.925","Text":"We have 2 special points here and here."},{"Start":"01:43.925 ","End":"01:46.670","Text":"This is like the top of the mountain,"},{"Start":"01:46.670 ","End":"01:48.785","Text":"like the bottom of the valley."},{"Start":"01:48.785 ","End":"01:53.180","Text":"These type of points are called extrema."},{"Start":"01:53.180 ","End":"01:54.589","Text":"The singular is extremum."},{"Start":"01:54.589 ","End":"01:57.695","Text":"This one is an extremum of type maximum,"},{"Start":"01:57.695 ","End":"02:03.035","Text":"for obvious reasons it\u0027s the top and this is an extremum of type minimum,"},{"Start":"02:03.035 ","End":"02:06.335","Text":"maximum and minimum top of the mountain,"},{"Start":"02:06.335 ","End":"02:07.610","Text":"bottom of the valley."},{"Start":"02:07.610 ","End":"02:12.185","Text":"Let\u0027s label this as max for maximum,"},{"Start":"02:12.185 ","End":"02:14.345","Text":"and this says min for minimum."},{"Start":"02:14.345 ","End":"02:20.150","Text":"The question is, how do we define the maximum and the minimum?"},{"Start":"02:20.150 ","End":"02:24.110","Text":"If you look at it before the maximum we\u0027re increasing and after it we\u0027re decreasing."},{"Start":"02:24.110 ","End":"02:26.240","Text":"That\u0027s basically the definition of a maximum,"},{"Start":"02:26.240 ","End":"02:30.050","Text":"is a point at which the direction,"},{"Start":"02:30.050 ","End":"02:33.645","Text":"the increasing or decreasing changes its trend."},{"Start":"02:33.645 ","End":"02:38.975","Text":"Specifically, it goes from increasing before the point to decreasing."},{"Start":"02:38.975 ","End":"02:42.385","Text":"We go up to the mountain and then down from it."},{"Start":"02:42.385 ","End":"02:50.210","Text":"For the minimum, we defined an extremum as minimum if to the left of the point,"},{"Start":"02:50.210 ","End":"02:53.620","Text":"we\u0027re on a decreasing trend."},{"Start":"02:53.620 ","End":"02:55.355","Text":"After it we are then"},{"Start":"02:55.355 ","End":"03:00.020","Text":"increasing down to the bottom of the valley and then up and away from it."},{"Start":"03:00.020 ","End":"03:06.200","Text":"We can even tell this without the picture once we have written where f is increasing and"},{"Start":"03:06.200 ","End":"03:12.875","Text":"decreasing because we can see that at the point 1 to the left of it,"},{"Start":"03:12.875 ","End":"03:18.259","Text":"f is increasing when x is less than one and to the right of it, it\u0027s decreasing."},{"Start":"03:18.259 ","End":"03:20.900","Text":"Right away I know that one is a maximum."},{"Start":"03:20.900 ","End":"03:24.180","Text":"On the other hand with 2."},{"Start":"03:24.700 ","End":"03:32.585","Text":"I see that on the left it\u0027s decreasing and on the right it\u0027s increasing."},{"Start":"03:32.585 ","End":"03:35.315","Text":"I know to say that 2 is the minimum."},{"Start":"03:35.315 ","End":"03:38.030","Text":"Now of course in general, we won\u0027t have a sketch where"},{"Start":"03:38.030 ","End":"03:40.700","Text":"we can easily see this and we\u0027ll just get the function as"},{"Start":"03:40.700 ","End":"03:43.760","Text":"a formula and we\u0027ll have to see how to do"},{"Start":"03:43.760 ","End":"03:47.105","Text":"it computationally from the formula of the function."},{"Start":"03:47.105 ","End":"03:50.630","Text":"First, let me just write some of the things I said to say that"},{"Start":"03:50.630 ","End":"03:55.010","Text":"an extremum of type maximum or I\u0027ll just write it as a maximum."},{"Start":"03:55.010 ","End":"03:57.985","Text":"A maximum is a point"},{"Start":"03:57.985 ","End":"04:06.565","Text":"where the function changes trend from increasing to decreasing."},{"Start":"04:06.565 ","End":"04:10.275","Text":"A minimum is just the opposite,"},{"Start":"04:10.275 ","End":"04:14.245","Text":"from decreasing to increasing. I\u0027ll just write it."},{"Start":"04:14.245 ","End":"04:19.235","Text":"Minimum is where the function changes from decreasing to increasing."},{"Start":"04:19.235 ","End":"04:25.685","Text":"In both these cases I\u0027m talking about when we go in the direction from left to right."},{"Start":"04:25.685 ","End":"04:27.980","Text":"Meaning here it\u0027s increasing to the left,"},{"Start":"04:27.980 ","End":"04:29.135","Text":"decreasing to the right,"},{"Start":"04:29.135 ","End":"04:30.710","Text":"or here decreasing to the left,"},{"Start":"04:30.710 ","End":"04:32.000","Text":"increasing to the right."},{"Start":"04:32.000 ","End":"04:34.730","Text":"Now we\u0027ve been relying on sketches a lot too."},{"Start":"04:34.730 ","End":"04:36.020","Text":"When you get an exercise,"},{"Start":"04:36.020 ","End":"04:38.300","Text":"you\u0027re going to get a question given a formula,"},{"Start":"04:38.300 ","End":"04:41.765","Text":"say f of x equals x cubed minus 3x."},{"Start":"04:41.765 ","End":"04:47.065","Text":"You\u0027re going to have to find these maximum and minimum points, these extrema."},{"Start":"04:47.065 ","End":"04:52.710","Text":"I\u0027m going to show you a way to do this using the derivative. What else?"},{"Start":"04:52.720 ","End":"05:02.060","Text":"Let\u0027s look at this sketch and notice that if we draw the tangent at the maximum point,"},{"Start":"05:02.060 ","End":"05:07.210","Text":"what we\u0027re going to get is a line that is completely flat,"},{"Start":"05:07.210 ","End":"05:12.080","Text":"I mean horizontal, except for my bad sketching."},{"Start":"05:12.080 ","End":"05:15.920","Text":"Same here. We\u0027re going to get the tangent line is"},{"Start":"05:15.920 ","End":"05:19.835","Text":"going to be horizontal if you draw it right."},{"Start":"05:19.835 ","End":"05:28.520","Text":"This is a very important fact because this means that the slope is 0."},{"Start":"05:28.520 ","End":"05:31.160","Text":"Because a horizontal line has slope 0, or in other words,"},{"Start":"05:31.160 ","End":"05:34.115","Text":"that the derivative is 0 at these points."},{"Start":"05:34.115 ","End":"05:40.130","Text":"This observation is so important that I would like to write it down."},{"Start":"05:40.130 ","End":"05:45.005","Text":"That is, and I say this at an extremum,"},{"Start":"05:45.005 ","End":"05:48.179","Text":"that\u0027s a min or max,"},{"Start":"05:48.490 ","End":"05:53.870","Text":"the derivative of the function, the function is 0."},{"Start":"05:53.870 ","End":"05:55.160","Text":"Very important."},{"Start":"05:55.160 ","End":"05:59.615","Text":"That will help us to find these points by comparing the derivative to 0."},{"Start":"05:59.615 ","End":"06:02.720","Text":"I would like to add a little warning though."},{"Start":"06:02.720 ","End":"06:05.060","Text":"In logic things go one way."},{"Start":"06:05.060 ","End":"06:08.359","Text":"What we\u0027re saying is that if we have an extremum,"},{"Start":"06:08.359 ","End":"06:09.530","Text":"a minimum or a maximum,"},{"Start":"06:09.530 ","End":"06:12.065","Text":"then the derivative of the function is 0."},{"Start":"06:12.065 ","End":"06:14.450","Text":"But it doesn\u0027t imply the opposite."},{"Start":"06:14.450 ","End":"06:17.990","Text":"It doesn\u0027t mean that if the derivative of the function is 0,"},{"Start":"06:17.990 ","End":"06:21.755","Text":"that you necessarily have a minimum or maximum."},{"Start":"06:21.755 ","End":"06:23.255","Text":"You may or may not."},{"Start":"06:23.255 ","End":"06:27.005","Text":"I want you to just write that down then I\u0027ll clarify."},{"Start":"06:27.005 ","End":"06:32.555","Text":"Note, if the derivative is 0,"},{"Start":"06:32.555 ","End":"06:38.555","Text":"we don\u0027t necessarily, the word necessarily is important and often we do,"},{"Start":"06:38.555 ","End":"06:42.290","Text":"but we don\u0027t necessarily have an extremum,"},{"Start":"06:42.290 ","End":"06:43.790","Text":"a minimum or a maximum."},{"Start":"06:43.790 ","End":"06:49.620","Text":"I\u0027ll give you some examples of how this could be."},{"Start":"06:49.620 ","End":"06:55.305","Text":"I can think of several ways it could go bad."},{"Start":"06:55.305 ","End":"06:59.555","Text":"Not bad, I mean, illustrate how things can happen."},{"Start":"06:59.555 ","End":"07:07.385","Text":"You can have a graph model and draw the axes where we\u0027re going down and down and down."},{"Start":"07:07.385 ","End":"07:11.720","Text":"Decrease like an escalator or something."},{"Start":"07:11.720 ","End":"07:15.680","Text":"We get to a point where suddenly we leveled out."},{"Start":"07:15.680 ","End":"07:19.460","Text":"We\u0027re not really increasing or decreasing this point."},{"Start":"07:19.460 ","End":"07:24.569","Text":"Then we continue, and then we\u0027re decreasing again."},{"Start":"07:25.670 ","End":"07:29.360","Text":"Here we have definitely a decrease."},{"Start":"07:29.360 ","End":"07:32.690","Text":"Here we have another decrease."},{"Start":"07:32.690 ","End":"07:36.670","Text":"But at this point, we sort of not increasing or decreasing,"},{"Start":"07:36.670 ","End":"07:39.364","Text":"at the moment we\u0027re just level."},{"Start":"07:39.364 ","End":"07:43.340","Text":"Neither increasing nor decreasing for a moment and then continue."},{"Start":"07:43.340 ","End":"07:45.775","Text":"Now if you look at the derivative,"},{"Start":"07:45.775 ","End":"07:52.685","Text":"at this point, you find that the derivative is horizontal."},{"Start":"07:52.685 ","End":"07:54.530","Text":"We flattened out for a moment,"},{"Start":"07:54.530 ","End":"07:58.920","Text":"which means that f prime is equal to 0,"},{"Start":"07:58.920 ","End":"08:00.830","Text":"but it\u0027s not an extremum,"},{"Start":"08:00.830 ","End":"08:02.480","Text":"it\u0027s not a maximum or a minimum."},{"Start":"08:02.480 ","End":"08:04.040","Text":"This is a bit crooked."},{"Start":"08:04.040 ","End":"08:05.270","Text":"Imagine it\u0027s horizontal."},{"Start":"08:05.270 ","End":"08:08.870","Text":"We are horizontal, but we didn\u0027t"},{"Start":"08:08.870 ","End":"08:12.470","Text":"go from increase to decrease or from decrease to increase."},{"Start":"08:12.470 ","End":"08:16.355","Text":"We went from decrease to decrease. This could happen."},{"Start":"08:16.355 ","End":"08:22.760","Text":"Another extreme example, just to illustrate what could be,"},{"Start":"08:22.760 ","End":"08:27.140","Text":"I could have a function which is just totally constant,"},{"Start":"08:27.140 ","End":"08:30.840","Text":"like y equals 4 or something."},{"Start":"08:30.840 ","End":"08:33.775","Text":"A constant function. Once again,"},{"Start":"08:33.775 ","End":"08:36.310","Text":"the derivative at every point would be 0,"},{"Start":"08:36.310 ","End":"08:38.230","Text":"but I don\u0027t see that it\u0027s an extremum."},{"Start":"08:38.230 ","End":"08:40.540","Text":"There\u0027s no minimum or maximum, it\u0027s totally flat."},{"Start":"08:40.540 ","End":"08:48.130","Text":"W e look for points of maximum or minimum where the derivative is 0,"},{"Start":"08:48.130 ","End":"08:51.910","Text":"but that will only give us suspects it won\u0027t guarantee,"},{"Start":"08:51.910 ","End":"08:54.880","Text":"we\u0027ll have to do an extra check to make sure that we really are."},{"Start":"08:54.880 ","End":"08:56.800","Text":"Even if the derivative is 0,"},{"Start":"08:56.800 ","End":"09:02.630","Text":"we have to somehow verify further whether we have a maximum or minimum or neither."},{"Start":"09:03.020 ","End":"09:07.390","Text":"Having said this and given all these warnings,"},{"Start":"09:07.390 ","End":"09:16.115","Text":"how do we go about and check whether we have a maximum or a minimum?"},{"Start":"09:16.115 ","End":"09:21.160","Text":"Well, the answer is, after we found where the derivative is 0."},{"Start":"09:21.160 ","End":"09:23.700","Text":"Let\u0027s say we find the derivative is 0 here."},{"Start":"09:23.700 ","End":"09:26.540","Text":"We check a couple of neighboring points."},{"Start":"09:26.540 ","End":"09:29.990","Text":"One a little bit to the left of this point and one,"},{"Start":"09:29.990 ","End":"09:33.320","Text":"a bit to the right of the point and see if we have here"},{"Start":"09:33.320 ","End":"09:38.310","Text":"increasing and here decrease angle or some other combination."},{"Start":"09:38.310 ","End":"09:39.710","Text":"That\u0027s the idea."},{"Start":"09:39.710 ","End":"09:45.275","Text":"You substitute for the derivative at the point where at 0,"},{"Start":"09:45.275 ","End":"09:47.630","Text":"but to the left and to the right."},{"Start":"09:47.630 ","End":"09:50.135","Text":"All this requires an example."},{"Start":"09:50.135 ","End":"09:51.770","Text":"It\u0027s a bit tricky here."},{"Start":"09:51.770 ","End":"09:54.530","Text":"Let\u0027s go back down."},{"Start":"09:54.530 ","End":"09:56.870","Text":"Here\u0027s our exercise."},{"Start":"09:56.870 ","End":"10:02.840","Text":"Find the extrema and the intervals of increase, decrease."},{"Start":"10:02.840 ","End":"10:05.480","Text":"Intervals of domains just means where the function is"},{"Start":"10:05.480 ","End":"10:08.060","Text":"increasing and where the function is decreasing of"},{"Start":"10:08.060 ","End":"10:13.990","Text":"the function and here it is f of x equals x cubed minus 3x."},{"Start":"10:13.990 ","End":"10:17.980","Text":"Let\u0027s get to the solution."},{"Start":"10:22.680 ","End":"10:26.290","Text":"Then I\u0027ll break the solution up into steps,"},{"Start":"10:26.290 ","End":"10:31.720","Text":"and also I want to remark that this method I\u0027m going to show you will"},{"Start":"10:31.720 ","End":"10:34.750","Text":"find at the same time the intervals"},{"Start":"10:34.750 ","End":"10:38.290","Text":"of increase/decrease on the extrema with no extra work."},{"Start":"10:38.290 ","End":"10:42.160","Text":"Let\u0027s start with the preparatory step,"},{"Start":"10:42.160 ","End":"10:44.245","Text":"I\u0027ll just call it preparation,"},{"Start":"10:44.245 ","End":"10:45.790","Text":"and then we\u0027ll go with step 1,"},{"Start":"10:45.790 ","End":"10:47.110","Text":"2, 3, and so on."},{"Start":"10:47.110 ","End":"10:50.680","Text":"The preparation is just to differentiate the function."},{"Start":"10:50.680 ","End":"10:55.150","Text":"It\u0027s a whole step, because sometimes a differentiation can be complicated."},{"Start":"10:55.150 ","End":"10:58.615","Text":"But in here we can just write it immediately. No work at all."},{"Start":"10:58.615 ","End":"11:03.745","Text":"F prime of x is equal to 3x squared minus 3,"},{"Start":"11:03.745 ","End":"11:06.520","Text":"gets the first pre-step."},{"Start":"11:06.520 ","End":"11:09.310","Text":"Then we\u0027ll start with step 1."},{"Start":"11:09.310 ","End":"11:13.180","Text":"We compare the derivative to 0."},{"Start":"11:13.180 ","End":"11:18.565","Text":"The reason we do that is because we\u0027re looking for extrema maximum or minimum"},{"Start":"11:18.565 ","End":"11:24.550","Text":"and we know that from this proposition that at an extremum,"},{"Start":"11:24.550 ","End":"11:26.920","Text":"the derivative of the function is 0."},{"Start":"11:26.920 ","End":"11:28.885","Text":"I\u0027m not saying the reverse,"},{"Start":"11:28.885 ","End":"11:32.170","Text":"but if we have any chance of finding an extremum,"},{"Start":"11:32.170 ","End":"11:34.525","Text":"it\u0027s where the derivative is 0,"},{"Start":"11:34.525 ","End":"11:36.070","Text":"and then it may or may not be,"},{"Start":"11:36.070 ","End":"11:38.740","Text":"but it certainly won\u0027t be found anywhere else."},{"Start":"11:38.740 ","End":"11:40.600","Text":"F prime of x is 0,"},{"Start":"11:40.600 ","End":"11:45.490","Text":"means that 3x squared minus 3 equals 0,"},{"Start":"11:45.490 ","End":"11:47.470","Text":"x squared equals 1."},{"Start":"11:47.470 ","End":"11:50.620","Text":"I\u0027ll just do it all in 1 line, x squared equals 1."},{"Start":"11:50.620 ","End":"11:54.715","Text":"That means that x can be 1 of 2 things."},{"Start":"11:54.715 ","End":"11:56.800","Text":"It\u0027s either plus 1,"},{"Start":"11:56.800 ","End":"11:59.830","Text":"the plus for emphasis, or minus 1."},{"Start":"11:59.830 ","End":"12:02.230","Text":"We have 2 points to check."},{"Start":"12:02.230 ","End":"12:09.400","Text":"Now, we\u0027re going to have to check for each 1 whether it\u0027s a minimum or maximum,"},{"Start":"12:09.400 ","End":"12:13.940","Text":"and the way we do that will be described in step 2."},{"Start":"12:14.580 ","End":"12:18.639","Text":"Step 2 is a bit more involved."},{"Start":"12:18.639 ","End":"12:20.695","Text":"We\u0027re going to make a table."},{"Start":"12:20.695 ","End":"12:23.830","Text":"This is a method I found very useful and helpful"},{"Start":"12:23.830 ","End":"12:27.625","Text":"in answering this kind of question for organized."},{"Start":"12:27.625 ","End":"12:29.095","Text":"I\u0027m going to make a table,"},{"Start":"12:29.095 ","End":"12:36.100","Text":"I\u0027ll put some rows in x,"},{"Start":"12:36.100 ","End":"12:38.720","Text":"f prime of x,"},{"Start":"12:39.720 ","End":"12:43.600","Text":"and f of x, or these are the rows."},{"Start":"12:43.600 ","End":"12:47.140","Text":"Anyway, we\u0027re going to have a row for values of x,"},{"Start":"12:47.140 ","End":"12:51.745","Text":"we\u0027re going to have a row for values for f of x."},{"Start":"12:51.745 ","End":"12:54.610","Text":"In fact, later there will be a 4th row."},{"Start":"12:54.610 ","End":"12:57.520","Text":"I\u0027ll leave room for it. Well, give it a name now why,"},{"Start":"12:57.520 ","End":"12:59.735","Text":"but that will come to later."},{"Start":"12:59.735 ","End":"13:02.400","Text":"What we do is the following."},{"Start":"13:02.400 ","End":"13:04.470","Text":"We put the values of x,"},{"Start":"13:04.470 ","End":"13:11.010","Text":"which are the solutions of the f prime equals 0 into the table in sequential order,"},{"Start":"13:11.010 ","End":"13:12.615","Text":"I mean, in order of size."},{"Start":"13:12.615 ","End":"13:16.480","Text":"The minus 1 would be here"},{"Start":"13:16.480 ","End":"13:20.485","Text":"and the 1 is here and leave space between them because we\u0027re going to add stuff."},{"Start":"13:20.485 ","End":"13:22.930","Text":"We do start filling in the table."},{"Start":"13:22.930 ","End":"13:27.730","Text":"Now, f prime of x at each of these is going to have to be 0 because I mean,"},{"Start":"13:27.730 ","End":"13:30.655","Text":"it came from the solution where we made it equals 0,"},{"Start":"13:30.655 ","End":"13:32.125","Text":"so this we know."},{"Start":"13:32.125 ","End":"13:38.950","Text":"Note that these points divide the x-axis into intervals."},{"Start":"13:38.950 ","End":"13:42.040","Text":"Like between minus 1 and 1,"},{"Start":"13:42.040 ","End":"13:48.640","Text":"I have the interval minus 1 less than x, less than 1."},{"Start":"13:48.640 ","End":"13:54.865","Text":"To the right of 1, I have the interval x bigger than 1,"},{"Start":"13:54.865 ","End":"13:57.415","Text":"and to the left of minus 1,"},{"Start":"13:57.415 ","End":"14:01.540","Text":"I have the interval x less than minus 1."},{"Start":"14:01.540 ","End":"14:05.440","Text":"Just think of it like you have the number line,"},{"Start":"14:05.440 ","End":"14:08.710","Text":"the x-axis, and when I write 2 points on it,"},{"Start":"14:08.710 ","End":"14:11.980","Text":"minus 1 and plus 1,"},{"Start":"14:11.980 ","End":"14:17.425","Text":"then naturally I get an interval here, that\u0027s 1 interval."},{"Start":"14:17.425 ","End":"14:20.635","Text":"Then I, get another interval here,"},{"Start":"14:20.635 ","End":"14:22.765","Text":"and I get another interval here."},{"Start":"14:22.765 ","End":"14:27.700","Text":"What we have to do next is just choose any number from this interval."},{"Start":"14:27.700 ","End":"14:30.250","Text":"For example, from this middle interval,"},{"Start":"14:30.250 ","End":"14:34.060","Text":"I\u0027d like to choose 0 from this interval x bigger than 1,"},{"Start":"14:34.060 ","End":"14:37.945","Text":"I\u0027ll choose 2, and here I\u0027ll choose minus 2,"},{"Start":"14:37.945 ","End":"14:39.355","Text":"and it could be any."},{"Start":"14:39.355 ","End":"14:43.915","Text":"Next, we substitute into f prime because that\u0027s what this row requires,"},{"Start":"14:43.915 ","End":"14:45.175","Text":"it requires f prime."},{"Start":"14:45.175 ","End":"14:46.840","Text":"I look at f prime."},{"Start":"14:46.840 ","End":"14:49.795","Text":"Let\u0027s start with an easy 1. Then it start with the 0."},{"Start":"14:49.795 ","End":"14:57.340","Text":"F prime of 0, 0 squared is 0 times 3 is still 0 minus 3 is minus 3."},{"Start":"14:57.340 ","End":"14:59.860","Text":"But I don\u0027t actually write minus 3 here"},{"Start":"14:59.860 ","End":"15:02.950","Text":"because all I care about is the sign positive or negative,"},{"Start":"15:02.950 ","End":"15:05.500","Text":"so it\u0027s definitely a minus."},{"Start":"15:05.500 ","End":"15:08.335","Text":"If I put x equals 2,"},{"Start":"15:08.335 ","End":"15:11.170","Text":"I get 2 squared is 4 times 3 is 12,"},{"Start":"15:11.170 ","End":"15:14.950","Text":"minus 3 is 9, and that\u0027s positive."},{"Start":"15:14.950 ","End":"15:18.145","Text":"If x is minus 1,"},{"Start":"15:18.145 ","End":"15:20.620","Text":"I\u0027ll get the same and also get 9,"},{"Start":"15:20.620 ","End":"15:22.180","Text":"and that will be positive."},{"Start":"15:22.180 ","End":"15:24.909","Text":"Different numbers, if I chose different numbers,"},{"Start":"15:24.909 ","End":"15:31.615","Text":"I wouldn\u0027t have got 9 here and I wouldn\u0027t have got maybe minus 3 here,"},{"Start":"15:31.615 ","End":"15:33.940","Text":"but the sign wouldn\u0027t change."},{"Start":"15:33.940 ","End":"15:37.630","Text":"Because as we saw, like the function here, this plus, minus,"},{"Start":"15:37.630 ","End":"15:40.780","Text":"plus means here I\u0027m in this interval and going up and"},{"Start":"15:40.780 ","End":"15:44.455","Text":"then I\u0027m starting to go down and then I\u0027m starting to go up again."},{"Start":"15:44.455 ","End":"15:48.670","Text":"All I need to know if it\u0027s going up or down is this plus or minus."},{"Start":"15:48.670 ","End":"15:50.200","Text":"In all this interval,"},{"Start":"15:50.200 ","End":"15:54.280","Text":"this means that f is increasing,"},{"Start":"15:54.280 ","End":"15:57.160","Text":"and though I haven\u0027t got the graph I know it\u0027s increasing,"},{"Start":"15:57.160 ","End":"16:03.115","Text":"and I indicate this here by writing an arrow going upwards here."},{"Start":"16:03.115 ","End":"16:06.280","Text":"Here, if f prime is negative,"},{"Start":"16:06.280 ","End":"16:08.500","Text":"that means the function is decreasing,"},{"Start":"16:08.500 ","End":"16:10.930","Text":"so I write it with an arrow like this to"},{"Start":"16:10.930 ","End":"16:14.170","Text":"indicate the function is decreasing in this interval."},{"Start":"16:14.170 ","End":"16:15.970","Text":"In this last interval,"},{"Start":"16:15.970 ","End":"16:18.670","Text":"also the function is increasing."},{"Start":"16:18.670 ","End":"16:25.299","Text":"This, for example, would immediately tell me the interval of increase and decrease,"},{"Start":"16:25.299 ","End":"16:27.325","Text":"and we\u0027ll write it in a moment."},{"Start":"16:27.325 ","End":"16:30.550","Text":"But I can say from less than minus 1 it\u0027s increasing,"},{"Start":"16:30.550 ","End":"16:34.825","Text":"between minus 1 and 1 is decreasing and after 1 it\u0027s increasing,"},{"Start":"16:34.825 ","End":"16:38.440","Text":"so we have that, at least in our heads answered."},{"Start":"16:38.440 ","End":"16:43.570","Text":"Remember that if we are between increasing and"},{"Start":"16:43.570 ","End":"16:49.975","Text":"decreasing and we always go from left to right, here it is."},{"Start":"16:49.975 ","End":"16:54.235","Text":"If I\u0027m going from increasing to decreasing,"},{"Start":"16:54.235 ","End":"16:56.590","Text":"then we have a maximum,"},{"Start":"16:56.590 ","End":"16:59.320","Text":"and if we go from decreasing to increasing,"},{"Start":"16:59.320 ","End":"17:00.730","Text":"we get a minimum."},{"Start":"17:00.730 ","End":"17:06.715","Text":"That means that here I have a maximum,"},{"Start":"17:06.715 ","End":"17:11.620","Text":"so I just write it in brief as max for maximum,"},{"Start":"17:11.620 ","End":"17:17.290","Text":"and here, decreasing to increasing change is a minimum."},{"Start":"17:17.290 ","End":"17:20.770","Text":"You can see as I go up and up and up and then suddenly I\u0027m going down,"},{"Start":"17:20.770 ","End":"17:22.720","Text":"that means that this point was the top,"},{"Start":"17:22.720 ","End":"17:23.890","Text":"the highest I could go."},{"Start":"17:23.890 ","End":"17:26.695","Text":"Similarly, if I\u0027m going down and suddenly I\u0027m going"},{"Start":"17:26.695 ","End":"17:30.145","Text":"up that means I hit the lowest point, so that\u0027s the minimum."},{"Start":"17:30.145 ","End":"17:34.255","Text":"Now we come to the last row, y."},{"Start":"17:34.255 ","End":"17:37.750","Text":"Just like the second row, f prime of x,"},{"Start":"17:37.750 ","End":"17:42.355","Text":"we substituted values in this formula here."},{"Start":"17:42.355 ","End":"17:48.400","Text":"For y, we substitute in the original f of"},{"Start":"17:48.400 ","End":"17:55.280","Text":"x. I want y and only need it for these 2 values where f prime was 0."},{"Start":"17:55.350 ","End":"17:59.755","Text":"Let\u0027s see. If x is minus 1,"},{"Start":"17:59.755 ","End":"18:03.040","Text":"then y, which is f of x,"},{"Start":"18:03.040 ","End":"18:07.885","Text":"is minus 1 cubed is minus 1,"},{"Start":"18:07.885 ","End":"18:11.140","Text":"less 3 times minus 1,"},{"Start":"18:11.140 ","End":"18:16.310","Text":"which is plus 3, and minus 1 plus 3 is 2."},{"Start":"18:17.040 ","End":"18:21.760","Text":"When x is 1, 1 minus 3 is minus 2."},{"Start":"18:21.760 ","End":"18:26.125","Text":"In step 3, I\u0027m going to just collect"},{"Start":"18:26.125 ","End":"18:28.750","Text":"the data or just write the answers to what we were"},{"Start":"18:28.750 ","End":"18:31.765","Text":"asked for because there\u0027s quite a few things to say."},{"Start":"18:31.765 ","End":"18:35.860","Text":"In step 3, I\u0027ll write for each extremum."},{"Start":"18:35.860 ","End":"18:40.630","Text":"For example, we have the 1 where x is minus 1,"},{"Start":"18:40.630 ","End":"18:42.835","Text":"and now you see why we need the y,"},{"Start":"18:42.835 ","End":"18:46.780","Text":"because I can say that minus 1,"},{"Start":"18:46.780 ","End":"18:51.325","Text":"2 is a maximum."},{"Start":"18:51.325 ","End":"18:55.010","Text":"Maximum is part of the extrema."},{"Start":"18:56.030 ","End":"19:00.675","Text":"The other thing I can say is that 1, minus 2,"},{"Start":"19:00.675 ","End":"19:04.880","Text":"1, minus 2 is a minimum."},{"Start":"19:04.880 ","End":"19:11.505","Text":"The next thing I can say is that the intervals of increase,"},{"Start":"19:11.505 ","End":"19:17.340","Text":"those will be where the arrow is going up,"},{"Start":"19:17.340 ","End":"19:21.730","Text":"so that will be x less than minus 1 will be 1 interval"},{"Start":"19:21.730 ","End":"19:26.500","Text":"of increase and the other interval of increase will be x bigger than 1."},{"Start":"19:26.500 ","End":"19:32.695","Text":"Then finally, the question asked for intervals of decrease,"},{"Start":"19:32.695 ","End":"19:34.760","Text":"there is only 1 of these."},{"Start":"19:34.760 ","End":"19:37.200","Text":"and that 1 is from here,"},{"Start":"19:37.200 ","End":"19:44.380","Text":"is minus 1 less than x, less than 1."},{"Start":"19:44.380 ","End":"19:46.870","Text":"This basically answers the question."},{"Start":"19:46.870 ","End":"19:51.490","Text":"But already with this little bit of information,"},{"Start":"19:51.490 ","End":"19:54.705","Text":"give a rough sketch of the graph,"},{"Start":"19:54.705 ","End":"19:56.415","Text":"and I\u0027ll show you."},{"Start":"19:56.415 ","End":"20:00.195","Text":"Let\u0027s do this quickly is not part of the question is just optional."},{"Start":"20:00.195 ","End":"20:07.330","Text":"Just wanted to show you how just knowing increase and decrease index extrema can help us."},{"Start":"20:07.410 ","End":"20:16.660","Text":"We have a graph. I\u0027ll draw the extrema minus 1, 2 somewhere here,"},{"Start":"20:16.660 ","End":"20:20.335","Text":"let\u0027s say, and"},{"Start":"20:20.335 ","End":"20:27.830","Text":"the minimum 1 minus 2 somewhere here."},{"Start":"20:30.050 ","End":"20:34.785","Text":"This is the area of increase we don\u0027t know quite enough yet."},{"Start":"20:34.785 ","End":"20:39.090","Text":"Perhaps if I just compute the intersection with the x-axis,"},{"Start":"20:39.090 ","End":"20:44.040","Text":"I\u0027ll have a little bit more to go on or if I compute this equals 0,"},{"Start":"20:44.040 ","End":"20:49.930","Text":"you will find that I get x equals 0 or plus or minus the square root of 3."},{"Start":"20:49.930 ","End":"20:56.840","Text":"Let\u0027s say this is the square root of 3 and this is minus the square root of 3."},{"Start":"20:56.840 ","End":"20:58.675","Text":"I don\u0027t want to get too deeply into this."},{"Start":"20:58.675 ","End":"21:04.795","Text":"But we know that we\u0027re up to this point, we\u0027re going up."},{"Start":"21:04.795 ","End":"21:09.100","Text":"We hit the maximum here and we start going down,"},{"Start":"21:09.100 ","End":"21:11.920","Text":"and then we get to the minimum,"},{"Start":"21:11.920 ","End":"21:14.305","Text":"and then we go up again."},{"Start":"21:14.305 ","End":"21:17.650","Text":"We get a very rough sketch of the graph of which we just supplemented with"},{"Start":"21:17.650 ","End":"21:21.910","Text":"a bit more information like where it crosses the axes or something."},{"Start":"21:21.910 ","End":"21:25.540","Text":"We\u0027re on our way to investigation of functions,"},{"Start":"21:25.540 ","End":"21:27.730","Text":"and this is a big part of it."},{"Start":"21:27.730 ","End":"21:34.960","Text":"In this example, we had 2 solutions to the derivative equals 0,"},{"Start":"21:34.960 ","End":"21:38.020","Text":"where we had 1 and minus 1,"},{"Start":"21:38.020 ","End":"21:40.660","Text":"and it turned out that both of them were extrema."},{"Start":"21:40.660 ","End":"21:47.380","Text":"I wanted to give you an example where it turns out that the derivative is 0,"},{"Start":"21:47.380 ","End":"21:49.465","Text":"but it\u0027s not a minimum or a maximum,"},{"Start":"21:49.465 ","End":"21:51.910","Text":"and that will be in the next clip."},{"Start":"21:51.910 ","End":"21:54.710","Text":"We\u0027re done here"}],"ID":1664},{"Watched":false,"Name":"Extrema with Example 2","Duration":"7m 23s","ChapterTopicVideoID":8271,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8271.jpeg","UploadDate":"2019-10-29T13:03:06.8670000","DurationForVideoObject":"PT7M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"This clip is a continuation clip from the previous on extrema."},{"Start":"00:04.260 ","End":"00:07.140","Text":"We talked about the way to find extrema,"},{"Start":"00:07.140 ","End":"00:14.774","Text":"which is a maximum or minimum by differentiating and solving for the derivative equals 0,"},{"Start":"00:14.774 ","End":"00:20.040","Text":"the solution Xs are the candidates or suspect to be extrema."},{"Start":"00:20.040 ","End":"00:22.200","Text":"Now, in the last exercise,"},{"Start":"00:22.200 ","End":"00:26.070","Text":"we had 2 of them and they both turned out to be extrema,"},{"Start":"00:26.070 ","End":"00:27.120","Text":"1 of them was a maximum,"},{"Start":"00:27.120 ","End":"00:28.590","Text":"1 of them was a minimum."},{"Start":"00:28.590 ","End":"00:31.770","Text":"I want to show here an example that not always,"},{"Start":"00:31.770 ","End":"00:34.275","Text":"when you find that the derivative is 0,"},{"Start":"00:34.275 ","End":"00:37.395","Text":"it doesn\u0027t always come out to be an extremum."},{"Start":"00:37.395 ","End":"00:41.050","Text":"Here, we\u0027re going to see 1 which is neither maximum nor minimum."},{"Start":"00:41.050 ","End":"00:44.960","Text":"We\u0027re going to do it in the same style as last time with all the steps;"},{"Start":"00:44.960 ","End":"00:48.470","Text":"there was preparation, and then 2 or 3 steps."},{"Start":"00:48.470 ","End":"00:52.070","Text":"In the preparation, we just derive the function,"},{"Start":"00:52.070 ","End":"00:54.440","Text":"which in our case is very simple,"},{"Start":"00:54.440 ","End":"01:03.329","Text":"f prime of x comes out to be 4x cubed minus 12x squared,"},{"Start":"01:03.329 ","End":"01:05.460","Text":"we call that the preparation stage."},{"Start":"01:05.460 ","End":"01:08.540","Text":"Then we had step number 1 where we said,"},{"Start":"01:08.540 ","End":"01:13.580","Text":"well, let\u0027s set f prime of x to be 0 and solve it."},{"Start":"01:13.580 ","End":"01:20.540","Text":"That gives us that 4x cubed minus 12x squared equals 0."},{"Start":"01:20.540 ","End":"01:21.950","Text":"If we factor it,"},{"Start":"01:21.950 ","End":"01:26.195","Text":"we could actually take 4x squared outside the brackets."},{"Start":"01:26.195 ","End":"01:31.595","Text":"So 4x squared times x minus 3 is equal to 0."},{"Start":"01:31.595 ","End":"01:34.225","Text":"There are only 2 possible solutions,"},{"Start":"01:34.225 ","End":"01:36.680","Text":"either the 4x squared is 0,"},{"Start":"01:36.680 ","End":"01:38.450","Text":"which means that x squared is 0,"},{"Start":"01:38.450 ","End":"01:41.930","Text":"so x is 0, that\u0027s 1 possibility."},{"Start":"01:41.930 ","End":"01:44.810","Text":"The other possibility that x is equal to 3."},{"Start":"01:44.810 ","End":"01:48.310","Text":"Like before, we\u0027re going to draw a table,"},{"Start":"01:48.310 ","End":"01:53.675","Text":"and as I recall in the table we had the first row was x,"},{"Start":"01:53.675 ","End":"01:57.140","Text":"f prime of x, f of x."},{"Start":"01:57.140 ","End":"02:00.985","Text":"Later on we added a final row, which was y."},{"Start":"02:00.985 ","End":"02:02.630","Text":"The values of x,"},{"Start":"02:02.630 ","End":"02:05.765","Text":"we start by putting in the ones from here,"},{"Start":"02:05.765 ","End":"02:10.070","Text":"put in x equals 0 and x equals 3 in this case."},{"Start":"02:10.070 ","End":"02:15.170","Text":"That divides the x-axis into 3 separate intervals."},{"Start":"02:15.170 ","End":"02:19.730","Text":"The first one would be x less than 0."},{"Start":"02:19.730 ","End":"02:25.280","Text":"Then you have a middle interval where 0 is less than x, less than 3,"},{"Start":"02:25.280 ","End":"02:30.980","Text":"meaning x is between the 2 points and the last one is just x above 3."},{"Start":"02:30.980 ","End":"02:37.160","Text":"The next thing we did is to choose arbitrarily a value from each of these ranges."},{"Start":"02:37.160 ","End":"02:41.270","Text":"From here, I\u0027ll choose x equals minus 1."},{"Start":"02:41.270 ","End":"02:43.130","Text":"You feel free to choose something else,"},{"Start":"02:43.130 ","End":"02:44.660","Text":"x between 0 and 3,"},{"Start":"02:44.660 ","End":"02:51.105","Text":"I\u0027ll just take x equals 1, and here, I\u0027ll choose x equals 4."},{"Start":"02:51.105 ","End":"02:55.020","Text":"F prime at 0 and at 3 in both case is, 0."},{"Start":"02:55.020 ","End":"02:58.985","Text":"This is how it always is because where did we get this 0 and 3 from?"},{"Start":"02:58.985 ","End":"03:03.565","Text":"After all, it was by comparing f prime to 0 so it has to be."},{"Start":"03:03.565 ","End":"03:09.170","Text":"What we do for these values here is we don\u0027t actually want the value of f prime of x,"},{"Start":"03:09.170 ","End":"03:11.090","Text":"but only whether it\u0027s positive or negative."},{"Start":"03:11.090 ","End":"03:15.470","Text":"Let\u0027s see what happens when we substitute them in here."},{"Start":"03:15.470 ","End":"03:17.810","Text":"If I put minus 1 in there,"},{"Start":"03:17.810 ","End":"03:20.960","Text":"I get minus 4, minus 12."},{"Start":"03:20.960 ","End":"03:22.745","Text":"But anyway it\u0027s negative."},{"Start":"03:22.745 ","End":"03:25.235","Text":"Let\u0027s go for x equals 1."},{"Start":"03:25.235 ","End":"03:30.080","Text":"Yet this time 4 minus 12, also negative."},{"Start":"03:30.080 ","End":"03:31.910","Text":"This is already ringing bells,"},{"Start":"03:31.910 ","End":"03:34.460","Text":"minus followed by a minus, but wait."},{"Start":"03:34.460 ","End":"03:36.904","Text":"Then when x is 4,"},{"Start":"03:36.904 ","End":"03:40.610","Text":"this is 16 times 16 and which is less than 12 times 16."},{"Start":"03:40.610 ","End":"03:42.355","Text":"This comes out positive."},{"Start":"03:42.355 ","End":"03:44.090","Text":"It doesn\u0027t matter by the way,"},{"Start":"03:44.090 ","End":"03:46.715","Text":"which point you choose in this range,"},{"Start":"03:46.715 ","End":"03:48.170","Text":"the numbers will be different,"},{"Start":"03:48.170 ","End":"03:50.360","Text":"but the plus minus will be the same."},{"Start":"03:50.360 ","End":"03:54.365","Text":"Now, what this means here is the situation we haven\u0027t had before."},{"Start":"03:54.365 ","End":"03:57.109","Text":"It\u0027s decreasing up to the point,"},{"Start":"03:57.109 ","End":"03:59.870","Text":"and it\u0027s decreasing after the point."},{"Start":"03:59.870 ","End":"04:01.760","Text":"Afterwards we have an increase."},{"Start":"04:01.760 ","End":"04:04.085","Text":"This point is conventional;"},{"Start":"04:04.085 ","End":"04:06.950","Text":"a decrease followed by an increase we know is a minimum."},{"Start":"04:06.950 ","End":"04:09.950","Text":"So here we have a minimum at x equals 3,"},{"Start":"04:09.950 ","End":"04:11.450","Text":"but at x equals 0,"},{"Start":"04:11.450 ","End":"04:12.880","Text":"we don\u0027t know what it is."},{"Start":"04:12.880 ","End":"04:16.475","Text":"I actually do know when it has a name but I\u0027m not going to give it now."},{"Start":"04:16.475 ","End":"04:23.289","Text":"The final row, y is just what happens when we plug in this value of x into the function."},{"Start":"04:23.289 ","End":"04:29.505","Text":"If x is 0, we get 0 minus 0 is 0,"},{"Start":"04:29.505 ","End":"04:35.050","Text":"and if x is 3, then we get 3^4,"},{"Start":"04:35.050 ","End":"04:40.220","Text":"minus 4 times 3 cubed minus 27 here."},{"Start":"04:40.220 ","End":"04:43.940","Text":"That\u0027s enough to get started in drawing a quick sketch,"},{"Start":"04:43.940 ","End":"04:47.330","Text":"the intersection with the axis actually comes out to be,"},{"Start":"04:47.330 ","End":"04:50.300","Text":"4 if you plug it in, it turns out to be 0."},{"Start":"04:50.300 ","End":"04:58.400","Text":"I did this already. The intersection with the axis is actually when x is 0 or 4."},{"Start":"04:58.400 ","End":"05:00.040","Text":"I\u0027ll just add this in here."},{"Start":"05:00.040 ","End":"05:02.040","Text":"This is not part of the exercise,"},{"Start":"05:02.040 ","End":"05:04.565","Text":"this is like extra doing this sketch."},{"Start":"05:04.565 ","End":"05:11.190","Text":"Anyway, we know that the intersection with the axis is at 0 and 4."},{"Start":"05:11.190 ","End":"05:15.800","Text":"At 3, we know we get minus 27."},{"Start":"05:15.800 ","End":"05:16.880","Text":"It\u0027s somewhere way down,"},{"Start":"05:16.880 ","End":"05:18.875","Text":"but I\u0027ll just sketch it over here."},{"Start":"05:18.875 ","End":"05:21.335","Text":"That\u0027s pretty much it."},{"Start":"05:21.335 ","End":"05:24.620","Text":"But what I know is that here I have a minimum point."},{"Start":"05:24.620 ","End":"05:30.210","Text":"Here, it\u0027s increasing and it\u0027s got to go through here so we\u0027ve got something like this."},{"Start":"05:30.210 ","End":"05:32.165","Text":"Here, we know it goes through here."},{"Start":"05:32.165 ","End":"05:34.205","Text":"But the derivative here is 0,"},{"Start":"05:34.205 ","End":"05:36.635","Text":"means the tangent is horizontal,"},{"Start":"05:36.635 ","End":"05:43.730","Text":"rests for an infinitesimal amount of time and then decreases and then keeps going up."},{"Start":"05:43.730 ","End":"05:45.710","Text":"This keeps going down,"},{"Start":"05:45.710 ","End":"05:49.865","Text":"rests for a moment and then continues downwards."},{"Start":"05:49.865 ","End":"05:52.885","Text":"That\u0027s what, briefly what it looks like."},{"Start":"05:52.885 ","End":"05:54.739","Text":"Here, we have a minimum."},{"Start":"05:54.739 ","End":"05:59.815","Text":"This thing, which we don\u0027t have a name for it is actually called a point of inflection."},{"Start":"05:59.815 ","End":"06:03.200","Text":"I\u0027m going to write it, but we don\u0027t have to know about this yet."},{"Start":"06:03.200 ","End":"06:05.710","Text":"It\u0027s not a minimum and not a maximum,"},{"Start":"06:05.710 ","End":"06:08.015","Text":"that\u0027s what this question mark is."},{"Start":"06:08.015 ","End":"06:10.970","Text":"Now, the only thing we haven\u0027t answered from the question,"},{"Start":"06:10.970 ","End":"06:12.710","Text":"we\u0027ve answered about the extrema,"},{"Start":"06:12.710 ","End":"06:14.015","Text":"we have a minimum here,"},{"Start":"06:14.015 ","End":"06:17.825","Text":"is the domain for the intervals of increase and decrease."},{"Start":"06:17.825 ","End":"06:19.810","Text":"Well, for the increase,"},{"Start":"06:19.810 ","End":"06:22.640","Text":"increase is just where x is bigger than 3."},{"Start":"06:22.640 ","End":"06:29.255","Text":"Now, the point is that the decrease wouldn\u0027t be very wrong if you wrote x is less than 0,"},{"Start":"06:29.255 ","End":"06:32.370","Text":"or 0 less than x, less than 3."},{"Start":"06:32.370 ","End":"06:35.415","Text":"In other words, if you took these 2 intervals separately,"},{"Start":"06:35.415 ","End":"06:38.290","Text":"the less than 0 and the between 0 and 3."},{"Start":"06:38.290 ","End":"06:41.600","Text":"But actually it\u0027s decreasing also at 0,"},{"Start":"06:41.600 ","End":"06:43.250","Text":"because here we\u0027re always positive,"},{"Start":"06:43.250 ","End":"06:45.500","Text":"this part\u0027s decreasing and still positive,"},{"Start":"06:45.500 ","End":"06:47.390","Text":"0 is less than positive,"},{"Start":"06:47.390 ","End":"06:48.770","Text":"so it\u0027s still decreasing,"},{"Start":"06:48.770 ","End":"06:51.964","Text":"and 0 is also a larger than all these negatives."},{"Start":"06:51.964 ","End":"06:53.690","Text":"So it\u0027s decreasing here,"},{"Start":"06:53.690 ","End":"06:55.100","Text":"even if you include the 0."},{"Start":"06:55.100 ","End":"06:56.839","Text":"In other words, it keeps decreasing."},{"Start":"06:56.839 ","End":"06:59.665","Text":"There\u0027s nowhere that it stays at the same height."},{"Start":"06:59.665 ","End":"07:05.075","Text":"So it\u0027s actually more correct even to write just that x is less than 3,"},{"Start":"07:05.075 ","End":"07:07.195","Text":"all the way down to minus infinity."},{"Start":"07:07.195 ","End":"07:08.685","Text":"From here to the left,"},{"Start":"07:08.685 ","End":"07:11.645","Text":"decreases and from here to the right, increasing."},{"Start":"07:11.645 ","End":"07:13.910","Text":"That answers the question and it actually"},{"Start":"07:13.910 ","End":"07:16.625","Text":"illustrates what I wanted is what we have a value,"},{"Start":"07:16.625 ","End":"07:21.530","Text":"which makes f prime of x 0 yet is neither a minimum nor maximum."},{"Start":"07:21.530 ","End":"07:23.910","Text":"Okay, I\u0027m done."}],"ID":8433},{"Watched":false,"Name":"Exercise 1","Duration":"4m ","ChapterTopicVideoID":4815,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4815.jpeg","UploadDate":"2015-07-15T04:30:41.4730000","DurationForVideoObject":"PT4M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:02.430 ","End":"00:06.450","Text":"f of x equals x squared minus 2x plus 5."},{"Start":"00:06.450 ","End":"00:11.070","Text":"We have to find all local extrema of the function."},{"Start":"00:11.070 ","End":"00:13.680","Text":"We also have to give the intervals where"},{"Start":"00:13.680 ","End":"00:16.780","Text":"the function is increasing and decreasing."},{"Start":"00:16.780 ","End":"00:19.335","Text":"These all follow as a routine,"},{"Start":"00:19.335 ","End":"00:21.060","Text":"always the same steps."},{"Start":"00:21.060 ","End":"00:22.230","Text":"The first thing to do"},{"Start":"00:22.230 ","End":"00:24.345","Text":"is to differentiate the function."},{"Start":"00:24.345 ","End":"00:28.620","Text":"Differentiating, we get f prime of x"},{"Start":"00:28.620 ","End":"00:32.850","Text":"is equal to 2x minus 2."},{"Start":"00:32.850 ","End":"00:34.665","Text":"That\u0027s the preparation stage."},{"Start":"00:34.665 ","End":"00:36.930","Text":"The next thing to do is to equate"},{"Start":"00:36.930 ","End":"00:40.150","Text":"the derivative to 0 and solve."},{"Start":"00:40.150 ","End":"00:43.655","Text":"I get f prime x equals 0."},{"Start":"00:43.655 ","End":"00:45.020","Text":"I\u0027m comparing it to 0,"},{"Start":"00:45.020 ","End":"00:49.305","Text":"which means 2x minus 2 equals 0."},{"Start":"00:49.305 ","End":"00:51.320","Text":"Then I solve the equation."},{"Start":"00:51.320 ","End":"00:53.930","Text":"Clearly, this gives me 2x equals 2,"},{"Start":"00:53.930 ","End":"00:55.970","Text":"so x equals 1."},{"Start":"00:55.970 ","End":"00:59.795","Text":"Now, next thing to do is to make a table."},{"Start":"00:59.795 ","End":"01:03.529","Text":"The first row is going to be x."},{"Start":"01:03.529 ","End":"01:07.220","Text":"Second row, I call f prime of x."},{"Start":"01:07.220 ","End":"01:10.790","Text":"The third row will be labeled f of x."},{"Start":"01:10.790 ","End":"01:13.900","Text":"Finally, there\u0027ll be a row called y."},{"Start":"01:13.900 ","End":"01:15.980","Text":"The values we put in here"},{"Start":"01:15.980 ","End":"01:18.380","Text":"are all the values we find for a solution"},{"Start":"01:18.380 ","End":"01:20.285","Text":"for f prime of x equals 0."},{"Start":"01:20.285 ","End":"01:22.595","Text":"In this case, there is only 1 value."},{"Start":"01:22.595 ","End":"01:24.755","Text":"That\u0027s the only 1 I put in here,"},{"Start":"01:24.755 ","End":"01:26.460","Text":"which is x is 1."},{"Start":"01:26.460 ","End":"01:29.540","Text":"At this point also f prime of x is 0,"},{"Start":"01:29.540 ","End":"01:31.040","Text":"because that\u0027s how we found the 1"},{"Start":"01:31.040 ","End":"01:32.885","Text":"by equating f prime to 0."},{"Start":"01:32.885 ","End":"01:33.950","Text":"Now, we leave spaces"},{"Start":"01:33.950 ","End":"01:35.180","Text":"if they may be more than 1,"},{"Start":"01:35.180 ","End":"01:36.305","Text":"and another exercise."},{"Start":"01:36.305 ","End":"01:37.333","Text":"Between the spaces,"},{"Start":"01:37.333 ","End":"01:38.525","Text":"we write intervals."},{"Start":"01:38.525 ","End":"01:40.370","Text":"For example, to the left of 1,"},{"Start":"01:40.370 ","End":"01:42.740","Text":"we have the interval which"},{"Start":"01:42.740 ","End":"01:45.950","Text":"I label x less than 1."},{"Start":"01:45.950 ","End":"01:47.150","Text":"On the other side,"},{"Start":"01:47.150 ","End":"01:49.430","Text":"we have the interval x bigger than."},{"Start":"01:49.430 ","End":"01:50.930","Text":"It\u0027s trivial here."},{"Start":"01:50.930 ","End":"01:52.400","Text":"But when there\u0027s more than 1 solution,"},{"Start":"01:52.400 ","End":"01:54.005","Text":"it\u0027s not so trivial."},{"Start":"01:54.005 ","End":"01:55.789","Text":"Now, from this interval,"},{"Start":"01:55.789 ","End":"01:57.935","Text":"you choose an arbitrary value,"},{"Start":"01:57.935 ","End":"02:00.030","Text":"any value that\u0027s less than 1."},{"Start":"02:00.030 ","End":"02:02.210","Text":"I\u0027ll pick 0, though you could"},{"Start":"02:02.210 ","End":"02:03.500","Text":"have picked something else."},{"Start":"02:03.500 ","End":"02:04.970","Text":"For x bigger than 1,"},{"Start":"02:04.970 ","End":"02:07.235","Text":"seems natural to pick 2."},{"Start":"02:07.235 ","End":"02:08.680","Text":"What we do with the 0 and 2"},{"Start":"02:08.680 ","End":"02:11.030","Text":"is substitute into f prime of x,"},{"Start":"02:11.030 ","End":"02:12.980","Text":"which is this function here."},{"Start":"02:12.980 ","End":"02:15.005","Text":"But we don\u0027t write the number down."},{"Start":"02:15.005 ","End":"02:16.320","Text":"Just substitute first,"},{"Start":"02:16.320 ","End":"02:18.075","Text":"and I\u0027ll tell you what to do with that."},{"Start":"02:18.075 ","End":"02:20.165","Text":"Substitute x equals 0,"},{"Start":"02:20.165 ","End":"02:22.910","Text":"twice 0 minus 2 is minus 2."},{"Start":"02:22.910 ","End":"02:24.710","Text":"I don\u0027t write minus 2,"},{"Start":"02:24.710 ","End":"02:25.880","Text":"I just write the sign,"},{"Start":"02:25.880 ","End":"02:27.520","Text":"whether it\u0027s plus or minus."},{"Start":"02:27.520 ","End":"02:29.750","Text":"Minus 2 is obviously negative."},{"Start":"02:29.750 ","End":"02:32.645","Text":"Similarly, if I put in x equals 2,"},{"Start":"02:32.645 ","End":"02:34.310","Text":"twice 2 minus 2 is 2,"},{"Start":"02:34.310 ","End":"02:35.614","Text":"and that\u0027s positive."},{"Start":"02:35.614 ","End":"02:37.400","Text":"I put a plus sign here,"},{"Start":"02:37.400 ","End":"02:38.720","Text":"and they\u0027re all worth f of x."},{"Start":"02:38.720 ","End":"02:41.180","Text":"I interpret what this minus means."},{"Start":"02:41.180 ","End":"02:43.295","Text":"When f prime is negative,"},{"Start":"02:43.295 ","End":"02:44.660","Text":"that means that the function"},{"Start":"02:44.660 ","End":"02:47.000","Text":"is decreasing and I just give that"},{"Start":"02:47.000 ","End":"02:50.810","Text":"a symbol of a diagonal arrow decreasing."},{"Start":"02:50.810 ","End":"02:52.910","Text":"Similarly, if f prime is positive,"},{"Start":"02:52.910 ","End":"02:54.530","Text":"the function is increasing,"},{"Start":"02:54.530 ","End":"02:55.910","Text":"and I also write a diagonal arrow,"},{"Start":"02:55.910 ","End":"02:57.050","Text":"but going the other way,"},{"Start":"02:57.050 ","End":"03:00.470","Text":"between the area interval of decrease"},{"Start":"03:00.470 ","End":"03:02.060","Text":"and an interval of increase."},{"Start":"03:02.060 ","End":"03:03.380","Text":"The point in between"},{"Start":"03:03.380 ","End":"03:06.020","Text":"is an extremum of type minimum."},{"Start":"03:06.020 ","End":"03:07.940","Text":"When we go down to the point"},{"Start":"03:07.940 ","End":"03:09.140","Text":"and then up away from it,"},{"Start":"03:09.140 ","End":"03:11.015","Text":"that\u0027s always a minimum."},{"Start":"03:11.015 ","End":"03:13.480","Text":"The minimum is where x equals 1."},{"Start":"03:13.480 ","End":"03:16.395","Text":"I\u0027d like to know what y is at that point."},{"Start":"03:16.395 ","End":"03:18.800","Text":"If I substitute in the original"},{"Start":"03:18.800 ","End":"03:20.480","Text":"function x equals 1,"},{"Start":"03:20.480 ","End":"03:23.360","Text":"I\u0027ll get y equals 4."},{"Start":"03:23.360 ","End":"03:24.140","Text":"At this point,"},{"Start":"03:24.140 ","End":"03:25.100","Text":"I\u0027m ready to answer"},{"Start":"03:25.100 ","End":"03:26.660","Text":"all the questions to draw"},{"Start":"03:26.660 ","End":"03:29.045","Text":"the conclusions that we have."},{"Start":"03:29.045 ","End":"03:30.950","Text":"This of course, is an extremum."},{"Start":"03:30.950 ","End":"03:33.755","Text":"When we\u0027re asked to find all the extrema,"},{"Start":"03:33.755 ","End":"03:35.490","Text":"there is only 1."},{"Start":"03:35.490 ","End":"03:36.859","Text":"That is the minimum"},{"Start":"03:36.859 ","End":"03:39.980","Text":"which occurs at the point 1, 4."},{"Start":"03:39.980 ","End":"03:41.390","Text":"When we\u0027re asked for areas"},{"Start":"03:41.390 ","End":"03:44.150","Text":"or intervals of increase,"},{"Start":"03:44.150 ","End":"03:45.690","Text":"we look for the up arrow,"},{"Start":"03:45.690 ","End":"03:47.870","Text":"and that gives us x bigger than 1,"},{"Start":"03:47.870 ","End":"03:49.560","Text":"just 1 interval."},{"Start":"03:49.560 ","End":"03:52.455","Text":"There\u0027s 1 interval of decrease."},{"Start":"03:52.455 ","End":"03:54.530","Text":"That\u0027s where the arrow goes down,"},{"Start":"03:54.530 ","End":"03:57.095","Text":"and that\u0027s x less than 1."},{"Start":"03:57.095 ","End":"03:58.610","Text":"That answers all the questions."},{"Start":"03:58.610 ","End":"04:00.720","Text":"We\u0027re done."}],"ID":4815},{"Watched":false,"Name":"Exercise 2","Duration":"4m 42s","ChapterTopicVideoID":4801,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4801.jpeg","UploadDate":"2015-07-15T04:17:30.5970000","DurationForVideoObject":"PT4M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"In this exercise, we\u0027re given a function f of x is equal to x cubed minus 3x."},{"Start":"00:05.415 ","End":"00:09.540","Text":"For this function, we have to find all the local extrema and also to"},{"Start":"00:09.540 ","End":"00:13.620","Text":"give the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:13.620 ","End":"00:17.180","Text":"This type of exercise is routine and always follows the same steps,"},{"Start":"00:17.180 ","End":"00:19.860","Text":"but first thing to do is to find the derivative."},{"Start":"00:19.860 ","End":"00:21.265","Text":"That\u0027s how we always start."},{"Start":"00:21.265 ","End":"00:25.160","Text":"This is immediate, 3x squared minus 3."},{"Start":"00:25.160 ","End":"00:27.690","Text":"The next thing we do is compare it to 0,"},{"Start":"00:27.690 ","End":"00:30.480","Text":"and then I get x squared equals 1."},{"Start":"00:30.480 ","End":"00:32.850","Text":"If I bring the 3 over and divide by 3,"},{"Start":"00:32.850 ","End":"00:37.860","Text":"and then I get 2 values of x. X is either equal to minus 1 or to 1."},{"Start":"00:37.860 ","End":"00:39.135","Text":"There\u0027s a 2 solutions."},{"Start":"00:39.135 ","End":"00:41.495","Text":"The next step is to make a table,"},{"Start":"00:41.495 ","End":"00:45.065","Text":"the table has the following rows x,"},{"Start":"00:45.065 ","End":"00:47.270","Text":"f prime of x,"},{"Start":"00:47.270 ","End":"00:50.590","Text":"f of x, and finally, y."},{"Start":"00:50.590 ","End":"00:54.395","Text":"We start off with the x. Here we put in all the values where we got"},{"Start":"00:54.395 ","End":"00:58.400","Text":"f prime of x equals 0 and leave some space between them."},{"Start":"00:58.400 ","End":"01:00.845","Text":"I\u0027ll put the minus 1 here and 1 here."},{"Start":"01:00.845 ","End":"01:03.320","Text":"We put them in increasing order always."},{"Start":"01:03.320 ","End":"01:06.770","Text":"The next thing we do is write f prime of x at these points is"},{"Start":"01:06.770 ","End":"01:11.210","Text":"0 and between these points and around them are intervals."},{"Start":"01:11.210 ","End":"01:14.150","Text":"I mean these points split the line up basically into 3 intervals."},{"Start":"01:14.150 ","End":"01:15.425","Text":"This, this, and this,"},{"Start":"01:15.425 ","End":"01:17.165","Text":"and I write what these intervals are."},{"Start":"01:17.165 ","End":"01:20.355","Text":"For example, here it\u0027s x less than minus 1."},{"Start":"01:20.355 ","End":"01:22.470","Text":"Here it\u0027s x bigger than 1,"},{"Start":"01:22.470 ","End":"01:25.595","Text":"and between them we have x is between minus 1 and 1."},{"Start":"01:25.595 ","End":"01:28.160","Text":"Next thing we do is choose a sample or"},{"Start":"01:28.160 ","End":"01:32.570","Text":"a representative number from each interval from x less than minus 1,"},{"Start":"01:32.570 ","End":"01:34.370","Text":"I\u0027ll just choose minus 2,"},{"Start":"01:34.370 ","End":"01:37.820","Text":"any value will do, x between minus 1 and 1."},{"Start":"01:37.820 ","End":"01:40.820","Text":"I\u0027ll pick 0, x bigger than 1, I\u0027ll pick 2."},{"Start":"01:40.820 ","End":"01:47.465","Text":"Using these values, we substitute into f prime of x. F prime of x is what we have here."},{"Start":"01:47.465 ","End":"01:49.070","Text":"We substitute these values,"},{"Start":"01:49.070 ","End":"01:50.690","Text":"but we don\u0027t write down the answer,"},{"Start":"01:50.690 ","End":"01:53.030","Text":"just the sign whether it\u0027s positive or negative."},{"Start":"01:53.030 ","End":"01:54.785","Text":"If I put in minus 2,"},{"Start":"01:54.785 ","End":"01:56.450","Text":"minus 2 squared is 4,"},{"Start":"01:56.450 ","End":"01:57.680","Text":"3 times 4 is 12,"},{"Start":"01:57.680 ","End":"02:00.275","Text":"minus 3 is 9, but I don\u0027t write 9 here,"},{"Start":"02:00.275 ","End":"02:01.595","Text":"I just write plus,"},{"Start":"02:01.595 ","End":"02:05.390","Text":"I put x equals 0 and I get 3 times 0 minus 3 is minus 3,"},{"Start":"02:05.390 ","End":"02:06.680","Text":"but I don\u0027t put the minus 3,"},{"Start":"02:06.680 ","End":"02:07.850","Text":"I just put the minus."},{"Start":"02:07.850 ","End":"02:10.340","Text":"By the way, it turns out that if I put any number in"},{"Start":"02:10.340 ","End":"02:12.830","Text":"this interval I may get different numerical answers,"},{"Start":"02:12.830 ","End":"02:13.940","Text":"but they\u0027ll all be minus."},{"Start":"02:13.940 ","End":"02:15.200","Text":"If I choose any number from here,"},{"Start":"02:15.200 ","End":"02:16.370","Text":"it will always be plus."},{"Start":"02:16.370 ","End":"02:18.230","Text":"It doesn\u0027t matter which one you choose."},{"Start":"02:18.230 ","End":"02:19.280","Text":"The numbers will be different,"},{"Start":"02:19.280 ","End":"02:21.080","Text":"but the plus-minus will be the same."},{"Start":"02:21.080 ","End":"02:22.220","Text":"Here if I put in 2,"},{"Start":"02:22.220 ","End":"02:23.930","Text":"I get the same as for minus 2."},{"Start":"02:23.930 ","End":"02:25.750","Text":"This will also be a plus."},{"Start":"02:25.750 ","End":"02:27.860","Text":"In this row where it says f of x,"},{"Start":"02:27.860 ","End":"02:29.930","Text":"I interpret these symbols,"},{"Start":"02:29.930 ","End":"02:32.720","Text":"plus and minus. A derivative is positive."},{"Start":"02:32.720 ","End":"02:35.420","Text":"That means that the function is increasing,"},{"Start":"02:35.420 ","End":"02:38.960","Text":"an increasing is indicated by an arrow up like this,"},{"Start":"02:38.960 ","End":"02:42.950","Text":"here is a minus, so f prime is negative here and f prime is negative,"},{"Start":"02:42.950 ","End":"02:44.330","Text":"the function is decreasing."},{"Start":"02:44.330 ","End":"02:46.220","Text":"I indicated by an arrow like this,"},{"Start":"02:46.220 ","End":"02:47.450","Text":"as I go from left to right,"},{"Start":"02:47.450 ","End":"02:48.979","Text":"I go down, that\u0027s decreasing."},{"Start":"02:48.979 ","End":"02:50.585","Text":"Here once again a plus,"},{"Start":"02:50.585 ","End":"02:55.265","Text":"so increasing and now the function is increasing and decreasing."},{"Start":"02:55.265 ","End":"02:57.140","Text":"I could already answer part of the question,"},{"Start":"02:57.140 ","End":"02:59.600","Text":"but I want to answer about the extrema first."},{"Start":"02:59.600 ","End":"03:04.115","Text":"When x is minus 1, it\u0027s between an area of increase and decrease."},{"Start":"03:04.115 ","End":"03:06.785","Text":"When it\u0027s between an increase and a decrease,"},{"Start":"03:06.785 ","End":"03:10.145","Text":"then it\u0027s always an extremum and specifically, it\u0027s a maximum."},{"Start":"03:10.145 ","End":"03:12.605","Text":"We go up to it and then down away from it."},{"Start":"03:12.605 ","End":"03:14.975","Text":"Conversely, at x equals 1,"},{"Start":"03:14.975 ","End":"03:18.410","Text":"it\u0027s between a decrease and an increase and so it\u0027s a minimum."},{"Start":"03:18.410 ","End":"03:22.715","Text":"The function goes down to the minimum and then up away from it. This is minimum."},{"Start":"03:22.715 ","End":"03:25.295","Text":"The last thing is that for these 2 values,"},{"Start":"03:25.295 ","End":"03:28.445","Text":"I\u0027d like to get the y-value minus 1."},{"Start":"03:28.445 ","End":"03:31.745","Text":"I put it into the original function, not the derivative."},{"Start":"03:31.745 ","End":"03:35.995","Text":"We get minus 1 cubed is minus 1,"},{"Start":"03:35.995 ","End":"03:38.420","Text":"minus 3x is plus 3."},{"Start":"03:38.420 ","End":"03:41.600","Text":"Minus 1 plus 3 seems to be plus 2."},{"Start":"03:41.600 ","End":"03:43.640","Text":"If I put x equals 1,"},{"Start":"03:43.640 ","End":"03:47.360","Text":"I get 1 cubed minus 3 times 1 is minus 2."},{"Start":"03:47.360 ","End":"03:48.770","Text":"Now, once we have this table,"},{"Start":"03:48.770 ","End":"03:50.315","Text":"we can write down everything."},{"Start":"03:50.315 ","End":"03:55.010","Text":"What we needed to say was where are the extrema and where is increasing,"},{"Start":"03:55.010 ","End":"03:56.330","Text":"and where it\u0027s decreasing."},{"Start":"03:56.330 ","End":"03:58.640","Text":"I\u0027ll write it here, extrema."},{"Start":"03:58.640 ","End":"04:00.425","Text":"We have 2 of these."},{"Start":"04:00.425 ","End":"04:02.330","Text":"We have minus 1,"},{"Start":"04:02.330 ","End":"04:04.265","Text":"2 and 1 minus 2,"},{"Start":"04:04.265 ","End":"04:05.930","Text":"but one of them is a maximum."},{"Start":"04:05.930 ","End":"04:12.090","Text":"Let\u0027s say we have a maximum at minus 1, 2."},{"Start":"04:12.090 ","End":"04:16.220","Text":"The other extremum we have is a minimum at 1 minus 2."},{"Start":"04:16.220 ","End":"04:19.250","Text":"Now, the intervals of increase and decrease first,"},{"Start":"04:19.250 ","End":"04:22.385","Text":"the increase, we have where the arrow goes up,"},{"Start":"04:22.385 ","End":"04:25.370","Text":"so that\u0027s x less than minus 1."},{"Start":"04:25.370 ","End":"04:27.980","Text":"Also, here where the arrow goes up,"},{"Start":"04:27.980 ","End":"04:29.870","Text":"which is x bigger than 1,"},{"Start":"04:29.870 ","End":"04:33.695","Text":"and the interval of decrease is only one of them,"},{"Start":"04:33.695 ","End":"04:39.155","Text":"which is here, which is when x is between minus 1 and 1."},{"Start":"04:39.155 ","End":"04:42.510","Text":"That answers all the questions and we\u0027re done."}],"ID":4801},{"Watched":false,"Name":"Exercise 3","Duration":"4m 1s","ChapterTopicVideoID":4802,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4802.jpeg","UploadDate":"2015-07-15T04:17:52.6670000","DurationForVideoObject":"PT4M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we\u0027re given a function of x,"},{"Start":"00:03.270 ","End":"00:07.005","Text":"2x cubed minus 15x squared plus 24x minus 1."},{"Start":"00:07.005 ","End":"00:10.650","Text":"We are asked to find all the local extrema of the function and"},{"Start":"00:10.650 ","End":"00:15.075","Text":"also to give the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:15.075 ","End":"00:18.600","Text":"This kind of exercise is routine and always follows the same steps."},{"Start":"00:18.600 ","End":"00:23.340","Text":"The first step is to differentiate the function f prime of x is equal"},{"Start":"00:23.340 ","End":"00:31.100","Text":"to 6x squared minus 30x plus 24, and that\u0027s it."},{"Start":"00:31.100 ","End":"00:35.960","Text":"The next thing to do is to equate this to 0 and to solve this equals 0,"},{"Start":"00:35.960 ","End":"00:38.255","Text":"let me divide by 6."},{"Start":"00:38.255 ","End":"00:43.655","Text":"x squared minus 5x plus 4 equals 0,"},{"Start":"00:43.655 ","End":"00:46.130","Text":"we get a nice quadratic equation."},{"Start":"00:46.130 ","End":"00:48.740","Text":"I won\u0027t waste time with the formula solving it."},{"Start":"00:48.740 ","End":"00:50.375","Text":"I\u0027ll just tell you the solutions,"},{"Start":"00:50.375 ","End":"00:54.845","Text":"x equals 1 and x equals 4."},{"Start":"00:54.845 ","End":"00:56.270","Text":"Once we\u0027ve got the 2 X\u0027s,"},{"Start":"00:56.270 ","End":"00:57.910","Text":"then we make a nice table."},{"Start":"00:57.910 ","End":"01:01.280","Text":"The table should contain the following 4 rows."},{"Start":"01:01.280 ","End":"01:02.975","Text":"First one will be called x,"},{"Start":"01:02.975 ","End":"01:05.270","Text":"then we\u0027ll have f prime of x,"},{"Start":"01:05.270 ","End":"01:08.210","Text":"then f of x, and then y."},{"Start":"01:08.210 ","End":"01:10.640","Text":"The first thing I do is to put in the row where"},{"Start":"01:10.640 ","End":"01:14.350","Text":"the values of x we found which make the derivative 0."},{"Start":"01:14.350 ","End":"01:17.495","Text":"We put them in increasing order from left to right,"},{"Start":"01:17.495 ","End":"01:19.700","Text":"so 1 here and 4 here."},{"Start":"01:19.700 ","End":"01:21.800","Text":"Also, we leave space in between because"},{"Start":"01:21.800 ","End":"01:23.945","Text":"there\u0027s going to be extra information to be added."},{"Start":"01:23.945 ","End":"01:26.150","Text":"These 2 points divide the line."},{"Start":"01:26.150 ","End":"01:27.560","Text":"If you imagine them on the x line,"},{"Start":"01:27.560 ","End":"01:29.840","Text":"they divide the line into intervals."},{"Start":"01:29.840 ","End":"01:32.075","Text":"The first one is x less than 1."},{"Start":"01:32.075 ","End":"01:35.210","Text":"The next interval would be x is between 1 and 4"},{"Start":"01:35.210 ","End":"01:38.610","Text":"and the next interval is where x is greater than 4."},{"Start":"01:38.610 ","End":"01:42.950","Text":"Then we choose a sample or a representative from each interval."},{"Start":"01:42.950 ","End":"01:46.205","Text":"For example, for x less than 1, I could choose 0."},{"Start":"01:46.205 ","End":"01:47.885","Text":"For x between 1 and 4,"},{"Start":"01:47.885 ","End":"01:49.854","Text":"I\u0027ll choose 2 arbitrary."},{"Start":"01:49.854 ","End":"01:52.335","Text":"For x bigger than 4, I\u0027ll choose 5."},{"Start":"01:52.335 ","End":"01:53.970","Text":"Now, these sample points,"},{"Start":"01:53.970 ","End":"01:56.290","Text":"I substitute into f prime of x,"},{"Start":"01:56.290 ","End":"01:58.055","Text":"which is over here."},{"Start":"01:58.055 ","End":"02:00.410","Text":"But I don\u0027t write down the answer at the end."},{"Start":"02:00.410 ","End":"02:02.780","Text":"I just write down whether I\u0027ve got plus or minus."},{"Start":"02:02.780 ","End":"02:05.560","Text":"Obviously, I won\u0027t get 0 because it was a 0,"},{"Start":"02:05.560 ","End":"02:06.960","Text":"it would be in black already."},{"Start":"02:06.960 ","End":"02:11.190","Text":"Put x equals 0 and we have f prime is 24,"},{"Start":"02:11.190 ","End":"02:14.025","Text":"24 is positive, so I put a plus here."},{"Start":"02:14.025 ","End":"02:19.060","Text":"Substitute 2, 6 times 4 is 24 plus 24 is 48,"},{"Start":"02:19.060 ","End":"02:22.025","Text":"48 minus 60 negative,"},{"Start":"02:22.025 ","End":"02:25.845","Text":"put x equals 5, 5 squared is 25,"},{"Start":"02:25.845 ","End":"02:29.655","Text":"this is 150, 150 minus 150,"},{"Start":"02:29.655 ","End":"02:32.340","Text":"0 plus 24 is positive."},{"Start":"02:32.340 ","End":"02:36.575","Text":"In f of x, underneath the orange symbols, I interpret them."},{"Start":"02:36.575 ","End":"02:38.270","Text":"If f prime is positive,"},{"Start":"02:38.270 ","End":"02:39.470","Text":"the derivative is positive,"},{"Start":"02:39.470 ","End":"02:41.165","Text":"then the function is increasing."},{"Start":"02:41.165 ","End":"02:42.875","Text":"If the derivative is negative,"},{"Start":"02:42.875 ","End":"02:47.570","Text":"then the function is decreasing and I use these arrows, here again increasing."},{"Start":"02:47.570 ","End":"02:51.080","Text":"Last thing I do in the table, these values of X,"},{"Start":"02:51.080 ","End":"02:56.405","Text":"I find the corresponding values of Y by substituting in the original function."},{"Start":"02:56.405 ","End":"03:03.365","Text":"When X is 1, I get 2 minus 15 is minus 13 plus 24,"},{"Start":"03:03.365 ","End":"03:07.280","Text":"makes it plus 11 minus 1 is 10."},{"Start":"03:07.280 ","End":"03:11.030","Text":"If I substitute the 4 in f of x, well,"},{"Start":"03:11.030 ","End":"03:15.565","Text":"I did it already at the side and I made the answer minus 17."},{"Start":"03:15.565 ","End":"03:18.695","Text":"Next, we just have to draw the conclusions,"},{"Start":"03:18.695 ","End":"03:22.625","Text":"which is what we\u0027re asked for about the extrema and the intervals."},{"Start":"03:22.625 ","End":"03:26.150","Text":"As for extrema, there are 2 of them."},{"Start":"03:26.150 ","End":"03:29.595","Text":"We have a maximum at 1, 10."},{"Start":"03:29.595 ","End":"03:32.630","Text":"We have a maximum, the point 1, 10,"},{"Start":"03:32.630 ","End":"03:39.180","Text":"and we have a minimum at 4, minus 17."},{"Start":"03:39.180 ","End":"03:40.410","Text":"Those are the extrema."},{"Start":"03:40.410 ","End":"03:44.040","Text":"Then the intervals of increase,"},{"Start":"03:44.040 ","End":"03:47.030","Text":"I look for the upward arrow, it\u0027s here and here."},{"Start":"03:47.030 ","End":"03:51.110","Text":"x less than 1 and x bigger than 4."},{"Start":"03:51.110 ","End":"03:54.980","Text":"Then the decrease just here where the arrow goes down is"},{"Start":"03:54.980 ","End":"04:00.330","Text":"when x is between 1 and 4. We\u0027re done."}],"ID":4802},{"Watched":false,"Name":"Exercise 4","Duration":"5m 7s","ChapterTopicVideoID":4803,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4803.jpeg","UploadDate":"2015-07-15T04:18:18.6600000","DurationForVideoObject":"PT5M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.685","Text":"In this exercise, we\u0027re given a function."},{"Start":"00:02.685 ","End":"00:06.240","Text":"Here it is, x^4 minus 2x cubed and we have to find"},{"Start":"00:06.240 ","End":"00:08.310","Text":"its local extrema as well as"},{"Start":"00:08.310 ","End":"00:11.810","Text":"the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:11.810 ","End":"00:15.054","Text":"Let\u0027s begin. This kind of exercise"},{"Start":"00:15.054 ","End":"00:18.675","Text":"is well known and it goes in the standard set of steps."},{"Start":"00:18.675 ","End":"00:22.375","Text":"There\u0027s a preparation and then there are 2 main steps,"},{"Start":"00:22.375 ","End":"00:23.745","Text":"and then there\u0027s a conclusion."},{"Start":"00:23.745 ","End":"00:28.755","Text":"For the preparation, we have to find the derivative f prime of x, that\u0027s all."},{"Start":"00:28.755 ","End":"00:32.400","Text":"Let\u0027s do it, f prime of x is equal to,"},{"Start":"00:32.400 ","End":"00:34.170","Text":"this is standard polynomial,"},{"Start":"00:34.170 ","End":"00:39.750","Text":"so 4x cubed minus 2 times 3 is 6x squared,"},{"Start":"00:39.750 ","End":"00:46.115","Text":"and I can take 2x squared out of the brackets and get 2x squared."},{"Start":"00:46.115 ","End":"00:50.535","Text":"What we\u0027re left with here is a 2 here and an x here, so that\u0027s 2x."},{"Start":"00:50.535 ","End":"00:52.380","Text":"Here we\u0027re left with just 3,"},{"Start":"00:52.380 ","End":"00:54.435","Text":"so it\u0027s 2x minus 3."},{"Start":"00:54.435 ","End":"00:56.775","Text":"That\u0027s the preparation step."},{"Start":"00:56.775 ","End":"01:02.660","Text":"The first main step is to solve the equation f prime of x equals 0."},{"Start":"01:02.660 ","End":"01:06.910","Text":"The solutions to this will give us our suspects for extrema."},{"Start":"01:06.910 ","End":"01:10.125","Text":"This gives us that this is 0,"},{"Start":"01:10.125 ","End":"01:16.455","Text":"so 2x squared times 2x minus 3 is equal to 0."},{"Start":"01:16.455 ","End":"01:19.940","Text":"If we look at this, there\u0027s only 2 ways that this could be 0,"},{"Start":"01:19.940 ","End":"01:24.075","Text":"either x is 0 or 2x minus 3 is 0."},{"Start":"01:24.075 ","End":"01:25.640","Text":"If the latter is the case,"},{"Start":"01:25.640 ","End":"01:28.130","Text":"then x is equal to 3 over 2,"},{"Start":"01:28.130 ","End":"01:32.825","Text":"or if you prefer in decimals 1.5. That\u0027s that step."},{"Start":"01:32.825 ","End":"01:34.549","Text":"The next step is the table."},{"Start":"01:34.549 ","End":"01:37.100","Text":"Here\u0027s a blank table with the usual rows,"},{"Start":"01:37.100 ","End":"01:40.145","Text":"x, f prime, f and y."},{"Start":"01:40.145 ","End":"01:45.590","Text":"For the x, we put in the values which we suspect to be an extremum,"},{"Start":"01:45.590 ","End":"01:47.420","Text":"and these will be 0,"},{"Start":"01:47.420 ","End":"01:49.025","Text":"and 1 and 1/2."},{"Start":"01:49.025 ","End":"01:53.820","Text":"We put them in increasing order 0 and 1.5,"},{"Start":"01:53.820 ","End":"01:55.800","Text":"and we leave a bit of space around."},{"Start":"01:55.800 ","End":"01:58.970","Text":"F prime at these 2 points is 0,"},{"Start":"01:58.970 ","End":"02:02.750","Text":"because that\u0027s how we found these points where the derivative is 0."},{"Start":"02:02.750 ","End":"02:05.930","Text":"That makes it automatically a suspect for extremum."},{"Start":"02:05.930 ","End":"02:09.725","Text":"Then here we put the intervals that these 2 points define,"},{"Start":"02:09.725 ","End":"02:12.755","Text":"so we have the interval where x is less than 0,"},{"Start":"02:12.755 ","End":"02:16.635","Text":"the interval where x is between 0 and 1 and 1/2,"},{"Start":"02:16.635 ","End":"02:19.625","Text":"and the interval where x is bigger than 1 and 1/2."},{"Start":"02:19.625 ","End":"02:23.285","Text":"We choose an x arbitrarily from each interval."},{"Start":"02:23.285 ","End":"02:25.490","Text":"From here I\u0027ll choose minus 1,"},{"Start":"02:25.490 ","End":"02:28.325","Text":"from here I\u0027ll choose x equals 1,"},{"Start":"02:28.325 ","End":"02:31.220","Text":"and from here I\u0027ll choose x equals 2."},{"Start":"02:31.220 ","End":"02:36.140","Text":"What I need to do is to plug these into the f prime function,"},{"Start":"02:36.140 ","End":"02:38.230","Text":"which is equal to this,"},{"Start":"02:38.230 ","End":"02:39.910","Text":"but we don\u0027t want the actual answer,"},{"Start":"02:39.910 ","End":"02:41.920","Text":"only whether it\u0027s positive or negative."},{"Start":"02:41.920 ","End":"02:43.645","Text":"If I put in minus 1,"},{"Start":"02:43.645 ","End":"02:46.194","Text":"I get this would be positive always,"},{"Start":"02:46.194 ","End":"02:48.315","Text":"and minus 1 times 2,"},{"Start":"02:48.315 ","End":"02:51.050","Text":"minus 3 is obviously negative."},{"Start":"02:51.050 ","End":"02:53.390","Text":"Next, I put in x equals 1."},{"Start":"02:53.390 ","End":"02:55.550","Text":"Again, this bit is positive,"},{"Start":"02:55.550 ","End":"02:58.775","Text":"twice 1 minus 3 is negative."},{"Start":"02:58.775 ","End":"03:00.500","Text":"We have a negative here."},{"Start":"03:00.500 ","End":"03:02.270","Text":"If I put in 2, again,"},{"Start":"03:02.270 ","End":"03:05.390","Text":"positive times 4 minus 3, it\u0027s plus,"},{"Start":"03:05.390 ","End":"03:08.600","Text":"which means that here the function is decreasing,"},{"Start":"03:08.600 ","End":"03:10.865","Text":"derivative negative means decreasing,"},{"Start":"03:10.865 ","End":"03:12.035","Text":"as well as here,"},{"Start":"03:12.035 ","End":"03:17.060","Text":"and here, derivative positive means that the function is increasing."},{"Start":"03:17.060 ","End":"03:21.860","Text":"When a function is between a decreasing and an increasing or the other way around,"},{"Start":"03:21.860 ","End":"03:23.180","Text":"then we have an extremum."},{"Start":"03:23.180 ","End":"03:25.010","Text":"But between decreasing and increasing,"},{"Start":"03:25.010 ","End":"03:27.680","Text":"I can specifically say that this is of type minimum,"},{"Start":"03:27.680 ","End":"03:29.660","Text":"which I abbreviate to MIN."},{"Start":"03:29.660 ","End":"03:31.745","Text":"But here between 2 decreasing,"},{"Start":"03:31.745 ","End":"03:33.365","Text":"we don\u0027t have an extremum."},{"Start":"03:33.365 ","End":"03:36.020","Text":"It\u0027s most likely a point of inflection,"},{"Start":"03:36.020 ","End":"03:37.840","Text":"but I don\u0027t want to say that yet."},{"Start":"03:37.840 ","End":"03:40.235","Text":"I don\u0027t know what it is or to say."},{"Start":"03:40.235 ","End":"03:44.885","Text":"All I still need from the table is the y value that belongs to"},{"Start":"03:44.885 ","End":"03:50.720","Text":"1.5 for x and y is just equal to f of x."},{"Start":"03:50.720 ","End":"03:54.145","Text":"We have to substitute 1 and 1/2 in here."},{"Start":"03:54.145 ","End":"03:56.415","Text":"Let me do it at the side."},{"Start":"03:56.415 ","End":"04:04.560","Text":"I compute it at minus 1.6875."},{"Start":"04:04.560 ","End":"04:07.460","Text":"We finished this table stage."},{"Start":"04:07.460 ","End":"04:11.900","Text":"Now the last stage is the conclusions which answer the questions that were asked,"},{"Start":"04:11.900 ","End":"04:16.970","Text":"and that means that we have the extrema, we just have 1,"},{"Start":"04:16.970 ","End":"04:24.480","Text":"and that is at the point 1.5 minus 1.6875,"},{"Start":"04:24.480 ","End":"04:26.370","Text":"and it\u0027s of type minimum,"},{"Start":"04:26.370 ","End":"04:32.000","Text":"and the intervals where the function is increasing is where I have"},{"Start":"04:32.000 ","End":"04:38.330","Text":"the orange up arrow and that is x bigger than 1.5."},{"Start":"04:38.330 ","End":"04:41.920","Text":"For decreasing, we have 2 intervals,"},{"Start":"04:41.920 ","End":"04:46.025","Text":"but there\u0027s a single missing point in between where the function is defined."},{"Start":"04:46.025 ","End":"04:50.975","Text":"In this case, you\u0027re allowed to combine the 2 intervals and make it 1 big interval,"},{"Start":"04:50.975 ","End":"04:56.925","Text":"so together it\u0027s x less than 1.5."},{"Start":"04:56.925 ","End":"05:03.680","Text":"Although you could say separately, x less than 0 or x between 0 and 1.5."},{"Start":"05:03.680 ","End":"05:08.010","Text":"That\u0027s it, that answers the questions, and we\u0027re done."}],"ID":4803},{"Watched":false,"Name":"Exercise 5","Duration":"7m 29s","ChapterTopicVideoID":4804,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4804.jpeg","UploadDate":"2015-07-15T04:18:58.9870000","DurationForVideoObject":"PT7M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.770","Text":"In this exercise, we\u0027re given the function f of x equals 3x^5 minus 20x cubed."},{"Start":"00:07.770 ","End":"00:11.280","Text":"What we have to do is to find all the local extrema,"},{"Start":"00:11.280 ","End":"00:15.420","Text":"and the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:15.420 ","End":"00:18.510","Text":"The first thing I do right after I\u0027ve copied"},{"Start":"00:18.510 ","End":"00:22.425","Text":"the exercise is to differentiate this function."},{"Start":"00:22.425 ","End":"00:24.450","Text":"We have to find f prime of x,"},{"Start":"00:24.450 ","End":"00:30.929","Text":"and that is equal to 15x^4 minus 60x squared,"},{"Start":"00:30.929 ","End":"00:35.595","Text":"and then we set it equal to 0 and bind the solutions of this equation."},{"Start":"00:35.595 ","End":"00:38.775","Text":"It\u0027ll be easier if I divide throughout by 15,"},{"Start":"00:38.775 ","End":"00:45.170","Text":"and which case I get x. I think I\u0027m going to also take x squared outside the brackets."},{"Start":"00:45.170 ","End":"00:47.760","Text":"In this case, I will divide by 15,"},{"Start":"00:47.760 ","End":"00:50.040","Text":"and then I\u0027ll get 1 here and 4 here,"},{"Start":"00:50.040 ","End":"00:52.440","Text":"then I\u0027ll take x squared outside the bracket,"},{"Start":"00:52.440 ","End":"00:57.650","Text":"so I\u0027ve got x squared times x squared minus 4 equals 0."},{"Start":"00:57.650 ","End":"00:59.255","Text":"I think we can do this in 1 step."},{"Start":"00:59.255 ","End":"01:01.010","Text":"Either x squared is 0,"},{"Start":"01:01.010 ","End":"01:04.385","Text":"which makes x equals also 0,"},{"Start":"01:04.385 ","End":"01:06.710","Text":"or x squared minus 4 is 0."},{"Start":"01:06.710 ","End":"01:08.510","Text":"If x squared minus 4 is 0,"},{"Start":"01:08.510 ","End":"01:13.145","Text":"then x squared is 4, so x is either minus 2 or 2."},{"Start":"01:13.145 ","End":"01:15.440","Text":"You could also factorize this x minus 2,"},{"Start":"01:15.440 ","End":"01:17.060","Text":"x plus 2, and so on."},{"Start":"01:17.060 ","End":"01:18.895","Text":"Now, with these 3 solutions,"},{"Start":"01:18.895 ","End":"01:20.605","Text":"we now make a table,"},{"Start":"01:20.605 ","End":"01:23.975","Text":"and the table contains 4 rows, as always,"},{"Start":"01:23.975 ","End":"01:26.375","Text":"x, f prime of x,"},{"Start":"01:26.375 ","End":"01:29.060","Text":"f of x, and y."},{"Start":"01:29.060 ","End":"01:32.960","Text":"In other way, the y is just f of x in the original equation."},{"Start":"01:32.960 ","End":"01:35.485","Text":"These are the 4 row headers,"},{"Start":"01:35.485 ","End":"01:37.550","Text":"and then make room for the rows."},{"Start":"01:37.550 ","End":"01:43.760","Text":"What we do is put these values into these rows in the increasing order."},{"Start":"01:43.760 ","End":"01:45.950","Text":"I would get minus 2,"},{"Start":"01:45.950 ","End":"01:49.085","Text":"and then 0, and then 2."},{"Start":"01:49.085 ","End":"01:54.410","Text":"The derivative at these 3 points is known because at all these 3 points it\u0027s 0."},{"Start":"01:54.410 ","End":"01:58.370","Text":"I mean, that\u0027s how we found these points by equating f prime to 0."},{"Start":"01:58.370 ","End":"02:04.440","Text":"The next thing is to write the intervals that these 3 points split the line into."},{"Start":"02:04.440 ","End":"02:06.920","Text":"We have the interval up to this point between"},{"Start":"02:06.920 ","End":"02:09.470","Text":"these 2 points between here, I\u0027ll just write it down."},{"Start":"02:09.470 ","End":"02:14.660","Text":"The line is separated here into an interval x less than minus 2,"},{"Start":"02:14.660 ","End":"02:17.090","Text":"between these, so minus 2,"},{"Start":"02:17.090 ","End":"02:18.910","Text":"less than x, less than 0."},{"Start":"02:18.910 ","End":"02:20.825","Text":"The x is between these 2,"},{"Start":"02:20.825 ","End":"02:23.570","Text":"0, less than x, less than 2,"},{"Start":"02:23.570 ","End":"02:26.105","Text":"and the x is to the right of 2,"},{"Start":"02:26.105 ","End":"02:27.695","Text":"x bigger than 2,"},{"Start":"02:27.695 ","End":"02:32.000","Text":"and then we take a sample x from each of these intervals,"},{"Start":"02:32.000 ","End":"02:33.380","Text":"4 of them in this case."},{"Start":"02:33.380 ","End":"02:35.915","Text":"It doesn\u0027t matter which arbitrary,"},{"Start":"02:35.915 ","End":"02:37.370","Text":"whatever is convenient for us,"},{"Start":"02:37.370 ","End":"02:41.045","Text":"so for x less than 2 I\u0027ll choose minus 3,"},{"Start":"02:41.045 ","End":"02:42.470","Text":"for x between these 2,"},{"Start":"02:42.470 ","End":"02:44.105","Text":"I\u0027ll choose minus 1,"},{"Start":"02:44.105 ","End":"02:46.490","Text":"for x between these 2, I\u0027ll choose 1,"},{"Start":"02:46.490 ","End":"02:49.630","Text":"and for x bigger than 2, I\u0027ll choose 3."},{"Start":"02:49.630 ","End":"02:53.900","Text":"Now what we do is we look at f prime of x and"},{"Start":"02:53.900 ","End":"02:58.460","Text":"we substitute all these values we got into f prime of x."},{"Start":"02:58.460 ","End":"03:01.130","Text":"Let me start with some of the easier ones."},{"Start":"03:01.130 ","End":"03:03.080","Text":"If x is 1,"},{"Start":"03:03.080 ","End":"03:09.080","Text":"then I get 15 minus 60 is minus 45,"},{"Start":"03:09.080 ","End":"03:11.060","Text":"and that comes out negative."},{"Start":"03:11.060 ","End":"03:12.380","Text":"All I need is not the number,"},{"Start":"03:12.380 ","End":"03:14.330","Text":"but just the sign that it\u0027s negative."},{"Start":"03:14.330 ","End":"03:20.345","Text":"If x is 3, 3^4 is 81, 81 times 15,"},{"Start":"03:20.345 ","End":"03:27.720","Text":"1,215, and 9 times 60 is 540."},{"Start":"03:27.720 ","End":"03:30.160","Text":"This is bigger than this, it\u0027s positive."},{"Start":"03:30.160 ","End":"03:31.710","Text":"Now, this is an even function,"},{"Start":"03:31.710 ","End":"03:34.475","Text":"if I put instead of x minus x, the same thing,"},{"Start":"03:34.475 ","End":"03:36.980","Text":"so this is also going to be minus,"},{"Start":"03:36.980 ","End":"03:38.855","Text":"and this is going to be plus."},{"Start":"03:38.855 ","End":"03:41.060","Text":"Now, how this affects f of x,"},{"Start":"03:41.060 ","End":"03:43.085","Text":"or what we can tell from this, here,"},{"Start":"03:43.085 ","End":"03:44.825","Text":"f prime is positive,"},{"Start":"03:44.825 ","End":"03:46.370","Text":"and if f prime is positive,"},{"Start":"03:46.370 ","End":"03:49.160","Text":"it means that the function is an increasing function,"},{"Start":"03:49.160 ","End":"03:50.570","Text":"which we write like this."},{"Start":"03:50.570 ","End":"03:53.400","Text":"Here, f prime is negative,"},{"Start":"03:53.400 ","End":"03:55.800","Text":"and so the function here is decreasing."},{"Start":"03:55.800 ","End":"03:58.095","Text":"Likewise it\u0027s decreasing here,"},{"Start":"03:58.095 ","End":"04:00.285","Text":"and here it\u0027s increasing."},{"Start":"04:00.285 ","End":"04:04.115","Text":"Now, when we have a point that\u0027s between an increase and a decrease,"},{"Start":"04:04.115 ","End":"04:06.169","Text":"that\u0027s assigned for an extremum,"},{"Start":"04:06.169 ","End":"04:08.990","Text":"specifically of type maximum,"},{"Start":"04:08.990 ","End":"04:12.380","Text":"that\u0027s when we go up to a point and down from it, that\u0027s a maximum."},{"Start":"04:12.380 ","End":"04:15.425","Text":"Likewise, here I see there\u0027s a down and an up,"},{"Start":"04:15.425 ","End":"04:18.545","Text":"decreasing and increasing, and between them it\u0027s a minimum."},{"Start":"04:18.545 ","End":"04:22.610","Text":"But between this and this is 2 decreases,"},{"Start":"04:22.610 ","End":"04:24.965","Text":"so this is not going to be an extremum."},{"Start":"04:24.965 ","End":"04:26.795","Text":"In fact, what corresponds to 0,"},{"Start":"04:26.795 ","End":"04:28.205","Text":"I don\u0027t really know what it is,"},{"Start":"04:28.205 ","End":"04:30.650","Text":"it\u0027s probably something called an inflection point,"},{"Start":"04:30.650 ","End":"04:32.090","Text":"and you may have heard of that or not."},{"Start":"04:32.090 ","End":"04:34.184","Text":"But in any event it\u0027s not an extremum,"},{"Start":"04:34.184 ","End":"04:37.520","Text":"because an extremum is either between increasing and decreasing,"},{"Start":"04:37.520 ","End":"04:40.220","Text":"where it\u0027s a maximum, or between decreasing and increasing,"},{"Start":"04:40.220 ","End":"04:41.410","Text":"in which case it\u0027s a minimum."},{"Start":"04:41.410 ","End":"04:43.425","Text":"Now, for the points of interest,"},{"Start":"04:43.425 ","End":"04:45.195","Text":"for the extremum, that\u0027s these 2,"},{"Start":"04:45.195 ","End":"04:46.885","Text":"this one\u0027s no longer of interest,"},{"Start":"04:46.885 ","End":"04:49.310","Text":"I\u0027m going to put what the value of y is,"},{"Start":"04:49.310 ","End":"04:52.580","Text":"which is by substituting in the original function,"},{"Start":"04:52.580 ","End":"04:55.175","Text":"I substitute the x and I\u0027ll get the y."},{"Start":"04:55.175 ","End":"04:57.215","Text":"If I plug in 2,"},{"Start":"04:57.215 ","End":"05:02.275","Text":"let\u0027s see what I get, 2^5 is 32, so that\u0027s 96."},{"Start":"05:02.275 ","End":"05:07.230","Text":"Here, 20 times 8 is 160,"},{"Start":"05:07.230 ","End":"05:13.700","Text":"96 minus 160 is minus 64."},{"Start":"05:13.700 ","End":"05:17.225","Text":"I make this minus 64,"},{"Start":"05:17.225 ","End":"05:18.995","Text":"and for minus 2,"},{"Start":"05:18.995 ","End":"05:20.750","Text":"well this is an odd function,"},{"Start":"05:20.750 ","End":"05:23.265","Text":"so it\u0027s going to be plus 64."},{"Start":"05:23.265 ","End":"05:25.760","Text":"Just quite of interest for 0,"},{"Start":"05:25.760 ","End":"05:28.100","Text":"although I don\u0027t need this anymore,"},{"Start":"05:28.100 ","End":"05:30.785","Text":"if I substituted, it would have been 0,"},{"Start":"05:30.785 ","End":"05:33.830","Text":"but 0, 0 is not an extremum,"},{"Start":"05:33.830 ","End":"05:38.585","Text":"it\u0027s probably an inflection and we won\u0027t write it in the answer, I was just curious."},{"Start":"05:38.585 ","End":"05:42.920","Text":"Now that we\u0027ve got this table out of the way, I mean, done,"},{"Start":"05:42.920 ","End":"05:47.350","Text":"we can use it to get all the information we need as what the question asked."},{"Start":"05:47.350 ","End":"05:49.400","Text":"We were asked about extrema,"},{"Start":"05:49.400 ","End":"05:53.960","Text":"and we were asked about areas where the function is increasing the interval."},{"Start":"05:53.960 ","End":"05:56.180","Text":"I\u0027ll just put here increasing,"},{"Start":"05:56.180 ","End":"06:00.705","Text":"and we\u0027ll also want to know where it\u0027s decreasing, in which interval."},{"Start":"06:00.705 ","End":"06:02.310","Text":"The extrema, we have 2 of them."},{"Start":"06:02.310 ","End":"06:05.115","Text":"We have minus 264,"},{"Start":"06:05.115 ","End":"06:06.540","Text":"where we have a maximum."},{"Start":"06:06.540 ","End":"06:10.985","Text":"If 1 of them is a maximum at minus 264,"},{"Start":"06:10.985 ","End":"06:13.580","Text":"and we also have this 1 which is a minimum,"},{"Start":"06:13.580 ","End":"06:17.165","Text":"and it occurs at 2 minus 64."},{"Start":"06:17.165 ","End":"06:21.455","Text":"As for increasing, I look for the arrows going upwards these 2,"},{"Start":"06:21.455 ","End":"06:25.550","Text":"so I have intervals x less than minus 2,"},{"Start":"06:25.550 ","End":"06:27.920","Text":"and I also have x bigger than 2."},{"Start":"06:27.920 ","End":"06:32.075","Text":"For decreasing, and this is the slightly tricky part."},{"Start":"06:32.075 ","End":"06:36.545","Text":"Instinctively, you would just say this interval and this interval."},{"Start":"06:36.545 ","End":"06:37.820","Text":"But in actual fact,"},{"Start":"06:37.820 ","End":"06:41.570","Text":"1 is a single point connecting 2 decreasing areas,"},{"Start":"06:41.570 ","End":"06:43.280","Text":"it\u0027s joins the decrease,"},{"Start":"06:43.280 ","End":"06:45.740","Text":"this whole thing combined with the 0,"},{"Start":"06:45.740 ","End":"06:47.315","Text":"if I take all this together,"},{"Start":"06:47.315 ","End":"06:50.040","Text":"it\u0027s like I get 1 interval minus 2,"},{"Start":"06:50.040 ","End":"06:52.095","Text":"less than x, less than 2."},{"Start":"06:52.095 ","End":"06:54.690","Text":"If I sketch it, for decreasing,"},{"Start":"06:54.690 ","End":"06:56.325","Text":"for 1 single moment,"},{"Start":"06:56.325 ","End":"06:59.480","Text":"we\u0027re not decreasing the tangent straightens out,"},{"Start":"06:59.480 ","End":"07:01.640","Text":"and then immediately we\u0027re decreasing again."},{"Start":"07:01.640 ","End":"07:05.240","Text":"But on the whole this function decreases as we go from left to right,"},{"Start":"07:05.240 ","End":"07:06.270","Text":"the function goes down."},{"Start":"07:06.270 ","End":"07:08.450","Text":"We should really combine these 2,"},{"Start":"07:08.450 ","End":"07:11.755","Text":"this area where it\u0027s decreasing here."},{"Start":"07:11.755 ","End":"07:13.720","Text":"For the moment there, it\u0027s not decreasing,"},{"Start":"07:13.720 ","End":"07:17.224","Text":"and then it\u0027s decreasing, so the whole area and the whole interval, it\u0027s decreasing."},{"Start":"07:17.224 ","End":"07:20.000","Text":"I\u0027ll write this as minus 2,"},{"Start":"07:20.000 ","End":"07:21.950","Text":"less than x, less than 2."},{"Start":"07:21.950 ","End":"07:25.099","Text":"That wouldn\u0027t be too bad if you wrote it as 2 separate intervals,"},{"Start":"07:25.099 ","End":"07:26.570","Text":"you\u0027ll just be missing a point."},{"Start":"07:26.570 ","End":"07:30.270","Text":"We\u0027ve answered all the questions, and we\u0027re done."}],"ID":4804},{"Watched":false,"Name":"Exercise 6","Duration":"7m 34s","ChapterTopicVideoID":4805,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4805.jpeg","UploadDate":"2015-07-15T04:19:39.0130000","DurationForVideoObject":"PT7M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.240","Text":"In this exercise, we\u0027re given a function f^x equals x over x squared plus 3."},{"Start":"00:06.240 ","End":"00:10.800","Text":"What we have to do is find all local extrema of the function,"},{"Start":"00:10.800 ","End":"00:14.640","Text":"and also to indicate which intervals is the function increasing,"},{"Start":"00:14.640 ","End":"00:16.530","Text":"and where is it decreasing."},{"Start":"00:16.530 ","End":"00:20.940","Text":"As usual, this type of exercise is done very methodical manner,"},{"Start":"00:20.940 ","End":"00:23.550","Text":"the same pattern, the same set of rules."},{"Start":"00:23.550 ","End":"00:26.520","Text":"The first 1 is to differentiate the function."},{"Start":"00:26.520 ","End":"00:27.720","Text":"That\u0027s where we start,"},{"Start":"00:27.720 ","End":"00:31.110","Text":"but I\u0027d like to copy it first. Here we are."},{"Start":"00:31.110 ","End":"00:34.125","Text":"F^x is x over x squared plus 3,"},{"Start":"00:34.125 ","End":"00:35.790","Text":"and now we want the derivative."},{"Start":"00:35.790 ","End":"00:38.010","Text":"Derivative, we\u0027ll use the quotient rule"},{"Start":"00:38.010 ","End":"00:40.635","Text":"because we have a function of x over a function of x,"},{"Start":"00:40.635 ","End":"00:43.935","Text":"so f over g derivative."},{"Start":"00:43.935 ","End":"00:45.200","Text":"Just to remind you,"},{"Start":"00:45.200 ","End":"00:47.360","Text":"this is over g squared."},{"Start":"00:47.360 ","End":"00:49.655","Text":"We have f prime g,"},{"Start":"00:49.655 ","End":"00:53.240","Text":"minus f_g prime over g squared,"},{"Start":"00:53.240 ","End":"00:56.125","Text":"and in our case, this comes to,"},{"Start":"00:56.125 ","End":"01:00.890","Text":"on the denominator, x squared plus 3 squared."},{"Start":"01:00.890 ","End":"01:02.630","Text":"Here this is f,"},{"Start":"01:02.630 ","End":"01:06.829","Text":"this is g. F prime is 1 times g,"},{"Start":"01:06.829 ","End":"01:11.105","Text":"which is x squared plus 3 minus f,"},{"Start":"01:11.105 ","End":"01:15.515","Text":"which is x, and g prime is 2_x."},{"Start":"01:15.515 ","End":"01:17.690","Text":"If I rewrite this,"},{"Start":"01:17.690 ","End":"01:22.800","Text":"I\u0027ll get x squared plus 3 minus 2_x squared."},{"Start":"01:22.800 ","End":"01:27.300","Text":"In the numerator, I have minus x squared plus 3,"},{"Start":"01:27.300 ","End":"01:29.120","Text":"and on the denominator,"},{"Start":"01:29.120 ","End":"01:32.435","Text":"x squared plus 3, all squared."},{"Start":"01:32.435 ","End":"01:37.550","Text":"After we\u0027ve differentiated, we have to set this equal to 0 and to solve."},{"Start":"01:37.550 ","End":"01:39.364","Text":"Let this equal 0."},{"Start":"01:39.364 ","End":"01:43.430","Text":"Now, this denominator, I can multiply both sides by it."},{"Start":"01:43.430 ","End":"01:48.350","Text":"By the way, it\u0027s never 0 because x squared plus 3 is positive."},{"Start":"01:48.350 ","End":"01:51.190","Text":"Squared is also positive, so it\u0027s never 0."},{"Start":"01:51.190 ","End":"01:55.400","Text":"What we get is that minus x squared plus 3 is 0,"},{"Start":"01:55.400 ","End":"01:57.050","Text":"and then x squared is 3,"},{"Start":"01:57.050 ","End":"02:00.620","Text":"so x is plus or minus the square root of 3."},{"Start":"02:00.620 ","End":"02:03.620","Text":"We have minus the square root of 3 is 1,"},{"Start":"02:03.620 ","End":"02:07.370","Text":"solution and square root of 3 is the other solution."},{"Start":"02:07.370 ","End":"02:11.360","Text":"That was all this work just to solve for f prime equal to 0."},{"Start":"02:11.360 ","End":"02:14.960","Text":"Next thing we do is to draw our famous table,"},{"Start":"02:14.960 ","End":"02:17.195","Text":"which we\u0027ve been doing in every exercise,"},{"Start":"02:17.195 ","End":"02:19.505","Text":"where I take 5 rows."},{"Start":"02:19.505 ","End":"02:21.755","Text":"The first is labeled x,"},{"Start":"02:21.755 ","End":"02:25.315","Text":"the next is labeled f prime of x,"},{"Start":"02:25.315 ","End":"02:28.080","Text":"the next row is labeled f^x,"},{"Start":"02:28.080 ","End":"02:30.354","Text":"and the last 1 is y."},{"Start":"02:30.354 ","End":"02:33.980","Text":"These are rows. I\u0027ll just separate them with horizontal lines."},{"Start":"02:33.980 ","End":"02:35.805","Text":"We first fill in the x\u0027s,"},{"Start":"02:35.805 ","End":"02:39.710","Text":"and these are the x\u0027s which are the solutions to f prime equals 0."},{"Start":"02:39.710 ","End":"02:42.420","Text":"We have 2 of those, we put them in increasing order."},{"Start":"02:42.420 ","End":"02:45.260","Text":"Here we have minus the square root of 3,"},{"Start":"02:45.260 ","End":"02:48.205","Text":"and here I have the square root of 3,"},{"Start":"02:48.205 ","End":"02:51.830","Text":"and the derivative of each of these is 0."},{"Start":"02:51.830 ","End":"02:54.590","Text":"That\u0027s how we got these points, after all."},{"Start":"02:54.590 ","End":"02:59.825","Text":"Then these 2 points on the line divide the line into 3 intervals, in this case."},{"Start":"02:59.825 ","End":"03:02.180","Text":"This interval, this interval, and this interval,"},{"Start":"03:02.180 ","End":"03:07.355","Text":"which I will write as x less than minus root 3."},{"Start":"03:07.355 ","End":"03:10.910","Text":"Here, x is between minus root 3,"},{"Start":"03:10.910 ","End":"03:13.760","Text":"and x is between that and root 3."},{"Start":"03:13.760 ","End":"03:18.800","Text":"The last interval is x bigger than root 3."},{"Start":"03:18.800 ","End":"03:20.300","Text":"What we have to do is,"},{"Start":"03:20.300 ","End":"03:21.634","Text":"for each of these intervals,"},{"Start":"03:21.634 ","End":"03:23.390","Text":"choose a sample point,"},{"Start":"03:23.390 ","End":"03:27.770","Text":"a representative point arbitrarily less than minus root 3,"},{"Start":"03:27.770 ","End":"03:30.035","Text":"and root 3 is roughly 1.7."},{"Start":"03:30.035 ","End":"03:32.270","Text":"I need less than minus 1.7,"},{"Start":"03:32.270 ","End":"03:34.485","Text":"so let\u0027s make it minus 2."},{"Start":"03:34.485 ","End":"03:37.380","Text":"Between minus root 3 and root 3, I\u0027ll go for 0."},{"Start":"03:37.380 ","End":"03:40.130","Text":"That\u0027s usually easiest to calculate with."},{"Start":"03:40.130 ","End":"03:42.110","Text":"Bigger than root 3, again,"},{"Start":"03:42.110 ","End":"03:43.845","Text":"this is 1.7 something,"},{"Start":"03:43.845 ","End":"03:45.165","Text":"so I\u0027ll take it as 2."},{"Start":"03:45.165 ","End":"03:47.840","Text":"Now, with each of these points, 3 of them here,"},{"Start":"03:47.840 ","End":"03:55.430","Text":"I have to substitute into f prime of x. F prime of x is any 1 of these."},{"Start":"03:55.430 ","End":"03:57.930","Text":"Let\u0027s say I\u0027ll take it as this 1,"},{"Start":"03:57.930 ","End":"04:00.320","Text":"will be my f prime of x."},{"Start":"04:00.320 ","End":"04:02.270","Text":"Here\u0027s f prime. As after that,"},{"Start":"04:02.270 ","End":"04:05.150","Text":"I\u0027ve been changing it by multiplying by both sides."},{"Start":"04:05.150 ","End":"04:07.580","Text":"This is still f prime in its simplest form."},{"Start":"04:07.580 ","End":"04:10.560","Text":"Now, if I substitute minus 2,"},{"Start":"04:10.560 ","End":"04:13.065","Text":"minus 2, squared is 4."},{"Start":"04:13.065 ","End":"04:16.685","Text":"This is minus 4 plus 3, which is negative."},{"Start":"04:16.685 ","End":"04:18.650","Text":"The denominator is positive, always,"},{"Start":"04:18.650 ","End":"04:21.560","Text":"so I get negative over positive is negative."},{"Start":"04:21.560 ","End":"04:26.440","Text":"All I need to write here is not the value but simply the sign, and it\u0027s negative."},{"Start":"04:26.440 ","End":"04:28.925","Text":"By the way, if you chose a different sample point,"},{"Start":"04:28.925 ","End":"04:30.530","Text":"you\u0027d get different values here,"},{"Start":"04:30.530 ","End":"04:31.790","Text":"but you\u0027d still get negative."},{"Start":"04:31.790 ","End":"04:33.320","Text":"The sign wouldn\u0027t change."},{"Start":"04:33.320 ","End":"04:36.425","Text":"Here I put in x equals 0."},{"Start":"04:36.425 ","End":"04:39.320","Text":"Minus 0 plus 3 is 3, it\u0027s positive,"},{"Start":"04:39.320 ","End":"04:41.060","Text":"and since the denominator is positive,"},{"Start":"04:41.060 ","End":"04:43.895","Text":"it stays positive, so I write a plus here."},{"Start":"04:43.895 ","End":"04:46.510","Text":"When x is bigger than root 3, x is 2."},{"Start":"04:46.510 ","End":"04:49.640","Text":"Here I have minus 4 plus 3,"},{"Start":"04:49.640 ","End":"04:51.620","Text":"it\u0027s minus 1 over positive,"},{"Start":"04:51.620 ","End":"04:53.030","Text":"so that\u0027s a minus."},{"Start":"04:53.030 ","End":"04:54.290","Text":"That\u0027s for f prime."},{"Start":"04:54.290 ","End":"04:57.605","Text":"Now, we interpret this as what it does to f itself."},{"Start":"04:57.605 ","End":"05:00.515","Text":"This means that the derivative is negative."},{"Start":"05:00.515 ","End":"05:04.505","Text":"Derivative negative is the sign for function decreasing."},{"Start":"05:04.505 ","End":"05:07.730","Text":"The function itself in this interval is decreasing."},{"Start":"05:07.730 ","End":"05:09.280","Text":"I write this with an arrow."},{"Start":"05:09.280 ","End":"05:13.095","Text":"For a plus, I know it\u0027s increasing, an arrow like this."},{"Start":"05:13.095 ","End":"05:18.455","Text":"Here again, decreasing and in-between at this point 2 or x is minus root 3,"},{"Start":"05:18.455 ","End":"05:22.060","Text":"because it was decreasing before and increasing after,"},{"Start":"05:22.060 ","End":"05:26.705","Text":"this means that this is an extremum and there\u0027s 2 types, minimum or maximum."},{"Start":"05:26.705 ","End":"05:29.285","Text":"The decreasing increasing is the minimum"},{"Start":"05:29.285 ","End":"05:32.090","Text":"because we go down to the minimum and up away from it."},{"Start":"05:32.090 ","End":"05:36.875","Text":"Similarly, this 1, because it\u0027s increasing and then decreasing in the middle,"},{"Start":"05:36.875 ","End":"05:38.710","Text":"it\u0027s got to be a maximum point."},{"Start":"05:38.710 ","End":"05:41.314","Text":"That\u0027s also an extremum of type maximum."},{"Start":"05:41.314 ","End":"05:43.250","Text":"Finally, what we\u0027re interested in is,"},{"Start":"05:43.250 ","End":"05:44.855","Text":"are these extremum points?"},{"Start":"05:44.855 ","End":"05:49.685","Text":"We have an x, but we\u0027d also like a y so we can say what the point is in full."},{"Start":"05:49.685 ","End":"05:51.200","Text":"To do this, of course,"},{"Start":"05:51.200 ","End":"05:53.180","Text":"we substitute in the original,"},{"Start":"05:53.180 ","End":"05:54.890","Text":"we may not get whole numbers here."},{"Start":"05:54.890 ","End":"05:57.530","Text":"If x is minus root 3,"},{"Start":"05:57.530 ","End":"05:59.720","Text":"x squared is 3,"},{"Start":"05:59.720 ","End":"06:01.600","Text":"3 plus 3, is 6."},{"Start":"06:01.600 ","End":"06:05.410","Text":"We have minus root 3, over 6."},{"Start":"06:06.660 ","End":"06:12.680","Text":"It\u0027s some decimal minus 1.7 over 6, roughly minus 0.3."},{"Start":"06:12.680 ","End":"06:15.110","Text":"It doesn\u0027t matter. I just leave it like this."},{"Start":"06:15.110 ","End":"06:18.470","Text":"When x is root 3, same denominator,"},{"Start":"06:18.470 ","End":"06:20.660","Text":"just going to get a numerator with a plus,"},{"Start":"06:20.660 ","End":"06:24.664","Text":"so we\u0027d get plus root 3, over 6."},{"Start":"06:24.664 ","End":"06:26.105","Text":"Now with all this information,"},{"Start":"06:26.105 ","End":"06:28.490","Text":"we can answer the questions that we were asked."},{"Start":"06:28.490 ","End":"06:32.810","Text":"We can draw conclusions which they were asked about the extrema"},{"Start":"06:32.810 ","End":"06:37.750","Text":"and the intervals where the function is increasing and decreasing."},{"Start":"06:37.750 ","End":"06:40.350","Text":"I\u0027ll first address the extrema."},{"Start":"06:40.350 ","End":"06:41.870","Text":"We have the extrema,"},{"Start":"06:41.870 ","End":"06:43.655","Text":"are the minimum and the maximum."},{"Start":"06:43.655 ","End":"06:44.990","Text":"We have 1 of them,"},{"Start":"06:44.990 ","End":"06:49.175","Text":"which is a minimum at the point minus root 3,"},{"Start":"06:49.175 ","End":"06:52.220","Text":"then minus root 3, over 6."},{"Start":"06:52.220 ","End":"06:54.860","Text":"The other 1 is this 1, is the maximum."},{"Start":"06:54.860 ","End":"06:58.055","Text":"We have a maximum at root 3,"},{"Start":"06:58.055 ","End":"07:00.230","Text":"root 3, over 6."},{"Start":"07:00.230 ","End":"07:02.315","Text":"That\u0027s the 2 extrema that we have."},{"Start":"07:02.315 ","End":"07:04.925","Text":"Now, as for intervals of increase,"},{"Start":"07:04.925 ","End":"07:09.500","Text":"so the function is increasing wherever the arrow is up,"},{"Start":"07:09.500 ","End":"07:11.355","Text":"and that\u0027s just this interval."},{"Start":"07:11.355 ","End":"07:14.000","Text":"That\u0027s on the interval where minus root 3,"},{"Start":"07:14.000 ","End":"07:17.150","Text":"less than x, less than root 3."},{"Start":"07:17.150 ","End":"07:19.580","Text":"For decreasing, we actually have"},{"Start":"07:19.580 ","End":"07:22.910","Text":"2 intervals because we have a down arrow here and a down arrow here,"},{"Start":"07:22.910 ","End":"07:29.250","Text":"so it\u0027s either x less than minus root 3 or it\u0027s x bigger than root 3."},{"Start":"07:29.250 ","End":"07:32.239","Text":"This answers all the questions that we were asked,"},{"Start":"07:32.239 ","End":"07:34.800","Text":"and so we\u0027re done."}],"ID":4805},{"Watched":false,"Name":"Exercise 7","Duration":"6m 22s","ChapterTopicVideoID":4806,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4806.jpeg","UploadDate":"2015-07-15T04:20:13.0500000","DurationForVideoObject":"PT6M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.870","Text":"In this exercise, we\u0027re given a function f of x equals x over x squared plus x plus 1."},{"Start":"00:06.870 ","End":"00:10.530","Text":"What we have to do is find all the local extrema and"},{"Start":"00:10.530 ","End":"00:14.760","Text":"also the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:14.760 ","End":"00:18.360","Text":"This kind of exercise is done by a recipe step by step."},{"Start":"00:18.360 ","End":"00:20.025","Text":"We\u0027ve done these before."},{"Start":"00:20.025 ","End":"00:22.800","Text":"The first step is to differentiate the function."},{"Start":"00:22.800 ","End":"00:27.120","Text":"Just, by the way, I\u0027ll write a y here because sometimes I need y and y is f of x."},{"Start":"00:27.120 ","End":"00:29.610","Text":"The first step is to differentiate."},{"Start":"00:29.610 ","End":"00:32.100","Text":"This is going to be the quotient rule."},{"Start":"00:32.100 ","End":"00:34.625","Text":"F prime of x equals,"},{"Start":"00:34.625 ","End":"00:38.090","Text":"and I\u0027ll just write the quotient rule at the side."},{"Start":"00:38.090 ","End":"00:43.115","Text":"That is that f over g prime is equal to"},{"Start":"00:43.115 ","End":"00:49.550","Text":"f prime g minus f g prime over g squared."},{"Start":"00:49.550 ","End":"00:54.545","Text":"In our case, denominator is g, numerator is f,"},{"Start":"00:54.545 ","End":"01:01.820","Text":"and what we get is x squared plus x plus 1 squared in the denominator, and here,"},{"Start":"01:01.820 ","End":"01:05.315","Text":"f prime is 1 times g,"},{"Start":"01:05.315 ","End":"01:11.915","Text":"which is x squared plus x plus 1 minus f g prime,"},{"Start":"01:11.915 ","End":"01:16.819","Text":"that\u0027s x and g prime is 2x plus 1."},{"Start":"01:16.819 ","End":"01:18.635","Text":"Let\u0027s see what we can do with this."},{"Start":"01:18.635 ","End":"01:22.354","Text":"I just collect together the x squared and the x\u0027s,"},{"Start":"01:22.354 ","End":"01:30.665","Text":"x squared minus 2x squared is minus x squared plus x minus x."},{"Start":"01:30.665 ","End":"01:34.520","Text":"That gives nothing and we\u0027re left with a plus 1."},{"Start":"01:34.520 ","End":"01:41.710","Text":"Minus x squared plus 1 over x squared plus x plus 1 squared."},{"Start":"01:41.710 ","End":"01:43.850","Text":"Once we have the derivative,"},{"Start":"01:43.850 ","End":"01:47.525","Text":"the next thing is to compare it to 0 and solve for x."},{"Start":"01:47.525 ","End":"01:49.595","Text":"If this quotient is 0,"},{"Start":"01:49.595 ","End":"01:52.520","Text":"the denominator is never 0."},{"Start":"01:52.520 ","End":"01:56.150","Text":"In fact, it\u0027s positive and to make this 0,"},{"Start":"01:56.150 ","End":"01:58.220","Text":"we have to have the numerator being 0."},{"Start":"01:58.220 ","End":"02:02.194","Text":"We get minus x squared plus 1 equals 0."},{"Start":"02:02.194 ","End":"02:05.145","Text":"That means that x squared is equal to 1."},{"Start":"02:05.145 ","End":"02:07.515","Text":"X is going to be plus or minus 1."},{"Start":"02:07.515 ","End":"02:09.140","Text":"There\u0027s only 2 possibilities,"},{"Start":"02:09.140 ","End":"02:12.530","Text":"x equals minus 1 and 1."},{"Start":"02:12.530 ","End":"02:14.670","Text":"Now that we have this,"},{"Start":"02:14.670 ","End":"02:17.085","Text":"we\u0027re going to make a table."},{"Start":"02:17.085 ","End":"02:20.045","Text":"Our table has 4 rows,"},{"Start":"02:20.045 ","End":"02:22.384","Text":"which I call x,"},{"Start":"02:22.384 ","End":"02:24.140","Text":"f prime of x,"},{"Start":"02:24.140 ","End":"02:26.284","Text":"f of x, and y."},{"Start":"02:26.284 ","End":"02:31.190","Text":"These are the row headers and here are the 4 rows."},{"Start":"02:31.190 ","End":"02:33.500","Text":"In here, we, first of all,"},{"Start":"02:33.500 ","End":"02:37.925","Text":"fill the x row with the solutions for f prime equals 0,"},{"Start":"02:37.925 ","End":"02:43.610","Text":"but in order, and they are in order minus 1 and then 1 and leave spaces in between."},{"Start":"02:43.610 ","End":"02:46.580","Text":"The derivative at each of these points is 0."},{"Start":"02:46.580 ","End":"02:48.860","Text":"That\u0027s how we found these points."},{"Start":"02:48.860 ","End":"02:51.920","Text":"What we want next is to write"},{"Start":"02:51.920 ","End":"02:55.865","Text":"some intervals that these 2 points break the line up into 3 intervals."},{"Start":"02:55.865 ","End":"03:01.175","Text":"The intervals are x less than minus 1 between these 2 points,"},{"Start":"03:01.175 ","End":"03:04.065","Text":"minus 1 less than x, less than 1,"},{"Start":"03:04.065 ","End":"03:05.595","Text":"and beyond this point,"},{"Start":"03:05.595 ","End":"03:07.470","Text":"x bigger than 1."},{"Start":"03:07.470 ","End":"03:10.025","Text":"Then we pick a sample,"},{"Start":"03:10.025 ","End":"03:12.560","Text":"an arbitrary point from each of the intervals."},{"Start":"03:12.560 ","End":"03:14.090","Text":"For less than minus 1,"},{"Start":"03:14.090 ","End":"03:15.880","Text":"I\u0027ll go with minus 2."},{"Start":"03:15.880 ","End":"03:17.905","Text":"Between minus 1 and 1,"},{"Start":"03:17.905 ","End":"03:20.270","Text":"I\u0027ll go for 0 and bigger than 1,"},{"Start":"03:20.270 ","End":"03:21.650","Text":"I\u0027ll go for 2."},{"Start":"03:21.650 ","End":"03:26.000","Text":"What I have to do with these orange numbers is to substitute"},{"Start":"03:26.000 ","End":"03:31.785","Text":"them into f prime of x and f prime of x will be,"},{"Start":"03:31.785 ","End":"03:34.550","Text":"that\u0027s this mess, but it\u0027s here,"},{"Start":"03:34.550 ","End":"03:37.540","Text":"this will be f prime of x."},{"Start":"03:37.540 ","End":"03:39.590","Text":"What I want is not the value,"},{"Start":"03:39.590 ","End":"03:41.960","Text":"but only whether it\u0027s positive or negative."},{"Start":"03:41.960 ","End":"03:44.900","Text":"Remember the denominator is always positive."},{"Start":"03:44.900 ","End":"03:48.340","Text":"Really I just have to substitute into the numerator."},{"Start":"03:48.340 ","End":"03:49.590","Text":"Let start with the easy 1."},{"Start":"03:49.590 ","End":"03:54.780","Text":"When x is 0 minus 0 plus 1 is 1 and that\u0027s positive."},{"Start":"03:54.780 ","End":"03:56.235","Text":"I put a plus here."},{"Start":"03:56.235 ","End":"04:00.825","Text":"For minus 2, I get minus 4."},{"Start":"04:00.825 ","End":"04:02.910","Text":"Minus 2 squared is 4 and there is a minus,"},{"Start":"04:02.910 ","End":"04:05.400","Text":"minus 4 plus 1 is minus 3."},{"Start":"04:05.400 ","End":"04:11.270","Text":"This is negative and you get the same thing if you substitute 2 also negative."},{"Start":"04:11.270 ","End":"04:16.880","Text":"What this means is that if the derivative is negative,"},{"Start":"04:16.880 ","End":"04:19.865","Text":"what it means for f is that f is decreasing."},{"Start":"04:19.865 ","End":"04:22.490","Text":"Whenever the derivative is negative, it\u0027s decreasing."},{"Start":"04:22.490 ","End":"04:26.465","Text":"Likewise, if the derivative is positive, the function\u0027s increasing."},{"Start":"04:26.465 ","End":"04:29.960","Text":"Here again, it\u0027s negative so once again, decreasing."},{"Start":"04:29.960 ","End":"04:33.095","Text":"Now at this value x equals minus 1,"},{"Start":"04:33.095 ","End":"04:38.145","Text":"it\u0027s between a decreasing and increasing interval."},{"Start":"04:38.145 ","End":"04:39.430","Text":"At this point itself,"},{"Start":"04:39.430 ","End":"04:43.635","Text":"we have an extremum and when we go from decreased to increase,"},{"Start":"04:43.635 ","End":"04:45.560","Text":"that is a minimum point."},{"Start":"04:45.560 ","End":"04:47.119","Text":"This is also an extremum,"},{"Start":"04:47.119 ","End":"04:50.635","Text":"but between increase and decrease, it\u0027s a maximum."},{"Start":"04:50.635 ","End":"04:52.380","Text":"Now for these 2 interesting points,"},{"Start":"04:52.380 ","End":"04:53.760","Text":"I have the x,"},{"Start":"04:53.760 ","End":"04:55.635","Text":"but I\u0027ll also like the y."},{"Start":"04:55.635 ","End":"04:59.720","Text":"What I do is to substitute into the original function,"},{"Start":"04:59.720 ","End":"05:02.215","Text":"which is this 1 here."},{"Start":"05:02.215 ","End":"05:05.550","Text":"What I get is that when x is 1,"},{"Start":"05:05.550 ","End":"05:08.955","Text":"I get 1 over 1 plus 1 plus 1,"},{"Start":"05:08.955 ","End":"05:10.890","Text":"and that is 1/3."},{"Start":"05:10.890 ","End":"05:13.380","Text":"The y for this point is 1/3."},{"Start":"05:13.380 ","End":"05:15.915","Text":"For the other 1, minus 1,"},{"Start":"05:15.915 ","End":"05:21.710","Text":"I get minus 1 over 1 minus 1 plus 1, which is 1,"},{"Start":"05:21.710 ","End":"05:26.090","Text":"so minus 1 over 1 is minus 1."},{"Start":"05:26.090 ","End":"05:30.680","Text":"Okay, now I have all the information I need to answer the questions about"},{"Start":"05:30.680 ","End":"05:36.290","Text":"extrema and increasing and decreasing intervals so I\u0027ll just write those conclusions."},{"Start":"05:36.290 ","End":"05:38.285","Text":"What we want, the extrema."},{"Start":"05:38.285 ","End":"05:41.630","Text":"That\u0027s this point here, which is a minimum."},{"Start":"05:41.630 ","End":"05:44.885","Text":"In other words, I have a minimum at the x and the y,"},{"Start":"05:44.885 ","End":"05:47.815","Text":"a minimum at minus 1, minus 1."},{"Start":"05:47.815 ","End":"05:49.430","Text":"I also have a maximum,"},{"Start":"05:49.430 ","End":"05:55.250","Text":"the other type of extremum and this maximum is at the point 1,"},{"Start":"05:55.250 ","End":"05:56.870","Text":"1/3. That\u0027s for extrema."},{"Start":"05:56.870 ","End":"06:00.100","Text":"Now for the increasing and decreasing."},{"Start":"06:00.100 ","End":"06:06.710","Text":"We have increasing where the arrow goes up and that is between minus 1 and 1."},{"Start":"06:06.710 ","End":"06:10.775","Text":"For decreasing, that\u0027s where the arrow goes downwards,"},{"Start":"06:10.775 ","End":"06:12.335","Text":"which is here and here."},{"Start":"06:12.335 ","End":"06:13.885","Text":"We have 2 intervals,"},{"Start":"06:13.885 ","End":"06:18.895","Text":"x less than minus 1 and x bigger than 1."},{"Start":"06:18.895 ","End":"06:22.680","Text":"That answers all the questions and we\u0027re done."}],"ID":4806},{"Watched":false,"Name":"Exercise 8","Duration":"7m 22s","ChapterTopicVideoID":4807,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4807.jpeg","UploadDate":"2015-07-15T04:20:52.3530000","DurationForVideoObject":"PT7M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this exercise, we\u0027re given a function,"},{"Start":"00:02.610 ","End":"00:05.160","Text":"this is it, x minus 1 over x squared."},{"Start":"00:05.160 ","End":"00:08.850","Text":"What we have to do is to find its local extrema"},{"Start":"00:08.850 ","End":"00:13.200","Text":"and the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:13.200 ","End":"00:15.470","Text":"I\u0027ve written the function out already."},{"Start":"00:15.470 ","End":"00:18.585","Text":"What we have to do is the standard recipe or protocol."},{"Start":"00:18.585 ","End":"00:21.240","Text":"We first differentiate the function."},{"Start":"00:21.240 ","End":"00:24.585","Text":"We get that f prime of x is equal to,"},{"Start":"00:24.585 ","End":"00:27.390","Text":"we\u0027re going to need the quotient rule in case you\u0027ve forgotten it."},{"Start":"00:27.390 ","End":"00:29.760","Text":"Here it is and following this,"},{"Start":"00:29.760 ","End":"00:33.045","Text":"we get g squared is x^4,"},{"Start":"00:33.045 ","End":"00:36.380","Text":"f prime is 1 times g,"},{"Start":"00:36.380 ","End":"00:39.560","Text":"which is x squared minus f,"},{"Start":"00:39.560 ","End":"00:44.250","Text":"which is x minus 1 times g prime, which is 2x."},{"Start":"00:44.250 ","End":"00:46.070","Text":"Here we have x squared,"},{"Start":"00:46.070 ","End":"00:49.640","Text":"and here we have minus 2x squared."},{"Start":"00:49.640 ","End":"00:51.965","Text":"It\u0027s minus x squared."},{"Start":"00:51.965 ","End":"00:53.210","Text":"We also have a minus,"},{"Start":"00:53.210 ","End":"00:58.000","Text":"a minus which is plus 2x over x^4."},{"Start":"00:58.000 ","End":"01:01.380","Text":"Then we let this equal 0 and solve."},{"Start":"01:01.380 ","End":"01:03.860","Text":"If this is going to be equal to 0,"},{"Start":"01:03.860 ","End":"01:07.699","Text":"then the denominator\u0027s never equal to 0."},{"Start":"01:07.699 ","End":"01:09.425","Text":"It\u0027s fact it\u0027s always positive."},{"Start":"01:09.425 ","End":"01:13.925","Text":"The only way a fraction can be 0 is if its numerator is 0."},{"Start":"01:13.925 ","End":"01:20.175","Text":"We get that minus x squared plus 2x equals 0."},{"Start":"01:20.175 ","End":"01:22.565","Text":"If we rewrite this,"},{"Start":"01:22.565 ","End":"01:24.650","Text":"we could take x outside the brackets,"},{"Start":"01:24.650 ","End":"01:30.110","Text":"we\u0027re left with x times minus x plus 2 equals 0."},{"Start":"01:30.110 ","End":"01:33.850","Text":"If either 1 of these is 0, x could be 0."},{"Start":"01:33.850 ","End":"01:37.880","Text":"Either 1 or 0 will do or minus x plus 2 is 0,"},{"Start":"01:37.880 ","End":"01:40.220","Text":"so x is equal to 2."},{"Start":"01:40.220 ","End":"01:42.440","Text":"Now there\u0027s something here that I forgot to do before,"},{"Start":"01:42.440 ","End":"01:43.775","Text":"but it\u0027s not too late."},{"Start":"01:43.775 ","End":"01:47.330","Text":"That is to look at the domain of the function."},{"Start":"01:47.330 ","End":"01:49.010","Text":"If we look at the function,"},{"Start":"01:49.010 ","End":"01:51.455","Text":"f of x equals x minus 1 over x squared,"},{"Start":"01:51.455 ","End":"01:55.370","Text":"we can see that there is a value which is not allowed,"},{"Start":"01:55.370 ","End":"01:58.990","Text":"and that is that x equals 0 is not in the domain."},{"Start":"01:58.990 ","End":"02:03.724","Text":"I\u0027m going to note that down that f of x is what we defined,"},{"Start":"02:03.724 ","End":"02:08.345","Text":"but that x is not equal to 0."},{"Start":"02:08.345 ","End":"02:09.530","Text":"That\u0027s important."},{"Start":"02:09.530 ","End":"02:13.415","Text":"Now, when we found these suspects for extrema,"},{"Start":"02:13.415 ","End":"02:15.530","Text":"we notice that x equals 0,"},{"Start":"02:15.530 ","End":"02:16.880","Text":"we got from the equation,"},{"Start":"02:16.880 ","End":"02:18.650","Text":"but it\u0027s not in the domain."},{"Start":"02:18.650 ","End":"02:20.560","Text":"I\u0027m going to cross this out."},{"Start":"02:20.560 ","End":"02:23.700","Text":"The only suspect for an extremum is x equals 2."},{"Start":"02:23.700 ","End":"02:26.030","Text":"But this is still going to be part of the process,"},{"Start":"02:26.030 ","End":"02:29.210","Text":"as you\u0027ll see, the x equals 0, remember it."},{"Start":"02:29.210 ","End":"02:35.045","Text":"After this stage, we get the stage of the table and let me just sketch it in."},{"Start":"02:35.045 ","End":"02:39.290","Text":"Here\u0027s the outline of the table and we have to fill it in."},{"Start":"02:39.290 ","End":"02:43.145","Text":"Now here\u0027s the thing about the values of x that we put in."},{"Start":"02:43.145 ","End":"02:46.100","Text":"We put in 2 kinds of values of x,"},{"Start":"02:46.100 ","End":"02:49.190","Text":"one is where the derivative was 0,"},{"Start":"02:49.190 ","End":"02:51.455","Text":"which has suspects for extrema,"},{"Start":"02:51.455 ","End":"02:53.750","Text":"and that would be x equals 2."},{"Start":"02:53.750 ","End":"02:58.175","Text":"But we also put in the table the values of x which are"},{"Start":"02:58.175 ","End":"03:03.010","Text":"out of the domain where f is undefined and that would be x equals 0."},{"Start":"03:03.010 ","End":"03:07.875","Text":"I have to put in 0 and 2 and in increasing order,"},{"Start":"03:07.875 ","End":"03:11.835","Text":"so we put in 0 here and 2 here."},{"Start":"03:11.835 ","End":"03:15.235","Text":"We also leave space because we\u0027re going to add some more information."},{"Start":"03:15.235 ","End":"03:18.920","Text":"At 2, the derivative is equal to 0,"},{"Start":"03:18.920 ","End":"03:22.115","Text":"which makes it a suspect for an extremum."},{"Start":"03:22.115 ","End":"03:27.140","Text":"But at 0 the function is not defined and usually it indicates an asymptote."},{"Start":"03:27.140 ","End":"03:30.080","Text":"I\u0027m just going to fill the rest of the values with"},{"Start":"03:30.080 ","End":"03:34.160","Text":"just a dotted line that this is not really part of the function,"},{"Start":"03:34.160 ","End":"03:37.955","Text":"but we still have to put it in the table because around the 0,"},{"Start":"03:37.955 ","End":"03:42.835","Text":"f could change from increasing to decreasing or vice versa is possible."},{"Start":"03:42.835 ","End":"03:48.845","Text":"We need to put this asymptote or x where f is undefined into the table too."},{"Start":"03:48.845 ","End":"03:50.870","Text":"Now these 2 points, 0 and 2,"},{"Start":"03:50.870 ","End":"03:53.660","Text":"divide the line into intervals,"},{"Start":"03:53.660 ","End":"03:56.315","Text":"I\u0027m going to write down what those intervals are."},{"Start":"03:56.315 ","End":"03:59.540","Text":"Here we have x less than 0."},{"Start":"03:59.540 ","End":"04:05.540","Text":"Here we have that x is between 0 and 2 and here we have x bigger than 2."},{"Start":"04:05.540 ","End":"04:10.985","Text":"Next stage is to choose a sample point from each of these intervals."},{"Start":"04:10.985 ","End":"04:13.310","Text":"For example, when x is less than 0,"},{"Start":"04:13.310 ","End":"04:16.270","Text":"I could choose x equals negative 1."},{"Start":"04:16.270 ","End":"04:20.440","Text":"It doesn\u0027t really matter which values you choose as long as they are from the range."},{"Start":"04:20.440 ","End":"04:22.445","Text":"When x is between 0 and 2,"},{"Start":"04:22.445 ","End":"04:24.380","Text":"I\u0027ll choose x equals 1."},{"Start":"04:24.380 ","End":"04:27.905","Text":"For x bigger than 2, I\u0027ll choose x equals 3."},{"Start":"04:27.905 ","End":"04:33.980","Text":"Next step is to substitute these values of x into f prime."},{"Start":"04:33.980 ","End":"04:36.610","Text":"I\u0027ve already circled this in orange."},{"Start":"04:36.610 ","End":"04:38.660","Text":"The other thing is I don\u0027t actually have to find"},{"Start":"04:38.660 ","End":"04:41.890","Text":"the value just whether it\u0027s positive or negative."},{"Start":"04:41.890 ","End":"04:44.360","Text":"If I put in minus 1,"},{"Start":"04:44.360 ","End":"04:50.370","Text":"the denominator is positive always so I can ignore that when I find."},{"Start":"04:50.370 ","End":"04:52.560","Text":"I just need the sign of the numerator."},{"Start":"04:52.560 ","End":"04:55.830","Text":"If it\u0027s minus 1 and minus 1 squared is 1,"},{"Start":"04:55.830 ","End":"04:58.995","Text":"that\u0027s minus 1 and then minus 2."},{"Start":"04:58.995 ","End":"05:01.230","Text":"That\u0027s negative."},{"Start":"05:01.230 ","End":"05:03.420","Text":"The derivative is negative,"},{"Start":"05:03.420 ","End":"05:04.865","Text":"I put with a minus."},{"Start":"05:04.865 ","End":"05:08.300","Text":"Then I\u0027ll try x equals 1."},{"Start":"05:08.300 ","End":"05:14.450","Text":"If x is 1, that\u0027s minus 1 plus 2, so that\u0027s positive."},{"Start":"05:14.450 ","End":"05:17.510","Text":"When x is equal to 3,"},{"Start":"05:17.510 ","End":"05:21.275","Text":"I have minus 9 plus 6,"},{"Start":"05:21.275 ","End":"05:24.560","Text":"that\u0027s minus 3, that\u0027s negative."},{"Start":"05:24.560 ","End":"05:29.465","Text":"Now we interpret these signs in terms of the function\u0027s behavior."},{"Start":"05:29.465 ","End":"05:31.655","Text":"If the derivative is negative,"},{"Start":"05:31.655 ","End":"05:33.815","Text":"the function is decreasing."},{"Start":"05:33.815 ","End":"05:35.135","Text":"Indicate with an arrow."},{"Start":"05:35.135 ","End":"05:37.730","Text":"When the derivative is positive,"},{"Start":"05:37.730 ","End":"05:41.140","Text":"the function is increasing, upward arrow."},{"Start":"05:41.140 ","End":"05:44.030","Text":"Here again, derivative is negative,"},{"Start":"05:44.030 ","End":"05:46.265","Text":"so function is decreasing."},{"Start":"05:46.265 ","End":"05:49.130","Text":"Now between an increasing and decreasing,"},{"Start":"05:49.130 ","End":"05:54.115","Text":"we know that x equals 2 has to be an extremum of type maximum."},{"Start":"05:54.115 ","End":"05:56.520","Text":"If it was defined at 0,"},{"Start":"05:56.520 ","End":"05:59.010","Text":"then I would put here a minimum but it\u0027s"},{"Start":"05:59.010 ","End":"06:02.315","Text":"outside the domain so I don\u0027t write anything at 0,"},{"Start":"06:02.315 ","End":"06:03.845","Text":"just the maximum here."},{"Start":"06:03.845 ","End":"06:08.705","Text":"The last stage is to write the y for all the extrema,"},{"Start":"06:08.705 ","End":"06:10.625","Text":"which case I just need it for 2."},{"Start":"06:10.625 ","End":"06:14.300","Text":"I substitute in the original function f,"},{"Start":"06:14.300 ","End":"06:22.370","Text":"I put x equals 2 in here and I get 2 minus 1 over 2 squared, which is 1/4."},{"Start":"06:22.370 ","End":"06:27.200","Text":"This is so I can talk about the extremum in terms of its x, y, 2,1/4."},{"Start":"06:27.200 ","End":"06:29.450","Text":"After filling the table,"},{"Start":"06:29.450 ","End":"06:34.010","Text":"we now get to the stage where we can draw conclusions which is essentially"},{"Start":"06:34.010 ","End":"06:39.155","Text":"answering the original question of extrema increasing and decreasing."},{"Start":"06:39.155 ","End":"06:41.365","Text":"Let\u0027s start with the extrema."},{"Start":"06:41.365 ","End":"06:47.435","Text":"We almost had 2, but we only have 1 and we have a maximum and they still call it max,"},{"Start":"06:47.435 ","End":"06:54.265","Text":"meaning this is a maximum point at the point where x is 2 and y is 1/4."},{"Start":"06:54.265 ","End":"06:58.120","Text":"Now the intervals of increase and decrease so"},{"Start":"06:58.120 ","End":"07:02.810","Text":"the function is increasing where the arrow goes up."},{"Start":"07:02.810 ","End":"07:10.070","Text":"That means between 0 and 2 and the function is decreasing when the arrow is going down,"},{"Start":"07:10.070 ","End":"07:18.015","Text":"which is here and here we have x is less than 0 and we also have x is bigger than 2."},{"Start":"07:18.015 ","End":"07:22.420","Text":"That answers all the questions. We\u0027re done."}],"ID":4807},{"Watched":false,"Name":"Exercise 9","Duration":"8m 5s","ChapterTopicVideoID":4808,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4808.jpeg","UploadDate":"2015-07-15T04:21:39.9900000","DurationForVideoObject":"PT8M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we\u0027re given a function f of x is"},{"Start":"00:03.360 ","End":"00:06.555","Text":"2x squared over x plus 1 squared and we have to find"},{"Start":"00:06.555 ","End":"00:09.465","Text":"all the local extrema and also"},{"Start":"00:09.465 ","End":"00:13.245","Text":"the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:13.245 ","End":"00:16.470","Text":"This type of exercise has a standard set of steps,"},{"Start":"00:16.470 ","End":"00:20.955","Text":"cookbook style and the first thing we do is to differentiate the function."},{"Start":"00:20.955 ","End":"00:23.985","Text":"So we have f prime of x equals."},{"Start":"00:23.985 ","End":"00:26.295","Text":"Now in case you\u0027ve forgotten the quotient rule,"},{"Start":"00:26.295 ","End":"00:28.740","Text":"I\u0027ll write it for you again. Here it is."},{"Start":"00:28.740 ","End":"00:32.355","Text":"I used u and v instead of f and g because f is already taken."},{"Start":"00:32.355 ","End":"00:35.490","Text":"Here we go, a derivative of the top,"},{"Start":"00:35.490 ","End":"00:44.660","Text":"which is 4x times the denominator as is x plus 1 squared minus the numerator,"},{"Start":"00:44.660 ","End":"00:49.805","Text":"as is 2x squared times derivative of the denominator,"},{"Start":"00:49.805 ","End":"00:54.440","Text":"which is twice x plus 1 times the internal derivative,"},{"Start":"00:54.440 ","End":"00:58.550","Text":"which is 1 and all over the denominator squared,"},{"Start":"00:58.550 ","End":"01:01.565","Text":"which is x plus 1 to the fourth."},{"Start":"01:01.565 ","End":"01:06.200","Text":"What I\u0027ll do is I\u0027ll just simplify the numerator here at the side in a scratch area,"},{"Start":"01:06.200 ","End":"01:08.540","Text":"we get here 4x."},{"Start":"01:08.540 ","End":"01:16.130","Text":"Now x plus 1 squared is x squared plus 2x plus 1 and here we have minus,"},{"Start":"01:16.130 ","End":"01:18.049","Text":"let\u0027s take all the numbers together,"},{"Start":"01:18.049 ","End":"01:20.600","Text":"2 times 2 times 1 is 4,"},{"Start":"01:20.600 ","End":"01:26.820","Text":"and x squared times x plus 1 and this is equal to,"},{"Start":"01:26.820 ","End":"01:31.395","Text":"let\u0027s see, 4x times x squared is 4x cubed."},{"Start":"01:31.395 ","End":"01:37.560","Text":"Then we get plus 8x squared plus 4x minus 4x squared times"},{"Start":"01:37.560 ","End":"01:43.860","Text":"x is minus 4x cubed and minus 4x squared,"},{"Start":"01:43.860 ","End":"01:47.460","Text":"which equals,4x cubed with 4x cubed cancels,"},{"Start":"01:47.460 ","End":"01:53.330","Text":"8x squared minus 4x squared is 4x squared and plus 4x."},{"Start":"01:53.330 ","End":"01:55.045","Text":"So back to here."},{"Start":"01:55.045 ","End":"01:58.910","Text":"So this is equal to 4x squared plus 4x."},{"Start":"01:58.910 ","End":"02:02.440","Text":"I can take the 4 outside the brackets here."},{"Start":"02:02.440 ","End":"02:12.455","Text":"It\u0027s 4 times x squared plus x over x plus 1 to the fourth."},{"Start":"02:12.455 ","End":"02:16.490","Text":"Now the next step is to equate the derivative to"},{"Start":"02:16.490 ","End":"02:21.200","Text":"0 and to solve for x. I\u0027m writing now equals 0."},{"Start":"02:21.200 ","End":"02:23.555","Text":"Now 4 is positive,"},{"Start":"02:23.555 ","End":"02:25.250","Text":"the denominator is positive,"},{"Start":"02:25.250 ","End":"02:27.755","Text":"I mean, we can just get rid of those."},{"Start":"02:27.755 ","End":"02:30.680","Text":"Let me, if I divide by 4 and multiply by this,"},{"Start":"02:30.680 ","End":"02:35.870","Text":"all I need for it to be 0 is for this x squared plus x to be equal to 0."},{"Start":"02:35.870 ","End":"02:41.555","Text":"I get from here that x squared plus x equals 0."},{"Start":"02:41.555 ","End":"02:45.530","Text":"Just divide it by 4 and multiply it by this denominator"},{"Start":"02:45.530 ","End":"02:50.765","Text":"that gives us x squared plus x is 0 and therefore if I factorize x,"},{"Start":"02:50.765 ","End":"02:53.360","Text":"x plus 1 equals 0,"},{"Start":"02:53.360 ","End":"02:59.880","Text":"and that means that x will equal either 0 or minus 1."},{"Start":"02:59.880 ","End":"03:01.730","Text":"There\u0027s something I forgot to do before,"},{"Start":"03:01.730 ","End":"03:05.645","Text":"but it\u0027s not too late and that is to note the domain of the function."},{"Start":"03:05.645 ","End":"03:09.200","Text":"If I look at this function which is 2x squared over x plus 1 squared,"},{"Start":"03:09.200 ","End":"03:11.570","Text":"we can\u0027t have a denominator being 0."},{"Start":"03:11.570 ","End":"03:16.130","Text":"So we have to rule out the possibility that x is equal to minus 1."},{"Start":"03:16.130 ","End":"03:22.040","Text":"In other words, the domain is x not equal to minus 1 and if we look over here,"},{"Start":"03:22.040 ","End":"03:25.880","Text":"we\u0027ve included minus 1 in the suspects,"},{"Start":"03:25.880 ","End":"03:28.850","Text":"so minus 1 is not a suspect for an extremum,"},{"Start":"03:28.850 ","End":"03:30.290","Text":"the function is not defined there."},{"Start":"03:30.290 ","End":"03:32.900","Text":"I\u0027m just going to cross it out so it\u0027s still visible."},{"Start":"03:32.900 ","End":"03:34.730","Text":"We\u0027re going to use the minus 1 somewhere,"},{"Start":"03:34.730 ","End":"03:36.335","Text":"but not as an extremum."},{"Start":"03:36.335 ","End":"03:41.210","Text":"The next stage here is to draw the table with its 4 rows,"},{"Start":"03:41.210 ","End":"03:42.650","Text":"I\u0027ll just put it in here."},{"Start":"03:42.650 ","End":"03:44.435","Text":"Now when we fill the table,"},{"Start":"03:44.435 ","End":"03:47.150","Text":"we need to put in 2 kinds of x,"},{"Start":"03:47.150 ","End":"03:51.270","Text":"1 which are suspects for extrema, in other the words,"},{"Start":"03:51.270 ","End":"03:54.080","Text":"x equals 0, but we also have to put into"},{"Start":"03:54.080 ","End":"03:58.250","Text":"the table the values of x for which the function is undefined."},{"Start":"03:58.250 ","End":"04:04.130","Text":"We have to put in both minus 1 and the 0 and in increasing order."},{"Start":"04:04.130 ","End":"04:08.090","Text":"First the minus 1 and then the 0,"},{"Start":"04:08.090 ","End":"04:11.065","Text":"and we leave space around them for extra information."},{"Start":"04:11.065 ","End":"04:15.920","Text":"These points divide the line into intervals and we\u0027re going to label the intervals."},{"Start":"04:15.920 ","End":"04:20.260","Text":"This interval is the interval where x is less than minus 1."},{"Start":"04:20.260 ","End":"04:24.395","Text":"Here x is between minus 1 and 0,"},{"Start":"04:24.395 ","End":"04:27.125","Text":"and here x is bigger than 0."},{"Start":"04:27.125 ","End":"04:30.485","Text":"Then we choose a sample point from each of the intervals."},{"Start":"04:30.485 ","End":"04:32.165","Text":"Arbitrary could be anything."},{"Start":"04:32.165 ","End":"04:34.490","Text":"I\u0027ll choose minus 2."},{"Start":"04:34.490 ","End":"04:36.530","Text":"For between minus 1 and 0,"},{"Start":"04:36.530 ","End":"04:42.430","Text":"I\u0027ll choose minus 1/2 and for bigger than 0, I\u0027ll choose 1."},{"Start":"04:42.430 ","End":"04:46.040","Text":"With these values, I have to substitute them into"},{"Start":"04:46.040 ","End":"04:51.590","Text":"f prime of x and our f prime of x is here."},{"Start":"04:51.590 ","End":"04:54.610","Text":"But we don\u0027t write down the value of the substitution,"},{"Start":"04:54.610 ","End":"04:56.510","Text":"just whether it\u0027s positive or negative."},{"Start":"04:56.510 ","End":"04:59.180","Text":"That\u0027s all we care about and I see that 4 is"},{"Start":"04:59.180 ","End":"05:02.075","Text":"positive and x plus 1 to the fourth is positive."},{"Start":"05:02.075 ","End":"05:05.645","Text":"The sign is basically determined by the x squared plus x."},{"Start":"05:05.645 ","End":"05:07.335","Text":"If I put minus 2 in,"},{"Start":"05:07.335 ","End":"05:09.720","Text":"I get plus 4 minus 2,"},{"Start":"05:09.720 ","End":"05:14.545","Text":"which is 2, which is positive and I write a plus sign for that."},{"Start":"05:14.545 ","End":"05:19.300","Text":"With minus a 1/2, minus a 1/2 squared is plus a 1/4,"},{"Start":"05:19.300 ","End":"05:22.010","Text":"but then minus a half is negative,"},{"Start":"05:22.010 ","End":"05:23.615","Text":"that\u0027s a minus here."},{"Start":"05:23.615 ","End":"05:27.935","Text":"For x equals 1, 1 squared plus 1, obviously positive."},{"Start":"05:27.935 ","End":"05:30.440","Text":"That\u0027s the signs."},{"Start":"05:30.440 ","End":"05:32.360","Text":"We also, at the beginning,"},{"Start":"05:32.360 ","End":"05:36.560","Text":"we should write down the f prime for the values that we can."},{"Start":"05:36.560 ","End":"05:38.764","Text":"Normally that would be 0 and 0,"},{"Start":"05:38.764 ","End":"05:42.890","Text":"but the minus 1 we didn\u0027t get from f prime equals 0."},{"Start":"05:42.890 ","End":"05:46.940","Text":"The minus 1 came from the function being not defined there,"},{"Start":"05:46.940 ","End":"05:50.195","Text":"so in the case of 0, the f prime is 0."},{"Start":"05:50.195 ","End":"05:51.440","Text":"In the case of minus 1,"},{"Start":"05:51.440 ","End":"05:53.689","Text":"the function or its derivative and not defined,"},{"Start":"05:53.689 ","End":"05:55.475","Text":"and it\u0027s usually an asymptote."},{"Start":"05:55.475 ","End":"05:59.480","Text":"What I do is I just put a dotted line through the rest of the table,"},{"Start":"05:59.480 ","End":"06:01.175","Text":"not going to put anything there."},{"Start":"06:01.175 ","End":"06:04.470","Text":"It just serves as a dividing line which might, for example, here,"},{"Start":"06:04.470 ","End":"06:08.570","Text":"separate between positive and negative or increasing and decreasing."},{"Start":"06:08.570 ","End":"06:11.165","Text":"Now we continue to the next row."},{"Start":"06:11.165 ","End":"06:13.580","Text":"When the derivative is positive,"},{"Start":"06:13.580 ","End":"06:16.429","Text":"what it says about the function is that it\u0027s increasing,"},{"Start":"06:16.429 ","End":"06:18.995","Text":"which I write with an upward arrow."},{"Start":"06:18.995 ","End":"06:21.500","Text":"When it\u0027s negative, the first derivative,"},{"Start":"06:21.500 ","End":"06:23.975","Text":"that means that the function is decreasing."},{"Start":"06:23.975 ","End":"06:26.280","Text":"When the derivative is positive,"},{"Start":"06:26.280 ","End":"06:28.880","Text":"once again, the function is increasing."},{"Start":"06:28.880 ","End":"06:32.075","Text":"Now 0, which is the suspect for extremum,"},{"Start":"06:32.075 ","End":"06:36.365","Text":"turns out is indeed an extremum and it\u0027s of type minimum."},{"Start":"06:36.365 ","End":"06:38.420","Text":"Because if I\u0027m decreasing and then increasing,"},{"Start":"06:38.420 ","End":"06:39.860","Text":"this must be a minimum."},{"Start":"06:39.860 ","End":"06:42.980","Text":"Here, if hadn\u0027t paid attention that it was not defined,"},{"Start":"06:42.980 ","End":"06:44.210","Text":"I would have written maximum,"},{"Start":"06:44.210 ","End":"06:46.910","Text":"but no function is just not defined there."},{"Start":"06:46.910 ","End":"06:48.200","Text":"So I only have 1 extra,"},{"Start":"06:48.200 ","End":"06:49.730","Text":"which is this minimum."},{"Start":"06:49.730 ","End":"06:55.580","Text":"Finally, I put in the value of y for the interesting points for minus 1,"},{"Start":"06:55.580 ","End":"06:58.115","Text":"there is no y because the function\u0027s not defined."},{"Start":"06:58.115 ","End":"07:02.630","Text":"Y of course is just f of x. I can get y for x equals"},{"Start":"07:02.630 ","End":"07:07.200","Text":"0 by substituting in the original function."},{"Start":"07:07.200 ","End":"07:09.350","Text":"If I put x equals 0,"},{"Start":"07:09.350 ","End":"07:11.030","Text":"I get, well, the numerator is 0,"},{"Start":"07:11.030 ","End":"07:14.090","Text":"so the whole thing 0, so y is also 0,"},{"Start":"07:14.090 ","End":"07:15.560","Text":"but I don\u0027t put anything here."},{"Start":"07:15.560 ","End":"07:17.000","Text":"Now that the table is complete,"},{"Start":"07:17.000 ","End":"07:19.700","Text":"we can draw conclusions which is essentially answering"},{"Start":"07:19.700 ","End":"07:24.020","Text":"the original questions about extrema increasing and decreasing."},{"Start":"07:24.020 ","End":"07:27.665","Text":"I\u0027ll just write those down here."},{"Start":"07:27.665 ","End":"07:30.320","Text":"First of all, the extrema, we almost had 2,"},{"Start":"07:30.320 ","End":"07:37.385","Text":"but we only have 1 and that is the minimum that we have at x equals 0, y equals 0."},{"Start":"07:37.385 ","End":"07:40.100","Text":"Then where is the function increasing?"},{"Start":"07:40.100 ","End":"07:42.890","Text":"It\u0027s increasing wherever the arrow goes up,"},{"Start":"07:42.890 ","End":"07:44.315","Text":"which is here and here,"},{"Start":"07:44.315 ","End":"07:51.065","Text":"which means x less than minus 1 and also at x bigger than 0,"},{"Start":"07:51.065 ","End":"07:56.870","Text":"and for decreasing, we just look where the arrow is going down, which is here."},{"Start":"07:56.870 ","End":"08:00.380","Text":"That gives us that x is between minus"},{"Start":"08:00.380 ","End":"08:05.940","Text":"1 and 0 and that basically answers all the questions and so we\u0027re done."}],"ID":4808},{"Watched":false,"Name":"Exercise 10","Duration":"9m 53s","ChapterTopicVideoID":4809,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4809.jpeg","UploadDate":"2015-07-15T04:22:35.3000000","DurationForVideoObject":"PT9M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"In this exercise, we\u0027re given a function f of x equals x cubed over x plus 1 squared,"},{"Start":"00:06.450 ","End":"00:09.720","Text":"and we have to find its local extrema and"},{"Start":"00:09.720 ","End":"00:13.365","Text":"the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:13.365 ","End":"00:15.675","Text":"I\u0027ve copied the function over here."},{"Start":"00:15.675 ","End":"00:19.020","Text":"This type of exercise has a standard set of steps."},{"Start":"00:19.020 ","End":"00:23.040","Text":"First thing we do for this exercise is to differentiate."},{"Start":"00:23.040 ","End":"00:27.150","Text":"We get f prime of x equals a quotient."},{"Start":"00:27.150 ","End":"00:30.105","Text":"You may have forgotten the quotient rule so let me write it down."},{"Start":"00:30.105 ","End":"00:39.675","Text":"U over v derivative is u prime v minus uv prime over v squared."},{"Start":"00:39.675 ","End":"00:45.410","Text":"I just chose to use u and v because if I used f over g, f\u0027s already taken."},{"Start":"00:45.410 ","End":"00:50.285","Text":"Here\u0027s the u and here is the v. What we get is u prime,"},{"Start":"00:50.285 ","End":"00:53.150","Text":"which is 3x squared times v,"},{"Start":"00:53.150 ","End":"00:57.800","Text":"which is x plus 1 squared minus u,"},{"Start":"00:57.800 ","End":"01:01.670","Text":"which is x cubed times v prime,"},{"Start":"01:01.670 ","End":"01:06.769","Text":"which is 2 times x plus 1 times internal derivative,"},{"Start":"01:06.769 ","End":"01:13.975","Text":"which is 1, all this over x plus 1 squared, squared, so x^4."},{"Start":"01:13.975 ","End":"01:19.025","Text":"This is now equal to- I would better do it at the side."},{"Start":"01:19.025 ","End":"01:24.740","Text":"Here we have 3x squared times x squared plus 2x plus 1."},{"Start":"01:24.740 ","End":"01:27.945","Text":"I just expanded this bit, minus,"},{"Start":"01:27.945 ","End":"01:29.235","Text":"put the 2 in front,"},{"Start":"01:29.235 ","End":"01:34.005","Text":"2x cubed times x plus 1."},{"Start":"01:34.005 ","End":"01:40.890","Text":"This equals 3x^4 plus 6x cubed plus 3x"},{"Start":"01:40.890 ","End":"01:47.640","Text":"squared minus 2x^4 and minus 2x cubed and this equals,"},{"Start":"01:47.640 ","End":"01:52.740","Text":"let\u0027s collect together 3x^4 minus 2x^4 is x^4."},{"Start":"01:52.740 ","End":"01:56.160","Text":"For x cubed, we have 6 minus 2,"},{"Start":"01:56.160 ","End":"01:58.335","Text":"that\u0027s 4 of them and x squared,"},{"Start":"01:58.335 ","End":"02:00.540","Text":"we just have 3 of them."},{"Start":"02:00.540 ","End":"02:04.365","Text":"Then I can factorize to x squared,"},{"Start":"02:04.365 ","End":"02:08.220","Text":"x squared plus 4x plus 3."},{"Start":"02:08.220 ","End":"02:10.380","Text":"I\u0027ll write that here,"},{"Start":"02:10.380 ","End":"02:19.240","Text":"x squared times x squared plus 4x plus 3 over x plus 1^4."},{"Start":"02:19.240 ","End":"02:23.540","Text":"Now, the next step after differentiating is to set it equal to"},{"Start":"02:23.540 ","End":"02:28.520","Text":"0 and to solve for x. I set now this equals 0,"},{"Start":"02:28.520 ","End":"02:30.260","Text":"and I want to find the solutions."},{"Start":"02:30.260 ","End":"02:31.910","Text":"Now if I look at it,"},{"Start":"02:31.910 ","End":"02:35.120","Text":"this can be 0, when the numerator 0,"},{"Start":"02:35.120 ","End":"02:36.710","Text":"the denominator doesn\u0027t matter."},{"Start":"02:36.710 ","End":"02:39.320","Text":"Fraction 0 when its numerator is 0."},{"Start":"02:39.320 ","End":"02:41.960","Text":"I have 2 possibilities because I have a product."},{"Start":"02:41.960 ","End":"02:49.260","Text":"I have that either x squared equals 0 or x squared plus 4x plus 3 equals 0."},{"Start":"02:49.260 ","End":"02:54.015","Text":"Now, x squared is 0 means that x equals 0."},{"Start":"02:54.015 ","End":"02:57.185","Text":"If I solve this quadratic equation,"},{"Start":"02:57.185 ","End":"03:01.355","Text":"it turns out that the solutions are minus 1 and minus 3."},{"Start":"03:01.355 ","End":"03:03.710","Text":"I\u0027m not going to solve it for you."},{"Start":"03:03.710 ","End":"03:11.405","Text":"You can do quadratic equations and here we get the solutions are minus 1 and minus 3."},{"Start":"03:11.405 ","End":"03:16.820","Text":"The answer altogether is that x equals 0 or x equals minus 1,"},{"Start":"03:16.820 ","End":"03:19.505","Text":"or x equals minus 3,"},{"Start":"03:19.505 ","End":"03:23.250","Text":"3 possible suspects for an extrema."},{"Start":"03:23.250 ","End":"03:25.650","Text":"Now, there\u0027s something I forgot to do earlier,"},{"Start":"03:25.650 ","End":"03:29.240","Text":"but it\u0027s not too late and that is to examine the domain of"},{"Start":"03:29.240 ","End":"03:33.995","Text":"the function f. We can see that its denominator mustn\u0027t be 0,"},{"Start":"03:33.995 ","End":"03:37.010","Text":"that the domain is every x except x equals 1."},{"Start":"03:37.010 ","End":"03:41.880","Text":"In other words, x is not equal to minus 1 because at minus 1,"},{"Start":"03:41.880 ","End":"03:43.830","Text":"the denominator is 0."},{"Start":"03:43.830 ","End":"03:46.370","Text":"When we look at our list of suspects,"},{"Start":"03:46.370 ","End":"03:51.290","Text":"x equals minus 1 is not a suspect because the function is not even defined there,"},{"Start":"03:51.290 ","End":"03:52.820","Text":"so I\u0027ll just cross it out,"},{"Start":"03:52.820 ","End":"03:54.170","Text":"but so it\u0027s still legible,"},{"Start":"03:54.170 ","End":"03:55.910","Text":"we\u0027ll still have use for the minus 1,"},{"Start":"03:55.910 ","End":"03:57.619","Text":"but just not as a suspect."},{"Start":"03:57.619 ","End":"04:00.169","Text":"Next, we\u0027ll go to the table,"},{"Start":"04:00.169 ","End":"04:03.515","Text":"and I\u0027ll explain what to do with the minus 1 there."},{"Start":"04:03.515 ","End":"04:08.480","Text":"Here\u0027s the table and what we do with the table is, first of all,"},{"Start":"04:08.480 ","End":"04:11.030","Text":"we put some values in the x row,"},{"Start":"04:11.030 ","End":"04:12.964","Text":"and we put 2 kinds of values,"},{"Start":"04:12.964 ","End":"04:18.655","Text":"one which is a suspect for extrema and that is x equals 0 and minus 3."},{"Start":"04:18.655 ","End":"04:23.180","Text":"We also have to put in values for which the function is not defined."},{"Start":"04:23.180 ","End":"04:26.690","Text":"In other words, the minus 1 does come in after all."},{"Start":"04:26.690 ","End":"04:28.700","Text":"But we put them in an increasing order,"},{"Start":"04:28.700 ","End":"04:36.470","Text":"so the order is minus 3 and then minus 1 and then 0."},{"Start":"04:36.470 ","End":"04:39.000","Text":"But minus 1 is different and that these 2 are"},{"Start":"04:39.000 ","End":"04:42.980","Text":"suspect and minus 1 actually is where the function is not defined,"},{"Start":"04:42.980 ","End":"04:45.140","Text":"and it\u0027s probably an asymptote there."},{"Start":"04:45.140 ","End":"04:49.130","Text":"I indicate that by just putting a dotted line for the rest"},{"Start":"04:49.130 ","End":"04:53.390","Text":"of the column here and I don\u0027t put anything under it."},{"Start":"04:53.390 ","End":"04:55.340","Text":"I also know that f prime,"},{"Start":"04:55.340 ","End":"04:57.020","Text":"but the others is 0."},{"Start":"04:57.020 ","End":"04:58.820","Text":"We thought that f prime was 0 here,"},{"Start":"04:58.820 ","End":"05:00.405","Text":"but actually, it\u0027s not defined."},{"Start":"05:00.405 ","End":"05:06.290","Text":"What I do with these values is that they separate the line into intervals here,"},{"Start":"05:06.290 ","End":"05:07.320","Text":"here, here, and here,"},{"Start":"05:07.320 ","End":"05:10.175","Text":"and I\u0027ll just write down what these intervals are."},{"Start":"05:10.175 ","End":"05:13.490","Text":"This is x less than minus 3,"},{"Start":"05:13.490 ","End":"05:18.860","Text":"between this we have x is in between minus 3 and minus 1."},{"Start":"05:18.860 ","End":"05:22.670","Text":"Here, x is between minus 1 and 0,"},{"Start":"05:22.670 ","End":"05:25.220","Text":"and here x is bigger than 0."},{"Start":"05:25.220 ","End":"05:29.300","Text":"As usual, we pick a sample point from each of the intervals."},{"Start":"05:29.300 ","End":"05:32.225","Text":"For this, I\u0027ll pick the point minus 4,"},{"Start":"05:32.225 ","End":"05:36.770","Text":"in this range I\u0027ll pick x is minus 2, between these 2,"},{"Start":"05:36.770 ","End":"05:39.230","Text":"I\u0027ll pick minus 1/2,"},{"Start":"05:39.230 ","End":"05:41.990","Text":"and here I\u0027ll pick x equals 1."},{"Start":"05:41.990 ","End":"05:47.375","Text":"Now what we have to do is to substitute these in f prime of x,"},{"Start":"05:47.375 ","End":"05:49.930","Text":"I\u0027ll just outline it."},{"Start":"05:49.930 ","End":"05:51.780","Text":"Let\u0027s put them in."},{"Start":"05:51.780 ","End":"05:55.580","Text":"Remember we don\u0027t have to get the exact value at all,"},{"Start":"05:55.580 ","End":"05:58.550","Text":"we just need to say whether it\u0027s positive or negative."},{"Start":"05:58.550 ","End":"05:59.990","Text":"If that\u0027s the case, well,"},{"Start":"05:59.990 ","End":"06:03.875","Text":"x squared and x plus 1^4 are both positive,"},{"Start":"06:03.875 ","End":"06:07.730","Text":"so all we need to do is put them in this bit here,"},{"Start":"06:07.730 ","End":"06:09.200","Text":"x squared plus 4x plus 3,"},{"Start":"06:09.200 ","End":"06:11.540","Text":"and see if we get positive or negative."},{"Start":"06:11.540 ","End":"06:14.500","Text":"If I put in minus 4,"},{"Start":"06:14.500 ","End":"06:17.820","Text":"I get minus 4 squared is 16."},{"Start":"06:17.820 ","End":"06:22.130","Text":"Minus 16 plus 3 is 3, is positive."},{"Start":"06:22.130 ","End":"06:23.765","Text":"Then the minus 2,"},{"Start":"06:23.765 ","End":"06:26.140","Text":"minus 2 squared is 4,"},{"Start":"06:26.140 ","End":"06:30.225","Text":"minus 8 plus 3, it\u0027s negative."},{"Start":"06:30.225 ","End":"06:37.050","Text":"The minus 1/2, minus 1/2 is 1/4,"},{"Start":"06:37.050 ","End":"06:42.164","Text":"1/4 minus 2 and plus 3,"},{"Start":"06:42.164 ","End":"06:46.860","Text":"so it\u0027s minus a 1/4 plus 1, 3/4 is positive."},{"Start":"06:46.860 ","End":"06:53.240","Text":"When x is 1, 1 plus 4 plus 3, certainly positive."},{"Start":"06:53.240 ","End":"06:59.210","Text":"Now let\u0027s interpret the first derivative in terms of the function."},{"Start":"06:59.210 ","End":"07:01.910","Text":"If the derivative is positive,"},{"Start":"07:01.910 ","End":"07:04.070","Text":"it means that the function is increasing,"},{"Start":"07:04.070 ","End":"07:06.110","Text":"which I indicate with an up arrow."},{"Start":"07:06.110 ","End":"07:10.010","Text":"Derivative negative, meaning the function is decreasing."},{"Start":"07:10.010 ","End":"07:13.550","Text":"Once again positive, so increasing."},{"Start":"07:13.550 ","End":"07:18.235","Text":"Here again, positive, which means increasing."},{"Start":"07:18.235 ","End":"07:20.270","Text":"What can we conclude?"},{"Start":"07:20.270 ","End":"07:24.320","Text":"Well, here we obviously have an extremum because an increase"},{"Start":"07:24.320 ","End":"07:28.700","Text":"followed by a decrease indicate an extremum of type maximum."},{"Start":"07:28.700 ","End":"07:30.230","Text":"Here the function is not defined,"},{"Start":"07:30.230 ","End":"07:32.140","Text":"so we can\u0027t talk about extremum."},{"Start":"07:32.140 ","End":"07:35.209","Text":"It would have been a minimum but not defined."},{"Start":"07:35.209 ","End":"07:37.430","Text":"Between increasing and increasing,"},{"Start":"07:37.430 ","End":"07:38.899","Text":"we don\u0027t have an extremum."},{"Start":"07:38.899 ","End":"07:41.540","Text":"I don\u0027t know what it is, it\u0027s probably an inflection point,"},{"Start":"07:41.540 ","End":"07:45.670","Text":"but I\u0027ll just write a question mark that it\u0027s not an extremum it\u0027s something else."},{"Start":"07:45.670 ","End":"07:53.420","Text":"Finally, we need to put the y in for the points of interest which are minus 3 and 0."},{"Start":"07:53.420 ","End":"07:56.645","Text":"I put those in the original function,"},{"Start":"07:56.645 ","End":"07:58.930","Text":"not in the derivative or anything,"},{"Start":"07:58.930 ","End":"08:01.155","Text":"and we need the actual value this time."},{"Start":"08:01.155 ","End":"08:03.845","Text":"With the easy one when x is 0,"},{"Start":"08:03.845 ","End":"08:06.410","Text":"numerator 0, so it\u0027s 0."},{"Start":"08:06.410 ","End":"08:08.600","Text":"When we have minus 3,"},{"Start":"08:08.600 ","End":"08:12.440","Text":"minus 3 cubed is minus 27,"},{"Start":"08:12.440 ","End":"08:16.740","Text":"and the denominator is minus 2 squared is 4,"},{"Start":"08:16.740 ","End":"08:18.750","Text":"so minus 27 over 4,"},{"Start":"08:18.750 ","End":"08:19.950","Text":"I\u0027ll leave it like that,"},{"Start":"08:19.950 ","End":"08:23.935","Text":"minus 27 over 4."},{"Start":"08:23.935 ","End":"08:27.200","Text":"I think we have everything we need for answering"},{"Start":"08:27.200 ","End":"08:31.220","Text":"the original question about the extrema and the increasing,"},{"Start":"08:31.220 ","End":"08:34.445","Text":"decreasing the conclusions from the table."},{"Start":"08:34.445 ","End":"08:37.185","Text":"We have for extrema,"},{"Start":"08:37.185 ","End":"08:39.855","Text":"we almost had 3, but we only have 1."},{"Start":"08:39.855 ","End":"08:43.340","Text":"1 was ruled out because the function was undefined and 1 was ruled out"},{"Start":"08:43.340 ","End":"08:46.975","Text":"because we didn\u0027t have a change of trend from increasing to decreasing."},{"Start":"08:46.975 ","End":"08:50.565","Text":"All we have is a single 1 which is a maximum,"},{"Start":"08:50.565 ","End":"08:56.660","Text":"and it occurs at the point where x is 0 and y is minus 27 over 4."},{"Start":"08:56.660 ","End":"09:00.185","Text":"Then the areas of intervals of increase and decrease,"},{"Start":"09:00.185 ","End":"09:03.660","Text":"the function is increasing when we see the up arrow."},{"Start":"09:03.660 ","End":"09:06.155","Text":"Actually, we have 3 areas of increase,"},{"Start":"09:06.155 ","End":"09:10.865","Text":"but this area is x less than minus 3."},{"Start":"09:10.865 ","End":"09:16.895","Text":"Here, when we have 2 intervals of increase which are joined except for this 1 point,"},{"Start":"09:16.895 ","End":"09:20.975","Text":"it turns out we can combine them into 1 single interval."},{"Start":"09:20.975 ","End":"09:23.880","Text":"It\u0027s been explained in the theory section."},{"Start":"09:23.880 ","End":"09:26.970","Text":"Anyway, we combine these 2 from minus 1-0,"},{"Start":"09:26.970 ","End":"09:30.990","Text":"and from 0 onwards it\u0027s all together we get from minus 1 onwards,"},{"Start":"09:30.990 ","End":"09:33.750","Text":"so x is bigger than minus 1."},{"Start":"09:33.750 ","End":"09:35.180","Text":"Just combining these 2,"},{"Start":"09:35.180 ","End":"09:38.360","Text":"the single point doesn\u0027t break the increasing trend."},{"Start":"09:38.360 ","End":"09:41.090","Text":"The decreasing is where we see"},{"Start":"09:41.090 ","End":"09:45.395","Text":"a down arrow and there\u0027s only 1 place where that is and that\u0027s here."},{"Start":"09:45.395 ","End":"09:50.280","Text":"Decreasing is x between minus 3 and minus 1."},{"Start":"09:50.280 ","End":"09:54.000","Text":"That answers the questions and we\u0027re done."}],"ID":4809},{"Watched":false,"Name":"Exercise 11","Duration":"7m 21s","ChapterTopicVideoID":4810,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4810.jpeg","UploadDate":"2015-07-15T04:23:14.2700000","DurationForVideoObject":"PT7M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this exercise, we\u0027re given a function f of x"},{"Start":"00:03.510 ","End":"00:07.255","Text":"is equal to x plus 1 over x minus 1 all cubed."},{"Start":"00:07.255 ","End":"00:10.755","Text":"What we have to do is find it\u0027s extrema"},{"Start":"00:10.755 ","End":"00:14.775","Text":"and the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.775 ","End":"00:18.660","Text":"I just like to say that the first thing we should do is look at"},{"Start":"00:18.660 ","End":"00:25.605","Text":"its domain and we see that the domain is where the denominator is not 0."},{"Start":"00:25.605 ","End":"00:30.150","Text":"The domain is x not equal to 1,"},{"Start":"00:30.150 ","End":"00:31.410","Text":"every other value is okay,"},{"Start":"00:31.410 ","End":"00:33.330","Text":"but we can\u0027t put x equals 1."},{"Start":"00:33.330 ","End":"00:36.829","Text":"Now, this exercise is standard exercise"},{"Start":"00:36.829 ","End":"00:40.520","Text":"with certain steps that we follow and we get to the solution."},{"Start":"00:40.520 ","End":"00:46.910","Text":"The first thing we do is to differentiate f. We get f prime of x."},{"Start":"00:46.910 ","End":"00:52.460","Text":"First of all, the chain rule so 3 x plus 1 over x minus"},{"Start":"00:52.460 ","End":"00:58.880","Text":"1 squared times the internal derivative of what\u0027s in here."},{"Start":"00:58.880 ","End":"01:02.300","Text":"That\u0027s a quotient and in case you have forgotten the quotient rule,"},{"Start":"01:02.300 ","End":"01:03.430","Text":"I\u0027ll write it down."},{"Start":"01:03.430 ","End":"01:05.820","Text":"Here we are u over v prime,"},{"Start":"01:05.820 ","End":"01:09.740","Text":"so applying it to the x plus 1 over x minus 1 we"},{"Start":"01:09.740 ","End":"01:13.985","Text":"multiply by denominator squared at the bottom."},{"Start":"01:13.985 ","End":"01:16.790","Text":"Then we have prime,"},{"Start":"01:16.790 ","End":"01:20.360","Text":"which is just 1, I won\u0027t bother writing it times v,"},{"Start":"01:20.360 ","End":"01:24.335","Text":"which is x minus 1, minus u,"},{"Start":"01:24.335 ","End":"01:27.070","Text":"which is x plus 1,"},{"Start":"01:27.070 ","End":"01:30.360","Text":"times v prime, which is also 1."},{"Start":"01:30.360 ","End":"01:32.325","Text":"I won\u0027t bother writing it."},{"Start":"01:32.325 ","End":"01:34.790","Text":"If we simplify this,"},{"Start":"01:34.790 ","End":"01:40.005","Text":"we get x minus 1 less x plus 1 is just minus 2,"},{"Start":"01:40.005 ","End":"01:44.145","Text":"the minus 2 with the 3 gives minus 6."},{"Start":"01:44.145 ","End":"01:53.465","Text":"It\u0027s minus 6 and then this I can write as x plus 1 squared over x minus 1 squared."},{"Start":"01:53.465 ","End":"01:56.630","Text":"Here I\u0027ve taken the minus 2 out already,"},{"Start":"01:56.630 ","End":"02:00.765","Text":"times 1 over x minus 1 squared."},{"Start":"02:00.765 ","End":"02:08.760","Text":"Altogether I get 6 x plus 1 squared over x minus 1 to the 4th."},{"Start":"02:08.760 ","End":"02:15.950","Text":"The next step after we found the derivative is to set it to 0 and solve for x,"},{"Start":"02:15.950 ","End":"02:18.875","Text":"so we have this thing equals 0."},{"Start":"02:18.875 ","End":"02:24.080","Text":"Now a fraction is 0 if and only if its numerator is 0."},{"Start":"02:24.080 ","End":"02:28.549","Text":"Basically what we get is x plus 1 squared is 0,"},{"Start":"02:28.549 ","End":"02:31.115","Text":"which means that x plus 1 is 0."},{"Start":"02:31.115 ","End":"02:35.800","Text":"Other words, we get x equals minus 1 only."},{"Start":"02:35.800 ","End":"02:42.515","Text":"That\u0027s the only suspect for an extremum and next we go to the table."},{"Start":"02:42.515 ","End":"02:45.505","Text":"I\u0027ll draw a standard table here."},{"Start":"02:45.505 ","End":"02:48.230","Text":"What we have to fill it with first,"},{"Start":"02:48.230 ","End":"02:52.715","Text":"we go to the next row is we have to put in 2 values of x."},{"Start":"02:52.715 ","End":"02:57.230","Text":"1 which is suspect to extrema from f prime equals 0,"},{"Start":"02:57.230 ","End":"02:59.020","Text":"that would be the minus 1."},{"Start":"02:59.020 ","End":"03:05.090","Text":"The other is where the function is not defined and that would be x equals 1."},{"Start":"03:05.090 ","End":"03:11.000","Text":"We put them in in increasing order so I\u0027ll put the minus 1 here and 1 here,"},{"Start":"03:11.000 ","End":"03:14.075","Text":"and we leave some space between them because these 2 values"},{"Start":"03:14.075 ","End":"03:17.720","Text":"chop up the line into 3 separate intervals."},{"Start":"03:17.720 ","End":"03:19.970","Text":"We have the first interval,"},{"Start":"03:19.970 ","End":"03:23.030","Text":"which is x less than minus 1"},{"Start":"03:23.030 ","End":"03:28.040","Text":"the middle interval minus 1 less than x less than 1 and finally,"},{"Start":"03:28.040 ","End":"03:30.280","Text":"the interval x bigger than 1."},{"Start":"03:30.280 ","End":"03:33.169","Text":"What we do next is we take the sample"},{"Start":"03:33.169 ","End":"03:36.680","Text":"x from each of the intervals so from x less than minus 1,"},{"Start":"03:36.680 ","End":"03:42.965","Text":"I\u0027m going to choose a value minus 2 from this interval I\u0027ll choose x equals 0,"},{"Start":"03:42.965 ","End":"03:45.770","Text":"and from this interval I\u0027ll choose x equals 2."},{"Start":"03:45.770 ","End":"03:50.060","Text":"The next thing we do is substitute these values of x,"},{"Start":"03:50.060 ","End":"03:55.250","Text":"the sample points into f prime of x and f prime of x."},{"Start":"03:55.250 ","End":"03:57.395","Text":"The simplest form of it is here,"},{"Start":"03:57.395 ","End":"03:59.450","Text":"and I\u0027ll just circle it."},{"Start":"03:59.450 ","End":"04:02.600","Text":"There\u0027s a missing minus here. There we go."},{"Start":"04:02.600 ","End":"04:05.990","Text":"Now when we substitute these values minus 2, 0 or 2,"},{"Start":"04:05.990 ","End":"04:08.060","Text":"I don\u0027t actually want the answer,"},{"Start":"04:08.060 ","End":"04:09.560","Text":"just the sine of the answer,"},{"Start":"04:09.560 ","End":"04:12.455","Text":"whether it\u0027s plus or minus, in that case,"},{"Start":"04:12.455 ","End":"04:17.435","Text":"looking at it actually it\u0027s always minus because x plus 1 squared is always positive."},{"Start":"04:17.435 ","End":"04:22.705","Text":"This thing to the 4th is also always positive and minus 6 is always negative."},{"Start":"04:22.705 ","End":"04:25.790","Text":"Minus, plus, plus, it\u0027s a minus."},{"Start":"04:25.790 ","End":"04:27.830","Text":"I don\u0027t even have to do any substitution."},{"Start":"04:27.830 ","End":"04:30.140","Text":"I can say this is negative here,"},{"Start":"04:30.140 ","End":"04:32.165","Text":"that it\u0027s negative over here,"},{"Start":"04:32.165 ","End":"04:35.455","Text":"and that it\u0027s negative over here, how strange."},{"Start":"04:35.455 ","End":"04:41.570","Text":"Also on the f prime of x row is where we write the values corresponding to these,"},{"Start":"04:41.570 ","End":"04:46.475","Text":"but minus 1 is a legitimate value and f prime is 0 there."},{"Start":"04:46.475 ","End":"04:49.580","Text":"1 was only included because it\u0027s not defined there."},{"Start":"04:49.580 ","End":"04:53.180","Text":"It\u0027s probably a vertical asymptote, usually is."},{"Start":"04:53.180 ","End":"04:55.580","Text":"But in any event, we\u0027re not going to put anything in here."},{"Start":"04:55.580 ","End":"04:57.140","Text":"I\u0027ll just put some dotted lines,"},{"Start":"04:57.140 ","End":"04:58.430","Text":"so I haven\u0027t forgotten it,"},{"Start":"04:58.430 ","End":"04:59.825","Text":"but it\u0027s out of bounds."},{"Start":"04:59.825 ","End":"05:04.414","Text":"Now I interpret these minuses in terms of the function."},{"Start":"05:04.414 ","End":"05:08.120","Text":"Derivative negative means that the function is decreasing,"},{"Start":"05:08.120 ","End":"05:11.570","Text":"so it\u0027s a down arrow and the same here and here,"},{"Start":"05:11.570 ","End":"05:14.660","Text":"the function appears to be always decreasing."},{"Start":"05:14.660 ","End":"05:18.235","Text":"Finally, I just put in the y for our suspect,"},{"Start":"05:18.235 ","End":"05:24.110","Text":"our suspect for an extremum is minus 1 and when x is minus 1,"},{"Start":"05:24.110 ","End":"05:29.395","Text":"y, which is just the f of x. so y is equal 2."},{"Start":"05:29.395 ","End":"05:33.125","Text":"Let\u0027s see if we put in minus 1,"},{"Start":"05:33.125 ","End":"05:36.185","Text":"we\u0027ll get, if we substitute in here,"},{"Start":"05:36.185 ","End":"05:39.905","Text":"minus 1 plus 1 is 0."},{"Start":"05:39.905 ","End":"05:42.920","Text":"That\u0027s also 0."},{"Start":"05:42.920 ","End":"05:44.315","Text":"As for this,"},{"Start":"05:44.315 ","End":"05:48.820","Text":"it\u0027s not an extremum because an extremum separates"},{"Start":"05:48.820 ","End":"05:53.600","Text":"between down and up or between increasing and decreasing or the other way around."},{"Start":"05:53.600 ","End":"05:55.955","Text":"This, I don\u0027t know what it is,"},{"Start":"05:55.955 ","End":"05:57.530","Text":"it\u0027s not an extremum."},{"Start":"05:57.530 ","End":"06:00.005","Text":"Actually, it\u0027s probably an inflection point,"},{"Start":"06:00.005 ","End":"06:02.360","Text":"but in any event, it doesn\u0027t interest us."},{"Start":"06:02.360 ","End":"06:05.900","Text":"Actually both things are fallen and 1 is a point"},{"Start":"06:05.900 ","End":"06:09.680","Text":"outside the definition and minus 1 is not an extremum either."},{"Start":"06:09.680 ","End":"06:11.829","Text":"Actually we don\u0027t have any extrema,"},{"Start":"06:11.829 ","End":"06:13.970","Text":"so when we draw our conclusions,"},{"Start":"06:13.970 ","End":"06:16.070","Text":"there\u0027s not that much to write."},{"Start":"06:16.070 ","End":"06:22.860","Text":"What I\u0027ll say is that as far as extrema go, there are none."},{"Start":"06:22.860 ","End":"06:27.785","Text":"As for the intervals where the function is increasing,"},{"Start":"06:27.785 ","End":"06:32.690","Text":"there\u0027s also non, for decreasing that\u0027s where we have plenty."},{"Start":"06:32.690 ","End":"06:36.035","Text":"The intervals 1 of them is"},{"Start":"06:36.035 ","End":"06:41.710","Text":"here above the eye where it\u0027s undefined and that\u0027s x bigger than 1."},{"Start":"06:41.710 ","End":"06:45.060","Text":"We apparently have 2 more intervals,"},{"Start":"06:45.060 ","End":"06:48.020","Text":"but in situations like these where there\u0027s a point in"},{"Start":"06:48.020 ","End":"06:51.260","Text":"the middle between a decreasing and decreasing."},{"Start":"06:51.260 ","End":"06:54.320","Text":"There\u0027s good reasons show that this is actually swallowed up in"},{"Start":"06:54.320 ","End":"06:58.280","Text":"1 big interval which is decreasing and that\u0027s these 2 combined,"},{"Start":"06:58.280 ","End":"07:02.490","Text":"which means x less than 1."},{"Start":"07:02.860 ","End":"07:06.600","Text":"Actually, I could probably combine these 2 and say"},{"Start":"07:06.600 ","End":"07:10.280","Text":"decreasing everywhere except where it\u0027s not defined."},{"Start":"07:10.280 ","End":"07:16.360","Text":"I could just say where x is not equal to 1 if I wanted to."},{"Start":"07:16.360 ","End":"07:19.320","Text":"That answers the question,"},{"Start":"07:19.320 ","End":"07:22.030","Text":"so we are done."}],"ID":4810},{"Watched":false,"Name":"Exercise 12","Duration":"6m 36s","ChapterTopicVideoID":4811,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4811.jpeg","UploadDate":"2015-07-15T04:23:50.2930000","DurationForVideoObject":"PT6M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.160","Text":"In this exercise, we\u0027re given a function f of x equals x minus 1 over x cubed."},{"Start":"00:06.160 ","End":"00:10.050","Text":"What we have to do is find its local extrema and"},{"Start":"00:10.050 ","End":"00:14.775","Text":"also the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.775 ","End":"00:18.465","Text":"This is 1 of those standard exercises which proceed"},{"Start":"00:18.465 ","End":"00:23.535","Text":"step-by-step in a cookbook style and just have to remember the steps."},{"Start":"00:23.535 ","End":"00:27.630","Text":"The first step is to differentiate the function."},{"Start":"00:27.630 ","End":"00:30.140","Text":"We get that the derivative of f,"},{"Start":"00:30.140 ","End":"00:34.895","Text":"this is a quotient and the quotient has its rule,"},{"Start":"00:34.895 ","End":"00:37.190","Text":"in case you\u0027ve forgotten it,"},{"Start":"00:37.190 ","End":"00:46.240","Text":"that u over v derivative is u prime v minus uv prime over v squared."},{"Start":"00:46.240 ","End":"00:50.010","Text":"That\u0027s what we\u0027ll apply here with u being the numerator,"},{"Start":"00:50.010 ","End":"00:51.230","Text":"v is the denominator."},{"Start":"00:51.230 ","End":"00:54.035","Text":"So u prime is 1,"},{"Start":"00:54.035 ","End":"01:00.210","Text":"v is x cubed minus u is x minus 1,"},{"Start":"01:00.210 ","End":"01:06.780","Text":"v prime is 3x squared and all over x^6."},{"Start":"01:06.780 ","End":"01:09.065","Text":"Let\u0027s simplify it a bit."},{"Start":"01:09.065 ","End":"01:11.420","Text":"What we get is, now let\u0027s see,"},{"Start":"01:11.420 ","End":"01:16.610","Text":"we have x cubed and from here we have minus x times 3x squared,"},{"Start":"01:16.610 ","End":"01:19.445","Text":"which is minus 3x cubed."},{"Start":"01:19.445 ","End":"01:24.800","Text":"Altogether, we\u0027re left with minus 2x cubed and the other thing is"},{"Start":"01:24.800 ","End":"01:30.850","Text":"minus the minus is plus 3x squared all over x^6."},{"Start":"01:30.850 ","End":"01:34.280","Text":"By the way, we should have noted the denominator can\u0027t be 0,"},{"Start":"01:34.280 ","End":"01:37.640","Text":"so the domain is x naught equal to 0."},{"Start":"01:37.640 ","End":"01:43.805","Text":"Continuing here, I can cancel the top and the bottom by x squared."},{"Start":"01:43.805 ","End":"01:53.245","Text":"So I will get minus 2x plus 3 over x^4."},{"Start":"01:53.245 ","End":"01:56.450","Text":"The next step after the differentiation is to set"},{"Start":"01:56.450 ","End":"02:01.085","Text":"f prime to be equal 0 and to solve for x."},{"Start":"02:01.085 ","End":"02:05.135","Text":"A fraction can only be 0 if its numerator is 0."},{"Start":"02:05.135 ","End":"02:08.915","Text":"So the numerator is 0 when minus 2x plus 3 is 0,"},{"Start":"02:08.915 ","End":"02:12.995","Text":"which is seen to be x equals minus 1 and a half."},{"Start":"02:12.995 ","End":"02:16.935","Text":"X equals 3/2, 2x equals 3,"},{"Start":"02:16.935 ","End":"02:20.030","Text":"x equals 1 and a half or 3/2,"},{"Start":"02:20.030 ","End":"02:22.055","Text":"whichever is more convenient for us."},{"Start":"02:22.055 ","End":"02:26.495","Text":"Now we come to the part where we draw the table."},{"Start":"02:26.495 ","End":"02:29.750","Text":"In the row for x\u0027s,"},{"Start":"02:29.750 ","End":"02:32.390","Text":"we put 2 kinds of x."},{"Start":"02:32.390 ","End":"02:37.520","Text":"One which is a suspect for the extrema where f prime was 0,"},{"Start":"02:37.520 ","End":"02:39.005","Text":"that\u0027s the 1 and a half."},{"Start":"02:39.005 ","End":"02:43.940","Text":"But we also include in the table the values of x for which f is undefined."},{"Start":"02:43.940 ","End":"02:47.540","Text":"So we have 0 and 1 and a half we put them in order,"},{"Start":"02:47.540 ","End":"02:52.970","Text":"let say here\u0027s 0 and here\u0027s 1 and a half and we leave space"},{"Start":"02:52.970 ","End":"02:55.430","Text":"in the middle because that\u0027s where we\u0027re going to write down"},{"Start":"02:55.430 ","End":"02:58.400","Text":"the intervals that they produce on the line."},{"Start":"02:58.400 ","End":"03:01.189","Text":"So here we have x less than 0."},{"Start":"03:01.189 ","End":"03:02.570","Text":"Here we have the x is"},{"Start":"03:02.570 ","End":"03:09.110","Text":"between 0 and 1 and a half and here we have x bigger than 1 and a half."},{"Start":"03:09.110 ","End":"03:13.640","Text":"Then we choose a sample point from each of these intervals,"},{"Start":"03:13.640 ","End":"03:15.095","Text":"and it could be anything."},{"Start":"03:15.095 ","End":"03:19.370","Text":"For here, I\u0027d like to choose minus 1 between here and here,"},{"Start":"03:19.370 ","End":"03:21.800","Text":"I\u0027ll choose x equals 1,"},{"Start":"03:21.800 ","End":"03:24.350","Text":"and here I\u0027ll choose x equals 2."},{"Start":"03:24.350 ","End":"03:31.145","Text":"Next is to substitute these 3 values into f prime of x,"},{"Start":"03:31.145 ","End":"03:33.170","Text":"which is, it\u0027s all of these,"},{"Start":"03:33.170 ","End":"03:34.840","Text":"I\u0027ll just use this 1."},{"Start":"03:34.840 ","End":"03:36.830","Text":"We substitute them in here,"},{"Start":"03:36.830 ","End":"03:38.510","Text":"but we don\u0027t want the actual value,"},{"Start":"03:38.510 ","End":"03:42.335","Text":"I just want to know whether it\u0027s positive or negative, plus or minus."},{"Start":"03:42.335 ","End":"03:44.090","Text":"Let\u0027s try them 1 after the other."},{"Start":"03:44.090 ","End":"03:46.040","Text":"The denominator is positive,"},{"Start":"03:46.040 ","End":"03:48.305","Text":"so I only have to care about the numerator."},{"Start":"03:48.305 ","End":"03:51.080","Text":"Let\u0027s put in x equals minus 1."},{"Start":"03:51.080 ","End":"03:52.460","Text":"If I put in minus 1,"},{"Start":"03:52.460 ","End":"03:54.815","Text":"I get plus 2 plus 3 is 5."},{"Start":"03:54.815 ","End":"03:56.705","Text":"That\u0027s certainly positive."},{"Start":"03:56.705 ","End":"03:58.940","Text":"If I put in x equals 1,"},{"Start":"03:58.940 ","End":"04:01.550","Text":"I get minus 2 plus 3,"},{"Start":"04:01.550 ","End":"04:04.990","Text":"which is 1, which is still positive."},{"Start":"04:04.990 ","End":"04:07.580","Text":"If I put in x equals 2,"},{"Start":"04:07.580 ","End":"04:10.070","Text":"I get minus 4 plus 3,"},{"Start":"04:10.070 ","End":"04:12.335","Text":"which happens to be negative."},{"Start":"04:12.335 ","End":"04:15.250","Text":"What this means in terms of the function itself,"},{"Start":"04:15.250 ","End":"04:17.210","Text":"when the derivative is positive,"},{"Start":"04:17.210 ","End":"04:19.040","Text":"the function is increasing,"},{"Start":"04:19.040 ","End":"04:22.370","Text":"which I indicate with an arrow and likewise here."},{"Start":"04:22.370 ","End":"04:24.470","Text":"But when the derivative is negative,"},{"Start":"04:24.470 ","End":"04:25.860","Text":"the function is decreasing,"},{"Start":"04:25.860 ","End":"04:27.830","Text":"so it\u0027s 1 of these arrows."},{"Start":"04:27.830 ","End":"04:32.790","Text":"I just forgot to write that place where f prime for x equals 1 and a half,"},{"Start":"04:32.790 ","End":"04:35.315","Text":"the derivative is 0."},{"Start":"04:35.315 ","End":"04:36.755","Text":"That\u0027s how we found this,"},{"Start":"04:36.755 ","End":"04:42.050","Text":"but 0 is not in the domain and it usually indicates a vertical asymptote."},{"Start":"04:42.050 ","End":"04:44.405","Text":"Anyway, I\u0027m just going to dot it out."},{"Start":"04:44.405 ","End":"04:45.860","Text":"Say that I know about 0,"},{"Start":"04:45.860 ","End":"04:47.860","Text":"but it\u0027s not in the domain."},{"Start":"04:47.860 ","End":"04:51.380","Text":"Then again, we\u0027re still looking at f of x and"},{"Start":"04:51.380 ","End":"04:54.590","Text":"I see that I\u0027m looking for to verify the suspects."},{"Start":"04:54.590 ","End":"04:56.720","Text":"Well, 0 is not a suspect is not in the domain,"},{"Start":"04:56.720 ","End":"04:58.625","Text":"1 and a half is a suspect and yes,"},{"Start":"04:58.625 ","End":"05:01.190","Text":"we\u0027ve got 1 because here it\u0027s increasing and"},{"Start":"05:01.190 ","End":"05:03.995","Text":"here it\u0027s decreasing and between these 2,"},{"Start":"05:03.995 ","End":"05:05.420","Text":"we have a maximum."},{"Start":"05:05.420 ","End":"05:08.795","Text":"So we found a maximum at x equals 1 and a half."},{"Start":"05:08.795 ","End":"05:10.985","Text":"I\u0027d like to know what y is though."},{"Start":"05:10.985 ","End":"05:16.205","Text":"So I\u0027ll just substitute that 1 and a half in the original f of x."},{"Start":"05:16.205 ","End":"05:18.395","Text":"Well, I make it 4/27."},{"Start":"05:18.395 ","End":"05:22.400","Text":"The next thing is to just draw conclusions,"},{"Start":"05:22.400 ","End":"05:25.850","Text":"which is basically just answering the questions that we were asked,"},{"Start":"05:25.850 ","End":"05:27.800","Text":"which is we were asked about"},{"Start":"05:27.800 ","End":"05:32.620","Text":"the local extrema and the increasing and decreasing intervals."},{"Start":"05:32.620 ","End":"05:34.995","Text":"Let\u0027s write that."},{"Start":"05:34.995 ","End":"05:36.620","Text":"For extrema,"},{"Start":"05:36.620 ","End":"05:38.000","Text":"we have 1 of them,"},{"Start":"05:38.000 ","End":"05:39.455","Text":"which is a maximum,"},{"Start":"05:39.455 ","End":"05:46.385","Text":"and it\u0027s located at 3/2, 4/27."},{"Start":"05:46.385 ","End":"05:51.350","Text":"As far as the increasing intervals where the function is increasing,"},{"Start":"05:51.350 ","End":"05:57.650","Text":"we just look at the arrow up on the orange and we have 2 separate intervals."},{"Start":"05:57.650 ","End":"06:04.294","Text":"We have x less than 0 and we have 0 less than x,"},{"Start":"06:04.294 ","End":"06:06.050","Text":"less than 1 and a half."},{"Start":"06:06.050 ","End":"06:09.965","Text":"I can\u0027t combine these 2 intervals into 1 like I sometimes do."},{"Start":"06:09.965 ","End":"06:11.960","Text":"If there was a point in between,"},{"Start":"06:11.960 ","End":"06:14.240","Text":"and it was defined here, that we could,"},{"Start":"06:14.240 ","End":"06:19.340","Text":"but the interval can\u0027t go over a place where the function is not defined,"},{"Start":"06:19.340 ","End":"06:21.590","Text":"can\u0027t include 0 in the interval,"},{"Start":"06:21.590 ","End":"06:23.135","Text":"is not in the domain."},{"Start":"06:23.135 ","End":"06:25.280","Text":"As for the decreasing part,"},{"Start":"06:25.280 ","End":"06:29.495","Text":"we just look at where the arrow is going down and that\u0027s here."},{"Start":"06:29.495 ","End":"06:33.240","Text":"So that would be x bigger than 1 and a half."},{"Start":"06:33.240 ","End":"06:37.180","Text":"That answers all the questions and so we\u0027re done."}],"ID":4811},{"Watched":false,"Name":"Exercise 13","Duration":"3m 47s","ChapterTopicVideoID":4812,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4812.jpeg","UploadDate":"2015-07-15T04:24:10.0700000","DurationForVideoObject":"PT3M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.930","Text":"In this exercise, we\u0027re given the function f of x is equal to x minus e^x."},{"Start":"00:06.930 ","End":"00:10.500","Text":"What we have to do is find its local extrema and"},{"Start":"00:10.500 ","End":"00:14.400","Text":"also the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.400 ","End":"00:18.135","Text":"Notice that this function is defined everywhere."},{"Start":"00:18.135 ","End":"00:20.910","Text":"There\u0027s no reason for anything not to be"},{"Start":"00:20.910 ","End":"00:23.610","Text":"defined as nothing in the denominator or anything."},{"Start":"00:23.610 ","End":"00:25.110","Text":"X is defined everywhere,"},{"Start":"00:25.110 ","End":"00:27.135","Text":"e^x is defined everywhere."},{"Start":"00:27.135 ","End":"00:32.070","Text":"This type of exercise has a standard set of steps to follow and we\u0027ll get to the answer."},{"Start":"00:32.070 ","End":"00:35.415","Text":"The first step is to differentiate the function."},{"Start":"00:35.415 ","End":"00:39.930","Text":"We get that f prime of x is equal to 1 minus e^x."},{"Start":"00:39.930 ","End":"00:43.940","Text":"The next step is to compare it to 0 and,"},{"Start":"00:43.940 ","End":"00:46.090","Text":"of course, solve for x."},{"Start":"00:46.090 ","End":"00:48.240","Text":"If 1 minus e^x equals 0,"},{"Start":"00:48.240 ","End":"00:51.305","Text":"then e^x is equal to 1."},{"Start":"00:51.305 ","End":"00:54.470","Text":"The only x where e to the power of x is 1,"},{"Start":"00:54.470 ","End":"00:56.360","Text":"is x equals 0."},{"Start":"00:56.360 ","End":"01:00.305","Text":"If you\u0027d like, you could take the natural logarithm of 1 as the answer."},{"Start":"01:00.305 ","End":"01:05.530","Text":"Anyway, it\u0027s x equals 0 and that\u0027s our candidate for being an extremum."},{"Start":"01:05.530 ","End":"01:09.525","Text":"What we do is draw our usual table, here it is."},{"Start":"01:09.525 ","End":"01:13.849","Text":"In the table we put in all suspects for extrema,"},{"Start":"01:13.849 ","End":"01:17.390","Text":"as well as points where the function is not defined, and there are none."},{"Start":"01:17.390 ","End":"01:21.095","Text":"All in all, we only put in x equals 0,"},{"Start":"01:21.095 ","End":"01:24.890","Text":"and this divides the line into 2 intervals only,"},{"Start":"01:24.890 ","End":"01:32.239","Text":"1 interval is where x is less than 0 and the other interval, x bigger than 0."},{"Start":"01:32.239 ","End":"01:35.165","Text":"Then we pick a sample point from each interval."},{"Start":"01:35.165 ","End":"01:37.520","Text":"For less than 0 I\u0027ll pick minus 1,"},{"Start":"01:37.520 ","End":"01:40.540","Text":"and for greater than 0 I\u0027ll pick x equals 1."},{"Start":"01:40.540 ","End":"01:45.080","Text":"I just forgot to say that f prime of x is 0 when x is 0,"},{"Start":"01:45.080 ","End":"01:49.000","Text":"because we got it from this equation where f prime is 0."},{"Start":"01:49.000 ","End":"01:55.400","Text":"What we\u0027re going to do here for f prime is to substitute these values in f prime,"},{"Start":"01:55.400 ","End":"01:57.875","Text":"which is this function here."},{"Start":"01:57.875 ","End":"01:59.480","Text":"But I don\u0027t want the actual answer,"},{"Start":"01:59.480 ","End":"02:01.579","Text":"only whether it\u0027s positive or negative."},{"Start":"02:01.579 ","End":"02:03.515","Text":"Let\u0027s put in minus 1,"},{"Start":"02:03.515 ","End":"02:06.620","Text":"1 minus e to the minus 1,"},{"Start":"02:06.620 ","End":"02:10.140","Text":"that\u0027s 1 minus 1 over e. Now 1 over e is less than 1,"},{"Start":"02:10.140 ","End":"02:12.990","Text":"it\u0027s 1 over 2.78 something."},{"Start":"02:12.990 ","End":"02:15.555","Text":"This is still going to be positive,"},{"Start":"02:15.555 ","End":"02:17.985","Text":"that means I\u0027ll write a plus here."},{"Start":"02:17.985 ","End":"02:20.705","Text":"If I put x equals 1,"},{"Start":"02:20.705 ","End":"02:23.720","Text":"I get 1 minus e and 1 is certainly less than e,"},{"Start":"02:23.720 ","End":"02:24.980","Text":"which is 2 point something,"},{"Start":"02:24.980 ","End":"02:26.620","Text":"so this is negative."},{"Start":"02:26.620 ","End":"02:29.120","Text":"If the derivative is positive,"},{"Start":"02:29.120 ","End":"02:31.160","Text":"that means that the function is increasing,"},{"Start":"02:31.160 ","End":"02:32.765","Text":"which I write like this."},{"Start":"02:32.765 ","End":"02:34.760","Text":"Here when the derivative is negative,"},{"Start":"02:34.760 ","End":"02:37.700","Text":"the function\u0027s decreasing, which I indicate like this."},{"Start":"02:37.700 ","End":"02:39.920","Text":"At the point where x equals 0,"},{"Start":"02:39.920 ","End":"02:42.860","Text":"because we\u0027re between an increasing and a decreasing,"},{"Start":"02:42.860 ","End":"02:45.500","Text":"we definitely have an extremum and more than that,"},{"Start":"02:45.500 ","End":"02:47.575","Text":"I can say it\u0027s of type maximum."},{"Start":"02:47.575 ","End":"02:48.900","Text":"Could be a maximum or minimum,"},{"Start":"02:48.900 ","End":"02:50.250","Text":"in this case it\u0027s a maximum."},{"Start":"02:50.250 ","End":"02:52.460","Text":"I also like to know the y at this point,"},{"Start":"02:52.460 ","End":"02:57.065","Text":"which means that I substitute x equals 0 in the original function."},{"Start":"02:57.065 ","End":"03:01.130","Text":"When x is 0, I get 0 minus e^0,"},{"Start":"03:01.130 ","End":"03:06.180","Text":"that\u0027s 1, so altogether I end up with minus 1 in the y column."},{"Start":"03:06.180 ","End":"03:11.794","Text":"I have all the information I need now to answer the questions about the extrema,"},{"Start":"03:11.794 ","End":"03:13.460","Text":"the increase and the decrease,"},{"Start":"03:13.460 ","End":"03:14.960","Text":"so let\u0027s write that."},{"Start":"03:14.960 ","End":"03:17.090","Text":"We have an extremum,"},{"Start":"03:17.090 ","End":"03:21.080","Text":"we only have the 1 which happens to be of type maximum,"},{"Start":"03:21.080 ","End":"03:24.350","Text":"and it\u0027s at the point 0 minus 1."},{"Start":"03:24.350 ","End":"03:27.905","Text":"As for the intervals where the function is increasing,"},{"Start":"03:27.905 ","End":"03:34.535","Text":"I just look at the upward orange arrow for increasing and that occurs at x less than 0."},{"Start":"03:34.535 ","End":"03:39.440","Text":"When I want to know the intervals at which the function is decreasing,"},{"Start":"03:39.440 ","End":"03:44.540","Text":"then I just look at the downward arrow and I get that x is bigger than 0."},{"Start":"03:44.540 ","End":"03:48.540","Text":"That answers all the questions and there we\u0027re done."}],"ID":4812},{"Watched":false,"Name":"Exercise 14","Duration":"4m 39s","ChapterTopicVideoID":4813,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4813.jpeg","UploadDate":"2015-07-15T04:24:35.2500000","DurationForVideoObject":"PT4M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:02.910 ","End":"00:05.355","Text":"f of x is e^1 over x."},{"Start":"00:05.355 ","End":"00:06.300","Text":"For this function,"},{"Start":"00:06.300 ","End":"00:09.210","Text":"we need to find its local extrema,"},{"Start":"00:09.210 ","End":"00:12.330","Text":"as well as the intervals where this function"},{"Start":"00:12.330 ","End":"00:15.000","Text":"is increasing and where it\u0027s decreasing."},{"Start":"00:15.000 ","End":"00:17.220","Text":"I\u0027ve copied the exercise out here,"},{"Start":"00:17.220 ","End":"00:18.930","Text":"and we\u0027re going to do it using"},{"Start":"00:18.930 ","End":"00:20.070","Text":"the cookbook approach,"},{"Start":"00:20.070 ","End":"00:22.965","Text":"the step-by-step, like in the tutorial."},{"Start":"00:22.965 ","End":"00:24.734","Text":"In the preparation phase,"},{"Start":"00:24.734 ","End":"00:26.310","Text":"I do a couple of things."},{"Start":"00:26.310 ","End":"00:29.610","Text":"I take a quick look at the domain of the function"},{"Start":"00:29.610 ","End":"00:32.145","Text":"and also differentiate a function."},{"Start":"00:32.145 ","End":"00:34.530","Text":"Let\u0027s see for the domain,"},{"Start":"00:34.530 ","End":"00:37.590","Text":"the only problem is x equals 0."},{"Start":"00:37.590 ","End":"00:41.220","Text":"The domain is all x except for 0."},{"Start":"00:41.220 ","End":"00:42.915","Text":"That\u0027s the domain."},{"Start":"00:42.915 ","End":"00:45.050","Text":"The other thing we do in the prep stage"},{"Start":"00:45.050 ","End":"00:47.195","Text":"is to differentiate the function."},{"Start":"00:47.195 ","End":"00:50.000","Text":"F prime of x is equal to,"},{"Start":"00:50.000 ","End":"00:51.455","Text":"use the chain rule."},{"Start":"00:51.455 ","End":"00:56.690","Text":"It\u0027s e^1 over x times the derivative of 1 over x"},{"Start":"00:56.690 ","End":"00:59.035","Text":"is minus 1 over x squared."},{"Start":"00:59.035 ","End":"01:03.030","Text":"It\u0027s nicer to write as e^1"},{"Start":"01:03.030 ","End":"01:07.845","Text":"over x minus over x squared."},{"Start":"01:07.845 ","End":"01:10.690","Text":"Then what we call step 1,"},{"Start":"01:10.690 ","End":"01:13.850","Text":"was to set the derivative equal 0"},{"Start":"01:13.850 ","End":"01:15.410","Text":"and solve the equations."},{"Start":"01:15.410 ","End":"01:17.270","Text":"We\u0027re going to solve the equation"},{"Start":"01:17.270 ","End":"01:19.970","Text":"f prime of x equals 0."},{"Start":"01:19.970 ","End":"01:21.515","Text":"Or in other words,"},{"Start":"01:21.515 ","End":"01:27.535","Text":"minus e^1 over x over x squared equals 0."},{"Start":"01:27.535 ","End":"01:29.390","Text":"We\u0027re doing this equation by the way,"},{"Start":"01:29.390 ","End":"01:33.655","Text":"because the answer suspects to be extrema,"},{"Start":"01:33.655 ","End":"01:35.195","Text":"and we\u0027re looking for extrema."},{"Start":"01:35.195 ","End":"01:38.285","Text":"Now, x is not 0 as not in the domain."},{"Start":"01:38.285 ","End":"01:41.600","Text":"All we have to do for a fraction to be 0"},{"Start":"01:41.600 ","End":"01:44.270","Text":"is for its numerator to be 0."},{"Start":"01:44.270 ","End":"01:46.010","Text":"The minus doesn\u0027t matter either."},{"Start":"01:46.010 ","End":"01:47.930","Text":"In other words, this is going to be 0"},{"Start":"01:47.930 ","End":"01:52.955","Text":"implies that e^1 over x is equal to 0."},{"Start":"01:52.955 ","End":"01:55.535","Text":"The only way this whole thing can be 0."},{"Start":"01:55.535 ","End":"01:58.640","Text":"but e to the power of anything"},{"Start":"01:58.640 ","End":"02:01.820","Text":"is always positive, it\u0027s never 0."},{"Start":"02:01.820 ","End":"02:05.090","Text":"For this equation there is no such x,"},{"Start":"02:05.090 ","End":"02:07.080","Text":"x means there\u0027s no solutions,"},{"Start":"02:07.080 ","End":"02:10.130","Text":"so it looks like we don\u0027t have any"},{"Start":"02:10.130 ","End":"02:12.850","Text":"candidate for extrema or suspects."},{"Start":"02:12.850 ","End":"02:15.735","Text":"In any event, we do draw a table,"},{"Start":"02:15.735 ","End":"02:17.790","Text":"that\u0027s next stage,"},{"Start":"02:17.790 ","End":"02:20.175","Text":"and I\u0027ll just draw it here second."},{"Start":"02:20.175 ","End":"02:22.320","Text":"Here we are. Here\u0027s the table."},{"Start":"02:22.320 ","End":"02:25.980","Text":"In this table, we put 2 kinds of x 1"},{"Start":"02:25.980 ","End":"02:28.160","Text":"where f prime is 0,"},{"Start":"02:28.160 ","End":"02:30.020","Text":"which are suspects for the extrema,"},{"Start":"02:30.020 ","End":"02:31.220","Text":"and there are none."},{"Start":"02:31.220 ","End":"02:33.155","Text":"I can\u0027t put any of those in here."},{"Start":"02:33.155 ","End":"02:35.510","Text":"But the other kind of these are points"},{"Start":"02:35.510 ","End":"02:37.590","Text":"where the function is not defined,"},{"Start":"02:37.590 ","End":"02:40.760","Text":"so I put x equals 0 in the table."},{"Start":"02:40.760 ","End":"02:42.395","Text":"That\u0027s all I can put in."},{"Start":"02:42.395 ","End":"02:46.040","Text":"Since the function is not defined there,"},{"Start":"02:46.040 ","End":"02:49.210","Text":"I indicate this with dotted lines."},{"Start":"02:49.210 ","End":"02:50.040","Text":"Nothing there."},{"Start":"02:50.040 ","End":"02:54.285","Text":"All it serves as a dividing line between intervals."},{"Start":"02:54.285 ","End":"02:56.570","Text":"There\u0027s an interval on this side"},{"Start":"02:56.570 ","End":"02:59.975","Text":"where x is less than 0,"},{"Start":"02:59.975 ","End":"03:01.940","Text":"and there\u0027s an interval on the other side"},{"Start":"03:01.940 ","End":"03:04.029","Text":"where x is bigger than 0."},{"Start":"03:04.029 ","End":"03:06.110","Text":"At least we can check where we have"},{"Start":"03:06.110 ","End":"03:08.200","Text":"increasing and decreasing intervals."},{"Start":"03:08.200 ","End":"03:10.925","Text":"The way to do this is just to take a sample point."},{"Start":"03:10.925 ","End":"03:12.530","Text":"Let\u0027s say for x less than 0,"},{"Start":"03:12.530 ","End":"03:13.790","Text":"I\u0027ll choose minus 1,"},{"Start":"03:13.790 ","End":"03:16.295","Text":"and for x bigger than 0, I\u0027ll choose 1,"},{"Start":"03:16.295 ","End":"03:19.580","Text":"and I substitute this 1 in f prime of x."},{"Start":"03:19.580 ","End":"03:22.525","Text":"F prime of x was here,"},{"Start":"03:22.525 ","End":"03:23.760","Text":"this form, whatever,"},{"Start":"03:23.760 ","End":"03:26.795","Text":"and e to the something is positive,"},{"Start":"03:26.795 ","End":"03:28.295","Text":"x squared is positive."},{"Start":"03:28.295 ","End":"03:32.170","Text":"In fact, this is always negative for whatever x is."},{"Start":"03:32.170 ","End":"03:35.630","Text":"Not only is this equal to a negative number,"},{"Start":"03:35.630 ","End":"03:37.870","Text":"but this will also be negative."},{"Start":"03:37.870 ","End":"03:40.220","Text":"When f prime is negative,"},{"Start":"03:40.220 ","End":"03:42.860","Text":"it says that the function is decreasing,"},{"Start":"03:42.860 ","End":"03:44.750","Text":"which I indicate with a down arrow,"},{"Start":"03:44.750 ","End":"03:47.435","Text":"and likewise here it\u0027s decreasing."},{"Start":"03:47.435 ","End":"03:49.910","Text":"The function is always decreasing,"},{"Start":"03:49.910 ","End":"03:51.739","Text":"and both these intervals."},{"Start":"03:51.739 ","End":"03:55.085","Text":"I don\u0027t need the y for anything."},{"Start":"03:55.085 ","End":"03:58.100","Text":"I can basically answer the questions"},{"Start":"03:58.100 ","End":"03:59.900","Text":"that were originally asked of us."},{"Start":"03:59.900 ","End":"04:01.040","Text":"This is the last stage,"},{"Start":"04:01.040 ","End":"04:02.557","Text":"which is the conclusions."},{"Start":"04:02.557 ","End":"04:04.235","Text":"We\u0027re asked about 3 things."},{"Start":"04:04.235 ","End":"04:06.979","Text":"Extrema. What are the extrema?"},{"Start":"04:06.979 ","End":"04:07.760","Text":"All of them."},{"Start":"04:07.760 ","End":"04:10.550","Text":"Well, there\u0027s none, there are no extrema"},{"Start":"04:10.550 ","End":"04:13.775","Text":"intervals where the function is increasing."},{"Start":"04:13.775 ","End":"04:15.260","Text":"I\u0027m looking for an up arrow,"},{"Start":"04:15.260 ","End":"04:17.765","Text":"there are none, also none."},{"Start":"04:17.765 ","End":"04:22.100","Text":"Finally, intervals where the function is decreasing,"},{"Start":"04:22.100 ","End":"04:24.050","Text":"I would say all x."},{"Start":"04:24.050 ","End":"04:26.090","Text":"But it\u0027s not true because at x equals 0,"},{"Start":"04:26.090 ","End":"04:27.575","Text":"the function is not defined."},{"Start":"04:27.575 ","End":"04:29.150","Text":"There are actually 2 intervals,"},{"Start":"04:29.150 ","End":"04:31.430","Text":"x bigger than 0,"},{"Start":"04:31.430 ","End":"04:35.795","Text":"and the other interval is x bigger than 0."},{"Start":"04:35.795 ","End":"04:36.960","Text":"That\u0027s the answer."},{"Start":"04:36.960 ","End":"04:38.530","Text":"We\u0027re done."}],"ID":4813},{"Watched":false,"Name":"Exercise 15","Duration":"7m 9s","ChapterTopicVideoID":4814,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4814.jpeg","UploadDate":"2015-07-15T04:25:14.1800000","DurationForVideoObject":"PT7M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.585","Text":"In this exercise, we\u0027re given a function f of x equals x e^minus 2x squared,"},{"Start":"00:06.585 ","End":"00:10.365","Text":"what we have to do is to find all the local extrema and"},{"Start":"00:10.365 ","End":"00:14.235","Text":"the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.235 ","End":"00:21.345","Text":"Let me start by copying the exercise f of x equals x e^minus 2x squared."},{"Start":"00:21.345 ","End":"00:25.350","Text":"This type of exercise is very cookbook style or"},{"Start":"00:25.350 ","End":"00:29.140","Text":"a certain set of steps that we have to follow when we get to the answer."},{"Start":"00:29.140 ","End":"00:32.325","Text":"The first step is to find the derivative of f,"},{"Start":"00:32.325 ","End":"00:35.770","Text":"so that\u0027s f prime of x and this equals,"},{"Start":"00:35.770 ","End":"00:40.565","Text":"first of all, we have a product and you should remember the product rule."},{"Start":"00:40.565 ","End":"00:42.065","Text":"I\u0027m not going to write it."},{"Start":"00:42.065 ","End":"00:47.725","Text":"What we have is derivative of this times this plus this times the derivative of this."},{"Start":"00:47.725 ","End":"00:54.300","Text":"Derivative of x is 1 times e^minus 2x squared plus x,"},{"Start":"00:54.300 ","End":"00:58.115","Text":"as it is times the derivative of e^minus 2x squared."},{"Start":"00:58.115 ","End":"01:00.215","Text":"Here we use the chain rule."},{"Start":"01:00.215 ","End":"01:02.705","Text":"We start off with e to the sum think,"},{"Start":"01:02.705 ","End":"01:07.370","Text":"it\u0027s just e to that sum think times that sum think prime."},{"Start":"01:07.370 ","End":"01:10.055","Text":"We need the derivative of minus 2x squared,"},{"Start":"01:10.055 ","End":"01:12.335","Text":"which is minus 4x."},{"Start":"01:12.335 ","End":"01:14.750","Text":"What I want to do is simplify this."},{"Start":"01:14.750 ","End":"01:20.495","Text":"I can take e^minus 2x squared outside the brackets e^minus 2x squared."},{"Start":"01:20.495 ","End":"01:22.884","Text":"What I\u0027m left with is 1,"},{"Start":"01:22.884 ","End":"01:25.785","Text":"and also x times minus 4x,"},{"Start":"01:25.785 ","End":"01:29.175","Text":"which is minus 4x squared."},{"Start":"01:29.175 ","End":"01:33.080","Text":"That\u0027s the preparation stage just to differentiate."},{"Start":"01:33.080 ","End":"01:37.240","Text":"Next step is to set it equal to 0 and to solve for x,"},{"Start":"01:37.240 ","End":"01:42.340","Text":"so I set this equal to 0 and have to see what x is."},{"Start":"01:42.340 ","End":"01:44.900","Text":"Now e to the sum think is always positive,"},{"Start":"01:44.900 ","End":"01:50.090","Text":"it\u0027s never 0, and so what we have is that this thing must be 0,"},{"Start":"01:50.090 ","End":"01:55.110","Text":"so we get 1 minus 4x squared must equal 0 and if"},{"Start":"01:55.110 ","End":"02:00.480","Text":"we rearrange we get x squared equals 1/4."},{"Start":"02:00.480 ","End":"02:07.495","Text":"There\u0027s 2 possible values for x. X can either equal minus 1/2 or 1/2."},{"Start":"02:07.495 ","End":"02:12.425","Text":"The next step is to draw a table and I\u0027ll do it for you."},{"Start":"02:12.425 ","End":"02:14.090","Text":"Here\u0027s the table."},{"Start":"02:14.090 ","End":"02:16.474","Text":"It always has the same rows,"},{"Start":"02:16.474 ","End":"02:18.080","Text":"x, f prime of x,"},{"Start":"02:18.080 ","End":"02:19.675","Text":"f of x, and y."},{"Start":"02:19.675 ","End":"02:23.870","Text":"The first thing we do is put stuff in the x row and what we"},{"Start":"02:23.870 ","End":"02:28.715","Text":"put in are these values where f prime is 0."},{"Start":"02:28.715 ","End":"02:31.175","Text":"These are suspects to be an extremum."},{"Start":"02:31.175 ","End":"02:35.285","Text":"We put them in in order and leave some space around them."},{"Start":"02:35.285 ","End":"02:40.295","Text":"We also put in values where the function is undefined,"},{"Start":"02:40.295 ","End":"02:43.370","Text":"but the domain of this is all x."},{"Start":"02:43.370 ","End":"02:46.905","Text":"There is no place that f is not defined."},{"Start":"02:46.905 ","End":"02:50.900","Text":"Minus 1/2 and 1/2 and then we put"},{"Start":"02:50.900 ","End":"02:54.690","Text":"in the intervals that these points divide the line into."},{"Start":"02:54.690 ","End":"02:59.150","Text":"We have an interval x less than minus 1/2."},{"Start":"02:59.150 ","End":"03:04.190","Text":"We have an interval minus 1/2 less than x, less than 1/2."},{"Start":"03:04.190 ","End":"03:10.325","Text":"We have this interval on the right where x is bigger than 1/2."},{"Start":"03:10.325 ","End":"03:13.940","Text":"Next step is to choose a value,"},{"Start":"03:13.940 ","End":"03:18.185","Text":"a representative any value from each of these intervals."},{"Start":"03:18.185 ","End":"03:24.195","Text":"From here, I\u0027ll choose minus 1 and between minus 1/2 and 1/2,"},{"Start":"03:24.195 ","End":"03:28.735","Text":"I\u0027ll choose 0, and for bigger than 1/2, I\u0027ll choose 1."},{"Start":"03:28.735 ","End":"03:33.650","Text":"Now we have the sample values we substitute into f prime of x,"},{"Start":"03:33.650 ","End":"03:38.060","Text":"which is, let\u0027s see, in the simplest form, it\u0027s here."},{"Start":"03:38.060 ","End":"03:43.475","Text":"But I don\u0027t actually want the value only whether it\u0027s positive or negative."},{"Start":"03:43.475 ","End":"03:48.420","Text":"Let\u0027s see, we\u0027ll take them 1 at a time and I noticed that e to anything is positive."},{"Start":"03:48.420 ","End":"03:50.750","Text":"To decide whether this is positive or negative,"},{"Start":"03:50.750 ","End":"03:55.030","Text":"all I have to do really is substitute into the 1 minus 4x squared."},{"Start":"03:55.030 ","End":"03:56.575","Text":"Take minus 1."},{"Start":"03:56.575 ","End":"04:01.745","Text":"1 minus comes out to be 1 minus 4 because minus 1 squared is 1,"},{"Start":"04:01.745 ","End":"04:04.415","Text":"it\u0027s minus 3, it\u0027s negative."},{"Start":"04:04.415 ","End":"04:08.660","Text":"I put a minus sign here to indicate that it\u0027s negative."},{"Start":"04:08.660 ","End":"04:10.865","Text":"Then I put in the 0."},{"Start":"04:10.865 ","End":"04:13.460","Text":"If x is 0, it is 1 minus 0,"},{"Start":"04:13.460 ","End":"04:15.650","Text":"it\u0027s 1, which is positive,"},{"Start":"04:15.650 ","End":"04:17.990","Text":"and then when I put in 1, again,"},{"Start":"04:17.990 ","End":"04:22.310","Text":"I get 1 minus 4 which is minus 3, which is negative."},{"Start":"04:22.310 ","End":"04:24.860","Text":"I forgot today that we also put in,"},{"Start":"04:24.860 ","End":"04:27.694","Text":"we can put in f prime for these 2 points,"},{"Start":"04:27.694 ","End":"04:28.955","Text":"for these 2 suspects,"},{"Start":"04:28.955 ","End":"04:34.220","Text":"the f prime is going to be 0 because that\u0027s how we got to it by comparing to 0."},{"Start":"04:34.220 ","End":"04:35.675","Text":"That\u0027s how we got these values,"},{"Start":"04:35.675 ","End":"04:42.045","Text":"and then we say what these signs mean as far as the function itself goes."},{"Start":"04:42.045 ","End":"04:46.760","Text":"The minus in the f prime means that the function is decreasing."},{"Start":"04:46.760 ","End":"04:48.500","Text":"If its derivative is negative,"},{"Start":"04:48.500 ","End":"04:50.585","Text":"then the function is decreasing."},{"Start":"04:50.585 ","End":"04:53.525","Text":"Likewise, if the derivative is positive,"},{"Start":"04:53.525 ","End":"04:55.085","Text":"the function is increasing,"},{"Start":"04:55.085 ","End":"04:56.945","Text":"which is an arrow like this."},{"Start":"04:56.945 ","End":"04:59.690","Text":"Once again, function is decreasing."},{"Start":"04:59.690 ","End":"05:01.415","Text":"It\u0027s an arrow going down."},{"Start":"05:01.415 ","End":"05:03.305","Text":"Now when we have a point,"},{"Start":"05:03.305 ","End":"05:04.985","Text":"the x equals minus 1/2,"},{"Start":"05:04.985 ","End":"05:08.930","Text":"which is in-between an area of decreasing and increasing,"},{"Start":"05:08.930 ","End":"05:10.385","Text":"then it is an extremum."},{"Start":"05:10.385 ","End":"05:13.025","Text":"The extremum is of type minimum,"},{"Start":"05:13.025 ","End":"05:17.450","Text":"which I just write as min for short and the x equals 1/2,"},{"Start":"05:17.450 ","End":"05:20.510","Text":"which is between increasing and decreasing."},{"Start":"05:20.510 ","End":"05:22.760","Text":"That\u0027s an extremum of type maximum,"},{"Start":"05:22.760 ","End":"05:30.755","Text":"which I just write as max for short and all I need now is the y value of these 2 extrema."},{"Start":"05:30.755 ","End":"05:35.885","Text":"I substitute in the function itself, which is here."},{"Start":"05:35.885 ","End":"05:38.600","Text":"If x is minus 1/2,"},{"Start":"05:38.600 ","End":"05:42.320","Text":"minus 1/2 squared is 1/4,"},{"Start":"05:42.320 ","End":"05:46.190","Text":"minus twice a 1/4 minus 1/2,"},{"Start":"05:46.190 ","End":"05:51.840","Text":"which is x e^minus 1/2 and if x is 1/2."},{"Start":"05:51.840 ","End":"05:56.865","Text":"I think we get the same 1/2 squared is 1/4 minus twice 1/4 is minus 1/2,"},{"Start":"05:56.865 ","End":"05:59.055","Text":"but here we get plus 1/2,"},{"Start":"05:59.055 ","End":"06:03.840","Text":"so here it\u0027s 1/2, e^minus 1/2."},{"Start":"06:03.840 ","End":"06:05.330","Text":"The table part is done."},{"Start":"06:05.330 ","End":"06:08.450","Text":"The last big stage is the conclusions,"},{"Start":"06:08.450 ","End":"06:10.730","Text":"which is essentially answering the questions,"},{"Start":"06:10.730 ","End":"06:14.810","Text":"what are the extrema and where is the function increasing and decreasing."},{"Start":"06:14.810 ","End":"06:16.965","Text":"Now I can write the answer."},{"Start":"06:16.965 ","End":"06:19.565","Text":"The extrema, there are 2 of them."},{"Start":"06:19.565 ","End":"06:22.265","Text":"There is a minimum at the point,"},{"Start":"06:22.265 ","End":"06:27.135","Text":"minus 1/2 and minus 1/2 e^minus 1/2."},{"Start":"06:27.135 ","End":"06:36.770","Text":"That\u0027s 1 of them, and we have also a maximum at the point 1/2 and 1/2 e^minus 1/2."},{"Start":"06:36.770 ","End":"06:39.530","Text":"Now where is the function increasing?"},{"Start":"06:39.530 ","End":"06:44.105","Text":"It\u0027s increasing when I see the orange arrow that points upwards,"},{"Start":"06:44.105 ","End":"06:48.030","Text":"which is here, and that belongs to this interval."},{"Start":"06:48.030 ","End":"06:50.735","Text":"It\u0027s minus 1/2 less than x,"},{"Start":"06:50.735 ","End":"06:54.980","Text":"less than 1/2 and the function is going to be decreasing,"},{"Start":"06:54.980 ","End":"06:58.835","Text":"where I have the orange arrow going down here and here."},{"Start":"06:58.835 ","End":"07:02.075","Text":"That\u0027s when x is less than minus 1/2,"},{"Start":"07:02.075 ","End":"07:06.365","Text":"and also when x is bigger than 1.2."},{"Start":"07:06.365 ","End":"07:10.080","Text":"That answers all the questions and we\u0027re done."}],"ID":4814},{"Watched":false,"Name":"Exercise 16","Duration":"7m 4s","ChapterTopicVideoID":4834,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4834.jpeg","UploadDate":"2015-07-21T02:40:39.0870000","DurationForVideoObject":"PT7M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:02.580 ","End":"00:04.905","Text":"f of x is x plus 2 e^1/x."},{"Start":"00:04.905 ","End":"00:06.750","Text":"For this function,"},{"Start":"00:06.750 ","End":"00:09.330","Text":"we have to find its local extrema,"},{"Start":"00:09.330 ","End":"00:11.340","Text":"as well as the intervals where the function"},{"Start":"00:11.340 ","End":"00:13.770","Text":"is increasing and where it\u0027s decreasing."},{"Start":"00:13.770 ","End":"00:15.900","Text":"This is a standard type of exercise"},{"Start":"00:15.900 ","End":"00:17.355","Text":"with a cookbook approach."},{"Start":"00:17.355 ","End":"00:20.010","Text":"First step is to differentiate"},{"Start":"00:20.010 ","End":"00:22.320","Text":"the function to find f-prime."},{"Start":"00:22.320 ","End":"00:23.520","Text":"There\u0027s a step before that"},{"Start":"00:23.520 ","End":"00:25.439","Text":"where I should note the domain,"},{"Start":"00:25.439 ","End":"00:27.720","Text":"and I note that here all x are okay,"},{"Start":"00:27.720 ","End":"00:29.955","Text":"except for x equals 0."},{"Start":"00:29.955 ","End":"00:31.770","Text":"X must not equal 0,"},{"Start":"00:31.770 ","End":"00:33.390","Text":"otherwise it won\u0027t be defined."},{"Start":"00:33.390 ","End":"00:35.025","Text":"Back to the derivative."},{"Start":"00:35.025 ","End":"00:36.480","Text":"We have a product here,"},{"Start":"00:36.480 ","End":"00:38.520","Text":"we use the product rule."},{"Start":"00:38.520 ","End":"00:40.290","Text":"It\u0027s the first times the derivative"},{"Start":"00:40.290 ","End":"00:42.890","Text":"of the second plus the derivative"},{"Start":"00:42.890 ","End":"00:44.150","Text":"of the first times the second,"},{"Start":"00:44.150 ","End":"00:45.530","Text":"doesn\u0027t matter in which order."},{"Start":"00:45.530 ","End":"00:47.270","Text":"Here, we\u0027ll take x plus 2 times"},{"Start":"00:47.270 ","End":"00:49.880","Text":"the derivative of this using the chain rule."},{"Start":"00:49.880 ","End":"00:53.180","Text":"It\u0027s e^1/x times the inner derivative,"},{"Start":"00:53.180 ","End":"00:55.960","Text":"which is minus 1/x squared."},{"Start":"00:55.960 ","End":"00:57.305","Text":"Then the other way round."},{"Start":"00:57.305 ","End":"00:58.850","Text":"I left this as is before."},{"Start":"00:58.850 ","End":"01:01.430","Text":"This time I differentiate this, which is 1,"},{"Start":"01:01.430 ","End":"01:03.470","Text":"and leave the other 1 as is."},{"Start":"01:03.470 ","End":"01:05.870","Text":"I want to simplify this a little bit."},{"Start":"01:05.870 ","End":"01:10.445","Text":"I\u0027ll take e^1/x outside the brackets,"},{"Start":"01:10.445 ","End":"01:17.580","Text":"and I\u0027m left with minus x plus 2/x squared plus 1."},{"Start":"01:17.580 ","End":"01:22.040","Text":"Next step is to set the derivative equal to 0"},{"Start":"01:22.040 ","End":"01:25.025","Text":"and find for which x is the derivative 0."},{"Start":"01:25.025 ","End":"01:28.370","Text":"Since this is not equal to 0,"},{"Start":"01:28.370 ","End":"01:30.470","Text":"e to the something is always positive,"},{"Start":"01:30.470 ","End":"01:33.860","Text":"I just have that this must be 0."},{"Start":"01:33.860 ","End":"01:36.230","Text":"If I bring this over to the other side,"},{"Start":"01:36.230 ","End":"01:42.425","Text":"I\u0027ll get that 1 equals x plus 2/x squared."},{"Start":"01:42.425 ","End":"01:45.905","Text":"In other words, x squared equals x plus 2."},{"Start":"01:45.905 ","End":"01:48.230","Text":"If x squared equals x plus 2,"},{"Start":"01:48.230 ","End":"01:52.450","Text":"we can just say minus x minus 2 equals 0,"},{"Start":"01:52.450 ","End":"01:54.875","Text":"and then we have a quadratic equation."},{"Start":"01:54.875 ","End":"01:59.600","Text":"X is equal to either minus 1 or 2."},{"Start":"01:59.600 ","End":"02:02.285","Text":"Then we go on to the next step,"},{"Start":"02:02.285 ","End":"02:05.555","Text":"which is to draw a table."},{"Start":"02:05.555 ","End":"02:08.255","Text":"Here\u0027s the blank table."},{"Start":"02:08.255 ","End":"02:12.350","Text":"The first thing I do is to put in some values of x."},{"Start":"02:12.350 ","End":"02:14.210","Text":"Now, there\u0027s 2 kinds of x"},{"Start":"02:14.210 ","End":"02:15.320","Text":"that I put in here."},{"Start":"02:15.320 ","End":"02:17.030","Text":"The first is the solutions"},{"Start":"02:17.030 ","End":"02:19.410","Text":"to f-prime equals 0,"},{"Start":"02:19.410 ","End":"02:21.490","Text":"which of these minus 1 and 2,"},{"Start":"02:21.490 ","End":"02:23.330","Text":"and also the points at which"},{"Start":"02:23.330 ","End":"02:25.115","Text":"the function is undefined."},{"Start":"02:25.115 ","End":"02:28.235","Text":"I need to put in 0 minus 1 and 2,"},{"Start":"02:28.235 ","End":"02:29.990","Text":"but in the correct order."},{"Start":"02:29.990 ","End":"02:32.990","Text":"First, we put in the minus 1."},{"Start":"02:32.990 ","End":"02:37.265","Text":"Then the 0, and then the 2."},{"Start":"02:37.265 ","End":"02:40.880","Text":"I can say that for minus 1 and 2,"},{"Start":"02:40.880 ","End":"02:42.544","Text":"the derivative is 0."},{"Start":"02:42.544 ","End":"02:45.260","Text":"That\u0027s how we found the minus 1 and 2"},{"Start":"02:45.260 ","End":"02:47.495","Text":"by setting the derivative to be 0."},{"Start":"02:47.495 ","End":"02:50.270","Text":"Now at 0, the function is undefined,"},{"Start":"02:50.270 ","End":"02:52.939","Text":"and it usually means an asymptote."},{"Start":"02:52.939 ","End":"02:54.770","Text":"Not always, but in any event,"},{"Start":"02:54.770 ","End":"02:56.540","Text":"I can\u0027t do anything with 0."},{"Start":"02:56.540 ","End":"02:58.340","Text":"I just put in some dotted lines"},{"Start":"02:58.340 ","End":"03:01.055","Text":"to say that 0 is undefined."},{"Start":"03:01.055 ","End":"03:04.295","Text":"But it\u0027s still useful to us in finding intervals,"},{"Start":"03:04.295 ","End":"03:05.750","Text":"so it\u0027s in the table."},{"Start":"03:05.750 ","End":"03:07.355","Text":"What are these intervals?"},{"Start":"03:07.355 ","End":"03:10.655","Text":"X is less than minus 1."},{"Start":"03:10.655 ","End":"03:14.310","Text":"The next is minus 1 less than x,"},{"Start":"03:14.310 ","End":"03:15.525","Text":"less than 0,"},{"Start":"03:15.525 ","End":"03:17.232","Text":"then between 0 and 2,"},{"Start":"03:17.232 ","End":"03:19.195","Text":"and then bigger than 2,"},{"Start":"03:19.195 ","End":"03:22.220","Text":"what we do is to choose a sample point"},{"Start":"03:22.220 ","End":"03:24.495","Text":"arbitrarily from each interval."},{"Start":"03:24.495 ","End":"03:25.950","Text":"For less than minus 1,"},{"Start":"03:25.950 ","End":"03:28.110","Text":"I\u0027ll choose minus 2."},{"Start":"03:28.110 ","End":"03:30.060","Text":"Between minus 1 and 0,"},{"Start":"03:30.060 ","End":"03:31.980","Text":"I\u0027ll choose minus 1/2."},{"Start":"03:31.980 ","End":"03:33.885","Text":"Here, I\u0027ll choose 1,"},{"Start":"03:33.885 ","End":"03:36.325","Text":"and here, I\u0027ll choose 3."},{"Start":"03:36.325 ","End":"03:37.940","Text":"What we do with these points"},{"Start":"03:37.940 ","End":"03:40.220","Text":"is to substitute them into f-prime"},{"Start":"03:40.220 ","End":"03:43.070","Text":"of x and f-prime of x."},{"Start":"03:43.070 ","End":"03:44.555","Text":"I\u0027ll take it from here."},{"Start":"03:44.555 ","End":"03:47.210","Text":"But we don\u0027t actually need the value."},{"Start":"03:47.210 ","End":"03:48.740","Text":"We only need to know whether"},{"Start":"03:48.740 ","End":"03:51.020","Text":"it comes out positive or negative."},{"Start":"03:51.020 ","End":"03:53.540","Text":"That makes things easier because"},{"Start":"03:53.540 ","End":"03:55.745","Text":"I don\u0027t have to look at e^1/x."},{"Start":"03:55.745 ","End":"04:04.550","Text":"It\u0027s always positive."},{"Start":"04:04.550 ","End":"04:04.551","Text":"E to the something is always positive."},{"Start":"04:04.551 ","End":"04:04.552","Text":"I just have to put the values in this bit,"},{"Start":"04:04.552 ","End":"04:04.553","Text":"and just enough to know whether"},{"Start":"04:04.553 ","End":"04:05.765","Text":"it\u0027s positive or negative."},{"Start":"04:05.765 ","End":"04:08.345","Text":"Let me take the minus 2."},{"Start":"04:08.345 ","End":"04:11.075","Text":"Minus 2, if I put it in here,"},{"Start":"04:11.075 ","End":"04:12.680","Text":"it comes out positive."},{"Start":"04:12.680 ","End":"04:15.560","Text":"Now, let\u0027s see what happens with minus 1/2."},{"Start":"04:15.560 ","End":"04:16.955","Text":"It\u0027s negative."},{"Start":"04:16.955 ","End":"04:22.020","Text":"Then when x equals 1, we get negative."},{"Start":"04:22.020 ","End":"04:26.940","Text":"For x equals 3, we have positive."},{"Start":"04:26.940 ","End":"04:28.580","Text":"Here, we put a plus."},{"Start":"04:28.580 ","End":"04:30.500","Text":"Now, we interpret these pluses"},{"Start":"04:30.500 ","End":"04:33.140","Text":"and minuses in terms of the function."},{"Start":"04:33.140 ","End":"04:35.615","Text":"If f-prime is positive,"},{"Start":"04:35.615 ","End":"04:37.760","Text":"it means the function is increasing."},{"Start":"04:37.760 ","End":"04:39.140","Text":"I\u0027ll put arrow."},{"Start":"04:39.140 ","End":"04:40.670","Text":"The same for minus,"},{"Start":"04:40.670 ","End":"04:42.470","Text":"if f-prime is negative,"},{"Start":"04:42.470 ","End":"04:46.295","Text":"then the function is decreasing, and so on."},{"Start":"04:46.295 ","End":"04:47.600","Text":"Here, we have decreasing"},{"Start":"04:47.600 ","End":"04:49.925","Text":"and here we have increasing."},{"Start":"04:49.925 ","End":"04:52.145","Text":"Now, for the suspects for extrema,"},{"Start":"04:52.145 ","End":"04:53.873","Text":"we can determine the answer."},{"Start":"04:53.873 ","End":"04:55.610","Text":"We can say that they both are extrema,"},{"Start":"04:55.610 ","End":"04:57.500","Text":"because each 1 of them is between"},{"Start":"04:57.500 ","End":"05:00.050","Text":"an increasing and a decreasing in some order."},{"Start":"05:00.050 ","End":"05:03.050","Text":"The minus 1 is between increasing and decreasing,"},{"Start":"05:03.050 ","End":"05:04.805","Text":"which makes it a maximum."},{"Start":"05:04.805 ","End":"05:07.880","Text":"The 2 is between a decreasing and an increasing,"},{"Start":"05:07.880 ","End":"05:09.305","Text":"which makes it a minimum."},{"Start":"05:09.305 ","End":"05:10.520","Text":"You go down to the minimum"},{"Start":"05:10.520 ","End":"05:11.615","Text":"and then up away from it,"},{"Start":"05:11.615 ","End":"05:13.040","Text":"like here, up to the maximum,"},{"Start":"05:13.040 ","End":"05:14.090","Text":"and then down from it."},{"Start":"05:14.090 ","End":"05:18.905","Text":"All we need now is the y-values for these extrema."},{"Start":"05:18.905 ","End":"05:20.480","Text":"To get these y-values,"},{"Start":"05:20.480 ","End":"05:23.540","Text":"I substitute in the original function."},{"Start":"05:23.540 ","End":"05:25.610","Text":"This time, I need to know the actual value,"},{"Start":"05:25.610 ","End":"05:27.590","Text":"not just the plus or minus."},{"Start":"05:27.590 ","End":"05:30.245","Text":"Let\u0027s try minus 1 first."},{"Start":"05:30.245 ","End":"05:36.050","Text":"This is e^minus 1 or 1/e."},{"Start":"05:36.050 ","End":"05:40.195","Text":"For the 2, I get 4e^1/2."},{"Start":"05:40.195 ","End":"05:42.655","Text":"The table is now filled."},{"Start":"05:42.655 ","End":"05:45.050","Text":"We can now go on to the last stage,"},{"Start":"05:45.050 ","End":"05:46.340","Text":"which is the conclusions,"},{"Start":"05:46.340 ","End":"05:48.320","Text":"where basically I\u0027m giving the answers"},{"Start":"05:48.320 ","End":"05:50.210","Text":"to the local extrema,"},{"Start":"05:50.210 ","End":"05:53.180","Text":"and the increasing and decreasing intervals."},{"Start":"05:53.180 ","End":"05:56.349","Text":"Extrema, we have 2 of them."},{"Start":"05:56.349 ","End":"06:01.580","Text":"We have a maximum at the point minus 1,1/e,"},{"Start":"06:01.580 ","End":"06:06.580","Text":"and I also have a minimum at the point 2,"},{"Start":"06:06.580 ","End":"06:10.460","Text":"4e^1/2, or if you prefer, square root of e."},{"Start":"06:10.460 ","End":"06:11.915","Text":"Then the answer is that."},{"Start":"06:11.915 ","End":"06:14.450","Text":"All we\u0027re left with now are the intervals"},{"Start":"06:14.450 ","End":"06:17.220","Text":"where the function is increasing."},{"Start":"06:17.220 ","End":"06:20.090","Text":"I get that from looking at the orange arrow up."},{"Start":"06:20.090 ","End":"06:23.015","Text":"I have an arrow up here and here,"},{"Start":"06:23.015 ","End":"06:24.380","Text":"which gives me 2 intervals."},{"Start":"06:24.380 ","End":"06:27.170","Text":"I get x less than minus 1,"},{"Start":"06:27.170 ","End":"06:29.630","Text":"and I also have increasing"},{"Start":"06:29.630 ","End":"06:32.995","Text":"when x is bigger than 2."},{"Start":"06:32.995 ","End":"06:34.320","Text":"Then I need to know where"},{"Start":"06:34.320 ","End":"06:36.200","Text":"the function is decreasing,"},{"Start":"06:36.200 ","End":"06:38.030","Text":"and it\u0027s decreasing where"},{"Start":"06:38.030 ","End":"06:40.580","Text":"the orange arrow is going down."},{"Start":"06:40.580 ","End":"06:42.260","Text":"Again, I have 2 of these."},{"Start":"06:42.260 ","End":"06:45.640","Text":"I have x between minus 1 and 0,"},{"Start":"06:45.640 ","End":"06:49.405","Text":"and I also have x between 0 and 2."},{"Start":"06:49.405 ","End":"06:51.770","Text":"But I can\u0027t combine these intervals"},{"Start":"06:51.770 ","End":"06:54.140","Text":"into say, minus 1 less than x less than 2,"},{"Start":"06:54.140 ","End":"06:55.910","Text":"because in the middle, we have a point"},{"Start":"06:55.910 ","End":"06:58.040","Text":"where the function is undefined,"},{"Start":"06:58.040 ","End":"06:59.060","Text":"so I can\u0027t combine."},{"Start":"06:59.060 ","End":"07:01.265","Text":"I\u0027ll have to leave them separately."},{"Start":"07:01.265 ","End":"07:04.980","Text":"We\u0027re done for this exercise."}],"ID":4820},{"Watched":false,"Name":"Exercise 17","Duration":"5m 14s","ChapterTopicVideoID":4820,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4820.jpeg","UploadDate":"2015-07-21T02:30:38.0030000","DurationForVideoObject":"PT5M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.955","Text":"In this exercise, we\u0027re given function of x,"},{"Start":"00:02.955 ","End":"00:04.860","Text":"natural log of x over x,"},{"Start":"00:04.860 ","End":"00:07.199","Text":"and we have to find for this function"},{"Start":"00:07.199 ","End":"00:10.620","Text":"it\u0027s extrema and the intervals where"},{"Start":"00:10.620 ","End":"00:12.915","Text":"it\u0027s increasing and decreasing."},{"Start":"00:12.915 ","End":"00:16.080","Text":"This type of exercise is cookbook style"},{"Start":"00:16.080 ","End":"00:18.150","Text":"as a standard set of steps that we follow"},{"Start":"00:18.150 ","End":"00:19.350","Text":"to get to the answer."},{"Start":"00:19.350 ","End":"00:21.540","Text":"The first thing I do is to note"},{"Start":"00:21.540 ","End":"00:24.165","Text":"the domain of the function."},{"Start":"00:24.165 ","End":"00:26.160","Text":"Here because of the natural log,"},{"Start":"00:26.160 ","End":"00:28.920","Text":"the domain will be x bigger than 0,"},{"Start":"00:28.920 ","End":"00:32.010","Text":"and that takes care of the x in the denominator,"},{"Start":"00:32.010 ","End":"00:34.620","Text":"so x bigger than 0 is the domain."},{"Start":"00:34.620 ","End":"00:37.440","Text":"Then we differentiate the function"},{"Start":"00:37.440 ","End":"00:40.410","Text":"f prime of x is equal to."},{"Start":"00:40.410 ","End":"00:42.920","Text":"It\u0027s a quotient, so I\u0027ll just write down"},{"Start":"00:42.920 ","End":"00:44.480","Text":"the quotient rule for u."},{"Start":"00:44.480 ","End":"00:50.090","Text":"U over v derivative is u prime v"},{"Start":"00:50.090 ","End":"00:53.465","Text":"minus uv prime over v squared."},{"Start":"00:53.465 ","End":"00:54.620","Text":"Now in this case,"},{"Start":"00:54.620 ","End":"00:55.870","Text":"this is u, this is v,"},{"Start":"00:55.870 ","End":"01:00.380","Text":"so we get u prime 1 over x times v,"},{"Start":"01:00.380 ","End":"01:03.770","Text":"which is x minus natural log of x"},{"Start":"01:03.770 ","End":"01:07.310","Text":"times 1 all over x squared."},{"Start":"01:07.310 ","End":"01:14.510","Text":"This equals 1 minus natural log of x over x squared."},{"Start":"01:14.510 ","End":"01:15.650","Text":"Then the next step"},{"Start":"01:15.650 ","End":"01:19.185","Text":"is to let the derivative equals 0,"},{"Start":"01:19.185 ","End":"01:22.460","Text":"to equate it to 0 and to solve the equation."},{"Start":"01:22.460 ","End":"01:25.610","Text":"If this is equal to 0, it\u0027s a fraction."},{"Start":"01:25.610 ","End":"01:27.605","Text":"The numerator has to be 0,"},{"Start":"01:27.605 ","End":"01:29.750","Text":"so we get that 1 minus"},{"Start":"01:29.750 ","End":"01:33.184","Text":"natural log of x is equal to 0."},{"Start":"01:33.184 ","End":"01:38.210","Text":"This gives us that natural log of x equals 1,"},{"Start":"01:38.210 ","End":"01:41.480","Text":"and that gives us that x equals e."},{"Start":"01:41.480 ","End":"01:42.830","Text":"That\u0027s the only candidate"},{"Start":"01:42.830 ","End":"01:44.885","Text":"for the extremum that we have."},{"Start":"01:44.885 ","End":"01:46.609","Text":"Now, we put all this information"},{"Start":"01:46.609 ","End":"01:48.260","Text":"in a standard table."},{"Start":"01:48.260 ","End":"01:49.850","Text":"The first thing we do"},{"Start":"01:49.850 ","End":"01:51.620","Text":"is to put in the values of x."},{"Start":"01:51.620 ","End":"01:52.670","Text":"Well, there\u0027s only 1 of them."},{"Start":"01:52.670 ","End":"01:54.410","Text":"The table is going to be a bit empty."},{"Start":"01:54.410 ","End":"01:57.050","Text":"But we put in x equals e,"},{"Start":"01:57.050 ","End":"02:00.605","Text":"and this value separates the interval."},{"Start":"02:00.605 ","End":"02:01.910","Text":"But remember, we were only"},{"Start":"02:01.910 ","End":"02:03.800","Text":"in x bigger than 0 anyway,"},{"Start":"02:03.800 ","End":"02:06.230","Text":"so this interval is just"},{"Start":"02:06.230 ","End":"02:10.020","Text":"going to be from 0 up to e."},{"Start":"02:10.020 ","End":"02:11.360","Text":"On the other side,"},{"Start":"02:11.360 ","End":"02:13.550","Text":"we have x bigger than e,"},{"Start":"02:13.550 ","End":"02:16.735","Text":"so the domain is split into 2 intervals."},{"Start":"02:16.735 ","End":"02:19.640","Text":"Then we pick an arbitrary value,"},{"Start":"02:19.640 ","End":"02:20.990","Text":"a sample, if you like,"},{"Start":"02:20.990 ","End":"02:22.220","Text":"from each of them."},{"Start":"02:22.220 ","End":"02:24.800","Text":"I\u0027ll pick 2 from here,"},{"Start":"02:24.800 ","End":"02:26.330","Text":"and 3 from here."},{"Start":"02:26.330 ","End":"02:28.845","Text":"We know that e is between 2 and 3."},{"Start":"02:28.845 ","End":"02:31.910","Text":"I also forgot to write that f prime of x,"},{"Start":"02:31.910 ","End":"02:33.950","Text":"we know for e it\u0027s 0."},{"Start":"02:33.950 ","End":"02:36.445","Text":"That\u0027s how we got the e after all."},{"Start":"02:36.445 ","End":"02:38.420","Text":"Here, we substitute these"},{"Start":"02:38.420 ","End":"02:41.705","Text":"sample values into f prime of x."},{"Start":"02:41.705 ","End":"02:43.460","Text":"But I don\u0027t want the actual value"},{"Start":"02:43.460 ","End":"02:46.190","Text":"only whether it\u0027s positive or negative."},{"Start":"02:46.190 ","End":"02:50.045","Text":"If I put in 2, I\u0027ve got a way of doing it."},{"Start":"02:50.045 ","End":"02:55.085","Text":"Natural log of 2 is less than natural log of e,"},{"Start":"02:55.085 ","End":"02:56.960","Text":"because 2 is less than e,"},{"Start":"02:56.960 ","End":"02:59.690","Text":"and natural log of e is 1."},{"Start":"02:59.690 ","End":"03:02.825","Text":"If natural log of 2 is less than 1,"},{"Start":"03:02.825 ","End":"03:05.600","Text":"then 1 minus natural log of 2"},{"Start":"03:05.600 ","End":"03:08.000","Text":"is bigger than 0."},{"Start":"03:08.000 ","End":"03:11.900","Text":"Likewise for 3, natural log of 3"},{"Start":"03:11.900 ","End":"03:16.250","Text":"is bigger than natural log of e, which is 1,"},{"Start":"03:16.250 ","End":"03:18.170","Text":"which means that 1 minus"},{"Start":"03:18.170 ","End":"03:21.575","Text":"natural log of 3 is less than 0."},{"Start":"03:21.575 ","End":"03:23.585","Text":"When I substitute in here,"},{"Start":"03:23.585 ","End":"03:25.190","Text":"the denominator is positive,"},{"Start":"03:25.190 ","End":"03:28.175","Text":"so I only care about the numerator,"},{"Start":"03:28.175 ","End":"03:31.805","Text":"so 1 minus natural log of 2 is positive."},{"Start":"03:31.805 ","End":"03:34.490","Text":"Here it is, positive after all."},{"Start":"03:34.490 ","End":"03:37.385","Text":"In the other 1 here, it\u0027s negative."},{"Start":"03:37.385 ","End":"03:38.360","Text":"This tells us something"},{"Start":"03:38.360 ","End":"03:40.295","Text":"about the function f of x."},{"Start":"03:40.295 ","End":"03:42.410","Text":"If the derivative is positive,"},{"Start":"03:42.410 ","End":"03:44.855","Text":"then the function is increasing."},{"Start":"03:44.855 ","End":"03:48.109","Text":"Conversely, if the derivative is negative,"},{"Start":"03:48.109 ","End":"03:49.880","Text":"then the function is decreasing,"},{"Start":"03:49.880 ","End":"03:51.985","Text":"which I indicate with arrows."},{"Start":"03:51.985 ","End":"03:54.870","Text":"Now, we can tell that e"},{"Start":"03:54.870 ","End":"03:56.755","Text":"is indeed an extremum,"},{"Start":"03:56.755 ","End":"03:59.330","Text":"because if it falls between the increasing"},{"Start":"03:59.330 ","End":"04:01.190","Text":"and decreasing or vice versa,"},{"Start":"04:01.190 ","End":"04:02.260","Text":"and it\u0027s an extremum."},{"Start":"04:02.260 ","End":"04:04.520","Text":"In this case, it\u0027s of type maximum,"},{"Start":"04:04.520 ","End":"04:06.200","Text":"which I abbreviate to max,"},{"Start":"04:06.200 ","End":"04:08.030","Text":"because it goes up to the maximum"},{"Start":"04:08.030 ","End":"04:09.905","Text":"and then down away from it."},{"Start":"04:09.905 ","End":"04:11.360","Text":"The only other thing I need"},{"Start":"04:11.360 ","End":"04:15.875","Text":"is the value of y when x is e,"},{"Start":"04:15.875 ","End":"04:17.570","Text":"y is f of x,"},{"Start":"04:17.570 ","End":"04:20.390","Text":"so I substitute e in here,"},{"Start":"04:20.390 ","End":"04:23.570","Text":"I get natural log of e over e."},{"Start":"04:23.570 ","End":"04:26.480","Text":"Natural log of e is just 1,"},{"Start":"04:26.480 ","End":"04:29.330","Text":"so I get 1 over e."},{"Start":"04:29.330 ","End":"04:31.610","Text":"I have all the information I need"},{"Start":"04:31.610 ","End":"04:33.920","Text":"to answer the questions about extrema"},{"Start":"04:33.920 ","End":"04:36.694","Text":"increasing and decreasing intervals."},{"Start":"04:36.694 ","End":"04:39.515","Text":"Conclusion is that extrema,"},{"Start":"04:39.515 ","End":"04:41.855","Text":"I only have 1 such."},{"Start":"04:41.855 ","End":"04:43.720","Text":"It happens to be a maximum,"},{"Start":"04:43.720 ","End":"04:45.770","Text":"and it occurs at the point where"},{"Start":"04:45.770 ","End":"04:49.090","Text":"x equals e and y is 1 over e."},{"Start":"04:49.090 ","End":"04:53.630","Text":"As for the intervals where the function is increasing,"},{"Start":"04:53.630 ","End":"04:56.060","Text":"I just have to look at the upward arrow"},{"Start":"04:56.060 ","End":"04:57.440","Text":"to know it\u0027s increasing,"},{"Start":"04:57.440 ","End":"04:59.340","Text":"and that happens when x"},{"Start":"04:59.340 ","End":"05:01.935","Text":"is between 0 and e."},{"Start":"05:01.935 ","End":"05:04.580","Text":"The interval where it\u0027s decreasing"},{"Start":"05:04.580 ","End":"05:07.505","Text":"is indicated by the downward arrow,"},{"Start":"05:07.505 ","End":"05:11.000","Text":"which means that x is bigger than e."},{"Start":"05:11.000 ","End":"05:13.740","Text":"That\u0027s it. We\u0027re done."}],"ID":4821},{"Watched":false,"Name":"Exercise 18","Duration":"5m 21s","ChapterTopicVideoID":4821,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4821.jpeg","UploadDate":"2015-07-21T02:31:05.8370000","DurationForVideoObject":"PT5M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:02.880 ","End":"00:06.045","Text":"f of x, which is x natural log of x."},{"Start":"00:06.045 ","End":"00:07.920","Text":"For this function, we have to find"},{"Start":"00:07.920 ","End":"00:10.470","Text":"the extrema and the intervals where"},{"Start":"00:10.470 ","End":"00:13.440","Text":"it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:13.440 ","End":"00:15.120","Text":"This type of exercises,"},{"Start":"00:15.120 ","End":"00:17.130","Text":"standard cookbook,"},{"Start":"00:17.130 ","End":"00:19.680","Text":"it\u0027s solved for the fixed recipe"},{"Start":"00:19.680 ","End":"00:20.955","Text":"or set of steps."},{"Start":"00:20.955 ","End":"00:24.120","Text":"I\u0027ll start, first thing I just note the domain."},{"Start":"00:24.120 ","End":"00:25.965","Text":"It\u0027s always good to do with a function."},{"Start":"00:25.965 ","End":"00:28.665","Text":"Since I have the natural log of x here,"},{"Start":"00:28.665 ","End":"00:32.810","Text":"the domain is x bigger than 0,"},{"Start":"00:32.810 ","End":"00:33.980","Text":"strictly bigger."},{"Start":"00:33.980 ","End":"00:37.630","Text":"The first thing to do is to differentiate f."},{"Start":"00:37.630 ","End":"00:39.785","Text":"We want f prime of x,"},{"Start":"00:39.785 ","End":"00:42.275","Text":"product rule, derivative of x,"},{"Start":"00:42.275 ","End":"00:45.170","Text":"which is 1 times natural log of x"},{"Start":"00:45.170 ","End":"00:46.520","Text":"plus the other way around,"},{"Start":"00:46.520 ","End":"00:51.050","Text":"x as is natural log of x differentiated."},{"Start":"00:51.050 ","End":"00:56.855","Text":"This is just equal to natural log of x plus 1."},{"Start":"00:56.855 ","End":"01:00.170","Text":"The next step is to set f prime"},{"Start":"01:00.170 ","End":"01:02.300","Text":"equal to 0 and solve for x."},{"Start":"01:02.300 ","End":"01:05.045","Text":"I\u0027m letting this equal 0."},{"Start":"01:05.045 ","End":"01:06.560","Text":"If I solve for x,"},{"Start":"01:06.560 ","End":"01:10.805","Text":"I get the natural log of x is minus 1,"},{"Start":"01:10.805 ","End":"01:15.300","Text":"and so x is e to the minus 1 or 1 over e."},{"Start":"01:15.300 ","End":"01:19.730","Text":"Now, we use our table in the x row."},{"Start":"01:19.730 ","End":"01:21.920","Text":"I only have 1 value to put in"},{"Start":"01:21.920 ","End":"01:24.755","Text":"my suspect for an extrema,"},{"Start":"01:24.755 ","End":"01:26.780","Text":"and that\u0027s 1 over e."},{"Start":"01:26.780 ","End":"01:30.030","Text":"Then the f prime for this point is 0,"},{"Start":"01:30.030 ","End":"01:31.940","Text":"because that\u0027s how we got the 1 over e."},{"Start":"01:31.940 ","End":"01:34.085","Text":"By setting f prime equals 0,"},{"Start":"01:34.085 ","End":"01:37.460","Text":"this divides the line into 2 intervals."},{"Start":"01:37.460 ","End":"01:40.580","Text":"But remember that x is only defined"},{"Start":"01:40.580 ","End":"01:41.990","Text":"for a positive numbers,"},{"Start":"01:41.990 ","End":"01:47.880","Text":"so this interval will be x between 0 and 1 over e."},{"Start":"01:47.880 ","End":"01:52.190","Text":"Here, we\u0027ll have x bigger than 1 over e."},{"Start":"01:52.190 ","End":"01:55.010","Text":"Now, we choose a sample an arbitrary"},{"Start":"01:55.010 ","End":"01:57.190","Text":"representative from each interval,"},{"Start":"01:57.190 ","End":"02:00.410","Text":"so 1 over e is somewhere"},{"Start":"02:00.410 ","End":"02:01.880","Text":"between a 1/2 and a 1/3,"},{"Start":"02:01.880 ","End":"02:04.475","Text":"since e is between 2 and 3."},{"Start":"02:04.475 ","End":"02:07.685","Text":"I\u0027ll just take it as 1 over 3."},{"Start":"02:07.685 ","End":"02:09.965","Text":"For x bigger than 1 over e,"},{"Start":"02:09.965 ","End":"02:11.970","Text":"I can pick x equals 1."},{"Start":"02:11.970 ","End":"02:13.850","Text":"Now, I need to find for each"},{"Start":"02:13.850 ","End":"02:15.680","Text":"of these f prime of x,"},{"Start":"02:15.680 ","End":"02:17.675","Text":"whether it\u0027s positive or negative,"},{"Start":"02:17.675 ","End":"02:20.585","Text":"f prime of x is here."},{"Start":"02:20.585 ","End":"02:22.880","Text":"I need to substitute 1/3"},{"Start":"02:22.880 ","End":"02:25.370","Text":"in natural log of x plus 1,"},{"Start":"02:25.370 ","End":"02:29.060","Text":"so I have natural log of 1/3"},{"Start":"02:29.060 ","End":"02:32.655","Text":"plus 1 is equal to 1."},{"Start":"02:32.655 ","End":"02:34.230","Text":"The natural log of 1/3"},{"Start":"02:34.230 ","End":"02:37.565","Text":"is minus the natural log of 3."},{"Start":"02:37.565 ","End":"02:40.490","Text":"Natural log of 1 minus natural log of 3,"},{"Start":"02:40.490 ","End":"02:42.350","Text":"natural log of 1 is 0."},{"Start":"02:42.350 ","End":"02:44.479","Text":"Now, natural log of 3"},{"Start":"02:44.479 ","End":"02:48.680","Text":"is bigger than natural log of e,"},{"Start":"02:48.680 ","End":"02:50.330","Text":"because 3 is bigger than e,"},{"Start":"02:50.330 ","End":"02:54.465","Text":"and so natural log of e is 1."},{"Start":"02:54.465 ","End":"02:57.165","Text":"Natural log of 3 is bigger than 1."},{"Start":"02:57.165 ","End":"02:59.295","Text":"If this thing is bigger than 1,"},{"Start":"02:59.295 ","End":"03:02.130","Text":"then 1 minus something"},{"Start":"03:02.130 ","End":"03:05.385","Text":"bigger than 1 is less than 0."},{"Start":"03:05.385 ","End":"03:07.130","Text":"All I need to know here"},{"Start":"03:07.130 ","End":"03:09.470","Text":"was not the actual value,"},{"Start":"03:09.470 ","End":"03:11.405","Text":"just whether it\u0027s positive or negative."},{"Start":"03:11.405 ","End":"03:14.405","Text":"I can conclude that this is negative."},{"Start":"03:14.405 ","End":"03:19.460","Text":"Now, if I take x as 1 and substituted,"},{"Start":"03:19.460 ","End":"03:21.410","Text":"so I get this time,"},{"Start":"03:21.410 ","End":"03:25.130","Text":"natural log of 1 plus 1"},{"Start":"03:25.130 ","End":"03:27.650","Text":"and natural log of 1 is 0,"},{"Start":"03:27.650 ","End":"03:30.385","Text":"so it\u0027s 1 which is bigger than 0."},{"Start":"03:30.385 ","End":"03:33.945","Text":"That means that I can write a plus here."},{"Start":"03:33.945 ","End":"03:36.850","Text":"Now, if the derivative is negative,"},{"Start":"03:36.850 ","End":"03:38.855","Text":"I need to know what it says about the function."},{"Start":"03:38.855 ","End":"03:41.255","Text":"Well, it says that the function\u0027s decreasing,"},{"Start":"03:41.255 ","End":"03:44.180","Text":"just like if the derivative is positive,"},{"Start":"03:44.180 ","End":"03:45.815","Text":"the function\u0027s increasing."},{"Start":"03:45.815 ","End":"03:48.155","Text":"I indicate that with these arrows."},{"Start":"03:48.155 ","End":"03:51.230","Text":"Now, I see that I can tell whether 1 over e"},{"Start":"03:51.230 ","End":"03:52.580","Text":"is an extrema or not."},{"Start":"03:52.580 ","End":"03:54.230","Text":"The answer is yes it is,"},{"Start":"03:54.230 ","End":"03:55.400","Text":"because if it comes between"},{"Start":"03:55.400 ","End":"03:56.810","Text":"increasing and decreasing"},{"Start":"03:56.810 ","End":"03:58.475","Text":"or decreasing and increasing,"},{"Start":"03:58.475 ","End":"04:00.739","Text":"then we know it\u0027s an extrema."},{"Start":"04:00.739 ","End":"04:03.260","Text":"This time, it\u0027s of type minimum because"},{"Start":"04:03.260 ","End":"04:06.284","Text":"decreasing and increasing is minimum."},{"Start":"04:06.284 ","End":"04:10.440","Text":"All I need now is this y, this value."},{"Start":"04:10.440 ","End":"04:13.970","Text":"For this, I have to substitute 1 over e"},{"Start":"04:13.970 ","End":"04:16.090","Text":"in the function itself which is here."},{"Start":"04:16.090 ","End":"04:18.845","Text":"If I put x equals 1 over e,"},{"Start":"04:18.845 ","End":"04:21.380","Text":"natural log of 1 over e"},{"Start":"04:21.380 ","End":"04:23.880","Text":"is minus natural log of e,"},{"Start":"04:23.880 ","End":"04:25.280","Text":"which is minus 1."},{"Start":"04:25.280 ","End":"04:27.560","Text":"Minus 1 times 1 over e"},{"Start":"04:27.560 ","End":"04:29.830","Text":"is just minus 1 over e."},{"Start":"04:29.830 ","End":"04:32.300","Text":"Now, we have all the information we need"},{"Start":"04:32.300 ","End":"04:36.800","Text":"to answer the question as to the local extrema,"},{"Start":"04:36.800 ","End":"04:39.655","Text":"and the increasing and decreasing intervals."},{"Start":"04:39.655 ","End":"04:43.640","Text":"The conclusion is that as for the extrema,"},{"Start":"04:43.640 ","End":"04:45.440","Text":"we only have 1 and that happens"},{"Start":"04:45.440 ","End":"04:47.975","Text":"to be a minimum which I abbreviate Min,"},{"Start":"04:47.975 ","End":"04:51.110","Text":"and it occurs at 1 over e"},{"Start":"04:51.110 ","End":"04:53.830","Text":"and minus 1 over e."},{"Start":"04:53.830 ","End":"04:56.290","Text":"That\u0027s the only extrema."},{"Start":"04:56.290 ","End":"05:00.185","Text":"As for the intervals of where it\u0027s increasing,"},{"Start":"05:00.185 ","End":"05:02.750","Text":"we just look at the orange arrow up"},{"Start":"05:02.750 ","End":"05:07.700","Text":"and we see that this is x bigger than 1 over e."},{"Start":"05:07.700 ","End":"05:09.855","Text":"For the decreasing,"},{"Start":"05:09.855 ","End":"05:12.090","Text":"just look at where I\u0027ve drawn"},{"Start":"05:12.090 ","End":"05:14.450","Text":"an arrow down for that comes out"},{"Start":"05:14.450 ","End":"05:19.670","Text":"to be 0 less than x, less than 1 over e."},{"Start":"05:19.670 ","End":"05:21.960","Text":"We\u0027re done."}],"ID":4822},{"Watched":false,"Name":"Exercise 19","Duration":"5m 9s","ChapterTopicVideoID":4822,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4822.jpeg","UploadDate":"2015-07-21T02:31:33.6770000","DurationForVideoObject":"PT5M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we\u0027re given a function of x,"},{"Start":"00:03.060 ","End":"00:07.670","Text":"natural log squared of x plus twice natural log of x minus 3."},{"Start":"00:07.670 ","End":"00:09.810","Text":"For this function, we have to find"},{"Start":"00:09.810 ","End":"00:15.120","Text":"its extrema and it\u0027s intervals of increase and decrease."},{"Start":"00:15.120 ","End":"00:16.860","Text":"I\u0027ve copied the exercise here."},{"Start":"00:16.860 ","End":"00:22.050","Text":"This type of exercise has a standard set of steps that if we follow cookbook style,"},{"Start":"00:22.050 ","End":"00:23.625","Text":"we\u0027ll get to the answer."},{"Start":"00:23.625 ","End":"00:26.190","Text":"The first thing to do, some preparation,"},{"Start":"00:26.190 ","End":"00:30.855","Text":"is just to note the domain and because there\u0027s a natural log here,"},{"Start":"00:30.855 ","End":"00:35.370","Text":"the domain would be x bigger than 0."},{"Start":"00:35.370 ","End":"00:39.135","Text":"Also in the preparation phase we differentiate."},{"Start":"00:39.135 ","End":"00:47.025","Text":"F prime of x is equal to 2 natural log of x times 1 over x"},{"Start":"00:47.025 ","End":"00:56.435","Text":"plus 2 times 1 over x. I can take 2 over x outside the brackets,"},{"Start":"00:56.435 ","End":"01:01.865","Text":"and I\u0027m left with natural log of x plus 1."},{"Start":"01:01.865 ","End":"01:06.740","Text":"Next step is to set f prime of x equals 0 and solve,"},{"Start":"01:06.740 ","End":"01:10.699","Text":"so let this equal 0 and solve for x."},{"Start":"01:10.699 ","End":"01:15.110","Text":"For this to be 0, I have to have natural log of x plus 1 equals 0,"},{"Start":"01:15.110 ","End":"01:19.700","Text":"so I need natural log of x to be equal to minus 1,"},{"Start":"01:19.700 ","End":"01:23.990","Text":"which means that x is e to the power of minus 1 or if"},{"Start":"01:23.990 ","End":"01:28.840","Text":"you prefer 1 over e. That\u0027s the only candidate we have for an extremum."},{"Start":"01:28.840 ","End":"01:33.050","Text":"The next step is to draw the standard table that we do"},{"Start":"01:33.050 ","End":"01:37.880","Text":"and first thing I do is put in my suspects for extremum,"},{"Start":"01:37.880 ","End":"01:42.350","Text":"just 1 over e. I note that the derivative here is 0."},{"Start":"01:42.350 ","End":"01:44.095","Text":"I mean, that\u0027s how I got to it."},{"Start":"01:44.095 ","End":"01:47.040","Text":"Then I indicate the intervals."},{"Start":"01:47.040 ","End":"01:51.830","Text":"1 over e gives me the interval 0 less than x"},{"Start":"01:51.830 ","End":"01:57.130","Text":"less than 1 over e. The bigger than 0 is because of the restriction on the domain."},{"Start":"01:57.130 ","End":"02:02.750","Text":"The other side I have x bigger than 1 over e unbounded on the other side."},{"Start":"02:02.750 ","End":"02:07.160","Text":"Then I pick candidate a sample point from each of the intervals."},{"Start":"02:07.160 ","End":"02:09.865","Text":"For x less than 1 over e,"},{"Start":"02:09.865 ","End":"02:14.130","Text":"I choose e is less than 3 so 1/3 will be less than"},{"Start":"02:14.130 ","End":"02:20.455","Text":"1 over e. For bigger than 1 over e, I\u0027ll take 1."},{"Start":"02:20.455 ","End":"02:23.119","Text":"Now I have to substitute these values,"},{"Start":"02:23.119 ","End":"02:27.530","Text":"these samples into the derivative,"},{"Start":"02:27.530 ","End":"02:30.290","Text":"which I can take from here,"},{"Start":"02:30.290 ","End":"02:32.930","Text":"but I don\u0027t want the actual numerical answer,"},{"Start":"02:32.930 ","End":"02:35.855","Text":"I only want the sign whether it\u0027s positive or negative."},{"Start":"02:35.855 ","End":"02:37.860","Text":"Let\u0027s try the easier 1 first."},{"Start":"02:37.860 ","End":"02:41.505","Text":"When x is 1, in any event it\u0027s positive."},{"Start":"02:41.505 ","End":"02:43.325","Text":"So I\u0027ll put a plus here,"},{"Start":"02:43.325 ","End":"02:48.350","Text":"and if x is 1/3 we\u0027ll do a computation here."},{"Start":"02:48.350 ","End":"02:56.135","Text":"The natural log of 1/3 is minus natural log of 3."},{"Start":"02:56.135 ","End":"03:00.190","Text":"Because natural log of 3 is bigger than 1,"},{"Start":"03:00.190 ","End":"03:03.965","Text":"because natural log of e is 1 so natural log of 3 is bigger than 1."},{"Start":"03:03.965 ","End":"03:06.830","Text":"When it\u0027s negative, everything is reversed,"},{"Start":"03:06.830 ","End":"03:09.470","Text":"so it\u0027s less than minus 1."},{"Start":"03:09.470 ","End":"03:12.185","Text":"If this is less than minus 1,"},{"Start":"03:12.185 ","End":"03:14.660","Text":"then when I add it to 1,"},{"Start":"03:14.660 ","End":"03:16.750","Text":"it will still be negative."},{"Start":"03:16.750 ","End":"03:20.330","Text":"Minus log of 3 plus 1,"},{"Start":"03:20.330 ","End":"03:25.050","Text":"if I add to both sides will still be less than 0 and so this is"},{"Start":"03:25.050 ","End":"03:30.995","Text":"less than 0 and so this here is going to be negative."},{"Start":"03:30.995 ","End":"03:34.160","Text":"What this says about the function itself is that here it\u0027s"},{"Start":"03:34.160 ","End":"03:37.430","Text":"decreasing because if the derivative is negative,"},{"Start":"03:37.430 ","End":"03:38.780","Text":"the function is decreasing."},{"Start":"03:38.780 ","End":"03:40.295","Text":"I\u0027m going to write that with an arrow."},{"Start":"03:40.295 ","End":"03:43.070","Text":"Likewise, when x is bigger than 1 over e,"},{"Start":"03:43.070 ","End":"03:45.980","Text":"derivative is positive so the function is increasing,"},{"Start":"03:45.980 ","End":"03:47.305","Text":"so an up arrow."},{"Start":"03:47.305 ","End":"03:51.620","Text":"When an extremum suspect is between them increasing and decreasing or"},{"Start":"03:51.620 ","End":"03:53.750","Text":"the other way round then it confirms that it is"},{"Start":"03:53.750 ","End":"03:56.420","Text":"indeed an extremum and this one is of type minimum."},{"Start":"03:56.420 ","End":"03:57.680","Text":"When it goes down then up,"},{"Start":"03:57.680 ","End":"03:58.864","Text":"then it\u0027s a minimum."},{"Start":"03:58.864 ","End":"04:01.489","Text":"All I need now is this value of y,"},{"Start":"04:01.489 ","End":"04:09.295","Text":"which is what I will get if I plug in 1 over e into the original function, which is this."},{"Start":"04:09.295 ","End":"04:14.400","Text":"Now natural log of x when x is 1 over e is minus 1."},{"Start":"04:14.400 ","End":"04:21.530","Text":"I have here minus 1 squared plus twice minus 1 minus 3."},{"Start":"04:21.530 ","End":"04:24.295","Text":"What does this give me? Minus 4."},{"Start":"04:24.295 ","End":"04:27.650","Text":"Now we have all the information we need to solve the questions"},{"Start":"04:27.650 ","End":"04:31.280","Text":"about the extrema and the increasing, decreasing."},{"Start":"04:31.280 ","End":"04:37.480","Text":"As far as extrema goes that I do have just 1 and it\u0027s a minimum."},{"Start":"04:37.480 ","End":"04:40.385","Text":"It happens at x equals 1 over e,"},{"Start":"04:40.385 ","End":"04:42.370","Text":"y equals minus 4."},{"Start":"04:42.370 ","End":"04:46.250","Text":"Then I need to say about where the function is increasing."},{"Start":"04:46.250 ","End":"04:50.930","Text":"For that, I just look at where they have the arrow up in orange here,"},{"Start":"04:50.930 ","End":"04:56.085","Text":"and that gives me x bigger than 1 over e. Finally,"},{"Start":"04:56.085 ","End":"04:58.460","Text":"where the function is decreasing,"},{"Start":"04:58.460 ","End":"05:03.305","Text":"I look at the arrow down and that gives me 0 less than"},{"Start":"05:03.305 ","End":"05:09.960","Text":"x less than 1 over e. That answers all the questions and we\u0027re done."}],"ID":4823},{"Watched":false,"Name":"Exercise 20","Duration":"4m 17s","ChapterTopicVideoID":4823,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4823.jpeg","UploadDate":"2015-07-21T02:31:56.3570000","DurationForVideoObject":"PT4M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this exercise, we\u0027re given f of x"},{"Start":"00:02.700 ","End":"00:04.410","Text":"is equal to this function,"},{"Start":"00:04.410 ","End":"00:06.560","Text":"1 over the square root of x squared plus 1,"},{"Start":"00:06.560 ","End":"00:10.559","Text":"and we have to find for f, it\u0027s local extrema,"},{"Start":"00:10.559 ","End":"00:12.480","Text":"as well as the intervals where"},{"Start":"00:12.480 ","End":"00:14.880","Text":"it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.880 ","End":"00:17.129","Text":"This is a particular exercise"},{"Start":"00:17.129 ","End":"00:18.270","Text":"that we know how to solve."},{"Start":"00:18.270 ","End":"00:20.294","Text":"It has a standard set of steps,"},{"Start":"00:20.294 ","End":"00:21.720","Text":"like a cookbook style."},{"Start":"00:21.720 ","End":"00:24.465","Text":"First thing is to note the domain,"},{"Start":"00:24.465 ","End":"00:27.165","Text":"no problems is defined for all x."},{"Start":"00:27.165 ","End":"00:29.730","Text":"Next thing, differentiate the function."},{"Start":"00:29.730 ","End":"00:31.770","Text":"I\u0027ve copied it here so I can write"},{"Start":"00:31.770 ","End":"00:34.560","Text":"that f prime of x is equal to."},{"Start":"00:34.560 ","End":"00:38.010","Text":"I prefer to look at this as x squared"},{"Start":"00:38.010 ","End":"00:40.050","Text":"plus 1 to the power of minus 1/2,"},{"Start":"00:40.050 ","End":"00:43.460","Text":"and differentiate it as an exponent"},{"Start":"00:43.460 ","End":"00:45.320","Text":"rather than using the quotient rule."},{"Start":"00:45.320 ","End":"00:49.370","Text":"We get minus a 1/2 x squared plus 1"},{"Start":"00:49.370 ","End":"00:51.875","Text":"to the power of minus 3 over 2."},{"Start":"00:51.875 ","End":"00:53.660","Text":"We don\u0027t want the negative exponent,"},{"Start":"00:53.660 ","End":"00:55.595","Text":"I\u0027ll write it on the denominator,"},{"Start":"00:55.595 ","End":"00:58.058","Text":"x squared plus 1 to the power of 3,"},{"Start":"00:58.058 ","End":"01:00.655","Text":"and then square root."},{"Start":"01:00.655 ","End":"01:02.930","Text":"Yeah, instead of the power of 3 over 2,"},{"Start":"01:02.930 ","End":"01:05.330","Text":"I can take the square root of this thing cubed,"},{"Start":"01:05.330 ","End":"01:08.060","Text":"and then the minus 1 over 2 is here."},{"Start":"01:08.060 ","End":"01:10.370","Text":"Then I have to multiply by the internal"},{"Start":"01:10.370 ","End":"01:13.750","Text":"derivative of x squared plus 1, which is 2x,"},{"Start":"01:13.750 ","End":"01:19.290","Text":"so I can just put this as minus 2x here."},{"Start":"01:19.290 ","End":"01:21.410","Text":"The 2 cancels, that\u0027s okay."},{"Start":"01:21.410 ","End":"01:25.130","Text":"Next thing is to solve f prime of x equals 0"},{"Start":"01:25.130 ","End":"01:28.300","Text":"and see which x will make this 0."},{"Start":"01:28.300 ","End":"01:31.640","Text":"There\u0027s only 1 possibility for this to be 0."},{"Start":"01:31.640 ","End":"01:32.990","Text":"The numerator is 0,"},{"Start":"01:32.990 ","End":"01:34.820","Text":"and that\u0027s only if x equals 0."},{"Start":"01:34.820 ","End":"01:37.400","Text":"This gives me x equals 0,"},{"Start":"01:37.400 ","End":"01:40.985","Text":"and that\u0027s my only candidate for an extremum."},{"Start":"01:40.985 ","End":"01:44.300","Text":"Next step is to draw the well-known table."},{"Start":"01:44.300 ","End":"01:48.110","Text":"In this table, we put in the suspects for extremum,"},{"Start":"01:48.110 ","End":"01:52.440","Text":"there\u0027s only 1 here, and so that\u0027s the 0,"},{"Start":"01:52.440 ","End":"01:56.615","Text":"and f prime of this was also going to be 0,"},{"Start":"01:56.615 ","End":"01:57.980","Text":"because that\u0027s where we set out"},{"Start":"01:57.980 ","End":"01:59.960","Text":"from f prime equals 0."},{"Start":"01:59.960 ","End":"02:02.030","Text":"Now, we write the intervals"},{"Start":"02:02.030 ","End":"02:04.075","Text":"that this point creates."},{"Start":"02:04.075 ","End":"02:06.395","Text":"It creates 2 intervals,"},{"Start":"02:06.395 ","End":"02:10.810","Text":"x less than 0 and x bigger than 0."},{"Start":"02:10.810 ","End":"02:12.950","Text":"I\u0027ll choose a representative"},{"Start":"02:12.950 ","End":"02:15.305","Text":"sample point from each of these."},{"Start":"02:15.305 ","End":"02:18.030","Text":"I\u0027ll choose minus 1 here,"},{"Start":"02:18.030 ","End":"02:19.710","Text":"and I\u0027ll choose 1 here."},{"Start":"02:19.710 ","End":"02:22.970","Text":"Then we substitute the sample points"},{"Start":"02:22.970 ","End":"02:28.255","Text":"in the function f prime, which is here."},{"Start":"02:28.255 ","End":"02:31.070","Text":"All I need to know is not the value,"},{"Start":"02:31.070 ","End":"02:33.350","Text":"but just whether it\u0027s positive or negative."},{"Start":"02:33.350 ","End":"02:35.360","Text":"Square root is always positive,"},{"Start":"02:35.360 ","End":"02:37.640","Text":"so the denominator is positive."},{"Start":"02:37.640 ","End":"02:40.250","Text":"If x is minus 1,"},{"Start":"02:40.250 ","End":"02:42.700","Text":"I\u0027ll get something positive here."},{"Start":"02:42.700 ","End":"02:45.660","Text":"I\u0027ve put plus sign to indicate that."},{"Start":"02:45.660 ","End":"02:49.770","Text":"When x is 1, here I get minus 2"},{"Start":"02:49.770 ","End":"02:51.390","Text":"over something positive."},{"Start":"02:51.390 ","End":"02:54.410","Text":"That\u0027s negative and now I interpret"},{"Start":"02:54.410 ","End":"02:56.420","Text":"this plus and minus in terms of f."},{"Start":"02:56.420 ","End":"02:58.175","Text":"If the derivative is positive,"},{"Start":"02:58.175 ","End":"03:00.590","Text":"that means that the function is increasing,"},{"Start":"03:00.590 ","End":"03:02.120","Text":"so an arrow like this."},{"Start":"03:02.120 ","End":"03:03.559","Text":"When it\u0027s negative,"},{"Start":"03:03.559 ","End":"03:05.030","Text":"the function\u0027s decreasing,"},{"Start":"03:05.030 ","End":"03:06.890","Text":"so an arrow like this."},{"Start":"03:06.890 ","End":"03:10.130","Text":"Now, the x equals 0 is confirmed"},{"Start":"03:10.130 ","End":"03:11.330","Text":"to be an extremum because"},{"Start":"03:11.330 ","End":"03:12.740","Text":"if it comes between increasing"},{"Start":"03:12.740 ","End":"03:14.390","Text":"and decreasing or vice versa,"},{"Start":"03:14.390 ","End":"03:15.410","Text":"then it\u0027s an extremum."},{"Start":"03:15.410 ","End":"03:17.705","Text":"In this case, it\u0027s of type maximum,"},{"Start":"03:17.705 ","End":"03:19.285","Text":"which I call max."},{"Start":"03:19.285 ","End":"03:20.940","Text":"Between increasing and decreasing,"},{"Start":"03:20.940 ","End":"03:21.930","Text":"that\u0027s a maximum,"},{"Start":"03:21.930 ","End":"03:25.580","Text":"and all I need now is this value of y,"},{"Start":"03:25.580 ","End":"03:28.460","Text":"which is what I get when I substitute 0"},{"Start":"03:28.460 ","End":"03:30.350","Text":"in the original function."},{"Start":"03:30.350 ","End":"03:34.220","Text":"If x is 0, then it\u0027s a square root of 0 plus 1 is 1,"},{"Start":"03:34.220 ","End":"03:36.355","Text":"1 over 1, it\u0027s 1."},{"Start":"03:36.355 ","End":"03:38.660","Text":"Now, but all the information I need"},{"Start":"03:38.660 ","End":"03:40.580","Text":"in order to answer the questions about"},{"Start":"03:40.580 ","End":"03:43.310","Text":"the extrema and the increasing, decreasing."},{"Start":"03:43.310 ","End":"03:46.790","Text":"Here goes the conclusion and the answer,"},{"Start":"03:46.790 ","End":"03:50.215","Text":"we have extrema only 1."},{"Start":"03:50.215 ","End":"03:52.010","Text":"It happens to be a maximum,"},{"Start":"03:52.010 ","End":"03:56.420","Text":"and it occurs at the point x is 0, y is 1."},{"Start":"03:56.420 ","End":"03:59.720","Text":"Intervals where the function is increasing,"},{"Start":"03:59.720 ","End":"04:01.460","Text":"I just look at the up arrow,"},{"Start":"04:01.460 ","End":"04:02.989","Text":"and see that it\u0027s increasing"},{"Start":"04:02.989 ","End":"04:05.510","Text":"for x less than 0."},{"Start":"04:05.510 ","End":"04:07.160","Text":"For decreasing,"},{"Start":"04:07.160 ","End":"04:09.710","Text":"I just look at the down arrow,"},{"Start":"04:09.710 ","End":"04:12.259","Text":"and see that this belongs to the interval"},{"Start":"04:12.259 ","End":"04:14.509","Text":"where x is bigger than 0."},{"Start":"04:14.509 ","End":"04:16.088","Text":"That answers all the questions,"},{"Start":"04:16.088 ","End":"04:18.030","Text":"and we\u0027re done."}],"ID":4824},{"Watched":false,"Name":"Exercise 21","Duration":"3m 26s","ChapterTopicVideoID":4824,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4824.jpeg","UploadDate":"2015-07-21T02:32:13.9570000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"In this exercise, we\u0027re given a function f of x is"},{"Start":"00:03.300 ","End":"00:06.750","Text":"equals x over the square root of x squared plus 1."},{"Start":"00:06.750 ","End":"00:09.465","Text":"We have to find its extrema,"},{"Start":"00:09.465 ","End":"00:12.000","Text":"and the intervals where it\u0027s increasing,"},{"Start":"00:12.000 ","End":"00:13.485","Text":"and where it\u0027s decreasing."},{"Start":"00:13.485 ","End":"00:17.250","Text":"This type of exercise is solved with a standard set of steps,"},{"Start":"00:17.250 ","End":"00:20.415","Text":"cookbook style, and let us go through them."},{"Start":"00:20.415 ","End":"00:24.015","Text":"Take a glance to see if there\u0027s any problems with the domain."},{"Start":"00:24.015 ","End":"00:27.960","Text":"I see it\u0027s defined for all x, no problem there."},{"Start":"00:27.960 ","End":"00:32.565","Text":"First thing to do is to differentiate f of x. I copied it here."},{"Start":"00:32.565 ","End":"00:35.895","Text":"So we have f prime of x equals."},{"Start":"00:35.895 ","End":"00:37.260","Text":"I see it\u0027s a quotient,"},{"Start":"00:37.260 ","End":"00:42.800","Text":"so I\u0027ll just remind you of the quotient rule in case you\u0027ve forgotten it that u over v"},{"Start":"00:42.800 ","End":"00:51.214","Text":"derivative is u prime v minus u v prime over v squared."},{"Start":"00:51.214 ","End":"00:55.885","Text":"I\u0027ll take u to be x and v to be the square root of x squared plus 1."},{"Start":"00:55.885 ","End":"00:58.760","Text":"V squared is the square root squared,"},{"Start":"00:58.760 ","End":"01:01.895","Text":"so it\u0027s just x squared plus 1."},{"Start":"01:01.895 ","End":"01:11.230","Text":"U prime is just 1 times v is square root of x squared plus 1 minus u,"},{"Start":"01:11.230 ","End":"01:14.670","Text":"and then v prime is 1 over"},{"Start":"01:14.670 ","End":"01:22.630","Text":"twice the square root of x squared plus 1,"},{"Start":"01:22.630 ","End":"01:27.145","Text":"times the internal derivative, which is 2x."},{"Start":"01:27.145 ","End":"01:29.195","Text":"Let\u0027s see if we can simplify it."},{"Start":"01:29.195 ","End":"01:35.495","Text":"If I multiply top and bottom by the square root of x squared plus 1,"},{"Start":"01:35.495 ","End":"01:36.980","Text":"and multiply it here,"},{"Start":"01:36.980 ","End":"01:39.440","Text":"I\u0027ll multiply it at the bottom also."},{"Start":"01:39.440 ","End":"01:41.140","Text":"What I\u0027ll get is,"},{"Start":"01:41.140 ","End":"01:44.340","Text":"here I\u0027ll have x squared plus 1,"},{"Start":"01:44.340 ","End":"01:47.745","Text":"square root of x squared plus 1,"},{"Start":"01:47.745 ","End":"01:52.110","Text":"and here I\u0027ll have x squared plus 1 minus,"},{"Start":"01:52.110 ","End":"01:54.255","Text":"and just x squared."},{"Start":"01:54.255 ","End":"01:57.395","Text":"The x squared and the x squared cancel."},{"Start":"01:57.395 ","End":"02:05.570","Text":"I have 1 over x squared plus 1 to the power of 1,"},{"Start":"02:05.570 ","End":"02:06.790","Text":"this is to the power of a half,"},{"Start":"02:06.790 ","End":"02:08.845","Text":"to the power of 3 over 2."},{"Start":"02:08.845 ","End":"02:12.940","Text":"The next step is to set this to be equal"},{"Start":"02:12.940 ","End":"02:17.650","Text":"to 0 and to solve and find which x could make it 0."},{"Start":"02:17.650 ","End":"02:23.340","Text":"Well, 1 over something can never be 0."},{"Start":"02:23.340 ","End":"02:29.260","Text":"The function, if it doesn\u0027t have any points where the derivative is 0,"},{"Start":"02:29.260 ","End":"02:32.710","Text":"the derivative is going to always be positive or always negative."},{"Start":"02:32.710 ","End":"02:34.570","Text":"I only have 1 single interval,"},{"Start":"02:34.570 ","End":"02:37.110","Text":"which is the whole line."},{"Start":"02:37.110 ","End":"02:40.730","Text":"All I have to do is take any point,"},{"Start":"02:40.730 ","End":"02:43.640","Text":"I want to check whether this function,"},{"Start":"02:43.640 ","End":"02:44.975","Text":"which is the derivative."},{"Start":"02:44.975 ","End":"02:48.680","Text":"If it\u0027s never 0, that means it\u0027s always positive or always negative."},{"Start":"02:48.680 ","End":"02:51.650","Text":"Well, let\u0027s check. Just substitute any value."},{"Start":"02:51.650 ","End":"02:53.790","Text":"Let\u0027s say x equals 0,"},{"Start":"02:53.790 ","End":"02:56.030","Text":"and I see that this equals to 1."},{"Start":"02:56.030 ","End":"02:59.435","Text":"In fact, for all x, this thing is positive,"},{"Start":"02:59.435 ","End":"03:01.699","Text":"to the power of 3 over 2 is still positive,"},{"Start":"03:01.699 ","End":"03:03.650","Text":"1 over positive is positive."},{"Start":"03:03.650 ","End":"03:07.630","Text":"This thing is always bigger than 0."},{"Start":"03:07.630 ","End":"03:14.270","Text":"Actually the solution comes out quite neatly is that the extrema,"},{"Start":"03:14.270 ","End":"03:17.705","Text":"there are none, there\u0027s no extrema."},{"Start":"03:17.705 ","End":"03:23.415","Text":"Then increasing, that would be all x and decreasing,"},{"Start":"03:23.415 ","End":"03:26.710","Text":"nowhere. That\u0027s the answer."}],"ID":4825},{"Watched":false,"Name":"Exercise 22","Duration":"3m 32s","ChapterTopicVideoID":4825,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4825.jpeg","UploadDate":"2015-07-21T02:32:32.3500000","DurationForVideoObject":"PT3M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.200","Text":"In this exercise, we\u0027re given f of x is x times square root of x squared plus 4."},{"Start":"00:06.200 ","End":"00:10.290","Text":"For this function, we have to find the local extrema"},{"Start":"00:10.290 ","End":"00:14.865","Text":"as well as the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:14.865 ","End":"00:19.305","Text":"This is a familiar type of exercise and it has certain standards steps,"},{"Start":"00:19.305 ","End":"00:22.245","Text":"a routine that we go through like a cookbook style."},{"Start":"00:22.245 ","End":"00:26.525","Text":"Let\u0027s first note the domain and see if there\u0027s any problems there."},{"Start":"00:26.525 ","End":"00:29.600","Text":"X squared is non-negative,"},{"Start":"00:29.600 ","End":"00:32.120","Text":"x squared plus 4 is definitely positive."},{"Start":"00:32.120 ","End":"00:34.535","Text":"The square root is defined."},{"Start":"00:34.535 ","End":"00:36.580","Text":"There are no problems with the domain."},{"Start":"00:36.580 ","End":"00:41.195","Text":"First proper thing we do is to differentiate the function."},{"Start":"00:41.195 ","End":"00:43.040","Text":"Let\u0027s find f prime of x."},{"Start":"00:43.040 ","End":"00:47.165","Text":"We\u0027ll use the product rule and I\u0027ll remind you of it."},{"Start":"00:47.165 ","End":"00:55.010","Text":"Uv derivative is u derivative times v plus u times v derivative."},{"Start":"00:55.010 ","End":"00:59.630","Text":"In our case we get x prime is"},{"Start":"00:59.630 ","End":"01:07.295","Text":"1 times square root of x squared plus 4 plus x as is."},{"Start":"01:07.295 ","End":"01:09.125","Text":"The derivative of this,"},{"Start":"01:09.125 ","End":"01:13.070","Text":"derivative of square root is 1 over twice the square root,"},{"Start":"01:13.070 ","End":"01:17.585","Text":"but the internal derivative is 2x."},{"Start":"01:17.585 ","End":"01:19.865","Text":"The 2 cancels with the 2."},{"Start":"01:19.865 ","End":"01:21.515","Text":"We can simplify this."},{"Start":"01:21.515 ","End":"01:27.020","Text":"If we put this over square root of x squared plus 4,"},{"Start":"01:27.020 ","End":"01:28.835","Text":"give it a common denominator,"},{"Start":"01:28.835 ","End":"01:31.925","Text":"square root of x squared plus 4."},{"Start":"01:31.925 ","End":"01:34.475","Text":"This side we have x times x,"},{"Start":"01:34.475 ","End":"01:36.685","Text":"which is x squared."},{"Start":"01:36.685 ","End":"01:39.350","Text":"Here, to get the common denominator,"},{"Start":"01:39.350 ","End":"01:45.215","Text":"it\u0027s just like I have to multiply the numerator also by x squared plus 4 square root."},{"Start":"01:45.215 ","End":"01:51.930","Text":"I just get x squared 4 without the square root plus x squared so that makes"},{"Start":"01:51.930 ","End":"01:59.705","Text":"it 2x squared plus 4 over the square root of x squared plus 4."},{"Start":"01:59.705 ","End":"02:02.060","Text":"Next thing to do is,"},{"Start":"02:02.060 ","End":"02:05.990","Text":"we have to solve the equation f prime of x equals"},{"Start":"02:05.990 ","End":"02:10.730","Text":"0 and see if we can get any suspects for extrema."},{"Start":"02:10.730 ","End":"02:17.750","Text":"2x squared plus 4 over this is never going to be 0 because for a fraction to be 0,"},{"Start":"02:17.750 ","End":"02:22.205","Text":"the numerator has to be 0 and the numerator is always positive."},{"Start":"02:22.205 ","End":"02:25.600","Text":"There are no solutions for x."},{"Start":"02:25.600 ","End":"02:27.785","Text":"This thing is never 0."},{"Start":"02:27.785 ","End":"02:32.240","Text":"That means that it\u0027s either always positive or always negative."},{"Start":"02:32.240 ","End":"02:36.890","Text":"All we have to do is take a sample x,"},{"Start":"02:36.890 ","End":"02:39.245","Text":"and if I put any x here,"},{"Start":"02:39.245 ","End":"02:43.850","Text":"say x equals 0, I get positive over positive."},{"Start":"02:43.850 ","End":"02:46.915","Text":"Positive over positive, it\u0027s always positive."},{"Start":"02:46.915 ","End":"02:52.010","Text":"In that steps have been abruptly brought to a halt."},{"Start":"02:52.010 ","End":"02:57.530","Text":"We can immediately write the answer to the 3 questions about the extrema,"},{"Start":"02:57.530 ","End":"02:59.585","Text":"the increasing and decreasing."},{"Start":"02:59.585 ","End":"03:02.705","Text":"Extrema, there are none."},{"Start":"03:02.705 ","End":"03:04.280","Text":"You don\u0027t have any extrema."},{"Start":"03:04.280 ","End":"03:07.240","Text":"I just wanted to write down here that it\u0027s always positive."},{"Start":"03:07.240 ","End":"03:10.015","Text":"Now, that means that for increasing,"},{"Start":"03:10.015 ","End":"03:12.290","Text":"intervals where the function is increasing,"},{"Start":"03:12.290 ","End":"03:15.680","Text":"it\u0027s increasing always for all x,"},{"Start":"03:15.680 ","End":"03:20.140","Text":"the whole line is 1 big interval where it\u0027s increasing."},{"Start":"03:20.140 ","End":"03:23.015","Text":"Since we were asked about decreasing,"},{"Start":"03:23.015 ","End":"03:25.130","Text":"I have to say nowhere."},{"Start":"03:25.130 ","End":"03:27.409","Text":"It\u0027s derivative is always positive,"},{"Start":"03:27.409 ","End":"03:33.180","Text":"it\u0027s nowhere negative and that\u0027s the solution. We\u0027re done."}],"ID":4826},{"Watched":false,"Name":"Exercise 23","Duration":"6m 41s","ChapterTopicVideoID":4826,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4826.jpeg","UploadDate":"2015-07-21T02:33:06.9700000","DurationForVideoObject":"PT6M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.395","Text":"In this exercise, we\u0027re given f of x as follows,"},{"Start":"00:04.395 ","End":"00:07.825","Text":"and we have to find the local extrema of"},{"Start":"00:07.825 ","End":"00:12.840","Text":"this f and also the intervals of where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:12.840 ","End":"00:16.320","Text":"This exercise follows a certain standard set"},{"Start":"00:16.320 ","End":"00:19.260","Text":"of steps which we\u0027re going to follow cookbook style."},{"Start":"00:19.260 ","End":"00:23.040","Text":"I first of all look and see if there\u0027s anything with the domain,"},{"Start":"00:23.040 ","End":"00:24.735","Text":"any problems, any limitation."},{"Start":"00:24.735 ","End":"00:28.830","Text":"I see that now this thing is defined for all x."},{"Start":"00:28.830 ","End":"00:32.190","Text":"A cube root is defined for every value,"},{"Start":"00:32.190 ","End":"00:33.735","Text":"so there\u0027s no problem."},{"Start":"00:33.735 ","End":"00:36.865","Text":"First proper thing we do is to differentiate."},{"Start":"00:36.865 ","End":"00:40.310","Text":"Let\u0027s differentiate f prime of x. I think it\u0027ll"},{"Start":"00:40.310 ","End":"00:43.774","Text":"be easier if we put it in terms of fractional exponent."},{"Start":"00:43.774 ","End":"00:45.755","Text":"This is x_2/3."},{"Start":"00:45.755 ","End":"00:49.250","Text":"We have x_2/3 minus x,"},{"Start":"00:49.250 ","End":"00:53.570","Text":"x_2/3, which is x_5/3."},{"Start":"00:53.570 ","End":"00:58.170","Text":"F prime of x is going to be 2/3 x to the power of"},{"Start":"00:58.170 ","End":"01:06.765","Text":"minus 1/3 less 5/3 x to the power of 2/3."},{"Start":"01:06.765 ","End":"01:08.360","Text":"What I would like to do,"},{"Start":"01:08.360 ","End":"01:09.860","Text":"and this is part of 1 of the steps,"},{"Start":"01:09.860 ","End":"01:13.295","Text":"is to set this thing equal to 0 and solve for x."},{"Start":"01:13.295 ","End":"01:17.205","Text":"Set this equal to 0,"},{"Start":"01:17.205 ","End":"01:19.040","Text":"and let\u0027s see what we get."},{"Start":"01:19.040 ","End":"01:26.210","Text":"Note by the way, that x cannot be 0 because 1 over x_1/3 would not be defined."},{"Start":"01:26.210 ","End":"01:29.600","Text":"I think what we\u0027ll do is multiply both sides"},{"Start":"01:29.600 ","End":"01:33.035","Text":"by x to the power of a 1/3 and get rid of this."},{"Start":"01:33.035 ","End":"01:38.730","Text":"I can also multiply by 3 to get rid of the fractions."},{"Start":"01:38.730 ","End":"01:41.630","Text":"If I multiply by 3x_1/3,"},{"Start":"01:41.630 ","End":"01:47.545","Text":"I\u0027ll be left with 2 times just 1 minus 5."},{"Start":"01:47.545 ","End":"01:49.670","Text":"If I multiply this by x_1/3,"},{"Start":"01:49.670 ","End":"01:52.820","Text":"I just get x is equal to 0,"},{"Start":"01:52.820 ","End":"01:57.160","Text":"and so x is equal to 2/5."},{"Start":"01:57.160 ","End":"02:00.335","Text":"That\u0027s our only candidate for an extremum."},{"Start":"02:00.335 ","End":"02:03.830","Text":"Let\u0027s make our table."},{"Start":"02:03.830 ","End":"02:07.235","Text":"In this table, we put the values of x,"},{"Start":"02:07.235 ","End":"02:09.440","Text":"which makes f prime 0,"},{"Start":"02:09.440 ","End":"02:12.015","Text":"but we also include special values."},{"Start":"02:12.015 ","End":"02:13.550","Text":"The 2/5 will go in here."},{"Start":"02:13.550 ","End":"02:17.810","Text":"Usually if there\u0027s a point where f is not defined, we put it there."},{"Start":"02:17.810 ","End":"02:23.510","Text":"But I also think that this 0 is suspicious because if f prime is not defined there,"},{"Start":"02:23.510 ","End":"02:26.090","Text":"it could change its sign around 0."},{"Start":"02:26.090 ","End":"02:28.760","Text":"I\u0027m going to put in both 0"},{"Start":"02:28.760 ","End":"02:35.330","Text":"and 2/5 to cover ourselves in case something funny happens around x equals 0."},{"Start":"02:35.330 ","End":"02:43.069","Text":"Here it\u0027s 0, f prime of x. I want to remark here that f prime of x is not defined."},{"Start":"02:43.069 ","End":"02:45.240","Text":"I don\u0027t know how to write that,"},{"Start":"02:45.240 ","End":"02:51.695","Text":"I\u0027ll put question mark here to say that something fishy is going on around there."},{"Start":"02:51.695 ","End":"02:53.900","Text":"Now we\u0027ve got the intervals,"},{"Start":"02:53.900 ","End":"02:56.750","Text":"which I like to do in orange."},{"Start":"02:56.750 ","End":"03:02.810","Text":"So x less than 0 and x"},{"Start":"03:02.810 ","End":"03:10.205","Text":"between 0 and 2/5 and x bigger than 2/5."},{"Start":"03:10.205 ","End":"03:15.065","Text":"Next, we pick a sample point from each of these ranges."},{"Start":"03:15.065 ","End":"03:18.950","Text":"I don\u0027t know what to pick for x bigger than 2/5."},{"Start":"03:18.950 ","End":"03:21.500","Text":"I\u0027ll pick x equals 1."},{"Start":"03:21.500 ","End":"03:23.885","Text":"For x between this and this,"},{"Start":"03:23.885 ","End":"03:26.600","Text":"I don\u0027t know, I\u0027ll pick 0.1."},{"Start":"03:26.600 ","End":"03:33.200","Text":"For x less than 0, I\u0027ll pick x equals minus 1."},{"Start":"03:33.200 ","End":"03:37.580","Text":"Let\u0027s plug them in and see what we get for f prime,"},{"Start":"03:37.580 ","End":"03:39.710","Text":"whether we get positive or negative."},{"Start":"03:39.710 ","End":"03:42.710","Text":"F prime is here."},{"Start":"03:42.710 ","End":"03:45.770","Text":"If x is minus 1,"},{"Start":"03:45.770 ","End":"03:48.230","Text":"I think it might be easier to substitute."},{"Start":"03:48.230 ","End":"03:52.585","Text":"If I write this as 2x minus 5,"},{"Start":"03:52.585 ","End":"03:57.689","Text":"if x is minus 1 then 2 plus 5 is 7,"},{"Start":"03:57.689 ","End":"04:00.260","Text":"but that\u0027s positive and 3 is positive,"},{"Start":"04:00.260 ","End":"04:02.420","Text":"but the cube root of minus 1 is negative."},{"Start":"04:02.420 ","End":"04:03.920","Text":"That makes this negative."},{"Start":"04:03.920 ","End":"04:07.250","Text":"If I take some small number like 0.1,"},{"Start":"04:07.250 ","End":"04:11.720","Text":"it\u0027s positive so the cube root of it is positive and 3 is positive,"},{"Start":"04:11.720 ","End":"04:16.415","Text":"and 2 minus 5 times 0.1 is still positive."},{"Start":"04:16.415 ","End":"04:20.030","Text":"We actually get a positive here. It\u0027s a good job."},{"Start":"04:20.030 ","End":"04:26.870","Text":"We put the 0 in the column because around this place where f prime is undefined,"},{"Start":"04:26.870 ","End":"04:28.655","Text":"at any event changes sign."},{"Start":"04:28.655 ","End":"04:31.295","Text":"Now, when we take x equals 1,"},{"Start":"04:31.295 ","End":"04:33.330","Text":"again, 3 is positive,"},{"Start":"04:33.330 ","End":"04:35.295","Text":"cube root of 1 is positive,"},{"Start":"04:35.295 ","End":"04:38.610","Text":"but 2 minus 5 times 1 is negative."},{"Start":"04:38.610 ","End":"04:40.040","Text":"Here it\u0027s negative."},{"Start":"04:40.040 ","End":"04:45.900","Text":"Now we interpret f prime in terms of f. We conclude that about f,"},{"Start":"04:45.900 ","End":"04:47.550","Text":"that this is negative,"},{"Start":"04:47.550 ","End":"04:49.610","Text":"so f is decreasing."},{"Start":"04:49.610 ","End":"04:51.380","Text":"When the derivative is positive,"},{"Start":"04:51.380 ","End":"04:55.820","Text":"the function is increasing and when the derivative is negative again,"},{"Start":"04:55.820 ","End":"04:58.250","Text":"the function is still decreasing,"},{"Start":"04:58.250 ","End":"04:59.845","Text":"which we write like this."},{"Start":"04:59.845 ","End":"05:05.465","Text":"Now here, at 2/5 we definitely have an extremum"},{"Start":"05:05.465 ","End":"05:11.600","Text":"because the derivative is 0 and it\u0027s between an area of increase and decrease."},{"Start":"05:11.600 ","End":"05:15.440","Text":"Here we definitely have an extremum of type maximum,"},{"Start":"05:15.440 ","End":"05:17.000","Text":"which I write as max."},{"Start":"05:17.000 ","End":"05:19.130","Text":"Here I don\u0027t have any extremum,"},{"Start":"05:19.130 ","End":"05:23.095","Text":"but at least I know where the intervals of increase and decrease are."},{"Start":"05:23.095 ","End":"05:28.130","Text":"Just let me finish the table by writing the value of y that belongs here,"},{"Start":"05:28.130 ","End":"05:34.790","Text":"which means we substitute the 2/5 in the original f of x,"},{"Start":"05:34.790 ","End":"05:36.770","Text":"either here or here."},{"Start":"05:36.770 ","End":"05:39.665","Text":"We might need a calculator here."},{"Start":"05:39.665 ","End":"05:42.830","Text":"I\u0027m not going to compute this as an actual number,"},{"Start":"05:42.830 ","End":"05:48.815","Text":"but that gives us all the information we need about extrema and increase and decrease."},{"Start":"05:48.815 ","End":"05:51.155","Text":"What 3 things we have to write?"},{"Start":"05:51.155 ","End":"05:58.920","Text":"Extrema, we only have 1 and that is a maximum point and it occurs at the point 2/5,"},{"Start":"05:58.920 ","End":"06:04.530","Text":"and 3/5 times the square root of 2/5 cubed,"},{"Start":"06:04.530 ","End":"06:08.660","Text":"and whoever has a calculator and wants to compute it, fine."},{"Start":"06:08.660 ","End":"06:12.500","Text":"Then we had the increasing and decreasing intervals."},{"Start":"06:12.500 ","End":"06:19.015","Text":"First the increasing, we look at the orange where it goes arrow upwards so it\u0027s here."},{"Start":"06:19.015 ","End":"06:24.380","Text":"We have the interval of 0 less than x, less than 2/5."},{"Start":"06:24.380 ","End":"06:31.460","Text":"For the decreasing, I have both places where we have orange arrows down."},{"Start":"06:31.460 ","End":"06:41.520","Text":"It\u0027s decreasing for x less than 0 and also for x bigger than 2/5. We\u0027re done."}],"ID":4827},{"Watched":false,"Name":"Exercise 24","Duration":"7m 38s","ChapterTopicVideoID":4827,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4827.jpeg","UploadDate":"2015-07-21T02:33:47.4270000","DurationForVideoObject":"PT7M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.135","Text":"In this exercise, we\u0027re given a function f of x,"},{"Start":"00:03.135 ","End":"00:06.870","Text":"cube root of x squared minus 1 all squared."},{"Start":"00:06.870 ","End":"00:10.740","Text":"What we have to do is find it\u0027s local extrema as"},{"Start":"00:10.740 ","End":"00:14.655","Text":"well as the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.655 ","End":"00:19.110","Text":"This type of exercise we\u0027ve done before and it has a certain standard,"},{"Start":"00:19.110 ","End":"00:22.410","Text":"set of steps that we follow, like cookbook style."},{"Start":"00:22.410 ","End":"00:25.620","Text":"I like to start with just examining the function to"},{"Start":"00:25.620 ","End":"00:29.220","Text":"see if its domain is in any way limited."},{"Start":"00:29.220 ","End":"00:31.740","Text":"I can see that this thing is defined for all x."},{"Start":"00:31.740 ","End":"00:35.190","Text":"If you look closely, there\u0027s no problem in letting x be anything positive,"},{"Start":"00:35.190 ","End":"00:37.230","Text":"negative, so that\u0027s no problem."},{"Start":"00:37.230 ","End":"00:40.665","Text":"The first proper thing we do is to differentiate."},{"Start":"00:40.665 ","End":"00:45.094","Text":"Let\u0027s differentiate and get f prime of x equals,"},{"Start":"00:45.094 ","End":"00:47.630","Text":"using the chain rule, we have something squared,"},{"Start":"00:47.630 ","End":"00:49.895","Text":"so we have twice that something,"},{"Start":"00:49.895 ","End":"00:57.290","Text":"which is twice cube root of x squared minus 1 times the inner derivative."},{"Start":"00:57.290 ","End":"00:59.015","Text":"Well, the inner derivative,"},{"Start":"00:59.015 ","End":"01:01.655","Text":"the cube root of x squared,"},{"Start":"01:01.655 ","End":"01:06.145","Text":"I can rewrite it as x to the power of 2/3."},{"Start":"01:06.145 ","End":"01:09.600","Text":"Then if I differentiate it, that\u0027s this prime,"},{"Start":"01:09.600 ","End":"01:15.915","Text":"then I have 2/3 x to the power of minus 1/3."},{"Start":"01:15.915 ","End":"01:21.425","Text":"The minus 1/3 I can write as 1 over the cube root of x."},{"Start":"01:21.425 ","End":"01:27.530","Text":"We have 2/3 times 1 over the cube root of x. I can just write it like this."},{"Start":"01:27.530 ","End":"01:29.960","Text":"Now I put this here,"},{"Start":"01:29.960 ","End":"01:33.415","Text":"and if I put this here, I\u0027ll get 2."},{"Start":"01:33.415 ","End":"01:37.180","Text":"The over, I can put all over here,"},{"Start":"01:37.180 ","End":"01:41.750","Text":"3 times the cube root of x."},{"Start":"01:41.750 ","End":"01:44.930","Text":"Now one of the things I notice is that the cube root"},{"Start":"01:44.930 ","End":"01:48.710","Text":"of x on the denominator and x cannot be 0."},{"Start":"01:48.710 ","End":"01:50.255","Text":"The cube root of 0 is 0,"},{"Start":"01:50.255 ","End":"01:52.370","Text":"that would give me 0 on the denominator."},{"Start":"01:52.370 ","End":"01:58.505","Text":"For f prime, I must have that x cannot be 0."},{"Start":"01:58.505 ","End":"02:01.140","Text":"For f, it could be 0, but for f prime it can\u0027t be,"},{"Start":"02:01.140 ","End":"02:03.140","Text":"so this is going to be an important point that we\u0027re going to"},{"Start":"02:03.140 ","End":"02:05.525","Text":"put in our table that we do later."},{"Start":"02:05.525 ","End":"02:09.905","Text":"Next thing to do is to set f prime equals 0."},{"Start":"02:09.905 ","End":"02:17.215","Text":"If f prime is equal to 0 then the numerator has to be 0,"},{"Start":"02:17.215 ","End":"02:20.290","Text":"and so you can see that what we have is the 2"},{"Start":"02:20.290 ","End":"02:23.450","Text":"and the 2 on the denominator don\u0027t come into play here."},{"Start":"02:23.450 ","End":"02:25.895","Text":"All I need is that this thing be 0."},{"Start":"02:25.895 ","End":"02:30.530","Text":"In other words, the cube root of x squared should equal 1."},{"Start":"02:30.530 ","End":"02:34.520","Text":"This part here brings us this and that gives us"},{"Start":"02:34.520 ","End":"02:39.620","Text":"immediately that x squared is equal to 1 by cubing both sides."},{"Start":"02:39.620 ","End":"02:45.650","Text":"If x squared is 1 then x is going to equal either plus or minus 1,"},{"Start":"02:45.650 ","End":"02:49.335","Text":"and these are candidates for the extrema."},{"Start":"02:49.335 ","End":"02:53.180","Text":"We also have this interesting point which we should put into our table."},{"Start":"02:53.180 ","End":"02:55.610","Text":"Here\u0027s a blank table with x,"},{"Start":"02:55.610 ","End":"02:58.070","Text":"f prime of x, f of x, and y."},{"Start":"02:58.070 ","End":"03:01.715","Text":"First of all, put in a few values for x, the interesting ones,"},{"Start":"03:01.715 ","End":"03:07.550","Text":"these two suspects for extrema and also this one where f prime is 0,"},{"Start":"03:07.550 ","End":"03:09.455","Text":"so it might change sign."},{"Start":"03:09.455 ","End":"03:11.315","Text":"We put them in an order, of course,"},{"Start":"03:11.315 ","End":"03:14.900","Text":"increasing order and leave a bit of space around each, so let\u0027s see."},{"Start":"03:14.900 ","End":"03:16.510","Text":"We have minus 1,"},{"Start":"03:16.510 ","End":"03:20.520","Text":"then we have 0, and then we have 1."},{"Start":"03:20.520 ","End":"03:23.780","Text":"For these two, we have f prime is 0."},{"Start":"03:23.780 ","End":"03:27.410","Text":"These are our suspects for extrema and"},{"Start":"03:27.410 ","End":"03:31.145","Text":"this one affects possibly the increasing and decreasing intervals."},{"Start":"03:31.145 ","End":"03:32.705","Text":"Now what are these intervals?"},{"Start":"03:32.705 ","End":"03:34.820","Text":"These points divide the line into intervals."},{"Start":"03:34.820 ","End":"03:37.720","Text":"Here we have x less than minus 1,"},{"Start":"03:37.720 ","End":"03:42.170","Text":"here we have that x is between minus 1 and 0,"},{"Start":"03:42.170 ","End":"03:45.980","Text":"here we have that x is between 0 and 1,"},{"Start":"03:45.980 ","End":"03:48.115","Text":"and here x is bigger than 1."},{"Start":"03:48.115 ","End":"03:49.880","Text":"Now with these intervals,"},{"Start":"03:49.880 ","End":"03:51.230","Text":"we choose a sample,"},{"Start":"03:51.230 ","End":"03:53.660","Text":"an arbitrary point from each interval."},{"Start":"03:53.660 ","End":"03:55.400","Text":"For less than minus 1,"},{"Start":"03:55.400 ","End":"03:57.410","Text":"let me choose minus 2,"},{"Start":"03:57.410 ","End":"03:59.030","Text":"between minus 1 and 0,"},{"Start":"03:59.030 ","End":"04:00.905","Text":"I\u0027m going to choose 1/2,"},{"Start":"04:00.905 ","End":"04:04.180","Text":"between 0 and 1, I\u0027ll choose 1/2."},{"Start":"04:04.180 ","End":"04:06.590","Text":"I meant to say here minus a half, of course."},{"Start":"04:06.590 ","End":"04:08.480","Text":"Here for x bigger than 1,"},{"Start":"04:08.480 ","End":"04:10.220","Text":"I\u0027ll choose the value 2."},{"Start":"04:10.220 ","End":"04:12.575","Text":"We have to check for each of these values,"},{"Start":"04:12.575 ","End":"04:17.085","Text":"whether this f prime of x gives us positive or negative."},{"Start":"04:17.085 ","End":"04:18.840","Text":"The numbers don\u0027t make any difference,"},{"Start":"04:18.840 ","End":"04:22.665","Text":"so I\u0027ll just circle where we have to substitute."},{"Start":"04:22.665 ","End":"04:25.775","Text":"Take them one at a time, minus 2."},{"Start":"04:25.775 ","End":"04:28.205","Text":"I have positive over negative,"},{"Start":"04:28.205 ","End":"04:30.695","Text":"which gives me negative."},{"Start":"04:30.695 ","End":"04:33.430","Text":"Now let\u0027s see, minus 1/2,"},{"Start":"04:33.430 ","End":"04:37.670","Text":"negative over negative will give me positive."},{"Start":"04:37.670 ","End":"04:40.020","Text":"Next, x equals 1/2,"},{"Start":"04:40.020 ","End":"04:45.095","Text":"negative over positive will give us negative here."},{"Start":"04:45.095 ","End":"04:47.510","Text":"F prime 0 is undefined."},{"Start":"04:47.510 ","End":"04:48.740","Text":"I don\u0027t know how to indicate that,"},{"Start":"04:48.740 ","End":"04:50.330","Text":"question mark or a cross."},{"Start":"04:50.330 ","End":"04:52.855","Text":"Maybe I\u0027ll use a cross meaning undefined."},{"Start":"04:52.855 ","End":"04:55.655","Text":"Then where x is equal to 2,"},{"Start":"04:55.655 ","End":"04:58.415","Text":"positive over positive is positive."},{"Start":"04:58.415 ","End":"05:03.140","Text":"That means that we can now determine for each of these subintervals,"},{"Start":"05:03.140 ","End":"05:06.005","Text":"whether the function is increasing or decreasing on them."},{"Start":"05:06.005 ","End":"05:10.380","Text":"Minus means derivative, negative means function,"},{"Start":"05:10.380 ","End":"05:13.160","Text":"decreasing indicate with an arrow like this."},{"Start":"05:13.160 ","End":"05:15.799","Text":"The same here, derivative positive,"},{"Start":"05:15.799 ","End":"05:17.120","Text":"so function is increasing,"},{"Start":"05:17.120 ","End":"05:18.260","Text":"I write it like this."},{"Start":"05:18.260 ","End":"05:23.180","Text":"Similarly here, we have decreasing and here we have increasing.."},{"Start":"05:23.180 ","End":"05:30.475","Text":"Anything I\u0027d like more from the table are the values of y for the special suspects,"},{"Start":"05:30.475 ","End":"05:31.740","Text":"minus 1 and the 1."},{"Start":"05:31.740 ","End":"05:36.225","Text":"Remember that y is just the function of x."},{"Start":"05:36.225 ","End":"05:41.640","Text":"When x is minus 1, y is 0."},{"Start":"05:41.640 ","End":"05:43.755","Text":"Let\u0027s see. When x is 1,"},{"Start":"05:43.755 ","End":"05:46.050","Text":"looks like the same thing."},{"Start":"05:46.050 ","End":"05:50.200","Text":"We have all the information we need to answer the questions,"},{"Start":"05:50.200 ","End":"05:51.290","Text":"to draw the conclusions,"},{"Start":"05:51.290 ","End":"05:53.960","Text":"which is the answers to these questions about the extrema,"},{"Start":"05:53.960 ","End":"05:56.135","Text":"the increasing and decreasing."},{"Start":"05:56.135 ","End":"05:59.575","Text":"I\u0027ll just give myself a bit more space here."},{"Start":"05:59.575 ","End":"06:03.450","Text":"What we can do is write the following."},{"Start":"06:04.540 ","End":"06:08.885","Text":"I forgot to write one more important thing in the table,"},{"Start":"06:08.885 ","End":"06:13.369","Text":"and that is that when I see suspect"},{"Start":"06:13.369 ","End":"06:18.035","Text":"for an extremum and it\u0027s between a decreasing and then increasing,"},{"Start":"06:18.035 ","End":"06:20.090","Text":"then it\u0027s definitely an extremum."},{"Start":"06:20.090 ","End":"06:22.715","Text":"Even more, I can say that subtype minimum."},{"Start":"06:22.715 ","End":"06:25.055","Text":"Here I have a minimum point."},{"Start":"06:25.055 ","End":"06:27.800","Text":"Here the derivative is not defined,"},{"Start":"06:27.800 ","End":"06:29.815","Text":"so there\u0027s no extremum."},{"Start":"06:29.815 ","End":"06:32.450","Text":"Here at x equals 1,"},{"Start":"06:32.450 ","End":"06:37.115","Text":"I also have an extremum because I\u0027m between decreasing and increasing,"},{"Start":"06:37.115 ","End":"06:39.410","Text":"but this time it\u0027s subtype maximum,"},{"Start":"06:39.410 ","End":"06:42.260","Text":"which I abbreviate to max."},{"Start":"06:42.260 ","End":"06:48.900","Text":"Now I can say definitely I have two extrema at minus 1 or specifically at minus 1,"},{"Start":"06:48.900 ","End":"06:56.600","Text":"0, I have a minimum and the other one is a maximum and it occurs at 1,0."},{"Start":"06:56.600 ","End":"06:58.805","Text":"Those are the two extrema."},{"Start":"06:58.805 ","End":"07:02.095","Text":"Now I need to know the areas where the function is"},{"Start":"07:02.095 ","End":"07:06.350","Text":"increasing and I do this by looking at the up arrows in orange,"},{"Start":"07:06.350 ","End":"07:08.300","Text":"so I have an up arrow here,"},{"Start":"07:08.300 ","End":"07:11.765","Text":"which gives me minus 1 less than x,"},{"Start":"07:11.765 ","End":"07:19.415","Text":"less than 0 and I also have increasing at x bigger than 1."},{"Start":"07:19.415 ","End":"07:23.300","Text":"The last thing, the interval where the function is decreasing,"},{"Start":"07:23.300 ","End":"07:28.580","Text":"I look for the down arrow and that gives me that x is less than minus"},{"Start":"07:28.580 ","End":"07:35.180","Text":"1 and I also have decreasing when 0 is less than x, less than 1."},{"Start":"07:35.180 ","End":"07:38.490","Text":"That answers all the questions, so we\u0027re done."}],"ID":4828},{"Watched":false,"Name":"Exercise 25","Duration":"8m 13s","ChapterTopicVideoID":4828,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4828.jpeg","UploadDate":"2015-07-21T02:34:30.4770000","DurationForVideoObject":"PT8M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.010","Text":"In this exercise, we\u0027re given a function,"},{"Start":"00:03.010 ","End":"00:07.070","Text":"f of x as cube root of x squared minus 1."},{"Start":"00:07.070 ","End":"00:10.614","Text":"Our task is to find its local extrema,"},{"Start":"00:10.614 ","End":"00:15.505","Text":"as well as the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:15.505 ","End":"00:18.760","Text":"This is 1 of those types of exercises you\u0027ve seen before."},{"Start":"00:18.760 ","End":"00:24.040","Text":"It has a certain standard set of steps that will take us to the answer, cookbook style."},{"Start":"00:24.040 ","End":"00:26.860","Text":"First of all I\u0027ll just make a note about the domain,"},{"Start":"00:26.860 ","End":"00:28.150","Text":"and I see this no problem."},{"Start":"00:28.150 ","End":"00:30.160","Text":"It\u0027s defined for all x."},{"Start":"00:30.160 ","End":"00:31.420","Text":"Because this is a cube root,"},{"Start":"00:31.420 ","End":"00:34.285","Text":"not a square root, so defined everywhere."},{"Start":"00:34.285 ","End":"00:40.060","Text":"The first thing we do is to find the derivative of f prime of x equals."},{"Start":"00:40.060 ","End":"00:45.560","Text":"I like to do this in the scratch area and say that if f of x is this, well,"},{"Start":"00:45.560 ","End":"00:50.990","Text":"I can also rewrite that as x squared minus 1 to the power of 1/3,"},{"Start":"00:50.990 ","End":"00:54.560","Text":"then use exponents so that f prime of x will"},{"Start":"00:54.560 ","End":"00:59.660","Text":"equal 1/3 times this thing to the power of minus 2/3."},{"Start":"00:59.660 ","End":"01:01.685","Text":"This is x squared minus 1,"},{"Start":"01:01.685 ","End":"01:06.505","Text":"and times in the derivative of this x squared minus 1 is 2x."},{"Start":"01:06.505 ","End":"01:12.650","Text":"This is more convenient for me to write instead of the negative exponent as a fraction,"},{"Start":"01:12.650 ","End":"01:16.160","Text":"where on the numerator I have 2x."},{"Start":"01:16.160 ","End":"01:19.960","Text":"On the denominator, I have the 3 from the 1/3,"},{"Start":"01:19.960 ","End":"01:24.105","Text":"and I have x squared minus 1 to the power of 2/3."},{"Start":"01:24.105 ","End":"01:29.269","Text":"The power of 2/3 means I can take the cube root of the things squared."},{"Start":"01:29.269 ","End":"01:33.020","Text":"The cube root of something squared is the power of 2/3."},{"Start":"01:33.020 ","End":"01:36.860","Text":"This is x squared minus 1."},{"Start":"01:36.860 ","End":"01:41.810","Text":"What I see is that it\u0027s possible for the denominator to be 0,"},{"Start":"01:41.810 ","End":"01:43.655","Text":"and that\u0027s what I have to watch out."},{"Start":"01:43.655 ","End":"01:46.264","Text":"Where could the denominator be 0?"},{"Start":"01:46.264 ","End":"01:49.710","Text":"It could be 0 if x squared minus 1 is 0."},{"Start":"01:49.710 ","End":"01:53.870","Text":"That would give me that x is equal to plus or minus 1."},{"Start":"01:53.870 ","End":"01:56.765","Text":"These are 2 points to watch out for."},{"Start":"01:56.765 ","End":"01:59.929","Text":"For these points, f prime is not defined."},{"Start":"01:59.929 ","End":"02:01.865","Text":"It has 0 in the denominator."},{"Start":"02:01.865 ","End":"02:03.980","Text":"Here, if x is plus or minus 1,"},{"Start":"02:03.980 ","End":"02:06.470","Text":"then f prime is undefined."},{"Start":"02:06.470 ","End":"02:08.570","Text":"So f prime of x is"},{"Start":"02:08.570 ","End":"02:13.894","Text":"2x over 3 times the cube root"},{"Start":"02:13.894 ","End":"02:18.905","Text":"of x squared minus 1 squared,"},{"Start":"02:18.905 ","End":"02:28.260","Text":"and I notice that this is not defined for x must not equal minus 1 or 1."},{"Start":"02:28.260 ","End":"02:32.840","Text":"Now the next step in the recipe is to set"},{"Start":"02:32.840 ","End":"02:39.390","Text":"f prime of x to equal 0 and find out which x gives us this."},{"Start":"02:39.970 ","End":"02:42.950","Text":"When something is 0,"},{"Start":"02:42.950 ","End":"02:45.700","Text":"it\u0027s a fraction and the numerator has to be 0."},{"Start":"02:45.700 ","End":"02:52.580","Text":"It immediately follows that x equals 0 and this is our candidate for the extremum."},{"Start":"02:52.580 ","End":"02:54.245","Text":"We only have 1 candidate,"},{"Start":"02:54.245 ","End":"02:56.485","Text":"and that is x equals 0."},{"Start":"02:56.485 ","End":"03:01.790","Text":"Then we have these 2 values where something peculiar is going on with f prime."},{"Start":"03:01.790 ","End":"03:04.910","Text":"Let\u0027s put these values in a table,"},{"Start":"03:04.910 ","End":"03:07.270","Text":"the standard table that we usually draw."},{"Start":"03:07.270 ","End":"03:09.770","Text":"What we put in these special values,"},{"Start":"03:09.770 ","End":"03:17.495","Text":"0 minus 1 and 1 in order of increase and with some space around each 1 is minus 1 here,"},{"Start":"03:17.495 ","End":"03:20.860","Text":"then 0, and then 1."},{"Start":"03:20.860 ","End":"03:24.570","Text":"Now the 0 is where f prime is 0."},{"Start":"03:24.570 ","End":"03:27.690","Text":"This is our suspect for extremum."},{"Start":"03:27.690 ","End":"03:34.430","Text":"The minus 1 and the 1 where f prime is undefined because it\u0027s got 0 in the denominator."},{"Start":"03:34.430 ","End":"03:35.865","Text":"I don\u0027t know how to write undefined."},{"Start":"03:35.865 ","End":"03:38.465","Text":"I\u0027ll just write it as an x, mean forbidden."},{"Start":"03:38.465 ","End":"03:41.480","Text":"The intervals I was mentioning,"},{"Start":"03:41.480 ","End":"03:45.350","Text":"these 3 points divide the line into 4 intervals here,"},{"Start":"03:45.350 ","End":"03:46.430","Text":"here, here, and here."},{"Start":"03:46.430 ","End":"03:48.560","Text":"We\u0027ll just label each 1."},{"Start":"03:48.560 ","End":"03:52.520","Text":"This 1 is x less than minus 1 here,"},{"Start":"03:52.520 ","End":"03:53.960","Text":"minus 1 less than x,"},{"Start":"03:53.960 ","End":"03:55.790","Text":"less than 0, here,"},{"Start":"03:55.790 ","End":"03:58.145","Text":"0 less than x less than 1,"},{"Start":"03:58.145 ","End":"04:01.260","Text":"and here x bigger than 1."},{"Start":"04:01.260 ","End":"04:03.770","Text":"Then according to the recipe,"},{"Start":"04:03.770 ","End":"04:09.935","Text":"we choose a value in each of the intervals arbitrarily, whatever\u0027s convenient."},{"Start":"04:09.935 ","End":"04:12.700","Text":"Less than minus 1, I\u0027ll take minus 2."},{"Start":"04:12.700 ","End":"04:14.490","Text":"Between minus 1 and 0,"},{"Start":"04:14.490 ","End":"04:16.980","Text":"I\u0027ll choose minus 1/2."},{"Start":"04:16.980 ","End":"04:18.510","Text":"Between 0 and 1,"},{"Start":"04:18.510 ","End":"04:22.740","Text":"I\u0027ll choose 1/2, and bigger than 1, I\u0027ll choose 2."},{"Start":"04:22.740 ","End":"04:30.020","Text":"Then what we have to do is to substitute the sample points into f prime of x,"},{"Start":"04:30.020 ","End":"04:31.700","Text":"which is in here."},{"Start":"04:31.700 ","End":"04:35.945","Text":"But I don\u0027t really care about the 2 and the 3 because all I"},{"Start":"04:35.945 ","End":"04:41.640","Text":"really want is not the value but the plus or minus the sine of the answer."},{"Start":"04:41.640 ","End":"04:44.580","Text":"Let\u0027s tackle them 1 at a time."},{"Start":"04:44.580 ","End":"04:46.020","Text":"Minus 2."},{"Start":"04:46.020 ","End":"04:48.215","Text":"We have negative over positive,"},{"Start":"04:48.215 ","End":"04:51.980","Text":"which is negative, and I indicate that with a minus here."},{"Start":"04:51.980 ","End":"04:54.110","Text":"Now let\u0027s take the minus 1/2."},{"Start":"04:54.110 ","End":"04:56.635","Text":"Here we have negative over positive."},{"Start":"04:56.635 ","End":"04:59.665","Text":"Again, we have negative."},{"Start":"04:59.665 ","End":"05:02.820","Text":"Now for x equals 1/2,"},{"Start":"05:02.820 ","End":"05:04.320","Text":"that\u0027s very easy actually."},{"Start":"05:04.320 ","End":"05:08.150","Text":"This needs to also going to be positive because 1/2 and 2 are going to be positive."},{"Start":"05:08.150 ","End":"05:11.270","Text":"The denominator is always positive,"},{"Start":"05:11.270 ","End":"05:13.070","Text":"so we\u0027ll have a plus, plus."},{"Start":"05:13.070 ","End":"05:16.010","Text":"Now what does this tell us about f of x?"},{"Start":"05:16.010 ","End":"05:18.560","Text":"That if its derivative is negative,"},{"Start":"05:18.560 ","End":"05:22.925","Text":"that it\u0027s decreasing, which I write like this, and here too."},{"Start":"05:22.925 ","End":"05:24.710","Text":"When the derivative is positive,"},{"Start":"05:24.710 ","End":"05:26.000","Text":"the function is increasing,"},{"Start":"05:26.000 ","End":"05:29.140","Text":"which I write like this, and here also."},{"Start":"05:29.140 ","End":"05:32.540","Text":"Now 0, which was our suspect for an extremum,"},{"Start":"05:32.540 ","End":"05:34.250","Text":"turns out indeed to be 1."},{"Start":"05:34.250 ","End":"05:39.815","Text":"Because when it\u0027s between an interval of decrease and an interval of increase,"},{"Start":"05:39.815 ","End":"05:41.600","Text":"that\u0027s always a minimum,"},{"Start":"05:41.600 ","End":"05:43.400","Text":"which I write for short as min."},{"Start":"05:43.400 ","End":"05:46.525","Text":"So this is an extremum of type minimum."},{"Start":"05:46.525 ","End":"05:51.230","Text":"I\u0027ll just write the y value for this point. It\u0027s important."},{"Start":"05:51.230 ","End":"05:55.250","Text":"Just substitute in the original function."},{"Start":"05:55.250 ","End":"05:59.900","Text":"This is equal to y, and here\u0027s where I want to put the 0."},{"Start":"05:59.900 ","End":"06:01.850","Text":"If x is 0,"},{"Start":"06:01.850 ","End":"06:04.280","Text":"we have minus 1."},{"Start":"06:04.280 ","End":"06:07.684","Text":"Now I have all the information in the table that I need,"},{"Start":"06:07.684 ","End":"06:11.885","Text":"and it\u0027s time to draw some conclusions."},{"Start":"06:11.885 ","End":"06:14.555","Text":"The conclusions are these extrema,"},{"Start":"06:14.555 ","End":"06:16.595","Text":"the increasing and decreasing."},{"Start":"06:16.595 ","End":"06:18.550","Text":"It\u0027s what we\u0027re asked for."},{"Start":"06:18.550 ","End":"06:20.705","Text":"Where do we have extrema?"},{"Start":"06:20.705 ","End":"06:23.060","Text":"We only have 1, that\u0027s this 1."},{"Start":"06:23.060 ","End":"06:29.555","Text":"In other words, we have a minimum at the point 0 minus 1."},{"Start":"06:29.555 ","End":"06:31.860","Text":"That\u0027s the only extremum I have."},{"Start":"06:31.860 ","End":"06:38.345","Text":"As for increasing, the intervals of increase are where the arrows go up,"},{"Start":"06:38.345 ","End":"06:39.905","Text":"which is this and this."},{"Start":"06:39.905 ","End":"06:42.740","Text":"But I claim that I can combine these"},{"Start":"06:42.740 ","End":"06:46.035","Text":"two because there\u0027s just 1 point in between that\u0027s missing."},{"Start":"06:46.035 ","End":"06:48.420","Text":"The function itself is continuous."},{"Start":"06:48.420 ","End":"06:52.340","Text":"If it\u0027s increasing and then at some point it\u0027s peculiar,"},{"Start":"06:52.340 ","End":"06:53.620","Text":"it doesn\u0027t have a derivative,"},{"Start":"06:53.620 ","End":"06:56.615","Text":"but then it\u0027s still increasing and the function\u0027s continuous."},{"Start":"06:56.615 ","End":"07:03.425","Text":"These 2 intervals could be combined into 1 as 1 single increasing interval."},{"Start":"07:03.425 ","End":"07:06.425","Text":"I\u0027ll write it as x bigger than 1,"},{"Start":"07:06.425 ","End":"07:08.030","Text":"x bigger than 0."},{"Start":"07:08.030 ","End":"07:10.535","Text":"Altogether, it\u0027s x bigger than 0."},{"Start":"07:10.535 ","End":"07:14.135","Text":"From my experience, this is probably a vertical tangent,"},{"Start":"07:14.135 ","End":"07:18.950","Text":"which might look something like here increasing and then for a moment,"},{"Start":"07:18.950 ","End":"07:21.020","Text":"f prime not defined,"},{"Start":"07:21.020 ","End":"07:23.580","Text":"but then continuing to increase."},{"Start":"07:23.580 ","End":"07:26.700","Text":"The increase is overall the interval."},{"Start":"07:26.700 ","End":"07:30.205","Text":"It doesn\u0027t have to write separately here and here, probably."},{"Start":"07:30.205 ","End":"07:33.029","Text":"In any event, we can combine and the function is continuous,"},{"Start":"07:33.029 ","End":"07:35.180","Text":"so it\u0027s made up of 2 increasing intervals."},{"Start":"07:35.180 ","End":"07:37.295","Text":"We can combine them into 1 interval."},{"Start":"07:37.295 ","End":"07:41.080","Text":"The same thing happens for the decreasing interval,"},{"Start":"07:41.080 ","End":"07:44.675","Text":"is that it\u0027s decreasing when the arrows go down,"},{"Start":"07:44.675 ","End":"07:46.295","Text":"and it\u0027s these 2 intervals,"},{"Start":"07:46.295 ","End":"07:48.950","Text":"but the function is continuous everywhere,"},{"Start":"07:48.950 ","End":"07:51.390","Text":"especially at minus 1."},{"Start":"07:51.390 ","End":"07:55.605","Text":"I can write just that x is less than 0."},{"Start":"07:55.605 ","End":"08:00.795","Text":"Here it\u0027s as if I could write these 2 together as 1 interval,"},{"Start":"08:00.795 ","End":"08:05.745","Text":"x less than 0, and these 2 as x bigger than 0."},{"Start":"08:05.745 ","End":"08:09.785","Text":"We\u0027re decreasing and then we have a minimum and then we\u0027re increasing."},{"Start":"08:09.785 ","End":"08:14.310","Text":"That answers the questions, and we\u0027re done."}],"ID":4829},{"Watched":false,"Name":"Exercise 26","Duration":"7m 9s","ChapterTopicVideoID":4829,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4829.jpeg","UploadDate":"2015-07-21T02:35:08.6730000","DurationForVideoObject":"PT7M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.225","Text":"In this exercise we\u0027re given the following function,"},{"Start":"00:03.225 ","End":"00:08.400","Text":"f of x is absolute value of x minus 3 over x minus 2."},{"Start":"00:08.400 ","End":"00:11.025","Text":"We have to find its local extrema"},{"Start":"00:11.025 ","End":"00:15.060","Text":"and intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:15.060 ","End":"00:19.305","Text":"I notice by the way that the domain is"},{"Start":"00:19.305 ","End":"00:24.855","Text":"x not equal to 2 because the denominator would then be 0."},{"Start":"00:24.855 ","End":"00:27.480","Text":"This exercise is solved cookbook style."},{"Start":"00:27.480 ","End":"00:29.309","Text":"There are certain standard steps."},{"Start":"00:29.309 ","End":"00:31.710","Text":"First of all I\u0027m just going to copy it, but you know what,"},{"Start":"00:31.710 ","End":"00:36.300","Text":"instead of just copying it since x minus 3 is piecewise defined,"},{"Start":"00:36.300 ","End":"00:40.700","Text":"I\u0027ll copy it in portions meaning I\u0027ll take"},{"Start":"00:40.700 ","End":"00:45.670","Text":"the part where x is bigger or equal to 3 and where x is less than 3."},{"Start":"00:45.670 ","End":"00:49.565","Text":"Here the absolute value of x minus 3 is itself,"},{"Start":"00:49.565 ","End":"00:53.600","Text":"so it\u0027s just x minus 3 over x minus 2."},{"Start":"00:53.600 ","End":"00:56.450","Text":"But when x is less than 3 this is negated,"},{"Start":"00:56.450 ","End":"01:01.710","Text":"so we have 3 minus x over x minus 2."},{"Start":"01:01.710 ","End":"01:05.690","Text":"The first step in the recipe is to"},{"Start":"01:05.690 ","End":"01:10.970","Text":"differentiate what is f prime of x so we get that f prime of x."},{"Start":"01:10.970 ","End":"01:15.540","Text":"Now this is a little bit tricky because I need an open interval,"},{"Start":"01:15.540 ","End":"01:16.920","Text":"I don\u0027t need endpoints,"},{"Start":"01:16.920 ","End":"01:21.030","Text":"x equals 3 might be problematic so I\u0027m going to put it separately."},{"Start":"01:21.030 ","End":"01:26.975","Text":"I\u0027m going to have a space for x bigger than 3 and a space for x less than 3."},{"Start":"01:26.975 ","End":"01:30.710","Text":"Separately, we\u0027re going to look at what happens when x equals 3,"},{"Start":"01:30.710 ","End":"01:32.565","Text":"it\u0027s a same point."},{"Start":"01:32.565 ","End":"01:38.295","Text":"For x is bigger than 3 we can do this by the quotient rule,"},{"Start":"01:38.295 ","End":"01:39.555","Text":"I won\u0027t rewrite it,"},{"Start":"01:39.555 ","End":"01:42.000","Text":"I\u0027ll just tell you the quotient for this,"},{"Start":"01:42.000 ","End":"01:43.350","Text":"I\u0027ll just write it at the side,"},{"Start":"01:43.350 ","End":"01:46.575","Text":"x minus 3 over x minus 2."},{"Start":"01:46.575 ","End":"01:52.220","Text":"Derivative is derivative of the numerator which is 1 times the denominator"},{"Start":"01:52.220 ","End":"01:59.370","Text":"minus the numerator times derivative of denominator over denominator squared."},{"Start":"01:59.370 ","End":"02:00.795","Text":"If you simplify this,"},{"Start":"02:00.795 ","End":"02:07.710","Text":"x minus 2 less x minus 3 it\u0027s just 1 over x minus 2 squared which I can put in here,"},{"Start":"02:07.710 ","End":"02:11.340","Text":"1 over x minus 2 squared."},{"Start":"02:11.340 ","End":"02:13.430","Text":"That\u0027s f prime for x bigger than 3."},{"Start":"02:13.430 ","End":"02:16.525","Text":"For x less than 3, we just change sign so,"},{"Start":"02:16.525 ","End":"02:17.880","Text":"don\u0027t need to re-compute it,"},{"Start":"02:17.880 ","End":"02:21.920","Text":"just put a minus 1 over the x minus 2 squared."},{"Start":"02:21.920 ","End":"02:24.290","Text":"Now for x equals 3 it\u0027s more tricky."},{"Start":"02:24.290 ","End":"02:27.590","Text":"We\u0027ll only have a limit if the limit of this from"},{"Start":"02:27.590 ","End":"02:31.325","Text":"the limit of f prime from the left and from the right are equal,"},{"Start":"02:31.325 ","End":"02:34.834","Text":"but they\u0027re not because the limit from the right,"},{"Start":"02:34.834 ","End":"02:40.800","Text":"if x goes to 3 from the right we have to take this is 1 over x minus 2 squared."},{"Start":"02:40.800 ","End":"02:45.345","Text":"If we put x equals 3 get 1 over 3 minus 2 squared which is 1."},{"Start":"02:45.345 ","End":"02:50.209","Text":"On the other hand if I take the limit as x goes to 3 from the left,"},{"Start":"02:50.209 ","End":"02:54.485","Text":"and this time it\u0027s minus 1 over x minus 2 squared,"},{"Start":"02:54.485 ","End":"02:57.740","Text":"we get minus 1 and these 2 are not equal."},{"Start":"02:57.740 ","End":"03:00.410","Text":"If they\u0027re not equal then there is no limit."},{"Start":"03:00.410 ","End":"03:03.375","Text":"There is no limit I\u0027ll just write undefined."},{"Start":"03:03.375 ","End":"03:08.295","Text":"That\u0027s 1 of those unusual cases that the function is defined for"},{"Start":"03:08.295 ","End":"03:13.485","Text":"x equals 3 but the derivative is undefined."},{"Start":"03:13.485 ","End":"03:17.075","Text":"That makes it also a suspect for an extremum."},{"Start":"03:17.075 ","End":"03:22.235","Text":"The next step in the cookbook is to set f prime equals 0."},{"Start":"03:22.235 ","End":"03:26.930","Text":"If I set f prime of x equals 0, what do I get?"},{"Start":"03:26.930 ","End":"03:32.630","Text":"I get that either this has got to be 0 or this has got to be 0 because this is undefined."},{"Start":"03:32.630 ","End":"03:34.220","Text":"But just by looking at it,"},{"Start":"03:34.220 ","End":"03:36.635","Text":"we can see that this is never 0."},{"Start":"03:36.635 ","End":"03:40.970","Text":"We get no such x or I can just write never,"},{"Start":"03:40.970 ","End":"03:42.755","Text":"f prime is never 0."},{"Start":"03:42.755 ","End":"03:44.870","Text":"It\u0027s undefined when x equals 3,"},{"Start":"03:44.870 ","End":"03:46.460","Text":"and that makes it a suspect."},{"Start":"03:46.460 ","End":"03:49.775","Text":"But we don\u0027t get any new suspects from assigning to 0."},{"Start":"03:49.775 ","End":"03:52.640","Text":"Let\u0027s put all the stuff so far in a table."},{"Start":"03:52.640 ","End":"03:55.165","Text":"We\u0027re going to put in some special points,"},{"Start":"03:55.165 ","End":"03:58.455","Text":"and in order 2 and 3 are special points,"},{"Start":"03:58.455 ","End":"04:03.905","Text":"2 was special because the function\u0027s not defined at that point."},{"Start":"04:03.905 ","End":"04:08.300","Text":"That means I denote that by dotted lines,"},{"Start":"04:08.300 ","End":"04:13.385","Text":"just not defined but it could be a borderline between increasing and decreasing."},{"Start":"04:13.385 ","End":"04:19.420","Text":"Then x equals 3 is special because f prime is undefined here."},{"Start":"04:19.420 ","End":"04:23.090","Text":"Let\u0027s write it as x, x here I\u0027ll mean undefined."},{"Start":"04:23.090 ","End":"04:26.045","Text":"When the function is defined but the"},{"Start":"04:26.045 ","End":"04:29.770","Text":"derivative is undefined then it could be an extremum."},{"Start":"04:29.770 ","End":"04:32.640","Text":"Now let\u0027s divide the intervals,"},{"Start":"04:32.640 ","End":"04:36.785","Text":"these 2 and 3 cut the x-axis into 3 intervals,"},{"Start":"04:36.785 ","End":"04:38.900","Text":"and they are as follows,"},{"Start":"04:38.900 ","End":"04:41.090","Text":"x less than 2,"},{"Start":"04:41.090 ","End":"04:45.275","Text":"x between 2 and 3 and x bigger than 3."},{"Start":"04:45.275 ","End":"04:48.440","Text":"The procedure is that we choose a number from"},{"Start":"04:48.440 ","End":"04:53.090","Text":"each interval arbitrary I\u0027ll choose x equals 1 here."},{"Start":"04:53.090 ","End":"04:56.240","Text":"I\u0027ll choose x equals 2 and a half here,"},{"Start":"04:56.240 ","End":"04:58.655","Text":"and I\u0027ll choose x equals 4 here."},{"Start":"04:58.655 ","End":"05:06.700","Text":"Then we have to plug these values into f prime and this is f prime."},{"Start":"05:06.700 ","End":"05:10.260","Text":"If I put in 1 into f prime,"},{"Start":"05:10.260 ","End":"05:12.470","Text":"1 belongs to this range."},{"Start":"05:12.470 ","End":"05:16.330","Text":"It comes out to be minus 1 but I don\u0027t need the actual value,"},{"Start":"05:16.330 ","End":"05:17.875","Text":"I only needed sign."},{"Start":"05:17.875 ","End":"05:22.220","Text":"This is negative and if it\u0027s negative that"},{"Start":"05:22.220 ","End":"05:24.500","Text":"means that the function is decreasing because"},{"Start":"05:24.500 ","End":"05:27.115","Text":"when derivative\u0027s negative function\u0027s decreasing."},{"Start":"05:27.115 ","End":"05:34.580","Text":"Next let\u0027s try 2 and 1/2 so we also get a minus here."},{"Start":"05:34.580 ","End":"05:36.300","Text":"But when x is 4,"},{"Start":"05:36.300 ","End":"05:40.820","Text":"we have to read off the top line because 4 is bigger than 3 so here we have a plus,"},{"Start":"05:40.820 ","End":"05:43.430","Text":"which means that we are also decreasing here."},{"Start":"05:43.430 ","End":"05:49.115","Text":"This undefined probably means a vertical asymptote but I don\u0027t know, I\u0027m just guessing."},{"Start":"05:49.115 ","End":"05:51.320","Text":"Here we are increasing,"},{"Start":"05:51.320 ","End":"05:53.270","Text":"so we\u0027ve got something,"},{"Start":"05:53.270 ","End":"05:56.015","Text":"we have an extremum because this suspect for"},{"Start":"05:56.015 ","End":"05:59.510","Text":"an extremum is between a decreasing and an increasing,"},{"Start":"05:59.510 ","End":"06:03.095","Text":"and that makes an extremum of type minimum."},{"Start":"06:03.095 ","End":"06:07.670","Text":"I\u0027d like to see what the y value is so I know where this minimum exactly occurs,"},{"Start":"06:07.670 ","End":"06:12.035","Text":"and y is equal to f of x is y."},{"Start":"06:12.035 ","End":"06:17.735","Text":"If I put in the value 3 then we get 0."},{"Start":"06:17.735 ","End":"06:21.230","Text":"We have all the information we need now for answering the questions,"},{"Start":"06:21.230 ","End":"06:25.995","Text":"this is the conclusion step of the recipe. Let\u0027s answer."},{"Start":"06:25.995 ","End":"06:27.200","Text":"We had 3 questions,"},{"Start":"06:27.200 ","End":"06:29.840","Text":"extremum, increasing and decreasing."},{"Start":"06:29.840 ","End":"06:31.970","Text":"Extrema, we have 1,"},{"Start":"06:31.970 ","End":"06:37.950","Text":"it\u0027s of type minimum and it occurs at the 0.3, 0."},{"Start":"06:38.000 ","End":"06:43.355","Text":"Intervals where x is increasing I have to look at the up arrow,"},{"Start":"06:43.355 ","End":"06:47.255","Text":"and I see that this belongs to x bigger than 3."},{"Start":"06:47.255 ","End":"06:52.055","Text":"The places where function is decreasing when the arrows are going down,"},{"Start":"06:52.055 ","End":"07:00.305","Text":"which means that x is less than 2 and it\u0027s also decreasing when x is between 2 and 3."},{"Start":"07:00.305 ","End":"07:04.940","Text":"I can\u0027t combine these because I have a gap in the middle which is the 2."},{"Start":"07:04.940 ","End":"07:10.290","Text":"The function\u0027s not defined so I leave it like that and we are done."}],"ID":4830},{"Watched":false,"Name":"Exercise 27","Duration":"6m 30s","ChapterTopicVideoID":4830,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4830.jpeg","UploadDate":"2015-07-21T02:35:42.4800000","DurationForVideoObject":"PT6M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"In this exercise, we\u0027re given a function of x,"},{"Start":"00:03.075 ","End":"00:06.405","Text":"x minus twice arctangent of x,"},{"Start":"00:06.405 ","End":"00:09.765","Text":"and we have to find its local extrema,"},{"Start":"00:09.765 ","End":"00:14.335","Text":"and also the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:14.335 ","End":"00:19.490","Text":"This is a familiar question which has a standard set of steps,"},{"Start":"00:19.490 ","End":"00:22.420","Text":"cookbook style that\u0027ll take us to the solution."},{"Start":"00:22.420 ","End":"00:23.840","Text":"But first, I give a glance,"},{"Start":"00:23.840 ","End":"00:27.230","Text":"see what\u0027s happening with the domain and I see there\u0027s no problems there."},{"Start":"00:27.230 ","End":"00:29.120","Text":"It\u0027s defined for all x."},{"Start":"00:29.120 ","End":"00:36.305","Text":"Next thing we do is differentiate the function f prime of x equals derivative of x is"},{"Start":"00:36.305 ","End":"00:45.120","Text":"1 and the derivative of the arctangent 2 from here is just 1 over 1 plus x squared."},{"Start":"00:45.120 ","End":"00:48.480","Text":"This is what I get for f prime."},{"Start":"00:48.480 ","End":"00:53.840","Text":"The next step is to set this equal to 0 and solve for x."},{"Start":"00:53.840 ","End":"01:01.975","Text":"If this is 0, then I can say that 1 equals 2 over 1 over x squared."},{"Start":"01:01.975 ","End":"01:07.275","Text":"That means that 1 plus x squared equals 2,"},{"Start":"01:07.275 ","End":"01:12.120","Text":"so 1 plus x squared equals 2."},{"Start":"01:12.120 ","End":"01:16.500","Text":"That means that x squared is 1."},{"Start":"01:16.500 ","End":"01:21.995","Text":"So x is equal to either minus 1 or 1."},{"Start":"01:21.995 ","End":"01:27.630","Text":"These are 2 points which are both suspects to be extrema,"},{"Start":"01:27.630 ","End":"01:29.315","Text":"so we have 2 suspects."},{"Start":"01:29.315 ","End":"01:33.125","Text":"Next, we draw our famous table."},{"Start":"01:33.125 ","End":"01:35.760","Text":"Here it is with rows x,"},{"Start":"01:35.760 ","End":"01:38.910","Text":"f prime of x, f of x, and y."},{"Start":"01:38.910 ","End":"01:47.065","Text":"In it, we put these values of x where f prime is 0 and in order minus 1 and 1."},{"Start":"01:47.065 ","End":"01:50.750","Text":"I left spaces because we\u0027re going to write some intervals here."},{"Start":"01:50.750 ","End":"01:54.560","Text":"I notice also that for these 2 values of x,"},{"Start":"01:54.560 ","End":"01:56.000","Text":"f prime is 0."},{"Start":"01:56.000 ","End":"01:59.885","Text":"I mean, that\u0027s how I found them by setting f prime to 0."},{"Start":"01:59.885 ","End":"02:06.560","Text":"The intervals, this here represents the interval where x is less than minus 1."},{"Start":"02:06.560 ","End":"02:10.715","Text":"Here I have x between minus 1 and 1,"},{"Start":"02:10.715 ","End":"02:13.610","Text":"and here I have x greater than 1."},{"Start":"02:13.610 ","End":"02:19.205","Text":"For each interval, I choose a sample point, just arbitrary."},{"Start":"02:19.205 ","End":"02:22.065","Text":"I\u0027ll take minus 2 here,"},{"Start":"02:22.065 ","End":"02:24.015","Text":"I\u0027ll take 0 here,"},{"Start":"02:24.015 ","End":"02:25.650","Text":"and I\u0027ll take 2 here."},{"Start":"02:25.650 ","End":"02:31.490","Text":"What we\u0027re going to do is substitute the sample points into the derivative,"},{"Start":"02:31.490 ","End":"02:34.175","Text":"which is this function here."},{"Start":"02:34.175 ","End":"02:36.260","Text":"But we\u0027re not going to take the value,"},{"Start":"02:36.260 ","End":"02:38.974","Text":"only the sign, whether it\u0027s plus or minus."},{"Start":"02:38.974 ","End":"02:42.440","Text":"It turns out that no matter which sample points you choose,"},{"Start":"02:42.440 ","End":"02:43.910","Text":"you may get different values,"},{"Start":"02:43.910 ","End":"02:45.760","Text":"but you\u0027ll get the same plus or minus,"},{"Start":"02:45.760 ","End":"02:48.335","Text":"so it doesn\u0027t matter which sample points you take."},{"Start":"02:48.335 ","End":"02:53.295","Text":"Okay, let\u0027s try x is 0, it\u0027s negative."},{"Start":"02:53.295 ","End":"02:55.910","Text":"Here I have decreasing."},{"Start":"02:55.910 ","End":"03:01.535","Text":"Let\u0027s try x equals 2 is positive."},{"Start":"03:01.535 ","End":"03:04.640","Text":"If I try minus 2,"},{"Start":"03:04.640 ","End":"03:08.240","Text":"I will get the same thing because x squared is the same for"},{"Start":"03:08.240 ","End":"03:12.325","Text":"2 and minus 2 so that will also be positive."},{"Start":"03:12.325 ","End":"03:15.425","Text":"When f prime is positive,"},{"Start":"03:15.425 ","End":"03:18.080","Text":"it means that the function is increasing."},{"Start":"03:18.080 ","End":"03:20.600","Text":"I\u0027ll write that with an arrow like this."},{"Start":"03:20.600 ","End":"03:22.520","Text":"For f prime negative,"},{"Start":"03:22.520 ","End":"03:24.140","Text":"the function is decreasing."},{"Start":"03:24.140 ","End":"03:26.930","Text":"I indicate like this and again positive,"},{"Start":"03:26.930 ","End":"03:28.280","Text":"so it\u0027s like this."},{"Start":"03:28.280 ","End":"03:32.510","Text":"This is good because that means that both our suspects,"},{"Start":"03:32.510 ","End":"03:37.245","Text":"are indeed extrema because they\u0027re between an increasing and a decreasing area."},{"Start":"03:37.245 ","End":"03:40.655","Text":"Now here if we first have the increasing and then the decreasing,"},{"Start":"03:40.655 ","End":"03:44.600","Text":"that means it\u0027s a maximum and I abbreviate to Max, and here,"},{"Start":"03:44.600 ","End":"03:46.730","Text":"when I\u0027m decreasing and then increasing,"},{"Start":"03:46.730 ","End":"03:49.955","Text":"it\u0027s a minimum which I abbreviate as Min."},{"Start":"03:49.955 ","End":"03:53.720","Text":"I\u0027d like to know what the maximum point is with this y also."},{"Start":"03:53.720 ","End":"03:56.285","Text":"This thing here is also y."},{"Start":"03:56.285 ","End":"03:59.390","Text":"If x is minus 1,"},{"Start":"03:59.390 ","End":"04:04.295","Text":"then I\u0027m going to get that f of x is equal to"},{"Start":"04:04.295 ","End":"04:11.970","Text":"minus 1 minus twice arctangent of minus 1."},{"Start":"04:11.970 ","End":"04:20.295","Text":"Let\u0027s see, the arctangent of minus 1 is going to be minus Pi over 4."},{"Start":"04:20.295 ","End":"04:26.249","Text":"This equals minus 1 minus twice arctangent"},{"Start":"04:26.249 ","End":"04:32.055","Text":"of minus 1 is minus Pi over 4."},{"Start":"04:32.055 ","End":"04:33.860","Text":"What do we have altogether?"},{"Start":"04:33.860 ","End":"04:36.685","Text":"Minus 1 minus Pi over 2."},{"Start":"04:36.685 ","End":"04:38.660","Text":"Oh, and I can write that in here,"},{"Start":"04:38.660 ","End":"04:43.205","Text":"minus 1 minus Pi over 2."},{"Start":"04:43.205 ","End":"04:45.425","Text":"For x equals 1,"},{"Start":"04:45.425 ","End":"04:52.570","Text":"I have 1 minus twice arctangent of 1."},{"Start":"04:55.760 ","End":"04:58.670","Text":"Oh, I see that here I made a mistake,"},{"Start":"04:58.670 ","End":"05:00.874","Text":"minus and minus is a plus."},{"Start":"05:00.874 ","End":"05:05.320","Text":"Here I have minus twice Pi over 4,"},{"Start":"05:05.320 ","End":"05:10.820","Text":"and this is 1 minus Pi over 2."},{"Start":"05:10.820 ","End":"05:13.510","Text":"This is, I can write by the 1."},{"Start":"05:13.510 ","End":"05:15.590","Text":"This plus which I fixed here,"},{"Start":"05:15.590 ","End":"05:17.060","Text":"I also fix it here."},{"Start":"05:17.060 ","End":"05:19.535","Text":"Now we come to the final step,"},{"Start":"05:19.535 ","End":"05:22.055","Text":"which is basically just to draw conclusions,"},{"Start":"05:22.055 ","End":"05:25.070","Text":"which is just to answer the question to give"},{"Start":"05:25.070 ","End":"05:29.575","Text":"the local extrema and the increasing and decreasing intervals."},{"Start":"05:29.575 ","End":"05:32.675","Text":"I\u0027ll write the conclusions."},{"Start":"05:32.675 ","End":"05:36.260","Text":"First of all, we were asked about the extrema."},{"Start":"05:36.260 ","End":"05:39.005","Text":"The extrema, we have 2 of them."},{"Start":"05:39.005 ","End":"05:41.760","Text":"We have a maximum."},{"Start":"05:41.760 ","End":"05:50.765","Text":"This occurs at the point where x is minus 1 and y is minus 1 plus Pi over 2."},{"Start":"05:50.765 ","End":"05:59.765","Text":"We also have a minimum and that happens at the point 1 and 1 minus Pi over 2."},{"Start":"05:59.765 ","End":"06:05.700","Text":"Then we have to write the increasing intervals or where the function is increasing."},{"Start":"06:05.700 ","End":"06:08.670","Text":"We look for the orange up arrow."},{"Start":"06:08.670 ","End":"06:16.145","Text":"That occurs when x is less than minus 1 and also increasing when x bigger than 1,"},{"Start":"06:16.145 ","End":"06:21.740","Text":"whereas the decreasing intervals occur with the down arrow,"},{"Start":"06:21.740 ","End":"06:26.810","Text":"which is where x is between minus 1 and 1."},{"Start":"06:26.810 ","End":"06:30.450","Text":"That answers the questions and there, we\u0027re done."}],"ID":4831},{"Watched":false,"Name":"Exercise 28","Duration":"7m 41s","ChapterTopicVideoID":4831,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4831.jpeg","UploadDate":"2015-07-21T02:36:23.2630000","DurationForVideoObject":"PT7M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we\u0027re given a trigonometric function f of x,"},{"Start":"00:04.350 ","End":"00:08.385","Text":"which is 8 cosine x plus 2 cosine 2x minus 3,"},{"Start":"00:08.385 ","End":"00:12.720","Text":"and we restrict it to the domain from 0-2 Pi,"},{"Start":"00:12.720 ","End":"00:14.700","Text":"the closed interval in degrees,"},{"Start":"00:14.700 ","End":"00:17.910","Text":"if you wanted it 0 degrees and 360 degrees."},{"Start":"00:17.910 ","End":"00:21.030","Text":"We have to find the local extrema and"},{"Start":"00:21.030 ","End":"00:25.380","Text":"also the intervals where the function is increasing and where the function is decreasing."},{"Start":"00:25.380 ","End":"00:28.530","Text":"This is a familiar style type of exercise,"},{"Start":"00:28.530 ","End":"00:30.960","Text":"and we know how to solve it in 4 stages."},{"Start":"00:30.960 ","End":"00:33.929","Text":"The first stage is a preparation stage,"},{"Start":"00:33.929 ","End":"00:36.270","Text":"and we just have to find f prime of x."},{"Start":"00:36.270 ","End":"00:37.510","Text":"Let\u0027s get started."},{"Start":"00:37.510 ","End":"00:42.380","Text":"F prime of x is equal to derivative of cosine is minus sine."},{"Start":"00:42.380 ","End":"00:51.234","Text":"We get here minus 8 sine x plus twice the derivative of cosine is minus sine."},{"Start":"00:51.234 ","End":"00:57.225","Text":"We have minus sine 2 x times inner derivative which is 2,"},{"Start":"00:57.225 ","End":"00:59.670","Text":"and the minus 3 just disappears."},{"Start":"00:59.670 ","End":"01:02.225","Text":"If I just slightly rewrite this,"},{"Start":"01:02.225 ","End":"01:11.060","Text":"I get that this is minus 8 sine x minus 4 sine 2 x,"},{"Start":"01:11.060 ","End":"01:18.065","Text":"and now I\u0027d like to expand it further by using the trigonometrical identity is that"},{"Start":"01:18.065 ","End":"01:25.905","Text":"the sine of 2 Alpha is 2 sine Alpha cosine Alpha."},{"Start":"01:25.905 ","End":"01:27.800","Text":"If I apply that here,"},{"Start":"01:27.800 ","End":"01:33.544","Text":"then I will get that this is equal to minus 8 sine x minus,"},{"Start":"01:33.544 ","End":"01:41.130","Text":"the 2 goes with the 4 to give 8 and we have sine x times cosine x."},{"Start":"01:41.130 ","End":"01:43.965","Text":"That\u0027s the first stage."},{"Start":"01:43.965 ","End":"01:50.179","Text":"The next stage is to solve an equation f prime of x equals 0."},{"Start":"01:50.179 ","End":"01:56.195","Text":"The reason we want to solve this equation is that its solutions are suspects for extrema."},{"Start":"01:56.195 ","End":"01:58.625","Text":"If we let f prime of x equals 0,"},{"Start":"01:58.625 ","End":"02:00.335","Text":"it means that this is 0."},{"Start":"02:00.335 ","End":"02:02.300","Text":"But let me get rid of the minus 8."},{"Start":"02:02.300 ","End":"02:05.585","Text":"I\u0027ll divide by minus 8, because if this is 0,"},{"Start":"02:05.585 ","End":"02:13.550","Text":"then it means that sine x plus sine x cosine x is also equal to 0,"},{"Start":"02:13.550 ","End":"02:17.570","Text":"and if I take sine x outside the brackets,"},{"Start":"02:17.570 ","End":"02:24.555","Text":"I get that sine x times 1 plus cosine x is equal to 0."},{"Start":"02:24.555 ","End":"02:26.595","Text":"When a product is 0,"},{"Start":"02:26.595 ","End":"02:29.360","Text":"it means that 1 of the 2 factors is 0."},{"Start":"02:29.360 ","End":"02:37.909","Text":"So either sine x equals 0 or 1 plus cosine x is equal to 0,"},{"Start":"02:37.909 ","End":"02:42.360","Text":"which means that cosine x is minus 1."},{"Start":"02:42.360 ","End":"02:46.720","Text":"Now, remember the solutions are between 0 and 2 Pi,"},{"Start":"02:46.720 ","End":"02:49.900","Text":"or if you like 0 and 360 degrees."},{"Start":"02:49.900 ","End":"02:54.320","Text":"There\u0027s actually 3 places where the sine function is 0."},{"Start":"02:54.320 ","End":"02:56.930","Text":"If we do it in degrees,"},{"Start":"02:56.930 ","End":"02:59.930","Text":"then sine of 0 degrees is 0,"},{"Start":"02:59.930 ","End":"03:02.540","Text":"sine of 180 degrees is 0,"},{"Start":"03:02.540 ","End":"03:05.360","Text":"and sine of 360 degrees is 0."},{"Start":"03:05.360 ","End":"03:11.455","Text":"But the cosine is only equal to minus 1 at 1 place and that\u0027s at 180 degrees,"},{"Start":"03:11.455 ","End":"03:14.675","Text":"and of course I should write that in radians too."},{"Start":"03:14.675 ","End":"03:17.525","Text":"This is 0, this is Pi,"},{"Start":"03:17.525 ","End":"03:19.250","Text":"and this is 2 Pi,"},{"Start":"03:19.250 ","End":"03:21.095","Text":"and this is just Pi."},{"Start":"03:21.095 ","End":"03:23.420","Text":"Basically, the solutions that we get,"},{"Start":"03:23.420 ","End":"03:25.025","Text":"if I write them in radians,"},{"Start":"03:25.025 ","End":"03:29.915","Text":"there are 3 possible ones because this 1 is repeated is we have either 0"},{"Start":"03:29.915 ","End":"03:35.045","Text":"or Pi or 2 Pi and those are the solutions to this."},{"Start":"03:35.045 ","End":"03:40.130","Text":"That means that we\u0027re done with this stage and the next stage is a table."},{"Start":"03:40.130 ","End":"03:42.290","Text":"We first put in some values for x."},{"Start":"03:42.290 ","End":"03:44.105","Text":"These will be our suspects,"},{"Start":"03:44.105 ","End":"03:46.360","Text":"0 Pi and 2 Pi,"},{"Start":"03:46.360 ","End":"03:51.005","Text":"0 and 2 Pi are actually the endpoints of the interval, the domain."},{"Start":"03:51.005 ","End":"03:53.855","Text":"We only actually get 2 intervals, not 4."},{"Start":"03:53.855 ","End":"03:56.855","Text":"We don\u0027t get anything beyond 2 Pi or beyond 0."},{"Start":"03:56.855 ","End":"04:05.075","Text":"We have basically 2 defined intervals and that is 0 less than or equal to x less than Pi,"},{"Start":"04:05.075 ","End":"04:06.505","Text":"and on the other side,"},{"Start":"04:06.505 ","End":"04:08.820","Text":"Pi less than x,"},{"Start":"04:08.820 ","End":"04:11.625","Text":"less than or equal to 2 Pi."},{"Start":"04:11.625 ","End":"04:17.000","Text":"As usual, we take a representative from each and from here I\u0027ll take"},{"Start":"04:17.000 ","End":"04:22.815","Text":"Pi over 2 and from here I\u0027ll take 3 Pi over 2,"},{"Start":"04:22.815 ","End":"04:24.870","Text":"which if you wanted in degrees,"},{"Start":"04:24.870 ","End":"04:32.190","Text":"this is 90 degrees and this is 270 degrees,"},{"Start":"04:32.190 ","End":"04:33.410","Text":"and as we already said before,"},{"Start":"04:33.410 ","End":"04:36.830","Text":"this is 360, 180, and 0."},{"Start":"04:36.830 ","End":"04:43.280","Text":"I also should note that the value of f prime at these points is 0,"},{"Start":"04:43.280 ","End":"04:48.035","Text":"which makes these 3 points suspect for an extrema."},{"Start":"04:48.035 ","End":"04:54.170","Text":"Now what we have to do with these sample points is to substitute them in f prime,"},{"Start":"04:54.170 ","End":"04:58.065","Text":"and this is it after we simplified it."},{"Start":"04:58.065 ","End":"05:02.100","Text":"Let\u0027s see. If we put in Pi over 2."},{"Start":"05:02.100 ","End":"05:04.200","Text":"We get minus 8,"},{"Start":"05:04.200 ","End":"05:06.140","Text":"but we don\u0027t want the actual value,"},{"Start":"05:06.140 ","End":"05:08.000","Text":"we only want the sign,"},{"Start":"05:08.000 ","End":"05:09.770","Text":"and it is negative."},{"Start":"05:09.770 ","End":"05:14.315","Text":"Then if we put in 3 Pi over 2, that\u0027s 270 degrees."},{"Start":"05:14.315 ","End":"05:17.450","Text":"Then 270 is like minus 90."},{"Start":"05:17.450 ","End":"05:20.810","Text":"We get plus 8 here and 0 here,"},{"Start":"05:20.810 ","End":"05:23.630","Text":"and all I need to know is that this is a plus,"},{"Start":"05:23.630 ","End":"05:28.280","Text":"because that means that in this interval the function is decreasing."},{"Start":"05:28.280 ","End":"05:32.660","Text":"Its derivative is negative and if the derivative is positive, it\u0027s increasing."},{"Start":"05:32.660 ","End":"05:36.050","Text":"We can straight away say that this is"},{"Start":"05:36.050 ","End":"05:41.210","Text":"a minimum and extremum of type minimum because it\u0027s between decreasing and increasing."},{"Start":"05:41.210 ","End":"05:48.725","Text":"But, and here\u0027s something exceptional when we have the end of a closed interval."},{"Start":"05:48.725 ","End":"05:50.450","Text":"We don\u0027t have anything on the left,"},{"Start":"05:50.450 ","End":"05:52.160","Text":"we only have a right-hand side."},{"Start":"05:52.160 ","End":"05:54.800","Text":"It\u0027s enough for this to be decreasing,"},{"Start":"05:54.800 ","End":"05:56.650","Text":"for this to be a maximum."},{"Start":"05:56.650 ","End":"05:59.030","Text":"Similarly, when we\u0027re on the right end point,"},{"Start":"05:59.030 ","End":"06:00.709","Text":"there\u0027s nowhere to go beyond."},{"Start":"06:00.709 ","End":"06:03.275","Text":"But if I\u0027m the left of the end point,"},{"Start":"06:03.275 ","End":"06:04.670","Text":"when we go to the endpoint,"},{"Start":"06:04.670 ","End":"06:06.590","Text":"we\u0027re increasing, then this is also"},{"Start":"06:06.590 ","End":"06:10.190","Text":"a maximum because we go up to it and then there\u0027s nowhere else to go."},{"Start":"06:10.190 ","End":"06:12.035","Text":"That\u0027s how it works here."},{"Start":"06:12.035 ","End":"06:16.145","Text":"Minimum in the usual way and maximum exceptionally for endpoints."},{"Start":"06:16.145 ","End":"06:18.830","Text":"Now, all I need more from this table,"},{"Start":"06:18.830 ","End":"06:22.505","Text":"are the values of y at these 3 points,"},{"Start":"06:22.505 ","End":"06:25.730","Text":"which means substituting in the original function."},{"Start":"06:25.730 ","End":"06:28.585","Text":"If I plug in x equals 0,"},{"Start":"06:28.585 ","End":"06:31.600","Text":"cosine of 0 is 1,"},{"Start":"06:31.600 ","End":"06:36.570","Text":"and so I get 8 plus 2 minus 3 is 7."},{"Start":"06:36.570 ","End":"06:39.520","Text":"I\u0027ll spare you the computations for the rest."},{"Start":"06:39.520 ","End":"06:42.845","Text":"This is minus 9 and this is again 7."},{"Start":"06:42.845 ","End":"06:47.380","Text":"Now we have all we need for the last stage,"},{"Start":"06:47.380 ","End":"06:49.855","Text":"which is the conclusion stage."},{"Start":"06:49.855 ","End":"06:52.735","Text":"The conclusion is just answering the questions."},{"Start":"06:52.735 ","End":"06:55.585","Text":"They were 3 things, there was extrema,"},{"Start":"06:55.585 ","End":"06:59.635","Text":"areas intervals of increase and intervals of decrease."},{"Start":"06:59.635 ","End":"07:02.665","Text":"First extrema, and then we have 3 of them."},{"Start":"07:02.665 ","End":"07:08.125","Text":"We have a maximum at 0, 7."},{"Start":"07:08.125 ","End":"07:13.035","Text":"We have a minimum at Pi, minus 9,"},{"Start":"07:13.035 ","End":"07:18.420","Text":"and another maximum at 2 Pi, 7."},{"Start":"07:18.420 ","End":"07:19.995","Text":"That\u0027s 3 of them."},{"Start":"07:19.995 ","End":"07:23.765","Text":"Then we want the intervals of increase,"},{"Start":"07:23.765 ","End":"07:25.940","Text":"increases the up arrow."},{"Start":"07:25.940 ","End":"07:29.420","Text":"It\u0027s x between Pi and 2 Pi,"},{"Start":"07:29.420 ","End":"07:34.835","Text":"and the interval of decrease is where the orange arrow goes down,"},{"Start":"07:34.835 ","End":"07:38.450","Text":"and that\u0027s between 0 and Pi."},{"Start":"07:38.450 ","End":"07:40.219","Text":"That answers the questions,"},{"Start":"07:40.219 ","End":"07:42.510","Text":"and we are done."}],"ID":4832},{"Watched":false,"Name":"Exercise 29","Duration":"10m 31s","ChapterTopicVideoID":4832,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4832.jpeg","UploadDate":"2015-07-21T02:37:22.6730000","DurationForVideoObject":"PT10M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.825","Text":"In this exercise, we\u0027re given a function f of x,"},{"Start":"00:03.825 ","End":"00:06.585","Text":"which is e to the minus x sine x."},{"Start":"00:06.585 ","End":"00:10.080","Text":"Note that this is not a trigonometrical function."},{"Start":"00:10.080 ","End":"00:12.870","Text":"It\u0027s mixed, it has some trig, but not all."},{"Start":"00:12.870 ","End":"00:16.409","Text":"Anyway, we have to find all its local extrema"},{"Start":"00:16.409 ","End":"00:20.640","Text":"as well as the intervals where it\u0027s increasing and where it\u0027s decreasing."},{"Start":"00:20.640 ","End":"00:26.120","Text":"This is a standard format of an exercise and it\u0027s done in 4 stages."},{"Start":"00:26.120 ","End":"00:31.040","Text":"The first stage is a preparation stage where we have to find the derivative,"},{"Start":"00:31.040 ","End":"00:33.890","Text":"so f prime of x is equal to."},{"Start":"00:33.890 ","End":"00:35.030","Text":"Let\u0027s see what we have here."},{"Start":"00:35.030 ","End":"00:36.560","Text":"First of all, we have a product,"},{"Start":"00:36.560 ","End":"00:39.140","Text":"so we\u0027ll use the product rule."},{"Start":"00:39.140 ","End":"00:40.820","Text":"I\u0027ll remind you what it is."},{"Start":"00:40.820 ","End":"00:46.370","Text":"Uv prime is u prime v plus uv prime,"},{"Start":"00:46.370 ","End":"00:48.085","Text":"just to jog your memory."},{"Start":"00:48.085 ","End":"00:55.490","Text":"Here we have the derivative of the first is e minus x with a minus and then sine x"},{"Start":"00:55.490 ","End":"01:03.665","Text":"untouched plus e to the minus x as is and sine x derived is cosine x."},{"Start":"01:03.665 ","End":"01:09.905","Text":"Basically, what we get is e to the minus x times minus sine plus cosine."},{"Start":"01:09.905 ","End":"01:11.030","Text":"Let\u0027s write it the other way."},{"Start":"01:11.030 ","End":"01:15.260","Text":"Cosine x minus sine of x."},{"Start":"01:15.260 ","End":"01:17.660","Text":"That\u0027s basically it for the preparation stage."},{"Start":"01:17.660 ","End":"01:19.325","Text":"This is the derivative."},{"Start":"01:19.325 ","End":"01:28.550","Text":"The next stage is to solve the equation f prime of x is equal to 0."},{"Start":"01:28.550 ","End":"01:32.240","Text":"We want solutions that are in our domain,"},{"Start":"01:32.240 ","End":"01:34.205","Text":"which note, is not all of x."},{"Start":"01:34.205 ","End":"01:39.635","Text":"We\u0027re restricting it to minus Pi less than or equal to x, less than or equal to 0."},{"Start":"01:39.635 ","End":"01:43.310","Text":"In other words, in degrees from minus a 180-0."},{"Start":"01:43.310 ","End":"01:47.390","Text":"We have to find the solutions of x in that domain."},{"Start":"01:47.390 ","End":"01:49.810","Text":"These will be our suspects for extrema."},{"Start":"01:49.810 ","End":"01:51.485","Text":"Let\u0027s see what we get."},{"Start":"01:51.485 ","End":"02:00.800","Text":"We get that e to the minus x times cosine x minus sine x is equal to 0,"},{"Start":"02:00.800 ","End":"02:05.005","Text":"but e to the minus x is never 0."},{"Start":"02:05.005 ","End":"02:09.725","Text":"We can divide by it or we can just say that the 0 has to come from here."},{"Start":"02:09.725 ","End":"02:12.830","Text":"If this thing is equal to 0,"},{"Start":"02:12.830 ","End":"02:16.940","Text":"then we get that cosine of x is equal to sine x."},{"Start":"02:16.940 ","End":"02:21.980","Text":"The easiest way, I think is just to divide both sides by cosine x."},{"Start":"02:21.980 ","End":"02:24.590","Text":"Remember, sine over cosine is tangent."},{"Start":"02:24.590 ","End":"02:28.480","Text":"We get that tangent x is equal to 1."},{"Start":"02:28.480 ","End":"02:31.325","Text":"Tangent x is equal to 1."},{"Start":"02:31.325 ","End":"02:34.415","Text":"Normally, we take it at Pi over 4,"},{"Start":"02:34.415 ","End":"02:37.280","Text":"but that\u0027s outside our range."},{"Start":"02:37.280 ","End":"02:43.440","Text":"The general solution is Pi over 4 plus n times the period which is Pi."},{"Start":"02:43.440 ","End":"02:49.460","Text":"But we also have to have x in this interval from 0 to Pi."},{"Start":"02:49.460 ","End":"02:52.580","Text":"If we take n is equal to minus 1,"},{"Start":"02:52.580 ","End":"02:55.690","Text":"then we\u0027ll get minus 3Pi over 4."},{"Start":"02:55.690 ","End":"02:57.560","Text":"That\u0027s the only 1 that will work."},{"Start":"02:57.560 ","End":"03:00.380","Text":"N equals minus 1 will take us into our interval,"},{"Start":"03:00.380 ","End":"03:07.040","Text":"so x is equal to minus 3Pi over 4 and if you want that in degrees,"},{"Start":"03:07.040 ","End":"03:11.475","Text":"that is minus 135 degrees."},{"Start":"03:11.475 ","End":"03:13.080","Text":"If we add a 180 degrees,"},{"Start":"03:13.080 ","End":"03:16.880","Text":"we get 45 and the tangent of that is 1."},{"Start":"03:16.880 ","End":"03:19.490","Text":"The way you do this with a calculator,"},{"Start":"03:19.490 ","End":"03:21.920","Text":"how did I get the Pi over 4?"},{"Start":"03:21.920 ","End":"03:26.540","Text":"Can do it with the calculator if you do shift tangent."},{"Start":"03:26.540 ","End":"03:28.185","Text":"In other words, basically,"},{"Start":"03:28.185 ","End":"03:30.450","Text":"x here is the arc tangent."},{"Start":"03:30.450 ","End":"03:32.115","Text":"That\u0027s the opposite of tangent,"},{"Start":"03:32.115 ","End":"03:33.825","Text":"arc tangent of 1."},{"Start":"03:33.825 ","End":"03:42.795","Text":"The calculator would give you 45 degrees plus n times a 180 degrees."},{"Start":"03:42.795 ","End":"03:44.530","Text":"When n is minus 1,"},{"Start":"03:44.530 ","End":"03:48.055","Text":"it brings us in our range of minus a 180-0."},{"Start":"03:48.055 ","End":"03:51.375","Text":"Sorry if I went on a bit too much there."},{"Start":"03:51.375 ","End":"03:58.354","Text":"We have a solution and that makes it a suspect for an extremum,"},{"Start":"03:58.354 ","End":"04:00.200","Text":"and now we go on to the next stage,"},{"Start":"04:00.200 ","End":"04:02.575","Text":"which is the table stage."},{"Start":"04:02.575 ","End":"04:07.775","Text":"We fill it in with the interesting values of x."},{"Start":"04:07.775 ","End":"04:13.130","Text":"The only interesting value really is this minus 3Pi over 4,"},{"Start":"04:13.130 ","End":"04:17.075","Text":"but I\u0027ll also put in the endpoints."},{"Start":"04:17.075 ","End":"04:23.420","Text":"That\u0027s often what is done to indicate that this thing goes down to minus Pi,"},{"Start":"04:23.420 ","End":"04:26.645","Text":"and that\u0027s it, and up to 0 and that\u0027s it,"},{"Start":"04:26.645 ","End":"04:30.515","Text":"which means that we only get 2 intervals here."},{"Start":"04:30.515 ","End":"04:36.665","Text":"Also, I want to just note that f prime is 0 here because it\u0027s a suspect."},{"Start":"04:36.665 ","End":"04:42.360","Text":"That\u0027s how we got the minus 3Pi over 4 from derivative 0."},{"Start":"04:43.640 ","End":"04:46.820","Text":"Let\u0027s see what intervals we get."},{"Start":"04:46.820 ","End":"04:52.535","Text":"Well, we get the interval minus Pi less than or equal to x,"},{"Start":"04:52.535 ","End":"04:56.685","Text":"less than minus 3Pi over 4."},{"Start":"04:56.685 ","End":"05:00.380","Text":"Here we get minus 3Pi over 4,"},{"Start":"05:00.380 ","End":"05:06.470","Text":"less than x, less than or equal to 0, 2 intervals."},{"Start":"05:06.470 ","End":"05:09.795","Text":"Let\u0027s take an angle in each 1."},{"Start":"05:09.795 ","End":"05:15.200","Text":"I\u0027ll just, again, write it in degrees with minus 135 degrees."},{"Start":"05:15.200 ","End":"05:20.960","Text":"This was minus a 180 degrees and this is 0 degrees."},{"Start":"05:21.980 ","End":"05:25.215","Text":"Let\u0027s choose it in degrees."},{"Start":"05:25.215 ","End":"05:28.610","Text":"Here\u0027s the thing. Because f is written in"},{"Start":"05:28.610 ","End":"05:32.375","Text":"terms of trigonometrical and non-trigonometrical,"},{"Start":"05:32.375 ","End":"05:36.140","Text":"in the non-trigonometrical, you can\u0027t substitute degrees."},{"Start":"05:36.140 ","End":"05:39.255","Text":"You can only substitute the x in radians."},{"Start":"05:39.255 ","End":"05:41.070","Text":"I have to write both."},{"Start":"05:41.070 ","End":"05:45.030","Text":"I\u0027ll choose minus a 150 degrees."},{"Start":"05:45.030 ","End":"05:51.815","Text":"Let\u0027s say x here is minus 90 degrees and in radians,"},{"Start":"05:51.815 ","End":"05:57.720","Text":"that comes out to minus 150 degrees is minus 5Pi over 6 and the other point,"},{"Start":"05:57.720 ","End":"06:01.395","Text":"minus 90 degrees is minus Pi over 2."},{"Start":"06:01.395 ","End":"06:07.760","Text":"What we need is to substitute this into the derivative."},{"Start":"06:07.760 ","End":"06:11.720","Text":"The derivative is this function here."},{"Start":"06:11.720 ","End":"06:14.630","Text":"All I want for these is the sign."},{"Start":"06:14.630 ","End":"06:16.100","Text":"I don\u0027t want the actual value,"},{"Start":"06:16.100 ","End":"06:18.335","Text":"just whether it\u0027s plus or minus."},{"Start":"06:18.335 ","End":"06:23.685","Text":"But remember, we can\u0027t substitute degrees except in the trigonometric bits."},{"Start":"06:23.685 ","End":"06:26.510","Text":"That\u0027s why this exercise is confusing"},{"Start":"06:26.510 ","End":"06:29.615","Text":"because it mixes non-trigonometric with trigonometric."},{"Start":"06:29.615 ","End":"06:32.240","Text":"Let\u0027s first of all take this 1."},{"Start":"06:32.240 ","End":"06:37.610","Text":"I mentioned before that when I substitute values in here for the cosine and sine,"},{"Start":"06:37.610 ","End":"06:39.260","Text":"I can use radians or degrees,"},{"Start":"06:39.260 ","End":"06:40.775","Text":"but for the e to the power of,"},{"Start":"06:40.775 ","End":"06:43.120","Text":"I have to use the gradients."},{"Start":"06:43.120 ","End":"06:47.075","Text":"In this case, we happen to be lucky because e to the power of anything is positive."},{"Start":"06:47.075 ","End":"06:52.640","Text":"Actually, I can even ignore the e to the power of if I only want the sign, plus or minus."},{"Start":"06:52.640 ","End":"06:56.030","Text":"I just have to look at cosine x minus sine x. I"},{"Start":"06:56.030 ","End":"07:00.545","Text":"checked for a 150 degrees or minus 5Pi over 6,"},{"Start":"07:00.545 ","End":"07:03.530","Text":"it comes out to be negative,"},{"Start":"07:03.530 ","End":"07:07.445","Text":"which means that the function here is decreasing."},{"Start":"07:07.445 ","End":"07:10.640","Text":"Here for minus 90 degrees,"},{"Start":"07:10.640 ","End":"07:15.855","Text":"I plugged it in and checked that this thing is positive."},{"Start":"07:15.855 ","End":"07:17.960","Text":"We have a plus here,"},{"Start":"07:17.960 ","End":"07:21.060","Text":"means that here it\u0027s increasing,"},{"Start":"07:21.060 ","End":"07:25.850","Text":"and so we can actually say that this point here is certainly"},{"Start":"07:25.850 ","End":"07:27.590","Text":"an extremum and it\u0027s certainly"},{"Start":"07:27.590 ","End":"07:30.890","Text":"a minimum because it\u0027s between a decreasing and increasing."},{"Start":"07:30.890 ","End":"07:34.235","Text":"Now in this peculiar exercise which has endpoints,"},{"Start":"07:34.235 ","End":"07:36.710","Text":"the endpoints we only have to check 1 side."},{"Start":"07:36.710 ","End":"07:40.685","Text":"In other words, if I\u0027m going up to the point,"},{"Start":"07:40.685 ","End":"07:43.370","Text":"it\u0027s a right end point and I\u0027m increasing on the left,"},{"Start":"07:43.370 ","End":"07:46.115","Text":"it has to be a maximum because I\u0027m going up to it."},{"Start":"07:46.115 ","End":"07:48.305","Text":"Likewise on the left side,"},{"Start":"07:48.305 ","End":"07:52.580","Text":"if there is only decreasing on the right but there is nowhere to go on the left,"},{"Start":"07:52.580 ","End":"07:53.750","Text":"we\u0027re decreasing from it,"},{"Start":"07:53.750 ","End":"07:55.745","Text":"so it has to also be a maximum."},{"Start":"07:55.745 ","End":"07:56.940","Text":"This is special case,"},{"Start":"07:56.940 ","End":"07:59.675","Text":"we only have to check 1 side at the endpoints."},{"Start":"07:59.675 ","End":"08:03.080","Text":"We actually have 3 extrema and what I need"},{"Start":"08:03.080 ","End":"08:06.545","Text":"now in the table is just the y-coordinate so I can just,"},{"Start":"08:06.545 ","End":"08:07.865","Text":"in the summary, give them."},{"Start":"08:07.865 ","End":"08:13.670","Text":"I need these 3 values of y and y is equal to f of x,"},{"Start":"08:13.670 ","End":"08:16.850","Text":"which is e to the minus x sine x. Let\u0027s see."},{"Start":"08:16.850 ","End":"08:19.220","Text":"As I say, we can\u0027t substitute degrees in"},{"Start":"08:19.220 ","End":"08:22.720","Text":"the e part and here we actually do need the value."},{"Start":"08:22.720 ","End":"08:25.550","Text":"Let\u0027s do the middle 1 first, it\u0027s the hardest."},{"Start":"08:25.550 ","End":"08:29.795","Text":"If I put in minus 3Pi over 4 and I can\u0027t use the degrees,"},{"Start":"08:29.795 ","End":"08:31.580","Text":"I get e to the minus,"},{"Start":"08:31.580 ","End":"08:33.440","Text":"minus 3Pi over 4."},{"Start":"08:33.440 ","End":"08:38.135","Text":"That\u0027s e to the power of plus 3Pi over 4."},{"Start":"08:38.135 ","End":"08:39.470","Text":"Then in the sine x there,"},{"Start":"08:39.470 ","End":"08:41.510","Text":"I can substitute degrees if I want to."},{"Start":"08:41.510 ","End":"08:49.190","Text":"The sine of minus 135 turns out to be minus 1 over the square root of 2."},{"Start":"08:49.190 ","End":"08:54.290","Text":"This is times minus 1 over the square root of 2."},{"Start":"08:54.290 ","End":"08:56.190","Text":"You can compute it with actual values,"},{"Start":"08:56.190 ","End":"08:58.820","Text":"but I\u0027d like to leave it just as in this form."},{"Start":"08:58.820 ","End":"09:02.890","Text":"Maybe I\u0027ll rewrite it as minus this over square root of 2. Now let\u0027s see."},{"Start":"09:02.890 ","End":"09:07.160","Text":"At minus Pi, which is minus a 180 degrees,"},{"Start":"09:07.160 ","End":"09:09.470","Text":"the sine is equal to 0,"},{"Start":"09:09.470 ","End":"09:13.360","Text":"so it doesn\u0027t matter about the e. This is equal to 0."},{"Start":"09:13.360 ","End":"09:15.440","Text":"Likewise, at 0 degrees,"},{"Start":"09:15.440 ","End":"09:16.640","Text":"sine of 0 is 0,"},{"Start":"09:16.640 ","End":"09:18.605","Text":"so it doesn\u0027t matter about the e part,"},{"Start":"09:18.605 ","End":"09:20.180","Text":"so this is 0."},{"Start":"09:20.180 ","End":"09:22.280","Text":"We\u0027ve got everything we want from the table."},{"Start":"09:22.280 ","End":"09:26.000","Text":"The last stage of the solution is just to draw conclusions,"},{"Start":"09:26.000 ","End":"09:28.700","Text":"which is essentially to answer the questions that we were asked."},{"Start":"09:28.700 ","End":"09:34.610","Text":"What we were asked was first of all about the extrema and we actually have 3."},{"Start":"09:34.610 ","End":"09:40.490","Text":"We have a maximum at minus Pi, 0,"},{"Start":"09:40.490 ","End":"09:44.840","Text":"we have another maximum at 0,0,"},{"Start":"09:44.840 ","End":"09:50.940","Text":"and we have a minimum at minus 3Pi over 4,"},{"Start":"09:50.940 ","End":"09:59.105","Text":"e to the 3Pi over 4 divided by the square root of 2."},{"Start":"09:59.105 ","End":"10:01.070","Text":"I think this was supposed to be minus."},{"Start":"10:01.070 ","End":"10:02.690","Text":"There\u0027s our minimum."},{"Start":"10:02.690 ","End":"10:05.875","Text":"As for the intervals of increase,"},{"Start":"10:05.875 ","End":"10:08.255","Text":"it\u0027s increasing when the arrow\u0027s going up,"},{"Start":"10:08.255 ","End":"10:12.650","Text":"which means it\u0027s this interval which is minus 3Pi over 4,"},{"Start":"10:12.650 ","End":"10:16.400","Text":"less than x, less than or equal to 0."},{"Start":"10:16.400 ","End":"10:20.600","Text":"The decrease is where the orange arrow is going down,"},{"Start":"10:20.600 ","End":"10:25.940","Text":"which is minus Pi less than or equal to x,"},{"Start":"10:25.940 ","End":"10:32.190","Text":"less than minus 3Pi over 4. We are done."}],"ID":4833},{"Watched":false,"Name":"Exercise 30","Duration":"9m 58s","ChapterTopicVideoID":4833,"CourseChapterTopicPlaylistID":65381,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4833.jpeg","UploadDate":"2015-07-21T02:38:16.3870000","DurationForVideoObject":"PT9M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we\u0027re given the following function,"},{"Start":"00:04.320 ","End":"00:10.875","Text":"f of x is 2 cosine squared x minus sine x and we have to find its local extrema,"},{"Start":"00:10.875 ","End":"00:16.320","Text":"as well as the intervals where the function is increasing and where it\u0027s decreasing."},{"Start":"00:16.320 ","End":"00:20.730","Text":"Notice that this is a trigonometric function and its domain is"},{"Start":"00:20.730 ","End":"00:25.650","Text":"restricted to the interval from 0 to Pi, the closed interval."},{"Start":"00:25.650 ","End":"00:28.470","Text":"This is a standard format of exercise."},{"Start":"00:28.470 ","End":"00:30.945","Text":"We know how to solve it in 4 stages."},{"Start":"00:30.945 ","End":"00:33.495","Text":"The first stage is the preparation."},{"Start":"00:33.495 ","End":"00:36.950","Text":"All you have to do is find the first derivative,"},{"Start":"00:36.950 ","End":"00:41.420","Text":"f prime of x. Differentiating, we get,"},{"Start":"00:41.420 ","End":"00:44.485","Text":"the 2 was there then cosine squared,"},{"Start":"00:44.485 ","End":"00:47.015","Text":"so we get 2 cosine x,"},{"Start":"00:47.015 ","End":"00:50.360","Text":"but times the inner derivative of the cosine,"},{"Start":"00:50.360 ","End":"00:52.760","Text":"which is minus sine x."},{"Start":"00:52.760 ","End":"00:53.840","Text":"That\u0027s the first term."},{"Start":"00:53.840 ","End":"00:59.180","Text":"The second term, we have a sine so the derivative is cosine of 2x,"},{"Start":"00:59.180 ","End":"01:03.035","Text":"but the inner derivative of 2x is 2."},{"Start":"01:03.035 ","End":"01:08.480","Text":"I\u0027m just going to rewrite this as minus the 2 with the 2, that\u0027s minus 4."},{"Start":"01:08.480 ","End":"01:10.010","Text":"Allow me to reverse the order."},{"Start":"01:10.010 ","End":"01:18.075","Text":"I prefer the sine first before the cosine and minus 2 cosine 2x."},{"Start":"01:18.075 ","End":"01:22.325","Text":"Now, I\u0027d like to remind you of a trigonometrical identity,"},{"Start":"01:22.325 ","End":"01:30.255","Text":"and that is that 2 sine Alpha cosine Alpha is sine of 2 Alpha."},{"Start":"01:30.255 ","End":"01:35.810","Text":"We can see that this is very useful to us because we have 2 sine Alpha cosine Alpha."},{"Start":"01:35.810 ","End":"01:38.149","Text":"In short, if we use this formula,"},{"Start":"01:38.149 ","End":"01:45.425","Text":"we\u0027ll get minus 2 sine of 2x minus 2 cosine 2x."},{"Start":"01:45.425 ","End":"01:48.080","Text":"That\u0027s about as simplified as I want,"},{"Start":"01:48.080 ","End":"01:51.780","Text":"and that\u0027s the end of the preparation stage."},{"Start":"01:51.780 ","End":"01:53.490","Text":"On to the next stage."},{"Start":"01:53.490 ","End":"02:00.905","Text":"The next stage, is to find the solutions of the equation f prime of x is equal to 0."},{"Start":"02:00.905 ","End":"02:03.200","Text":"The x is that we find as a solution,"},{"Start":"02:03.200 ","End":"02:05.935","Text":"we\u0027re going to be suspects for extrema."},{"Start":"02:05.935 ","End":"02:08.550","Text":"F prime of x is this."},{"Start":"02:08.550 ","End":"02:10.160","Text":"If I let this equals 0,"},{"Start":"02:10.160 ","End":"02:13.190","Text":"I can just put the first term on the left,"},{"Start":"02:13.190 ","End":"02:21.605","Text":"and so I get 2 sine 2x equals minus 2 cosine of 2x."},{"Start":"02:21.605 ","End":"02:26.360","Text":"If I divide both sides by 2 cosine 2x,"},{"Start":"02:26.360 ","End":"02:31.070","Text":"then the 2 with the 2 will cancel and I\u0027ll get tangent of 2x,"},{"Start":"02:31.070 ","End":"02:33.339","Text":"because sine over cosine is tangent,"},{"Start":"02:33.339 ","End":"02:35.660","Text":"is equal to minus 1."},{"Start":"02:35.660 ","End":"02:38.765","Text":"Now, the equation for tangent,"},{"Start":"02:38.765 ","End":"02:42.740","Text":"we first of all have to find 1 solution and then it\u0027s periodic."},{"Start":"02:42.740 ","End":"02:44.015","Text":"I\u0027ll write it in a moment."},{"Start":"02:44.015 ","End":"02:49.440","Text":"What we get is that 2x is what is called the arctangent,"},{"Start":"02:49.440 ","End":"02:51.375","Text":"the reverse of the tangent."},{"Start":"02:51.375 ","End":"02:53.600","Text":"On most pocket calculators,"},{"Start":"02:53.600 ","End":"03:00.635","Text":"you compute this by doing Shift tangent of minus 1,and this comes out to be,"},{"Start":"03:00.635 ","End":"03:02.240","Text":"if it\u0027s in radians,"},{"Start":"03:02.240 ","End":"03:05.050","Text":"you will get 3 Pi over 4,"},{"Start":"03:05.050 ","End":"03:06.815","Text":"and if you did it in degree,"},{"Start":"03:06.815 ","End":"03:09.785","Text":"it is a 135 degrees."},{"Start":"03:09.785 ","End":"03:12.170","Text":"But the general solution is not this,"},{"Start":"03:12.170 ","End":"03:20.800","Text":"the general solution is 2x equals 3 Pi over 4 plus n times Pi,"},{"Start":"03:20.800 ","End":"03:23.480","Text":"where n is any integer, a whole number,"},{"Start":"03:23.480 ","End":"03:26.870","Text":"because the tangent is periodic with the period of Pi."},{"Start":"03:26.870 ","End":"03:29.780","Text":"Then if we divide by 2,"},{"Start":"03:29.780 ","End":"03:31.370","Text":"we get what x is."},{"Start":"03:31.370 ","End":"03:39.185","Text":"So x is 3 Pi over 8 plus n times Pi over 2."},{"Start":"03:39.185 ","End":"03:42.320","Text":"I\u0027ll now write this also in degrees,"},{"Start":"03:42.320 ","End":"03:44.015","Text":"in case it helps someone,"},{"Start":"03:44.015 ","End":"03:51.215","Text":"it is 67 and 1/2 degrees plus n times 90 degrees,"},{"Start":"03:51.215 ","End":"03:52.730","Text":"which is Pi over 2."},{"Start":"03:52.730 ","End":"03:58.205","Text":"Now, our solutions have to be restricted to the domain,"},{"Start":"03:58.205 ","End":"04:01.670","Text":"and our domain was from 0 to Pi."},{"Start":"04:01.670 ","End":"04:04.820","Text":"Or if we take it in degrees,"},{"Start":"04:04.820 ","End":"04:10.355","Text":"we have that the interval 0 to Pi is 0 to 180 degrees,"},{"Start":"04:10.355 ","End":"04:12.380","Text":"and that\u0027s where x has to be."},{"Start":"04:12.380 ","End":"04:16.655","Text":"There\u0027s only 2 possible values of n we could take if you think about it."},{"Start":"04:16.655 ","End":"04:18.725","Text":"If n is negative 1,"},{"Start":"04:18.725 ","End":"04:21.080","Text":"we\u0027re ready less than 0, so that\u0027s no good."},{"Start":"04:21.080 ","End":"04:24.765","Text":"If n is 0, we have 67 and 1/2 degrees and that\u0027s good."},{"Start":"04:24.765 ","End":"04:30.315","Text":"If n is 1, then we have 67 and 1/2 plus 90 equals 157."},{"Start":"04:30.315 ","End":"04:33.570","Text":"We can only take 0 and 1 basically."},{"Start":"04:33.570 ","End":"04:37.210","Text":"Our solutions, if we go back to radians,"},{"Start":"04:37.210 ","End":"04:41.960","Text":"in radians we have x is equal to either 3 Pi over 8,"},{"Start":"04:41.960 ","End":"04:43.565","Text":"or if we add 1,"},{"Start":"04:43.565 ","End":"04:45.755","Text":"3 Pi plus Pi over 2,"},{"Start":"04:45.755 ","End":"04:50.045","Text":"that comes out to be 7 Pi over 8."},{"Start":"04:50.045 ","End":"04:53.270","Text":"Once again, I\u0027ll write that in degrees,"},{"Start":"04:53.270 ","End":"04:57.935","Text":"it\u0027s either we have 67 and 1/2 degrees"},{"Start":"04:57.935 ","End":"05:03.765","Text":"or a 157 and 1/2 degrees."},{"Start":"05:03.765 ","End":"05:05.535","Text":"These are our 2 solutions,"},{"Start":"05:05.535 ","End":"05:09.494","Text":"and this makes these 2 our suspects for extrema."},{"Start":"05:09.494 ","End":"05:11.580","Text":"That ends this stage."},{"Start":"05:11.580 ","End":"05:17.225","Text":"Next, we go to the stage of the table."},{"Start":"05:17.225 ","End":"05:21.410","Text":"Here\u0027s a blank table with rows x,"},{"Start":"05:21.410 ","End":"05:24.410","Text":"f prime f and y as usual."},{"Start":"05:24.410 ","End":"05:28.055","Text":"The values of x that we put in are the interesting values,"},{"Start":"05:28.055 ","End":"05:31.895","Text":"mainly the suspect, so I\u0027m going to put it in radians."},{"Start":"05:31.895 ","End":"05:34.430","Text":"We put them in order, or increasing order, of course,"},{"Start":"05:34.430 ","End":"05:36.425","Text":"3 Pi over 8,"},{"Start":"05:36.425 ","End":"05:38.225","Text":"and then over here,"},{"Start":"05:38.225 ","End":"05:43.845","Text":"7 Pi over 8 and the value of f prime,"},{"Start":"05:43.845 ","End":"05:45.435","Text":"these 2 points is 0."},{"Start":"05:45.435 ","End":"05:49.100","Text":"That\u0027s how we found these 2 points by setting f prime to 0"},{"Start":"05:49.100 ","End":"05:53.800","Text":"and they divide our domain into 3 sub-intervals."},{"Start":"05:53.800 ","End":"05:56.630","Text":"The intervals we get are 0,"},{"Start":"05:56.630 ","End":"06:05.430","Text":"which is the edge of the domain less than or equal to x less than 3 Pi over 8,"},{"Start":"06:05.430 ","End":"06:10.775","Text":"and second interval we have is where x is between the 2 values,"},{"Start":"06:10.775 ","End":"06:17.060","Text":"between 7 Pi over 8 above and 3 Pi over 8 below."},{"Start":"06:17.060 ","End":"06:24.440","Text":"Then we have from 7 Pi over 8 up to the edge of the limit,"},{"Start":"06:24.440 ","End":"06:27.635","Text":"which is just Pi less than or equal to."},{"Start":"06:27.635 ","End":"06:32.000","Text":"Then we have to choose a value in each of these ranges."},{"Start":"06:32.000 ","End":"06:35.775","Text":"I think I\u0027ll go with degrees for a while."},{"Start":"06:35.775 ","End":"06:38.670","Text":"Let\u0027s choose from this range,"},{"Start":"06:38.670 ","End":"06:42.510","Text":"about 45 degrees, Pi over 4."},{"Start":"06:42.510 ","End":"06:44.405","Text":"From here to here,"},{"Start":"06:44.405 ","End":"06:46.655","Text":"90 degrees is a nice angle."},{"Start":"06:46.655 ","End":"06:48.095","Text":"From here to here,"},{"Start":"06:48.095 ","End":"06:50.995","Text":"I\u0027ll take a 170 degrees."},{"Start":"06:50.995 ","End":"06:57.005","Text":"Now, I\u0027m missing f prime of x and f of x. I\u0027m going to rewrite them."},{"Start":"06:57.005 ","End":"07:00.695","Text":"Here I\u0027ve copied f and f prime, so they\u0027re handy."},{"Start":"07:00.695 ","End":"07:02.270","Text":"Let\u0027s take 1 of them."},{"Start":"07:02.270 ","End":"07:06.739","Text":"I\u0027ll take the 45 degrees and see what happens with f prime."},{"Start":"07:06.739 ","End":"07:08.780","Text":"If x is 45 degrees,"},{"Start":"07:08.780 ","End":"07:11.569","Text":"then 2x is 90 degrees,"},{"Start":"07:11.569 ","End":"07:13.640","Text":"cosine of 90 is 0,"},{"Start":"07:13.640 ","End":"07:15.580","Text":"and sine of 90 is 1."},{"Start":"07:15.580 ","End":"07:19.440","Text":"This bit is 0, so we have minus 2 times 1, which is minus 2."},{"Start":"07:19.440 ","End":"07:20.730","Text":"All I need is the sine,"},{"Start":"07:20.730 ","End":"07:21.885","Text":"not the actual value."},{"Start":"07:21.885 ","End":"07:23.269","Text":"Here it\u0027s negative."},{"Start":"07:23.269 ","End":"07:26.450","Text":"I\u0027m not going to waste your time with substituting everyone,"},{"Start":"07:26.450 ","End":"07:33.485","Text":"I\u0027ll just say that this comes out to be positive and this comes out to be negative also."},{"Start":"07:33.485 ","End":"07:38.675","Text":"What this means when the derivative is negative is that the function is decreasing,"},{"Start":"07:38.675 ","End":"07:41.495","Text":"when it\u0027s positive, it\u0027s increasing,"},{"Start":"07:41.495 ","End":"07:45.335","Text":"and negative means again decreasing."},{"Start":"07:45.335 ","End":"07:49.279","Text":"Now, this point, which is between the decreasing"},{"Start":"07:49.279 ","End":"07:53.210","Text":"and the increasing is automatically a minimum point."},{"Start":"07:53.210 ","End":"07:54.770","Text":"That\u0027s how we define a minimum."},{"Start":"07:54.770 ","End":"07:58.630","Text":"It\u0027s an extremum of type minimum, which I abbreviate."},{"Start":"07:58.630 ","End":"08:00.830","Text":"Between an increasing and decreasing,"},{"Start":"08:00.830 ","End":"08:02.614","Text":"we have a maximum extremum,"},{"Start":"08:02.614 ","End":"08:04.745","Text":"which I call max."},{"Start":"08:04.745 ","End":"08:11.080","Text":"The only other thing we need from the table is the value of y for the extrema."},{"Start":"08:11.080 ","End":"08:12.980","Text":"I had to put something here and here,"},{"Start":"08:12.980 ","End":"08:18.365","Text":"and that\u0027s where I substitute in the original function."},{"Start":"08:18.365 ","End":"08:20.960","Text":"Here I put this and this."},{"Start":"08:20.960 ","End":"08:24.395","Text":"I will not bore you with the computations,"},{"Start":"08:24.395 ","End":"08:25.940","Text":"I did it at the side,"},{"Start":"08:25.940 ","End":"08:31.845","Text":"and this comes out to be 1 minus the square root of 2,"},{"Start":"08:31.845 ","End":"08:36.530","Text":"and this 1 comes out to be 1 plus the square root of 2."},{"Start":"08:36.530 ","End":"08:38.130","Text":"I\u0027m going to delete these,"},{"Start":"08:38.130 ","End":"08:39.590","Text":"they\u0027re just in the way."},{"Start":"08:39.590 ","End":"08:43.055","Text":"This basically ends the table stage."},{"Start":"08:43.055 ","End":"08:44.570","Text":"Now, we come to the last stage,"},{"Start":"08:44.570 ","End":"08:49.715","Text":"which is the conclusions that we can draw and which answer the original question."},{"Start":"08:49.715 ","End":"08:54.105","Text":"I\u0027ll just put a line here to separate the work area."},{"Start":"08:54.105 ","End":"08:57.060","Text":"We have 3 things that we were asked,"},{"Start":"08:57.060 ","End":"08:59.940","Text":"extrema, increasing and decreasing."},{"Start":"08:59.940 ","End":"09:02.235","Text":"The extrema, there are 2 of them."},{"Start":"09:02.235 ","End":"09:07.280","Text":"1 of them is at 3 Pi over 8,"},{"Start":"09:07.280 ","End":"09:09.835","Text":"1 minus the square root of 2,"},{"Start":"09:09.835 ","End":"09:12.100","Text":"and that is a minimum."},{"Start":"09:12.100 ","End":"09:17.575","Text":"The second 1 occurs at 7 Pi over 8,"},{"Start":"09:17.575 ","End":"09:21.675","Text":"1 plus the square root of 2."},{"Start":"09:21.675 ","End":"09:25.704","Text":"Here we have a maximum extremum."},{"Start":"09:25.704 ","End":"09:28.395","Text":"Now, the intervals of increase,"},{"Start":"09:28.395 ","End":"09:31.445","Text":"that\u0027s, we look for the arrow going upwards, that\u0027s this 1."},{"Start":"09:31.445 ","End":"09:39.390","Text":"This interval is 3 Pi over 8 less than x less than 7 Pi over 8."},{"Start":"09:39.390 ","End":"09:41.645","Text":"The interval of decrease,"},{"Start":"09:41.645 ","End":"09:43.835","Text":"there\u0027s 2 of those, the arrow down,"},{"Start":"09:43.835 ","End":"09:49.625","Text":"so we have x between 0 and 3 Pi over 8,"},{"Start":"09:49.625 ","End":"09:56.615","Text":"but also the x between 7 Pi over 8 and Pi."},{"Start":"09:56.615 ","End":"09:59.250","Text":"That\u0027s it. We\u0027re done."}],"ID":4834}],"Thumbnail":null,"ID":65381},{"Name":"Inflection, Convex, Concave","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Concavity \u0026 Convexity","Duration":"18m 37s","ChapterTopicVideoID":1654,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1654.jpeg","UploadDate":"2019-10-29T12:56:53.1770000","DurationForVideoObject":"PT18M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.205","Text":"In this clip, I\u0027ll be talking about concavity and convexity of a function."},{"Start":"00:08.205 ","End":"00:13.560","Text":"Let me draw a couple of examples to show you what I mean."},{"Start":"00:13.560 ","End":"00:19.039","Text":"Let\u0027s take 1 function in blue over here,"},{"Start":"00:19.039 ","End":"00:25.590","Text":"and I\u0027ll take another 1 and I\u0027ll color it in red over here."},{"Start":"00:26.840 ","End":"00:31.935","Text":"Now, both these functions are decreasing."},{"Start":"00:31.935 ","End":"00:33.575","Text":"As I go from left to right,"},{"Start":"00:33.575 ","End":"00:35.750","Text":"the value of y gets smaller and smaller."},{"Start":"00:35.750 ","End":"00:38.180","Text":"Also here as I go from left to right,"},{"Start":"00:38.180 ","End":"00:40.145","Text":"the y values get smaller."},{"Start":"00:40.145 ","End":"00:42.455","Text":"But although they\u0027re both decreasing,"},{"Start":"00:42.455 ","End":"00:43.820","Text":"they look quite different,"},{"Start":"00:43.820 ","End":"00:46.860","Text":"their basic shape is not the same."},{"Start":"00:47.020 ","End":"00:52.035","Text":"There is a word to describe what we intuitively see,"},{"Start":"00:52.035 ","End":"00:59.910","Text":"and the words to describe this function is that this function is convex,"},{"Start":"01:01.580 ","End":"01:06.550","Text":"and this function is concave."},{"Start":"01:10.580 ","End":"01:18.460","Text":"The way to remember which is which is to look at the word cave."},{"Start":"01:18.890 ","End":"01:24.180","Text":"Now, there was some confusion at the time and in fact,"},{"Start":"01:24.180 ","End":"01:28.870","Text":"even at one point this was called convex and this was called concave."},{"Start":"01:28.870 ","End":"01:31.155","Text":"The idea is this,"},{"Start":"01:31.155 ","End":"01:34.760","Text":"when you approach this curve from this side,"},{"Start":"01:34.760 ","End":"01:36.665","Text":"it looks like a cave."},{"Start":"01:36.665 ","End":"01:38.705","Text":"But you could also say that, \"Yeah,"},{"Start":"01:38.705 ","End":"01:41.630","Text":"this 1 is also like a cave if you go from here.\""},{"Start":"01:41.630 ","End":"01:45.825","Text":"1 point, these are also called"},{"Start":"01:45.825 ","End":"01:53.865","Text":"concave down or downwards,"},{"Start":"01:53.865 ","End":"01:56.730","Text":"and this was called concave up."},{"Start":"01:56.730 ","End":"01:58.890","Text":"By focusing on the cave part,"},{"Start":"01:58.890 ","End":"02:00.920","Text":"so here the cave part is at the top side,"},{"Start":"02:00.920 ","End":"02:03.380","Text":"here the cave part is at the bottom side."},{"Start":"02:03.380 ","End":"02:05.675","Text":"This is concave up or upward."},{"Start":"02:05.675 ","End":"02:10.480","Text":"But mostly we\u0027ll just be using these 2 terms, convex and concave,"},{"Start":"02:10.480 ","End":"02:15.875","Text":"to describe this basic bend that makes these 2 qualitatively different,"},{"Start":"02:15.875 ","End":"02:18.590","Text":"even though they\u0027re both decreasing."},{"Start":"02:18.590 ","End":"02:23.600","Text":"What I\u0027d like to do is to make these concepts a bit more precise."},{"Start":"02:23.600 ","End":"02:27.350","Text":"For example, here we\u0027re working with sketches,"},{"Start":"02:27.350 ","End":"02:32.150","Text":"so it\u0027s quite easy to talk about cave shapes and so on."},{"Start":"02:32.150 ","End":"02:35.900","Text":"But when we don\u0027t have a sketch and we just get a formula,"},{"Start":"02:35.900 ","End":"02:42.750","Text":"such as f of x is x squared minus natural log of x plus x^4 and so on,"},{"Start":"02:42.750 ","End":"02:49.790","Text":"then I want something more of concrete by way of defining what is convex and concave."},{"Start":"02:49.790 ","End":"02:54.950","Text":"There are actually 2 main approaches to this definition,"},{"Start":"02:54.950 ","End":"02:58.070","Text":"and we\u0027re going to favor 1 of them, the 1st."},{"Start":"02:58.070 ","End":"03:04.000","Text":"The 1st one relates to increasing and decreasing,"},{"Start":"03:04.000 ","End":"03:07.294","Text":"and tangents and derivatives."},{"Start":"03:07.294 ","End":"03:09.635","Text":"It goes like this."},{"Start":"03:09.635 ","End":"03:15.160","Text":"First of all, let\u0027s draw some little tangents all along."},{"Start":"03:15.160 ","End":"03:18.315","Text":"A tangent here, tangent here."},{"Start":"03:18.315 ","End":"03:20.449","Text":"I\u0027m not doing it very precisely,"},{"Start":"03:20.449 ","End":"03:22.385","Text":"but you get the idea."},{"Start":"03:22.385 ","End":"03:26.929","Text":"If I draw a bunch of tangents here and I look at their slopes,"},{"Start":"03:26.929 ","End":"03:29.869","Text":"this one might be minus 6,"},{"Start":"03:29.869 ","End":"03:32.830","Text":"this one maybe minus 4,"},{"Start":"03:32.830 ","End":"03:36.475","Text":"minus 2, minus a 1/2."},{"Start":"03:36.475 ","End":"03:43.295","Text":"You can see that the slopes are all negative here, but they\u0027re increasing."},{"Start":"03:43.295 ","End":"03:47.570","Text":"Even if we were to continue this curve up here,"},{"Start":"03:47.570 ","End":"03:49.370","Text":"we would get into the positive."},{"Start":"03:49.370 ","End":"03:51.320","Text":"We\u0027d have slope 1,"},{"Start":"03:51.320 ","End":"03:53.315","Text":"slope 2, and so on."},{"Start":"03:53.315 ","End":"03:58.365","Text":"The property of convex is that the slope is increasing,"},{"Start":"03:58.365 ","End":"04:00.690","Text":"and in a minute I\u0027ll write that."},{"Start":"04:00.690 ","End":"04:06.435","Text":"For concave, if we look at these slopes,"},{"Start":"04:06.435 ","End":"04:09.350","Text":"here it starts almost flat, maybe even 0,"},{"Start":"04:09.350 ","End":"04:13.080","Text":"or minus 1/2, minus 1,"},{"Start":"04:13.080 ","End":"04:15.480","Text":"minus 2, I\u0027m not exactly sure,"},{"Start":"04:15.480 ","End":"04:20.050","Text":"but they\u0027re all negative but they\u0027re decreasing."},{"Start":"04:20.480 ","End":"04:26.690","Text":"Basically, this is the characteristic that distinguishes convex from concave,"},{"Start":"04:26.690 ","End":"04:31.835","Text":"is that here slopes are increasing and here they\u0027re decreasing, and I\u0027ll write that."},{"Start":"04:31.835 ","End":"04:39.510","Text":"Approach number 1 is that for convex"},{"Start":"04:39.970 ","End":"04:51.450","Text":"means that the slope is increasing."},{"Start":"04:57.550 ","End":"05:03.440","Text":"The definition of concave is where"},{"Start":"05:03.440 ","End":"05:12.090","Text":"the slope is decreasing."},{"Start":"05:12.500 ","End":"05:14.730","Text":"I don\u0027t have much room here."},{"Start":"05:14.730 ","End":"05:22.480","Text":"The slope here,"},{"Start":"05:22.480 ","End":"05:25.385","Text":"though it\u0027s negative is getting less and less negative,"},{"Start":"05:25.385 ","End":"05:27.604","Text":"going towards the positive is increasing,"},{"Start":"05:27.604 ","End":"05:32.170","Text":"here the slope gets more and more negative, it\u0027s decreasing."},{"Start":"05:32.170 ","End":"05:35.785","Text":"That\u0027s definition number 1,"},{"Start":"05:35.785 ","End":"05:38.825","Text":"the convex and concave."},{"Start":"05:38.825 ","End":"05:42.030","Text":"Definition number 2."},{"Start":"05:42.200 ","End":"05:44.970","Text":"I\u0027ll need a sketch for that,"},{"Start":"05:44.970 ","End":"05:46.650","Text":"I don\u0027t want to use these again."},{"Start":"05:46.650 ","End":"05:51.875","Text":"But let\u0027s say I have here a convex function,"},{"Start":"05:51.875 ","End":"05:55.229","Text":"just like the blue there,"},{"Start":"05:55.229 ","End":"05:57.995","Text":"but I\u0027m going to continue with even more."},{"Start":"05:57.995 ","End":"06:04.640","Text":"This is convex, and here I\u0027ll draw a concave,"},{"Start":"06:04.640 ","End":"06:06.780","Text":"and if I continue it."},{"Start":"06:08.990 ","End":"06:13.300","Text":"Concave or concave down,"},{"Start":"06:13.430 ","End":"06:16.630","Text":"convex or concave up."},{"Start":"06:16.630 ","End":"06:18.545","Text":"The fact is this,"},{"Start":"06:18.545 ","End":"06:24.360","Text":"that if I take a convex function and I join any 2 points on it,"},{"Start":"06:24.970 ","End":"06:28.505","Text":"like here or here."},{"Start":"06:28.505 ","End":"06:32.060","Text":"These lines are called chords, by the way."},{"Start":"06:32.060 ","End":"06:37.200","Text":"It\u0027s spelled C-H-O-R-D, a line joining 2 points."},{"Start":"06:37.200 ","End":"06:40.890","Text":"Just like a tangent only touches 1 point a chord connects to."},{"Start":"06:40.890 ","End":"06:46.660","Text":"All these chords are above the curve, notice."},{"Start":"06:46.660 ","End":"06:50.255","Text":"Whereas in this , if I draw some chords,"},{"Start":"06:50.255 ","End":"06:51.890","Text":"like from here to here,"},{"Start":"06:51.890 ","End":"06:53.855","Text":"or from here to here,"},{"Start":"06:53.855 ","End":"06:56.375","Text":"or from here to here,"},{"Start":"06:56.375 ","End":"07:02.800","Text":"and so on, these chords are all below the curve."},{"Start":"07:03.160 ","End":"07:06.539","Text":"I\u0027ll just write something."},{"Start":"07:07.430 ","End":"07:15.390","Text":"Convex means chords above the curve,"},{"Start":"07:17.270 ","End":"07:20.280","Text":"and here in concave,"},{"Start":"07:20.280 ","End":"07:27.070","Text":"the chords are below the curve."},{"Start":"07:27.140 ","End":"07:30.750","Text":"This is not so easy to work with,"},{"Start":"07:30.750 ","End":"07:34.355","Text":"mainly we are going to be using this definition."},{"Start":"07:34.355 ","End":"07:38.060","Text":"In fact, soon we\u0027ll even actually get a 3rd definition which"},{"Start":"07:38.060 ","End":"07:43.530","Text":"is a lot more practical relating to the 2nd derivative, but patience."},{"Start":"07:46.280 ","End":"07:53.840","Text":"Now I\u0027m going to write a proposition that will give us, in a way,"},{"Start":"07:53.840 ","End":"07:57.920","Text":"a 3rd definition of convex and concave,"},{"Start":"07:57.920 ","End":"08:00.170","Text":"but before I write the proposition,"},{"Start":"08:00.170 ","End":"08:02.435","Text":"I just want to give a bit of background."},{"Start":"08:02.435 ","End":"08:06.520","Text":"We\u0027re often looking for geometrical meanings,"},{"Start":"08:07.850 ","End":"08:11.490","Text":"for example, the 1st derivative,"},{"Start":"08:11.490 ","End":"08:14.850","Text":"we used to ask, \"What does a derivative mean?\""},{"Start":"08:14.850 ","End":"08:20.320","Text":"One of the meanings of a derivative was the slope of the tangent,"},{"Start":"08:20.320 ","End":"08:22.760","Text":"but when we asked what does it mean for a derivative to be"},{"Start":"08:22.760 ","End":"08:25.460","Text":"positive or negative, we had an answer."},{"Start":"08:25.460 ","End":"08:28.640","Text":"Derivative positive, function is increasing,"},{"Start":"08:28.640 ","End":"08:31.595","Text":"derivative negative, function is decreasing,"},{"Start":"08:31.595 ","End":"08:33.950","Text":"and that\u0027s how we distinguish between increasing and"},{"Start":"08:33.950 ","End":"08:38.000","Text":"decreasing by looking at the derivative and seeing if it\u0027s positive or negative."},{"Start":"08:38.000 ","End":"08:42.380","Text":"Now, I asked the same question about the geometrical meaning of the 2nd derivative,"},{"Start":"08:42.380 ","End":"08:47.060","Text":"or specifically, what does it mean for the 2nd derivative to be positive or negative?"},{"Start":"08:47.060 ","End":"08:51.035","Text":"Well, it turns out this exactly relates to our matter in hand,"},{"Start":"08:51.035 ","End":"08:53.135","Text":"is that for convex functions,"},{"Start":"08:53.135 ","End":"08:55.775","Text":"the 2nd derivative is positive,"},{"Start":"08:55.775 ","End":"08:57.560","Text":"and for concave functions,"},{"Start":"08:57.560 ","End":"08:59.539","Text":"2nd derivative is negative."},{"Start":"08:59.539 ","End":"09:02.330","Text":"Let me first write this down."},{"Start":"09:03.530 ","End":"09:06.045","Text":"The fact is this,"},{"Start":"09:06.045 ","End":"09:14.680","Text":"if f-double-prime of x is bigger than 0,"},{"Start":"09:14.900 ","End":"09:20.320","Text":"then f is convex,"},{"Start":"09:21.410 ","End":"09:28.140","Text":"and if f-double-prime is less than 0,"},{"Start":"09:28.140 ","End":"09:34.000","Text":"then f is concave."},{"Start":"09:34.610 ","End":"09:37.380","Text":"That actually gives me a 3rd way,"},{"Start":"09:37.380 ","End":"09:38.860","Text":"and this is a practical way,"},{"Start":"09:38.860 ","End":"09:44.870","Text":"of knowing where I am convex and where I am concave."},{"Start":"09:46.810 ","End":"09:51.215","Text":"I am giving this proposition without explanation at the moment,"},{"Start":"09:51.215 ","End":"09:53.450","Text":"but at the end of the clip for those who want,"},{"Start":"09:53.450 ","End":"09:58.070","Text":"I will explain a proof of why this is so."},{"Start":"09:58.070 ","End":"10:00.410","Text":"Meanwhile, we\u0027ll just take it as given,"},{"Start":"10:00.410 ","End":"10:05.340","Text":"and I\u0027ll do an exercise to help illustrate the concept."},{"Start":"10:05.940 ","End":"10:08.990","Text":"Exercise."},{"Start":"10:13.860 ","End":"10:17.185","Text":"The exercise is to check whether this function,"},{"Start":"10:17.185 ","End":"10:20.935","Text":"f of x equals x to the fourth plus x squared minus 2x plus 5,"},{"Start":"10:20.935 ","End":"10:23.005","Text":"is convex or concave?"},{"Start":"10:23.005 ","End":"10:24.895","Text":"I\u0027m not going to draw it or sketch it."},{"Start":"10:24.895 ","End":"10:27.280","Text":"I\u0027m going to use the proposition and"},{"Start":"10:27.280 ","End":"10:31.190","Text":"the first thing I\u0027ll do is find what is f double-prime."},{"Start":"10:38.430 ","End":"10:42.610","Text":"Before I get to f double-prime,"},{"Start":"10:42.610 ","End":"10:46.420","Text":"I have to get to prime so I\u0027ll just write it below here."},{"Start":"10:46.420 ","End":"10:55.525","Text":"F prime of x is going to be 4x cubed plus 2x minus 2"},{"Start":"10:55.525 ","End":"11:06.860","Text":"and f double prime is equal to 12x squared plus 2."},{"Start":"11:06.900 ","End":"11:09.565","Text":"Now if I look at this,"},{"Start":"11:09.565 ","End":"11:11.680","Text":"x squared is non-negative,"},{"Start":"11:11.680 ","End":"11:15.610","Text":"it\u0027s 0 or positive and so all this 12x squared cannot be negative,"},{"Start":"11:15.610 ","End":"11:16.960","Text":"it\u0027s at least 0."},{"Start":"11:16.960 ","End":"11:20.650","Text":"If I add 2, then I\u0027m already at least 2 so"},{"Start":"11:20.650 ","End":"11:24.790","Text":"certainly all of these values for any x are positive,"},{"Start":"11:24.790 ","End":"11:32.740","Text":"positive even bigger than 2 so this is certainly bigger than 0 and therefore,"},{"Start":"11:32.740 ","End":"11:39.520","Text":"that means that f is therefore bigger than 0,"},{"Start":"11:39.520 ","End":"11:45.430","Text":"convex and that\u0027s it."},{"Start":"11:45.430 ","End":"11:51.580","Text":"The next point I want to make here is that life isn\u0027t always as"},{"Start":"11:51.580 ","End":"11:57.565","Text":"simple for a function to be always convex like this 1 or always concave."},{"Start":"11:57.565 ","End":"11:59.485","Text":"It could actually be mixed."},{"Start":"11:59.485 ","End":"12:03.715","Text":"I\u0027ll draw a little sketch to show you what I mean."},{"Start":"12:03.715 ","End":"12:09.415","Text":"Some axes here and"},{"Start":"12:09.415 ","End":"12:16.045","Text":"here and the function might go something"},{"Start":"12:16.045 ","End":"12:23.780","Text":"like this and then deep a bit and then go like this."},{"Start":"12:32.850 ","End":"12:36.265","Text":"Let\u0027s see, is this concave or convex?"},{"Start":"12:36.265 ","End":"12:39.670","Text":"Concave downwards because it\u0027s cave shaped when you look at it from"},{"Start":"12:39.670 ","End":"12:43.330","Text":"the bottom so this is concave and this part of"},{"Start":"12:43.330 ","End":"12:50.610","Text":"I\u0027ll say is blue because that\u0027s how we did the concave earlier but at some point,"},{"Start":"12:50.610 ","End":"12:52.950","Text":"here we see that trend is reversed,"},{"Start":"12:52.950 ","End":"12:54.255","Text":"that it\u0027s like this,"},{"Start":"12:54.255 ","End":"12:58.840","Text":"which is more like this and it starts to be convex."},{"Start":"12:58.840 ","End":"13:06.370","Text":"Convex we did in red so something like this."},{"Start":"13:06.370 ","End":"13:16.480","Text":"What we have is at some point it switches from being concave to convex."},{"Start":"13:16.480 ","End":"13:21.010","Text":"Let\u0027s see. Here we were concave or"},{"Start":"13:21.010 ","End":"13:27.730","Text":"concave down and here we are convex or concave up."},{"Start":"13:27.730 ","End":"13:32.650","Text":"This point actually has a name for kind of"},{"Start":"13:32.650 ","End":"13:38.800","Text":"point where it changes from concavity to convexity or the other way around."},{"Start":"13:38.800 ","End":"13:42.500","Text":"This is actually called an inflection point."},{"Start":"13:46.920 ","End":"13:49.510","Text":"In the next clip,"},{"Start":"13:49.510 ","End":"13:52.645","Text":"we\u0027ll be learning how to determine"},{"Start":"13:52.645 ","End":"13:58.060","Text":"inflection points and also intervals of concavity and convexity."},{"Start":"13:58.060 ","End":"14:03.880","Text":"It\u0027s actually going to be very similar to when we did increase and decrease."},{"Start":"14:03.880 ","End":"14:09.370","Text":"We found that the function is sometimes increasing and sometimes decreasing."},{"Start":"14:09.370 ","End":"14:13.420","Text":"For example, this might be the point and I know where"},{"Start":"14:13.420 ","End":"14:17.125","Text":"x equals 4 and then we would say, okay,"},{"Start":"14:17.125 ","End":"14:19.165","Text":"for x less than 4,"},{"Start":"14:19.165 ","End":"14:21.730","Text":"we have an interval of concavity,"},{"Start":"14:21.730 ","End":"14:28.000","Text":"from 4 to infinity we have convexity and at 4 itself we have an inflection point."},{"Start":"14:28.000 ","End":"14:31.150","Text":"There could be several inflection points."},{"Start":"14:31.150 ","End":"14:34.240","Text":"I mean, the curve could be like a snake or something."},{"Start":"14:34.240 ","End":"14:39.580","Text":"That we\u0027ll do in the next clip but at least you\u0027ve been introduced to it and the fact"},{"Start":"14:39.580 ","End":"14:45.200","Text":"that any given function could have both concavity and convexity in it."},{"Start":"14:46.250 ","End":"14:53.370","Text":"The only thing left is the explanation of the proposition,"},{"Start":"14:53.370 ","End":"15:01.775","Text":"which is certainly optional so those of you who want to skip that can end now."},{"Start":"15:01.775 ","End":"15:08.425","Text":"Very well. I\u0027m going to go ahead now and explain the proposition."},{"Start":"15:08.425 ","End":"15:11.140","Text":"Let\u0027s start with a sketch of"},{"Start":"15:11.140 ","End":"15:18.770","Text":"a convex function and then it\u0027ll be similarly for the concave."},{"Start":"15:23.490 ","End":"15:30.460","Text":"There\u0027s my convex function and what we said was that 1 of"},{"Start":"15:30.460 ","End":"15:36.715","Text":"the definitions of convex is that the slopes are increasing."},{"Start":"15:36.715 ","End":"15:39.325","Text":"If I take a bunch of slopes,"},{"Start":"15:39.325 ","End":"15:41.995","Text":"slopes of the tangent, I mean."},{"Start":"15:41.995 ","End":"15:43.780","Text":"When I say slope of a function at a point,"},{"Start":"15:43.780 ","End":"15:46.840","Text":"I mean the slope of the tangent to the function."},{"Start":"15:46.840 ","End":"15:52.600","Text":"At each point, I have a slope and the slopes themselves,"},{"Start":"15:52.600 ","End":"15:56.305","Text":"their values are sort of a function in themselves and this function is here,"},{"Start":"15:56.305 ","End":"15:58.780","Text":"may be minus 6,"},{"Start":"15:58.780 ","End":"16:02.080","Text":"minus 3, minus 2,"},{"Start":"16:02.080 ","End":"16:06.175","Text":"minus 1, 0, 1/2,"},{"Start":"16:06.175 ","End":"16:08.680","Text":"3/4, 1, 2, and so on."},{"Start":"16:08.680 ","End":"16:11.180","Text":"They are increasing."},{"Start":"16:11.640 ","End":"16:14.935","Text":"In the case of convex,"},{"Start":"16:14.935 ","End":"16:17.770","Text":"this will be my f of x."},{"Start":"16:17.770 ","End":"16:20.260","Text":"If f is convex,"},{"Start":"16:20.260 ","End":"16:31.580","Text":"then that means that the slope which is f prime is increasing."},{"Start":"16:32.790 ","End":"16:40.135","Text":"By definition, the slopes are increasing and the slopes are just described by f prime."},{"Start":"16:40.135 ","End":"16:47.215","Text":"Now, in general, if any function say g,"},{"Start":"16:47.215 ","End":"16:48.970","Text":"think of it as a g for the moment."},{"Start":"16:48.970 ","End":"16:51.500","Text":"If g is increasing,"},{"Start":"16:52.020 ","End":"16:55.750","Text":"means that G prime is positive."},{"Start":"16:55.750 ","End":"17:01.390","Text":"I\u0027ll just write that if g is increasing for any function g,"},{"Start":"17:01.390 ","End":"17:06.955","Text":"g is increasing, we said that means that g prime is positive."},{"Start":"17:06.955 ","End":"17:10.420","Text":"In our case g is just the slopes,"},{"Start":"17:10.420 ","End":"17:16.030","Text":"which is f prime so that means that f prime is our g so f prime,"},{"Start":"17:16.030 ","End":"17:21.100","Text":"prime is positive but what is f prime, prime?"},{"Start":"17:21.100 ","End":"17:23.800","Text":"The derivative of f prime is f double prime."},{"Start":"17:23.800 ","End":"17:30.190","Text":"That means that f double prime is bigger than 0 and similarly,"},{"Start":"17:30.190 ","End":"17:34.315","Text":"if we had that f was concave,"},{"Start":"17:34.315 ","End":"17:42.040","Text":"our definition was that the f prime is decreasing."},{"Start":"17:42.040 ","End":"17:45.130","Text":"Well, we really missed a step here and so the slope is"},{"Start":"17:45.130 ","End":"17:51.320","Text":"increasing so here the slope is decreasing."},{"Start":"17:51.510 ","End":"17:55.330","Text":"Maybe I\u0027ll write it all. Why am I being lazy?"},{"Start":"17:55.330 ","End":"17:59.440","Text":"That f prime is decreasing,"},{"Start":"17:59.440 ","End":"18:05.440","Text":"which is the slope and that means that f prime,"},{"Start":"18:05.440 ","End":"18:11.350","Text":"same logic as before, if it\u0027s decreasing,"},{"Start":"18:11.350 ","End":"18:14.920","Text":"then its derivative is negative and the derivative of"},{"Start":"18:14.920 ","End":"18:19.495","Text":"the derivative is simply the second derivative is negative."},{"Start":"18:19.495 ","End":"18:22.855","Text":"That\u0027s basically what our proposition says."},{"Start":"18:22.855 ","End":"18:28.015","Text":"That f double-prime, bigger than 0 for convex,"},{"Start":"18:28.015 ","End":"18:34.975","Text":"f double-prime smaller than 0 for concave and that\u0027s it."},{"Start":"18:34.975 ","End":"18:37.100","Text":"The end of the clip."}],"ID":1666},{"Watched":false,"Name":"Inflection Points","Duration":"18m 42s","ChapterTopicVideoID":1655,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1655.jpeg","UploadDate":"2019-10-29T13:07:05.3730000","DurationForVideoObject":"PT18M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"In this section, I\u0027m going to talk about inflection points"},{"Start":"00:03.990 ","End":"00:09.580","Text":"and their relation to concavity and convexity."},{"Start":"00:09.740 ","End":"00:13.395","Text":"I\u0027ll start straight away with an example."},{"Start":"00:13.395 ","End":"00:16.289","Text":"I\u0027ll sketch a graph."},{"Start":"00:16.289 ","End":"00:19.455","Text":"Just take a bit of a y-axis,"},{"Start":"00:19.455 ","End":"00:22.620","Text":"a bit of an x-axis."},{"Start":"00:22.620 ","End":"00:28.350","Text":"Let\u0027s take something that dips down here,"},{"Start":"00:28.350 ","End":"00:30.210","Text":"and then goes up here,"},{"Start":"00:30.210 ","End":"00:33.060","Text":"and maybe dips down again,"},{"Start":"00:33.060 ","End":"00:39.610","Text":"and continues somewhere I\u0027m not sure where, but continues."},{"Start":"00:40.160 ","End":"00:47.640","Text":"If we look at it, we see that here it looks like from a cave, from below."},{"Start":"00:47.640 ","End":"00:54.470","Text":"This is a concave part of the curve and so is this but here we are convex."},{"Start":"00:54.470 ","End":"01:01.395","Text":"In fact, let me just highlight where it\u0027s convex."},{"Start":"01:01.395 ","End":"01:07.810","Text":"Convex would be something where the concave is upwards."},{"Start":"01:07.810 ","End":"01:13.760","Text":"Looks like around here it\u0027s convex up to some point,"},{"Start":"01:13.760 ","End":"01:16.040","Text":"hard to say exactly."},{"Start":"01:16.040 ","End":"01:25.895","Text":"Here it\u0027s concave, and here it\u0027s concave."},{"Start":"01:25.895 ","End":"01:29.280","Text":"Oops, missed a bit there."},{"Start":"01:32.090 ","End":"01:35.430","Text":"Where the concave meets the convex,"},{"Start":"01:35.430 ","End":"01:39.800","Text":"there are special points to"},{"Start":"01:39.800 ","End":"01:45.395","Text":"indicate the change of trend from concave to convex or from convex to concave,"},{"Start":"01:45.395 ","End":"01:48.845","Text":"and these points are called inflection points."},{"Start":"01:48.845 ","End":"01:52.800","Text":"Over here and over here I"},{"Start":"01:52.800 ","End":"02:00.820","Text":"have an inflection point or 2 inflection points."},{"Start":"02:06.050 ","End":"02:12.380","Text":"If I look at the x values on the x-axis of these 2 points,"},{"Start":"02:12.380 ","End":"02:14.630","Text":"this will tell me something."},{"Start":"02:14.630 ","End":"02:19.415","Text":"This actually will divide the line up into different intervals,"},{"Start":"02:19.415 ","End":"02:21.950","Text":"in this case 3 intervals up to here."},{"Start":"02:21.950 ","End":"02:24.960","Text":"From here to here and from here to here."},{"Start":"02:31.400 ","End":"02:36.470","Text":"We can say that the function is concave when x is here,"},{"Start":"02:36.470 ","End":"02:40.624","Text":"we can say that the function is convex when x is in this interval,"},{"Start":"02:40.624 ","End":"02:44.460","Text":"and that it\u0027s again concave when x is in this interval."},{"Start":"02:45.340 ","End":"02:49.054","Text":"I often draw a little picture for concave,"},{"Start":"02:49.054 ","End":"02:51.185","Text":"I indicate it like this,"},{"Start":"02:51.185 ","End":"02:53.515","Text":"only I got it wrong."},{"Start":"02:53.515 ","End":"02:57.450","Text":"Convex I illustrate like this,"},{"Start":"02:57.450 ","End":"03:00.285","Text":"and concave again like this."},{"Start":"03:00.285 ","End":"03:02.325","Text":"Just saves writing."},{"Start":"03:02.325 ","End":"03:05.330","Text":"Just like for increasing and decreasing functions,"},{"Start":"03:05.330 ","End":"03:08.465","Text":"we wrote a little diagonal arrow and so on."},{"Start":"03:08.465 ","End":"03:16.480","Text":"This enables us to actually now determine the intervals for concavity and convexity."},{"Start":"03:17.750 ","End":"03:20.495","Text":"If our function is f of x,"},{"Start":"03:20.495 ","End":"03:26.545","Text":"we can write that f is concave"},{"Start":"03:26.545 ","End":"03:31.850","Text":"for x less than"},{"Start":"03:31.850 ","End":"03:39.775","Text":"4 and for x bigger than 10."},{"Start":"03:39.775 ","End":"03:50.880","Text":"Similarly, f is convex for or when x is between 4 and 10."},{"Start":"03:50.880 ","End":"03:55.215","Text":"4 less than x, less than 10."},{"Start":"03:55.215 ","End":"03:57.499","Text":"As I already mentioned,"},{"Start":"03:57.499 ","End":"04:02.590","Text":"the points on the border of concavity and convexity whether"},{"Start":"04:02.590 ","End":"04:08.750","Text":"you look at the point on the graph or more customary the points on the x-axis."},{"Start":"04:08.750 ","End":"04:11.900","Text":"We\u0027ve mentioned this, but let\u0027s write it down"},{"Start":"04:11.900 ","End":"04:16.770","Text":"anyway as a little bit more formal definition."},{"Start":"04:16.780 ","End":"04:22.140","Text":"Like this, an inflection point"},{"Start":"04:25.750 ","End":"04:30.320","Text":"is a point at which the function changes its trend"},{"Start":"04:30.320 ","End":"04:34.610","Text":"from concavity to convexity or vice-versa,"},{"Start":"04:34.610 ","End":"04:39.610","Text":"means the other way around from convexity to concavity."},{"Start":"04:40.490 ","End":"04:47.720","Text":"In our case, the inflection points happen to be x equals 4 and x equals 10."},{"Start":"04:47.720 ","End":"04:54.340","Text":"Now, we have taken a sketch and deduced all these things from the sketch,"},{"Start":"04:54.340 ","End":"04:56.930","Text":"but of course in practice you\u0027re not going to get a sketch,"},{"Start":"04:56.930 ","End":"04:59.059","Text":"you\u0027re going to get a formula, an equation."},{"Start":"04:59.059 ","End":"05:01.310","Text":"You\u0027ll get a typical question like,"},{"Start":"05:01.310 ","End":"05:07.475","Text":"consider the function f of x equals x^4 minus x squared plus 10."},{"Start":"05:07.475 ","End":"05:09.530","Text":"Find its inflection points,"},{"Start":"05:09.530 ","End":"05:13.195","Text":"and find the intervals of concavity and convexity."},{"Start":"05:13.195 ","End":"05:17.130","Text":"We need a technique to do that without the sketches."},{"Start":"05:19.610 ","End":"05:27.170","Text":"Our main tool will be the second derivative from the previous clip,"},{"Start":"05:27.170 ","End":"05:31.755","Text":"because there we learnt that when a function is concave"},{"Start":"05:31.755 ","End":"05:37.675","Text":"that second derivative is negative."},{"Start":"05:37.675 ","End":"05:44.295","Text":"When it\u0027s convex, I don\u0027t have much room here I\u0027ll write it here,"},{"Start":"05:44.295 ","End":"05:47.910","Text":"f double prime is positive."},{"Start":"05:47.910 ","End":"05:54.610","Text":"Again, concave f double prime is negative."},{"Start":"05:54.610 ","End":"05:58.775","Text":"It would make sense to expect that an inflection point"},{"Start":"05:58.775 ","End":"06:03.335","Text":"f double prime is exactly 0 and that is the case."},{"Start":"06:03.335 ","End":"06:07.939","Text":"I\u0027m going to write that down as a proposition, or a claim,"},{"Start":"06:07.939 ","End":"06:14.230","Text":"or not a theorem but like a proposition."},{"Start":"06:26.600 ","End":"06:31.370","Text":"This reminds me a bit of the proposition about extrema,"},{"Start":"06:31.370 ","End":"06:36.090","Text":"where at an extremum f single prime is 0."},{"Start":"06:36.090 ","End":"06:38.750","Text":"When we looked for an extremum,"},{"Start":"06:38.750 ","End":"06:44.060","Text":"we searched amongst suspects where f prime was 0."},{"Start":"06:44.060 ","End":"06:45.710","Text":"Now just as there,"},{"Start":"06:45.710 ","End":"06:51.555","Text":"it wasn\u0027t a guarantee that if f prime is 0 we have an extremum,"},{"Start":"06:51.555 ","End":"06:53.720","Text":"the same here just the fact that"},{"Start":"06:53.720 ","End":"06:57.320","Text":"f double prime being 0 doesn\u0027t guarantee an inflection point."},{"Start":"06:57.320 ","End":"07:01.019","Text":"I\u0027ll write that as a note,"},{"Start":"07:03.650 ","End":"07:09.245","Text":"it is possible that f double prime is 0 at a point,"},{"Start":"07:09.245 ","End":"07:11.870","Text":"but the point is not an inflection point."},{"Start":"07:11.870 ","End":"07:13.990","Text":"It\u0027s not a guarantee."},{"Start":"07:13.990 ","End":"07:17.580","Text":"We know that at an inflection point f double prime is 0,"},{"Start":"07:17.580 ","End":"07:19.770","Text":"but if this holds and it were f,"},{"Start":"07:19.770 ","End":"07:25.595","Text":"the point is just a suspect or candidate to be an inflection point."},{"Start":"07:25.595 ","End":"07:29.039","Text":"Very similar to what we had with extrema."},{"Start":"07:29.110 ","End":"07:34.775","Text":"Actually, I can easily give an example where this is true."},{"Start":"07:34.775 ","End":"07:38.750","Text":"For example, a straight line function."},{"Start":"07:38.750 ","End":"07:40.985","Text":"I\u0027ll give an example of the note."},{"Start":"07:40.985 ","End":"07:44.150","Text":"If we take f of x equals,"},{"Start":"07:44.150 ","End":"07:48.540","Text":"I don\u0027t know, 4x plus 10 as a straight line,"},{"Start":"07:48.540 ","End":"07:50.145","Text":"then what we get is,"},{"Start":"07:50.145 ","End":"07:53.170","Text":"if we figure it in our heads,"},{"Start":"07:53.840 ","End":"07:57.045","Text":"f prime is 4,"},{"Start":"07:57.045 ","End":"07:59.725","Text":"f double prime is 0."},{"Start":"07:59.725 ","End":"08:03.550","Text":"F double prime equals 0 always,"},{"Start":"08:03.550 ","End":"08:06.030","Text":"for every x I mean."},{"Start":"08:06.030 ","End":"08:09.180","Text":"Yet it doesn\u0027t have any inflection points,"},{"Start":"08:09.180 ","End":"08:11.565","Text":"there\u0027s no concavity or convexity."},{"Start":"08:11.565 ","End":"08:13.500","Text":"It\u0027s a straight line."},{"Start":"08:13.500 ","End":"08:16.550","Text":"In this case you will definitely have an example,"},{"Start":"08:16.550 ","End":"08:20.690","Text":"all the points will have f prime 0 and none of them will be inflection."},{"Start":"08:20.690 ","End":"08:23.220","Text":"This is not a guarantee."},{"Start":"08:23.780 ","End":"08:29.765","Text":"In general we\u0027ll look for inflection points by taking the function,"},{"Start":"08:29.765 ","End":"08:32.660","Text":"deriving it twice to get f double prime,"},{"Start":"08:32.660 ","End":"08:37.130","Text":"setting that equals to 0, finding the solutions."},{"Start":"08:37.130 ","End":"08:42.950","Text":"Amongst those solutions we\u0027ll check if it\u0027s inflection by looking if it\u0027s concave"},{"Start":"08:42.950 ","End":"08:48.080","Text":"on the left and convex on the right or well, the other way around."},{"Start":"08:48.080 ","End":"08:49.610","Text":"If so we do have,"},{"Start":"08:49.610 ","End":"08:51.575","Text":"and if not then we don\u0027t."},{"Start":"08:51.575 ","End":"08:58.820","Text":"I think an example will really be welcome here and you\u0027ll understand everything I hope."},{"Start":"08:58.820 ","End":"09:02.130","Text":"Let\u0027s take an example."},{"Start":"09:07.800 ","End":"09:16.870","Text":"Let\u0027s take f of x to be x to"},{"Start":"09:16.870 ","End":"09:24.790","Text":"the 4th minus 10x cubed"},{"Start":"09:24.790 ","End":"09:33.175","Text":"plus 24x squared plus x plus 1."},{"Start":"09:33.175 ","End":"09:40.780","Text":"What we have to do, find"},{"Start":"09:40.780 ","End":"09:46.270","Text":"the inflection points and"},{"Start":"09:46.270 ","End":"09:55.120","Text":"the intervals of convexity and concavity."},{"Start":"09:55.120 ","End":"10:00.400","Text":"We\u0027ll tackle this pretty much in the same way as we did"},{"Start":"10:00.400 ","End":"10:05.935","Text":"for extrema and intervals of increase and decrease."},{"Start":"10:05.935 ","End":"10:09.100","Text":"We\u0027ll do it by means of a table and the same steps."},{"Start":"10:09.100 ","End":"10:12.160","Text":"We\u0027ll do it in steps with the first step,"},{"Start":"10:12.160 ","End":"10:16.765","Text":"let me write solution first, solution."},{"Start":"10:16.765 ","End":"10:18.235","Text":"Let\u0027s do the first step,"},{"Start":"10:18.235 ","End":"10:23.050","Text":"there we called it preparation and same thing I\u0027ll do here."},{"Start":"10:23.050 ","End":"10:24.970","Text":"I won\u0027t write the names of the steps,"},{"Start":"10:24.970 ","End":"10:31.105","Text":"but step preparation was finding the derivative that was there."},{"Start":"10:31.105 ","End":"10:32.830","Text":"Here we have a bit more work."},{"Start":"10:32.830 ","End":"10:35.920","Text":"We have to find the second derivative."},{"Start":"10:35.920 ","End":"10:38.575","Text":"Here we have f of x."},{"Start":"10:38.575 ","End":"10:41.680","Text":"We have f prime of x,"},{"Start":"10:41.680 ","End":"10:43.420","Text":"that go through that first,"},{"Start":"10:43.420 ","End":"10:55.410","Text":"is equal to 4x cubed minus 30x squared plus"},{"Start":"10:55.410 ","End":"11:06.100","Text":"48x plus 1 and"},{"Start":"11:06.100 ","End":"11:11.440","Text":"that makes f double-prime of x equal to 12x"},{"Start":"11:11.440 ","End":"11:19.570","Text":"squared minus 60x plus 48."},{"Start":"11:19.570 ","End":"11:21.595","Text":"That\u0027s the preparation stage."},{"Start":"11:21.595 ","End":"11:24.160","Text":"Then the second stage,"},{"Start":"11:24.160 ","End":"11:29.395","Text":"the next stage, stage 1 or whatever,"},{"Start":"11:29.395 ","End":"11:32.949","Text":"was to set f double prime of x equal to 0."},{"Start":"11:32.949 ","End":"11:36.984","Text":"If we let f double prime of x equals 0,"},{"Start":"11:36.984 ","End":"11:47.510","Text":"then we get 12x squared minus 60x plus 48 equals 0."},{"Start":"11:47.510 ","End":"11:49.320","Text":"We can cancel."},{"Start":"11:49.320 ","End":"11:51.540","Text":"I see the 12 goes into all of these,"},{"Start":"11:51.540 ","End":"11:53.430","Text":"so we divide by 12,"},{"Start":"11:53.430 ","End":"12:00.385","Text":"we get x squared minus 5x plus 4 equals 0."},{"Start":"12:00.385 ","End":"12:04.105","Text":"In this step we also have to solve the quadratic equation."},{"Start":"12:04.105 ","End":"12:07.240","Text":"Now I\u0027m not going to waste time solving quadratic equations."},{"Start":"12:07.240 ","End":"12:10.300","Text":"You can do these, so I can just tell you the answer that"},{"Start":"12:10.300 ","End":"12:16.900","Text":"x can be equal to 1 and x could be equal to 4,"},{"Start":"12:16.900 ","End":"12:22.790","Text":"these are the 2 solutions where f double prime of x is 0."},{"Start":"12:22.920 ","End":"12:27.790","Text":"The next step is to draw a table pretty much"},{"Start":"12:27.790 ","End":"12:32.440","Text":"like the 1 we had for increase, decrease in extrema."},{"Start":"12:32.440 ","End":"12:35.440","Text":"Except that instead of the row for f prime,"},{"Start":"12:35.440 ","End":"12:37.135","Text":"we\u0027re going to have f double prime."},{"Start":"12:37.135 ","End":"12:39.820","Text":"If I remember, we had several rows."},{"Start":"12:39.820 ","End":"12:42.370","Text":"The first row was called x,"},{"Start":"12:42.370 ","End":"12:46.165","Text":"over there in the increased, decreased."},{"Start":"12:46.165 ","End":"12:48.220","Text":"Then we had f prime of x,"},{"Start":"12:48.220 ","End":"12:51.925","Text":"but here we\u0027re going to have f double-prime of x."},{"Start":"12:51.925 ","End":"12:55.880","Text":"Then we had the row called f of x."},{"Start":"12:56.460 ","End":"12:58.540","Text":"Later at the end,"},{"Start":"12:58.540 ","End":"13:00.010","Text":"or I could put it in now,"},{"Start":"13:00.010 ","End":"13:02.720","Text":"we had a row called y."},{"Start":"13:02.940 ","End":"13:08.815","Text":"Let me just write some separating lines between the rows."},{"Start":"13:08.815 ","End":"13:12.040","Text":"Have 1 here, 1 here,"},{"Start":"13:12.040 ","End":"13:15.400","Text":"1 here that\u0027ll do."},{"Start":"13:15.400 ","End":"13:19.885","Text":"For x, we take the values that we found here,"},{"Start":"13:19.885 ","End":"13:23.080","Text":"which is 1 and 4"},{"Start":"13:23.080 ","End":"13:27.160","Text":"and we leave some space because we are going to be writing things in the middle."},{"Start":"13:27.160 ","End":"13:30.940","Text":"We know that f double-prime at both these points is 0,"},{"Start":"13:30.940 ","End":"13:33.415","Text":"so that gets us started off."},{"Start":"13:33.415 ","End":"13:39.560","Text":"Then what we do is we find intervals."},{"Start":"13:39.660 ","End":"13:47.045","Text":"This blank area represents the interval where x is less than 1."},{"Start":"13:47.045 ","End":"13:50.520","Text":"Here we have x between 1 and 4."},{"Start":"13:50.520 ","End":"13:54.480","Text":"Write it as 1 less than x, less than 4."},{"Start":"13:54.480 ","End":"13:58.835","Text":"This part is x bigger than 4."},{"Start":"13:58.835 ","End":"14:03.715","Text":"The next thing to do is to take a sample point from each of these intervals."},{"Start":"14:03.715 ","End":"14:06.415","Text":"It could be anything you want."},{"Start":"14:06.415 ","End":"14:09.835","Text":"Here I\u0027ll just take 0, it\u0027s a convenient number,"},{"Start":"14:09.835 ","End":"14:13.755","Text":"here I\u0027ll choose say 2 as my sample point,"},{"Start":"14:13.755 ","End":"14:17.360","Text":"and here I\u0027ll choose 5 as my sample point."},{"Start":"14:17.360 ","End":"14:19.000","Text":"What do we do with these?"},{"Start":"14:19.000 ","End":"14:22.855","Text":"We substitute into f double-prime of x."},{"Start":"14:22.855 ","End":"14:25.360","Text":"But I actually don\u0027t want the value,"},{"Start":"14:25.360 ","End":"14:29.260","Text":"I only care if it\u0027s positive or negative."},{"Start":"14:29.260 ","End":"14:35.795","Text":"If I substitute x equals 0,"},{"Start":"14:35.795 ","End":"14:40.950","Text":"here, I get f double prime of x is 48."},{"Start":"14:40.950 ","End":"14:44.945","Text":"Let me just highlight this to say where I\u0027m substituting."},{"Start":"14:44.945 ","End":"14:47.650","Text":"Yeah, x is 0,"},{"Start":"14:47.650 ","End":"14:51.860","Text":"f double-prime is 48."},{"Start":"14:53.370 ","End":"15:00.230","Text":"But as I said, I don\u0027t really want the values,"},{"Start":"15:02.040 ","End":"15:07.255","Text":"all I need is the plus or minus, the sign."},{"Start":"15:07.255 ","End":"15:10.855","Text":"This is plus and when it\u0027s plus,"},{"Start":"15:10.855 ","End":"15:16.645","Text":"that means that I have the convex shape like this,"},{"Start":"15:16.645 ","End":"15:18.625","Text":"the function behaves like this."},{"Start":"15:18.625 ","End":"15:22.600","Text":"I\u0027m not going to do all the computations for you, take my word for it."},{"Start":"15:22.600 ","End":"15:28.540","Text":"This comes out a minus and so here we have concave,"},{"Start":"15:28.540 ","End":"15:33.380","Text":"and here we get a plus again, so convex again."},{"Start":"15:33.480 ","End":"15:37.540","Text":"Now we check what happens at these points here."},{"Start":"15:37.540 ","End":"15:41.725","Text":"Are they inflection points or not?"},{"Start":"15:41.725 ","End":"15:47.515","Text":"Because this changes from convex to concave,"},{"Start":"15:47.515 ","End":"15:48.805","Text":"we definitely have a change,"},{"Start":"15:48.805 ","End":"15:51.700","Text":"then this really is an inflection point."},{"Start":"15:51.700 ","End":"15:57.865","Text":"I\u0027ll write inflection here,"},{"Start":"15:57.865 ","End":"16:01.075","Text":"and I\u0027ll write inflection here."},{"Start":"16:01.075 ","End":"16:03.655","Text":"Let\u0027s try through a long word,"},{"Start":"16:03.655 ","End":"16:06.410","Text":"make an abbreviation for it."},{"Start":"16:06.570 ","End":"16:15.985","Text":"Then finally, we substitute the value of y just for these 2 points,"},{"Start":"16:15.985 ","End":"16:19.300","Text":"these that were suspects and we now know the inflection points,"},{"Start":"16:19.300 ","End":"16:22.030","Text":"but there\u0027s still an x and a y to a point."},{"Start":"16:22.030 ","End":"16:29.529","Text":"If I plug in 1 this time in the function itself,"},{"Start":"16:29.529 ","End":"16:32.905","Text":"now what happened to my original function?"},{"Start":"16:32.905 ","End":"16:35.990","Text":"Must be up there somewhere."},{"Start":"16:37.140 ","End":"16:44.060","Text":"There it is. Let\u0027s see if I plug in 1."},{"Start":"16:44.610 ","End":"16:48.490","Text":"I went and did it at the side and here\u0027s the answer;"},{"Start":"16:48.490 ","End":"16:53.150","Text":"here it\u0027s 35 and here it\u0027s 5."},{"Start":"16:55.890 ","End":"17:03.110","Text":"You pretty much have all that we can get out of this f double-prime."},{"Start":"17:03.540 ","End":"17:06.820","Text":"I don\u0027t have enough maybe for a sketch,"},{"Start":"17:06.820 ","End":"17:09.475","Text":"but I certainly have enough to answer the question"},{"Start":"17:09.475 ","End":"17:14.110","Text":"and the question set to ask for inflection points."},{"Start":"17:14.110 ","End":"17:23.860","Text":"I have inflection points,"},{"Start":"17:23.860 ","End":"17:34.525","Text":"1, 35 and 4, 5."},{"Start":"17:34.525 ","End":"17:41.050","Text":"The other thing that we\u0027re asked were the intervals of concavity and convexity."},{"Start":"17:41.050 ","End":"17:47.540","Text":"I have concavity;"},{"Start":"17:47.760 ","End":"17:52.510","Text":"Concave is concave down is this,"},{"Start":"17:52.510 ","End":"17:59.935","Text":"and that\u0027s between 1 and 4 so 1 less than x, less than 4."},{"Start":"17:59.935 ","End":"18:05.480","Text":"The intervals of convexity,"},{"Start":"18:11.430 ","End":"18:16.090","Text":"2 of those, x less than"},{"Start":"18:16.090 ","End":"18:23.480","Text":"1 and x greater than 4."},{"Start":"18:24.750 ","End":"18:28.190","Text":"That answers the question."},{"Start":"18:28.290 ","End":"18:34.370","Text":"You should practice; there are lots and lots of exercises besides this tutorial."},{"Start":"18:34.370 ","End":"18:39.080","Text":"I suggest you go ahead and start working on them."},{"Start":"18:39.080 ","End":"18:41.730","Text":"That\u0027s all. Bye."}],"ID":1667},{"Watched":false,"Name":"Exercise 1","Duration":"2m 26s","ChapterTopicVideoID":4848,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4848.jpeg","UploadDate":"2016-05-01T17:10:55.8000000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.864","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:02.864 ","End":"00:06.945","Text":"f of x equals x^2 minus 2x plus 5."},{"Start":"00:06.945 ","End":"00:11.325","Text":"What we have to do is find all of its inflection points and"},{"Start":"00:11.325 ","End":"00:16.005","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:16.005 ","End":"00:20.669","Text":"This kind of exercise has a certain cookbook-style routine."},{"Start":"00:20.669 ","End":"00:24.270","Text":"There\u0027s a standard set of steps that we follow to get to the answer."},{"Start":"00:24.270 ","End":"00:28.185","Text":"I first give a quick glance to see if there\u0027s any problems with the domain."},{"Start":"00:28.185 ","End":"00:30.060","Text":"But I see there\u0027s no problems there."},{"Start":"00:30.060 ","End":"00:32.010","Text":"It\u0027s defined for all x."},{"Start":"00:32.010 ","End":"00:36.585","Text":"Then we have the preparation phase which is to find f double prime."},{"Start":"00:36.585 ","End":"00:39.005","Text":"But first, we have to find the first derivative."},{"Start":"00:39.005 ","End":"00:44.270","Text":"So f prime of x is 2x minus 2,"},{"Start":"00:44.270 ","End":"00:49.565","Text":"and then f double prime of x is 2 constant."},{"Start":"00:49.565 ","End":"00:53.195","Text":"Then we come to the first proper, step 1,"},{"Start":"00:53.195 ","End":"00:58.399","Text":"which is to set f double prime of x to be equal to 0."},{"Start":"00:58.399 ","End":"01:03.490","Text":"We\u0027re looking for suspects for inflection points."},{"Start":"01:03.490 ","End":"01:09.515","Text":"This won\u0027t give us any solutions because this will give us that 2 equals 0,"},{"Start":"01:09.515 ","End":"01:12.335","Text":"and so there\u0027s no solutions."},{"Start":"01:12.335 ","End":"01:16.160","Text":"Whenever x is, 2 is not going to be equal to 0."},{"Start":"01:16.160 ","End":"01:18.020","Text":"No solution there."},{"Start":"01:18.020 ","End":"01:20.495","Text":"Then we get to our table."},{"Start":"01:20.495 ","End":"01:23.720","Text":"In this case, the table is going to be rather empty"},{"Start":"01:23.720 ","End":"01:28.550","Text":"because there are no points where f double prime is 0"},{"Start":"01:28.550 ","End":"01:32.240","Text":"or no points where the function is undefined and"},{"Start":"01:32.240 ","End":"01:36.320","Text":"no points where the second derivative is undefined, it\u0027s just nothing."},{"Start":"01:36.320 ","End":"01:39.950","Text":"It\u0027s all one big interval which is all x."},{"Start":"01:39.950 ","End":"01:42.665","Text":"In any event, we can pick a random point."},{"Start":"01:42.665 ","End":"01:47.675","Text":"Let\u0027s say x equals 0 because all x is what goes here."},{"Start":"01:47.675 ","End":"01:50.300","Text":"f double prime is 2"},{"Start":"01:50.300 ","End":"01:53.090","Text":"but all we care about is that it\u0027s positive."},{"Start":"01:53.090 ","End":"01:57.650","Text":"Then f of x is convex or concave up."},{"Start":"01:57.650 ","End":"02:02.420","Text":"Basically, that\u0027s all we need to know in order to answer the questions."},{"Start":"02:02.420 ","End":"02:05.585","Text":"We\u0027re asked for inflection points."},{"Start":"02:05.585 ","End":"02:06.770","Text":"This is the last stage,"},{"Start":"02:06.770 ","End":"02:10.190","Text":"which is the conclusion stage of the exercise."},{"Start":"02:10.190 ","End":"02:12.380","Text":"Our inflection point\u0027s none."},{"Start":"02:12.380 ","End":"02:16.845","Text":"Then we need to know where it\u0027s concave up or convex,"},{"Start":"02:16.845 ","End":"02:19.133","Text":"and that is all x."},{"Start":"02:19.133 ","End":"02:26.800","Text":"Concave or concave down is nowhere. We\u0027re done."}],"ID":4848},{"Watched":false,"Name":"Exercise 2","Duration":"3m 59s","ChapterTopicVideoID":4849,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4849.jpeg","UploadDate":"2016-05-01T17:11:21.6570000","DurationForVideoObject":"PT3M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.279","Text":"In this exercise, we\u0027re given a function f of x equals x cube minus 3x,"},{"Start":"00:05.279 ","End":"00:09.330","Text":"and we have to find all of its inflection points and"},{"Start":"00:09.330 ","End":"00:13.485","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.485 ","End":"00:17.025","Text":"This type of exercise is well-known and there are"},{"Start":"00:17.025 ","End":"00:20.850","Text":"the standard set of steps to solving it, cookbook style,"},{"Start":"00:20.850 ","End":"00:24.540","Text":"and there\u0027s a preparation stage where, first of all,"},{"Start":"00:24.540 ","End":"00:28.755","Text":"I just take a look and see if there\u0027s any problems with the domain."},{"Start":"00:28.755 ","End":"00:30.090","Text":"No, all is well there."},{"Start":"00:30.090 ","End":"00:31.890","Text":"It\u0027s defined for all x."},{"Start":"00:31.890 ","End":"00:38.340","Text":"The first proper thing to do is to find the second derivative."},{"Start":"00:38.340 ","End":"00:39.860","Text":"The second derivative, well,"},{"Start":"00:39.860 ","End":"00:41.660","Text":"we have to go through the first derivative,"},{"Start":"00:41.660 ","End":"00:45.065","Text":"so let\u0027s figure out what\u0027s f prime of x."},{"Start":"00:45.065 ","End":"00:48.715","Text":"Straightforward enough, 3x squared minus 3,"},{"Start":"00:48.715 ","End":"00:54.235","Text":"and then the second derivative which is just 6x."},{"Start":"00:54.235 ","End":"00:58.725","Text":"We\u0027re looking for suspect for inflection points,"},{"Start":"00:58.725 ","End":"01:03.840","Text":"and we get suspect mainly from setting f double prime equal 0."},{"Start":"01:03.840 ","End":"01:05.880","Text":"That\u0027s the first proper step,"},{"Start":"01:05.880 ","End":"01:10.655","Text":"so let\u0027s set f double prime of x equal 0,"},{"Start":"01:10.655 ","End":"01:14.465","Text":"which means that 6x is 0,"},{"Start":"01:14.465 ","End":"01:16.820","Text":"so x equals 0,"},{"Start":"01:16.820 ","End":"01:21.050","Text":"and that\u0027s our only suspect for inflection points."},{"Start":"01:21.050 ","End":"01:23.630","Text":"Others could come from points where the function is not"},{"Start":"01:23.630 ","End":"01:27.170","Text":"defined or where the second derivative is undefined,"},{"Start":"01:27.170 ","End":"01:28.370","Text":"but we don\u0027t have those,"},{"Start":"01:28.370 ","End":"01:29.960","Text":"so x equals 0."},{"Start":"01:29.960 ","End":"01:33.260","Text":"Then the second step is to draw a table."},{"Start":"01:33.260 ","End":"01:36.155","Text":"There\u0027s only one value I can put in there,"},{"Start":"01:36.155 ","End":"01:39.890","Text":"which is for x is 0."},{"Start":"01:39.890 ","End":"01:45.075","Text":"This divides the x-axis into 2 intervals."},{"Start":"01:45.075 ","End":"01:48.345","Text":"This one here where x is less than 0."},{"Start":"01:48.345 ","End":"01:51.210","Text":"This one here where x is bigger than 0."},{"Start":"01:51.210 ","End":"01:53.870","Text":"We choose an arbitrary point in each interval,"},{"Start":"01:53.870 ","End":"01:57.245","Text":"so I\u0027ll choose minus 1 for this,"},{"Start":"01:57.245 ","End":"01:59.855","Text":"and I\u0027ll choose 1 for this."},{"Start":"01:59.855 ","End":"02:03.710","Text":"F double prime of x, I forgot to write, was 0 here."},{"Start":"02:03.710 ","End":"02:12.195","Text":"For here, I have to put in minus 1 into f double prime of x, here it is."},{"Start":"02:12.195 ","End":"02:14.110","Text":"If I put in minus 1,"},{"Start":"02:14.110 ","End":"02:15.410","Text":"I get minus 6."},{"Start":"02:15.410 ","End":"02:17.270","Text":"But I don\u0027t need the actual value,"},{"Start":"02:17.270 ","End":"02:19.070","Text":"I only need the sign,"},{"Start":"02:19.070 ","End":"02:21.025","Text":"so here it\u0027s negative."},{"Start":"02:21.025 ","End":"02:23.610","Text":"If I put x equals 1 into here,"},{"Start":"02:23.610 ","End":"02:27.170","Text":"I get 6, but what\u0027s important is that it\u0027s positive."},{"Start":"02:27.170 ","End":"02:30.485","Text":"When the second derivative is negative,"},{"Start":"02:30.485 ","End":"02:38.360","Text":"then that means that the function is concave or concave down."},{"Start":"02:38.360 ","End":"02:42.320","Text":"When it\u0027s positive, were convex or concave up,"},{"Start":"02:42.320 ","End":"02:44.515","Text":"any event it shaped like this."},{"Start":"02:44.515 ","End":"02:46.830","Text":"Yes, we are in luck."},{"Start":"02:46.830 ","End":"02:51.020","Text":"Our suspect turns out to be indeed an inflection point"},{"Start":"02:51.020 ","End":"02:55.730","Text":"because it\u0027s between concave down and concave up."},{"Start":"02:55.730 ","End":"02:58.115","Text":"Yes, it\u0027s an inflection."},{"Start":"02:58.115 ","End":"03:03.170","Text":"I\u0027d also like to know the value of y for the inflection so I can say exactly where it is."},{"Start":"03:03.170 ","End":"03:04.925","Text":"When x equals 0,"},{"Start":"03:04.925 ","End":"03:06.865","Text":"y is just f of x."},{"Start":"03:06.865 ","End":"03:10.085","Text":"We have to substitute here the x equals 0."},{"Start":"03:10.085 ","End":"03:13.490","Text":"Well, 0 cubed minus 3 times 0 is just 0."},{"Start":"03:13.490 ","End":"03:16.655","Text":"Now we have everything that we need in the table,"},{"Start":"03:16.655 ","End":"03:23.345","Text":"so I can now get to the last stage of the solution which is the conclusion stage,"},{"Start":"03:23.345 ","End":"03:26.000","Text":"which answers basically the questions that were asked."},{"Start":"03:26.000 ","End":"03:29.775","Text":"Other words, where do we have inflection points?"},{"Start":"03:29.775 ","End":"03:31.851","Text":"Answer is, there\u0027s just one"},{"Start":"03:31.851 ","End":"03:36.095","Text":"and it occurs when x is 0 and y is 0."},{"Start":"03:36.095 ","End":"03:37.565","Text":"That\u0027s the only one."},{"Start":"03:37.565 ","End":"03:40.840","Text":"As for convex and concave,"},{"Start":"03:40.840 ","End":"03:43.440","Text":"convex is concave up also."},{"Start":"03:43.440 ","End":"03:47.355","Text":"We have all of x is less than 0,"},{"Start":"03:47.355 ","End":"03:49.395","Text":"that\u0027s the concave part,"},{"Start":"03:49.395 ","End":"03:51.405","Text":"x less than 0."},{"Start":"03:51.405 ","End":"03:54.160","Text":"Here, this part, the convex,"},{"Start":"03:54.160 ","End":"03:56.390","Text":"is where x is bigger than 0,"},{"Start":"03:56.390 ","End":"04:00.150","Text":"and that\u0027s all there is. We\u0027re done."}],"ID":4849},{"Watched":false,"Name":"Exercise 3","Duration":"3m 50s","ChapterTopicVideoID":4850,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4850.jpeg","UploadDate":"2016-05-01T17:11:47.6770000","DurationForVideoObject":"PT3M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this exercise, we\u0027re given the following function,"},{"Start":"00:02.850 ","End":"00:05.385","Text":"f of x equals this polynomial,"},{"Start":"00:05.385 ","End":"00:09.030","Text":"and we have to find its inflection points and"},{"Start":"00:09.030 ","End":"00:13.020","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.020 ","End":"00:18.225","Text":"This kind of exercise is familiar and has a standard set of steps that we follow."},{"Start":"00:18.225 ","End":"00:25.305","Text":"Cookbook style depth 0 or preparation is to find the second derivative."},{"Start":"00:25.305 ","End":"00:30.000","Text":"But before that, take a look at the domain and see if there\u0027s anything unusual there."},{"Start":"00:30.000 ","End":"00:31.590","Text":"Well, no, it\u0027s a polynomial,"},{"Start":"00:31.590 ","End":"00:33.780","Text":"it\u0027s defined for all x, no problems."},{"Start":"00:33.780 ","End":"00:36.060","Text":"Now, about the second derivative, of course,"},{"Start":"00:36.060 ","End":"00:40.260","Text":"we have to pass through the first derivative. Let\u0027s do that."},{"Start":"00:40.260 ","End":"00:47.145","Text":"This is equal to 6x^2 minus 30x plus 24."},{"Start":"00:47.145 ","End":"00:49.965","Text":"Now the second derivative,"},{"Start":"00:49.965 ","End":"00:55.460","Text":"just differentiate again, so we get 12x minus 30."},{"Start":"00:55.460 ","End":"01:02.460","Text":"Step 1 is to set f double prime to equals 0 and solve for x."},{"Start":"01:02.460 ","End":"01:07.175","Text":"We let f double prime of x equals 0,"},{"Start":"01:07.175 ","End":"01:12.374","Text":"meaning that 12x minus 30 is equal to 0,"},{"Start":"01:12.374 ","End":"01:16.715","Text":"and there\u0027s just one solution, it\u0027s 30 over 12,"},{"Start":"01:16.715 ","End":"01:18.485","Text":"which is 2 and 1/2,"},{"Start":"01:18.485 ","End":"01:21.335","Text":"so x is equal to 2 and 1/2."},{"Start":"01:21.335 ","End":"01:25.405","Text":"This is our only suspect for an inflection point."},{"Start":"01:25.405 ","End":"01:29.070","Text":"Step 2 is to fill in a table."},{"Start":"01:29.070 ","End":"01:32.959","Text":"There\u0027s only one value that I can put in the table,"},{"Start":"01:32.959 ","End":"01:35.270","Text":"that\u0027s 2 and 1/2 for x,"},{"Start":"01:35.270 ","End":"01:37.340","Text":"and that\u0027s our only suspect."},{"Start":"01:37.340 ","End":"01:41.620","Text":"This breaks up the x-axis into 2 intervals,"},{"Start":"01:41.620 ","End":"01:43.860","Text":"x less than 2 and 1/2,"},{"Start":"01:43.860 ","End":"01:46.785","Text":"and here, x bigger than 2 and 1/2."},{"Start":"01:46.785 ","End":"01:49.700","Text":"Then we choose a sample point from each."},{"Start":"01:49.700 ","End":"01:53.820","Text":"Quite arbitrary, I\u0027ll just choose x equals 2 for here"},{"Start":"01:53.820 ","End":"01:56.000","Text":"and x equals 3 for here."},{"Start":"01:56.000 ","End":"02:00.315","Text":"Then I look at f double prime for x equals 2 and 1/2,"},{"Start":"02:00.315 ","End":"02:01.650","Text":"f double prime is 0,"},{"Start":"02:01.650 ","End":"02:03.595","Text":"that\u0027s how we got our 2 and 1/2."},{"Start":"02:03.595 ","End":"02:05.375","Text":"For x equals 2,"},{"Start":"02:05.375 ","End":"02:08.165","Text":"I need to substitute it in here."},{"Start":"02:08.165 ","End":"02:10.910","Text":"12 times 2 is 24"},{"Start":"02:10.910 ","End":"02:12.905","Text":"minus 30 is minus 6,"},{"Start":"02:12.905 ","End":"02:14.630","Text":"but I don\u0027t need the actual value,"},{"Start":"02:14.630 ","End":"02:17.855","Text":"I just need the sign. It is minus."},{"Start":"02:17.855 ","End":"02:20.360","Text":"When f double prime is minus,"},{"Start":"02:20.360 ","End":"02:28.520","Text":"it means that f is this shape which you can call concave or concave down."},{"Start":"02:28.520 ","End":"02:30.650","Text":"Then when x is 3,"},{"Start":"02:30.650 ","End":"02:34.220","Text":"12 times 3 minus 30 is positive."},{"Start":"02:34.220 ","End":"02:36.590","Text":"Positive means it\u0027s shaped like this,"},{"Start":"02:36.590 ","End":"02:39.470","Text":"which is convex or concave up."},{"Start":"02:39.470 ","End":"02:44.510","Text":"When we have a point that\u0027s between concave and convex,"},{"Start":"02:44.510 ","End":"02:46.205","Text":"then it is an inflection point."},{"Start":"02:46.205 ","End":"02:47.480","Text":"We found one."},{"Start":"02:47.480 ","End":"02:50.155","Text":"This is indeed an inflection."},{"Start":"02:50.155 ","End":"02:52.320","Text":"I\u0027d like to know its y value"},{"Start":"02:52.320 ","End":"02:55.025","Text":"so I can say where the inflection is."},{"Start":"02:55.025 ","End":"02:56.880","Text":"To find the y,"},{"Start":"02:56.880 ","End":"03:00.460","Text":"y is, after all, just f of x."},{"Start":"03:00.460 ","End":"03:03.300","Text":"I put in 2 and 1/2,"},{"Start":"03:03.300 ","End":"03:06.690","Text":"I make it minus 3 and 1/2,"},{"Start":"03:06.690 ","End":"03:08.460","Text":"I haven\u0027t made a mistake."},{"Start":"03:08.460 ","End":"03:10.910","Text":"In any event, we have all the information we need for"},{"Start":"03:10.910 ","End":"03:14.075","Text":"the last step which is the conclusion,"},{"Start":"03:14.075 ","End":"03:16.939","Text":"which answers the following questions"},{"Start":"03:16.939 ","End":"03:20.450","Text":"about the inflection and the concave and the convex."},{"Start":"03:20.450 ","End":"03:23.725","Text":"Let\u0027s write that. Inflection points?"},{"Start":"03:23.725 ","End":"03:25.450","Text":"Yes, I have one."},{"Start":"03:25.450 ","End":"03:30.645","Text":"One only at the point 2 and 1/2, minus 3 and 1/2."},{"Start":"03:30.645 ","End":"03:37.586","Text":"Convex or concave up is all this range,"},{"Start":"03:37.586 ","End":"03:39.660","Text":"x bigger than 2 and 1/2."},{"Start":"03:39.660 ","End":"03:43.117","Text":"Concave is this shape,"},{"Start":"03:43.117 ","End":"03:47.970","Text":"and that\u0027s where x is less than 2 and 1/2."},{"Start":"03:47.970 ","End":"03:50.890","Text":"That\u0027s all there is. We\u0027re done."}],"ID":4850},{"Watched":false,"Name":"Exercise 4","Duration":"4m 33s","ChapterTopicVideoID":4851,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4851.jpeg","UploadDate":"2016-05-01T17:12:19.0970000","DurationForVideoObject":"PT4M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.000","Text":"In this exercise, we\u0027re given a function f of x equals x^4 minus 2x^3."},{"Start":"00:06.000 ","End":"00:09.690","Text":"We have to find its inflection points and"},{"Start":"00:09.690 ","End":"00:13.620","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.620 ","End":"00:16.484","Text":"This is a standard type of exercise"},{"Start":"00:16.484 ","End":"00:21.265","Text":"and we have fixed set of steps that we follow to get to the answer."},{"Start":"00:21.265 ","End":"00:24.390","Text":"There\u0027s a preparation step where we"},{"Start":"00:24.390 ","End":"00:28.200","Text":"differentiate f of x twice to get the second derivative,"},{"Start":"00:28.200 ","End":"00:34.020","Text":"but we also take a look at the domain and I see that it\u0027s defined for all x,"},{"Start":"00:34.020 ","End":"00:35.570","Text":"so no problems there."},{"Start":"00:35.570 ","End":"00:39.485","Text":"Let\u0027s get to the second derivative via the first derivative."},{"Start":"00:39.485 ","End":"00:45.170","Text":"f prime of x is 4x^3 minus 6x^2,"},{"Start":"00:45.170 ","End":"00:53.945","Text":"so f double prime of x is 12x^2 minus 12x."},{"Start":"00:53.945 ","End":"00:56.330","Text":"That\u0027s the first preparation stage."},{"Start":"00:56.330 ","End":"01:02.495","Text":"Then stage 1 or step 1 is to compare f double prime to 0 and look for the solutions."},{"Start":"01:02.495 ","End":"01:07.000","Text":"The reason we\u0027re doing this is these solutions are going to be suspects for inflection."},{"Start":"01:07.000 ","End":"01:16.145","Text":"So f double prime of x equals 0 gives us that 12x ^2 minus 12x is 0,"},{"Start":"01:16.145 ","End":"01:19.145","Text":"divide both sides by 12."},{"Start":"01:19.145 ","End":"01:23.420","Text":"In fact, we can even take 12x out the brackets."},{"Start":"01:23.420 ","End":"01:33.575","Text":"We\u0027ve got 12x times x minus 1 is equal to 0 and that gives us 2 solutions."},{"Start":"01:33.575 ","End":"01:36.710","Text":"Solution either x is 0 or x equals 1."},{"Start":"01:36.710 ","End":"01:40.260","Text":"So x is either 0 or 1."},{"Start":"01:40.260 ","End":"01:44.865","Text":"That\u0027s 2 solutions and that\u0027s the end of this stage."},{"Start":"01:44.865 ","End":"01:47.580","Text":"Next step is to draw a table."},{"Start":"01:47.580 ","End":"01:54.290","Text":"In the x, I put all my suspects for inflection and these are x equals 0 and x equals 1."},{"Start":"01:54.290 ","End":"01:57.845","Text":"You put them in an increasing order with some space in between"},{"Start":"01:57.845 ","End":"02:03.680","Text":"f double prime here and here 0 because that\u0027s how we found these points."},{"Start":"02:03.680 ","End":"02:08.765","Text":"Then we draw the intervals that these 2 points define, 3 intervals:"},{"Start":"02:08.765 ","End":"02:11.005","Text":"x less than 0,"},{"Start":"02:11.005 ","End":"02:13.750","Text":"x between 0 and 1,"},{"Start":"02:13.750 ","End":"02:15.605","Text":"and x bigger than 1."},{"Start":"02:15.605 ","End":"02:19.460","Text":"We take a sample point from each interval arbitrarily."},{"Start":"02:19.460 ","End":"02:21.635","Text":"I\u0027ll choose minus 1 here,"},{"Start":"02:21.635 ","End":"02:24.578","Text":"I\u0027ll choose 1/2 between 0 and 1,"},{"Start":"02:24.578 ","End":"02:26.360","Text":"and I\u0027ll choose 2 here."},{"Start":"02:26.360 ","End":"02:29.345","Text":"Then we substitute these in f double prime of x."},{"Start":"02:29.345 ","End":"02:32.704","Text":"That\u0027s this function here, second derivative."},{"Start":"02:32.704 ","End":"02:37.393","Text":"If I put in minus 1, then I get plus 12"},{"Start":"02:37.393 ","End":"02:40.985","Text":"and plus 12 in any event, it\u0027s positive,"},{"Start":"02:40.985 ","End":"02:43.205","Text":"and that\u0027s all I need is the sign."},{"Start":"02:43.205 ","End":"02:49.340","Text":"When it\u0027s positive, function is like this which is convex or concave up."},{"Start":"02:49.340 ","End":"02:51.755","Text":"When x is 1/2,"},{"Start":"02:51.755 ","End":"02:59.539","Text":"you can check that we get negative and when x is 2, we get positive."},{"Start":"02:59.539 ","End":"03:01.610","Text":"I won\u0027t go through all the details."},{"Start":"03:01.610 ","End":"03:06.355","Text":"The function like this here, like this here, and like this here;"},{"Start":"03:06.355 ","End":"03:09.405","Text":"convex, concave, convex."},{"Start":"03:09.405 ","End":"03:13.895","Text":"This is good because that means that out of our 2 suspects,"},{"Start":"03:13.895 ","End":"03:16.940","Text":"we found 2 inflection points."},{"Start":"03:16.940 ","End":"03:22.435","Text":"This is an inflection and this is also an inflection."},{"Start":"03:22.435 ","End":"03:26.780","Text":"All I want now is the y value of the inflection points."},{"Start":"03:26.780 ","End":"03:32.655","Text":"So I look at the function itself which is x^4 minus 2x^3."},{"Start":"03:32.655 ","End":"03:34.250","Text":"That\u0027s my y."},{"Start":"03:34.250 ","End":"03:37.340","Text":"When I put x equals 0,"},{"Start":"03:37.340 ","End":"03:40.535","Text":"I get that y equals 0."},{"Start":"03:40.535 ","End":"03:43.505","Text":"When I put x equals 1,"},{"Start":"03:43.505 ","End":"03:48.125","Text":"I get 1 minus 2 which is minus 1."},{"Start":"03:48.125 ","End":"03:54.050","Text":"That\u0027s all the information I need in order to solve or to answer these questions."},{"Start":"03:54.050 ","End":"03:57.680","Text":"It\u0027s my conclusion phase of the recipe."},{"Start":"03:57.680 ","End":"03:59.465","Text":"Let\u0027s write them down."},{"Start":"03:59.465 ","End":"04:01.550","Text":"Inflection, I have 2 of them."},{"Start":"04:01.550 ","End":"04:04.555","Text":"I have at 0,0,"},{"Start":"04:04.555 ","End":"04:09.795","Text":"and another one at 1, minus 1."},{"Start":"04:09.795 ","End":"04:13.590","Text":"Now, for convex and for concave."},{"Start":"04:13.590 ","End":"04:16.215","Text":"The convex is this shape,"},{"Start":"04:16.215 ","End":"04:22.020","Text":"so I get that at x less than 0 and at x bigger than 1."},{"Start":"04:22.020 ","End":"04:28.050","Text":"The concave, this one is for 0 less than x less than 1,"},{"Start":"04:28.050 ","End":"04:29.910","Text":"x between 0 and 1."},{"Start":"04:29.910 ","End":"04:33.820","Text":"That answers everything, and we\u0027re done."}],"ID":4851},{"Watched":false,"Name":"Exercise 5","Duration":"7m 36s","ChapterTopicVideoID":4852,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4852.jpeg","UploadDate":"2018-03-01T11:33:55.6300000","DurationForVideoObject":"PT7M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"In this exercise, we\u0027re given this function,"},{"Start":"00:03.255 ","End":"00:06.840","Text":"f of x is 3x^5 minus 20x cubed,"},{"Start":"00:06.840 ","End":"00:10.320","Text":"and we have to find the inflection points as well as the intervals where"},{"Start":"00:10.320 ","End":"00:13.845","Text":"the function is concave up and concave down."},{"Start":"00:13.845 ","End":"00:18.015","Text":"This is a standard exercise with a fixed set of steps."},{"Start":"00:18.015 ","End":"00:23.250","Text":"In the beginning, we check the domain to see if anything unusual is happening."},{"Start":"00:23.250 ","End":"00:26.865","Text":"It\u0027s a polynomial defined for all x, nothing special there."},{"Start":"00:26.865 ","End":"00:29.460","Text":"Then we have to do the preparation,"},{"Start":"00:29.460 ","End":"00:32.580","Text":"which is to find the second derivative."},{"Start":"00:32.580 ","End":"00:34.890","Text":"So we\u0027ll start with the first derivative,"},{"Start":"00:34.890 ","End":"00:43.665","Text":"f prime of x is 15x^4 minus 3 times 20 is 60x squared,"},{"Start":"00:43.665 ","End":"00:52.870","Text":"and then f double prime of x will be 4 times 15 is 60x cubed minus 120x."},{"Start":"00:54.530 ","End":"00:57.005","Text":"That\u0027s the preparation."},{"Start":"00:57.005 ","End":"01:04.220","Text":"Now first thing we do is find inflection points by setting the second derivative to 0,"},{"Start":"01:04.220 ","End":"01:06.280","Text":"and then we get our suspects."},{"Start":"01:06.280 ","End":"01:10.100","Text":"If f prime of x is equal to 0,"},{"Start":"01:10.100 ","End":"01:18.660","Text":"then that means that 60x cubed minus 120x is 0."},{"Start":"01:18.660 ","End":"01:22.620","Text":"We can take 60x outside the brackets,"},{"Start":"01:22.620 ","End":"01:31.745","Text":"60x times x squared minus 2 equals 0."},{"Start":"01:31.745 ","End":"01:35.165","Text":"This gives 3 possible values of x."},{"Start":"01:35.165 ","End":"01:38.705","Text":"Either x is 0 from here,"},{"Start":"01:38.705 ","End":"01:41.300","Text":"or x squared minus 2 is 0,"},{"Start":"01:41.300 ","End":"01:44.050","Text":"so x squared equals 2,"},{"Start":"01:44.050 ","End":"01:48.935","Text":"so x is plus or minus the square root of 2."},{"Start":"01:48.935 ","End":"01:51.290","Text":"So there are 3 suspects,"},{"Start":"01:51.290 ","End":"01:52.540","Text":"and then we make a table."},{"Start":"01:52.540 ","End":"01:55.820","Text":"I\u0027ve reused it from the previous exercise."},{"Start":"01:55.820 ","End":"01:58.725","Text":"I put in the table,"},{"Start":"01:58.725 ","End":"02:04.760","Text":"first of all, the values of x, and these will be 0, square root of 2,"},{"Start":"02:04.760 ","End":"02:06.950","Text":"leave also space between"},{"Start":"02:06.950 ","End":"02:12.230","Text":"the values for the intervals, and we have minus the square root of 2."},{"Start":"02:12.230 ","End":"02:15.080","Text":"Now these 3 points actually define 4 intervals."},{"Start":"02:15.080 ","End":"02:19.265","Text":"We have the interval where x is less than minus root 2."},{"Start":"02:19.265 ","End":"02:25.460","Text":"Between these 2, we have minus root 2 less than x less than 0,"},{"Start":"02:25.460 ","End":"02:31.580","Text":"and then here we have 0 less than x less than root 2,"},{"Start":"02:31.580 ","End":"02:36.290","Text":"and here we have x greater than root 2."},{"Start":"02:36.290 ","End":"02:40.625","Text":"What we want to do is take a sample point in each of these intervals"},{"Start":"02:40.625 ","End":"02:42.290","Text":"because, in each of these intervals,"},{"Start":"02:42.290 ","End":"02:44.450","Text":"the sign of f double prime won\u0027t change,"},{"Start":"02:44.450 ","End":"02:47.150","Text":"and we just want to know if it\u0027s positive or negative."},{"Start":"02:47.150 ","End":"02:49.480","Text":"Root 2 is about 1.4."},{"Start":"02:49.480 ","End":"02:53.625","Text":"So I\u0027ll take here minus 2 as my representative,"},{"Start":"02:53.625 ","End":"02:55.770","Text":"here, I\u0027ll take minus 1,"},{"Start":"02:55.770 ","End":"02:59.970","Text":"here, 1, and here I\u0027ll take 2."},{"Start":"02:59.970 ","End":"03:03.800","Text":"What we want to do is find the sign of f double prime of x,"},{"Start":"03:03.800 ","End":"03:05.470","Text":"not the actual value."},{"Start":"03:05.470 ","End":"03:07.210","Text":"Let\u0027s take the minus 2."},{"Start":"03:07.210 ","End":"03:08.540","Text":"Now, 60 is positive,"},{"Start":"03:08.540 ","End":"03:11.390","Text":"so I just have to plug it in here and see what I get."},{"Start":"03:11.390 ","End":"03:16.040","Text":"Minus 2 is negative, and then minus 2 squared is 4,"},{"Start":"03:16.040 ","End":"03:18.320","Text":"minus 2 is 2, it\u0027s positive."},{"Start":"03:18.320 ","End":"03:21.275","Text":"So negative times positive, negative."},{"Start":"03:21.275 ","End":"03:26.090","Text":"The second derivative, of course, at these 3 points, is 0."},{"Start":"03:26.090 ","End":"03:27.350","Text":"That\u0027s how we found them."},{"Start":"03:27.350 ","End":"03:30.650","Text":"I need to put a plus or a minus in the intervals."},{"Start":"03:30.650 ","End":"03:33.570","Text":"Next one is minus 1."},{"Start":"03:33.570 ","End":"03:35.765","Text":"Minus 1 is negative."},{"Start":"03:35.765 ","End":"03:39.395","Text":"Minus 1 squared is 1 minus 2 is also negative,"},{"Start":"03:39.395 ","End":"03:42.665","Text":"negative times negative, so it\u0027s positive."},{"Start":"03:42.665 ","End":"03:45.914","Text":"I put x equals 1, 1 is positive,"},{"Start":"03:45.914 ","End":"03:48.080","Text":"1 squared minus 2 is negative,"},{"Start":"03:48.080 ","End":"03:55.070","Text":"and 2, 2^2 minus 2 is, 2 is positive and 2 is positive, so it\u0027s positive."},{"Start":"03:55.070 ","End":"03:58.100","Text":"What this means, where I write f of x,"},{"Start":"03:58.100 ","End":"03:59.270","Text":"I don\u0027t mean the actual value,"},{"Start":"03:59.270 ","End":"04:00.995","Text":"I just mean its behavior."},{"Start":"04:00.995 ","End":"04:09.450","Text":"Minus means that it\u0027s concave down, so it\u0027s like this,"},{"Start":"04:09.450 ","End":"04:14.695","Text":"and then a plus means that the function is concave up,"},{"Start":"04:14.695 ","End":"04:21.395","Text":"and here, again, concave down and concave up."},{"Start":"04:21.395 ","End":"04:25.310","Text":"When the concavity switches direction,"},{"Start":"04:25.310 ","End":"04:28.460","Text":"at the border, there\u0027s an inflection point."},{"Start":"04:28.460 ","End":"04:31.655","Text":"At x equals minus root 2,"},{"Start":"04:31.655 ","End":"04:35.080","Text":"we have an inflection point."},{"Start":"04:35.080 ","End":"04:37.910","Text":"Here, concave up to down,"},{"Start":"04:37.910 ","End":"04:40.115","Text":"also an inflection point,"},{"Start":"04:40.115 ","End":"04:43.025","Text":"and here we have an inflection point."},{"Start":"04:43.025 ","End":"04:46.130","Text":"Now let\u0027s actually compute the inflection point."},{"Start":"04:46.130 ","End":"04:48.535","Text":"We have the x, we want the y also."},{"Start":"04:48.535 ","End":"04:51.140","Text":"Sure, yeah, y equals f of x."},{"Start":"04:51.140 ","End":"04:52.790","Text":"But here, when I say y,"},{"Start":"04:52.790 ","End":"04:54.140","Text":"I mean the actual value,"},{"Start":"04:54.140 ","End":"04:57.550","Text":"so let\u0027s plug in minus root 2."},{"Start":"04:57.550 ","End":"04:58.640","Text":"To make it easier,"},{"Start":"04:58.640 ","End":"04:59.790","Text":"I could rewrite this,"},{"Start":"04:59.790 ","End":"05:06.465","Text":"I could take x cubed out and write 3x squared minus 20."},{"Start":"05:06.465 ","End":"05:10.020","Text":"Now if we plug in minus root 2,"},{"Start":"05:10.020 ","End":"05:12.885","Text":"this is minus 2, root 2,"},{"Start":"05:12.885 ","End":"05:16.070","Text":"and root 2 squared is 2,"},{"Start":"05:16.070 ","End":"05:18.650","Text":"3 times 2 is 6,"},{"Start":"05:18.650 ","End":"05:25.875","Text":"minus 20 is minus 14 times minus 2 root 2."},{"Start":"05:25.875 ","End":"05:32.520","Text":"It comes out 28 root 2. That\u0027s easy."},{"Start":"05:32.520 ","End":"05:36.945","Text":"When x is 0, y is 0,"},{"Start":"05:36.945 ","End":"05:39.725","Text":"and when x is root 2,"},{"Start":"05:39.725 ","End":"05:43.100","Text":"we\u0027re just going to get the opposite sign as here."},{"Start":"05:43.100 ","End":"05:49.000","Text":"Here it\u0027s going to be minus 28 root 2."},{"Start":"05:49.000 ","End":"05:52.520","Text":"We\u0027re basically done, we just have to collect the information together."},{"Start":"05:52.520 ","End":"05:53.960","Text":"I want 3 things."},{"Start":"05:53.960 ","End":"05:55.850","Text":"I want, first of all,"},{"Start":"05:55.850 ","End":"05:58.900","Text":"is the inflection points,"},{"Start":"05:58.900 ","End":"06:01.430","Text":"and we have 3 of those."},{"Start":"06:01.430 ","End":"06:07.715","Text":"We have minus root 2, 28 root 2 is the first."},{"Start":"06:07.715 ","End":"06:11.860","Text":"The second is the origin 0,0,"},{"Start":"06:11.860 ","End":"06:18.450","Text":"and the third is root 2, minus 28 root 2."},{"Start":"06:18.450 ","End":"06:22.250","Text":"Now we want concave up."},{"Start":"06:22.250 ","End":"06:24.580","Text":"There are 2 such intervals,"},{"Start":"06:24.580 ","End":"06:28.760","Text":"that would be this one and this one."},{"Start":"06:28.760 ","End":"06:35.690","Text":"Just to remind myself that this is up and this is concave up and this is concave down,"},{"Start":"06:35.690 ","End":"06:38.180","Text":"and this is concave down."},{"Start":"06:38.180 ","End":"06:43.805","Text":"Let\u0027s see. Up is this interval. It\u0027s what\u0027s written here."},{"Start":"06:43.805 ","End":"06:46.205","Text":"I\u0027ll write it in interval notation."},{"Start":"06:46.205 ","End":"06:51.725","Text":"X goes from minus root 2 to 0,"},{"Start":"06:51.725 ","End":"06:54.620","Text":"it doesn\u0027t matter if you include the endpoints or not,"},{"Start":"06:54.620 ","End":"06:58.939","Text":"and we also have x bigger than root 2,"},{"Start":"06:58.939 ","End":"07:05.240","Text":"which I can also write as the interval from root 2 to infinity."},{"Start":"07:05.240 ","End":"07:10.955","Text":"Then lastly, we want the intervals where it\u0027s concave down."},{"Start":"07:10.955 ","End":"07:12.545","Text":"Again, there\u0027s 2 of them,"},{"Start":"07:12.545 ","End":"07:14.360","Text":"this one and this one,"},{"Start":"07:14.360 ","End":"07:17.330","Text":"x less than minus root 2,"},{"Start":"07:17.330 ","End":"07:22.560","Text":"I\u0027ll write that as minus infinity to minus root 2."},{"Start":"07:22.560 ","End":"07:24.830","Text":"The other interval is here,"},{"Start":"07:24.830 ","End":"07:27.185","Text":"x between 0 and root 2."},{"Start":"07:27.185 ","End":"07:33.705","Text":"I\u0027ll write this as from 0 to root 2."},{"Start":"07:33.705 ","End":"07:37.560","Text":"This summarizes it all, and we\u0027re done."}],"ID":4852},{"Watched":false,"Name":"Exercise 6","Duration":"8m 21s","ChapterTopicVideoID":4853,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4853.jpeg","UploadDate":"2016-05-01T17:13:58.7270000","DurationForVideoObject":"PT8M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.405","Text":"In this exercise, we\u0027re given the following function,"},{"Start":"00:03.405 ","End":"00:06.840","Text":"f of x equals x over x squared plus 3,"},{"Start":"00:06.840 ","End":"00:09.600","Text":"we have to find its inflection points and"},{"Start":"00:09.600 ","End":"00:13.995","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.995 ","End":"00:18.030","Text":"This is a familiar type of exercise and we know how to solve it"},{"Start":"00:18.030 ","End":"00:22.590","Text":"in certain steps that will lead us to the solution cookbook style."},{"Start":"00:22.590 ","End":"00:28.935","Text":"The first step is the preparation step where I find the second derivative,"},{"Start":"00:28.935 ","End":"00:32.590","Text":"but I also take a look and see what\u0027s with the domain of the function,"},{"Start":"00:32.590 ","End":"00:35.345","Text":"if there\u0027s any problems there and I can see that"},{"Start":"00:35.345 ","End":"00:40.010","Text":"the denominator of this quotient is always positive, so there\u0027s no problem."},{"Start":"00:40.010 ","End":"00:41.450","Text":"It\u0027s defined for all x."},{"Start":"00:41.450 ","End":"00:44.570","Text":"Let\u0027s get to the computation of f double-prime."},{"Start":"00:44.570 ","End":"00:46.880","Text":"Of course we need f prime first."},{"Start":"00:46.880 ","End":"00:50.000","Text":"Also, I would like to remind you of the quotient rule for those"},{"Start":"00:50.000 ","End":"00:53.540","Text":"who haven\u0027t memorized it yet that u over"},{"Start":"00:53.540 ","End":"01:01.685","Text":"v derivative is u prime v minus uv prime over v squared,"},{"Start":"01:01.685 ","End":"01:05.390","Text":"where we have a fraction, u is the denominator in this case x and"},{"Start":"01:05.390 ","End":"01:09.320","Text":"v in this case is x squared plus 3, the denominator."},{"Start":"01:09.320 ","End":"01:13.940","Text":"F prime is, I would like to start with the denominator, it\u0027s the easiest,"},{"Start":"01:13.940 ","End":"01:19.070","Text":"x squared plus 3 squared and here we have u prime,"},{"Start":"01:19.070 ","End":"01:21.470","Text":"which is 1 times v,"},{"Start":"01:21.470 ","End":"01:30.380","Text":"which is x squared plus 3 minus x as is and the derivative of x squared plus 3 is 2x."},{"Start":"01:30.380 ","End":"01:32.915","Text":"Altogether, what we get,"},{"Start":"01:32.915 ","End":"01:35.104","Text":"if we just open the brackets,"},{"Start":"01:35.104 ","End":"01:39.195","Text":"we get minus x squared plus 3."},{"Start":"01:39.195 ","End":"01:40.365","Text":"We have the 3 from here,"},{"Start":"01:40.365 ","End":"01:45.120","Text":"x squared minus 2x squared over x squared plus 3 squared."},{"Start":"01:45.120 ","End":"01:47.990","Text":"That finishes step 1."},{"Start":"01:47.990 ","End":"01:51.050","Text":"Next on to f double-prime of x,"},{"Start":"01:51.050 ","End":"01:54.560","Text":"which we have to differentiate f prime."},{"Start":"01:54.560 ","End":"01:56.885","Text":"Once again, the quotient rule."},{"Start":"01:56.885 ","End":"02:02.735","Text":"On the denominator, we now have x squared plus 3 to the power"},{"Start":"02:02.735 ","End":"02:08.870","Text":"of 4 and we have the derivative of the numerator u prime,"},{"Start":"02:08.870 ","End":"02:16.535","Text":"which is minus 2x times x squared plus 3 squared minus u,"},{"Start":"02:16.535 ","End":"02:20.075","Text":"which is the minus x squared plus 3,"},{"Start":"02:20.075 ","End":"02:23.915","Text":"times the derivative of the denominator."},{"Start":"02:23.915 ","End":"02:29.600","Text":"The derivative of the denominator will be from the chain rule twice x"},{"Start":"02:29.600 ","End":"02:35.680","Text":"squared plus 3 times inner derivative, which is 2x."},{"Start":"02:35.680 ","End":"02:38.930","Text":"Let\u0027s see if we can simplify this a bit."},{"Start":"02:38.930 ","End":"02:45.990","Text":"I notice that x squared plus 3 can be taken out of the numerator and denominator."},{"Start":"02:45.990 ","End":"02:50.960","Text":"Here I\u0027m left with x squared plus 3 to the power of"},{"Start":"02:50.960 ","End":"02:56.480","Text":"3 and here I\u0027ve taken out 1 of the x squared plus 3,"},{"Start":"02:56.480 ","End":"03:00.135","Text":"so I\u0027m left with minus 2x,"},{"Start":"03:00.135 ","End":"03:02.775","Text":"x squared plus 3."},{"Start":"03:02.775 ","End":"03:05.580","Text":"I\u0027ve taken this x squared plus 3 out,"},{"Start":"03:05.580 ","End":"03:08.265","Text":"so I\u0027m left with minus,"},{"Start":"03:08.265 ","End":"03:15.290","Text":"and I\u0027ll write the 4x first and minus x squared plus 3."},{"Start":"03:15.290 ","End":"03:17.405","Text":"Let\u0027s see what we can do with this."},{"Start":"03:17.405 ","End":"03:24.345","Text":"We get minus 2x cubed and from here plus 4x cubed,"},{"Start":"03:24.345 ","End":"03:32.145","Text":"so altogether 2x cubed and then minus 6x minus 12x,"},{"Start":"03:32.145 ","End":"03:39.945","Text":"so it\u0027s minus 18x over x squared plus 3 cubed."},{"Start":"03:39.945 ","End":"03:42.900","Text":"This is the end of the first step."},{"Start":"03:42.900 ","End":"03:47.030","Text":"Next step is to set f double prime equal to 0."},{"Start":"03:47.030 ","End":"03:51.195","Text":"That\u0027s how we find suspects for inflection points."},{"Start":"03:51.195 ","End":"03:55.400","Text":"F double prime of x equals 0."},{"Start":"03:55.400 ","End":"03:59.224","Text":"Now the denominator is not going to be 0."},{"Start":"03:59.224 ","End":"04:03.475","Text":"I mean, a fraction is 0 when its numerator is 0."},{"Start":"04:03.475 ","End":"04:07.865","Text":"We have to set 2x cubed minus 18x equals 0,"},{"Start":"04:07.865 ","End":"04:09.980","Text":"which I can divide by 2."},{"Start":"04:09.980 ","End":"04:13.490","Text":"In fact, I can take 2x outside the brackets. Let\u0027s do that."},{"Start":"04:13.490 ","End":"04:15.080","Text":"We get that the numerator,"},{"Start":"04:15.080 ","End":"04:21.994","Text":"which is 2x times x squared minus 9, is equal to 0."},{"Start":"04:21.994 ","End":"04:25.535","Text":"Like I said, the fraction is 0 means its numerator is 0."},{"Start":"04:25.535 ","End":"04:28.670","Text":"This is can be factorized into x plus 3,"},{"Start":"04:28.670 ","End":"04:35.120","Text":"x minus 3 or we can also see this part gives us x squared minus 9 is 0,"},{"Start":"04:35.120 ","End":"04:37.010","Text":"so x is plus or minus 3,"},{"Start":"04:37.010 ","End":"04:38.720","Text":"and x could be 0 here."},{"Start":"04:38.720 ","End":"04:40.310","Text":"In short, we have 3 values."},{"Start":"04:40.310 ","End":"04:44.095","Text":"We have 0 and we have plus or minus 3."},{"Start":"04:44.095 ","End":"04:47.750","Text":"These are 3 suspects for inflection points."},{"Start":"04:47.750 ","End":"04:50.660","Text":"Let\u0027s draw our usual table."},{"Start":"04:50.660 ","End":"04:54.440","Text":"We put in the 3 values of x in increasing order,"},{"Start":"04:54.440 ","End":"04:58.800","Text":"minus 3, 0, and 3."},{"Start":"04:58.800 ","End":"05:00.590","Text":"At all these 3 points,"},{"Start":"05:00.590 ","End":"05:02.375","Text":"f double prime is 0."},{"Start":"05:02.375 ","End":"05:06.610","Text":"After all, that\u0027s how we got these points by setting f double prime to be 0."},{"Start":"05:06.610 ","End":"05:10.595","Text":"These points break up the line into intervals,"},{"Start":"05:10.595 ","End":"05:13.415","Text":"x less than minus 3,"},{"Start":"05:13.415 ","End":"05:16.685","Text":"x between minus 3 and 0,"},{"Start":"05:16.685 ","End":"05:19.475","Text":"x between 0 and 3,"},{"Start":"05:19.475 ","End":"05:22.010","Text":"and x bigger than 3."},{"Start":"05:22.010 ","End":"05:25.610","Text":"We choose an arbitrary sample point from each interval."},{"Start":"05:25.610 ","End":"05:28.115","Text":"From here I\u0027ll take minus 4."},{"Start":"05:28.115 ","End":"05:32.300","Text":"From here I\u0027ll take minus 1, from here I\u0027ll take 1,"},{"Start":"05:32.300 ","End":"05:34.280","Text":"and from here I\u0027ll take 4,"},{"Start":"05:34.280 ","End":"05:37.100","Text":"and we substitute into f double prime."},{"Start":"05:37.100 ","End":"05:41.630","Text":"F double prime is here after we simplified it,"},{"Start":"05:41.630 ","End":"05:46.875","Text":"but we don\u0027t need the actual answer only the sign, positive or negative."},{"Start":"05:46.875 ","End":"05:50.015","Text":"If I put in x equals minus 4,"},{"Start":"05:50.015 ","End":"05:52.669","Text":"the denominator is always positive notice,"},{"Start":"05:52.669 ","End":"05:54.815","Text":"because x squared plus 3 is positive."},{"Start":"05:54.815 ","End":"05:58.460","Text":"We just need to find the sign of the numerator."},{"Start":"05:58.460 ","End":"06:01.115","Text":"If I put x equals minus 4,"},{"Start":"06:01.115 ","End":"06:05.340","Text":"I\u0027ll get, this is negative and I won\u0027t do all of them,"},{"Start":"06:05.340 ","End":"06:07.970","Text":"I\u0027ll just tell you that this comes up positive,"},{"Start":"06:07.970 ","End":"06:10.310","Text":"this is negative and this is positive,"},{"Start":"06:10.310 ","End":"06:14.270","Text":"which makes this concave and this also,"},{"Start":"06:14.270 ","End":"06:17.950","Text":"and here convex and here convex."},{"Start":"06:17.950 ","End":"06:21.575","Text":"Each of these 3 points that were suspect are indeed"},{"Start":"06:21.575 ","End":"06:24.949","Text":"inflection points because each is between a concave and convex."},{"Start":"06:24.949 ","End":"06:26.920","Text":"This is an inflection,"},{"Start":"06:26.920 ","End":"06:29.539","Text":"this is also an inflection point,"},{"Start":"06:29.539 ","End":"06:34.145","Text":"and this is an inflection point, 3 inflection points."},{"Start":"06:34.145 ","End":"06:42.365","Text":"These inflection points I need the values of y. Y is over here the original f of x."},{"Start":"06:42.365 ","End":"06:47.210","Text":"Let\u0027s see, if I put y as minus 3,"},{"Start":"06:47.210 ","End":"06:51.125","Text":"I\u0027ll get minus 3 squared is 9,"},{"Start":"06:51.125 ","End":"06:53.975","Text":"plus 3 is 12 and on the numerator,"},{"Start":"06:53.975 ","End":"06:59.855","Text":"minus 3 over 12 is minus 1/4."},{"Start":"06:59.855 ","End":"07:01.855","Text":"This is minus 1/4."},{"Start":"07:01.855 ","End":"07:04.110","Text":"When x is 0,"},{"Start":"07:04.110 ","End":"07:10.050","Text":"y comes out to be 0 and when x is 3,"},{"Start":"07:10.050 ","End":"07:13.065","Text":"we get 3 over 3 squared plus 3,"},{"Start":"07:13.065 ","End":"07:15.870","Text":"3 over 12 is 1/4."},{"Start":"07:15.870 ","End":"07:18.365","Text":"That\u0027s the end of this step."},{"Start":"07:18.365 ","End":"07:22.355","Text":"The last step is the conclusion phase,"},{"Start":"07:22.355 ","End":"07:25.865","Text":"where we just basically answered the questions that were asked."},{"Start":"07:25.865 ","End":"07:30.170","Text":"Other words, where do we have an inflection ad the answer is that we were lucky."},{"Start":"07:30.170 ","End":"07:31.325","Text":"We got 3 of them."},{"Start":"07:31.325 ","End":"07:35.285","Text":"We got 1 at minus 3, minus 1/4."},{"Start":"07:35.285 ","End":"07:38.495","Text":"We got 1 at the origin, 0, 0,"},{"Start":"07:38.495 ","End":"07:42.890","Text":"and we got 1 at 3, 1/4."},{"Start":"07:42.890 ","End":"07:45.710","Text":"As for the convex intervals,"},{"Start":"07:45.710 ","End":"07:48.545","Text":"the convex is this 1 and this 1,"},{"Start":"07:48.545 ","End":"07:56.760","Text":"it\u0027s x between minus 3 and 0 as well as x bigger than 3."},{"Start":"07:56.760 ","End":"08:01.340","Text":"For the concave, we look for the concave shape, which is this,"},{"Start":"08:01.340 ","End":"08:09.840","Text":"and this concave up and this comes out at x less than minus 3 is the first concave 1."},{"Start":"08:09.840 ","End":"08:16.020","Text":"The second concave 1 is where x is between 0 and 3."},{"Start":"08:16.020 ","End":"08:21.540","Text":"That\u0027s the answer to all the questions we were asked and so we\u0027re done."}],"ID":4853},{"Watched":false,"Name":"Exercise 7","Duration":"7m 6s","ChapterTopicVideoID":4854,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4854.jpeg","UploadDate":"2016-05-01T17:14:46.3670000","DurationForVideoObject":"PT7M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.005","Text":"In this exercise, we\u0027re given a function f of x equals 2x squared over x plus 1 squared,"},{"Start":"00:07.005 ","End":"00:09.855","Text":"we have to find its inflection points and"},{"Start":"00:09.855 ","End":"00:13.830","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.830 ","End":"00:15.360","Text":"For this type of exercise,"},{"Start":"00:15.360 ","End":"00:21.030","Text":"there is a standard step-by-step cookbook style solution and we\u0027ll follow it."},{"Start":"00:21.030 ","End":"00:23.925","Text":"First, we have a preparation step,"},{"Start":"00:23.925 ","End":"00:27.615","Text":"which is basically to find f double prime of x,"},{"Start":"00:27.615 ","End":"00:31.185","Text":"but also take a look at the domain of the function."},{"Start":"00:31.185 ","End":"00:35.235","Text":"Looking at it, I see that there is 1 thing about the domain,"},{"Start":"00:35.235 ","End":"00:40.100","Text":"x must not equal minus 1 because that would make the denominator 0."},{"Start":"00:40.100 ","End":"00:43.955","Text":"For this function, I would say x is not equal to minus 1,"},{"Start":"00:43.955 ","End":"00:46.445","Text":"and this minus 1 will play a part later."},{"Start":"00:46.445 ","End":"00:49.310","Text":"Now let\u0027s go for the second derivative."},{"Start":"00:49.310 ","End":"00:52.550","Text":"Of course, we have to begin with the first derivative and I\u0027ll"},{"Start":"00:52.550 ","End":"00:56.510","Text":"remind you of the chain rule that you should know it by now."},{"Start":"00:56.510 ","End":"01:06.290","Text":"U over v prime equals u prime v minus uv prime over v squared."},{"Start":"01:06.290 ","End":"01:11.305","Text":"Here u is 2x squared and v is x plus 1 squared."},{"Start":"01:11.305 ","End":"01:18.620","Text":"We get x plus 1 to the fourth because this was squared and now it\u0027s squared again,"},{"Start":"01:18.620 ","End":"01:25.895","Text":"the derivative of the numerator is 4x times the denominator as is x plus 1"},{"Start":"01:25.895 ","End":"01:34.190","Text":"squared minus the numerator as is 2x squared times the derivative of the denominator,"},{"Start":"01:34.190 ","End":"01:39.560","Text":"which is twice x plus 1 times the inner derivative, which is 1."},{"Start":"01:39.560 ","End":"01:42.305","Text":"Let\u0027s see if we can simplify this."},{"Start":"01:42.305 ","End":"01:43.710","Text":"Well, for 1 thing,"},{"Start":"01:43.710 ","End":"01:49.880","Text":"we can get rid of 1 of the x plus 1\u0027s because it\u0027s common to numerator and denominator."},{"Start":"01:49.880 ","End":"01:51.320","Text":"Let\u0027s do that."},{"Start":"01:51.320 ","End":"01:55.375","Text":"That will bring us down to x plus 1 to the power of 3."},{"Start":"01:55.375 ","End":"01:57.010","Text":"Here we\u0027ll have 4x,"},{"Start":"01:57.010 ","End":"02:05.519","Text":"x plus 1 minus 2x squared with 2 is 4x squared and the x plus 1 is gone."},{"Start":"02:05.519 ","End":"02:07.095","Text":"Let\u0027s see where that leaves us."},{"Start":"02:07.095 ","End":"02:09.990","Text":"4x squared minus 4x squared is nothing."},{"Start":"02:09.990 ","End":"02:14.899","Text":"We\u0027re left with just 4x over x plus 1 cubed."},{"Start":"02:14.899 ","End":"02:19.580","Text":"F double-prime of x is equal to, again,"},{"Start":"02:19.580 ","End":"02:24.620","Text":"I\u0027m going to use the quotient rule and we will get this time,"},{"Start":"02:24.620 ","End":"02:29.510","Text":"x plus 1 to the 6th for the denominator squared."},{"Start":"02:29.510 ","End":"02:34.410","Text":"Then u prime, which is 4 times v x plus"},{"Start":"02:34.410 ","End":"02:40.490","Text":"1 cubed minus 4x times derivative of this,"},{"Start":"02:40.490 ","End":"02:44.270","Text":"which is 3x plus 1 squared."},{"Start":"02:44.270 ","End":"02:47.030","Text":"Again, in a derivative is just 1."},{"Start":"02:47.030 ","End":"02:48.680","Text":"What can we do with this?"},{"Start":"02:48.680 ","End":"02:53.555","Text":"This time we can divide top and bottom by x plus 1 squared."},{"Start":"02:53.555 ","End":"02:58.055","Text":"That will leave me with just 4x plus 1."},{"Start":"02:58.055 ","End":"03:00.440","Text":"If I take away x plus 1 squared,"},{"Start":"03:00.440 ","End":"03:04.740","Text":"I\u0027m just left with 12x of 4x plus 1"},{"Start":"03:04.740 ","End":"03:10.160","Text":"minus 12x because we\u0027re taking out the x plus 1 squared."},{"Start":"03:10.160 ","End":"03:15.025","Text":"On the denominator will be left with x plus 1 to the 4th."},{"Start":"03:15.025 ","End":"03:18.300","Text":"If we just simplify this,"},{"Start":"03:18.300 ","End":"03:25.020","Text":"we\u0027ll get 4x minus 12x is minus 8x plus 4"},{"Start":"03:25.020 ","End":"03:32.625","Text":"minus 8x plus 4 over x plus 1 to the 4th."},{"Start":"03:32.625 ","End":"03:35.460","Text":"That\u0027s our f double-prime."},{"Start":"03:35.460 ","End":"03:38.975","Text":"Next step is to set f double prime equals"},{"Start":"03:38.975 ","End":"03:44.350","Text":"0 and to solve for x so f double prime of x equals 0."},{"Start":"03:44.350 ","End":"03:47.780","Text":"That gives us, since this is a fraction,"},{"Start":"03:47.780 ","End":"03:51.560","Text":"the only way it can be 0 is if the numerator is 0."},{"Start":"03:51.560 ","End":"03:56.734","Text":"That gives us that minus 8x plus 4 equals 0,"},{"Start":"03:56.734 ","End":"04:00.495","Text":"which gives us that x equals 1/2"},{"Start":"04:00.495 ","End":"04:02.885","Text":"4 over 8, or 1/2."},{"Start":"04:02.885 ","End":"04:04.490","Text":"This step, a short step,"},{"Start":"04:04.490 ","End":"04:08.120","Text":"and the next 1 is to make it table."},{"Start":"04:08.120 ","End":"04:14.780","Text":"In this table, we\u0027re going to put 2 values, 1 is x equals 1/2,"},{"Start":"04:14.780 ","End":"04:18.895","Text":"which is a suspect for being an inflection."},{"Start":"04:18.895 ","End":"04:22.490","Text":"Also, we\u0027re going to put in x equals minus 1,"},{"Start":"04:22.490 ","End":"04:26.660","Text":"where the function is not defined we\u0027ll put them in an order."},{"Start":"04:26.660 ","End":"04:37.100","Text":"Here we have minus 1 and here we have 1/2 and at 1/2 is where f double prime is 0,"},{"Start":"04:37.100 ","End":"04:41.600","Text":"but minus 1 is a point which is not in the domain."},{"Start":"04:41.600 ","End":"04:43.915","Text":"Everything\u0027s undefined here."},{"Start":"04:43.915 ","End":"04:47.895","Text":"These 2 points define intervals."},{"Start":"04:47.895 ","End":"04:53.125","Text":"The intervals are x less than minus 1,"},{"Start":"04:53.125 ","End":"04:56.610","Text":"x between minus 1/2,"},{"Start":"04:56.610 ","End":"04:59.405","Text":"and x bigger than 1/2."},{"Start":"04:59.405 ","End":"05:02.030","Text":"We choose sample points from each."},{"Start":"05:02.030 ","End":"05:10.250","Text":"I\u0027ll choose minus 2 as the sample point here and I\u0027ll choose 0 as a sample point here,"},{"Start":"05:10.250 ","End":"05:13.475","Text":"and I\u0027ll choose 1 as a sample point here."},{"Start":"05:13.475 ","End":"05:19.985","Text":"These we substitute in f double prime of x,"},{"Start":"05:19.985 ","End":"05:22.145","Text":"f double prime of x is here,"},{"Start":"05:22.145 ","End":"05:25.325","Text":"but we don\u0027t need the actual answer only the sign,"},{"Start":"05:25.325 ","End":"05:27.410","Text":"and since the denominator is positive,"},{"Start":"05:27.410 ","End":"05:28.610","Text":"I can ignore it."},{"Start":"05:28.610 ","End":"05:31.850","Text":"If I put in minus 2 here,"},{"Start":"05:31.850 ","End":"05:34.400","Text":"then at any event it\u0027s positive."},{"Start":"05:34.400 ","End":"05:38.975","Text":"When it\u0027s positive, we have 1 of these and when x is 0,"},{"Start":"05:38.975 ","End":"05:41.375","Text":"we also get positive,"},{"Start":"05:41.375 ","End":"05:44.090","Text":"another plus and 1 of these,"},{"Start":"05:44.090 ","End":"05:45.980","Text":"and when x is 1,"},{"Start":"05:45.980 ","End":"05:47.660","Text":"we get minus 8 plus 4,"},{"Start":"05:47.660 ","End":"05:51.530","Text":"that\u0027s negative so 1 of these the word here,"},{"Start":"05:51.530 ","End":"05:54.440","Text":"we have convex, convex and concave."},{"Start":"05:54.440 ","End":"05:58.490","Text":"This 1/2, which was our suspect for an inflection,"},{"Start":"05:58.490 ","End":"06:03.305","Text":"is indeed an inflection because it\u0027s between a convex and a concave."},{"Start":"06:03.305 ","End":"06:06.145","Text":"It\u0027s an inflection."},{"Start":"06:06.145 ","End":"06:08.405","Text":"I\u0027d like to know its y value,"},{"Start":"06:08.405 ","End":"06:12.214","Text":"which means substituting in y,"},{"Start":"06:12.214 ","End":"06:17.760","Text":"I mean y is just f of x. I have to substitute in here."},{"Start":"06:17.760 ","End":"06:20.345","Text":"If I put in 1/2,"},{"Start":"06:20.345 ","End":"06:27.440","Text":"a quick computation shows that I get y equals 2/9,"},{"Start":"06:27.440 ","End":"06:29.734","Text":"and that\u0027s it with the table."},{"Start":"06:29.734 ","End":"06:32.540","Text":"The last stage is the conclusion"},{"Start":"06:32.540 ","End":"06:35.570","Text":"where we basically answer the questions that we were asked,"},{"Start":"06:35.570 ","End":"06:40.350","Text":"that we have an inflection point only at 1/2,"},{"Start":"06:40.350 ","End":"06:49.250","Text":"2/9 that the convex intervals here and here, in other words,"},{"Start":"06:49.250 ","End":"06:55.700","Text":"x less than minus 1 as well as minus 1 less than x,"},{"Start":"06:55.700 ","End":"07:00.995","Text":"less than 1/2 and concave when I have this shape,"},{"Start":"07:00.995 ","End":"07:04.325","Text":"which is x bigger than 1/2."},{"Start":"07:04.325 ","End":"07:06.750","Text":"Now we\u0027re done."}],"ID":4854},{"Watched":false,"Name":"Exercise 8","Duration":"6m 14s","ChapterTopicVideoID":5831,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5831.jpeg","UploadDate":"2016-05-01T17:15:26.9900000","DurationForVideoObject":"PT6M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.965","Text":"In this exercise, we have the function f of x is x minus 1 over x squared."},{"Start":"00:04.965 ","End":"00:08.250","Text":"What we have to do is find its inflection points,"},{"Start":"00:08.250 ","End":"00:13.485","Text":"as well as the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.485 ","End":"00:18.210","Text":"Note that the domain of this function is x not equal to 0,"},{"Start":"00:18.210 ","End":"00:20.220","Text":"that would make the denominator 0."},{"Start":"00:20.220 ","End":"00:26.595","Text":"Now, this type of exercise is familiar and has a cookbook type solution in 4 steps."},{"Start":"00:26.595 ","End":"00:29.340","Text":"The first step is the preparation."},{"Start":"00:29.340 ","End":"00:32.970","Text":"The preparation is to find the second derivative."},{"Start":"00:32.970 ","End":"00:34.800","Text":"So we want to find f double-prime."},{"Start":"00:34.800 ","End":"00:37.470","Text":"Of course, we first have to go through f prime."},{"Start":"00:37.470 ","End":"00:40.790","Text":"Now f prime of x equal to,"},{"Start":"00:40.790 ","End":"00:46.340","Text":"what I can do to make my life easier is to rewrite this in the form f of x is"},{"Start":"00:46.340 ","End":"00:52.610","Text":"equal to dividing out I get 1 over x minus 1 over x squared."},{"Start":"00:52.610 ","End":"00:56.690","Text":"Now it\u0027ll be easier to differentiate because this is also"},{"Start":"00:56.690 ","End":"01:01.430","Text":"equal to x to the minus 1 minus x to the minus 2."},{"Start":"01:01.430 ","End":"01:07.610","Text":"So using exponents, the first term is minus x to the minus 2,"},{"Start":"01:07.610 ","End":"01:10.100","Text":"minus 1 x to the minus 2,"},{"Start":"01:10.100 ","End":"01:16.075","Text":"and then minus minus 2 plus 2 x to the minus 3."},{"Start":"01:16.075 ","End":"01:20.450","Text":"Then we can easily get f double prime of x, because again,"},{"Start":"01:20.450 ","End":"01:21.470","Text":"we\u0027re still in the exponent,"},{"Start":"01:21.470 ","End":"01:26.524","Text":"so minus 2 times minus 1 is plus 2 x to the minus 3."},{"Start":"01:26.524 ","End":"01:33.055","Text":"Minus 3 times 2 is minus 6 times x to the minus 4."},{"Start":"01:33.055 ","End":"01:39.200","Text":"If I take outside the brackets, x to the minus 4,"},{"Start":"01:39.200 ","End":"01:47.705","Text":"I can get that this is equal to x to the minus 4 times 2 x minus 6."},{"Start":"01:47.705 ","End":"01:50.295","Text":"That\u0027s the end of this stage,"},{"Start":"01:50.295 ","End":"01:53.750","Text":"so the next thing we have to do is solve the equation,"},{"Start":"01:53.750 ","End":"01:56.585","Text":"f double prime of x equals 0."},{"Start":"01:56.585 ","End":"01:58.730","Text":"I want f double prime to be 0,"},{"Start":"01:58.730 ","End":"02:02.705","Text":"that will give me my suspects for inflection."},{"Start":"02:02.705 ","End":"02:06.475","Text":"This gives me that this is 0,"},{"Start":"02:06.475 ","End":"02:14.600","Text":"so 2x minus 6 and I can write the x to the 4th in the denominator better equals 0."},{"Start":"02:14.600 ","End":"02:17.480","Text":"Now, when a fraction is 0,"},{"Start":"02:17.480 ","End":"02:19.850","Text":"it means that its numerator is 0,"},{"Start":"02:19.850 ","End":"02:25.865","Text":"so what we get is that 2 x minus 6 equals 0,"},{"Start":"02:25.865 ","End":"02:29.195","Text":"and that gives us finally that x equals 3."},{"Start":"02:29.195 ","End":"02:34.280","Text":"So x equals 3 is a suspect for an inflection point."},{"Start":"02:34.280 ","End":"02:35.660","Text":"That\u0027s this step."},{"Start":"02:35.660 ","End":"02:40.955","Text":"Next step is the table and the values we want to put in here,"},{"Start":"02:40.955 ","End":"02:45.760","Text":"there\u0027s 2 kinds is the 1 which are suspect for an inflection, that\u0027s x equals 3,"},{"Start":"02:45.760 ","End":"02:49.490","Text":"but we also put in the point x equals 0,"},{"Start":"02:49.490 ","End":"02:53.045","Text":"where the function\u0027s undefined because it can split up the intervals."},{"Start":"02:53.045 ","End":"02:57.260","Text":"So I\u0027m going to put in x equals 0 and x equals 3."},{"Start":"02:57.260 ","End":"02:59.060","Text":"We put them in increasing order."},{"Start":"02:59.060 ","End":"03:01.610","Text":"At this point, f double prime is 0,"},{"Start":"03:01.610 ","End":"03:03.020","Text":"that\u0027s how we found it."},{"Start":"03:03.020 ","End":"03:06.710","Text":"But here, f is undefined, everything\u0027s undefined."},{"Start":"03:06.710 ","End":"03:09.515","Text":"I put some dotted lines here to say,"},{"Start":"03:09.515 ","End":"03:11.720","Text":"don\u0027t put anything here, it\u0027s undefined."},{"Start":"03:11.720 ","End":"03:15.720","Text":"These points split up the x-axis into intervals;"},{"Start":"03:15.720 ","End":"03:18.375","Text":"there\u0027s an interval x less than 0,"},{"Start":"03:18.375 ","End":"03:22.640","Text":"there\u0027s an interval where x is between 0 and 3."},{"Start":"03:22.640 ","End":"03:25.720","Text":"There\u0027s an interval where x is bigger than 3."},{"Start":"03:25.720 ","End":"03:31.175","Text":"We choose a point arbitrarily from each interval or whatever\u0027s convenient."},{"Start":"03:31.175 ","End":"03:34.130","Text":"For example, I could choose x equals 1 here,"},{"Start":"03:34.130 ","End":"03:37.205","Text":"and I could choose x equals 4 here,"},{"Start":"03:37.205 ","End":"03:40.620","Text":"and I could choose x equals minus 1 here and then"},{"Start":"03:40.620 ","End":"03:44.195","Text":"for these points I have to compute f double-prime,"},{"Start":"03:44.195 ","End":"03:45.920","Text":"but not exactly f double prime."},{"Start":"03:45.920 ","End":"03:48.590","Text":"I just want the sign plus or minus."},{"Start":"03:48.590 ","End":"03:51.805","Text":"F double prime is here."},{"Start":"03:51.805 ","End":"03:53.710","Text":"This is really the denominator."},{"Start":"03:53.710 ","End":"03:59.740","Text":"I should really have written it as 2x minus 6 over x to the 4th. You can see it better."},{"Start":"03:59.740 ","End":"04:02.575","Text":"The denominator is always positive when it\u0027s defined,"},{"Start":"04:02.575 ","End":"04:05.810","Text":"so all I need is the sign of the numerator."},{"Start":"04:05.810 ","End":"04:07.325","Text":"When x is minus 1,"},{"Start":"04:07.325 ","End":"04:09.070","Text":"this comes out negative."},{"Start":"04:09.070 ","End":"04:10.345","Text":"Put a minus sign."},{"Start":"04:10.345 ","End":"04:13.915","Text":"When x is 1, it\u0027s still negative."},{"Start":"04:13.915 ","End":"04:16.730","Text":"2 minus 6 minus 4, it\u0027s still negative,"},{"Start":"04:16.730 ","End":"04:20.980","Text":"and when x is 4 then I get a positive,"},{"Start":"04:20.980 ","End":"04:27.525","Text":"which means the negative means that the function is concave like this."},{"Start":"04:27.525 ","End":"04:32.080","Text":"Negative again means that here also the function is"},{"Start":"04:32.080 ","End":"04:37.075","Text":"concave and plus means that it\u0027s convex or concave up."},{"Start":"04:37.075 ","End":"04:42.370","Text":"Now this point where x is 3 turns out to be between concave and convex,"},{"Start":"04:42.370 ","End":"04:46.085","Text":"and that makes it not just the suspect, it\u0027s guilty."},{"Start":"04:46.085 ","End":"04:48.350","Text":"It is an inflection point,"},{"Start":"04:48.350 ","End":"04:51.609","Text":"but 0 is just a point where the function is undefined,"},{"Start":"04:51.609 ","End":"04:53.375","Text":"so we have 1 inflection point."},{"Start":"04:53.375 ","End":"04:56.270","Text":"What I\u0027d still like from the table is this value here of"},{"Start":"04:56.270 ","End":"04:59.555","Text":"y when x is 3 at our inflection point."},{"Start":"04:59.555 ","End":"05:05.180","Text":"So I need to substitute x equals 3 in the original function,"},{"Start":"05:05.180 ","End":"05:09.185","Text":"which would be either I could take it from here or I could take it from here."},{"Start":"05:09.185 ","End":"05:13.160","Text":"I could put x equals 3, let\u0027s say here,"},{"Start":"05:13.160 ","End":"05:18.364","Text":"which is f of x, and that\u0027s also by the way, equals y."},{"Start":"05:18.364 ","End":"05:19.910","Text":"So if you put x equals 3,"},{"Start":"05:19.910 ","End":"05:23.930","Text":"we get 1 over 3 minus 1 over 9,"},{"Start":"05:23.930 ","End":"05:26.360","Text":"which makes it 2/9."},{"Start":"05:26.360 ","End":"05:29.710","Text":"So here we\u0027ll have 2/9."},{"Start":"05:29.710 ","End":"05:31.440","Text":"That\u0027s the table phase."},{"Start":"05:31.440 ","End":"05:35.710","Text":"The last step is the conclusions in which basically you\u0027re answering"},{"Start":"05:35.710 ","End":"05:38.975","Text":"the original questions that they were asking us about"},{"Start":"05:38.975 ","End":"05:42.890","Text":"the inflection points and the intervals of convex and concave."},{"Start":"05:42.890 ","End":"05:47.510","Text":"First of all, I\u0027ll do inflection and we found there is exactly 1"},{"Start":"05:47.510 ","End":"05:52.610","Text":"and it is at the point x is 3, y is 2/9."},{"Start":"05:52.610 ","End":"05:55.655","Text":"Then we want the convex intervals."},{"Start":"05:55.655 ","End":"05:58.385","Text":"So it\u0027s convex just here."},{"Start":"05:58.385 ","End":"06:00.650","Text":"When x is bigger than 3,"},{"Start":"06:00.650 ","End":"06:05.285","Text":"that\u0027s where it\u0027s convex and it\u0027s concave in 2 places"},{"Start":"06:05.285 ","End":"06:10.675","Text":"where x is less than 0 and also when x is between 0 and 3."},{"Start":"06:10.675 ","End":"06:15.150","Text":"That answers the questions and we\u0027re done."}],"ID":5829},{"Watched":false,"Name":"Exercise 9","Duration":"7m 6s","ChapterTopicVideoID":5832,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5832.jpeg","UploadDate":"2016-05-01T17:16:13.9130000","DurationForVideoObject":"PT7M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.305","Text":"In this exercise, we\u0027re given a function f of x equals 2x squared over x plus 1 squared."},{"Start":"00:07.305 ","End":"00:10.140","Text":"We have to find its inflection points,"},{"Start":"00:10.140 ","End":"00:12.780","Text":"and the intervals where it\u0027s concave up,"},{"Start":"00:12.780 ","End":"00:14.715","Text":"and where it\u0027s concave down."},{"Start":"00:14.715 ","End":"00:16.410","Text":"For this type of exercise,"},{"Start":"00:16.410 ","End":"00:20.790","Text":"there is a standard step-by-step cookbook style solution,"},{"Start":"00:20.790 ","End":"00:22.440","Text":"and we\u0027ll follow it."},{"Start":"00:22.440 ","End":"00:25.560","Text":"First we have a preparation step,"},{"Start":"00:25.560 ","End":"00:29.280","Text":"which is basically to find f double prime of x,"},{"Start":"00:29.280 ","End":"00:33.159","Text":"but also to take a look at the domain of the function."},{"Start":"00:33.159 ","End":"00:37.260","Text":"Looking at it, I see that there is 1 thing about the domain."},{"Start":"00:37.260 ","End":"00:39.780","Text":"X must not equal minus 1,"},{"Start":"00:39.780 ","End":"00:42.060","Text":"because that would make the denominator 0,"},{"Start":"00:42.060 ","End":"00:45.845","Text":"but this function, I would say x is not equal to minus 1,"},{"Start":"00:45.845 ","End":"00:48.245","Text":"and this minus 1 will play a part later."},{"Start":"00:48.245 ","End":"00:50.885","Text":"Now let\u0027s go for the second derivative."},{"Start":"00:50.885 ","End":"00:53.555","Text":"Of course we have to begin with the first derivative,"},{"Start":"00:53.555 ","End":"00:56.180","Text":"and I\u0027ll remind you of the chain rule,"},{"Start":"00:56.180 ","End":"00:57.785","Text":"though you should know it by now,"},{"Start":"00:57.785 ","End":"01:06.665","Text":"u over v prime equals u prime v minus u v prime over v squared."},{"Start":"01:06.665 ","End":"01:08.525","Text":"Here, u is 2x squared,"},{"Start":"01:08.525 ","End":"01:10.835","Text":"and v is x plus 1 squared,"},{"Start":"01:10.835 ","End":"01:16.925","Text":"so we get, put the denominator first, x plus 1 to the fourth,"},{"Start":"01:16.925 ","End":"01:18.680","Text":"because this was squared,"},{"Start":"01:18.680 ","End":"01:20.210","Text":"and now it\u0027s squared again,"},{"Start":"01:20.210 ","End":"01:28.235","Text":"the derivative of the numerator is 4x times the denominator as is x plus 1 squared,"},{"Start":"01:28.235 ","End":"01:36.440","Text":"minus the numerator as is 2x squared times the derivative of the denominator,"},{"Start":"01:36.440 ","End":"01:39.200","Text":"which is twice x plus 1,"},{"Start":"01:39.200 ","End":"01:41.810","Text":"times the inner derivative, which is 1."},{"Start":"01:41.810 ","End":"01:44.785","Text":"Let\u0027s see if we can simplify this."},{"Start":"01:44.785 ","End":"01:46.230","Text":"Well, for 1 thing,"},{"Start":"01:46.230 ","End":"01:49.555","Text":"we can get rid of 1 of the x plus 1s,"},{"Start":"01:49.555 ","End":"01:52.880","Text":"because it\u0027s common to numerator and denominator."},{"Start":"01:52.880 ","End":"01:54.050","Text":"Let\u0027s do that."},{"Start":"01:54.050 ","End":"01:58.085","Text":"That will bring us down to x plus 1 to the power of 3,"},{"Start":"01:58.085 ","End":"01:59.855","Text":"and here we\u0027ll have 4x,"},{"Start":"01:59.855 ","End":"02:06.235","Text":"x plus 1 minus 2x squared with 2 is 4x squared,"},{"Start":"02:06.235 ","End":"02:08.625","Text":"and the x plus 1 is gone,"},{"Start":"02:08.625 ","End":"02:10.490","Text":"so let\u0027s see where that leaves us."},{"Start":"02:10.490 ","End":"02:13.370","Text":"4x squared minus 4x squared is nothing,"},{"Start":"02:13.370 ","End":"02:17.975","Text":"we\u0027re left with just 4x over x plus 1 cubed."},{"Start":"02:17.975 ","End":"02:22.085","Text":"F double prime of x is equal to,"},{"Start":"02:22.085 ","End":"02:24.925","Text":"I\u0027m going to use the quotient rule,"},{"Start":"02:24.925 ","End":"02:27.370","Text":"and we will get this time,"},{"Start":"02:27.370 ","End":"02:33.320","Text":"x plus 1 to the 6th for the denominator squared, then u prime,"},{"Start":"02:33.320 ","End":"02:38.200","Text":"which is 4 times v x plus 1 cubed,"},{"Start":"02:38.200 ","End":"02:42.590","Text":"minus 4x times derivative of this,"},{"Start":"02:42.590 ","End":"02:45.720","Text":"which is 3x plus 1 squared,"},{"Start":"02:45.720 ","End":"02:48.725","Text":"and any derivative is just 1."},{"Start":"02:48.725 ","End":"02:50.255","Text":"What can we do with this?"},{"Start":"02:50.255 ","End":"02:54.515","Text":"This time we can divide top and bottom by x plus 1 squared."},{"Start":"02:54.515 ","End":"02:58.759","Text":"That will leave me with just 4x plus 1,"},{"Start":"02:58.759 ","End":"03:01.340","Text":"and if I take away x plus 1 squared,"},{"Start":"03:01.340 ","End":"03:03.715","Text":"I\u0027m just left with 12x."},{"Start":"03:03.715 ","End":"03:07.915","Text":"4x plus 1 minus 12x,"},{"Start":"03:07.915 ","End":"03:10.730","Text":"because we\u0027re taking out the x plus 1 squared,"},{"Start":"03:10.730 ","End":"03:12.530","Text":"and on the denominator,"},{"Start":"03:12.530 ","End":"03:15.470","Text":"we\u0027ll be left with x plus 1 to the fourth."},{"Start":"03:15.470 ","End":"03:18.900","Text":"If we just simplify this,"},{"Start":"03:18.900 ","End":"03:24.540","Text":"we\u0027ll get 4x minus 12x is minus 8x plus"},{"Start":"03:24.540 ","End":"03:31.810","Text":"4 minus 8x plus 4 over x plus 1 to the 4th."},{"Start":"03:31.810 ","End":"03:35.270","Text":"Okay, that\u0027s our f double prime."},{"Start":"03:35.270 ","End":"03:36.755","Text":"Now the next step,"},{"Start":"03:36.755 ","End":"03:39.455","Text":"this is the end of 1 step."},{"Start":"03:39.455 ","End":"03:43.850","Text":"The next step is to set f double prime equal 0,"},{"Start":"03:43.850 ","End":"03:48.385","Text":"and to solve for x. F double prime of x equals 0,"},{"Start":"03:48.385 ","End":"03:50.255","Text":"and that gives us,"},{"Start":"03:50.255 ","End":"03:51.965","Text":"since this is a fraction,"},{"Start":"03:51.965 ","End":"03:55.390","Text":"the only way it can be 0 is if the numerator is 0,"},{"Start":"03:55.390 ","End":"04:00.829","Text":"so that gives us that minus 8x plus 4 equals 0,"},{"Start":"04:00.829 ","End":"04:04.475","Text":"which gives us that x equals 1/2,"},{"Start":"04:04.475 ","End":"04:06.785","Text":"4 over 8, or 1/2."},{"Start":"04:06.785 ","End":"04:08.570","Text":"This step, a short step,"},{"Start":"04:08.570 ","End":"04:11.930","Text":"and the next 1 is to make a table."},{"Start":"04:11.930 ","End":"04:15.860","Text":"In this table, we\u0027re going to put 2 values."},{"Start":"04:15.860 ","End":"04:18.500","Text":"1 is x equals 1/2,"},{"Start":"04:18.500 ","End":"04:22.300","Text":"which is a suspect for being an inflection,"},{"Start":"04:22.300 ","End":"04:28.190","Text":"and also we\u0027re going to put in x equals minus 1 where the function is not defined."},{"Start":"04:28.190 ","End":"04:29.750","Text":"We put them in order."},{"Start":"04:29.750 ","End":"04:32.440","Text":"Here we have minus 1,"},{"Start":"04:32.440 ","End":"04:35.130","Text":"and here we have 1/2,"},{"Start":"04:35.130 ","End":"04:39.725","Text":"and at 1/2 is where f double prime is 0,"},{"Start":"04:39.725 ","End":"04:44.210","Text":"but minus 1 is a point which is not in the domain."},{"Start":"04:44.210 ","End":"04:50.280","Text":"Everything\u0027s undefined here, and these 2 points define intervals,"},{"Start":"04:50.280 ","End":"04:55.570","Text":"and the Intervals are x less than minus 1,"},{"Start":"04:55.570 ","End":"04:59.075","Text":"x between minus 1 and a 1/2,"},{"Start":"04:59.075 ","End":"05:01.895","Text":"and x bigger than 1/2,"},{"Start":"05:01.895 ","End":"05:04.260","Text":"we choose sample points from each,"},{"Start":"05:04.260 ","End":"05:08.435","Text":"so I\u0027ll choose minus 2 as the sample point here,"},{"Start":"05:08.435 ","End":"05:12.125","Text":"I\u0027ll choose 0 as a sample point here,"},{"Start":"05:12.125 ","End":"05:15.335","Text":"and I\u0027ll choose 1 as the sample point here."},{"Start":"05:15.335 ","End":"05:19.405","Text":"These, we substitute in f double prime of x,"},{"Start":"05:19.405 ","End":"05:21.570","Text":"f double prime of x is here,"},{"Start":"05:21.570 ","End":"05:23.730","Text":"but we don\u0027t need the actual answer,"},{"Start":"05:23.730 ","End":"05:26.930","Text":"only the sign, and since the denominator is positive,"},{"Start":"05:26.930 ","End":"05:28.220","Text":"I can ignore it."},{"Start":"05:28.220 ","End":"05:31.595","Text":"If I put in minus 2 here,"},{"Start":"05:31.595 ","End":"05:35.220","Text":"then it\u0027ll be plus 16 plus 4."},{"Start":"05:35.220 ","End":"05:36.995","Text":"In any event, it\u0027s positive."},{"Start":"05:36.995 ","End":"05:39.440","Text":"When it\u0027s positive, we have 1 of these,"},{"Start":"05:39.440 ","End":"05:41.690","Text":"and when x is 0,"},{"Start":"05:41.690 ","End":"05:44.505","Text":"we also get positive,"},{"Start":"05:44.505 ","End":"05:47.420","Text":"so another plus on 1 of these,"},{"Start":"05:47.420 ","End":"05:49.310","Text":"and when x is 1,"},{"Start":"05:49.310 ","End":"05:50.990","Text":"we get minus 8 plus 4,"},{"Start":"05:50.990 ","End":"05:54.035","Text":"that\u0027s negative, so 1 of these,"},{"Start":"05:54.035 ","End":"05:58.100","Text":"and the words, here we have convex, convex and concave."},{"Start":"05:58.100 ","End":"06:02.120","Text":"This 1/2, which was our suspect for an inflection,"},{"Start":"06:02.120 ","End":"06:03.890","Text":"is indeed an inflection,"},{"Start":"06:03.890 ","End":"06:06.950","Text":"because it\u0027s between a convex and concave,"},{"Start":"06:06.950 ","End":"06:08.755","Text":"so it\u0027s an inflection,"},{"Start":"06:08.755 ","End":"06:11.270","Text":"and I\u0027d like to know its y value,"},{"Start":"06:11.270 ","End":"06:16.490","Text":"which means substituting in y, which is here,"},{"Start":"06:16.490 ","End":"06:19.550","Text":"and if I put in 1/2,"},{"Start":"06:19.550 ","End":"06:26.990","Text":"a quick computation shows that I get y equals 2/9,"},{"Start":"06:26.990 ","End":"06:29.134","Text":"and that\u0027s it with the table."},{"Start":"06:29.134 ","End":"06:31.970","Text":"The last stage is the conclusion,"},{"Start":"06:31.970 ","End":"06:34.955","Text":"where we basically answer the questions that we\u0027re asked,"},{"Start":"06:34.955 ","End":"06:41.290","Text":"that we have an inflection point only at 1/2 comma 2/9,"},{"Start":"06:41.290 ","End":"06:48.770","Text":"that the convex intervals here and here, in other words,"},{"Start":"06:48.770 ","End":"06:55.550","Text":"x less than minus 1 as well as minus 1 less than x,"},{"Start":"06:55.550 ","End":"07:00.785","Text":"less than 1/2, and concave when I have this shape,"},{"Start":"07:00.785 ","End":"07:03.920","Text":"which is x bigger than 1/2,"},{"Start":"07:03.920 ","End":"07:06.390","Text":"and now we\u0027re done."}],"ID":5830},{"Watched":false,"Name":"Exercise 10","Duration":"7m 46s","ChapterTopicVideoID":5833,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5833.jpeg","UploadDate":"2016-05-01T17:17:06.0570000","DurationForVideoObject":"PT7M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.345","Text":"In this exercise, we\u0027re given the function f of x equals"},{"Start":"00:04.345 ","End":"00:07.225","Text":"x^3 over x plus 1 squared,"},{"Start":"00:07.225 ","End":"00:09.825","Text":"and we have to find its inflection points,"},{"Start":"00:09.825 ","End":"00:14.685","Text":"as well as the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:14.685 ","End":"00:18.090","Text":"This is a familiar type of exercise and it has"},{"Start":"00:18.090 ","End":"00:22.230","Text":"a certain set of steps that we follow to get to the solution,"},{"Start":"00:22.230 ","End":"00:23.520","Text":"cookbook style."},{"Start":"00:23.520 ","End":"00:26.459","Text":"There\u0027s the preparation step first,"},{"Start":"00:26.459 ","End":"00:29.325","Text":"where we find f double prime,"},{"Start":"00:29.325 ","End":"00:34.880","Text":"but we also take a look at the domain of the function to see if there\u0027s any odd points."},{"Start":"00:34.880 ","End":"00:40.700","Text":"Yes, indeed, we note that for x equals minus 1,"},{"Start":"00:40.700 ","End":"00:46.580","Text":"the function is undefined and we\u0027ll use this point later in the process."},{"Start":"00:46.580 ","End":"00:49.085","Text":"Now, as for the second derivative,"},{"Start":"00:49.085 ","End":"00:52.490","Text":"of course, we have to find the first derivative first."},{"Start":"00:52.490 ","End":"00:55.665","Text":"I\u0027m going to use the quotient rule here,"},{"Start":"00:55.665 ","End":"00:56.925","Text":"I\u0027ll just remind you."},{"Start":"00:56.925 ","End":"01:02.700","Text":"Here it is. I\u0027ve used u and v instead of f and g because f is taken. Let\u0027s see."},{"Start":"01:02.700 ","End":"01:05.625","Text":"I\u0027ll start with the denominator v squared."},{"Start":"01:05.625 ","End":"01:08.070","Text":"If it\u0027s x plus 1 squared squared,"},{"Start":"01:08.070 ","End":"01:09.990","Text":"it\u0027s x plus 1 to the 4."},{"Start":"01:09.990 ","End":"01:17.090","Text":"u prime is 3x^2 times, this as is, x plus 1 squared"},{"Start":"01:17.090 ","End":"01:22.750","Text":"minus x^3 times derivative of x plus 1 squared,"},{"Start":"01:22.750 ","End":"01:27.680","Text":"which is twice x plus 1 times the inner derivative,"},{"Start":"01:27.680 ","End":"01:29.210","Text":"which is just 1."},{"Start":"01:29.210 ","End":"01:31.160","Text":"Let\u0027s simplify this a bit."},{"Start":"01:31.160 ","End":"01:34.865","Text":"I can divide top and bottom by x plus 1,"},{"Start":"01:34.865 ","End":"01:40.690","Text":"so what I get is 3x^2 times just x plus 1."},{"Start":"01:40.690 ","End":"01:43.790","Text":"Minus here, if I take out the x plus 1, it\u0027s gone"},{"Start":"01:43.790 ","End":"01:49.100","Text":"and we\u0027re left with 2x^3 over x plus 1 only cubed."},{"Start":"01:49.100 ","End":"01:51.710","Text":"This if I collect terms together,"},{"Start":"01:51.710 ","End":"01:55.265","Text":"I have 3x^3 minus 2x^3,"},{"Start":"01:55.265 ","End":"01:57.095","Text":"which is just x^3,"},{"Start":"01:57.095 ","End":"02:02.795","Text":"plus 3x^2 over same x plus 1 cubed."},{"Start":"02:02.795 ","End":"02:07.555","Text":"I can simplify it a bit more if I write it as,"},{"Start":"02:07.555 ","End":"02:09.480","Text":"I take out x^2,"},{"Start":"02:09.480 ","End":"02:16.135","Text":"and I\u0027m left with x plus 3 over x plus 1 cubed."},{"Start":"02:16.135 ","End":"02:18.785","Text":"That\u0027s only the first derivative."},{"Start":"02:18.785 ","End":"02:20.780","Text":"We have to differentiate again."},{"Start":"02:20.780 ","End":"02:25.830","Text":"f double prime of x. You know what?"},{"Start":"02:25.830 ","End":"02:27.020","Text":"I\u0027ve changed my mind,"},{"Start":"02:27.020 ","End":"02:28.120","Text":"I won\u0027t simplify this,"},{"Start":"02:28.120 ","End":"02:30.280","Text":"it might be easier this way."},{"Start":"02:30.280 ","End":"02:32.760","Text":"Again, using the same formula,"},{"Start":"02:32.760 ","End":"02:35.390","Text":"I\u0027ll start with the denominator,"},{"Start":"02:35.390 ","End":"02:37.580","Text":"where I have this thing squared,"},{"Start":"02:37.580 ","End":"02:40.475","Text":"which makes it x plus 1 to the 6."},{"Start":"02:40.475 ","End":"02:44.100","Text":"Now, derivative of the numerator,"},{"Start":"02:44.100 ","End":"02:50.265","Text":"we have 3x^2 plus 6x in brackets,"},{"Start":"02:50.265 ","End":"02:55.624","Text":"times the denominator, x plus 1 cubed,"},{"Start":"02:55.624 ","End":"03:03.685","Text":"minus the numerator as is, x^3 plus 3x^2,"},{"Start":"03:03.685 ","End":"03:07.159","Text":"times the derivative of the denominator"},{"Start":"03:07.159 ","End":"03:15.160","Text":"which is 3 times x plus 1 squared times 1, the inner derivative."},{"Start":"03:15.160 ","End":"03:17.835","Text":"Again, we need to simplify, it looks a mess."},{"Start":"03:17.835 ","End":"03:23.825","Text":"But I think we can take x plus 1 squared out of the top and out of the bottom."},{"Start":"03:23.825 ","End":"03:29.865","Text":"It\u0027s 3x^2 plus 6x times just x plus 1."},{"Start":"03:29.865 ","End":"03:31.860","Text":"Taking out x plus 1 squared"},{"Start":"03:31.860 ","End":"03:34.140","Text":"and take it out of here, this is all gone,"},{"Start":"03:34.140 ","End":"03:39.885","Text":"so I\u0027m left with minus 3 x^3 plus 3x^2"},{"Start":"03:39.885 ","End":"03:46.010","Text":"and over, this time only to the power of 4, x plus 1 to the 4."},{"Start":"03:46.010 ","End":"03:49.745","Text":"Now, I\u0027ll just clean up the numerator a bit,"},{"Start":"03:49.745 ","End":"03:52.405","Text":"collect like terms. Let\u0027s see."},{"Start":"03:52.405 ","End":"03:56.535","Text":"We get 3x^2 plus x is 3x^3."},{"Start":"03:56.535 ","End":"03:58.875","Text":"I\u0027m collecting together all the x^3 terms."},{"Start":"03:58.875 ","End":"04:01.920","Text":"We have 3x^3 minus 3x^3,"},{"Start":"04:01.920 ","End":"04:03.435","Text":"so there\u0027s no x^3."},{"Start":"04:03.435 ","End":"04:07.830","Text":"Next, the x^2s, the 6x^2 plus 3x^2,"},{"Start":"04:07.830 ","End":"04:12.120","Text":"that\u0027s 9x^2 minus 9x^2, there\u0027s no x^2 terms."},{"Start":"04:12.120 ","End":"04:16.200","Text":"6x times 1, so it\u0027s just 6x."},{"Start":"04:16.200 ","End":"04:20.565","Text":"6x over x plus 1 to the 1/4."},{"Start":"04:20.565 ","End":"04:23.340","Text":"That\u0027s it for the preparation phase."},{"Start":"04:23.340 ","End":"04:27.398","Text":"Next is to set f double prime to be 0 and solve for x."},{"Start":"04:27.398 ","End":"04:32.560","Text":"I\u0027ll let f double prime of x equals 0."},{"Start":"04:32.560 ","End":"04:34.910","Text":"If f double prime of x is 0,"},{"Start":"04:34.910 ","End":"04:38.270","Text":"since the denominators doesn\u0027t take any part,"},{"Start":"04:38.270 ","End":"04:41.180","Text":"we have a fraction, we only care about the numerator."},{"Start":"04:41.180 ","End":"04:44.829","Text":"This is the same as saying that 6x equals 0,"},{"Start":"04:44.829 ","End":"04:49.500","Text":"and 6x equals 0 only when x equals 0,"},{"Start":"04:49.500 ","End":"04:53.705","Text":"that\u0027s the only suspect for an inflection point."},{"Start":"04:53.705 ","End":"04:57.815","Text":"We put all this information into a table,"},{"Start":"04:57.815 ","End":"05:00.350","Text":"so I\u0027ll draw the table here."},{"Start":"05:00.350 ","End":"05:02.555","Text":"What we put in the table,"},{"Start":"05:02.555 ","End":"05:07.830","Text":"first of all, in the x row are the 2 points that are unusual."},{"Start":"05:07.830 ","End":"05:10.695","Text":"I mean, we have the point minus 1,"},{"Start":"05:10.695 ","End":"05:13.170","Text":"where the function is not defined,"},{"Start":"05:13.170 ","End":"05:14.660","Text":"and we have x equals 0,"},{"Start":"05:14.660 ","End":"05:18.220","Text":"which is a suspect for being an inflection point."},{"Start":"05:18.220 ","End":"05:19.590","Text":"Put them in an order,"},{"Start":"05:19.590 ","End":"05:23.750","Text":"so I have minus 1 and I have 0."},{"Start":"05:23.750 ","End":"05:26.870","Text":"I also leave some space around,"},{"Start":"05:26.870 ","End":"05:30.045","Text":"so I can write the intervals, these 2 form."},{"Start":"05:30.045 ","End":"05:33.380","Text":"We have an interval x less than minus 1,"},{"Start":"05:33.380 ","End":"05:37.595","Text":"we have an interval where x is between minus 1 and 0,"},{"Start":"05:37.595 ","End":"05:40.895","Text":"and an interval where x is bigger than 0."},{"Start":"05:40.895 ","End":"05:44.375","Text":"Minus 1 is where the function is undefined, of course,"},{"Start":"05:44.375 ","End":"05:47.410","Text":"and 0 is where f double prime is 0."},{"Start":"05:47.410 ","End":"05:50.110","Text":"Now, we have these intervals"},{"Start":"05:50.110 ","End":"05:53.210","Text":"and we choose a representative from each, it doesn\u0027t matter."},{"Start":"05:53.210 ","End":"05:55.430","Text":"I\u0027ll choose minus 2 here,"},{"Start":"05:55.430 ","End":"05:58.325","Text":"I\u0027ll choose minus 1/2 here,"},{"Start":"05:58.325 ","End":"06:00.185","Text":"and I\u0027ll choose 1 here."},{"Start":"06:00.185 ","End":"06:06.050","Text":"These we substitute into f double prime which we simplified over here."},{"Start":"06:06.050 ","End":"06:07.550","Text":"We don\u0027t want the actual value,"},{"Start":"06:07.550 ","End":"06:09.575","Text":"just whether it\u0027s positive or negative."},{"Start":"06:09.575 ","End":"06:11.285","Text":"If I put in minus 2,"},{"Start":"06:11.285 ","End":"06:14.630","Text":"the denominator\u0027s positive, so I only have to look at the numerator."},{"Start":"06:14.630 ","End":"06:17.240","Text":"Minus 2 here makes it negative,"},{"Start":"06:17.240 ","End":"06:18.575","Text":"and when it\u0027s negative,"},{"Start":"06:18.575 ","End":"06:21.435","Text":"the function is this way."},{"Start":"06:21.435 ","End":"06:23.940","Text":"When it\u0027s minus 1/2,"},{"Start":"06:23.940 ","End":"06:26.835","Text":"also we get negative number,"},{"Start":"06:26.835 ","End":"06:29.760","Text":"so also this way around."},{"Start":"06:29.760 ","End":"06:32.480","Text":"When x is 1, we get a positive,"},{"Start":"06:32.480 ","End":"06:34.395","Text":"which means that it\u0027s this way."},{"Start":"06:34.395 ","End":"06:39.090","Text":"This is concave, concave and convex, so concave up."},{"Start":"06:39.090 ","End":"06:43.125","Text":"Yes, we are lucky we have an inflection point,"},{"Start":"06:43.125 ","End":"06:47.270","Text":"because at 0 we have on 1 side we have concave,"},{"Start":"06:47.270 ","End":"06:48.750","Text":"on 1 side convex,"},{"Start":"06:48.750 ","End":"06:51.210","Text":"so this is an inflection point."},{"Start":"06:51.210 ","End":"06:54.665","Text":"At the inflection point, I\u0027d like to know what the value of y is."},{"Start":"06:54.665 ","End":"06:56.525","Text":"y is f of x,"},{"Start":"06:56.525 ","End":"07:00.005","Text":"so I have to substitute x equals 0 in here,"},{"Start":"07:00.005 ","End":"07:02.780","Text":"and we see that it comes out 0."},{"Start":"07:02.780 ","End":"07:04.610","Text":"I\u0027ve got the x, y of the inflection,"},{"Start":"07:04.610 ","End":"07:07.160","Text":"and that\u0027s all the information I need from the table."},{"Start":"07:07.160 ","End":"07:09.230","Text":"Now I can draw the conclusions."},{"Start":"07:09.230 ","End":"07:10.430","Text":"We were asked for 3 things:"},{"Start":"07:10.430 ","End":"07:12.955","Text":"inflection, concave, and convex."},{"Start":"07:12.955 ","End":"07:15.900","Text":"Inflection, we have one such point"},{"Start":"07:15.900 ","End":"07:18.670","Text":"and it\u0027s at the point 0,0."},{"Start":"07:18.670 ","End":"07:22.355","Text":"As for convex or concave up,"},{"Start":"07:22.355 ","End":"07:23.945","Text":"this is the convex one,"},{"Start":"07:23.945 ","End":"07:27.335","Text":"so that happens at x bigger than 0."},{"Start":"07:27.335 ","End":"07:32.940","Text":"For concave, it\u0027s where I\u0027ve marked this shape, this 2 intervals,"},{"Start":"07:32.940 ","End":"07:35.185","Text":"1 is x less than minus 1,"},{"Start":"07:35.185 ","End":"07:39.000","Text":"and the other is where x is between minus 1 and 0."},{"Start":"07:39.000 ","End":"07:42.360","Text":"Can\u0027t combine them because minus 1 is a definite gap,"},{"Start":"07:42.360 ","End":"07:44.090","Text":"its function\u0027s not defined there."},{"Start":"07:44.090 ","End":"07:47.340","Text":"Anyway, that\u0027s all and we\u0027re done."}],"ID":5831},{"Watched":false,"Name":"Exercise 11","Duration":"10m 52s","ChapterTopicVideoID":5834,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5834.jpeg","UploadDate":"2016-05-01T17:18:21.9270000","DurationForVideoObject":"PT10M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we\u0027re given f of x as follows,"},{"Start":"00:04.020 ","End":"00:06.930","Text":"x plus 1 over x minus 1 cubed."},{"Start":"00:06.930 ","End":"00:10.575","Text":"We have to find its inflection points and"},{"Start":"00:10.575 ","End":"00:14.685","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:14.685 ","End":"00:17.250","Text":"This is a familiar type of exercise and it has"},{"Start":"00:17.250 ","End":"00:20.549","Text":"a standard set of steps that will take us to the solution."},{"Start":"00:20.549 ","End":"00:21.600","Text":"But just before that,"},{"Start":"00:21.600 ","End":"00:25.500","Text":"I\u0027d like to look at the domain and see what\u0027s happening here."},{"Start":"00:25.500 ","End":"00:29.010","Text":"I noticed that there\u0027s an x minus 1 in the denominator,"},{"Start":"00:29.010 ","End":"00:32.690","Text":"so x cannot be equal to 1,"},{"Start":"00:32.690 ","End":"00:36.030","Text":"and that\u0027s the only limitation on the domain."},{"Start":"00:36.030 ","End":"00:39.630","Text":"Now as to the steps which we do cookbook style,"},{"Start":"00:39.630 ","End":"00:41.800","Text":"first, we have a preparation step,"},{"Start":"00:41.800 ","End":"00:47.645","Text":"and there we differentiate f of x twice to get the second derivative."},{"Start":"00:47.645 ","End":"00:50.000","Text":"Of course, we start with the first derivative,"},{"Start":"00:50.000 ","End":"00:52.730","Text":"so f prime of x is equal to,"},{"Start":"00:52.730 ","End":"00:54.560","Text":"and I\u0027m going to use the chain rule."},{"Start":"00:54.560 ","End":"00:56.915","Text":"We have something to the power of 3,"},{"Start":"00:56.915 ","End":"00:58.970","Text":"so we have, in the derivative,"},{"Start":"00:58.970 ","End":"01:03.175","Text":"3 times that something to the power of 2 first,"},{"Start":"01:03.175 ","End":"01:07.580","Text":"and what\u0027s inside here is the same x plus 1 over x minus 1."},{"Start":"01:07.580 ","End":"01:11.030","Text":"By the chain rule, we have to multiply by the inner derivative,"},{"Start":"01:11.030 ","End":"01:14.450","Text":"which is the derivative of this quotient."},{"Start":"01:14.450 ","End":"01:16.325","Text":"Just in case you forgotten,"},{"Start":"01:16.325 ","End":"01:18.365","Text":"I\u0027ll write down the chain rule."},{"Start":"01:18.365 ","End":"01:21.710","Text":"The chain rule says that if I have a fraction,"},{"Start":"01:21.710 ","End":"01:26.570","Text":"say, u over v, then its derivative is given by the following formula,"},{"Start":"01:26.570 ","End":"01:30.430","Text":"u prime v minus uv prime over v squared."},{"Start":"01:30.430 ","End":"01:32.045","Text":"This is u, this is v."},{"Start":"01:32.045 ","End":"01:36.230","Text":"So we get u prime is 1 times, v, x minus 1,"},{"Start":"01:36.230 ","End":"01:39.770","Text":"minus the numerator as is,"},{"Start":"01:39.770 ","End":"01:44.785","Text":"times derivative of denominator over denominator squared."},{"Start":"01:44.785 ","End":"01:47.705","Text":"This equals, I can simplify a bit."},{"Start":"01:47.705 ","End":"01:51.665","Text":"If you look at this, it\u0027s x minus 1 less x plus 1,"},{"Start":"01:51.665 ","End":"01:54.305","Text":"so that gives us minus 2."},{"Start":"01:54.305 ","End":"01:59.380","Text":"We have minus 2 and that combines with the 3 to give us minus 6,"},{"Start":"01:59.380 ","End":"02:01.620","Text":"minus 6 times this thing,"},{"Start":"02:01.620 ","End":"02:05.970","Text":"x plus 1 over x minus 1 squared,"},{"Start":"02:05.970 ","End":"02:11.505","Text":"and here, 1 over x minus 1 squared."},{"Start":"02:11.505 ","End":"02:17.475","Text":"Altogether, I could combine this x minus 1 and this x minus 1,"},{"Start":"02:17.475 ","End":"02:24.200","Text":"what we will get is minus 6 over x plus 1 squared from here,"},{"Start":"02:24.200 ","End":"02:28.520","Text":"and altogether, we have x plus 1 squared twice,"},{"Start":"02:28.520 ","End":"02:31.225","Text":"so that\u0027s x minus 1 to the 4,"},{"Start":"02:31.225 ","End":"02:33.320","Text":"that\u0027s the first derivative."},{"Start":"02:33.320 ","End":"02:35.750","Text":"Let\u0027s go for f double prime now."},{"Start":"02:35.750 ","End":"02:39.700","Text":"f double prime of x is equal to,"},{"Start":"02:39.700 ","End":"02:43.460","Text":"on the denominator, x minus 1 to the 8."},{"Start":"02:43.460 ","End":"02:46.715","Text":"Again, I\u0027m using the quotient rule here."},{"Start":"02:46.715 ","End":"02:52.450","Text":"But you know what, I think I\u0027ll put this minus 6 right in front so it won\u0027t bother us."},{"Start":"02:52.450 ","End":"02:54.645","Text":"I\u0027ll put the minus 6 in front here."},{"Start":"02:54.645 ","End":"02:57.465","Text":"Here it just stays as a constant,"},{"Start":"02:57.465 ","End":"03:02.450","Text":"and what I have here is the derivative of the numerator,"},{"Start":"03:02.450 ","End":"03:05.540","Text":"which is twice x plus 1,"},{"Start":"03:05.540 ","End":"03:07.280","Text":"internal derivative is 1,"},{"Start":"03:07.280 ","End":"03:08.765","Text":"I won\u0027t bother writing it,"},{"Start":"03:08.765 ","End":"03:16.365","Text":"times the denominator, x minus 1 to the 4 minus the other way around,"},{"Start":"03:16.365 ","End":"03:18.665","Text":"this one stays as is,"},{"Start":"03:18.665 ","End":"03:21.490","Text":"x plus 1 squared."},{"Start":"03:21.490 ","End":"03:25.025","Text":"The derivative of this using the chain rule is"},{"Start":"03:25.025 ","End":"03:30.499","Text":"4x minus 1 to the power of 3 times the inner derivative,"},{"Start":"03:30.499 ","End":"03:32.885","Text":"which is 1, which again I won\u0027t write."},{"Start":"03:32.885 ","End":"03:34.730","Text":"Now let\u0027s simplify this."},{"Start":"03:34.730 ","End":"03:38.765","Text":"I see that I have x minus 1 to the power of 3,"},{"Start":"03:38.765 ","End":"03:40.415","Text":"that\u0027s the smallest power,"},{"Start":"03:40.415 ","End":"03:44.420","Text":"and I can cancel it from the numerator and from the denominator."},{"Start":"03:44.420 ","End":"03:51.000","Text":"I get minus 6 twice x plus 1, x minus 1,"},{"Start":"03:51.000 ","End":"03:54.390","Text":"I\u0027ve removed power of 3 from it, and here,"},{"Start":"03:54.390 ","End":"03:57.435","Text":"minus 4x plus 1 squared,"},{"Start":"03:57.435 ","End":"04:01.970","Text":"and the denominator is just x minus 1 to the 5."},{"Start":"04:01.970 ","End":"04:03.935","Text":"Now we have to simplify."},{"Start":"04:03.935 ","End":"04:08.060","Text":"I\u0027ll simplify the numerator by opening the brackets."},{"Start":"04:08.060 ","End":"04:12.890","Text":"We get twice this times this is x squared minus 1,"},{"Start":"04:12.890 ","End":"04:16.195","Text":"so it\u0027s 2x squared minus 2,"},{"Start":"04:16.195 ","End":"04:20.700","Text":"and here we have x squared plus 2x plus 1."},{"Start":"04:20.700 ","End":"04:22.395","Text":"But for the minus 4,"},{"Start":"04:22.395 ","End":"04:30.435","Text":"it\u0027s 4x squared minus 8x minus 4 over x minus 1 to the 5."},{"Start":"04:30.435 ","End":"04:32.880","Text":"Let\u0027s continue."},{"Start":"04:32.880 ","End":"04:40.970","Text":"I\u0027m going to take 2 outside the brackets here and make this equal to minus 12,"},{"Start":"04:40.970 ","End":"04:42.695","Text":"but I\u0027m also going to collect terms."},{"Start":"04:42.695 ","End":"04:44.870","Text":"Now remember each of these is 1/2 what it was,"},{"Start":"04:44.870 ","End":"04:47.600","Text":"x squared minus 1 minus 2x squared."},{"Start":"04:47.600 ","End":"04:53.415","Text":"I\u0027m left with minus x squared and minus 4x,"},{"Start":"04:53.415 ","End":"05:01.770","Text":"and the minus 6 is really only minus 3 over x minus 1 to the 5."},{"Start":"05:02.350 ","End":"05:09.140","Text":"I can combine this minus with this minus 12 and get 12."},{"Start":"05:09.140 ","End":"05:12.605","Text":"Here I\u0027ll keep the x minus 1 to the 5."},{"Start":"05:12.605 ","End":"05:17.730","Text":"At the side, I\u0027ll do x squared plus 4x plus 3."},{"Start":"05:18.460 ","End":"05:24.335","Text":"I can factorize it because if I solve the equation, this equals 0,"},{"Start":"05:24.335 ","End":"05:30.545","Text":"I\u0027ll get x is equal to minus 1 and minus 3."},{"Start":"05:30.545 ","End":"05:32.420","Text":"By the rules of factorization,"},{"Start":"05:32.420 ","End":"05:35.652","Text":"I can factorize this as x plus 1, x plus 3,"},{"Start":"05:35.652 ","End":"05:37.725","Text":"and I\u0027ll write this over here."},{"Start":"05:37.725 ","End":"05:41.232","Text":"We have x plus 1, x plus 3"},{"Start":"05:41.232 ","End":"05:46.030","Text":"over x minus 1 to the 5."},{"Start":"05:46.030 ","End":"05:48.350","Text":"That\u0027s the end of this step of the solution."},{"Start":"05:48.350 ","End":"05:55.325","Text":"Now the next step is to compare f double prime to 0 and solve the equation."},{"Start":"05:55.325 ","End":"05:59.405","Text":"Let\u0027s write f double prime of x equals 0"},{"Start":"05:59.405 ","End":"06:02.720","Text":"and see what we get as a solution for this equation."},{"Start":"06:02.720 ","End":"06:10.654","Text":"I can forget about the 12 and write that x plus 1, x plus 3."},{"Start":"06:10.654 ","End":"06:17.720","Text":"In fact, I can settle for this because if a fraction is equal to 0,"},{"Start":"06:17.720 ","End":"06:20.690","Text":"then its numerator has to be 0,"},{"Start":"06:20.690 ","End":"06:24.020","Text":"that\u0027s the only way a fraction can be 0."},{"Start":"06:24.020 ","End":"06:26.720","Text":"So we get 2 values of x,"},{"Start":"06:26.720 ","End":"06:32.660","Text":"x equals minus 1 and x equals minus 3."},{"Start":"06:32.660 ","End":"06:34.990","Text":"The next step is to draw the table,"},{"Start":"06:34.990 ","End":"06:41.195","Text":"and in it, we put these 2 points which are suspects for inflection points."},{"Start":"06:41.195 ","End":"06:42.560","Text":"But that\u0027s not all,"},{"Start":"06:42.560 ","End":"06:45.969","Text":"we also have to put in the value x equals 1"},{"Start":"06:45.969 ","End":"06:48.920","Text":"because that\u0027s the place where the function is undefined."},{"Start":"06:48.920 ","End":"06:52.565","Text":"It\u0027s like a whole and that affects the intervals that we have."},{"Start":"06:52.565 ","End":"06:59.405","Text":"We put 1, minus 1, and minus 3 into the table in increasing order."},{"Start":"06:59.405 ","End":"07:06.935","Text":"That makes it minus 3 and then minus 1 and then 1,"},{"Start":"07:06.935 ","End":"07:12.845","Text":"and these 2 are the points where f double prime is 0,"},{"Start":"07:12.845 ","End":"07:15.910","Text":"but 1 is the place where the function is not defined,"},{"Start":"07:15.910 ","End":"07:19.820","Text":"so I indicate this with the dotted lines."},{"Start":"07:19.820 ","End":"07:22.505","Text":"Now we get to the intervals."},{"Start":"07:22.505 ","End":"07:25.115","Text":"We have 4 intervals."},{"Start":"07:25.115 ","End":"07:28.640","Text":"We have x less than minus 3,"},{"Start":"07:28.640 ","End":"07:32.075","Text":"x between 3 and minus 1,"},{"Start":"07:32.075 ","End":"07:35.442","Text":"x between minus 1 and 1,"},{"Start":"07:35.442 ","End":"07:38.625","Text":"and x bigger than 1."},{"Start":"07:38.625 ","End":"07:41.570","Text":"I choose a sample point from each of these."},{"Start":"07:41.570 ","End":"07:43.820","Text":"I\u0027ll choose minus 4,"},{"Start":"07:43.820 ","End":"07:45.740","Text":"here I\u0027ll choose minus 2,"},{"Start":"07:45.740 ","End":"07:47.975","Text":"here I\u0027ll choose 0,"},{"Start":"07:47.975 ","End":"07:49.730","Text":"and here I\u0027ll choose 2."},{"Start":"07:49.730 ","End":"07:52.685","Text":"I\u0027ve to substitute these in f double prime,"},{"Start":"07:52.685 ","End":"07:56.720","Text":"but I only care about the sign, plus or minus,"},{"Start":"07:56.720 ","End":"07:58.665","Text":"so let\u0027s see then."},{"Start":"07:58.665 ","End":"08:02.235","Text":"If I substitute in f prime,"},{"Start":"08:02.235 ","End":"08:04.705","Text":"let\u0027s take the minus 4."},{"Start":"08:04.705 ","End":"08:06.710","Text":"Let\u0027s look at the denominator."},{"Start":"08:06.710 ","End":"08:10.750","Text":"Minus 4 minus 1 is a negative to the 5, this is negative."},{"Start":"08:10.750 ","End":"08:14.150","Text":"Minus 4 plus 1 is still negative."},{"Start":"08:14.150 ","End":"08:16.520","Text":"Minus 4 plus 3 is also negative."},{"Start":"08:16.520 ","End":"08:20.375","Text":"Negative times negative over negative is negative,"},{"Start":"08:20.375 ","End":"08:23.975","Text":"which means that the function goes this way."},{"Start":"08:23.975 ","End":"08:26.630","Text":"If I put in minus 2,"},{"Start":"08:26.630 ","End":"08:28.805","Text":"this will be negative,"},{"Start":"08:28.805 ","End":"08:31.280","Text":"this will be negative,"},{"Start":"08:31.280 ","End":"08:33.185","Text":"but this will be positive."},{"Start":"08:33.185 ","End":"08:35.180","Text":"We have a positive here."},{"Start":"08:35.180 ","End":"08:43.215","Text":"Put in 0, we get positive, positive, negative, so that makes it negative."},{"Start":"08:43.215 ","End":"08:51.530","Text":"If I put x equals 2, positive, positive, positive, so we have positive."},{"Start":"08:51.530 ","End":"08:54.950","Text":"We have this area\u0027s concave, and this is concave,"},{"Start":"08:54.950 ","End":"08:57.004","Text":"this is convex, and this is convex."},{"Start":"08:57.004 ","End":"08:59.255","Text":"But if we look at our 2 suspects,"},{"Start":"08:59.255 ","End":"09:01.475","Text":"which were the minus 1 and minus 3,"},{"Start":"09:01.475 ","End":"09:04.670","Text":"we see that they are both inflection points because"},{"Start":"09:04.670 ","End":"09:08.195","Text":"they separate between concave and convex,"},{"Start":"09:08.195 ","End":"09:09.845","Text":"or convex and concave,"},{"Start":"09:09.845 ","End":"09:14.150","Text":"so this negative 3 will be an inflection point."},{"Start":"09:14.150 ","End":"09:15.440","Text":"For the inflection points,"},{"Start":"09:15.440 ","End":"09:17.234","Text":"I\u0027d like to know the x and the y."},{"Start":"09:17.234 ","End":"09:21.370","Text":"I\u0027d like these 2 values here and here."},{"Start":"09:21.370 ","End":"09:25.460","Text":"To get these, I have to substitute in the original function"},{"Start":"09:25.460 ","End":"09:29.240","Text":"because y is the original function,"},{"Start":"09:29.240 ","End":"09:32.075","Text":"and I just put these values in."},{"Start":"09:32.075 ","End":"09:34.520","Text":"Let\u0027s see, if it\u0027s minus 3"},{"Start":"09:34.520 ","End":"09:36.080","Text":"and I put it in here,"},{"Start":"09:36.080 ","End":"09:40.940","Text":"I get minus 2 over minus 4,"},{"Start":"09:40.940 ","End":"09:46.070","Text":"which is 1/2, but cubed is 1/8,"},{"Start":"09:46.070 ","End":"09:48.020","Text":"so here I have 1/8,"},{"Start":"09:48.020 ","End":"09:50.430","Text":"and if I put minus 1,"},{"Start":"09:50.430 ","End":"09:54.020","Text":"I get minus 1 plus 1 is 0,"},{"Start":"09:54.020 ","End":"09:56.689","Text":"and that need go no further, this will be 0."},{"Start":"09:56.689 ","End":"10:01.370","Text":"I have all the information I need to answer the questions."},{"Start":"10:01.370 ","End":"10:03.320","Text":"This is the conclusion step."},{"Start":"10:03.320 ","End":"10:10.970","Text":"What I was asked to find was the inflection points and the concave and convex intervals."},{"Start":"10:10.970 ","End":"10:13.210","Text":"First of all, the inflection,"},{"Start":"10:13.210 ","End":"10:14.700","Text":"we have 2 of those."},{"Start":"10:14.700 ","End":"10:17.690","Text":"We have one at minus 3, 1/8,"},{"Start":"10:17.690 ","End":"10:22.280","Text":"and we have another one at minus 1, 0."},{"Start":"10:22.280 ","End":"10:26.555","Text":"For the convex or concave up."},{"Start":"10:26.555 ","End":"10:29.225","Text":"That\u0027s the convex, this one and this one,"},{"Start":"10:29.225 ","End":"10:31.030","Text":"so we have 2 intervals."},{"Start":"10:31.030 ","End":"10:34.700","Text":"We have minus 3 less than x less than minus 1,"},{"Start":"10:34.700 ","End":"10:38.270","Text":"and we also have x bigger than 1."},{"Start":"10:38.270 ","End":"10:43.025","Text":"For the concave, I look for this shape here and here,"},{"Start":"10:43.025 ","End":"10:46.220","Text":"so it\u0027s x less than minus 3"},{"Start":"10:46.220 ","End":"10:51.368","Text":"or minus 1 less than x less than 1,"},{"Start":"10:51.368 ","End":"10:53.580","Text":"and we\u0027re done."}],"ID":5832},{"Watched":false,"Name":"Exercise 12","Duration":"5m 26s","ChapterTopicVideoID":5835,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5835.jpeg","UploadDate":"2016-05-01T17:18:57.8570000","DurationForVideoObject":"PT5M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.510","Text":"In this exercise, we\u0027re given f of x equals x minus 1 over x cubed and we have to find"},{"Start":"00:06.510 ","End":"00:10.080","Text":"its inflection points and the intervals where"},{"Start":"00:10.080 ","End":"00:14.100","Text":"the function is concave up and where it\u0027s concave down."},{"Start":"00:14.100 ","End":"00:16.950","Text":"This is a familiar type of exercise and there are"},{"Start":"00:16.950 ","End":"00:21.645","Text":"certain standard steps that we can follow to the solution cookbook style,"},{"Start":"00:21.645 ","End":"00:23.220","Text":"and we\u0027ll do that in a moment."},{"Start":"00:23.220 ","End":"00:28.410","Text":"But before that, I\u0027d like to just take a look at the domain of the function."},{"Start":"00:28.410 ","End":"00:34.409","Text":"Looking at it, we see immediately that the domain is all x except for x equals 0."},{"Start":"00:34.409 ","End":"00:36.570","Text":"We\u0027ll use this fact later."},{"Start":"00:36.570 ","End":"00:40.225","Text":"Now to the steps in solving the question."},{"Start":"00:40.225 ","End":"00:44.920","Text":"There\u0027s a preparation phase where we have to find f double prime."},{"Start":"00:44.920 ","End":"00:46.815","Text":"Of course to find f double prime,"},{"Start":"00:46.815 ","End":"00:48.695","Text":"we start with f prime."},{"Start":"00:48.695 ","End":"00:50.555","Text":"Now, to make things easier,"},{"Start":"00:50.555 ","End":"00:54.500","Text":"I think I would like to rewrite this and divide it out."},{"Start":"00:54.500 ","End":"00:55.794","Text":"It might be simpler."},{"Start":"00:55.794 ","End":"00:59.675","Text":"So wait a minute for the differentiation and just first rewrite this."},{"Start":"00:59.675 ","End":"01:06.020","Text":"X over x cubed is 1 over x squared and minus 1 over x cubed."},{"Start":"01:06.020 ","End":"01:09.875","Text":"This is also equal to x to the minus 2,"},{"Start":"01:09.875 ","End":"01:12.770","Text":"minus x to the minus 3."},{"Start":"01:12.770 ","End":"01:15.140","Text":"Now let\u0027s get onto the differentiation."},{"Start":"01:15.140 ","End":"01:25.350","Text":"F prime of x will equal minus 2x to the minus 3 plus 3x to the minus 4."},{"Start":"01:25.350 ","End":"01:28.215","Text":"Now we can do f double prime."},{"Start":"01:28.215 ","End":"01:34.635","Text":"This will equal minus 3 times minus 2 is 6x to the minus 4,"},{"Start":"01:34.635 ","End":"01:40.125","Text":"minus 12x to the minus 5."},{"Start":"01:40.125 ","End":"01:42.695","Text":"If I rewrite this,"},{"Start":"01:42.695 ","End":"01:45.995","Text":"I can put this x to the minus 5."},{"Start":"01:45.995 ","End":"01:48.140","Text":"I can take outside the brackets."},{"Start":"01:48.140 ","End":"01:52.960","Text":"In fact, x to the minus 5 goes into the denominator as x to the fifth."},{"Start":"01:52.960 ","End":"01:55.640","Text":"What we\u0027re left with is just an extra x here,"},{"Start":"01:55.640 ","End":"02:01.645","Text":"so it\u0027s 6x minus 12 over x to the fifth."},{"Start":"02:01.645 ","End":"02:04.575","Text":"That\u0027s the end of this step."},{"Start":"02:04.575 ","End":"02:10.595","Text":"The next step is to set f prime of x equals 0."},{"Start":"02:10.595 ","End":"02:18.005","Text":"The end of this step and f double prime of x equals 0 is what we have to solve for x."},{"Start":"02:18.005 ","End":"02:22.760","Text":"So it\u0027s 6 x minus 12 is 0 because the numerator"},{"Start":"02:22.760 ","End":"02:28.250","Text":"has to be 0 and x equals 12 over 6, x equals 2."},{"Start":"02:28.250 ","End":"02:30.080","Text":"That\u0027s the only solution,"},{"Start":"02:30.080 ","End":"02:33.920","Text":"and this is a suspect for being an inflection point."},{"Start":"02:33.920 ","End":"02:36.305","Text":"Next, we\u0027re going to make a table."},{"Start":"02:36.305 ","End":"02:42.740","Text":"In the x row, we put the suspects for inflection and that would be x equals 2."},{"Start":"02:42.740 ","End":"02:48.530","Text":"But we also have to put in a point where the function is undefined."},{"Start":"02:48.530 ","End":"02:53.225","Text":"So we need x equals 0 and x equals 2 in that order."},{"Start":"02:53.225 ","End":"02:58.825","Text":"X equals 0 just helps contribute to the way we divide the line up into intervals,"},{"Start":"02:58.825 ","End":"03:04.850","Text":"and x equals 2 is a suspect because this came from f double prime being 0."},{"Start":"03:04.850 ","End":"03:07.490","Text":"But this is a point where the function\u0027s not"},{"Start":"03:07.490 ","End":"03:10.925","Text":"defined and I indicate this with dotted lines here."},{"Start":"03:10.925 ","End":"03:14.960","Text":"We divide the line up into intervals according to these points."},{"Start":"03:14.960 ","End":"03:18.485","Text":"So we get x less than 0,"},{"Start":"03:18.485 ","End":"03:22.955","Text":"x between 0 and 2 and x bigger than 2."},{"Start":"03:22.955 ","End":"03:26.150","Text":"We choose a sample point from each, quite arbitrary,"},{"Start":"03:26.150 ","End":"03:28.010","Text":"I\u0027ll choose minus 1 here,"},{"Start":"03:28.010 ","End":"03:29.360","Text":"I\u0027ll choose 1 here,"},{"Start":"03:29.360 ","End":"03:31.055","Text":"and I\u0027ll choose 3 here."},{"Start":"03:31.055 ","End":"03:36.500","Text":"This we need to substitute in f double prime, which is here."},{"Start":"03:36.500 ","End":"03:40.700","Text":"Let\u0027s see, when x is minus 1 in the denominator,"},{"Start":"03:40.700 ","End":"03:43.385","Text":"we get minus 1 to the fifth is negative."},{"Start":"03:43.385 ","End":"03:48.500","Text":"In the numerator, we get 6 minus 12, which is negative."},{"Start":"03:48.500 ","End":"03:51.950","Text":"So negative over negative is positive."},{"Start":"03:51.950 ","End":"03:53.795","Text":"When x is 1,"},{"Start":"03:53.795 ","End":"04:01.760","Text":"we get 6 minus 12 is negative and 1 to the fifth is now positive."},{"Start":"04:01.760 ","End":"04:03.860","Text":"So we get minus over plus,"},{"Start":"04:03.860 ","End":"04:06.065","Text":"so this is minus."},{"Start":"04:06.065 ","End":"04:07.925","Text":"For x equals 3,"},{"Start":"04:07.925 ","End":"04:13.655","Text":"we get positive over positive, which is positive."},{"Start":"04:13.655 ","End":"04:16.070","Text":"So if this is how f double prime is,"},{"Start":"04:16.070 ","End":"04:21.540","Text":"the shape of the function is convex, concave, convex."},{"Start":"04:21.540 ","End":"04:23.565","Text":"Our suspect, x equals 2,"},{"Start":"04:23.565 ","End":"04:29.745","Text":"turns out to be an inflection because it\u0027s between concave and convex."},{"Start":"04:29.745 ","End":"04:34.910","Text":"So all I\u0027d like more to add to the table is the y value when x equals 2,"},{"Start":"04:34.910 ","End":"04:39.755","Text":"and that we can get from the original function."},{"Start":"04:39.755 ","End":"04:42.380","Text":"So if I put x equals 2,"},{"Start":"04:42.380 ","End":"04:48.515","Text":"I get 2 minus 1 over 2 cubed, which is 1/8."},{"Start":"04:48.515 ","End":"04:50.510","Text":"That\u0027s all I need from the table."},{"Start":"04:50.510 ","End":"04:52.820","Text":"Now I\u0027m ready to get to the last phase,"},{"Start":"04:52.820 ","End":"04:54.320","Text":"which is the conclusion phase,"},{"Start":"04:54.320 ","End":"04:58.910","Text":"which answers the questions about where do we have an inflection?"},{"Start":"04:58.910 ","End":"05:04.740","Text":"The answer is that we only have 1 and it\u0027s at the point 2 comma 1/8."},{"Start":"05:04.740 ","End":"05:11.780","Text":"Then we answer, where is the function convex in which intervals convex is this and this."},{"Start":"05:11.780 ","End":"05:17.860","Text":"So x less than 0 and also x greater than 2."},{"Start":"05:17.860 ","End":"05:21.335","Text":"As for concave, that\u0027s the other shape, that\u0027s this."},{"Start":"05:21.335 ","End":"05:27.300","Text":"That\u0027s when x is between 0 and 2 and that\u0027s it. We\u0027re done."}],"ID":5833},{"Watched":false,"Name":"Exercise 13","Duration":"2m 51s","ChapterTopicVideoID":5836,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5836.jpeg","UploadDate":"2016-05-01T17:19:15.6100000","DurationForVideoObject":"PT2M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.030","Text":"Given a function f of x equals x minus e^x and we have to"},{"Start":"00:06.030 ","End":"00:08.790","Text":"find its inflection points as well as"},{"Start":"00:08.790 ","End":"00:13.125","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.125 ","End":"00:19.850","Text":"This is a standard type of exercise and we have a solution in steps, cookbook style."},{"Start":"00:19.850 ","End":"00:25.105","Text":"But first, let\u0027s just take a look at the domain that comes before the steps."},{"Start":"00:25.105 ","End":"00:28.760","Text":"I see that there\u0027s no problem with the domain."},{"Start":"00:28.760 ","End":"00:31.775","Text":"It\u0027s defined everywhere, everything\u0027s fine."},{"Start":"00:31.775 ","End":"00:33.290","Text":"Now let\u0027s get to the steps,"},{"Start":"00:33.290 ","End":"00:36.889","Text":"there\u0027s a preparation phase and I should have copied the exercise"},{"Start":"00:36.889 ","End":"00:40.895","Text":"first where we have to find the second derivative."},{"Start":"00:40.895 ","End":"00:44.185","Text":"Of course, we need to find the first derivative first,"},{"Start":"00:44.185 ","End":"00:46.110","Text":"so that\u0027s straightforward enough."},{"Start":"00:46.110 ","End":"00:53.780","Text":"1 minus e^x and f double prime of x will therefore be just minus e^x."},{"Start":"00:53.780 ","End":"00:58.670","Text":"That was quick work with the preparation then we call the first step,"},{"Start":"00:58.670 ","End":"01:04.670","Text":"which is to set the second derivative to 0 and solve for x."},{"Start":"01:04.670 ","End":"01:08.600","Text":"If we have f double prime of x equals 0,"},{"Start":"01:08.600 ","End":"01:13.549","Text":"we get that minus e^x is equal to 0,"},{"Start":"01:13.549 ","End":"01:16.410","Text":"but e to the power of something is never 0,"},{"Start":"01:16.410 ","End":"01:18.260","Text":"it\u0027s always positive in fact,"},{"Start":"01:18.260 ","End":"01:19.580","Text":"so this is always negative."},{"Start":"01:19.580 ","End":"01:21.230","Text":"In any event, it\u0027s not 0."},{"Start":"01:21.230 ","End":"01:24.515","Text":"There is no such x or no solution."},{"Start":"01:24.515 ","End":"01:27.500","Text":"The next step is to make a table,"},{"Start":"01:27.500 ","End":"01:29.300","Text":"although the table\u0027s going to be pretty empty,"},{"Start":"01:29.300 ","End":"01:31.820","Text":"nevertheless, that\u0027s 1 of the steps."},{"Start":"01:31.820 ","End":"01:35.150","Text":"There are no special values of x to put in the table."},{"Start":"01:35.150 ","End":"01:38.270","Text":"We use these x\u0027s to break the line up into intervals"},{"Start":"01:38.270 ","End":"01:41.310","Text":"so the whole line now is 1 big interval,"},{"Start":"01:41.310 ","End":"01:44.825","Text":"it\u0027s just x from minus infinity to infinity."},{"Start":"01:44.825 ","End":"01:48.065","Text":"I\u0027ll just write it as minus infinity less than x,"},{"Start":"01:48.065 ","End":"01:51.305","Text":"less than infinity is just 1 continuous line."},{"Start":"01:51.305 ","End":"01:53.840","Text":"We choose the sample points from each interval."},{"Start":"01:53.840 ","End":"01:57.335","Text":"I\u0027ll just choose any value of x whatsoever."},{"Start":"01:57.335 ","End":"02:01.910","Text":"I\u0027ll choose x equals 0 and I\u0027ll check for it,"},{"Start":"02:01.910 ","End":"02:05.765","Text":"what is f double-prime of x minus e^0?"},{"Start":"02:05.765 ","End":"02:07.775","Text":"Is minus 1."},{"Start":"02:07.775 ","End":"02:10.640","Text":"All we care about is that it\u0027s negative."},{"Start":"02:10.640 ","End":"02:14.960","Text":"That gives us the shape of the function negative is like this,"},{"Start":"02:14.960 ","End":"02:17.915","Text":"which is what we call concave."},{"Start":"02:17.915 ","End":"02:22.519","Text":"This means, and since we\u0027re all 1 big interval that the function is always"},{"Start":"02:22.519 ","End":"02:26.565","Text":"concave and there are no inflection points,"},{"Start":"02:26.565 ","End":"02:28.910","Text":"so I can do the last stage,"},{"Start":"02:28.910 ","End":"02:32.670","Text":"which is the conclusions or answering the question."},{"Start":"02:32.670 ","End":"02:35.240","Text":"As for inflection points,"},{"Start":"02:35.240 ","End":"02:39.910","Text":"we have none. Convex intervals, none."},{"Start":"02:39.910 ","End":"02:42.380","Text":"Concave is everywhere,"},{"Start":"02:42.380 ","End":"02:46.070","Text":"meaning let\u0027s say all x or x between"},{"Start":"02:46.070 ","End":"02:49.940","Text":"minus infinity and infinity and not very interesting exercise,"},{"Start":"02:49.940 ","End":"02:52.080","Text":"but there it is."}],"ID":5834},{"Watched":false,"Name":"Exercise 14","Duration":"6m 29s","ChapterTopicVideoID":5837,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5837.jpeg","UploadDate":"2016-05-01T17:19:57.3800000","DurationForVideoObject":"PT6M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.370","Text":"In this exercise, we\u0027re given a function f of x equals e to the 1 over x,"},{"Start":"00:05.370 ","End":"00:09.120","Text":"and we have to find its inflection points and"},{"Start":"00:09.120 ","End":"00:13.125","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:13.125 ","End":"00:16.370","Text":"This is a standard familiar type of exercise and is"},{"Start":"00:16.370 ","End":"00:19.800","Text":"solved in a set of steps which we know"},{"Start":"00:19.800 ","End":"00:23.100","Text":"and we\u0027ll follow except that the steps don\u0027t cover"},{"Start":"00:23.100 ","End":"00:26.610","Text":"the stage where we check the domain of the function."},{"Start":"00:26.610 ","End":"00:29.745","Text":"I\u0027m just going to look at the domain and see that"},{"Start":"00:29.745 ","End":"00:31.950","Text":"exponent is defined everywhere."},{"Start":"00:31.950 ","End":"00:35.515","Text":"It\u0027s just that the 1 over x is not defined for x equals 0."},{"Start":"00:35.515 ","End":"00:39.209","Text":"This function is okay but x must not be 0,"},{"Start":"00:39.209 ","End":"00:42.360","Text":"and we\u0027ll use this value 0 later on."},{"Start":"00:42.360 ","End":"00:48.170","Text":"Now, the steps is a preparation step where we have to find the second derivative."},{"Start":"00:48.170 ","End":"00:52.850","Text":"Let\u0027s find the first derivative first and then go to the second."},{"Start":"00:52.850 ","End":"00:55.114","Text":"F prime of x equals,"},{"Start":"00:55.114 ","End":"00:56.525","Text":"using the chain rule,"},{"Start":"00:56.525 ","End":"01:05.605","Text":"it\u0027s e to the power of 1 over x times inner derivative which is minus 1 over x squared."},{"Start":"01:05.605 ","End":"01:08.080","Text":"Now, let\u0027s differentiate again,"},{"Start":"01:08.080 ","End":"01:10.205","Text":"f double prime of x."},{"Start":"01:10.205 ","End":"01:13.535","Text":"We can do it by making it a quotient or a product."},{"Start":"01:13.535 ","End":"01:16.505","Text":"You know what? I\u0027ll leave it as a product."},{"Start":"01:16.505 ","End":"01:19.730","Text":"Well, the product rule you\u0027d better know by now."},{"Start":"01:19.730 ","End":"01:23.420","Text":"It\u0027s each 1 times the derivative of the other and we add."},{"Start":"01:23.420 ","End":"01:26.975","Text":"Derivative of the first which we\u0027ve already done,"},{"Start":"01:26.975 ","End":"01:33.005","Text":"it\u0027s e to the 1 over x times minus 1 over x squared,"},{"Start":"01:33.005 ","End":"01:36.430","Text":"but times minus 1 over x squared again,"},{"Start":"01:36.430 ","End":"01:39.890","Text":"then plus this as is,"},{"Start":"01:39.890 ","End":"01:45.489","Text":"which is e to the 1 over x times the derivative of minus 1 over x squared."},{"Start":"01:45.489 ","End":"01:51.335","Text":"This is plus 2 over x cubed."},{"Start":"01:51.335 ","End":"01:52.595","Text":"How do I know this?"},{"Start":"01:52.595 ","End":"01:55.130","Text":"This is minus x to the minus 2,"},{"Start":"01:55.130 ","End":"01:59.060","Text":"and the derivative of x to the minus 2 is minus 2x to the minus 3."},{"Start":"01:59.060 ","End":"02:02.000","Text":"The minus with the minus cancels which is 2x"},{"Start":"02:02.000 ","End":"02:05.510","Text":"to the minus 3 which is same as what I wrote here."},{"Start":"02:05.510 ","End":"02:10.295","Text":"If we take out e to the 1 over x from both,"},{"Start":"02:10.295 ","End":"02:15.320","Text":"and here we get on the denominator x^4 and here x cubed."},{"Start":"02:15.320 ","End":"02:19.490","Text":"Let\u0027s make it all over x^4."},{"Start":"02:19.490 ","End":"02:21.450","Text":"From here we get 1,"},{"Start":"02:21.450 ","End":"02:23.760","Text":"minus 1 to the minus 1 is 1."},{"Start":"02:23.760 ","End":"02:28.455","Text":"From here we get 2 over x cubed is like 2x over x^4."},{"Start":"02:28.455 ","End":"02:30.615","Text":"This is our second derivative,"},{"Start":"02:30.615 ","End":"02:34.250","Text":"and that ends the preparation step."},{"Start":"02:34.250 ","End":"02:39.850","Text":"Next step is to set f double-prime to equals 0 and to solve for x."},{"Start":"02:39.850 ","End":"02:42.860","Text":"This will give us our suspects for inflection."},{"Start":"02:42.860 ","End":"02:46.850","Text":"The only way f double-prime is going to be 0,"},{"Start":"02:46.850 ","End":"02:52.310","Text":"this is never 0, and the fraction is 0 when its numerator is 0."},{"Start":"02:52.310 ","End":"02:56.965","Text":"We get 1 plus 2x equals 0."},{"Start":"02:56.965 ","End":"03:02.250","Text":"That gives us that x equals minus a 1/2."},{"Start":"03:02.250 ","End":"03:06.335","Text":"We have a suspect for an inflection minus a 1/2."},{"Start":"03:06.335 ","End":"03:08.905","Text":"The next step is the table."},{"Start":"03:08.905 ","End":"03:10.445","Text":"What do we put in the table?"},{"Start":"03:10.445 ","End":"03:16.100","Text":"We put in the 1/2 because it\u0027s the suspect for inflection."},{"Start":"03:16.100 ","End":"03:21.290","Text":"We also put in the value 0 because it\u0027s a point where the function is undefined and that"},{"Start":"03:21.290 ","End":"03:24.080","Text":"also goes in the table but we write them in"},{"Start":"03:24.080 ","End":"03:27.155","Text":"increasing order and with some space in between."},{"Start":"03:27.155 ","End":"03:32.640","Text":"Here\u0027s minus 1/2 and here we have 0."},{"Start":"03:32.640 ","End":"03:36.060","Text":"The minus 1/2 was from double-prime being 0."},{"Start":"03:36.060 ","End":"03:37.199","Text":"That\u0027s our suspect."},{"Start":"03:37.199 ","End":"03:43.120","Text":"0 was because function is not defined there and we indicate that with dotted lines."},{"Start":"03:43.120 ","End":"03:45.410","Text":"Next we separate these 2 points,"},{"Start":"03:45.410 ","End":"03:47.300","Text":"break up the line into intervals."},{"Start":"03:47.300 ","End":"03:49.610","Text":"I write down what those intervals are."},{"Start":"03:49.610 ","End":"03:53.010","Text":"That is x less than minus 1/2,"},{"Start":"03:53.010 ","End":"03:56.070","Text":"x between minus 1/2 and 0,"},{"Start":"03:56.070 ","End":"04:00.725","Text":"and after the point where we\u0027re undefined we have x bigger than 0."},{"Start":"04:00.725 ","End":"04:02.660","Text":"We choose a sample point from each,"},{"Start":"04:02.660 ","End":"04:04.520","Text":"so I\u0027ll choose minus 1."},{"Start":"04:04.520 ","End":"04:05.785","Text":"It\u0027s whatever we want."},{"Start":"04:05.785 ","End":"04:08.205","Text":"Between minus 1/2 and 0, well,"},{"Start":"04:08.205 ","End":"04:09.990","Text":"I\u0027ll go with minus a 1/4,"},{"Start":"04:09.990 ","End":"04:12.460","Text":"and for bigger than 0 I\u0027ll go with 1."},{"Start":"04:12.460 ","End":"04:19.430","Text":"We have to substitute these 3 into the second derivative which is this function here."},{"Start":"04:19.430 ","End":"04:23.070","Text":"But we don\u0027t want the actual answer only whether it\u0027s positive or negative."},{"Start":"04:23.070 ","End":"04:25.785","Text":"When x is minus 1,"},{"Start":"04:25.785 ","End":"04:27.275","Text":"well, this is positive."},{"Start":"04:27.275 ","End":"04:30.460","Text":"This thing is always positive because it\u0027s in the power of 4."},{"Start":"04:30.460 ","End":"04:32.700","Text":"I just put in minus 1 in here,"},{"Start":"04:32.700 ","End":"04:36.495","Text":"and it\u0027s 1 minus 2, it\u0027s negative here."},{"Start":"04:36.495 ","End":"04:39.170","Text":"If I put in minus a 1/4, again,"},{"Start":"04:39.170 ","End":"04:40.865","Text":"I just care about this part,"},{"Start":"04:40.865 ","End":"04:44.695","Text":"it\u0027s 1 minus 1/2, it\u0027s positive."},{"Start":"04:44.695 ","End":"04:48.035","Text":"I put x equals 1, then also positive,"},{"Start":"04:48.035 ","End":"04:51.110","Text":"which means that the function is shaped like this,"},{"Start":"04:51.110 ","End":"04:55.215","Text":"this, and this, concave, convex, convex."},{"Start":"04:55.215 ","End":"04:58.565","Text":"The minus 1/2 which was a suspect is really guilty."},{"Start":"04:58.565 ","End":"05:03.530","Text":"I mean it is an inflection point because it\u0027s between concave and a convex."},{"Start":"05:03.530 ","End":"05:09.625","Text":"It\u0027s an inflection point and I\u0027ll write that down here, inflection."},{"Start":"05:09.625 ","End":"05:14.660","Text":"All I need to know still is the value of y here."},{"Start":"05:14.660 ","End":"05:18.175","Text":"That\u0027s only extra information I need to get this point,"},{"Start":"05:18.175 ","End":"05:23.805","Text":"so y is just f of x. I have to put in into here,"},{"Start":"05:23.805 ","End":"05:27.405","Text":"x equals minus 1/2."},{"Start":"05:27.405 ","End":"05:28.635","Text":"What I get is,"},{"Start":"05:28.635 ","End":"05:30.075","Text":"if x is minus 1/2,"},{"Start":"05:30.075 ","End":"05:34.320","Text":"1 over minus 1/2 is minus 2,"},{"Start":"05:34.320 ","End":"05:42.240","Text":"and e^minus 2 is just e^minus 2 or 1 over e squared, whatever."},{"Start":"05:42.240 ","End":"05:46.700","Text":"Now we can go to the last step which is the conclusions."},{"Start":"05:46.700 ","End":"05:49.700","Text":"For the conclusions, we need 3 things."},{"Start":"05:49.700 ","End":"05:54.650","Text":"We need the inflection as well as the concave up and concave down."},{"Start":"05:54.650 ","End":"05:59.170","Text":"For inflection we do have 1 at the point minus 1/2,"},{"Start":"05:59.170 ","End":"06:06.260","Text":"1 over e squared and we have concave up or convex."},{"Start":"06:06.260 ","End":"06:09.770","Text":"That\u0027s these 2 here, that\u0027s minus 1/2,"},{"Start":"06:09.770 ","End":"06:14.194","Text":"less than x, less than 0 and x bigger than 0."},{"Start":"06:14.194 ","End":"06:16.460","Text":"I can\u0027t combine these intervals because there\u0027s"},{"Start":"06:16.460 ","End":"06:19.815","Text":"a gap in the middle where the function is not defined."},{"Start":"06:19.815 ","End":"06:23.900","Text":"The concave is, this is the concave and that\u0027s"},{"Start":"06:23.900 ","End":"06:30.000","Text":"where x is less than minus 1/2, and we\u0027re done."}],"ID":5835},{"Watched":false,"Name":"Exercise 15","Duration":"9m 15s","ChapterTopicVideoID":5838,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5838.jpeg","UploadDate":"2016-05-01T17:21:34.5770000","DurationForVideoObject":"PT9M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.775","Text":"In this exercise, we\u0027re given"},{"Start":"00:01.775 ","End":"00:06.474","Text":"f of x equals x times e to the minus 2x squared."},{"Start":"00:06.474 ","End":"00:07.740","Text":"For this function,"},{"Start":"00:07.740 ","End":"00:10.800","Text":"we have to find the inflection points and"},{"Start":"00:10.800 ","End":"00:15.090","Text":"the intervals where the function is convex and where it\u0027s concave."},{"Start":"00:15.090 ","End":"00:20.820","Text":"This is a familiar type of exercise and it has a certain standard set of steps,"},{"Start":"00:20.820 ","End":"00:23.280","Text":"cookbook style, which we\u0027ll follow."},{"Start":"00:23.280 ","End":"00:26.280","Text":"But first, I\u0027d just like to look at"},{"Start":"00:26.280 ","End":"00:30.165","Text":"the domain of the function that\u0027s not covered in the steps."},{"Start":"00:30.165 ","End":"00:32.400","Text":"Looking at it, no problems."},{"Start":"00:32.400 ","End":"00:34.440","Text":"It\u0027s defined for all x."},{"Start":"00:34.440 ","End":"00:39.475","Text":"Let\u0027s proceed with the steps of the solution."},{"Start":"00:39.475 ","End":"00:45.180","Text":"The first step is a preparation stage where we have to find f double prime."},{"Start":"00:45.180 ","End":"00:48.060","Text":"I just copied the f itself."},{"Start":"00:48.060 ","End":"00:51.210","Text":"Next thing is just f prime,"},{"Start":"00:51.210 ","End":"00:53.190","Text":"and then we\u0027ll do f double prime."},{"Start":"00:53.190 ","End":"00:55.850","Text":"Using the product rule here,"},{"Start":"00:55.850 ","End":"00:58.660","Text":"and you should know the product rule now, I expect it,"},{"Start":"00:58.660 ","End":"01:00.650","Text":"we have the derivative of x,"},{"Start":"01:00.650 ","End":"01:09.155","Text":"which is 1 times e to the minus 2x squared plus x times the derivative of this."},{"Start":"01:09.155 ","End":"01:10.955","Text":"Using the chain rule,"},{"Start":"01:10.955 ","End":"01:15.470","Text":"it\u0027s e to the minus 2x squared times the derivative of this,"},{"Start":"01:15.470 ","End":"01:18.720","Text":"which is minus 4x."},{"Start":"01:18.720 ","End":"01:23.940","Text":"We can take e to the minus 2x squared outside the brackets."},{"Start":"01:23.940 ","End":"01:25.604","Text":"What we\u0027re left with is 1,"},{"Start":"01:25.604 ","End":"01:30.780","Text":"and we have this with this gives us minus 4x squared."},{"Start":"01:30.780 ","End":"01:34.635","Text":"Now let\u0027s differentiate again."},{"Start":"01:34.635 ","End":"01:38.315","Text":"We get f double prime of x is equal to."},{"Start":"01:38.315 ","End":"01:42.469","Text":"Again, we\u0027ll use the product rule this time."},{"Start":"01:42.469 ","End":"01:51.570","Text":"We have the derivative of this is e to the minus 2x squared times minus 4x."},{"Start":"01:51.570 ","End":"01:59.385","Text":"This was here, 1 minus 4x squared plus this 1 as is,"},{"Start":"01:59.385 ","End":"02:04.715","Text":"which is e to the minus 2x squared times the derivative of this,"},{"Start":"02:04.715 ","End":"02:07.955","Text":"which is minus 8x."},{"Start":"02:07.955 ","End":"02:14.195","Text":"Again, I\u0027ll take e to the minus 2x squared outside the brackets."},{"Start":"02:14.195 ","End":"02:16.010","Text":"What are we left with?"},{"Start":"02:16.010 ","End":"02:19.280","Text":"We\u0027re left with minus 4x."},{"Start":"02:19.280 ","End":"02:27.540","Text":"This with this gives us plus 16x cubed and here we have minus 8x."},{"Start":"02:27.770 ","End":"02:32.470","Text":"I\u0027ll combine these 2 into minus 12x."},{"Start":"02:32.470 ","End":"02:35.460","Text":"I don\u0027t want to write an extra step."},{"Start":"02:35.830 ","End":"02:40.250","Text":"I\u0027m thinking I could take more outside the brackets."},{"Start":"02:40.250 ","End":"02:43.805","Text":"I could take 4x outside the brackets,"},{"Start":"02:43.805 ","End":"02:45.954","Text":"simplify even more."},{"Start":"02:45.954 ","End":"02:50.205","Text":"4x e to the minus 2x squared."},{"Start":"02:50.205 ","End":"02:57.645","Text":"Then all we\u0027re left with is minus 3 plus 4x squared."},{"Start":"02:57.645 ","End":"03:01.455","Text":"That finishes this step."},{"Start":"03:01.455 ","End":"03:04.595","Text":"Now just put a little line here and the step."},{"Start":"03:04.595 ","End":"03:10.625","Text":"Now the next step is to set f double prime to 0 and solve for x."},{"Start":"03:10.625 ","End":"03:14.075","Text":"Let\u0027s say f double prime of x equals 0."},{"Start":"03:14.075 ","End":"03:20.495","Text":"The solutions will be suspects for inflection points. So let\u0027s see."},{"Start":"03:20.495 ","End":"03:24.560","Text":"If f double prime, and this is it, is equal to 0,"},{"Start":"03:24.560 ","End":"03:27.545","Text":"then either x is 0,"},{"Start":"03:27.545 ","End":"03:29.480","Text":"this thing is never 0,"},{"Start":"03:29.480 ","End":"03:31.670","Text":"or this could be 0."},{"Start":"03:31.670 ","End":"03:33.770","Text":"Let\u0027s see what solutions we can get."},{"Start":"03:33.770 ","End":"03:39.110","Text":"We can get x equals 0 or from here,"},{"Start":"03:39.110 ","End":"03:41.045","Text":"let me do it at the side,"},{"Start":"03:41.045 ","End":"03:44.030","Text":"we get minus 3 plus 4x squared is 0."},{"Start":"03:44.030 ","End":"03:46.475","Text":"So 4x squared equals 3,"},{"Start":"03:46.475 ","End":"03:49.645","Text":"x squared equals 3/4."},{"Start":"03:49.645 ","End":"03:58.295","Text":"x equals plus or minus the square root of 3 over the square root of 4, which is that."},{"Start":"03:58.295 ","End":"04:04.972","Text":"We have minus square root of 3 over 2 and plus square root of 3 over 2."},{"Start":"04:04.972 ","End":"04:08.280","Text":"3 solutions for this equation,"},{"Start":"04:08.280 ","End":"04:11.565","Text":"3 suspects for inflection points."},{"Start":"04:11.565 ","End":"04:16.390","Text":"We\u0027ll sort these out with our usual table, which I\u0027ll draw."},{"Start":"04:16.390 ","End":"04:19.243","Text":"Here\u0027s a blank table with the usual rows:"},{"Start":"04:19.243 ","End":"04:21.685","Text":"x, f double prime, f, and y."},{"Start":"04:21.685 ","End":"04:26.285","Text":"In the x row, we\u0027ll put in these values in order."},{"Start":"04:26.285 ","End":"04:31.740","Text":"Minus square root of 3 over 2, 0,"},{"Start":"04:31.740 ","End":"04:35.660","Text":"and square root of 3 over 2,"},{"Start":"04:35.660 ","End":"04:39.560","Text":"which chops up the x-axis into intervals."},{"Start":"04:39.560 ","End":"04:46.040","Text":"The intervals are x less than minus root 3 over 2,"},{"Start":"04:46.040 ","End":"04:51.985","Text":"minus root 3 over 2 less than x less than 0,"},{"Start":"04:51.985 ","End":"04:56.649","Text":"x between root 3 over 2 and 0,"},{"Start":"04:56.649 ","End":"05:01.175","Text":"and x bigger than root 3 over 2."},{"Start":"05:01.175 ","End":"05:06.785","Text":"What we do here is just choose a sample point arbitrary from each of the intervals."},{"Start":"05:06.785 ","End":"05:11.255","Text":"For example, here, I could choose minus 1."},{"Start":"05:11.255 ","End":"05:13.820","Text":"Here, I can put in,"},{"Start":"05:13.820 ","End":"05:16.010","Text":"say, minus a 1/2,"},{"Start":"05:16.010 ","End":"05:18.310","Text":"here I can put 1/2,"},{"Start":"05:18.310 ","End":"05:23.985","Text":"and here I can put 1 because root 3 over 2 is less than 1."},{"Start":"05:23.985 ","End":"05:29.435","Text":"Then I plug these into f double prime, which is here."},{"Start":"05:29.435 ","End":"05:33.185","Text":"But I don\u0027t need the value just whether it\u0027s positive or negative."},{"Start":"05:33.185 ","End":"05:38.251","Text":"If I put in minus 1, now this bit is negative"},{"Start":"05:38.251 ","End":"05:43.845","Text":"and 4x squared minus 3 will be positive."},{"Start":"05:43.845 ","End":"05:47.480","Text":"Altogether, I will get negative."},{"Start":"05:47.480 ","End":"05:52.235","Text":"Then for minus a 1/2, this will be negative,"},{"Start":"05:52.235 ","End":"05:57.015","Text":"and 4x squared minus 3, it\u0027s negative."},{"Start":"05:57.015 ","End":"06:00.755","Text":"We get negative, negative is positive."},{"Start":"06:00.755 ","End":"06:04.610","Text":"When x is a 1/2, this is positive."},{"Start":"06:04.610 ","End":"06:07.865","Text":"This comes out to be negative."},{"Start":"06:07.865 ","End":"06:11.260","Text":"The last one, that\u0027s positive."},{"Start":"06:11.260 ","End":"06:14.435","Text":"That means that the function is shaped like this,"},{"Start":"06:14.435 ","End":"06:17.980","Text":"like this, like this, like this."},{"Start":"06:17.980 ","End":"06:23.025","Text":"Concave, convex, concave, convex."},{"Start":"06:23.025 ","End":"06:27.350","Text":"This alternation shows us that each one of these suspect is"},{"Start":"06:27.350 ","End":"06:31.235","Text":"between a concave and a convex or vice versa."},{"Start":"06:31.235 ","End":"06:35.150","Text":"All 3 suspects are inflection points."},{"Start":"06:35.150 ","End":"06:39.110","Text":"This is an inflection point, I\u0027ll just write the letter I."},{"Start":"06:39.110 ","End":"06:40.340","Text":"This is an inflection,"},{"Start":"06:40.340 ","End":"06:42.235","Text":"this is an inflection."},{"Start":"06:42.235 ","End":"06:46.190","Text":"What\u0027s missing from the table is I like the coordinates of the inflection,"},{"Start":"06:46.190 ","End":"06:47.964","Text":"so I need the values of y,"},{"Start":"06:47.964 ","End":"06:50.380","Text":"and to get the values of y,"},{"Start":"06:50.380 ","End":"06:55.655","Text":"I substitute in the original function which is here."},{"Start":"06:55.655 ","End":"06:57.260","Text":"Let\u0027s do the easy one."},{"Start":"06:57.260 ","End":"06:59.360","Text":"When x is 0,"},{"Start":"06:59.360 ","End":"07:02.185","Text":"I just get for y, 0."},{"Start":"07:02.185 ","End":"07:07.350","Text":"When x is minus root 3 over 2, let\u0027s see."},{"Start":"07:07.350 ","End":"07:14.985","Text":"x squared is 3/4 minus twice 3/4 is minus 3 over 2."},{"Start":"07:14.985 ","End":"07:17.210","Text":"If this is minus 3 over 2,"},{"Start":"07:17.210 ","End":"07:23.245","Text":"I\u0027ll just write it as x is minus root 3 over 2"},{"Start":"07:23.245 ","End":"07:25.540","Text":"e to the minus 3 over 2."},{"Start":"07:25.540 ","End":"07:28.160","Text":"I\u0027m not going to mess with calculators and the like."},{"Start":"07:28.160 ","End":"07:29.810","Text":"That\u0027s the expression."},{"Start":"07:29.810 ","End":"07:32.105","Text":"Square root of 3 over 2."},{"Start":"07:32.105 ","End":"07:35.820","Text":"Again, I get minus 3 over 2 here"},{"Start":"07:35.820 ","End":"07:37.440","Text":"but there\u0027s a plus here."},{"Start":"07:37.440 ","End":"07:40.470","Text":"It\u0027s root 3 over 2"},{"Start":"07:40.470 ","End":"07:43.045","Text":"e to the minus 3 over 2."},{"Start":"07:43.045 ","End":"07:46.370","Text":"Now I have all the information I need for the last step,"},{"Start":"07:46.370 ","End":"07:47.840","Text":"which is the conclusion,"},{"Start":"07:47.840 ","End":"07:50.540","Text":"or I basically answer the question that was asked,"},{"Start":"07:50.540 ","End":"07:55.585","Text":"and that is to find all the inflection points."},{"Start":"07:55.585 ","End":"07:56.970","Text":"I have 3."},{"Start":"07:56.970 ","End":"08:02.195","Text":"I have one at minus root 3 over 2,"},{"Start":"08:02.195 ","End":"08:08.505","Text":"minus root 3 over 2 e to the minus 3 over 2."},{"Start":"08:08.505 ","End":"08:14.565","Text":"Next one, slightly simpler to write, 0, 0."},{"Start":"08:14.565 ","End":"08:17.940","Text":"The last one, square root of 3 over 2,"},{"Start":"08:17.940 ","End":"08:24.360","Text":"square root of 3 over 2 e to the minus 3 over 2."},{"Start":"08:24.360 ","End":"08:27.260","Text":"3 inflection points, areas,"},{"Start":"08:27.260 ","End":"08:31.025","Text":"intervals where the function is convex."},{"Start":"08:31.025 ","End":"08:33.305","Text":"Convex is this one."},{"Start":"08:33.305 ","End":"08:37.880","Text":"This interval, and this interval, and this interval."},{"Start":"08:37.880 ","End":"08:40.510","Text":"x bigger than 3 over 2."},{"Start":"08:40.510 ","End":"08:46.550","Text":"Concave, x less than minus root 3 over 2,"},{"Start":"08:46.550 ","End":"08:54.260","Text":"as well as x between 0 and square root of 3 over 2."},{"Start":"08:54.260 ","End":"08:58.280","Text":"I didn\u0027t do the table quite according to standard."},{"Start":"08:58.280 ","End":"09:02.250","Text":"I should have put the inflection down here,"},{"Start":"09:02.250 ","End":"09:05.780","Text":"and here, I should have written the value of f double prime,"},{"Start":"09:05.780 ","End":"09:08.960","Text":"which was 0, 0, and 0."},{"Start":"09:08.960 ","End":"09:10.490","Text":"Everything is basically correct."},{"Start":"09:10.490 ","End":"09:13.130","Text":"I just didn\u0027t do the table quite right."},{"Start":"09:13.130 ","End":"09:16.440","Text":"We\u0027re done."}],"ID":5836},{"Watched":false,"Name":"Exercise 16","Duration":"8m 20s","ChapterTopicVideoID":5839,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5839.jpeg","UploadDate":"2016-05-01T17:22:30.9070000","DurationForVideoObject":"PT8M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.640","Text":"In this exercise, we\u0027re given f of x is equal to x plus 2 times e to the 1 over"},{"Start":"00:05.640 ","End":"00:11.250","Text":"x and we have to find its inflection points and also the intervals,"},{"Start":"00:11.250 ","End":"00:14.550","Text":"where it\u0027s concave and where it\u0027s convex."},{"Start":"00:14.550 ","End":"00:20.760","Text":"This type of exercise is familiar and it has a fixed set of steps that we follow,"},{"Start":"00:20.760 ","End":"00:23.400","Text":"cookbook style to get to the answer."},{"Start":"00:23.400 ","End":"00:27.690","Text":"But before that, I just want to take a look at the domain,"},{"Start":"00:27.690 ","End":"00:29.220","Text":"and in this case,"},{"Start":"00:29.220 ","End":"00:34.530","Text":"I noticed that the domain is all x except for x equals 0."},{"Start":"00:34.530 ","End":"00:38.070","Text":"X cannot equal 0."},{"Start":"00:38.070 ","End":"00:43.520","Text":"The first step is the preparation phase where we just differentiate f of x."},{"Start":"00:43.520 ","End":"00:46.340","Text":"When I said the first step is to find f prime of x,"},{"Start":"00:46.340 ","End":"00:50.315","Text":"I meant we have to find f double prime of x, second derivative."},{"Start":"00:50.315 ","End":"00:52.445","Text":"But to get to the second derivative,"},{"Start":"00:52.445 ","End":"00:54.400","Text":"we have to start with the first derivative,"},{"Start":"00:54.400 ","End":"00:56.640","Text":"so we\u0027re heading for the second derivative."},{"Start":"00:56.640 ","End":"00:58.760","Text":"But using the product rule here,"},{"Start":"00:58.760 ","End":"01:01.220","Text":"I get the derivative of x plus 2,"},{"Start":"01:01.220 ","End":"01:05.590","Text":"which is just 1 times e^1 over x."},{"Start":"01:05.590 ","End":"01:10.320","Text":"Then x plus 2 times the derivative of e^1 over x,"},{"Start":"01:10.320 ","End":"01:15.590","Text":"which is e^1 over x by the chain rule times the derivative of 1 over x,"},{"Start":"01:15.590 ","End":"01:17.810","Text":"and that\u0027s minus 1 over x squared."},{"Start":"01:17.810 ","End":"01:22.940","Text":"If I simplify this by taking e^1 over x outside the brackets,"},{"Start":"01:22.940 ","End":"01:24.290","Text":"what I\u0027m left with,"},{"Start":"01:24.290 ","End":"01:30.280","Text":"1 from here and then minus the x plus 2 over x squared."},{"Start":"01:30.280 ","End":"01:37.465","Text":"Let\u0027s see. Minus x over x squared is 1 over x and minus 2 over x squared."},{"Start":"01:37.465 ","End":"01:39.510","Text":"That gives us f prime."},{"Start":"01:39.510 ","End":"01:43.565","Text":"Now we can go on for the second derivative and that will equal,"},{"Start":"01:43.565 ","End":"01:47.255","Text":"again, we\u0027re going to use the product rule, so let\u0027s see."},{"Start":"01:47.255 ","End":"01:50.560","Text":"The first 1 is e to the 1 over x."},{"Start":"01:50.560 ","End":"01:52.425","Text":"The first 1 derived I mean,"},{"Start":"01:52.425 ","End":"01:56.835","Text":"times minus 1 over x squared."},{"Start":"01:56.835 ","End":"02:00.420","Text":"Then the second as is plus."},{"Start":"02:00.420 ","End":"02:03.865","Text":"This time we differentiate a second."},{"Start":"02:03.865 ","End":"02:07.745","Text":"We take e^1 over x as is,"},{"Start":"02:07.745 ","End":"02:10.710","Text":"and the derivative of this is, let\u0027s see."},{"Start":"02:10.710 ","End":"02:12.920","Text":"The derivative of 1 is nothing."},{"Start":"02:12.920 ","End":"02:15.675","Text":"A derivative of minus 1 over x,"},{"Start":"02:15.675 ","End":"02:19.395","Text":"this is minus x^minus 1,"},{"Start":"02:19.395 ","End":"02:23.130","Text":"so the derivative is plus x^minus 2."},{"Start":"02:23.130 ","End":"02:26.795","Text":"X^minus 2, I\u0027ll write it as 1 over x squared."},{"Start":"02:26.795 ","End":"02:31.565","Text":"The next 1, here we have minus 2x^minus 2."},{"Start":"02:31.565 ","End":"02:32.780","Text":"If we differentiate it,"},{"Start":"02:32.780 ","End":"02:35.555","Text":"we get plus 4 x^minus 3."},{"Start":"02:35.555 ","End":"02:40.510","Text":"4x^minus 3, I\u0027ll just write it as 4 over x cubed."},{"Start":"02:40.510 ","End":"02:42.630","Text":"Let\u0027s see if we can simplify this."},{"Start":"02:42.630 ","End":"02:48.260","Text":"We\u0027ll take e^1 over x outside the brackets and collect together like terms."},{"Start":"02:48.260 ","End":"02:57.375","Text":"From here I have minus 1 over x squared and from here plus 1 over x squared."},{"Start":"02:57.375 ","End":"03:00.215","Text":"I don\u0027t have any 1 over x squared at all,"},{"Start":"03:00.215 ","End":"03:01.430","Text":"minus and the plus."},{"Start":"03:01.430 ","End":"03:02.765","Text":"Let\u0027s see now."},{"Start":"03:02.765 ","End":"03:05.090","Text":"How about 1 over x cubed?"},{"Start":"03:05.090 ","End":"03:06.410","Text":"How many of those do we have?"},{"Start":"03:06.410 ","End":"03:11.195","Text":"Well, we have minus and minus is plus 1 over x cubed from here,"},{"Start":"03:11.195 ","End":"03:15.830","Text":"and plus 4 over x cubed from here."},{"Start":"03:15.830 ","End":"03:21.465","Text":"That will give us 5 over x cubed. What else?"},{"Start":"03:21.465 ","End":"03:25.205","Text":"We\u0027ll get 1 more term from this and this,"},{"Start":"03:25.205 ","End":"03:29.260","Text":"which will give us plus 2 over x to the fourth."},{"Start":"03:29.260 ","End":"03:31.665","Text":"That\u0027s about it."},{"Start":"03:31.665 ","End":"03:33.920","Text":"This is the second derivative,"},{"Start":"03:33.920 ","End":"03:37.490","Text":"but I think it will go better if I put a common denominator,"},{"Start":"03:37.490 ","End":"03:40.360","Text":"the x^4 could be a common denominator."},{"Start":"03:40.360 ","End":"03:43.025","Text":"I\u0027ll take e^1 over x,"},{"Start":"03:43.025 ","End":"03:46.645","Text":"I\u0027ll take the over x to the fourth outside."},{"Start":"03:46.645 ","End":"03:52.110","Text":"What I\u0027m left with is this would be 5x over x^4."},{"Start":"03:52.110 ","End":"03:57.095","Text":"I have 5x. This already is over x^4 plus 2."},{"Start":"03:57.095 ","End":"04:00.740","Text":"This is how I prefer to have the second derivative."},{"Start":"04:00.740 ","End":"04:04.310","Text":"That ends this preparatory stage."},{"Start":"04:04.310 ","End":"04:10.190","Text":"The second step is to solve the equation f double prime of x equals 0."},{"Start":"04:10.190 ","End":"04:15.915","Text":"The solutions will be those x which are suspect for being inflection."},{"Start":"04:15.915 ","End":"04:18.570","Text":"Let\u0027s see. If we solve this equal to 0,"},{"Start":"04:18.570 ","End":"04:20.885","Text":"the numerator has to be 0."},{"Start":"04:20.885 ","End":"04:24.590","Text":"This is always positive when it\u0027s defined."},{"Start":"04:24.590 ","End":"04:28.865","Text":"The only way we\u0027ll get a 0 is when 5x plus 2 is 0."},{"Start":"04:28.865 ","End":"04:32.990","Text":"We\u0027ll get x is minus 2/5."},{"Start":"04:32.990 ","End":"04:36.314","Text":"If x equals minus 2/5, that\u0027s our suspect,"},{"Start":"04:36.314 ","End":"04:41.920","Text":"and we can now draw a table and put the interesting values in."},{"Start":"04:41.920 ","End":"04:43.715","Text":"Here\u0027s a blank table."},{"Start":"04:43.715 ","End":"04:45.425","Text":"We need 2 values here,"},{"Start":"04:45.425 ","End":"04:49.995","Text":"minus 2/5 and 0 in increasing order."},{"Start":"04:49.995 ","End":"04:53.090","Text":"2/5 here and 0 here."},{"Start":"04:53.090 ","End":"04:56.180","Text":"The 0 is where the function\u0027s not defined,"},{"Start":"04:56.180 ","End":"05:00.920","Text":"so the rest of the table I just put dotted lines, nothing\u0027s defined here."},{"Start":"05:00.920 ","End":"05:05.240","Text":"The minus 2/5 is because of the second derivative being 0,"},{"Start":"05:05.240 ","End":"05:07.929","Text":"and it\u0027s our suspect for an extremum."},{"Start":"05:07.929 ","End":"05:10.505","Text":"Now, these 2 points define 3 intervals,"},{"Start":"05:10.505 ","End":"05:15.125","Text":"which we can describe as x less than minus 2/5,"},{"Start":"05:15.125 ","End":"05:18.410","Text":"x between minus 2/5 and 0,"},{"Start":"05:18.410 ","End":"05:20.590","Text":"and x bigger than 0."},{"Start":"05:20.590 ","End":"05:22.830","Text":"We have to choose arbitrary values,"},{"Start":"05:22.830 ","End":"05:24.630","Text":"1 from each interval."},{"Start":"05:24.630 ","End":"05:26.639","Text":"For x less than 2/5,"},{"Start":"05:26.639 ","End":"05:28.875","Text":"I\u0027ll choose minus 1,"},{"Start":"05:28.875 ","End":"05:32.060","Text":"for x between minus 2/5 and 0,"},{"Start":"05:32.060 ","End":"05:35.580","Text":"I\u0027ll choose the halfway minus 1/5,"},{"Start":"05:35.580 ","End":"05:38.270","Text":"and for x bigger than 0, I\u0027ll choose 1."},{"Start":"05:38.270 ","End":"05:42.200","Text":"It\u0027s a whole number. Then we have to put these values,"},{"Start":"05:42.200 ","End":"05:46.070","Text":"these 3 into f double-prime,"},{"Start":"05:46.070 ","End":"05:48.440","Text":"which when I simplify it,"},{"Start":"05:48.440 ","End":"05:50.284","Text":"this is the form I\u0027ll choose."},{"Start":"05:50.284 ","End":"05:53.600","Text":"But I don\u0027t want the value that I get when I substitute."},{"Start":"05:53.600 ","End":"05:55.865","Text":"Only the sign, whether it\u0027s plus or minus."},{"Start":"05:55.865 ","End":"05:59.975","Text":"That should make work easier because x^4 will always be positive"},{"Start":"05:59.975 ","End":"06:05.315","Text":"and e^1 over x will also be positive whenever they\u0027re defined, which is x not 0."},{"Start":"06:05.315 ","End":"06:08.035","Text":"All I need to look at is 5 x plus 2."},{"Start":"06:08.035 ","End":"06:10.510","Text":"When x is minus 1,"},{"Start":"06:10.510 ","End":"06:12.635","Text":"this comes out to be negative."},{"Start":"06:12.635 ","End":"06:15.035","Text":"When x is minus 1/5,"},{"Start":"06:15.035 ","End":"06:17.030","Text":"that comes out minus 1 plus 2,"},{"Start":"06:17.030 ","End":"06:20.060","Text":"it\u0027s positive, and when x is bigger than 0,"},{"Start":"06:20.060 ","End":"06:23.490","Text":"x is 1, that also comes out positive."},{"Start":"06:23.490 ","End":"06:28.790","Text":"Here we have concave, convex, and convex."},{"Start":"06:28.790 ","End":"06:32.060","Text":"Our suspect, which is minus 2/5,"},{"Start":"06:32.060 ","End":"06:37.940","Text":"turns out indeed to be an inflection point because it\u0027s between concave and convex."},{"Start":"06:37.940 ","End":"06:40.565","Text":"Here I have an inflection."},{"Start":"06:40.565 ","End":"06:41.780","Text":"At the inflection."},{"Start":"06:41.780 ","End":"06:46.025","Text":"I\u0027d like to know what the value of y is that goes with this x."},{"Start":"06:46.025 ","End":"06:49.080","Text":"Well, y is just the f of x,"},{"Start":"06:49.080 ","End":"06:55.740","Text":"so I just have to put in here minus 2/5 to see what it is."},{"Start":"06:55.740 ","End":"06:58.410","Text":"If I put in minus 2/5,"},{"Start":"06:58.410 ","End":"07:03.570","Text":"I get 8/5. e^1 over x,"},{"Start":"07:03.570 ","End":"07:08.940","Text":"1 over x would be minus 5 over 2."},{"Start":"07:08.940 ","End":"07:11.095","Text":"I\u0027ll just leave it like that,"},{"Start":"07:11.095 ","End":"07:15.030","Text":"e^minus 5 over 2."},{"Start":"07:15.030 ","End":"07:17.400","Text":"This ends this stage,"},{"Start":"07:17.400 ","End":"07:19.475","Text":"and now we come to the final stage,"},{"Start":"07:19.475 ","End":"07:21.740","Text":"which is the stage of conclusions."},{"Start":"07:21.740 ","End":"07:23.705","Text":"We\u0027re actually answering the question,"},{"Start":"07:23.705 ","End":"07:26.675","Text":"and so we have 3 things we have to write."},{"Start":"07:26.675 ","End":"07:28.520","Text":"The inflection points."},{"Start":"07:28.520 ","End":"07:33.290","Text":"We have to write where the function is convex"},{"Start":"07:33.290 ","End":"07:38.330","Text":"or concave up and where the function is concave or concave down."},{"Start":"07:38.330 ","End":"07:40.355","Text":"The inflection we get from here,"},{"Start":"07:40.355 ","End":"07:44.255","Text":"that\u0027s just 1 of them and it\u0027s at the point minus 2/5,"},{"Start":"07:44.255 ","End":"07:49.610","Text":"8/5 e^minus 5 over 2 in fractions."},{"Start":"07:49.610 ","End":"07:52.010","Text":"If you want, you could convert it to decimals."},{"Start":"07:52.010 ","End":"07:55.100","Text":"The convex is this shape here."},{"Start":"07:55.100 ","End":"08:00.530","Text":"There\u0027s 2 intervals which I can\u0027t combine because there\u0027s a gap in between,"},{"Start":"08:00.530 ","End":"08:03.655","Text":"but from minus 2/5 to 0,"},{"Start":"08:03.655 ","End":"08:07.305","Text":"like this, and also from 0 on."},{"Start":"08:07.305 ","End":"08:09.160","Text":"In other words, x bigger than 0."},{"Start":"08:09.160 ","End":"08:16.870","Text":"The concave is this shape and that goes with the interval x less than minus 2/5."},{"Start":"08:16.870 ","End":"08:21.570","Text":"That answers the questions and we\u0027re done."}],"ID":5837},{"Watched":false,"Name":"Exercise 17","Duration":"6m 51s","ChapterTopicVideoID":5840,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5840.jpeg","UploadDate":"2016-05-01T17:23:16.0470000","DurationForVideoObject":"PT6M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.050","Text":"In this exercise, we\u0027re given a function"},{"Start":"00:03.050 ","End":"00:05.900","Text":"f of x equals natural log of x over x."},{"Start":"00:05.900 ","End":"00:08.660","Text":"We have to find its inflection points"},{"Start":"00:08.660 ","End":"00:11.297","Text":"and on which intervals it\u0027s concave"},{"Start":"00:11.297 ","End":"00:15.660","Text":"and on which intervals it\u0027s convex or concave up, concave down."},{"Start":"00:15.660 ","End":"00:22.965","Text":"This is a familiar format of an exercise and there\u0027s a standard set of steps to solve it."},{"Start":"00:22.965 ","End":"00:25.230","Text":"We\u0027ll just follow those steps."},{"Start":"00:25.230 ","End":"00:26.865","Text":"But just before that,"},{"Start":"00:26.865 ","End":"00:29.370","Text":"we should take a look at the domain."},{"Start":"00:29.370 ","End":"00:33.465","Text":"The domain for the natural logarithm is x bigger than 0,"},{"Start":"00:33.465 ","End":"00:36.420","Text":"and for the denominator, x must be 0."},{"Start":"00:36.420 ","End":"00:42.920","Text":"Altogether, we have that x is bigger than 0 is our domain."},{"Start":"00:42.920 ","End":"00:47.330","Text":"Now, the first step in the solution is the preparation step,"},{"Start":"00:47.330 ","End":"00:50.300","Text":"and that\u0027s where we have to find f double prime."},{"Start":"00:50.300 ","End":"00:53.825","Text":"Of course, we have to start with f prime and that\u0027s equal,"},{"Start":"00:53.825 ","End":"00:57.624","Text":"using the quotient rule, denominator squared here"},{"Start":"00:57.624 ","End":"01:00.620","Text":"and at the top, the derivative of the numerator,"},{"Start":"01:00.620 ","End":"01:08.210","Text":"which is 1/x times the denominator minus the other way around the numerator,"},{"Start":"01:08.210 ","End":"01:13.025","Text":"as is, times the derivative of the denominator, which is 1."},{"Start":"01:13.025 ","End":"01:20.120","Text":"All in all, we get 1 minus natural log of x over x^2."},{"Start":"01:20.120 ","End":"01:22.910","Text":"Now, onto f double prime, again,"},{"Start":"01:22.910 ","End":"01:25.904","Text":"quotient rule, denominator squared"},{"Start":"01:25.904 ","End":"01:29.615","Text":"and here, the derivative of the numerator,"},{"Start":"01:29.615 ","End":"01:36.167","Text":"which is minus 1/x times the denominator, x^2,"},{"Start":"01:36.167 ","End":"01:40.734","Text":"minus the numerator, 1 minus log x,"},{"Start":"01:40.734 ","End":"01:45.215","Text":"times the derivative of the denominator, which is 2x."},{"Start":"01:45.215 ","End":"01:47.000","Text":"Let\u0027s try to simplify."},{"Start":"01:47.000 ","End":"01:49.865","Text":"We can divide top and bottom by x."},{"Start":"01:49.865 ","End":"01:51.380","Text":"If we divide this by x,"},{"Start":"01:51.380 ","End":"01:54.395","Text":"we get minus 1/x times x,"},{"Start":"01:54.395 ","End":"01:57.075","Text":"which is just minus 1."},{"Start":"01:57.075 ","End":"01:58.920","Text":"If we get rid of this x,"},{"Start":"01:58.920 ","End":"02:00.900","Text":"then we just got 2 here."},{"Start":"02:00.900 ","End":"02:08.374","Text":"It\u0027s minus 2 plus natural log of x over x^4,"},{"Start":"02:08.374 ","End":"02:12.525","Text":"and I\u0027ll just replace this minus 1 minus 2 with minus 3."},{"Start":"02:12.525 ","End":"02:14.855","Text":"Also, correct a little mistake I made."},{"Start":"02:14.855 ","End":"02:17.060","Text":"I forgot to divide the denominator by x,"},{"Start":"02:17.060 ","End":"02:18.575","Text":"so this is only x^3."},{"Start":"02:18.575 ","End":"02:21.950","Text":"That\u0027s the end of the preparation step."},{"Start":"02:21.950 ","End":"02:24.890","Text":"Next, we have what I call step 1,"},{"Start":"02:24.890 ","End":"02:28.645","Text":"which is to compare f double prime to 0."},{"Start":"02:28.645 ","End":"02:30.080","Text":"Let\u0027s write an equation."},{"Start":"02:30.080 ","End":"02:31.610","Text":"I\u0027ll continue over here."},{"Start":"02:31.610 ","End":"02:34.656","Text":"f double prime of x equals 0,"},{"Start":"02:34.656 ","End":"02:37.230","Text":"and I have to solve this for x."},{"Start":"02:37.230 ","End":"02:40.310","Text":"That will give me some suspects for inflection point."},{"Start":"02:40.310 ","End":"02:43.084","Text":"Hang on, I just realized I made another little mistake."},{"Start":"02:43.084 ","End":"02:45.970","Text":"I forgot about the 2 with the natural log of x."},{"Start":"02:45.970 ","End":"02:47.975","Text":"So f double prime equals 0."},{"Start":"02:47.975 ","End":"02:51.935","Text":"When a fraction is 0, it means its numerator is 0."},{"Start":"02:51.935 ","End":"02:57.714","Text":"We have that 2 natural log of x equals 3"},{"Start":"02:57.714 ","End":"03:02.520","Text":"and so natural log of x is 3/2,"},{"Start":"03:02.520 ","End":"03:07.931","Text":"and that gives us that x is e to the power of 3/2"},{"Start":"03:07.931 ","End":"03:11.785","Text":"by taking the exponent of both sides."},{"Start":"03:11.785 ","End":"03:15.755","Text":"This gives us a suspect for an inflection point"},{"Start":"03:15.755 ","End":"03:18.740","Text":"and that\u0027s the only interesting value we have."},{"Start":"03:18.740 ","End":"03:22.400","Text":"We\u0027re going to make our table and put this value in the table."},{"Start":"03:22.400 ","End":"03:26.045","Text":"The first thing we do is put in the value of x,"},{"Start":"03:26.045 ","End":"03:29.835","Text":"which is e to the 3/2 in the table."},{"Start":"03:29.835 ","End":"03:34.470","Text":"This divides the domain into two intervals."},{"Start":"03:34.470 ","End":"03:38.120","Text":"Now, remember, our domain is not all of x"},{"Start":"03:38.120 ","End":"03:40.860","Text":"but just x bigger than 0."},{"Start":"03:40.860 ","End":"03:48.432","Text":"This divides it up into x between 0 and e to the 3/2,"},{"Start":"03:48.432 ","End":"03:53.805","Text":"and here we have x bigger than e to the 3/2."},{"Start":"03:53.805 ","End":"03:57.585","Text":"We choose a sample point from each."},{"Start":"03:57.585 ","End":"04:01.065","Text":"I\u0027ll choose x equals 1 here."},{"Start":"04:01.065 ","End":"04:04.238","Text":"For example, 1 is certainly less than this,"},{"Start":"04:04.238 ","End":"04:08.600","Text":"and e to the 3/2, I\u0027m not sure exactly what it is,"},{"Start":"04:08.600 ","End":"04:12.045","Text":"but certainly less than 10."},{"Start":"04:12.045 ","End":"04:15.148","Text":"I\u0027m going to substitute into f double prime."},{"Start":"04:15.148 ","End":"04:19.600","Text":"But first, let me just write that for e to the 3/2,"},{"Start":"04:19.600 ","End":"04:22.070","Text":"then I get f double prime is 0."},{"Start":"04:22.070 ","End":"04:24.065","Text":"After all, that\u0027s how we found it."},{"Start":"04:24.065 ","End":"04:27.215","Text":"Here, if I put in x equals 1,"},{"Start":"04:27.215 ","End":"04:29.050","Text":"let me say, where is f double prime?"},{"Start":"04:29.050 ","End":"04:30.470","Text":"Where am I substituting?"},{"Start":"04:30.470 ","End":"04:34.385","Text":"I\u0027m substituting in this function here."},{"Start":"04:34.385 ","End":"04:36.440","Text":"If I put in x equals 1,"},{"Start":"04:36.440 ","End":"04:41.840","Text":"the denominator\u0027s positive, natural log of 1 is 0."},{"Start":"04:41.840 ","End":"04:47.520","Text":"I have minus 3 over something positive or 1 which is minus 3."},{"Start":"04:47.520 ","End":"04:50.045","Text":"In any event, it\u0027s negative."},{"Start":"04:50.045 ","End":"04:55.835","Text":"I changed my mind about the 10 and I\u0027m going to just make it e^2."},{"Start":"04:55.835 ","End":"04:59.695","Text":"I\u0027ll take as bigger than e to the 3/2 and that\u0027ll be sure."},{"Start":"04:59.695 ","End":"05:04.115","Text":"This is easier to substitute because the logarithm of e^2 is just 2."},{"Start":"05:04.115 ","End":"05:07.640","Text":"Minus 3 plus 2 times 2 is minus 3 plus 4 is 1,"},{"Start":"05:07.640 ","End":"05:09.995","Text":"which is positive over positive."},{"Start":"05:09.995 ","End":"05:11.540","Text":"Now when this is minus,"},{"Start":"05:11.540 ","End":"05:14.672","Text":"we have a concave situation,"},{"Start":"05:14.672 ","End":"05:17.315","Text":"and when we have a positive,"},{"Start":"05:17.315 ","End":"05:18.500","Text":"then for the function,"},{"Start":"05:18.500 ","End":"05:21.560","Text":"that means the convex situation or concave up."},{"Start":"05:21.560 ","End":"05:25.655","Text":"That means that this suspect really is an inflection point"},{"Start":"05:25.655 ","End":"05:29.345","Text":"because it\u0027s between the concave and convex."},{"Start":"05:29.345 ","End":"05:30.590","Text":"I can write that here,"},{"Start":"05:30.590 ","End":"05:32.300","Text":"I have an inflection."},{"Start":"05:32.300 ","End":"05:35.150","Text":"For the inflection, I\u0027d like to know what the exact point is."},{"Start":"05:35.150 ","End":"05:37.190","Text":"Here\u0027s the x, what\u0027s the y?"},{"Start":"05:37.190 ","End":"05:41.465","Text":"I have to substitute in the original function."},{"Start":"05:41.465 ","End":"05:46.022","Text":"This is y, which is the original function, natural log of x over x."},{"Start":"05:46.022 ","End":"05:52.865","Text":"Log of x is just 3/2 and x is e to the 3/2."},{"Start":"05:52.865 ","End":"05:58.140","Text":"This gives us 3/2 over e to the 3/2,"},{"Start":"05:58.140 ","End":"06:00.424","Text":"which is e to the minus 3/2,"},{"Start":"06:00.424 ","End":"06:03.795","Text":"and that\u0027s the end of the second step."},{"Start":"06:03.795 ","End":"06:06.395","Text":"Then the last step is the conclusion,"},{"Start":"06:06.395 ","End":"06:11.855","Text":"which is to basically answer the question about inflection and concave and convex."},{"Start":"06:11.855 ","End":"06:17.405","Text":"Let\u0027s do it over here and we\u0027ll write the conclusion that we have an inflection,"},{"Start":"06:17.405 ","End":"06:27.860","Text":"just one, and it occurs at the point e to the 3/2, 3/2 e to the minus 3/2."},{"Start":"06:27.860 ","End":"06:31.335","Text":"As for the convex,"},{"Start":"06:31.335 ","End":"06:34.620","Text":"the convex is this shape here."},{"Start":"06:34.620 ","End":"06:38.730","Text":"That\u0027s x bigger than e to the 3/2, that\u0027s the interval."},{"Start":"06:38.730 ","End":"06:41.750","Text":"For concave, I go with the other symbol,"},{"Start":"06:41.750 ","End":"06:44.371","Text":"this symbol here, and that gives me the"},{"Start":"06:44.371 ","End":"06:48.829","Text":"interval between 0 and e to the 3/2."},{"Start":"06:48.829 ","End":"06:51.870","Text":"That\u0027s it. We\u0027re done."}],"ID":5838},{"Watched":false,"Name":"Exercise 18","Duration":"2m 56s","ChapterTopicVideoID":5841,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5841.jpeg","UploadDate":"2016-05-01T17:23:36.7670000","DurationForVideoObject":"PT2M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.450","Text":"In this exercise, we\u0027re given the function f of x equals x times natural log of x."},{"Start":"00:06.450 ","End":"00:10.530","Text":"We have to find its inflection points and"},{"Start":"00:10.530 ","End":"00:14.625","Text":"the intervals where it\u0027s concave up and where it\u0027s concave down,"},{"Start":"00:14.625 ","End":"00:17.250","Text":"or if you like convex and concave."},{"Start":"00:17.250 ","End":"00:24.750","Text":"This type of exercise is familiar and has a standard set of steps to the solution."},{"Start":"00:24.750 ","End":"00:29.075","Text":"Just before that, I\u0027d like to note what the domain is."},{"Start":"00:29.075 ","End":"00:32.645","Text":"The domain of the function because of the natural logarithm,"},{"Start":"00:32.645 ","End":"00:36.610","Text":"is x bigger than 0."},{"Start":"00:36.610 ","End":"00:39.440","Text":"The first step is the preparation step."},{"Start":"00:39.440 ","End":"00:43.305","Text":"What we have to do here is to find f double-prime,"},{"Start":"00:43.305 ","End":"00:46.160","Text":"of course, we have to start with f prime."},{"Start":"00:46.160 ","End":"00:49.560","Text":"We can\u0027t go directly to the second derivative."},{"Start":"00:49.560 ","End":"00:52.625","Text":"This f prime is by the product rule,"},{"Start":"00:52.625 ","End":"01:00.500","Text":"1 times natural log of x plus this times the derivative of that 1 over x,"},{"Start":"01:00.500 ","End":"01:04.200","Text":"which is basically natural log of x plus 1."},{"Start":"01:04.200 ","End":"01:07.740","Text":"Then f double-prime is going to equal,"},{"Start":"01:07.740 ","End":"01:11.870","Text":"for this, we have 1 over x, and that\u0027s it."},{"Start":"01:11.870 ","End":"01:12.920","Text":"The 1 gives nothing,"},{"Start":"01:12.920 ","End":"01:15.050","Text":"so here\u0027s f double-prime."},{"Start":"01:15.050 ","End":"01:17.705","Text":"That\u0027s the end of the preparation step."},{"Start":"01:17.705 ","End":"01:19.745","Text":"Next is Step 1."},{"Start":"01:19.745 ","End":"01:24.560","Text":"We solve the equation f double prime of x equals 0."},{"Start":"01:24.560 ","End":"01:29.135","Text":"We look for solutions which would be then suspects for inflection points."},{"Start":"01:29.135 ","End":"01:31.850","Text":"But this doesn\u0027t give us anything, unfortunately,"},{"Start":"01:31.850 ","End":"01:35.780","Text":"because 1 over x equals 0 cannot be 0,"},{"Start":"01:35.780 ","End":"01:37.160","Text":"so there\u0027s no solution."},{"Start":"01:37.160 ","End":"01:41.390","Text":"So, we have no suspects for an inflection point."},{"Start":"01:41.390 ","End":"01:43.145","Text":"That\u0027s the end of this step."},{"Start":"01:43.145 ","End":"01:47.060","Text":"The next step is the table which is going to be a bit empty,"},{"Start":"01:47.060 ","End":"01:48.170","Text":"but here it is."},{"Start":"01:48.170 ","End":"01:50.090","Text":"Since there are no interesting points,"},{"Start":"01:50.090 ","End":"01:52.310","Text":"the x is all 1 big interval."},{"Start":"01:52.310 ","End":"01:55.375","Text":"The whole thing is x bigger than 0."},{"Start":"01:55.375 ","End":"01:57.680","Text":"But we still can choose a sample point."},{"Start":"01:57.680 ","End":"01:59.870","Text":"I\u0027m going to choose a sample point."},{"Start":"01:59.870 ","End":"02:03.635","Text":"Let\u0027s take x equals 1 as a sample point,"},{"Start":"02:03.635 ","End":"02:05.600","Text":"why not, it\u0027s bigger than 0."},{"Start":"02:05.600 ","End":"02:08.825","Text":"F double prime is here,"},{"Start":"02:08.825 ","End":"02:10.550","Text":"and that\u0027s 1 over x."},{"Start":"02:10.550 ","End":"02:12.065","Text":"That\u0027s also 1."},{"Start":"02:12.065 ","End":"02:13.400","Text":"But I don\u0027t need the 1."},{"Start":"02:13.400 ","End":"02:15.680","Text":"I just need the fact that it\u0027s positive."},{"Start":"02:15.680 ","End":"02:19.340","Text":"When it\u0027s positive, that means it\u0027s convex,"},{"Start":"02:19.340 ","End":"02:22.175","Text":"concave up like this."},{"Start":"02:22.175 ","End":"02:25.249","Text":"This means that since there\u0027s only 1 interval,"},{"Start":"02:25.249 ","End":"02:29.735","Text":"that it\u0027s convex for the whole domain."},{"Start":"02:29.735 ","End":"02:33.800","Text":"In conclusion, that\u0027s the last step is the conclusion."},{"Start":"02:33.800 ","End":"02:36.304","Text":"What I can write is the following."},{"Start":"02:36.304 ","End":"02:39.980","Text":"For inflection points of the function,"},{"Start":"02:39.980 ","End":"02:41.395","Text":"I can write none."},{"Start":"02:41.395 ","End":"02:44.725","Text":"For convex or concave up,"},{"Start":"02:44.725 ","End":"02:49.535","Text":"it\u0027s all the x in the domain which is x bigger than 0,"},{"Start":"02:49.535 ","End":"02:52.910","Text":"and concave is nowhere."},{"Start":"02:52.910 ","End":"02:54.560","Text":"It\u0027s always convex."},{"Start":"02:54.560 ","End":"02:57.870","Text":"That\u0027s how I write the answer, there we were done."}],"ID":5839},{"Watched":false,"Name":"Exercise 19","Duration":"5m 31s","ChapterTopicVideoID":5842,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5842.jpeg","UploadDate":"2016-05-01T17:24:13.9470000","DurationForVideoObject":"PT5M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, we\u0027re given a function f of x, as follows,"},{"Start":"00:04.800 ","End":"00:09.525","Text":"natural log of x squared plus twice natural log of x minus 3."},{"Start":"00:09.525 ","End":"00:12.390","Text":"What we have to do is find its inflection points"},{"Start":"00:12.390 ","End":"00:16.515","Text":"as well as the intervals where it\u0027s concave and where it\u0027s convex."},{"Start":"00:16.515 ","End":"00:19.920","Text":"I copied the exercise over here and note that its"},{"Start":"00:19.920 ","End":"00:24.520","Text":"domain is x is bigger than 0 because of the natural log."},{"Start":"00:24.520 ","End":"00:30.240","Text":"Now this type of exercise has a standard cookbook style solution in several steps:"},{"Start":"00:30.240 ","End":"00:34.390","Text":"a preliminary step, stage 1, stage 2, and a conclusion."},{"Start":"00:34.390 ","End":"00:39.515","Text":"Let\u0027s get to it. The preliminary step is to find the second derivative f double prime."},{"Start":"00:39.515 ","End":"00:45.005","Text":"But, of course, we have to find f prime first before we can get to the second derivative."},{"Start":"00:45.005 ","End":"00:50.750","Text":"This is equal to, by the chain rule, twice the natural log of x times inner derivative,"},{"Start":"00:50.750 ","End":"00:52.205","Text":"which is 1 over x."},{"Start":"00:52.205 ","End":"00:55.040","Text":"Here we have just twice and the derivative of"},{"Start":"00:55.040 ","End":"00:58.775","Text":"natural log is 1 over x and the 3 gives nothing."},{"Start":"00:58.775 ","End":"01:03.459","Text":"Basically, what we get is if we take the 2 outside the brackets,"},{"Start":"01:03.459 ","End":"01:10.340","Text":"in fact, the 2 over x, we get 2 over x times natural log of x plus 1."},{"Start":"01:10.340 ","End":"01:13.505","Text":"Now, f double prime is equal to,"},{"Start":"01:13.505 ","End":"01:20.171","Text":"we can use the product rule and get the derivative of 2 over x,"},{"Start":"01:20.171 ","End":"01:25.730","Text":"minus 2 over x squared times, this as is,"},{"Start":"01:25.730 ","End":"01:31.069","Text":"natural log of x plus 1 plus 2 over x as is,"},{"Start":"01:31.069 ","End":"01:33.740","Text":"times the derivative of the second bit,"},{"Start":"01:33.740 ","End":"01:35.810","Text":"which is just 1 over x."},{"Start":"01:35.810 ","End":"01:37.415","Text":"If we simplify this,"},{"Start":"01:37.415 ","End":"01:40.890","Text":"you can see that the minus 2 over x^2"},{"Start":"01:40.890 ","End":"01:44.695","Text":"cancels with the plus 2 over x squared from here."},{"Start":"01:44.695 ","End":"01:52.105","Text":"All we\u0027re left with is minus 2 natural log of x over x^2,"},{"Start":"01:52.105 ","End":"01:55.070","Text":"and that finishes this step."},{"Start":"01:55.070 ","End":"02:02.974","Text":"The next step is to solve the equation where the second derivative of x is equal to 0."},{"Start":"02:02.974 ","End":"02:08.300","Text":"The solutions to this will give us suspects for inflection point."},{"Start":"02:08.300 ","End":"02:10.835","Text":"If f double prime is 0,"},{"Start":"02:10.835 ","End":"02:13.955","Text":"means that this thing is 0."},{"Start":"02:13.955 ","End":"02:17.830","Text":"The numerator has to be 0 for a fraction to be 0."},{"Start":"02:17.830 ","End":"02:22.294","Text":"So basically, we just get that natural log of x is 0,"},{"Start":"02:22.294 ","End":"02:27.425","Text":"which means that x is equal to e to the power of 0,"},{"Start":"02:27.425 ","End":"02:30.125","Text":"which equals 1 and that\u0027s the only solution,"},{"Start":"02:30.125 ","End":"02:31.965","Text":"and that\u0027s our suspect."},{"Start":"02:31.965 ","End":"02:34.265","Text":"That\u0027s the end of this step."},{"Start":"02:34.265 ","End":"02:37.250","Text":"Next step is the table."},{"Start":"02:37.250 ","End":"02:41.420","Text":"We draw a table and there\u0027s only one value to put in here,"},{"Start":"02:41.420 ","End":"02:43.565","Text":"and that is x equals 1,"},{"Start":"02:43.565 ","End":"02:46.040","Text":"where the second derivative was 0,"},{"Start":"02:46.040 ","End":"02:48.320","Text":"which makes it a suspect for inflection."},{"Start":"02:48.320 ","End":"02:52.520","Text":"This divides the domain into two intervals,"},{"Start":"02:52.520 ","End":"02:54.200","Text":"and these intervals are:"},{"Start":"02:54.200 ","End":"02:56.765","Text":"over here, we have x bigger than 1"},{"Start":"02:56.765 ","End":"02:59.540","Text":"and here, not just x less than 1"},{"Start":"02:59.540 ","End":"03:04.400","Text":"but remember that, also, because of the domain that it has to be bigger than 0,"},{"Start":"03:04.400 ","End":"03:07.674","Text":"so 0 less than x less than 1."},{"Start":"03:07.674 ","End":"03:09.620","Text":"We have to choose"},{"Start":"03:09.620 ","End":"03:14.420","Text":"a sample point in each of these two intervals arbitrarily."},{"Start":"03:14.420 ","End":"03:18.500","Text":"For here, I\u0027ll choose e to the minus 1."},{"Start":"03:18.500 ","End":"03:22.565","Text":"That way when I substitute it will come out easier."},{"Start":"03:22.565 ","End":"03:25.910","Text":"So let\u0027s choose e to the minus 1, 1 over e,"},{"Start":"03:25.910 ","End":"03:27.666","Text":"which is certainly less than 1,"},{"Start":"03:27.666 ","End":"03:31.565","Text":"and for x bigger than 1, I\u0027ll choose just e."},{"Start":"03:31.565 ","End":"03:33.620","Text":"Again, I\u0027m choosing these numbers,"},{"Start":"03:33.620 ","End":"03:37.450","Text":"e which is e to the 1 so that the natural log will come out simple."},{"Start":"03:37.450 ","End":"03:41.210","Text":"What I need is to substitute them in f double prime, which is here,"},{"Start":"03:41.210 ","End":"03:45.020","Text":"but I don\u0027t need the actual value, just whether it\u0027s positive or negative."},{"Start":"03:45.020 ","End":"03:47.150","Text":"Let\u0027s take e to the minus 1."},{"Start":"03:47.150 ","End":"03:49.745","Text":"It\u0027s natural log is just the minus 1."},{"Start":"03:49.745 ","End":"03:54.680","Text":"Minus 1 times minus 2 is plus 2 and this thing is positive."},{"Start":"03:54.680 ","End":"03:56.930","Text":"All in all I get something positive,"},{"Start":"03:56.930 ","End":"03:58.320","Text":"that\u0027s all I care about,"},{"Start":"03:58.320 ","End":"03:59.690","Text":"and when x is e,"},{"Start":"03:59.690 ","End":"04:01.130","Text":"natural log of e is 1,"},{"Start":"04:01.130 ","End":"04:04.970","Text":"so I get minus 2 over something positive, in short, negative."},{"Start":"04:04.970 ","End":"04:07.370","Text":"What this tells us about the function,"},{"Start":"04:07.370 ","End":"04:09.290","Text":"if second derivative is positive,"},{"Start":"04:09.290 ","End":"04:12.516","Text":"then it\u0027s shaped like this or convex,"},{"Start":"04:12.516 ","End":"04:15.266","Text":"and for the negative second derivative,"},{"Start":"04:15.266 ","End":"04:18.451","Text":"it comes out like this or concave."},{"Start":"04:18.451 ","End":"04:22.057","Text":"The fact that this value, x equals 1,"},{"Start":"04:22.057 ","End":"04:27.580","Text":"comes between convex and concave means that it is an inflection point."},{"Start":"04:27.580 ","End":"04:32.630","Text":"So we found an inflection point at x equals 1."},{"Start":"04:32.630 ","End":"04:35.830","Text":"I\u0027d like to know what the value of y is here."},{"Start":"04:35.830 ","End":"04:39.494","Text":"For that, I have to substitute in y which is f of x,"},{"Start":"04:39.494 ","End":"04:41.740","Text":"and if I put in x equals 1,"},{"Start":"04:41.740 ","End":"04:43.955","Text":"natural log of 1 is 0,"},{"Start":"04:43.955 ","End":"04:48.575","Text":"so 0 squared plus twice 0 minus 3,"},{"Start":"04:48.575 ","End":"04:50.546","Text":"all in all gives us minus 3,"},{"Start":"04:50.546 ","End":"04:52.955","Text":"and now the table is complete."},{"Start":"04:52.955 ","End":"04:54.860","Text":"I have all what I need to know."},{"Start":"04:54.860 ","End":"04:58.010","Text":"The last stage of the recipe,"},{"Start":"04:58.010 ","End":"05:01.085","Text":"so to speak, is the conclusion phase,"},{"Start":"05:01.085 ","End":"05:04.025","Text":"which is to answer the original questions."},{"Start":"05:04.025 ","End":"05:06.680","Text":"We\u0027re asked, where are the inflection points?"},{"Start":"05:06.680 ","End":"05:08.555","Text":"Find all the inflection points."},{"Start":"05:08.555 ","End":"05:13.650","Text":"There is only one and that occurs at 1, minus 3."},{"Start":"05:13.650 ","End":"05:17.130","Text":"Then we need to say where it\u0027s concave up and down."},{"Start":"05:17.130 ","End":"05:21.030","Text":"I call that one, convex and the other one, we\u0027ll call concave."},{"Start":"05:21.030 ","End":"05:22.785","Text":"Convex is this shape,"},{"Start":"05:22.785 ","End":"05:26.264","Text":"so it\u0027s 0 less than x less than 1,"},{"Start":"05:26.264 ","End":"05:29.514","Text":"and concave is this shape, x bigger than 1."},{"Start":"05:29.514 ","End":"05:32.290","Text":"That\u0027s all, we\u0027re done."}],"ID":5840},{"Watched":false,"Name":"Exercise 20","Duration":"7m 57s","ChapterTopicVideoID":5843,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5843.jpeg","UploadDate":"2016-05-01T17:25:11.1100000","DurationForVideoObject":"PT7M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.659","Text":"In this exercise, we\u0027re given the function of x,"},{"Start":"00:03.659 ","End":"00:06.666","Text":"1 over the square root of x^2 plus 1,"},{"Start":"00:06.666 ","End":"00:11.752","Text":"and we have to say where are the points of inflection of this function"},{"Start":"00:11.752 ","End":"00:16.455","Text":"and in which intervals is the function concave or convex."},{"Start":"00:16.455 ","End":"00:24.080","Text":"I\u0027d like to also rewrite it as x^2 plus 1 to the power of minus 1/2."},{"Start":"00:24.080 ","End":"00:26.960","Text":"There\u0027s a standard set of steps, cookbook style,"},{"Start":"00:26.960 ","End":"00:29.750","Text":"4-steps, in fact, to solve this question."},{"Start":"00:29.750 ","End":"00:33.080","Text":"But before that, we have to also note the domain,"},{"Start":"00:33.080 ","End":"00:34.175","Text":"and if you look at it,"},{"Start":"00:34.175 ","End":"00:39.230","Text":"this function is defined for all x because x^2 plus 1 is strictly positive."},{"Start":"00:39.230 ","End":"00:42.565","Text":"There\u0027s no problem with the square root or with the denominator."},{"Start":"00:42.565 ","End":"00:44.255","Text":"That\u0027s the domain is all x."},{"Start":"00:44.255 ","End":"00:46.129","Text":"Now, as for the recipe,"},{"Start":"00:46.129 ","End":"00:48.080","Text":"the first step is the preparation,"},{"Start":"00:48.080 ","End":"00:51.140","Text":"which is finding f double prime."},{"Start":"00:51.140 ","End":"00:54.380","Text":"Let\u0027s go ahead with that, but, of course, we have to do it in 2 steps."},{"Start":"00:54.380 ","End":"01:00.395","Text":"First of all, we find f prime and this is going to equal if I use this form,"},{"Start":"01:00.395 ","End":"01:05.330","Text":"2 minus 1/2 times this thing to the minus 3/2,"},{"Start":"01:05.330 ","End":"01:06.925","Text":"that\u0027s with the exponents."},{"Start":"01:06.925 ","End":"01:08.585","Text":"Because of the chain rule,"},{"Start":"01:08.585 ","End":"01:12.755","Text":"we have to multiply by the inner derivative, which is 2x."},{"Start":"01:12.755 ","End":"01:22.670","Text":"Altogether, this is just equal to minus x times x^2 plus 1 to the minus 3/2."},{"Start":"01:22.670 ","End":"01:24.050","Text":"Now we have f prime,"},{"Start":"01:24.050 ","End":"01:26.225","Text":"we can go onto f double prime,"},{"Start":"01:26.225 ","End":"01:31.804","Text":"and we\u0027ll do this with the product rule as well as the rules we used before."},{"Start":"01:31.804 ","End":"01:33.630","Text":"A product rule you should know by now."},{"Start":"01:33.630 ","End":"01:36.515","Text":"Let\u0027s leave the minus outside the brackets"},{"Start":"01:36.515 ","End":"01:41.115","Text":"and just take it as x as 1 function times the other bit."},{"Start":"01:41.115 ","End":"01:47.150","Text":"x derived is 1 and the other part as is, x^2 plus 1"},{"Start":"01:47.150 ","End":"01:54.340","Text":"to the minus 3/2 plus the other way around, x as is times the derivative of this."},{"Start":"01:54.340 ","End":"01:58.500","Text":"Derivative of this is minus 3/2,"},{"Start":"01:58.500 ","End":"02:00.210","Text":"x^2 plus 1,"},{"Start":"02:00.210 ","End":"02:02.313","Text":"and subtract 1 from here,"},{"Start":"02:02.313 ","End":"02:04.135","Text":"we get minus 5/2."},{"Start":"02:04.135 ","End":"02:05.960","Text":"We have to multiply, of course,"},{"Start":"02:05.960 ","End":"02:09.665","Text":"by the 2x, which is the inner derivative here."},{"Start":"02:09.665 ","End":"02:13.205","Text":"We\u0027re taking this outside the brackets."},{"Start":"02:13.205 ","End":"02:14.660","Text":"What we\u0027re left with,"},{"Start":"02:14.660 ","End":"02:17.255","Text":"we have minus times this thing here,"},{"Start":"02:17.255 ","End":"02:21.665","Text":"x^2 plus 1 to the minus 5/2."},{"Start":"02:21.665 ","End":"02:27.260","Text":"What we have here is just x^2 plus 1 to the power of 1,"},{"Start":"02:27.260 ","End":"02:32.660","Text":"which, again, you can check by multiplying this by this, minus 5/2 plus 1 is minus 3/2."},{"Start":"02:32.660 ","End":"02:33.850","Text":"That\u0027s fine."},{"Start":"02:33.850 ","End":"02:38.930","Text":"What we have here is x times x is x^2"},{"Start":"02:38.930 ","End":"02:43.250","Text":"and 2 times minus 3/2 is minus 3."},{"Start":"02:43.250 ","End":"02:45.945","Text":"We have minus 3x^2,"},{"Start":"02:45.945 ","End":"02:48.710","Text":"and if I simplify this,"},{"Start":"02:48.710 ","End":"02:50.765","Text":"well, if you look at this bit,"},{"Start":"02:50.765 ","End":"02:53.467","Text":"this is x^2 minus 3x^2,"},{"Start":"02:53.467 ","End":"03:00.475","Text":"is just this bit is going to be minus 2x^2 and then plus 1 from here."},{"Start":"03:00.475 ","End":"03:07.850","Text":"What I can do is write this as just reverse the sign on this and cancel with this,"},{"Start":"03:07.850 ","End":"03:11.815","Text":"so I get 2x^2 minus 1."},{"Start":"03:11.815 ","End":"03:15.440","Text":"I think I\u0027ll put this in the denominator instead of having"},{"Start":"03:15.440 ","End":"03:21.980","Text":"a minus exponent, x^2 plus 1 to the power of 5/2."},{"Start":"03:21.980 ","End":"03:24.505","Text":"I\u0027ll go over to decimal at this point."},{"Start":"03:24.505 ","End":"03:29.030","Text":"That\u0027s the second derivative and that ends this step."},{"Start":"03:29.030 ","End":"03:35.660","Text":"The next step is to set f double prime equal to 0 and to solve the equation for x."},{"Start":"03:35.660 ","End":"03:39.815","Text":"This will give us our suspects for inflection points."},{"Start":"03:39.815 ","End":"03:41.870","Text":"Now, if a fraction is 0,"},{"Start":"03:41.870 ","End":"03:44.135","Text":"that means its numerator is 0."},{"Start":"03:44.135 ","End":"03:50.546","Text":"This gives us that 2x^2 minus 1 is equal to 0,"},{"Start":"03:50.546 ","End":"03:55.250","Text":"and if we isolate x^2, we get x^2 is 1/2,"},{"Start":"03:55.250 ","End":"04:02.160","Text":"which means that x is either plus or minus the square root of 1/2"},{"Start":"04:02.160 ","End":"04:05.025","Text":"and let\u0027s call it 0.5."},{"Start":"04:05.025 ","End":"04:08.705","Text":"Those are the 2 suspects for inflection points."},{"Start":"04:08.705 ","End":"04:10.820","Text":"The next step is to make a table."},{"Start":"04:10.820 ","End":"04:15.680","Text":"We, first of all, put in the interesting values of x and these are the 2 suspects,"},{"Start":"04:15.680 ","End":"04:24.005","Text":"which is minus the square root of 0.5 and plus the square root of 0.5."},{"Start":"04:24.005 ","End":"04:28.280","Text":"We put them in increasing order and we leave some space in between."},{"Start":"04:28.280 ","End":"04:30.230","Text":"Here, f double prime is 0."},{"Start":"04:30.230 ","End":"04:34.700","Text":"That\u0027s how we got these points and they separate the domain,"},{"Start":"04:34.700 ","End":"04:38.600","Text":"which is the whole x-axis into 3 bits."},{"Start":"04:38.600 ","End":"04:44.045","Text":"We have where x is less than minus the square root of 0.5,"},{"Start":"04:44.045 ","End":"04:48.395","Text":"we have x in between minus the square root of 0.5"},{"Start":"04:48.395 ","End":"04:50.942","Text":"and the square root of 0.5,"},{"Start":"04:50.942 ","End":"04:56.090","Text":"and the last interval is x bigger than root 0.5."},{"Start":"04:56.090 ","End":"04:59.690","Text":"What we do is take a sample point from each of these intervals."},{"Start":"04:59.690 ","End":"05:02.720","Text":"For example, here, I\u0027ll take a sample point of 0."},{"Start":"05:02.720 ","End":"05:04.505","Text":"Choose nice round numbers."},{"Start":"05:04.505 ","End":"05:10.425","Text":"I could choose 1 as being bigger than that and minus 1 being less than this."},{"Start":"05:10.425 ","End":"05:15.110","Text":"What we have to do is substitute these in f double prime."},{"Start":"05:15.110 ","End":"05:17.480","Text":"That\u0027s this function here."},{"Start":"05:17.480 ","End":"05:21.365","Text":"But we don\u0027t want the actual value only whether it\u0027s positive or negative."},{"Start":"05:21.365 ","End":"05:23.479","Text":"Now this thing, the denominator is positive,"},{"Start":"05:23.479 ","End":"05:25.670","Text":"so we only need the numerator."},{"Start":"05:25.670 ","End":"05:28.025","Text":"If I put in x equals minus 1,"},{"Start":"05:28.025 ","End":"05:30.980","Text":"I\u0027ll get 2 minus 1 is 1, it\u0027s positive."},{"Start":"05:30.980 ","End":"05:33.650","Text":"Exactly the same as if I put in x equals 1,"},{"Start":"05:33.650 ","End":"05:36.665","Text":"I get the same answer so it\u0027s also positive."},{"Start":"05:36.665 ","End":"05:40.130","Text":"If I put in 0, I just got minus 1 here, which is negative."},{"Start":"05:40.130 ","End":"05:43.430","Text":"It\u0027s negative, which means that this is an area which"},{"Start":"05:43.430 ","End":"05:46.730","Text":"is concave up or convex and so is this,"},{"Start":"05:46.730 ","End":"05:49.670","Text":"and this one is concave or concave down,"},{"Start":"05:49.670 ","End":"05:51.170","Text":"which looks like this."},{"Start":"05:51.170 ","End":"05:56.480","Text":"That means that our 2 suspects are between convex and concave or vice versa,"},{"Start":"05:56.480 ","End":"06:00.115","Text":"which makes them both definitely inflection points."},{"Start":"06:00.115 ","End":"06:06.410","Text":"I\u0027m going to write that this is an inflection and this is also an inflection."},{"Start":"06:06.410 ","End":"06:10.070","Text":"The last thing I need to put in the table is just the y value for"},{"Start":"06:10.070 ","End":"06:13.998","Text":"these two x values so I can get the actual point."},{"Start":"06:13.998 ","End":"06:15.570","Text":"This time I want the actual value,"},{"Start":"06:15.570 ","End":"06:17.250","Text":"not just the sign."},{"Start":"06:17.250 ","End":"06:22.902","Text":"What I do is I plugged these y after all this here is just f of x,"},{"Start":"06:22.902 ","End":"06:28.880","Text":"and if I put in x equals either plus 0.5 or minus 0.5,"},{"Start":"06:28.880 ","End":"06:34.095","Text":"I get a denominator 0.5 plus 1 is 1.5."},{"Start":"06:34.095 ","End":"06:37.950","Text":"I get 1 over the square root of 1.5,"},{"Start":"06:37.950 ","End":"06:40.667","Text":"1 over square root of 1.5,"},{"Start":"06:40.667 ","End":"06:48.050","Text":"and here I also get the same thing, 1 over the square root of 1.5."},{"Start":"06:48.050 ","End":"06:51.455","Text":"This now gives us all that we need to know."},{"Start":"06:51.455 ","End":"06:56.180","Text":"The last step is the conclusion where I write down what was asked of us."},{"Start":"06:56.180 ","End":"06:59.495","Text":"In other words, which are the inflection points?"},{"Start":"06:59.495 ","End":"07:03.800","Text":"Where is the function convex or concave up?"},{"Start":"07:03.800 ","End":"07:05.915","Text":"Where is it concave?"},{"Start":"07:05.915 ","End":"07:07.730","Text":"Here we have 2 points."},{"Start":"07:07.730 ","End":"07:17.175","Text":"We have this one which is minus square root of 0.5, 1 over square root of 1.5, that\u0027s one,"},{"Start":"07:17.175 ","End":"07:20.220","Text":"and the other one, same but without the minus there,"},{"Start":"07:20.220 ","End":"07:27.290","Text":"square root of 0.5, that\u0027s the x and the y of it is, 1 over the square root of 1.5."},{"Start":"07:27.290 ","End":"07:28.670","Text":"Those 2 inflection points."},{"Start":"07:28.670 ","End":"07:31.625","Text":"Convex is this shape here."},{"Start":"07:31.625 ","End":"07:33.845","Text":"That\u0027s 2 intervals."},{"Start":"07:33.845 ","End":"07:38.450","Text":"First one is x less than minus square root of 0.5"},{"Start":"07:38.450 ","End":"07:44.825","Text":"and the other one over here, x bigger than square root of 0.5."},{"Start":"07:44.825 ","End":"07:48.295","Text":"Concave, this shape here, is the middle interval,"},{"Start":"07:48.295 ","End":"07:54.813","Text":"is the one where x is between minus root 0.5 and plus root 0.5,"},{"Start":"07:54.813 ","End":"07:58.560","Text":"and that answers all of the questions, and we\u0027re done."}],"ID":5841},{"Watched":false,"Name":"Exercise 21","Duration":"6m ","ChapterTopicVideoID":5844,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5844.jpeg","UploadDate":"2016-05-01T17:25:52.0730000","DurationForVideoObject":"PT6M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this exercise, we have a function f of x,"},{"Start":"00:03.270 ","End":"00:06.225","Text":"which is x over the square root of x^2 plus 1."},{"Start":"00:06.225 ","End":"00:09.675","Text":"What we have to do is find its inflection points"},{"Start":"00:09.675 ","End":"00:13.545","Text":"and the intervals where it\u0027s convex and where it\u0027s concave."},{"Start":"00:13.545 ","End":"00:20.130","Text":"This is a familiar kind of exercise which is solvable in 4-steps in a cookbook style."},{"Start":"00:20.130 ","End":"00:22.520","Text":"The first step is a preparation phase,"},{"Start":"00:22.520 ","End":"00:24.100","Text":"then there\u0027s couple of other steps,"},{"Start":"00:24.100 ","End":"00:25.275","Text":"and then a conclusion."},{"Start":"00:25.275 ","End":"00:29.190","Text":"But we also have to make a note of what the domain is."},{"Start":"00:29.190 ","End":"00:31.950","Text":"In this case, the domain is all of x"},{"Start":"00:31.950 ","End":"00:34.980","Text":"because x^2 plus 1 is always strictly positive,"},{"Start":"00:34.980 ","End":"00:37.440","Text":"and therefore, its square root is also positive,"},{"Start":"00:37.440 ","End":"00:41.595","Text":"and so the denominator is not 0, so we\u0027re all set."},{"Start":"00:41.595 ","End":"00:46.250","Text":"The first preparation step is to find f double prime."},{"Start":"00:46.250 ","End":"00:48.080","Text":"But to find f double prime, of course,"},{"Start":"00:48.080 ","End":"00:51.470","Text":"we have to first find f prime, the first derivative."},{"Start":"00:51.470 ","End":"00:55.430","Text":"This equals, according to the quotient rule, in case you\u0027ve forgotten it,"},{"Start":"00:55.430 ","End":"00:56.795","Text":"I better write it again,"},{"Start":"00:56.795 ","End":"01:05.885","Text":"and that is that u/v derivative is u prime v minus uv prime over v^2,"},{"Start":"01:05.885 ","End":"01:08.240","Text":"which in our case gives us,"},{"Start":"01:08.240 ","End":"01:10.800","Text":"I\u0027ll start with the denominator, this is v,"},{"Start":"01:10.800 ","End":"01:14.210","Text":"then v^2 is just x squared plus 1 without the square root."},{"Start":"01:14.210 ","End":"01:21.141","Text":"Now, u prime is 1 times v as is square root of x^2 plus 1"},{"Start":"01:21.141 ","End":"01:24.886","Text":"minus and then numerator as is"},{"Start":"01:24.886 ","End":"01:27.965","Text":"times the derivative of the denominator."},{"Start":"01:27.965 ","End":"01:34.685","Text":"The derivative of the square root is 1 over twice the square root of the same thing,"},{"Start":"01:34.685 ","End":"01:38.390","Text":"x^2 plus 1 times the inner derivative,"},{"Start":"01:38.390 ","End":"01:40.835","Text":"of course, which is 2x."},{"Start":"01:40.835 ","End":"01:42.995","Text":"Let\u0027s see if we can tidy this up a bit."},{"Start":"01:42.995 ","End":"01:45.860","Text":"What I suggest is multiplying top and bottom"},{"Start":"01:45.860 ","End":"01:49.220","Text":"by the square root and that will get rid of it in the numerator."},{"Start":"01:49.220 ","End":"01:54.905","Text":"What we have is x^2 plus 1 here times square root of x^2 plus 1,"},{"Start":"01:54.905 ","End":"01:56.570","Text":"which altogether, if you figure it,"},{"Start":"01:56.570 ","End":"02:00.460","Text":"comes out to x^2 plus 1 to the power of 1 1/2."},{"Start":"02:00.460 ","End":"02:04.910","Text":"Then the numerator, the square root times the square root, is the thing itself."},{"Start":"02:04.910 ","End":"02:06.770","Text":"It\u0027s just x^2 plus 1."},{"Start":"02:06.770 ","End":"02:11.285","Text":"Here, the square root is eliminated because we multiplied by it."},{"Start":"02:11.285 ","End":"02:15.540","Text":"We\u0027re left with minus x/2 times 2x."},{"Start":"02:15.540 ","End":"02:17.430","Text":"Well, the 2 with the 2 cancels"},{"Start":"02:17.430 ","End":"02:20.175","Text":"so we\u0027re left with minus x^2."},{"Start":"02:20.175 ","End":"02:23.115","Text":"Altogether, the x^2 now cancels,"},{"Start":"02:23.115 ","End":"02:29.420","Text":"so what we\u0027re left with is x^2 plus 1 to the power of minus 3/2."},{"Start":"02:29.420 ","End":"02:32.000","Text":"This time, instead of putting it in the denominator,"},{"Start":"02:32.000 ","End":"02:35.390","Text":"I\u0027ll put it in the numerator with a minus sign and it\u0027ll be easy to"},{"Start":"02:35.390 ","End":"02:40.280","Text":"differentiate again using the chain rule and the exponent rule."},{"Start":"02:40.280 ","End":"02:45.350","Text":"f double prime of x is equal to minus 3/2"},{"Start":"02:45.350 ","End":"02:51.140","Text":"times this thing to the power of minus 5/2 times the inner derivative,"},{"Start":"02:51.140 ","End":"02:55.200","Text":"just 2x, so just need to put the x^2 plus 1 here."},{"Start":"02:55.200 ","End":"02:57.215","Text":"I\u0027ll just simplify a little bit."},{"Start":"02:57.215 ","End":"02:59.120","Text":"We can write this as,"},{"Start":"02:59.120 ","End":"03:02.425","Text":"let me take the minus 3/2 with the 2,"},{"Start":"03:02.425 ","End":"03:05.440","Text":"that gives minus 3 and the x."},{"Start":"03:05.440 ","End":"03:09.297","Text":"I\u0027ll put this negative exponent in the denominator"},{"Start":"03:09.297 ","End":"03:14.495","Text":"of the positive exponent x^2 plus 1 to the power of 5/2."},{"Start":"03:14.495 ","End":"03:17.900","Text":"5/2, I\u0027ll put in decimal 2.5."},{"Start":"03:17.900 ","End":"03:19.925","Text":"That\u0027s the second derivative."},{"Start":"03:19.925 ","End":"03:22.205","Text":"That\u0027s the end of this step."},{"Start":"03:22.205 ","End":"03:25.235","Text":"The next step is to solve an equation."},{"Start":"03:25.235 ","End":"03:28.670","Text":"f double prime of x is equal to 0."},{"Start":"03:28.670 ","End":"03:32.810","Text":"The solutions will be suspect for inflection points."},{"Start":"03:32.810 ","End":"03:36.965","Text":"If the second derivative is 0, and here it is,"},{"Start":"03:36.965 ","End":"03:38.930","Text":"it must be that the numerator is 0."},{"Start":"03:38.930 ","End":"03:41.660","Text":"I mean, when a fraction is 0, the numerator has to be 0."},{"Start":"03:41.660 ","End":"03:47.285","Text":"We get that minus 3x is equal to 0, and hence, x equals 0."},{"Start":"03:47.285 ","End":"03:52.220","Text":"That\u0027s our only solution and that\u0027ll be our suspect for an inflection point."},{"Start":"03:52.220 ","End":"03:55.235","Text":"I\u0027ll put that in the table which I\u0027m about to draw."},{"Start":"03:55.235 ","End":"03:57.800","Text":"The interesting values to put in the tables,"},{"Start":"03:57.800 ","End":"04:00.074","Text":"just this one, x equals 0,"},{"Start":"04:00.074 ","End":"04:03.710","Text":"and here, f double prime is 0,"},{"Start":"04:03.710 ","End":"04:06.005","Text":"so it\u0027s a suspect for an inflection,"},{"Start":"04:06.005 ","End":"04:11.870","Text":"and it also point 0 divides the line into two intervals."},{"Start":"04:11.870 ","End":"04:16.955","Text":"The interval x less than 0 and the interval x greater than 0."},{"Start":"04:16.955 ","End":"04:20.105","Text":"For each interval, I choose a representative point."},{"Start":"04:20.105 ","End":"04:24.943","Text":"How about I\u0027ll choose minus 1 here"},{"Start":"04:24.943 ","End":"04:26.175","Text":"and 1 here."},{"Start":"04:26.175 ","End":"04:28.100","Text":"For each of these two sample points,"},{"Start":"04:28.100 ","End":"04:30.255","Text":"I need to compute f double prime"},{"Start":"04:30.255 ","End":"04:32.900","Text":"or at least, I need to know if it\u0027s plus or minus."},{"Start":"04:32.900 ","End":"04:35.750","Text":"If I put minus 1 in f double prime,"},{"Start":"04:35.750 ","End":"04:37.340","Text":"here\u0027s f double prime."},{"Start":"04:37.340 ","End":"04:40.040","Text":"If I put minus 1 on the numerator,"},{"Start":"04:40.040 ","End":"04:42.350","Text":"I\u0027ll get plus 3, which is positive."},{"Start":"04:42.350 ","End":"04:44.750","Text":"The denominator is always positive."},{"Start":"04:44.750 ","End":"04:47.285","Text":"f double prime is positive,"},{"Start":"04:47.285 ","End":"04:52.360","Text":"which means that the function is convex or concave up."},{"Start":"04:52.360 ","End":"04:54.284","Text":"When the x is 1,"},{"Start":"04:54.284 ","End":"04:56.525","Text":"then we have a minus on the top,"},{"Start":"04:56.525 ","End":"04:58.970","Text":"plus on the bottom, altogether, a minus,"},{"Start":"04:58.970 ","End":"05:03.785","Text":"which means that it is concave or concave down."},{"Start":"05:03.785 ","End":"05:09.380","Text":"This happens to be an inflection point because it\u0027s between convex and concave."},{"Start":"05:09.380 ","End":"05:11.525","Text":"So this is an inflection."},{"Start":"05:11.525 ","End":"05:12.950","Text":"Yes, we found one."},{"Start":"05:12.950 ","End":"05:16.070","Text":"For the inflection, I\u0027d like to know the full coordinates,"},{"Start":"05:16.070 ","End":"05:17.990","Text":"the x and also the y."},{"Start":"05:17.990 ","End":"05:20.450","Text":"The y is just f of x."},{"Start":"05:20.450 ","End":"05:24.260","Text":"If I put in x equals 0 here, 0 on the top,"},{"Start":"05:24.260 ","End":"05:26.105","Text":"so the whole thing is 0."},{"Start":"05:26.105 ","End":"05:28.030","Text":"That\u0027s this step."},{"Start":"05:28.030 ","End":"05:30.105","Text":"Now, when it comes to the conclusion step,"},{"Start":"05:30.105 ","End":"05:32.360","Text":"I just write down what I needed to know,"},{"Start":"05:32.360 ","End":"05:35.621","Text":"which is where are the inflection points,"},{"Start":"05:35.621 ","End":"05:38.450","Text":"and where is the function convex,"},{"Start":"05:38.450 ","End":"05:39.645","Text":"find the intervals,"},{"Start":"05:39.645 ","End":"05:41.855","Text":"and the intervals where it\u0027s concave."},{"Start":"05:41.855 ","End":"05:43.655","Text":"We can answer all these questions."},{"Start":"05:43.655 ","End":"05:48.638","Text":"The inflection, like we said, happens at 0, 0, the only one inflection point."},{"Start":"05:48.638 ","End":"05:52.023","Text":"It\u0027s convex, which is this shape or concave up,"},{"Start":"05:52.023 ","End":"05:54.194","Text":"when x is less than 0,"},{"Start":"05:54.194 ","End":"05:57.935","Text":"and it\u0027s concave when x is bigger than 0."},{"Start":"05:57.935 ","End":"06:00.630","Text":"That\u0027s all. We\u0027re done."}],"ID":5842},{"Watched":false,"Name":"Exercise 22","Duration":"7m 26s","ChapterTopicVideoID":5845,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5845.jpeg","UploadDate":"2016-05-01T17:26:41.2470000","DurationForVideoObject":"PT7M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this exercise, we\u0027re given a function f of"},{"Start":"00:02.760 ","End":"00:06.360","Text":"x which is x times the square root of x squared plus 4."},{"Start":"00:06.360 ","End":"00:08.790","Text":"We have to find its inflection points and"},{"Start":"00:08.790 ","End":"00:12.285","Text":"the intervals where it\u0027s convex and where it\u0027s concave."},{"Start":"00:12.285 ","End":"00:15.330","Text":"This exercise is of a familiar format,"},{"Start":"00:15.330 ","End":"00:18.510","Text":"we know how to solve it in 4 steps,"},{"Start":"00:18.510 ","End":"00:19.875","Text":"and we\u0027ll get to it."},{"Start":"00:19.875 ","End":"00:23.745","Text":"We just have to do 1 thing first is to find its domain."},{"Start":"00:23.745 ","End":"00:26.910","Text":"Looking at it, we see there\u0027s no problem in x"},{"Start":"00:26.910 ","End":"00:29.940","Text":"being anything because x squared plus 4 is always positive,"},{"Start":"00:29.940 ","End":"00:31.470","Text":"has a square root, in short,"},{"Start":"00:31.470 ","End":"00:33.840","Text":"domain is all of x. Let\u0027s get started."},{"Start":"00:33.840 ","End":"00:35.830","Text":"The first step is the preparation step,"},{"Start":"00:35.830 ","End":"00:39.845","Text":"and there we have to find f double prime of x, the second derivative."},{"Start":"00:39.845 ","End":"00:41.765","Text":"Let\u0027s start with the first derivative,"},{"Start":"00:41.765 ","End":"00:46.010","Text":"f prime of x. I\u0027m using first the product rule,"},{"Start":"00:46.010 ","End":"00:49.610","Text":"so it\u0027s the derivative of the first, which is 1,"},{"Start":"00:49.610 ","End":"00:55.340","Text":"times the second square root of x squared plus 4 plus the first as is,"},{"Start":"00:55.340 ","End":"00:57.020","Text":"and the derivative of the second."},{"Start":"00:57.020 ","End":"00:59.495","Text":"The derivative of this, we\u0027ll use the chain rule."},{"Start":"00:59.495 ","End":"01:02.510","Text":"The derivative of square root is 1"},{"Start":"01:02.510 ","End":"01:12.020","Text":"over twice the square root of x squared plus 4 times the inner derivative,"},{"Start":"01:12.020 ","End":"01:16.070","Text":"which is 2x, which I\u0027ll just put in the numerator in place of the 1."},{"Start":"01:16.070 ","End":"01:18.320","Text":"We just want to simplify this a bit,"},{"Start":"01:18.320 ","End":"01:25.330","Text":"so what I\u0027ll do is I\u0027ll put it all over the square root of x squared plus 4."},{"Start":"01:25.330 ","End":"01:28.040","Text":"Make it as a quotient on the bottom,"},{"Start":"01:28.040 ","End":"01:31.754","Text":"the square root of x squared plus 4,"},{"Start":"01:31.754 ","End":"01:33.410","Text":"so for the first part,"},{"Start":"01:33.410 ","End":"01:39.290","Text":"I have to multiply by x squared plus 4 square root over the same thing."},{"Start":"01:39.290 ","End":"01:43.430","Text":"To multiply this by this over itself on the numerator,"},{"Start":"01:43.430 ","End":"01:47.925","Text":"I\u0027ll get x squared plus 4 without the square root."},{"Start":"01:47.925 ","End":"01:50.900","Text":"You can see that this over this is just the square root,"},{"Start":"01:50.900 ","End":"01:52.490","Text":"so everything\u0027s fine here."},{"Start":"01:52.490 ","End":"01:54.770","Text":"Here the 2 cancels with 2,"},{"Start":"01:54.770 ","End":"01:56.690","Text":"this thing has gone to the denominator,"},{"Start":"01:56.690 ","End":"01:59.239","Text":"so we\u0027re left with plus x squared."},{"Start":"01:59.239 ","End":"02:09.135","Text":"In short, what we have is 2x squared plus 4 over the square root of x squared plus 4,"},{"Start":"02:09.135 ","End":"02:10.875","Text":"that\u0027s the first derivative."},{"Start":"02:10.875 ","End":"02:12.680","Text":"Now that we have the first derivative,"},{"Start":"02:12.680 ","End":"02:17.915","Text":"we can go on to find the second derivative by differentiating the first derivative."},{"Start":"02:17.915 ","End":"02:20.950","Text":"This time we\u0027ll use the quotient rule,"},{"Start":"02:20.950 ","End":"02:24.305","Text":"but to remind you not everyone remembers the quotient rule."},{"Start":"02:24.305 ","End":"02:26.045","Text":"I\u0027ll write it down,"},{"Start":"02:26.045 ","End":"02:30.410","Text":"u over v derivative is u"},{"Start":"02:30.410 ","End":"02:36.825","Text":"prime v minus uv prime all over v squared."},{"Start":"02:36.825 ","End":"02:39.840","Text":"For a fraction u over v here\u0027s u, here\u0027s v,"},{"Start":"02:39.840 ","End":"02:43.925","Text":"the v squared will just be x squared plus 4 without the square root."},{"Start":"02:43.925 ","End":"02:47.150","Text":"Then numerator derived, which is"},{"Start":"02:47.150 ","End":"02:54.560","Text":"just 4x times the denominator as is square root of x squared plus 4,"},{"Start":"02:54.560 ","End":"02:58.610","Text":"and minus the numerator as is,"},{"Start":"02:58.610 ","End":"03:04.385","Text":"which is 2x squared plus 4 times the derivative of the denominator."},{"Start":"03:04.385 ","End":"03:08.225","Text":"Just like before, the derivative of x squared plus"},{"Start":"03:08.225 ","End":"03:14.995","Text":"4 square root is 1 over twice the square root x squared plus 4,"},{"Start":"03:14.995 ","End":"03:18.140","Text":"and times the inner derivative,"},{"Start":"03:18.140 ","End":"03:19.670","Text":"which is 2x."},{"Start":"03:19.670 ","End":"03:22.055","Text":"Let\u0027s see if we can simplify this again,"},{"Start":"03:22.055 ","End":"03:25.730","Text":"I suggest using the same trick again is multiplying top and"},{"Start":"03:25.730 ","End":"03:29.620","Text":"bottom by the square root of x squared plus 4."},{"Start":"03:29.620 ","End":"03:31.085","Text":"If I take this,"},{"Start":"03:31.085 ","End":"03:33.560","Text":"which is to the power of 1 times the square root,"},{"Start":"03:33.560 ","End":"03:35.180","Text":"which is to the power of 1/2,"},{"Start":"03:35.180 ","End":"03:38.930","Text":"altogether, I have x squared plus 4 to the power of 1"},{"Start":"03:38.930 ","End":"03:43.430","Text":"and 1/2 or 3 over 2 or let\u0027s do decimal this time 1.5."},{"Start":"03:43.430 ","End":"03:45.800","Text":"Now we multiply the bottom,"},{"Start":"03:45.800 ","End":"03:47.390","Text":"we have to multiply the top."},{"Start":"03:47.390 ","End":"03:50.690","Text":"We have 4x times the square root times the square root,"},{"Start":"03:50.690 ","End":"03:53.620","Text":"which is just x squared plus 4,"},{"Start":"03:53.620 ","End":"03:58.904","Text":"and here we have minus 2x squared plus 4,"},{"Start":"03:58.904 ","End":"04:00.920","Text":"2 with the 2 cancels,"},{"Start":"04:00.920 ","End":"04:04.505","Text":"we just multiply it by the square root of x squared plus 4,"},{"Start":"04:04.505 ","End":"04:06.530","Text":"so we\u0027re left with x."},{"Start":"04:06.530 ","End":"04:10.860","Text":"Now, let\u0027s see if we open up the top and expand it,"},{"Start":"04:10.860 ","End":"04:15.990","Text":"we shall get 4x cubed minus 2x cubed."},{"Start":"04:15.990 ","End":"04:18.600","Text":"That gives us 2x cubed,"},{"Start":"04:18.600 ","End":"04:25.170","Text":"we\u0027ll get this with this will give us plus 16x but less 4x,"},{"Start":"04:25.170 ","End":"04:33.085","Text":"so that will be 12x all over x squared plus 4^1 and 1/2."},{"Start":"04:33.085 ","End":"04:36.440","Text":"That\u0027s the end of this step, the preparation."},{"Start":"04:36.440 ","End":"04:42.910","Text":"The next thing we have to do is to solve the equation f double prime of x equals 0,"},{"Start":"04:42.910 ","End":"04:47.150","Text":"we do this so this will give us our suspects for inflection points."},{"Start":"04:47.150 ","End":"04:49.805","Text":"Now, if a fraction is 0,"},{"Start":"04:49.805 ","End":"04:53.530","Text":"then it must mean that its numerator is 0."},{"Start":"04:53.530 ","End":"04:54.945","Text":"I\u0027ll write it again,"},{"Start":"04:54.945 ","End":"04:59.270","Text":"2x cubed plus 12x equals 0,"},{"Start":"04:59.270 ","End":"05:03.485","Text":"and I can factorize it by take 2x out of the brackets,"},{"Start":"05:03.485 ","End":"05:07.825","Text":"I\u0027ll get x squared plus 6 is equal to 0."},{"Start":"05:07.825 ","End":"05:11.630","Text":"But x squared plus 6 cannot be 0 because that"},{"Start":"05:11.630 ","End":"05:15.455","Text":"would mean x squared is minus 6 and square can\u0027t be negative."},{"Start":"05:15.455 ","End":"05:18.890","Text":"We have to have that x equals 0 and this will be"},{"Start":"05:18.890 ","End":"05:22.480","Text":"our only suspect for an inflection point."},{"Start":"05:22.480 ","End":"05:23.970","Text":"That\u0027s the end of this step."},{"Start":"05:23.970 ","End":"05:25.460","Text":"Now, let\u0027s go to the next step,"},{"Start":"05:25.460 ","End":"05:26.960","Text":"which is to draw a table,"},{"Start":"05:26.960 ","End":"05:30.260","Text":"and we fill it with the interesting points is only 1,"},{"Start":"05:30.260 ","End":"05:31.790","Text":"and that is x equals 0,"},{"Start":"05:31.790 ","End":"05:37.104","Text":"which is our suspect for inflection because f double prime at that point is 0,"},{"Start":"05:37.104 ","End":"05:42.890","Text":"and this 0 divides the line into 2 intervals."},{"Start":"05:42.890 ","End":"05:48.650","Text":"We have an interval where x is less than 0 and an interval where x is bigger than 0."},{"Start":"05:48.650 ","End":"05:51.770","Text":"As usual, we take a sample point from each interval,"},{"Start":"05:51.770 ","End":"05:57.525","Text":"so I\u0027ll take x equals minus 1 from here and x equals 1 from here."},{"Start":"05:57.525 ","End":"06:02.880","Text":"Plug it into f double-prime but all I want is the sign plus or minus."},{"Start":"06:02.880 ","End":"06:05.085","Text":"Here\u0027s f double prime."},{"Start":"06:05.085 ","End":"06:07.370","Text":"The denominator is positive,"},{"Start":"06:07.370 ","End":"06:09.935","Text":"so I just need to see what the numerator is."},{"Start":"06:09.935 ","End":"06:11.420","Text":"If x is minus 1,"},{"Start":"06:11.420 ","End":"06:14.450","Text":"x cubed is negative and x is negative,"},{"Start":"06:14.450 ","End":"06:17.020","Text":"and so the whole thing comes out negative,"},{"Start":"06:17.020 ","End":"06:18.590","Text":"so I put a minus here,"},{"Start":"06:18.590 ","End":"06:22.250","Text":"which means we\u0027re shaped like this, concave down."},{"Start":"06:22.250 ","End":"06:28.830","Text":"If x is 1, everything\u0027s positive so we are convex or concave up."},{"Start":"06:28.830 ","End":"06:32.750","Text":"That means that this x equals 0 is"},{"Start":"06:32.750 ","End":"06:38.780","Text":"an inflection point because it\u0027s between convex and concave or the other way around."},{"Start":"06:38.780 ","End":"06:44.810","Text":"All I need more in this table is the value of y that belongs to x equals 0."},{"Start":"06:44.810 ","End":"06:50.055","Text":"For this, I can substitute in the original function,"},{"Start":"06:50.055 ","End":"06:56.000","Text":"so I need to substitute x equals 0 in here to get y,"},{"Start":"06:56.000 ","End":"06:58.250","Text":"x is 0, so the whole thing is 0,"},{"Start":"06:58.250 ","End":"07:00.490","Text":"so we have a 0 here."},{"Start":"07:00.490 ","End":"07:02.850","Text":"That\u0027s the end of the table step,"},{"Start":"07:02.850 ","End":"07:08.180","Text":"and the final step is the conclusion is where I actually write what I found."},{"Start":"07:08.180 ","End":"07:11.360","Text":"I have an inflection point at 0,"},{"Start":"07:11.360 ","End":"07:13.480","Text":"0, it\u0027s the only 1."},{"Start":"07:13.480 ","End":"07:19.670","Text":"The function is convex on the interval x bigger than 0,"},{"Start":"07:19.670 ","End":"07:27.480","Text":"and it\u0027s concave on the interval where x is less than 0, and that\u0027s it. We\u0027re done."}],"ID":5843},{"Watched":false,"Name":"Exercise 23","Duration":"9m 30s","ChapterTopicVideoID":5846,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5846.jpeg","UploadDate":"2016-05-01T17:27:44.1230000","DurationForVideoObject":"PT9M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.310","Text":"In this exercise, we\u0027re given the function"},{"Start":"00:03.310 ","End":"00:07.785","Text":"f of x equals the cube root of x^2 times 1 minus x."},{"Start":"00:07.785 ","End":"00:11.760","Text":"What we have to do is to find its points of inflection and"},{"Start":"00:11.760 ","End":"00:16.005","Text":"also the intervals where it\u0027s convex and where it\u0027s concave."},{"Start":"00:16.005 ","End":"00:18.979","Text":"This is a familiar type of exercise"},{"Start":"00:18.979 ","End":"00:21.774","Text":"and there\u0027s a known recipe for a solution"},{"Start":"00:21.774 ","End":"00:26.370","Text":"involving 4-steps: preparation, 2 more steps, and a conclusion."},{"Start":"00:26.370 ","End":"00:27.957","Text":"We can apply it here,"},{"Start":"00:27.957 ","End":"00:32.460","Text":"but, first, there\u0027s one small other thing we have to do is to find its domain."},{"Start":"00:32.460 ","End":"00:33.840","Text":"Looking at this function,"},{"Start":"00:33.840 ","End":"00:35.830","Text":"it\u0027s defined for every x."},{"Start":"00:35.830 ","End":"00:37.160","Text":"Unlike the square root,"},{"Start":"00:37.160 ","End":"00:39.740","Text":"the cube root is defined for positive and negative so"},{"Start":"00:39.740 ","End":"00:42.770","Text":"there\u0027s no problems substituting any x here."},{"Start":"00:42.770 ","End":"00:44.360","Text":"Okay, now let\u0027s get to it."},{"Start":"00:44.360 ","End":"00:49.100","Text":"The preparation phase is to find the second derivative f double prime."},{"Start":"00:49.100 ","End":"00:51.650","Text":"Of course, we have to start with the first derivative."},{"Start":"00:51.650 ","End":"00:53.990","Text":"So let\u0027s see what f prime of x is."},{"Start":"00:53.990 ","End":"00:55.580","Text":"Just to make it easier,"},{"Start":"00:55.580 ","End":"01:00.245","Text":"I\u0027m going to rewrite this in terms of fractional exponents."},{"Start":"01:00.245 ","End":"01:03.535","Text":"This here is x to the power of 2 over 3."},{"Start":"01:03.535 ","End":"01:05.520","Text":"So if I multiply it out,"},{"Start":"01:05.520 ","End":"01:09.455","Text":"x to the 2 over 3 times 1 is x to the 2 over 3."},{"Start":"01:09.455 ","End":"01:11.390","Text":"If I multiply it by x,"},{"Start":"01:11.390 ","End":"01:17.320","Text":"which is x to the 1 and I get minus x to the 1 plus 2/3."},{"Start":"01:17.320 ","End":"01:19.470","Text":"1 plus 2/3 is 5/3."},{"Start":"01:19.470 ","End":"01:27.660","Text":"So now the derivative is quite easy because we just get 2/3 x and then subtract 1,"},{"Start":"01:27.660 ","End":"01:29.520","Text":"so it\u0027s minus 1/3,"},{"Start":"01:29.520 ","End":"01:33.840","Text":"minus 5/3 x, and we subtract 1,"},{"Start":"01:33.840 ","End":"01:37.295","Text":"so we get 2/3 and we can continue."},{"Start":"01:37.295 ","End":"01:43.370","Text":"f double prime of x is equal to minus 1/3 times 2/3,"},{"Start":"01:43.370 ","End":"01:47.970","Text":"which is minus 2/9 x to the, subtract 1,"},{"Start":"01:47.970 ","End":"01:51.450","Text":"so we get minus 4/3"},{"Start":"01:51.450 ","End":"01:58.320","Text":"and take away 2/3 times 5/3 is 10/9 x to the,"},{"Start":"01:58.320 ","End":"02:00.760","Text":"subtracting 1, minus 1/3."},{"Start":"02:00.760 ","End":"02:07.220","Text":"What we can do is simplify this by taking outside of the brackets."},{"Start":"02:07.220 ","End":"02:11.210","Text":"If I take x to the minus 4/3 outside the brackets,"},{"Start":"02:11.210 ","End":"02:14.815","Text":"I will get x to the minus 4/3,"},{"Start":"02:14.815 ","End":"02:17.860","Text":"which is 1 over x to the 4/3."},{"Start":"02:17.860 ","End":"02:20.150","Text":"So in other words, I\u0027m taking this outside the bracket"},{"Start":"02:20.150 ","End":"02:22.220","Text":"and also putting it in the denominator,"},{"Start":"02:22.220 ","End":"02:23.635","Text":"2 steps in 1."},{"Start":"02:23.635 ","End":"02:28.820","Text":"What else I can do is I can take the common 9 and also put it in the denominator."},{"Start":"02:28.820 ","End":"02:30.800","Text":"So here I\u0027ll stick the 9."},{"Start":"02:30.800 ","End":"02:33.130","Text":"What I\u0027m left with now,"},{"Start":"02:33.560 ","End":"02:39.145","Text":"I took x to the minus 4/3 out, here, I just have minus 2."},{"Start":"02:39.145 ","End":"02:45.840","Text":"Here, I have x to the minus 1/3 is just x times x to the minus 4/3."},{"Start":"02:45.840 ","End":"02:50.540","Text":"If I do minus 1/3, minus, minus 4/3, it comes out 1."},{"Start":"02:50.540 ","End":"02:53.670","Text":"You can check it afterwards by multiplying out,"},{"Start":"02:53.670 ","End":"02:55.685","Text":"I\u0027m claiming this is just x."},{"Start":"02:55.685 ","End":"03:00.470","Text":"If you look at it here, x over x to the 4/3 is x to the minus 1/3,"},{"Start":"03:00.470 ","End":"03:01.835","Text":"and so on, it\u0027ll work out."},{"Start":"03:01.835 ","End":"03:06.905","Text":"So minus 2/9 and then I have 1 plus 5x over"},{"Start":"03:06.905 ","End":"03:12.071","Text":"x to the power of 1 1/3 or 4/3,"},{"Start":"03:12.071 ","End":"03:15.755","Text":"and that\u0027s the end of this preparation phase."},{"Start":"03:15.755 ","End":"03:17.615","Text":"We\u0027ve got f double prime."},{"Start":"03:17.615 ","End":"03:21.935","Text":"Next step is to equate f double prime to 0."},{"Start":"03:21.935 ","End":"03:25.400","Text":"What we\u0027re doing here is we\u0027re looking for solutions because solutions to"},{"Start":"03:25.400 ","End":"03:29.375","Text":"this are going to be suspects for an inflection point."},{"Start":"03:29.375 ","End":"03:32.030","Text":"So if this thing is 0,"},{"Start":"03:32.030 ","End":"03:33.875","Text":"the denominator could be 0,"},{"Start":"03:33.875 ","End":"03:36.365","Text":"and that\u0027s something that we have to watch out for."},{"Start":"03:36.365 ","End":"03:43.010","Text":"So what I have to add is that f double prime is not defined for x equals 0."},{"Start":"03:43.010 ","End":"03:45.530","Text":"This is provided x is not 0,"},{"Start":"03:45.530 ","End":"03:49.130","Text":"but x equals 0 will come into play later on,"},{"Start":"03:49.130 ","End":"03:51.410","Text":"you\u0027ll see, so I just noting this down,"},{"Start":"03:51.410 ","End":"03:53.520","Text":"so I\u0027m eliminating 0 for the moment."},{"Start":"03:53.520 ","End":"03:54.920","Text":"If I\u0027ve eliminated 0,"},{"Start":"03:54.920 ","End":"03:58.355","Text":"then a fraction is 0 when the numerator is 0."},{"Start":"03:58.355 ","End":"04:06.950","Text":"So I get 1 plus 5x is equal to 0 and x is equal to minus 1/5."},{"Start":"04:06.950 ","End":"04:14.130","Text":"That brings us to the end of this step where we have suspects for inflection from here."},{"Start":"04:14.130 ","End":"04:16.305","Text":"We\u0027re ready for the next step."},{"Start":"04:16.305 ","End":"04:20.765","Text":"I\u0027ll return to this thing about the x naught equals 0 in that step."},{"Start":"04:20.765 ","End":"04:23.255","Text":"Okay, so here\u0027s a table,"},{"Start":"04:23.255 ","End":"04:26.165","Text":"and the question is which values of x to put in here?"},{"Start":"04:26.165 ","End":"04:27.395","Text":"Well, first of all,"},{"Start":"04:27.395 ","End":"04:32.750","Text":"we put x equals minus 1/5 in there because that\u0027s a suspect"},{"Start":"04:32.750 ","End":"04:38.270","Text":"for an inflection but we also are going to include x equals 0."},{"Start":"04:38.270 ","End":"04:39.980","Text":"This is a delicate point."},{"Start":"04:39.980 ","End":"04:42.995","Text":"If the function is defined at x equals 0,"},{"Start":"04:42.995 ","End":"04:47.105","Text":"and it is because we said that this is defined for all x"},{"Start":"04:47.105 ","End":"04:51.590","Text":"but the second derivative doesn\u0027t exist at that point,"},{"Start":"04:51.590 ","End":"04:56.135","Text":"this also makes it a suspect for an inflection point."},{"Start":"04:56.135 ","End":"04:59.660","Text":"So we put in both minus 1/5 and 0."},{"Start":"04:59.660 ","End":"05:02.680","Text":"I\u0027ll repeat what I just said because it\u0027s important."},{"Start":"05:02.680 ","End":"05:06.202","Text":"If we have values of x for which f is defined"},{"Start":"05:06.202 ","End":"05:11.690","Text":"and either f double prime is 0 or f double prime is undefined,"},{"Start":"05:11.690 ","End":"05:15.964","Text":"in both cases, it makes the x or candidate for an inflection."},{"Start":"05:15.964 ","End":"05:21.572","Text":"So I\u0027m putting in minus 1/5 because at f double prime, this is 0,"},{"Start":"05:21.572 ","End":"05:27.770","Text":"and I\u0027m also going to put in 0 because f double prime is undefined,"},{"Start":"05:27.770 ","End":"05:29.865","Text":"but f is defined."},{"Start":"05:29.865 ","End":"05:32.240","Text":"So now that we\u0027ve done this,"},{"Start":"05:32.240 ","End":"05:36.695","Text":"we have to check them both to see whether they are indeed inflection and they would be"},{"Start":"05:36.695 ","End":"05:41.300","Text":"if they are on the border between concave and convex or vice versa."},{"Start":"05:41.300 ","End":"05:46.490","Text":"So let\u0027s get back here and denote the intervals which these two define."},{"Start":"05:46.490 ","End":"05:54.380","Text":"It breaks up the line into 3 intervals and these intervals are x less than minus 1/5."},{"Start":"05:54.380 ","End":"05:55.490","Text":"I\u0027ll make it decimal."},{"Start":"05:55.490 ","End":"05:59.900","Text":"So x less than minus 0.2."},{"Start":"05:59.900 ","End":"06:05.366","Text":"This interval is where minus 0.2 less than x less than 0,"},{"Start":"06:05.366 ","End":"06:08.180","Text":"and here we have x bigger than 0."},{"Start":"06:08.180 ","End":"06:13.205","Text":"As usual, we choose a sample point for each of these intervals."},{"Start":"06:13.205 ","End":"06:17.017","Text":"So here, I could choose, let\u0027s say, x equals minus 1,"},{"Start":"06:17.017 ","End":"06:21.149","Text":"between this and this, I\u0027ll choose minus 0.1,"},{"Start":"06:21.149 ","End":"06:25.490","Text":"and bigger than 0, I\u0027ll choose x equals 1,"},{"Start":"06:25.490 ","End":"06:31.745","Text":"and now I have to plug them in to f double prime and see what I get."},{"Start":"06:31.745 ","End":"06:35.675","Text":"But I don\u0027t actually care about the answer as such,"},{"Start":"06:35.675 ","End":"06:36.980","Text":"only about its sign,"},{"Start":"06:36.980 ","End":"06:38.675","Text":"whether it\u0027s plus or minus."},{"Start":"06:38.675 ","End":"06:40.100","Text":"Let\u0027s try them one at a time."},{"Start":"06:40.100 ","End":"06:41.945","Text":"When x is minus 1,"},{"Start":"06:41.945 ","End":"06:48.590","Text":"the denominator here is always positive because it\u0027s x to the 4/3 and x^4 is positive,"},{"Start":"06:48.590 ","End":"06:49.940","Text":"so its cube root\u0027s positive."},{"Start":"06:49.940 ","End":"06:53.196","Text":"So I can concentrate mainly on the 1 plus 5x."},{"Start":"06:53.196 ","End":"06:55.790","Text":"The minus 2/9 is also always negative."},{"Start":"06:55.790 ","End":"06:57.965","Text":"So the only variable is 1 plus 5x,"},{"Start":"06:57.965 ","End":"07:00.185","Text":"and when x is minus 1,"},{"Start":"07:00.185 ","End":"07:02.255","Text":"this is clearly negative."},{"Start":"07:02.255 ","End":"07:03.755","Text":"But I have the negative here,"},{"Start":"07:03.755 ","End":"07:06.305","Text":"so altogether, it is positive."},{"Start":"07:06.305 ","End":"07:08.975","Text":"Now, for the minus 0.1,"},{"Start":"07:08.975 ","End":"07:10.610","Text":"I just have to look at this part."},{"Start":"07:10.610 ","End":"07:13.265","Text":"That\u0027s 1 minus 0.5,"},{"Start":"07:13.265 ","End":"07:18.520","Text":"which is positive times this negative gives me a negative here."},{"Start":"07:18.520 ","End":"07:20.845","Text":"When x is 1,"},{"Start":"07:20.845 ","End":"07:23.405","Text":"I also get positive here,"},{"Start":"07:23.405 ","End":"07:24.890","Text":"1 plus 5 is 6,"},{"Start":"07:24.890 ","End":"07:26.160","Text":"but there\u0027s a negative here,"},{"Start":"07:26.160 ","End":"07:28.744","Text":"so here we also have a negative."},{"Start":"07:28.744 ","End":"07:32.165","Text":"This means that here it\u0027s convex or concave up,"},{"Start":"07:32.165 ","End":"07:36.185","Text":"and here and here, it\u0027s concave or concave down."},{"Start":"07:36.185 ","End":"07:42.232","Text":"That means that this suspect turns out indeed to be an inflection."},{"Start":"07:42.232 ","End":"07:47.870","Text":"But this one, the x equals 0, is between concave and concave"},{"Start":"07:47.870 ","End":"07:51.530","Text":"so it is not, not an inflection, just nothing."},{"Start":"07:51.530 ","End":"07:52.655","Text":"Well, maybe it\u0027s something"},{"Start":"07:52.655 ","End":"07:53.995","Text":"but it\u0027s not an inflection."},{"Start":"07:53.995 ","End":"07:55.620","Text":"For the inflection point,"},{"Start":"07:55.620 ","End":"08:01.400","Text":"you would like to know the value of y just so I can write it down."},{"Start":"08:01.400 ","End":"08:05.420","Text":"So let\u0027s go and fill in the y value here. Let\u0027s see."},{"Start":"08:05.420 ","End":"08:12.710","Text":"We have to substitute minus 0.2 in here and I\u0027ll just write it like that."},{"Start":"08:12.710 ","End":"08:14.465","Text":"Whoever has a calculator can."},{"Start":"08:14.465 ","End":"08:16.550","Text":"So 0.8 from here,"},{"Start":"08:16.550 ","End":"08:22.205","Text":"cube root of 0.2 squared is 0.04,"},{"Start":"08:22.205 ","End":"08:28.575","Text":"0.8 from here, cube root of 0.04,"},{"Start":"08:28.575 ","End":"08:31.850","Text":"whoever has a calculator can figure it out."},{"Start":"08:31.850 ","End":"08:34.130","Text":"That\u0027s all I need from the table."},{"Start":"08:34.130 ","End":"08:35.830","Text":"That\u0027s the end of this step."},{"Start":"08:35.830 ","End":"08:38.255","Text":"The last step is just to interpret,"},{"Start":"08:38.255 ","End":"08:41.855","Text":"to draw conclusions which will be the answers to what we\u0027re looking for."},{"Start":"08:41.855 ","End":"08:45.890","Text":"Otherwise, we\u0027re looking for this function\u0027s inflection points."},{"Start":"08:45.890 ","End":"08:50.258","Text":"So we have an inflection only one at this point here,"},{"Start":"08:50.258 ","End":"08:59.835","Text":"at minus 0.2 times 0.8 cube root of 0.04 to be calculated."},{"Start":"08:59.835 ","End":"09:03.245","Text":"We have convex intervals."},{"Start":"09:03.245 ","End":"09:10.101","Text":"Convex is this shape here so we have it at x less than 0.2,"},{"Start":"09:10.101 ","End":"09:14.570","Text":"and the concave and the concave down are these two,"},{"Start":"09:14.570 ","End":"09:21.065","Text":"so we have minus 0.2 less than x less than 0"},{"Start":"09:21.065 ","End":"09:27.094","Text":"and we also have concave at x bigger than 0."},{"Start":"09:27.094 ","End":"09:28.970","Text":"This answers the question,"},{"Start":"09:28.970 ","End":"09:31.380","Text":"and so we\u0027re done"}],"ID":5844},{"Watched":false,"Name":"Exercise 24","Duration":"6m 9s","ChapterTopicVideoID":5847,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5847.jpeg","UploadDate":"2016-05-01T17:28:24.4330000","DurationForVideoObject":"PT6M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this exercise, we\u0027re given the following function of"},{"Start":"00:03.150 ","End":"00:06.240","Text":"x cube root of x squared minus 1 all squared."},{"Start":"00:06.240 ","End":"00:09.690","Text":"What we have to do is find its inflection points and also"},{"Start":"00:09.690 ","End":"00:13.860","Text":"the intervals where it\u0027s convex and where it\u0027s concave."},{"Start":"00:13.860 ","End":"00:17.130","Text":"This is a familiar form of an exercise and"},{"Start":"00:17.130 ","End":"00:20.220","Text":"we know the solution in 4 steps as a preparation,"},{"Start":"00:20.220 ","End":"00:22.360","Text":"2 proper steps and a conclusion,"},{"Start":"00:22.360 ","End":"00:26.615","Text":"we just have to do 1 other thing first is to note the domain."},{"Start":"00:26.615 ","End":"00:28.820","Text":"There\u0027s no restriction on x."},{"Start":"00:28.820 ","End":"00:33.130","Text":"The cube root is defined for every number and no problem there."},{"Start":"00:33.130 ","End":"00:35.900","Text":"Let\u0027s get on to the first preparation stage,"},{"Start":"00:35.900 ","End":"00:38.735","Text":"which is to find f double prime."},{"Start":"00:38.735 ","End":"00:41.494","Text":"Of course we have to find f prime first,"},{"Start":"00:41.494 ","End":"00:47.420","Text":"but I think it\u0027ll be easier if we expand this and use fractional exponents."},{"Start":"00:47.420 ","End":"00:53.020","Text":"What I mean is like this cube root of x squared is x^2 over 3."},{"Start":"00:53.020 ","End":"00:56.390","Text":"We can also expand a minus b squared is"},{"Start":"00:56.390 ","End":"00:59.870","Text":"a squared minus 2ab plus b squared, well-known formula."},{"Start":"00:59.870 ","End":"01:06.755","Text":"I will write this as x^2/3 squared, which is x^4/3,"},{"Start":"01:06.755 ","End":"01:11.135","Text":"minus twice x^2/3 times 1,"},{"Start":"01:11.135 ","End":"01:13.295","Text":"times 1 I don\u0027t have to do anything,"},{"Start":"01:13.295 ","End":"01:16.520","Text":"plus 1 squared, which is 1."},{"Start":"01:16.520 ","End":"01:26.360","Text":"If this is f of x, then f prime of x will equal using the exponents 4/3x and take away 1,"},{"Start":"01:26.360 ","End":"01:31.250","Text":"so we\u0027re left with 1/3 minus 2 times 2/3,"},{"Start":"01:31.250 ","End":"01:36.880","Text":"Which is 4/3x^minus 1/3,"},{"Start":"01:36.880 ","End":"01:40.250","Text":"and the derivative of this is just 0."},{"Start":"01:40.250 ","End":"01:41.660","Text":"Now that we have f prime,"},{"Start":"01:41.660 ","End":"01:45.080","Text":"we can figure out what is f double-prime."},{"Start":"01:45.080 ","End":"01:54.200","Text":"Differentiate again, 1/3 times 4/3 is 4/9, x^minus 2/3,"},{"Start":"01:54.200 ","End":"02:00.210","Text":"taking away 1 minus 4/3 times minus 1/3 will make"},{"Start":"02:00.210 ","End":"02:06.690","Text":"it plus 4/9 and x^minus 4/3."},{"Start":"02:06.690 ","End":"02:11.810","Text":"That\u0027s our f double-prime except for possible simplification."},{"Start":"02:11.810 ","End":"02:15.770","Text":"Let\u0027s rewrite it as 4/9,"},{"Start":"02:15.770 ","End":"02:18.110","Text":"and I\u0027ll take the 4/9 outside the brackets."},{"Start":"02:18.110 ","End":"02:23.360","Text":"Now, this thing is 1 over the cube root of x squared."},{"Start":"02:23.360 ","End":"02:26.225","Text":"If I play with the exponents and this thing,"},{"Start":"02:26.225 ","End":"02:31.700","Text":"similar expression, 1 over the cube root of x^4."},{"Start":"02:31.700 ","End":"02:33.545","Text":"If you look at this,"},{"Start":"02:33.545 ","End":"02:39.800","Text":"x squared is always positive for every x and so is x^4."},{"Start":"02:39.800 ","End":"02:44.710","Text":"The cube roots are positive and 1 over positive is positive."},{"Start":"02:44.710 ","End":"02:46.909","Text":"Also, the 4/9 is positive."},{"Start":"02:46.909 ","End":"02:48.470","Text":"In other words, if you look at this,"},{"Start":"02:48.470 ","End":"02:53.855","Text":"you\u0027ll see that f double prime of x is always bigger than 0."},{"Start":"02:53.855 ","End":"02:56.255","Text":"That\u0027s 1 point I want to make."},{"Start":"02:56.255 ","End":"02:58.250","Text":"However, there\u0027s another fine point,"},{"Start":"02:58.250 ","End":"03:02.495","Text":"because notice that when x is 0,"},{"Start":"03:02.495 ","End":"03:04.789","Text":"this is not positive, the denominator,"},{"Start":"03:04.789 ","End":"03:08.125","Text":"it\u0027s actually 0, it\u0027s undefined."},{"Start":"03:08.125 ","End":"03:12.920","Text":"Really, it\u0027s bigger than 0 for when x is not equal to"},{"Start":"03:12.920 ","End":"03:18.064","Text":"0 and it\u0027s undefined for x equals 0."},{"Start":"03:18.064 ","End":"03:23.780","Text":"Next step is to solve the equation f double-prime of x is equal to 0."},{"Start":"03:23.780 ","End":"03:25.655","Text":"Now, from what I just said,"},{"Start":"03:25.655 ","End":"03:28.174","Text":"this thing is never equal to 0."},{"Start":"03:28.174 ","End":"03:30.365","Text":"When x is 0, it\u0027s undefined,"},{"Start":"03:30.365 ","End":"03:31.790","Text":"and when x is not 0,"},{"Start":"03:31.790 ","End":"03:36.755","Text":"it\u0027s positive so the answer will be no solution or no such x."},{"Start":"03:36.755 ","End":"03:39.920","Text":"There\u0027s only 1 interesting point in all of this,"},{"Start":"03:39.920 ","End":"03:42.125","Text":"and that\u0027s the point x equals 0,"},{"Start":"03:42.125 ","End":"03:44.550","Text":"where f of x is defined,"},{"Start":"03:44.550 ","End":"03:46.955","Text":"because 0 can go in here."},{"Start":"03:46.955 ","End":"03:48.800","Text":"I mean f was defined for all x,"},{"Start":"03:48.800 ","End":"03:51.365","Text":"but f double prime is not defined."},{"Start":"03:51.365 ","End":"03:56.030","Text":"This also makes 0 a candidate or suspect,"},{"Start":"03:56.030 ","End":"03:58.865","Text":"should I say, for an inflection point."},{"Start":"03:58.865 ","End":"04:00.365","Text":"Let\u0027s get to the table."},{"Start":"04:00.365 ","End":"04:03.830","Text":"Usually put in several points."},{"Start":"04:03.830 ","End":"04:08.509","Text":"Suspects for inflection where f double prime is 0,"},{"Start":"04:08.509 ","End":"04:12.905","Text":"but another suspect where f is defined,"},{"Start":"04:12.905 ","End":"04:15.050","Text":"but f double-prime is not defined,"},{"Start":"04:15.050 ","End":"04:16.970","Text":"and that\u0027s 1 that we have here."},{"Start":"04:16.970 ","End":"04:25.010","Text":"We have x equals 0 as our only possibility and f double-prime here is undefined."},{"Start":"04:25.010 ","End":"04:27.260","Text":"What this 0 does is first of all,"},{"Start":"04:27.260 ","End":"04:29.420","Text":"it breaks up the intervals,"},{"Start":"04:29.420 ","End":"04:32.870","Text":"the 1 long interval into 2 separate intervals."},{"Start":"04:32.870 ","End":"04:40.115","Text":"We have an interval where x is less than 0 and an interval where x is bigger than 0."},{"Start":"04:40.115 ","End":"04:42.200","Text":"At both of these,"},{"Start":"04:42.200 ","End":"04:43.480","Text":"we already talked about that,"},{"Start":"04:43.480 ","End":"04:47.825","Text":"that the f double prime is positive except for this point where it\u0027s undefined."},{"Start":"04:47.825 ","End":"04:51.095","Text":"Here it\u0027s positive and here it\u0027s positive,"},{"Start":"04:51.095 ","End":"04:55.129","Text":"which means that f is convex or concave"},{"Start":"04:55.129 ","End":"04:59.780","Text":"up for less than 0 and the same thing for bigger than 0."},{"Start":"04:59.780 ","End":"05:03.535","Text":"This point is between convex and convex,"},{"Start":"05:03.535 ","End":"05:06.680","Text":"so it is not going to be an inflection point."},{"Start":"05:06.680 ","End":"05:10.310","Text":"An inflection point has to be between convex and concave or the other way around."},{"Start":"05:10.310 ","End":"05:13.715","Text":"This is not an inflection."},{"Start":"05:13.715 ","End":"05:16.550","Text":"Basically we haven\u0027t found an inflection point."},{"Start":"05:16.550 ","End":"05:19.130","Text":"There was no points where f double prime was 0."},{"Start":"05:19.130 ","End":"05:23.150","Text":"The only chance we had where f double-prime was undefined,"},{"Start":"05:23.150 ","End":"05:27.020","Text":"but f was defined also turned out not to be an inflection."},{"Start":"05:27.020 ","End":"05:29.875","Text":"All in all, there\u0027s no inflection points."},{"Start":"05:29.875 ","End":"05:32.090","Text":"Now we can go to the last step,"},{"Start":"05:32.090 ","End":"05:36.380","Text":"which is to interpret the table and view of what was asked."},{"Start":"05:36.380 ","End":"05:39.549","Text":"We were asked for 3 things: the inflection points;"},{"Start":"05:39.549 ","End":"05:42.380","Text":"the inflection points I can just say there are none."},{"Start":"05:42.380 ","End":"05:46.820","Text":"Then we were asked for concave and convex intervals."},{"Start":"05:46.820 ","End":"05:51.230","Text":"The convex intervals is everywhere except x equals"},{"Start":"05:51.230 ","End":"05:57.080","Text":"0 or I can write it as x less than 0 and x bigger than 0,"},{"Start":"05:57.080 ","End":"05:59.480","Text":"or I could combine to say x not 0."},{"Start":"05:59.480 ","End":"06:02.950","Text":"Concave, also no such interval,"},{"Start":"06:02.950 ","End":"06:05.070","Text":"so nowhere or none."},{"Start":"06:05.070 ","End":"06:07.500","Text":"I\u0027ll just write nowhere concave."},{"Start":"06:07.500 ","End":"06:09.940","Text":"That answers the question."}],"ID":5845},{"Watched":false,"Name":"Exercise 25","Duration":"9m 2s","ChapterTopicVideoID":5848,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5848.jpeg","UploadDate":"2016-05-01T17:29:24.3170000","DurationForVideoObject":"PT9M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we\u0027re given a function f of"},{"Start":"00:03.360 ","End":"00:06.690","Text":"x equals the cube root of x squared minus 1,"},{"Start":"00:06.690 ","End":"00:10.170","Text":"we have to find its inflection points and the intervals"},{"Start":"00:10.170 ","End":"00:13.830","Text":"where the function is convex and where it\u0027s concave."},{"Start":"00:13.830 ","End":"00:19.440","Text":"This is familiar form of exercise and we have a fixed set of steps for a solution,"},{"Start":"00:19.440 ","End":"00:21.500","Text":"4 steps in fact, preparation,"},{"Start":"00:21.500 ","End":"00:24.380","Text":"step 1, step 2, and conclusions."},{"Start":"00:24.380 ","End":"00:28.565","Text":"Just before that, I\u0027d like to look at the domain."},{"Start":"00:28.565 ","End":"00:32.950","Text":"If I look at it, I see it\u0027s defined for all x because it\u0027s the cube root,"},{"Start":"00:32.950 ","End":"00:36.400","Text":"cube root is defined everywhere and so is x squared minus 1,"},{"Start":"00:36.400 ","End":"00:38.335","Text":"so no problems there."},{"Start":"00:38.335 ","End":"00:44.360","Text":"Now, the first step is the preparation is to find f double prime, the 2nd derivative."},{"Start":"00:44.360 ","End":"00:46.850","Text":"Of course we have to start with the 1st derivative."},{"Start":"00:46.850 ","End":"00:49.970","Text":"Let\u0027s see if we can figure out what is f prime of x."},{"Start":"00:49.970 ","End":"00:53.720","Text":"What I suggest here is imagining that instead of the cube root,"},{"Start":"00:53.720 ","End":"00:58.610","Text":"it\u0027s written as to the power of a 1/3 and then we can differentiate it by saying"},{"Start":"00:58.610 ","End":"01:04.035","Text":"it\u0027s 1/3 times this thing to power of minus 2/3,"},{"Start":"01:04.035 ","End":"01:07.520","Text":"that\u0027s 1/3 minus 1 times the inner derivative,"},{"Start":"01:07.520 ","End":"01:12.030","Text":"which is 2x and this equals"},{"Start":"01:12.550 ","End":"01:20.770","Text":"2/3x times x squared minus 1 to the power of minus 2/3."},{"Start":"01:20.770 ","End":"01:21.885","Text":"Now, let\u0027s continue."},{"Start":"01:21.885 ","End":"01:22.980","Text":"We\u0027ve got f prime,"},{"Start":"01:22.980 ","End":"01:25.130","Text":"let\u0027s find f double-prime."},{"Start":"01:25.130 ","End":"01:31.220","Text":"I\u0027ll keep the 2/3 outside the brackets and use the product rule here."},{"Start":"01:31.220 ","End":"01:33.950","Text":"I have derivative of x,"},{"Start":"01:33.950 ","End":"01:36.470","Text":"which is just 1, I won\u0027t even write it,"},{"Start":"01:36.470 ","End":"01:46.350","Text":"times x squared minus 1 to the minus 2/3 plus x times this 1 derived,"},{"Start":"01:46.350 ","End":"01:54.260","Text":"which is minus 2/3x squared minus 1 to the power of minus 5 over 3,"},{"Start":"01:54.260 ","End":"01:56.525","Text":"just subtracted 1, and again,"},{"Start":"01:56.525 ","End":"01:59.554","Text":"the inner derivative to x."},{"Start":"01:59.554 ","End":"02:01.325","Text":"Let\u0027s simplify a bit."},{"Start":"02:01.325 ","End":"02:06.350","Text":"What I\u0027ll do is I\u0027ll take the smallest power outside the square brackets,"},{"Start":"02:06.350 ","End":"02:11.210","Text":"that will be the x squared minus 1 to the minus 5 over 3,"},{"Start":"02:11.210 ","End":"02:18.870","Text":"so I get 2/3x squared minus 1 to the minus 5 over 3."},{"Start":"02:18.870 ","End":"02:24.880","Text":"What I\u0027m left with is here just x squared minus 1 to the power of 1."},{"Start":"02:24.880 ","End":"02:26.900","Text":"If you check it, if you multiply out,"},{"Start":"02:26.900 ","End":"02:31.535","Text":"you get minus 5 over 3 plus 1 is minus 2 over 3."},{"Start":"02:31.535 ","End":"02:36.320","Text":"Here, what we\u0027re left with after we\u0027ve taken this out is 2x,"},{"Start":"02:36.320 ","End":"02:38.735","Text":"x of minus 2 over 3,"},{"Start":"02:38.735 ","End":"02:43.760","Text":"so it looks like it\u0027s minus 4x squared over 3."},{"Start":"02:43.760 ","End":"02:47.270","Text":"Let\u0027s see what happens if we combine this."},{"Start":"02:47.270 ","End":"02:51.410","Text":"We get x squared minus 4/3x squared."},{"Start":"02:51.410 ","End":"02:58.180","Text":"Just this bit here is minus 1/3x squared."},{"Start":"02:58.180 ","End":"03:02.110","Text":"From here we get minus 1."},{"Start":"03:02.110 ","End":"03:07.205","Text":"Let\u0027s see if I take another 3 outside the brackets,"},{"Start":"03:07.205 ","End":"03:11.590","Text":"what I can get is 2 over 9,"},{"Start":"03:11.590 ","End":"03:14.885","Text":"I\u0027ll divide this by 3 and then I\u0027ll multiply here by 3,"},{"Start":"03:14.885 ","End":"03:16.640","Text":"so 2 over 9,"},{"Start":"03:16.640 ","End":"03:22.020","Text":"and I\u0027ll take the minus also outside the brackets because I have 2 negatives,"},{"Start":"03:22.020 ","End":"03:27.410","Text":"so let\u0027s make it minus 2/9 and then I can make this 3x squared plus 1."},{"Start":"03:27.410 ","End":"03:30.380","Text":"I\u0027ll write it in the numerator because I have"},{"Start":"03:30.380 ","End":"03:33.200","Text":"the intention of putting this in the denominator with"},{"Start":"03:33.200 ","End":"03:41.265","Text":"a positive exponent over x squared minus 1 to the power of 5 over 3."},{"Start":"03:41.265 ","End":"03:43.805","Text":"This is f double-prime."},{"Start":"03:43.805 ","End":"03:47.990","Text":"This is the end of this step, the preparation step."},{"Start":"03:47.990 ","End":"03:50.840","Text":"The next step is to take"},{"Start":"03:50.840 ","End":"03:56.735","Text":"f double prime of x and set it to 0 and see if we can solve for x."},{"Start":"03:56.735 ","End":"03:58.790","Text":"Actually, I\u0027d like to expand on this."},{"Start":"03:58.790 ","End":"04:03.710","Text":"There\u0027s really 2 kinds of x we\u0027re looking for: 1 is where the 2nd derivative is"},{"Start":"04:03.710 ","End":"04:10.055","Text":"0 and the other kind is where f double prime of x is undefined."},{"Start":"04:10.055 ","End":"04:13.505","Text":"Undefined, however, not just undefined,"},{"Start":"04:13.505 ","End":"04:15.290","Text":"but f of x should be defined,"},{"Start":"04:15.290 ","End":"04:18.400","Text":"but f of x is defined."},{"Start":"04:18.400 ","End":"04:21.890","Text":"Now, in our case, we have already,"},{"Start":"04:21.890 ","End":"04:26.735","Text":"I can tell you, we have some values of x where f double prime is undefined."},{"Start":"04:26.735 ","End":"04:28.685","Text":"If you look at the denominator,"},{"Start":"04:28.685 ","End":"04:31.680","Text":"if x squared minus 1 is 0,"},{"Start":"04:31.680 ","End":"04:34.590","Text":"then we\u0027re going to get 0 in the denominator."},{"Start":"04:34.590 ","End":"04:36.710","Text":"We certainly can get some of those,"},{"Start":"04:36.710 ","End":"04:39.575","Text":"but the function was defined everywhere."},{"Start":"04:39.575 ","End":"04:43.060","Text":"Let\u0027s see, for this value,"},{"Start":"04:43.060 ","End":"04:47.330","Text":"I can get x squared minus 1 equals 0,"},{"Start":"04:47.330 ","End":"04:50.750","Text":"x equals minus 1 or 1."},{"Start":"04:50.750 ","End":"04:53.750","Text":"For f double prime of x equals 0,"},{"Start":"04:53.750 ","End":"04:56.210","Text":"I have to get the numerator 0,"},{"Start":"04:56.210 ","End":"04:59.105","Text":"3x squared plus 1 equals 0,"},{"Start":"04:59.105 ","End":"05:04.550","Text":"but there\u0027s no such x because then something x squared would be something negative,"},{"Start":"05:04.550 ","End":"05:07.900","Text":"so there is no solution, no such x."},{"Start":"05:07.900 ","End":"05:09.410","Text":"We don\u0027t get anything from here,"},{"Start":"05:09.410 ","End":"05:11.060","Text":"but we\u0027ve got 2 points from here,"},{"Start":"05:11.060 ","End":"05:13.970","Text":"and these are also suspects for inflection."},{"Start":"05:13.970 ","End":"05:16.010","Text":"That\u0027s this step."},{"Start":"05:16.010 ","End":"05:22.820","Text":"The next step is the table and we\u0027re going to put the interesting values of x in here."},{"Start":"05:22.820 ","End":"05:25.015","Text":"At the moment we only have 2 of them,"},{"Start":"05:25.015 ","End":"05:28.525","Text":"and that is minus 1 and 1,"},{"Start":"05:28.525 ","End":"05:34.100","Text":"which are suspect because f double-prime here is undefined,"},{"Start":"05:34.100 ","End":"05:36.770","Text":"and here it\u0027s also undefined."},{"Start":"05:36.770 ","End":"05:39.800","Text":"We didn\u0027t get any with f double prime 0,"},{"Start":"05:39.800 ","End":"05:42.470","Text":"which we would have also thrown in as suspect."},{"Start":"05:42.470 ","End":"05:46.360","Text":"These 2 points divide the line into 3 intervals"},{"Start":"05:46.360 ","End":"05:50.975","Text":"and these intervals are x less than minus 1,"},{"Start":"05:50.975 ","End":"05:55.130","Text":"x between minus 1 and 1 and x bigger than 1,"},{"Start":"05:55.130 ","End":"05:57.575","Text":"we choose a sample points from each."},{"Start":"05:57.575 ","End":"06:00.500","Text":"From this, I\u0027ll choose minus 2, here,"},{"Start":"06:00.500 ","End":"06:02.060","Text":"I\u0027ll choose 0, here,"},{"Start":"06:02.060 ","End":"06:03.785","Text":"I\u0027ll choose minus 2."},{"Start":"06:03.785 ","End":"06:09.050","Text":"The idea is to substitute these in f double-prime but we only care about the sign,"},{"Start":"06:09.050 ","End":"06:11.165","Text":"we just want to write plus or minus."},{"Start":"06:11.165 ","End":"06:13.009","Text":"Let\u0027s take the first 1,"},{"Start":"06:13.009 ","End":"06:18.410","Text":"which is minus 1 and substitute it into f double-prime here."},{"Start":"06:18.410 ","End":"06:23.510","Text":"Before that, we should note that this is made up of 3 bits: the minus 2/9,"},{"Start":"06:23.510 ","End":"06:26.630","Text":"which is always negative, the denominator,"},{"Start":"06:26.630 ","End":"06:31.515","Text":"which here\u0027s where we could be positive or negative,"},{"Start":"06:31.515 ","End":"06:34.460","Text":"it all depends on if x squared minus 1 could be"},{"Start":"06:34.460 ","End":"06:37.800","Text":"negative or positive and then to the power of 5 over 3,"},{"Start":"06:37.800 ","End":"06:39.615","Text":"it would still be the same sign,"},{"Start":"06:39.615 ","End":"06:42.485","Text":"and then there\u0027s this numerator here which is always positive."},{"Start":"06:42.485 ","End":"06:44.389","Text":"We have a positive, a negative,"},{"Start":"06:44.389 ","End":"06:47.060","Text":"and an unknown depending on the x."},{"Start":"06:47.060 ","End":"06:48.890","Text":"When x is minus 2,"},{"Start":"06:48.890 ","End":"06:51.605","Text":"minus 2 squared is 4,"},{"Start":"06:51.605 ","End":"06:53.390","Text":"4 minus 1 is 3,"},{"Start":"06:53.390 ","End":"06:56.480","Text":"3 to this power is positive, so positive,"},{"Start":"06:56.480 ","End":"06:59.870","Text":"positive, negative, we get a negative."},{"Start":"06:59.870 ","End":"07:04.080","Text":"If x is 0, we have 0 minus 1 is negative,"},{"Start":"07:04.080 ","End":"07:08.325","Text":"minus 1 to the 5 over 3 is also minus 1 is negative,"},{"Start":"07:08.325 ","End":"07:11.670","Text":"so positive, negative, negative that makes it positive."},{"Start":"07:11.670 ","End":"07:17.210","Text":"For minus 2, minus 2 squared is 4, minus 1 is 3,"},{"Start":"07:17.210 ","End":"07:19.670","Text":"so this becomes positive, so you have positive,"},{"Start":"07:19.670 ","End":"07:23.700","Text":"positive, negative, and that makes it negative."},{"Start":"07:23.700 ","End":"07:30.365","Text":"This is interesting because although the f double prime is undefined here,"},{"Start":"07:30.365 ","End":"07:32.675","Text":"these points, both minus 1 and 1,"},{"Start":"07:32.675 ","End":"07:35.560","Text":"they lie in between concave."},{"Start":"07:35.560 ","End":"07:39.229","Text":"If so, if it\u0027s between concave and convex,"},{"Start":"07:39.229 ","End":"07:45.515","Text":"this point minus 1 is an inflection and so indeed is x equals 1,"},{"Start":"07:45.515 ","End":"07:48.815","Text":"that\u0027s also inflection, just missing 1 small thing."},{"Start":"07:48.815 ","End":"07:51.275","Text":"X equals minus 1 is not a complete point."},{"Start":"07:51.275 ","End":"07:53.825","Text":"I want to know what the y of this is."},{"Start":"07:53.825 ","End":"07:58.640","Text":"I just want to put in both minus 1 and 1 into y,"},{"Start":"07:58.640 ","End":"08:01.470","Text":"which is this thing here."},{"Start":"08:01.470 ","End":"08:04.880","Text":"If I put x equals 1 or minus 1,"},{"Start":"08:04.880 ","End":"08:06.980","Text":"x squared will give me 1,"},{"Start":"08:06.980 ","End":"08:08.570","Text":"1 minus 1 is 0,"},{"Start":"08:08.570 ","End":"08:10.465","Text":"cube root of 0 is 0,"},{"Start":"08:10.465 ","End":"08:13.670","Text":"so I will get 0 here and I will get 0 here."},{"Start":"08:13.670 ","End":"08:16.145","Text":"Now, I have all the information I need,"},{"Start":"08:16.145 ","End":"08:17.390","Text":"and this step is over,"},{"Start":"08:17.390 ","End":"08:20.030","Text":"so the next step is the conclusion phase."},{"Start":"08:20.030 ","End":"08:23.840","Text":"The conclusion I want to know just what was asked in the question."},{"Start":"08:23.840 ","End":"08:30.995","Text":"What are the inflection points and where is the function convex on which intervals,"},{"Start":"08:30.995 ","End":"08:33.410","Text":"and likewise for concave."},{"Start":"08:33.410 ","End":"08:35.015","Text":"We can answer this."},{"Start":"08:35.015 ","End":"08:38.780","Text":"We have 2 of those. We have 1 at minus 1,"},{"Start":"08:38.780 ","End":"08:43.555","Text":"0 and we have another 1 at 1, 0."},{"Start":"08:43.555 ","End":"08:46.650","Text":"As for convex, convex is this shape,"},{"Start":"08:46.650 ","End":"08:51.120","Text":"it\u0027s x between minus 1 and 1, and for concave,"},{"Start":"08:51.120 ","End":"08:55.955","Text":"we have 2 bits. We have x less than minus 1,"},{"Start":"08:55.955 ","End":"08:59.630","Text":"and we also have x bigger than 1."},{"Start":"08:59.630 ","End":"09:03.180","Text":"That answers the questions and we\u0027re done."}],"ID":5846},{"Watched":false,"Name":"Exercise 26","Duration":"8m 28s","ChapterTopicVideoID":5849,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5849.jpeg","UploadDate":"2016-05-01T17:30:21.9470000","DurationForVideoObject":"PT8M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.465","Text":"In this exercise, we\u0027re given a function f of x,"},{"Start":"00:03.465 ","End":"00:08.715","Text":"which is equal to the absolute value of x minus 3 over x minus 2."},{"Start":"00:08.715 ","End":"00:12.150","Text":"What we have to do is find the inflection points and"},{"Start":"00:12.150 ","End":"00:16.785","Text":"the intervals where the function is convex and where it\u0027s concave."},{"Start":"00:16.785 ","End":"00:21.960","Text":"This is a familiar type of exercise and we have a 4-step solution:"},{"Start":"00:21.960 ","End":"00:26.850","Text":"preparation step, 2 proper steps, and a conclusion to solve these."},{"Start":"00:26.850 ","End":"00:31.799","Text":"Just one thing we need to do first is to look at the domain of the function,"},{"Start":"00:31.799 ","End":"00:35.145","Text":"and we see that the domain of the function is"},{"Start":"00:35.145 ","End":"00:39.780","Text":"every x except x equals 2 because that would make the denominator 0."},{"Start":"00:39.780 ","End":"00:42.145","Text":"Now we\u0027ll do the 4 steps."},{"Start":"00:42.145 ","End":"00:48.770","Text":"The first preparation step is to compute f double prime of x, second derivative."},{"Start":"00:48.770 ","End":"00:53.120","Text":"I think that I\u0027d better write this as in split form."},{"Start":"00:53.120 ","End":"00:59.180","Text":"Split meaning it\u0027s also called piecewise-defined because of the absolute value."},{"Start":"00:59.180 ","End":"01:02.495","Text":"I\u0027ll write it as f of x is equal to."},{"Start":"01:02.495 ","End":"01:07.000","Text":"Now it\u0027s one thing for when this thing is positive or 0,"},{"Start":"01:07.000 ","End":"01:08.705","Text":"so it\u0027s just as it is."},{"Start":"01:08.705 ","End":"01:10.340","Text":"It\u0027s equal to x minus 3,"},{"Start":"01:10.340 ","End":"01:12.650","Text":"when x minus 3 is bigger or equal to 0,"},{"Start":"01:12.650 ","End":"01:16.160","Text":"which means that x is bigger or equal to 3,"},{"Start":"01:16.160 ","End":"01:20.690","Text":"and it\u0027s equal to the minus of that over the x minus 2,"},{"Start":"01:20.690 ","End":"01:22.120","Text":"and it\u0027s the minus of that,"},{"Start":"01:22.120 ","End":"01:26.180","Text":"so I\u0027ll write it as 3 minus x over x minus 2,"},{"Start":"01:26.180 ","End":"01:29.470","Text":"whenever x is less than 3."},{"Start":"01:29.470 ","End":"01:32.509","Text":"Now we can get to the derivative."},{"Start":"01:32.509 ","End":"01:35.960","Text":"But we have to be careful with this endpoint,"},{"Start":"01:35.960 ","End":"01:39.170","Text":"this x equals 3, because it\u0027s like a seam line."},{"Start":"01:39.170 ","End":"01:43.310","Text":"We can\u0027t just automatically differentiate the formula,"},{"Start":"01:43.310 ","End":"01:45.600","Text":"we have to check especially there."},{"Start":"01:45.600 ","End":"01:49.500","Text":"I have to write it as 3 possible values."},{"Start":"01:49.500 ","End":"01:55.960","Text":"I have to take into account both x bigger than 3, and x equals 3,"},{"Start":"01:55.960 ","End":"02:00.095","Text":"and the third possibility of x less than 3."},{"Start":"02:00.095 ","End":"02:02.300","Text":"Now, for x bigger than 3,"},{"Start":"02:02.300 ","End":"02:06.795","Text":"we just have to differentiate from this formula."},{"Start":"02:06.795 ","End":"02:11.660","Text":"I\u0027ll remind you of the quotient rule that"},{"Start":"02:11.660 ","End":"02:20.645","Text":"u/v derivative is u prime v minus uv prime over v^2,"},{"Start":"02:20.645 ","End":"02:26.090","Text":"and if I do that for x minus 3 over x minus 2,"},{"Start":"02:26.090 ","End":"02:31.399","Text":"I get derivative of this is 1 times x minus 2,"},{"Start":"02:31.399 ","End":"02:36.580","Text":"minus x minus 3 times 1,"},{"Start":"02:36.580 ","End":"02:40.425","Text":"over x minus 2 squared."},{"Start":"02:40.425 ","End":"02:45.525","Text":"This thing comes out x minus 2 less x minus 3 is just 1,"},{"Start":"02:45.525 ","End":"02:48.815","Text":"so it comes out to be 1 over x minus 2 squared."},{"Start":"02:48.815 ","End":"02:50.119","Text":"That\u0027s what I\u0027ll write here."},{"Start":"02:50.119 ","End":"02:54.645","Text":"1 over x minus 2 squared."},{"Start":"02:54.645 ","End":"02:57.045","Text":"Here, I\u0027ll leave it blank for a moment,"},{"Start":"02:57.045 ","End":"03:01.175","Text":"and for x less than 3, since this is just the minus of this,"},{"Start":"03:01.175 ","End":"03:02.795","Text":"we\u0027ll get the minus of this,"},{"Start":"03:02.795 ","End":"03:10.055","Text":"so it\u0027ll be minus 1 over x minus 2 squared."},{"Start":"03:10.055 ","End":"03:11.630","Text":"Now, as far as this goes,"},{"Start":"03:11.630 ","End":"03:14.015","Text":"I\u0027m not going to go too much into detail,"},{"Start":"03:14.015 ","End":"03:19.670","Text":"but if you take the limit as x goes to 3 from the right using this"},{"Start":"03:19.670 ","End":"03:21.350","Text":"or from the left using this,"},{"Start":"03:21.350 ","End":"03:23.210","Text":"you don\u0027t get the same value."},{"Start":"03:23.210 ","End":"03:24.650","Text":"In one case, you get 1,"},{"Start":"03:24.650 ","End":"03:26.465","Text":"in the other case, you get minus 1,"},{"Start":"03:26.465 ","End":"03:30.070","Text":"then it turns out that it doesn\u0027t exist here."},{"Start":"03:30.070 ","End":"03:33.680","Text":"It\u0027s undefined or there is no limit,"},{"Start":"03:33.680 ","End":"03:35.510","Text":"there is no derivative."},{"Start":"03:35.510 ","End":"03:37.370","Text":"If there\u0027s no first derivative,"},{"Start":"03:37.370 ","End":"03:41.630","Text":"there\u0027s certainly not going to be a second derivative here."},{"Start":"03:41.630 ","End":"03:47.074","Text":"In any event, we\u0027re done with the preparation step of computing f double prime,"},{"Start":"03:47.074 ","End":"03:49.910","Text":"and we\u0027ll just put a line here to indicate that."},{"Start":"03:49.910 ","End":"03:57.995","Text":"The next step is to see if we can find any solutions for f double prime of x equals 0"},{"Start":"03:57.995 ","End":"04:01.865","Text":"that will give us some suspects for inflection points."},{"Start":"04:01.865 ","End":"04:04.175","Text":"Unfortunately, if we look at it,"},{"Start":"04:04.175 ","End":"04:06.785","Text":"f double prime can never be 0."},{"Start":"04:06.785 ","End":"04:09.815","Text":"This is not 0, and this is not 0,"},{"Start":"04:09.815 ","End":"04:11.420","Text":"and undefined is not 0,"},{"Start":"04:11.420 ","End":"04:13.100","Text":"so this will never be 0,"},{"Start":"04:13.100 ","End":"04:15.710","Text":"so there are no solutions."},{"Start":"04:15.710 ","End":"04:21.255","Text":"That was one possible source of suspects for inflection,"},{"Start":"04:21.255 ","End":"04:23.900","Text":"but it\u0027s not the only source because you might"},{"Start":"04:23.900 ","End":"04:28.565","Text":"remember that we have a chance with this x equals 3,"},{"Start":"04:28.565 ","End":"04:32.240","Text":"because if the function is defined at 3"},{"Start":"04:32.240 ","End":"04:36.800","Text":"and the function we said was defined everywhere except x equals 2,"},{"Start":"04:36.800 ","End":"04:40.760","Text":"so there is certainly no problem in substituting x equals 3 here,"},{"Start":"04:40.760 ","End":"04:43.410","Text":"but it doesn\u0027t have a second derivative."},{"Start":"04:43.410 ","End":"04:45.680","Text":"That also makes it a candidate."},{"Start":"04:45.680 ","End":"04:48.300","Text":"The function is defined and the second derivative is not,"},{"Start":"04:48.300 ","End":"04:51.745","Text":"that also becomes a candidate for inflection,"},{"Start":"04:51.745 ","End":"04:53.770","Text":"and we put it in our table."},{"Start":"04:53.770 ","End":"04:55.730","Text":"We\u0027ll put x equals 3."},{"Start":"04:55.730 ","End":"05:00.409","Text":"It\u0027s a suspect because f is defined and f double prime is not."},{"Start":"05:00.409 ","End":"05:03.725","Text":"But we\u0027ll also put in x equals 2"},{"Start":"05:03.725 ","End":"05:06.900","Text":"because if we have an odd point where the function is not defined,"},{"Start":"05:06.900 ","End":"05:08.570","Text":"that also goes in the table."},{"Start":"05:08.570 ","End":"05:10.610","Text":"It\u0027s not a suspect for an inflection"},{"Start":"05:10.610 ","End":"05:15.865","Text":"but it still goes in the table because it splits up the intervals, the range."},{"Start":"05:15.865 ","End":"05:17.610","Text":"We put them in over there."},{"Start":"05:17.610 ","End":"05:19.890","Text":"Let\u0027s say, 2 is here,"},{"Start":"05:19.890 ","End":"05:21.870","Text":"and 3 is here,"},{"Start":"05:21.870 ","End":"05:28.505","Text":"and f double prime at x equals 3 is undefined."},{"Start":"05:28.505 ","End":"05:30.680","Text":"But with x equals 2,"},{"Start":"05:30.680 ","End":"05:32.480","Text":"it\u0027s not in the domain even"},{"Start":"05:32.480 ","End":"05:35.420","Text":"so I put dotted lines to indicate this."},{"Start":"05:35.420 ","End":"05:38.500","Text":"Now let\u0027s get onto the intervals."},{"Start":"05:38.500 ","End":"05:41.390","Text":"Here, we have x less than 2,"},{"Start":"05:41.390 ","End":"05:45.019","Text":"here, we have x between 2 and 3,"},{"Start":"05:45.019 ","End":"05:47.600","Text":"and here we have x bigger than 3,"},{"Start":"05:47.600 ","End":"05:50.180","Text":"and as usual we choose a sample point."},{"Start":"05:50.180 ","End":"05:52.900","Text":"I\u0027m going to choose x equals 1 here,"},{"Start":"05:52.900 ","End":"05:57.450","Text":"and I\u0027m going to choose x equals 2.5 here,"},{"Start":"05:57.450 ","End":"06:01.830","Text":"and I\u0027m going to choose x equals 4 over here."},{"Start":"06:01.830 ","End":"06:05.919","Text":"Then we have to put these values into f double prime,"},{"Start":"06:05.919 ","End":"06:10.595","Text":"and f double prime is what\u0027s in this box."},{"Start":"06:10.595 ","End":"06:12.416","Text":"But I don\u0027t need to know the actual value,"},{"Start":"06:12.416 ","End":"06:15.495","Text":"only whether it\u0027s plus or minus."},{"Start":"06:15.495 ","End":"06:17.595","Text":"I\u0027ve put in x equals 1."},{"Start":"06:17.595 ","End":"06:20.384","Text":"x equals 1 comes into this part,"},{"Start":"06:20.384 ","End":"06:25.650","Text":"and 1 minus 2 is negative 1"},{"Start":"06:25.650 ","End":"06:28.310","Text":"to the power of 3 is negative 1,"},{"Start":"06:28.310 ","End":"06:31.955","Text":"positive over negative is negative,"},{"Start":"06:31.955 ","End":"06:33.740","Text":"so here it\u0027s minus,"},{"Start":"06:33.740 ","End":"06:37.130","Text":"which means that the function is concave."},{"Start":"06:37.130 ","End":"06:40.840","Text":"If I put x equals 2.5,"},{"Start":"06:40.840 ","End":"06:43.368","Text":"I\u0027m still below 3."},{"Start":"06:43.368 ","End":"06:48.030","Text":"2.5 minus 2^3."},{"Start":"06:48.030 ","End":"06:50.415","Text":"This time it\u0027s positive,"},{"Start":"06:50.415 ","End":"06:53.210","Text":"so positive over positive is positive."},{"Start":"06:53.210 ","End":"06:57.245","Text":"Here we have a plus and that makes it convex,"},{"Start":"06:57.245 ","End":"07:05.150","Text":"and when x is 4, this time we go with x bigger than 3,"},{"Start":"07:05.150 ","End":"07:07.670","Text":"so 4 minus 2^3,"},{"Start":"07:07.670 ","End":"07:10.325","Text":"that\u0027s positive but there\u0027s a negative here,"},{"Start":"07:10.325 ","End":"07:16.160","Text":"so that makes it negative and that means it\u0027s also shaped like this."},{"Start":"07:16.160 ","End":"07:19.955","Text":"What we have here is concave, convex, concave,"},{"Start":"07:19.955 ","End":"07:23.480","Text":"but our suspect turns out to really"},{"Start":"07:23.480 ","End":"07:27.275","Text":"be an inflection point because it\u0027s between concave and convex."},{"Start":"07:27.275 ","End":"07:35.225","Text":"Here, I have an inflection and all I need now from the table is the value of y for this x,"},{"Start":"07:35.225 ","End":"07:39.335","Text":"and the value of y is just here."},{"Start":"07:39.335 ","End":"07:44.430","Text":"All we have to do is put x equals 3 in here"},{"Start":"07:44.430 ","End":"07:47.810","Text":"and I can see already that it\u0027s 0 from the numerator."},{"Start":"07:47.810 ","End":"07:51.520","Text":"Here we have a 0, and now we have everything we need in the table"},{"Start":"07:51.520 ","End":"07:53.735","Text":"so we can go onto the next step,"},{"Start":"07:53.735 ","End":"07:55.775","Text":"which is the conclusion phase,"},{"Start":"07:55.775 ","End":"07:59.420","Text":"where I write down all the things I wanted to find out."},{"Start":"07:59.420 ","End":"08:01.585","Text":"Start with the inflection."},{"Start":"08:01.585 ","End":"08:06.555","Text":"We have one and it\u0027s at the point 3, 0."},{"Start":"08:06.555 ","End":"08:08.873","Text":"Convex intervals."},{"Start":"08:08.873 ","End":"08:10.455","Text":"Convex is this,"},{"Start":"08:10.455 ","End":"08:14.415","Text":"so we have the x between 2 and 3 interval."},{"Start":"08:14.415 ","End":"08:17.375","Text":"Concave, we have 2 of them."},{"Start":"08:17.375 ","End":"08:23.660","Text":"We have x less than 2 and we also have x greater than 3."},{"Start":"08:23.660 ","End":"08:28.830","Text":"That answers the question, so we\u0027re done."}],"ID":5847},{"Watched":false,"Name":"Exercise 27","Duration":"4m 29s","ChapterTopicVideoID":5850,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5850.jpeg","UploadDate":"2016-05-01T17:30:51.4000000","DurationForVideoObject":"PT4M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this exercise, we\u0027re given the following function:"},{"Start":"00:03.480 ","End":"00:07.215","Text":"f of x equals x minus twice arctangent of x."},{"Start":"00:07.215 ","End":"00:09.705","Text":"We have to find it\u0027s inflection points"},{"Start":"00:09.705 ","End":"00:13.680","Text":"and the intervals where it\u0027s convex and where it\u0027s concave."},{"Start":"00:13.680 ","End":"00:16.830","Text":"This type of exercise is familiar and there is"},{"Start":"00:16.830 ","End":"00:22.590","Text":"a cookbook style 4-step solution to this kind of problem."},{"Start":"00:22.590 ","End":"00:30.000","Text":"We\u0027ll start it right away after we just note that the domain of this function is all x."},{"Start":"00:30.000 ","End":"00:34.644","Text":"The first step is the preparation phase where we have to find f double prime."},{"Start":"00:34.644 ","End":"00:36.260","Text":"To find f double prime,"},{"Start":"00:36.260 ","End":"00:38.405","Text":"we have to start with f prime."},{"Start":"00:38.405 ","End":"00:40.055","Text":"This is equal to,"},{"Start":"00:40.055 ","End":"00:43.880","Text":"the derivative of x is 1 minus twice."},{"Start":"00:43.880 ","End":"00:46.270","Text":"Now, the derivative of arctangent,"},{"Start":"00:46.270 ","End":"00:48.030","Text":"I\u0027ll just write it at the side,"},{"Start":"00:48.030 ","End":"00:54.435","Text":"the arctangent of x derivative is 1 over 1 plus x squared."},{"Start":"00:54.435 ","End":"01:01.380","Text":"Here we have 1 minus 2 over 1 plus x squared."},{"Start":"01:01.380 ","End":"01:04.110","Text":"We\u0027ll go on to f double prime."},{"Start":"01:04.110 ","End":"01:09.985","Text":"f double prime of x or equal to 1 goes to nothing."},{"Start":"01:09.985 ","End":"01:14.915","Text":"We can solve this with a quotient rule."},{"Start":"01:14.915 ","End":"01:17.770","Text":"I should write it down for you in case you forgotten it."},{"Start":"01:17.770 ","End":"01:26.734","Text":"u over v derivative is u prime v minus uv prime over v squared."},{"Start":"01:26.734 ","End":"01:33.140","Text":"If I use that over here with 2 being u and 1 plus x part being v,"},{"Start":"01:33.140 ","End":"01:37.310","Text":"I get the derivative of this is 0,"},{"Start":"01:37.310 ","End":"01:39.370","Text":"so it doesn\u0027t matter."},{"Start":"01:39.370 ","End":"01:43.235","Text":"The next term, the minus uv prime,"},{"Start":"01:43.235 ","End":"01:52.545","Text":"is minus 2 times the v prime is 2x over 1 plus x squared,"},{"Start":"01:52.545 ","End":"01:54.675","Text":"squared, which is v squared."},{"Start":"01:54.675 ","End":"01:56.655","Text":"Altogether, what do I get?"},{"Start":"01:56.655 ","End":"02:02.680","Text":"I get 4x over 1 plus x squared, squared."},{"Start":"02:02.680 ","End":"02:05.090","Text":"This ends the first step."},{"Start":"02:05.090 ","End":"02:08.660","Text":"Then the next step is to solve the equation:"},{"Start":"02:08.660 ","End":"02:11.765","Text":"f double prime of x is equal to 0."},{"Start":"02:11.765 ","End":"02:17.345","Text":"The idea is that the solutions will be suspect for inflection points."},{"Start":"02:17.345 ","End":"02:20.285","Text":"If we take a look at f double prime,"},{"Start":"02:20.285 ","End":"02:23.449","Text":"we see it\u0027s a fraction and the only way it\u0027s going to be 0,"},{"Start":"02:23.449 ","End":"02:25.520","Text":"is if the numerator is 0."},{"Start":"02:25.520 ","End":"02:32.565","Text":"This immediately gives us that the only solution is that x is equal to 0."},{"Start":"02:32.565 ","End":"02:35.435","Text":"This will go into the table,"},{"Start":"02:35.435 ","End":"02:36.995","Text":"which is the next step."},{"Start":"02:36.995 ","End":"02:43.255","Text":"The only value that\u0027s interesting that we can put in the table is x equals 0,"},{"Start":"02:43.255 ","End":"02:45.580","Text":"because at this point f double prime is 0,"},{"Start":"02:45.580 ","End":"02:47.540","Text":"and it might just be an inflection."},{"Start":"02:47.540 ","End":"02:50.825","Text":"It separates the line into 2 intervals,"},{"Start":"02:50.825 ","End":"02:56.645","Text":"the intervals x less than 0 and x greater than 0."},{"Start":"02:56.645 ","End":"03:00.680","Text":"We choose a sample point quite arbitrary in each interval."},{"Start":"03:00.680 ","End":"03:03.120","Text":"I\u0027ll choose mine as 1 here,"},{"Start":"03:03.120 ","End":"03:06.050","Text":"and I\u0027ll choose 1 here."},{"Start":"03:06.050 ","End":"03:11.750","Text":"I take these sample values and plug them into f double prime, which is this."},{"Start":"03:11.750 ","End":"03:13.580","Text":"But I don\u0027t want the actual value,"},{"Start":"03:13.580 ","End":"03:15.755","Text":"just whether it\u0027s positive or negative."},{"Start":"03:15.755 ","End":"03:17.390","Text":"Put in minus 1 here,"},{"Start":"03:17.390 ","End":"03:18.890","Text":"the denominator is positive,"},{"Start":"03:18.890 ","End":"03:21.755","Text":"the numerator\u0027s minus 4, negative."},{"Start":"03:21.755 ","End":"03:24.325","Text":"That makes this negative,"},{"Start":"03:24.325 ","End":"03:26.520","Text":"so f is 1 of these."},{"Start":"03:26.520 ","End":"03:28.475","Text":"If I put in x equals 1,"},{"Start":"03:28.475 ","End":"03:31.070","Text":"positive over positive is positive,"},{"Start":"03:31.070 ","End":"03:32.465","Text":"so I get 1 of these."},{"Start":"03:32.465 ","End":"03:35.720","Text":"This is the convex and this is the concave."},{"Start":"03:35.720 ","End":"03:38.045","Text":"Value of x, our suspect,"},{"Start":"03:38.045 ","End":"03:40.010","Text":"is indeed between the 2 types,"},{"Start":"03:40.010 ","End":"03:42.215","Text":"between concave and convex,"},{"Start":"03:42.215 ","End":"03:45.800","Text":"and that makes it an inflection point, so we have 1."},{"Start":"03:45.800 ","End":"03:49.715","Text":"For the inflection besides the x and also like the y of the point,"},{"Start":"03:49.715 ","End":"03:55.820","Text":"so I take this x equals 0 and put it in where y is,"},{"Start":"03:55.820 ","End":"04:01.500","Text":"which is this, so 0 is 0 and arctangent of 0 is 0."},{"Start":"04:01.500 ","End":"04:04.020","Text":"Altogether, y is 0."},{"Start":"04:04.020 ","End":"04:07.175","Text":"Now I have everything I need to answer the questions."},{"Start":"04:07.175 ","End":"04:09.035","Text":"This is the conclusion phase."},{"Start":"04:09.035 ","End":"04:11.720","Text":"Is that inflection points?"},{"Start":"04:11.720 ","End":"04:15.430","Text":"Yes, I do have one and it\u0027s at 0,0."},{"Start":"04:15.430 ","End":"04:19.005","Text":"Convex intervals, convex is this,"},{"Start":"04:19.005 ","End":"04:23.610","Text":"x bigger than 0, and concave is this."},{"Start":"04:23.610 ","End":"04:26.490","Text":"It\u0027s x less than 0."},{"Start":"04:26.490 ","End":"04:30.340","Text":"That answers all the questions, so we\u0027re done."}],"ID":5848},{"Watched":false,"Name":"Exercise 28","Duration":"9m 32s","ChapterTopicVideoID":5851,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5851.jpeg","UploadDate":"2016-05-01T17:31:55.8770000","DurationForVideoObject":"PT9M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we\u0027re given a function and we have to find its inflection points,"},{"Start":"00:04.950 ","End":"00:09.780","Text":"as well as the intervals where it\u0027s concave up and where it\u0027s concave down."},{"Start":"00:09.780 ","End":"00:13.155","Text":"The function happens to be a trigonometric function this time,"},{"Start":"00:13.155 ","End":"00:17.895","Text":"and it\u0027s usual to limit the domain because it\u0027s periodic,"},{"Start":"00:17.895 ","End":"00:22.080","Text":"and this time it\u0027s limited to the closed interval from 0-2Pi."},{"Start":"00:22.080 ","End":"00:26.535","Text":"This type of exercise is familiar and has a 4-stage solution."},{"Start":"00:26.535 ","End":"00:28.725","Text":"The 1st is the preparation step"},{"Start":"00:28.725 ","End":"00:32.775","Text":"and we have to find the 2nd derivative, f double prime."},{"Start":"00:32.775 ","End":"00:35.115","Text":"Of course, we start with the 1st derivative,"},{"Start":"00:35.115 ","End":"00:39.300","Text":"and we get that this is derivative of cosine is minus sine,"},{"Start":"00:39.300 ","End":"00:42.200","Text":"so minus 8 sine x,"},{"Start":"00:42.200 ","End":"00:46.300","Text":"and the derivative of cosine again is minus sine,"},{"Start":"00:46.300 ","End":"00:52.055","Text":"but this time we have to multiply by the inner derivative, which is 2."},{"Start":"00:52.055 ","End":"00:56.300","Text":"I\u0027m going to change this plus to a minus so that we"},{"Start":"00:56.300 ","End":"00:59.810","Text":"can also include the minus and the 2 from here,"},{"Start":"00:59.810 ","End":"01:03.525","Text":"so it\u0027s minus 4 sine 2x."},{"Start":"01:03.525 ","End":"01:05.900","Text":"Now that we\u0027ve got the 1st derivative,"},{"Start":"01:05.900 ","End":"01:09.710","Text":"we can go to the 2nd derivative. Let\u0027s see."},{"Start":"01:09.710 ","End":"01:12.620","Text":"What we have here is the derivative of sine is cosine,"},{"Start":"01:12.620 ","End":"01:19.145","Text":"so it\u0027s minus 8 cosine x derivative of sine again is cosine,"},{"Start":"01:19.145 ","End":"01:21.800","Text":"but we have the inner derivative which is 2,"},{"Start":"01:21.800 ","End":"01:26.665","Text":"so we also get minus 8 cosine of 2x,"},{"Start":"01:26.665 ","End":"01:30.170","Text":"and that\u0027s the end of the 1st step."},{"Start":"01:30.170 ","End":"01:31.805","Text":"We\u0027ve got the 2nd derivative."},{"Start":"01:31.805 ","End":"01:37.505","Text":"The next step is to solve the equation f double prime of x equals 0."},{"Start":"01:37.505 ","End":"01:40.370","Text":"When we find the solutions, the values of x,"},{"Start":"01:40.370 ","End":"01:43.310","Text":"they will be suspect for inflection points."},{"Start":"01:43.310 ","End":"01:44.600","Text":"Let\u0027s get started."},{"Start":"01:44.600 ","End":"01:54.360","Text":"First of all, we have the minus 8 cosine x minus 8 cosine 2x is equal to 0."},{"Start":"01:54.360 ","End":"01:57.080","Text":"If we divide by minus 8,"},{"Start":"01:57.080 ","End":"02:02.789","Text":"we can get the cosine x plus cosine 2x equals 0."},{"Start":"02:02.789 ","End":"02:06.098","Text":"I would like to remind you of the trigonometrical identity that"},{"Start":"02:06.098 ","End":"02:12.270","Text":"cosine 2x is 2 cosine squared x minus 1."},{"Start":"02:12.270 ","End":"02:17.885","Text":"If I put this instead of cosine 2x and I rearrange the order a bit,"},{"Start":"02:17.885 ","End":"02:26.990","Text":"I\u0027ll get that 2 cosine squared x plus cosine x minus 1 is equal to 0."},{"Start":"02:26.990 ","End":"02:30.620","Text":"This is a quadratic equation in cosine x."},{"Start":"02:30.620 ","End":"02:33.590","Text":"In fact, if I try solving it with a substitution,"},{"Start":"02:33.590 ","End":"02:36.120","Text":"let\u0027s say that t equals cosine x,"},{"Start":"02:36.120 ","End":"02:41.750","Text":"then I get a quadratic t squared plus t minus 1 equals 0,"},{"Start":"02:41.750 ","End":"02:44.375","Text":"and I can actually factor this out."},{"Start":"02:44.375 ","End":"02:48.080","Text":"It comes out to be t plus 1 2t minus 1,"},{"Start":"02:48.080 ","End":"02:53.955","Text":"which means that the solutions are that t equals minus 1 or 1/2,"},{"Start":"02:53.955 ","End":"02:56.000","Text":"which means that we have 2 possibilities."},{"Start":"02:56.000 ","End":"02:57.680","Text":"If we go back to cosine x,"},{"Start":"02:57.680 ","End":"03:06.995","Text":"we either have that cosine x is minus 1 or that cosine x is equal to 1/2."},{"Start":"03:06.995 ","End":"03:13.040","Text":"Remember that we\u0027re in the interval from 0-360 degrees or 0-2Pi that the only place"},{"Start":"03:13.040 ","End":"03:19.455","Text":"the cosine is minus 1 is when x equals Pi or 180 degrees,"},{"Start":"03:19.455 ","End":"03:22.730","Text":"and for the cosine x to be 1/2,"},{"Start":"03:22.730 ","End":"03:24.990","Text":"we have 2 solutions in general."},{"Start":"03:24.990 ","End":"03:29.170","Text":"It\u0027s cosine of 60 degrees is 1/2."},{"Start":"03:29.170 ","End":"03:30.830","Text":"If you do it on the calculator,"},{"Start":"03:30.830 ","End":"03:32.930","Text":"you can do shift cosine"},{"Start":"03:32.930 ","End":"03:37.184","Text":"and that should give you 60 degrees or Pi over 3,"},{"Start":"03:37.184 ","End":"03:41.104","Text":"but it\u0027s not the only solution, also 300 degrees."},{"Start":"03:41.104 ","End":"03:44.245","Text":"In other words, it would be 5Pi over 3."},{"Start":"03:44.245 ","End":"03:45.945","Text":"I\u0027ll just write those again."},{"Start":"03:45.945 ","End":"03:48.830","Text":"This is 180 degrees,"},{"Start":"03:48.830 ","End":"03:52.175","Text":"this is 60 degrees,"},{"Start":"03:52.175 ","End":"03:56.740","Text":"and this is 300 degrees for those who want it in degrees."},{"Start":"03:56.740 ","End":"03:59.550","Text":"Now we\u0027ve got the important points,"},{"Start":"03:59.550 ","End":"04:01.030","Text":"let\u0027s make a table,"},{"Start":"04:01.030 ","End":"04:06.360","Text":"and I\u0027m going to put in here the values that we got here,"},{"Start":"04:06.360 ","End":"04:08.795","Text":"but I\u0027m also going to put in the endpoints."},{"Start":"04:08.795 ","End":"04:12.245","Text":"Remember, we\u0027re going from 0-2Pi."},{"Start":"04:12.245 ","End":"04:14.840","Text":"First, we have 0, we have to put them in order,"},{"Start":"04:14.840 ","End":"04:17.470","Text":"then we have Pi over 3,"},{"Start":"04:17.470 ","End":"04:19.905","Text":"then I have Pi,"},{"Start":"04:19.905 ","End":"04:22.800","Text":"then 5Pi over 3,"},{"Start":"04:22.800 ","End":"04:26.235","Text":"and then the other endpoint, which is 2Pi."},{"Start":"04:26.235 ","End":"04:29.300","Text":"I\u0027ll just write that in degrees for those who want it."},{"Start":"04:29.300 ","End":"04:30.935","Text":"This is 0 degrees,"},{"Start":"04:30.935 ","End":"04:32.870","Text":"this is 60 degrees,180 degrees,"},{"Start":"04:32.870 ","End":"04:39.580","Text":"300 degrees, 360 degrees."},{"Start":"04:39.580 ","End":"04:47.145","Text":"We have that the 2nd derivative is 0 here, here, and here."},{"Start":"04:47.145 ","End":"04:53.810","Text":"Now, these points divide the interval from 0 to 2Pi into 3 separate intervals."},{"Start":"04:53.810 ","End":"04:55.385","Text":"I\u0027ll write down what they are."},{"Start":"04:55.385 ","End":"04:59.360","Text":"Here, I have 0 less than or equal to x,"},{"Start":"04:59.360 ","End":"05:02.060","Text":"which is less than Pi over 3."},{"Start":"05:02.060 ","End":"05:09.080","Text":"Here, I have Pi over 3 less than x less than Pi."},{"Start":"05:09.080 ","End":"05:15.005","Text":"Pi less than x less than 5Pi over 3."},{"Start":"05:15.005 ","End":"05:20.530","Text":"5Pi over 3 less than x less than or equal to 2Pi."},{"Start":"05:20.530 ","End":"05:23.480","Text":"We\u0027ve got to pick a sample point in each of these."},{"Start":"05:23.480 ","End":"05:26.530","Text":"Let me pick it in degrees."},{"Start":"05:26.530 ","End":"05:27.740","Text":"Let\u0027s say for this,"},{"Start":"05:27.740 ","End":"05:31.965","Text":"I\u0027ll choose 45 degrees or Pi over 4."},{"Start":"05:31.965 ","End":"05:36.890","Text":"Let\u0027s say for here between 60 and 180,"},{"Start":"05:36.890 ","End":"05:38.952","Text":"I\u0027ll choose 90 degrees;"},{"Start":"05:38.952 ","End":"05:43.425","Text":"here, I\u0027ll choose 270 degrees;"},{"Start":"05:43.425 ","End":"05:45.405","Text":"and here, I\u0027ll choose,"},{"Start":"05:45.405 ","End":"05:49.050","Text":"let\u0027s say, 315 degrees."},{"Start":"05:49.050 ","End":"05:54.720","Text":"Now, we\u0027re going to substitute this in f double prime. There it is."},{"Start":"05:54.720 ","End":"05:57.990","Text":"We don\u0027t actually want the value,"},{"Start":"05:57.990 ","End":"06:01.445","Text":"only the sign of whether it\u0027s plus or whether it\u0027s minus."},{"Start":"06:01.445 ","End":"06:04.505","Text":"Well, let\u0027s not waste time in the substitution."},{"Start":"06:04.505 ","End":"06:07.070","Text":"You should be able to do this with your calculator."},{"Start":"06:07.070 ","End":"06:10.170","Text":"This one comes out to be minus,"},{"Start":"06:10.170 ","End":"06:12.750","Text":"this one comes out to be plus,"},{"Start":"06:12.750 ","End":"06:15.120","Text":"this one comes out to be plus,"},{"Start":"06:15.120 ","End":"06:17.910","Text":"and this one comes out to be minus."},{"Start":"06:17.910 ","End":"06:22.810","Text":"One thing is clear that we have inflection points"},{"Start":"06:22.810 ","End":"06:25.130","Text":"here and here because an inflection is when it"},{"Start":"06:25.130 ","End":"06:28.055","Text":"changes from minus to plus or from plus to minus."},{"Start":"06:28.055 ","End":"06:30.890","Text":"I know that here, I have an inflection,"},{"Start":"06:30.890 ","End":"06:33.320","Text":"and here, I have an inflection,"},{"Start":"06:33.320 ","End":"06:39.890","Text":"and here, I don\u0027t have an inflection because it\u0027s between convex and convex."},{"Start":"06:39.890 ","End":"06:42.560","Text":"In actual fact, from another exercise,"},{"Start":"06:42.560 ","End":"06:45.920","Text":"I happen to know that this is in fact a minimum point,"},{"Start":"06:45.920 ","End":"06:50.095","Text":"but I\u0027ll just put that in brackets because I happen to remember it,"},{"Start":"06:50.095 ","End":"06:52.835","Text":"and that\u0027s not something I deduced here."},{"Start":"06:52.835 ","End":"06:58.040","Text":"Although you could check in the 1st degree if you put in Pi and f prime,"},{"Start":"06:58.040 ","End":"06:59.675","Text":"you will get 0."},{"Start":"06:59.675 ","End":"07:01.190","Text":"This is an inflection,"},{"Start":"07:01.190 ","End":"07:04.515","Text":"this is an inflection, and that\u0027s all,"},{"Start":"07:04.515 ","End":"07:08.045","Text":"except that I want the coordinates of the inflection points,"},{"Start":"07:08.045 ","End":"07:12.800","Text":"so I\u0027m also going to compute the value of y here and here."},{"Start":"07:12.800 ","End":"07:19.475","Text":"In which case, if I substitute Pi over 3 or 60 degrees,"},{"Start":"07:19.475 ","End":"07:24.635","Text":"I will get cosine of 60 is root 3 over 2."},{"Start":"07:24.635 ","End":"07:28.540","Text":"I\u0027ll just compute it for you and write you the answer."},{"Start":"07:28.540 ","End":"07:30.310","Text":"Here\u0027s what I make it."},{"Start":"07:30.310 ","End":"07:33.470","Text":"I make it that the answer here is"},{"Start":"07:33.470 ","End":"07:37.754","Text":"minus 5 root 3 minus 3,"},{"Start":"07:37.754 ","End":"07:39.905","Text":"and the same thing here,"},{"Start":"07:39.905 ","End":"07:44.355","Text":"minus 5 root 3 minus 3,"},{"Start":"07:44.355 ","End":"07:47.780","Text":"and here, I make it minus 9."},{"Start":"07:47.780 ","End":"07:50.765","Text":"The table is filled as much as I need,"},{"Start":"07:50.765 ","End":"07:53.810","Text":"and as far as the answers go,"},{"Start":"07:53.810 ","End":"07:59.160","Text":"I can summarize and we were asked about the inflection points,"},{"Start":"07:59.160 ","End":"08:04.010","Text":"so I can say that I have an inflection at 2 points,"},{"Start":"08:04.010 ","End":"08:07.494","Text":"the point Pi over 3,"},{"Start":"08:07.494 ","End":"08:12.060","Text":"minus 5 root 3 minus 3,"},{"Start":"08:12.060 ","End":"08:17.430","Text":"and another inflection point at 5Pi over 3,"},{"Start":"08:17.430 ","End":"08:21.915","Text":"and again, minus 5 root 3 minus 3."},{"Start":"08:21.915 ","End":"08:27.830","Text":"Now for the convex intervals where the function is convex or concave up,"},{"Start":"08:27.830 ","End":"08:32.089","Text":"that is, when the 2nd derivative is minus,"},{"Start":"08:32.089 ","End":"08:37.760","Text":"then that means that it\u0027s concave down and likewise here,"},{"Start":"08:37.760 ","End":"08:41.235","Text":"but here, it\u0027s convex or concave up,"},{"Start":"08:41.235 ","End":"08:42.590","Text":"so it is here."},{"Start":"08:42.590 ","End":"08:47.180","Text":"The convex, normally you would say there are 2 intervals, this and this."},{"Start":"08:47.180 ","End":"08:49.430","Text":"One is a single point in between and for"},{"Start":"08:49.430 ","End":"08:52.235","Text":"the point where the function is undefined or anything."},{"Start":"08:52.235 ","End":"08:53.900","Text":"We just join these 2 together;"},{"Start":"08:53.900 ","End":"08:57.310","Text":"1 odd point in a continuous function is not going to change."},{"Start":"08:57.310 ","End":"09:00.035","Text":"I can say that in all of this here,"},{"Start":"09:00.035 ","End":"09:05.384","Text":"these 2 combined together will give me from Pi over 3 to 5Pi over 3."},{"Start":"09:05.384 ","End":"09:07.020","Text":"Wrong color."},{"Start":"09:07.020 ","End":"09:10.820","Text":"There it is. These 2 combined and the concave"},{"Start":"09:10.820 ","End":"09:14.930","Text":"is where it\u0027s this shape, 2nd derivative negative,"},{"Start":"09:14.930 ","End":"09:18.286","Text":"and that gives us 2 separate intervals:"},{"Start":"09:18.286 ","End":"09:22.630","Text":"0 less than x less than Pi over 3,"},{"Start":"09:22.630 ","End":"09:24.155","Text":"as well as this one,"},{"Start":"09:24.155 ","End":"09:30.420","Text":"5Pi over 3 less than x less than 2Pi,"},{"Start":"09:30.420 ","End":"09:33.040","Text":"and we are done."}],"ID":5849},{"Watched":false,"Name":"Exercise 29","Duration":"8m 25s","ChapterTopicVideoID":5852,"CourseChapterTopicPlaylistID":65382,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5852.jpeg","UploadDate":"2016-05-01T17:32:54.7470000","DurationForVideoObject":"PT8M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.920","Text":"In this exercise, we\u0027re given a function f of x is e to"},{"Start":"00:04.920 ","End":"00:10.330","Text":"the minus x times sine x in a certain closed interval from minus Pi to 0."},{"Start":"00:10.330 ","End":"00:12.795","Text":"We have to find its inflection points,"},{"Start":"00:12.795 ","End":"00:17.400","Text":"as well as the intervals where the function is convex and where it\u0027s concave."},{"Start":"00:17.400 ","End":"00:22.200","Text":"As usual, this kind of exercise we solve in 4 stages."},{"Start":"00:22.200 ","End":"00:24.795","Text":"Stage 1 is the preparation step."},{"Start":"00:24.795 ","End":"00:27.915","Text":"We have to find f double-prime. Let\u0027s start."},{"Start":"00:27.915 ","End":"00:31.905","Text":"Of course, we have to start with f prime before we can get to f double prime."},{"Start":"00:31.905 ","End":"00:33.930","Text":"This equals by the product rule,"},{"Start":"00:33.930 ","End":"00:37.129","Text":"it\u0027s the derivative of e to the minus x,"},{"Start":"00:37.129 ","End":"00:40.550","Text":"which is minus e to the minus x and sine x untouched."},{"Start":"00:40.550 ","End":"00:45.079","Text":"Then the other way around e to the minus x untouched and sine x derived,"},{"Start":"00:45.079 ","End":"00:53.420","Text":"which is cosine of x. I could rewrite this as e to the minus x. I\u0027ll put the plus first,"},{"Start":"00:53.420 ","End":"00:57.350","Text":"cosine x minus sine x."},{"Start":"00:57.350 ","End":"00:59.395","Text":"We can now derive again,"},{"Start":"00:59.395 ","End":"01:01.020","Text":"second derivative."},{"Start":"01:01.020 ","End":"01:02.630","Text":"Again product rule."},{"Start":"01:02.630 ","End":"01:10.895","Text":"Derivative of the first minus e to the minus x times this as is cosine x"},{"Start":"01:10.895 ","End":"01:15.200","Text":"minus sine x plus e to the minus x"},{"Start":"01:15.200 ","End":"01:20.665","Text":"as is times the derivative of cosine x is minus sine x."},{"Start":"01:20.665 ","End":"01:25.460","Text":"The derivative of minus sine x is minus cosine x."},{"Start":"01:25.460 ","End":"01:30.485","Text":"Now, if we open up the brackets and take e to the minus x out,"},{"Start":"01:30.485 ","End":"01:37.175","Text":"we have minus cosine x plus sine x minus sine x minus cosine x."},{"Start":"01:37.175 ","End":"01:40.660","Text":"Basically this term cancels with this term."},{"Start":"01:40.660 ","End":"01:43.250","Text":"This is a minus minus and this is just a minus."},{"Start":"01:43.250 ","End":"01:47.030","Text":"What we\u0027re left with is minus 2 cosine x,"},{"Start":"01:47.030 ","End":"01:51.740","Text":"e to the minus x, minus 2 cosine x."},{"Start":"01:51.740 ","End":"01:53.930","Text":"That\u0027s our f double-prime,"},{"Start":"01:53.930 ","End":"01:57.860","Text":"and that\u0027s the end of our first preparation step."},{"Start":"01:57.860 ","End":"02:04.550","Text":"Second step is to solve an equation f double-prime of x equals 0 to solve for x,"},{"Start":"02:04.550 ","End":"02:08.240","Text":"these will be our suspects for inflection."},{"Start":"02:08.240 ","End":"02:11.660","Text":"If f double prime is 0,"},{"Start":"02:11.660 ","End":"02:18.785","Text":"then we get that minus 2e to the minus x cosine x is equal to 0."},{"Start":"02:18.785 ","End":"02:20.755","Text":"E to the power of is never 0,"},{"Start":"02:20.755 ","End":"02:22.225","Text":"minus 2 is never 0."},{"Start":"02:22.225 ","End":"02:26.299","Text":"It must be that the cosine x is equal to 0."},{"Start":"02:26.299 ","End":"02:32.345","Text":"But we must also remember that x is still got to be between minus Pi and 0."},{"Start":"02:32.345 ","End":"02:36.020","Text":"Now let\u0027s see where is cosine 0?"},{"Start":"02:36.020 ","End":"02:41.270","Text":"Cosine is 0 only when x is a multiple of x,"},{"Start":"02:41.270 ","End":"02:42.365","Text":"z the 0."},{"Start":"02:42.365 ","End":"02:44.060","Text":"Then every Pi."},{"Start":"02:44.060 ","End":"02:46.430","Text":"Let\u0027s work in degrees for a moment, it\u0027ll be simpler."},{"Start":"02:46.430 ","End":"02:49.910","Text":"If you check when cosine of something is 0,"},{"Start":"02:49.910 ","End":"02:52.715","Text":"you can do the arccosine of that something."},{"Start":"02:52.715 ","End":"02:54.320","Text":"The arccosine of something,"},{"Start":"02:54.320 ","End":"02:59.690","Text":"which you do as shift cosine should come out on the calculator as 90 degrees."},{"Start":"02:59.690 ","End":"03:04.445","Text":"Or you can just know it from the shape of the function that at 90 degrees it\u0027s 0."},{"Start":"03:04.445 ","End":"03:07.250","Text":"In fact, not only at 90 degrees,"},{"Start":"03:07.250 ","End":"03:11.175","Text":"but also at minus 90 or 270."},{"Start":"03:11.175 ","End":"03:14.105","Text":"In general, it has a period of 180 degrees."},{"Start":"03:14.105 ","End":"03:18.590","Text":"Every n times 180 degrees will also give us where cosine is 0."},{"Start":"03:18.590 ","End":"03:24.055","Text":"We want to restrict it to be between minus 180 and 0."},{"Start":"03:24.055 ","End":"03:28.355","Text":"The only value of n that could make it be in this interval is"},{"Start":"03:28.355 ","End":"03:32.690","Text":"n is minus 1 and then we get x equals minus 90 degrees."},{"Start":"03:32.690 ","End":"03:39.020","Text":"The only one that applies is x equals minus 90 degrees."},{"Start":"03:39.020 ","End":"03:42.155","Text":"In other words, if we go back to radians,"},{"Start":"03:42.155 ","End":"03:45.290","Text":"x equals minus Pi over 2."},{"Start":"03:45.290 ","End":"03:51.290","Text":"This is our suspect for an inflection in this interval we have."},{"Start":"03:51.290 ","End":"03:52.565","Text":"This is the end of the step,"},{"Start":"03:52.565 ","End":"03:55.989","Text":"and we next continue with the table."},{"Start":"03:55.989 ","End":"03:58.230","Text":"Here is our blank table."},{"Start":"03:58.230 ","End":"04:00.095","Text":"In the row for x,"},{"Start":"04:00.095 ","End":"04:02.150","Text":"I\u0027m going to put the endpoints first."},{"Start":"04:02.150 ","End":"04:03.510","Text":"It\u0027s minus Pi to 0."},{"Start":"04:03.510 ","End":"04:09.755","Text":"In the middle, we have minus Pi over 2 which is where f double prime is 0."},{"Start":"04:09.755 ","End":"04:17.900","Text":"Let\u0027s split the integral into 2 and the intervals are minus Pi less than x,"},{"Start":"04:17.900 ","End":"04:21.035","Text":"less than minus Pi over 2,"},{"Start":"04:21.035 ","End":"04:26.745","Text":"and also between minus Pi over 2 and 0."},{"Start":"04:26.745 ","End":"04:30.050","Text":"We pick a representative from each of these."},{"Start":"04:30.050 ","End":"04:34.260","Text":"To help us, I\u0027ll just write them in degrees again."},{"Start":"04:34.260 ","End":"04:37.790","Text":"Here we have minus 90 degrees,"},{"Start":"04:37.790 ","End":"04:40.414","Text":"here we have minus a 180 degrees,"},{"Start":"04:40.414 ","End":"04:42.475","Text":"and here we have just 0 degrees."},{"Start":"04:42.475 ","End":"04:46.520","Text":"What I\u0027m going to choose between minus 90 and 0,"},{"Start":"04:46.520 ","End":"04:49.895","Text":"I\u0027ll choose to say minus 45 degrees."},{"Start":"04:49.895 ","End":"04:54.270","Text":"What will I choose between minus 90 and minus 1?"},{"Start":"04:54.270 ","End":"04:58.280","Text":"I\u0027ll choose minus 135 degrees."},{"Start":"04:58.280 ","End":"05:04.065","Text":"But in radians, that\u0027s minus Pi over 4."},{"Start":"05:04.065 ","End":"05:07.930","Text":"This is minus 3 Pi over 4."},{"Start":"05:07.930 ","End":"05:13.205","Text":"What I want to do is to substitute these into the second derivative, which is this."},{"Start":"05:13.205 ","End":"05:17.585","Text":"But only to know whether I am positive or negative."},{"Start":"05:17.585 ","End":"05:21.725","Text":"If I take minus Pi over 4 or minus 45 degrees,"},{"Start":"05:21.725 ","End":"05:23.810","Text":"cosine being an even function,"},{"Start":"05:23.810 ","End":"05:27.860","Text":"cosine of minus 45 is the same as cosine of plus 45."},{"Start":"05:27.860 ","End":"05:31.780","Text":"It\u0027s 1 over the square root of 2, and that\u0027s positive."},{"Start":"05:31.780 ","End":"05:34.220","Text":"Here we have e to the anything is positive."},{"Start":"05:34.220 ","End":"05:35.780","Text":"All we have is this minus,"},{"Start":"05:35.780 ","End":"05:37.745","Text":"and this makes this negative."},{"Start":"05:37.745 ","End":"05:39.645","Text":"If we take minus 3 Pi over 2,"},{"Start":"05:39.645 ","End":"05:42.050","Text":"which is minus 135 degrees,"},{"Start":"05:42.050 ","End":"05:45.845","Text":"the cosine is the same as the cosine of plus 135,"},{"Start":"05:45.845 ","End":"05:49.730","Text":"and it\u0027s in the second quadrant, so negative."},{"Start":"05:49.730 ","End":"05:55.519","Text":"If it\u0027s negative, then with this negative it becomes positive."},{"Start":"05:55.519 ","End":"05:58.685","Text":"We\u0027ll be positive in this interval,"},{"Start":"05:58.685 ","End":"06:04.970","Text":"which means that here we are convex and here we are concave."},{"Start":"06:04.970 ","End":"06:07.940","Text":"Which means that this is"},{"Start":"06:07.940 ","End":"06:12.350","Text":"an inflection point because it\u0027s between concave and convex or vice versa."},{"Start":"06:12.350 ","End":"06:15.310","Text":"This is an inflection."},{"Start":"06:15.310 ","End":"06:18.410","Text":"That\u0027s got us, we\u0027ll figure out except that I want"},{"Start":"06:18.410 ","End":"06:21.200","Text":"to know the y coordinate of the inflection."},{"Start":"06:21.200 ","End":"06:26.045","Text":"I have to substitute minus Pi over 2 in the original function."},{"Start":"06:26.045 ","End":"06:30.220","Text":"Let\u0027s see now what was the original function?"},{"Start":"06:30.220 ","End":"06:33.845","Text":"I want to put in minus Pi over 2."},{"Start":"06:33.845 ","End":"06:36.695","Text":"Now, if you decide to use degrees,"},{"Start":"06:36.695 ","End":"06:39.685","Text":"you can only use degrees in the trigonometric part."},{"Start":"06:39.685 ","End":"06:42.965","Text":"The e to the power of can\u0027t use degrees."},{"Start":"06:42.965 ","End":"06:45.500","Text":"This is a mixed exercise when doing trig or not,"},{"Start":"06:45.500 ","End":"06:46.730","Text":"and you have to be very careful,"},{"Start":"06:46.730 ","End":"06:50.930","Text":"and recommend if you can\u0027t just don\u0027t use degrees in these questions."},{"Start":"06:50.930 ","End":"06:53.505","Text":"But anyway, the sine of minus Pi over 2,"},{"Start":"06:53.505 ","End":"06:54.740","Text":"and if you must think about it,"},{"Start":"06:54.740 ","End":"07:01.370","Text":"a sine of minus 90 is actually equal to minus 1."},{"Start":"07:01.370 ","End":"07:04.475","Text":"Here we have minus 1."},{"Start":"07:04.475 ","End":"07:06.950","Text":"We don\u0027t want the sine, we want the actual value."},{"Start":"07:06.950 ","End":"07:11.545","Text":"It\u0027s the sine of x of Pi over 2 is minus 1."},{"Start":"07:11.545 ","End":"07:18.110","Text":"That\u0027s minus and e to the minus x is e to the power of Pi over 2,"},{"Start":"07:18.110 ","End":"07:19.730","Text":"because this is x is minus,"},{"Start":"07:19.730 ","End":"07:23.670","Text":"so the minus x makes it plus Pi over 2."},{"Start":"07:23.670 ","End":"07:27.515","Text":"That\u0027s the x, y of the inflection,"},{"Start":"07:27.515 ","End":"07:30.950","Text":"and that finishes this phase with the table."},{"Start":"07:30.950 ","End":"07:33.290","Text":"All that\u0027s left is the conclusion phase where"},{"Start":"07:33.290 ","End":"07:35.660","Text":"we answer the questions that were originally asked."},{"Start":"07:35.660 ","End":"07:37.190","Text":"It asked for one thing,"},{"Start":"07:37.190 ","End":"07:40.545","Text":"where are the inflection points to find them all?"},{"Start":"07:40.545 ","End":"07:42.510","Text":"The answer there is only 1."},{"Start":"07:42.510 ","End":"07:46.890","Text":"It happens at the point minus Pi over 2,"},{"Start":"07:46.890 ","End":"07:51.915","Text":"e minus e to the Pi over 2."},{"Start":"07:51.915 ","End":"07:56.225","Text":"Then the intervals where the function is"},{"Start":"07:56.225 ","End":"08:02.600","Text":"convex is where it shaped like this plus second derivative."},{"Start":"08:02.600 ","End":"08:10.750","Text":"That is this interval minus Pi less than or equal to x less than minus Pi over 2."},{"Start":"08:10.750 ","End":"08:13.700","Text":"The concave is the other shape,"},{"Start":"08:13.700 ","End":"08:16.730","Text":"this one concave down and that occurs here,"},{"Start":"08:16.730 ","End":"08:21.605","Text":"which I\u0027ll copy, minus Pi over 2 less than x,"},{"Start":"08:21.605 ","End":"08:24.630","Text":"less than or equal to 0."}],"ID":5850}],"Thumbnail":null,"ID":65382},{"Name":"Vertical Asymptotes","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Vertical Asymptotes","Duration":"14m 10s","ChapterTopicVideoID":1849,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/1849.jpeg","UploadDate":"2019-10-29T12:54:36.9730000","DurationForVideoObject":"PT14M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.270","Text":"In this clip, I\u0027m going to talk about vertical asymptotes of a function."},{"Start":"00:06.270 ","End":"00:09.930","Text":"I\u0027ll begin right away with an example."},{"Start":"00:09.930 ","End":"00:15.195","Text":"You want to find vertical asymptotes for the function"},{"Start":"00:15.195 ","End":"00:21.540","Text":"y equals 4 over x minus 1."},{"Start":"00:21.540 ","End":"00:26.324","Text":"A function could have none or 1 or more vertical asymptotes."},{"Start":"00:26.324 ","End":"00:29.430","Text":"The place we look for a vertical asymptote is"},{"Start":"00:29.430 ","End":"00:33.705","Text":"a value of x where the function is not defined."},{"Start":"00:33.705 ","End":"00:38.295","Text":"In this case, there\u0027s only 1 such value and that is x equals 1."},{"Start":"00:38.295 ","End":"00:41.010","Text":"If there is any asymptote at all,"},{"Start":"00:41.010 ","End":"00:46.225","Text":"then it will be found at x equals 1."},{"Start":"00:46.225 ","End":"00:49.249","Text":"But there might not be an asymptote."},{"Start":"00:49.249 ","End":"00:53.600","Text":"Now what we do, to check if this is an asymptote,"},{"Start":"00:53.600 ","End":"00:57.815","Text":"is we check a couple of limits."},{"Start":"00:57.815 ","End":"01:07.730","Text":"We check the limit as x goes to 1 from the right."},{"Start":"01:07.730 ","End":"01:12.180","Text":"Let\u0027s call this f of x also,"},{"Start":"01:14.200 ","End":"01:18.130","Text":"of f of x."},{"Start":"01:18.130 ","End":"01:23.285","Text":"If this turns out to be infinity or minus infinity,"},{"Start":"01:23.285 ","End":"01:26.210","Text":"then we have a vertical asymptote."},{"Start":"01:26.210 ","End":"01:30.830","Text":"The vertical asymptote is the line x equals 1."},{"Start":"01:30.830 ","End":"01:33.020","Text":"Why is this a line?"},{"Start":"01:33.020 ","End":"01:35.930","Text":"Because if we look at the graph,"},{"Start":"01:35.930 ","End":"01:37.400","Text":"and I let, say,"},{"Start":"01:37.400 ","End":"01:39.905","Text":"this is the value 1,"},{"Start":"01:39.905 ","End":"01:47.130","Text":"then the vertical asymptote is the vertical line,"},{"Start":"01:47.500 ","End":"01:51.240","Text":"through the point x equals 1."},{"Start":"01:52.840 ","End":"01:56.010","Text":"It\u0027s a vertical line."},{"Start":"01:56.150 ","End":"02:00.980","Text":"As I say, if this turns out to be infinity or minus infinity,"},{"Start":"02:00.980 ","End":"02:03.590","Text":"then this is a vertical asymptote."},{"Start":"02:03.590 ","End":"02:05.510","Text":"But not only that,"},{"Start":"02:05.510 ","End":"02:07.130","Text":"we get another chance,"},{"Start":"02:07.130 ","End":"02:14.570","Text":"if we also check the limit as x goes to 1 from the left of f of x."},{"Start":"02:14.570 ","End":"02:20.210","Text":"If this is what I wanted to be as plus or minus infinity,"},{"Start":"02:20.210 ","End":"02:24.470","Text":"what I want this to be is infinity or minus infinity."},{"Start":"02:24.470 ","End":"02:26.660","Text":"If any 1 of these 4 holds,"},{"Start":"02:26.660 ","End":"02:28.100","Text":"if this limit is infinity,"},{"Start":"02:28.100 ","End":"02:29.570","Text":"or this limit is minus infinity,"},{"Start":"02:29.570 ","End":"02:30.620","Text":"or this limit is infinity,"},{"Start":"02:30.620 ","End":"02:32.195","Text":"or this limit is minus infinity,"},{"Start":"02:32.195 ","End":"02:33.785","Text":"or more than 1 of those,"},{"Start":"02:33.785 ","End":"02:37.500","Text":"then this line x equals 1,"},{"Start":"02:37.500 ","End":"02:40.275","Text":"becomes a vertical asymptote."},{"Start":"02:40.275 ","End":"02:43.265","Text":"Let\u0027s see what we have in our case."},{"Start":"02:43.265 ","End":"02:45.020","Text":"Let\u0027s check the top limit."},{"Start":"02:45.020 ","End":"02:51.050","Text":"I\u0027m just going to erase what I am expecting to find."},{"Start":"02:51.050 ","End":"02:55.470","Text":"I don\u0027t know that I will have this yet."},{"Start":"02:55.470 ","End":"02:57.260","Text":"Just bear in mind that\u0027s what we\u0027re looking for,"},{"Start":"02:57.260 ","End":"02:58.925","Text":"infinity or minus infinity,"},{"Start":"02:58.925 ","End":"03:01.370","Text":"and the limit as we go to this point,"},{"Start":"03:01.370 ","End":"03:03.230","Text":"but from either side."},{"Start":"03:03.230 ","End":"03:11.280","Text":"Let\u0027s try the limit as x goes to 1 from the right of f of x."},{"Start":"03:11.280 ","End":"03:16.680","Text":"If I substitute x,"},{"Start":"03:16.680 ","End":"03:18.780","Text":"I have to substitute x equals,"},{"Start":"03:18.780 ","End":"03:22.400","Text":"I can\u0027t substitute 1, but I can substitute something slightly"},{"Start":"03:22.400 ","End":"03:26.915","Text":"larger than 1, say, 1.000001."},{"Start":"03:26.915 ","End":"03:30.010","Text":"If x is slightly bigger than 1,"},{"Start":"03:30.010 ","End":"03:33.640","Text":"then what we have is,"},{"Start":"03:34.640 ","End":"03:40.370","Text":"it\u0027s 4 over, and something slightly bigger than 1 I\u0027ll just call it 1 plus."},{"Start":"03:40.370 ","End":"03:44.670","Text":"I think of it as 1 point a lot of 0s and a 1 or something."},{"Start":"03:45.160 ","End":"03:48.185","Text":"Now, something slightly bigger than 1,"},{"Start":"03:48.185 ","End":"03:50.030","Text":"tiniest bigger than 1."},{"Start":"03:50.030 ","End":"03:53.795","Text":"Less 1, is the tiniest bigger than 0."},{"Start":"03:53.795 ","End":"03:57.470","Text":"In other words, 1 plus minus 1 is just 0 plus,"},{"Start":"03:57.470 ","End":"04:00.325","Text":"which is tiny as positive number."},{"Start":"04:00.325 ","End":"04:04.560","Text":"If something is very tiny and positive then 4 over it,"},{"Start":"04:04.560 ","End":"04:06.035","Text":"is going to be huge."},{"Start":"04:06.035 ","End":"04:08.600","Text":"It\u0027s going to be, in fact, this is infinity,"},{"Start":"04:08.600 ","End":"04:11.600","Text":"plus, and I\u0027m just writing plus infinity for emphasis."},{"Start":"04:11.600 ","End":"04:15.090","Text":"But if this was 0 point 6 0s and a 1,"},{"Start":"04:15.090 ","End":"04:16.470","Text":"this would be a 1,000,000,"},{"Start":"04:16.470 ","End":"04:18.540","Text":"and we could get large as we liked."},{"Start":"04:18.540 ","End":"04:21.130","Text":"It goes up unbounded."},{"Start":"04:21.170 ","End":"04:26.000","Text":"That already confirms x equals 1 as an asymptote."},{"Start":"04:26.000 ","End":"04:33.825","Text":"What it also tells us that if we go along here to 1 from the right,"},{"Start":"04:33.825 ","End":"04:37.220","Text":"the function, I don\u0027t how it starts out,"},{"Start":"04:37.220 ","End":"04:39.770","Text":"but it ends up getting very large,"},{"Start":"04:39.770 ","End":"04:41.749","Text":"going to plus infinity."},{"Start":"04:41.749 ","End":"04:43.595","Text":"This goes up to infinity,"},{"Start":"04:43.595 ","End":"04:46.800","Text":"but never passes this line."},{"Start":"04:47.140 ","End":"04:49.965","Text":"Now, out of curiosity,"},{"Start":"04:49.965 ","End":"04:52.945","Text":"and maybe I also want to find out more about what the graph looks like."},{"Start":"04:52.945 ","End":"04:54.280","Text":"I\u0027ll also do the second 1,"},{"Start":"04:54.280 ","End":"04:58.720","Text":"although we\u0027ve already established that x equals 1 is an asymptote, I\u0027m curious."},{"Start":"04:58.720 ","End":"05:01.565","Text":"Let\u0027s see if x goes to 1 from the right,"},{"Start":"05:01.565 ","End":"05:05.199","Text":"basically, I substitute x equals 1 minus,"},{"Start":"05:05.199 ","End":"05:12.685","Text":"by which I mean that x in my head I think of has 0.9999999, a lot of 9s."},{"Start":"05:12.685 ","End":"05:19.270","Text":"1 minus this 0.999, less 1,"},{"Start":"05:19.270 ","End":"05:26.530","Text":"is equal to something slightly negative."},{"Start":"05:26.530 ","End":"05:29.955","Text":"I\u0027ll get minus point 0.00001."},{"Start":"05:29.955 ","End":"05:34.455","Text":"This slightly negative, and 4 over it,"},{"Start":"05:34.455 ","End":"05:36.180","Text":"is going to be minus infinity,"},{"Start":"05:36.180 ","End":"05:39.450","Text":"because 4 over minus 0 point whatever number of"},{"Start":"05:39.450 ","End":"05:43.760","Text":"0s and a 1 is going to be minus several million."},{"Start":"05:43.760 ","End":"05:49.485","Text":"Then you write this as 4 over 0 minus which is minus infinity."},{"Start":"05:49.485 ","End":"05:53.060","Text":"In fact, I also have some idea of what\u0027s happening on the left side."},{"Start":"05:53.060 ","End":"05:55.925","Text":"The function, I don\u0027t know how it starts out,"},{"Start":"05:55.925 ","End":"06:02.885","Text":"but I do know that as we get closer to 1,"},{"Start":"06:02.885 ","End":"06:08.960","Text":"here, the function gets more and more negative unbounded,"},{"Start":"06:08.960 ","End":"06:13.070","Text":"and this actually goes to minus infinity."},{"Start":"06:13.070 ","End":"06:15.955","Text":"This is an asymptote on both sides,"},{"Start":"06:15.955 ","End":"06:18.600","Text":"but sometimes you can get just a 1-sided asymptote,"},{"Start":"06:18.600 ","End":"06:20.065","Text":"and it\u0027s still an asymptote,"},{"Start":"06:20.065 ","End":"06:22.760","Text":"a vertical 1, that is."},{"Start":"06:23.690 ","End":"06:27.640","Text":"Hope this explains the concept."},{"Start":"06:27.860 ","End":"06:35.530","Text":"I think we ought to write a slightly more formal definition of vertical asymptote."},{"Start":"06:35.530 ","End":"06:40.574","Text":"I\u0027ll just cordon off this area here,"},{"Start":"06:40.574 ","End":"06:42.070","Text":"and in it I\u0027ll write"},{"Start":"06:42.070 ","End":"06:44.270","Text":"the definition"},{"Start":"06:49.880 ","End":"06:59.040","Text":"that the line x equals a,"},{"Start":"06:59.040 ","End":"07:00.330","Text":"a is some constant,"},{"Start":"07:00.330 ","End":"07:02.275","Text":"in this case, 1."},{"Start":"07:02.275 ","End":"07:04.970","Text":"The line x equals a is"},{"Start":"07:04.970 ","End":"07:08.730","Text":"a vertical asymptote"},{"Start":"07:13.240 ","End":"07:15.320","Text":"to the function,"},{"Start":"07:15.320 ","End":"07:23.185","Text":"f of"},{"Start":"07:23.185 ","End":"07:30.320","Text":"x if"},{"Start":"07:30.320 ","End":"07:36.240","Text":"the following holds, that the limit."}],"ID":1827},{"Watched":false,"Name":"Exercise 1","Duration":"26s","ChapterTopicVideoID":5853,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5853.jpeg","UploadDate":"2016-05-02T06:49:02.1970000","DurationForVideoObject":"PT26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.575","Text":"We\u0027re given this function,"},{"Start":"00:01.575 ","End":"00:04.590","Text":"f of x equals x squared minus 2x plus 5,"},{"Start":"00:04.590 ","End":"00:08.055","Text":"and we have to find its vertical asymptotes if any."},{"Start":"00:08.055 ","End":"00:12.030","Text":"Well, this function is a well-behaved function."},{"Start":"00:12.030 ","End":"00:16.095","Text":"It\u0027s defined everywhere for all x, it\u0027s even continuous."},{"Start":"00:16.095 ","End":"00:19.380","Text":"You can\u0027t expect it to have any asymptotes."},{"Start":"00:19.380 ","End":"00:23.430","Text":"The solution simply is, there are none."},{"Start":"00:23.430 ","End":"00:27.100","Text":"There are no asymptotes, and that\u0027s it."}],"ID":5851},{"Watched":false,"Name":"Exercise 2","Duration":"28s","ChapterTopicVideoID":5854,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5854.jpeg","UploadDate":"2016-05-02T06:49:04.8730000","DurationForVideoObject":"PT28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.445","Text":"In this exercise, we have to find"},{"Start":"00:02.445 ","End":"00:07.830","Text":"the vertical asymptotes of the function x cubed minus 3x."},{"Start":"00:07.830 ","End":"00:11.265","Text":"Well, this is a function which is a polynomial."},{"Start":"00:11.265 ","End":"00:13.980","Text":"Any event it\u0027s defined for all x,"},{"Start":"00:13.980 ","End":"00:16.695","Text":"it\u0027s well-behaved, it\u0027s even continuous,"},{"Start":"00:16.695 ","End":"00:18.585","Text":"it doesn\u0027t have any asymptotes."},{"Start":"00:18.585 ","End":"00:23.540","Text":"In fact, I can tell you more that all polynomials don\u0027t have asymptotes."},{"Start":"00:23.540 ","End":"00:26.160","Text":"To the solution, I\u0027ll just write the word none,"},{"Start":"00:26.160 ","End":"00:29.710","Text":"there are no asymptotes, and we\u0027re done."}],"ID":5852},{"Watched":false,"Name":"Exercise 3","Duration":"26s","ChapterTopicVideoID":5855,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5855.jpeg","UploadDate":"2016-05-02T06:49:07.5300000","DurationForVideoObject":"PT26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we\u0027re asked to find the asymptotes of the function f"},{"Start":"00:04.650 ","End":"00:09.045","Text":"of x is 2x cubed minus 15x squared plus 24x minus 1."},{"Start":"00:09.045 ","End":"00:10.995","Text":"This function is a polynomial,"},{"Start":"00:10.995 ","End":"00:12.570","Text":"it\u0027s defined for all x,"},{"Start":"00:12.570 ","End":"00:15.120","Text":"it\u0027s well-behaved, it\u0027s even continuous,"},{"Start":"00:15.120 ","End":"00:16.950","Text":"and so there are no asymptotes."},{"Start":"00:16.950 ","End":"00:20.805","Text":"In fact, all polynomials don\u0027t have asymptotes."},{"Start":"00:20.805 ","End":"00:23.185","Text":"I\u0027ll just settle for the word, \"NONE\"."},{"Start":"00:23.185 ","End":"00:26.890","Text":"There are no asymptotes here. We\u0027re done."}],"ID":5853},{"Watched":false,"Name":"Exercise 4","Duration":"26s","ChapterTopicVideoID":5856,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5856.jpeg","UploadDate":"2016-05-02T06:49:10.1830000","DurationForVideoObject":"PT26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"We\u0027re asked to find the vertical asymptotes of f of x,"},{"Start":"00:04.350 ","End":"00:07.170","Text":"which is x to the 4th minus 2x cubed."},{"Start":"00:07.170 ","End":"00:09.870","Text":"This function is a polynomial."},{"Start":"00:09.870 ","End":"00:12.075","Text":"It\u0027s defined for all x."},{"Start":"00:12.075 ","End":"00:15.210","Text":"It\u0027s even continuous, very well-behaved."},{"Start":"00:15.210 ","End":"00:17.895","Text":"It does not have any vertical asymptotes."},{"Start":"00:17.895 ","End":"00:20.350","Text":"I\u0027m just going to write the word none,"},{"Start":"00:20.350 ","End":"00:21.570","Text":"and I can tell you that,"},{"Start":"00:21.570 ","End":"00:27.280","Text":"in general, polynomials do not have vertical asymptotes. That\u0027s it."}],"ID":5854},{"Watched":false,"Name":"Exercise 5","Duration":"28s","ChapterTopicVideoID":5857,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5857.jpeg","UploadDate":"2016-05-02T06:49:13.0730000","DurationForVideoObject":"PT28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.740","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of f of x,"},{"Start":"00:04.740 ","End":"00:07.950","Text":"which is 3x^5 minus 20x cubed."},{"Start":"00:07.950 ","End":"00:10.455","Text":"Notice that this is just a polynomial,"},{"Start":"00:10.455 ","End":"00:14.550","Text":"and it\u0027s defined for all x, well behaved,"},{"Start":"00:14.550 ","End":"00:18.630","Text":"even continuous, and so it doesn\u0027t have any vertical asymptotes,"},{"Start":"00:18.630 ","End":"00:21.150","Text":"and I\u0027m just going to write the word none,"},{"Start":"00:21.150 ","End":"00:22.635","Text":"there are no asymptotes."},{"Start":"00:22.635 ","End":"00:24.630","Text":"I can tell you that in general,"},{"Start":"00:24.630 ","End":"00:27.510","Text":"polynomials do not have vertical asymptotes."},{"Start":"00:27.510 ","End":"00:29.350","Text":"We\u0027re done."}],"ID":5855},{"Watched":false,"Name":"Exercise 6","Duration":"30s","ChapterTopicVideoID":5858,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5858.jpeg","UploadDate":"2016-05-02T06:49:16.1830000","DurationForVideoObject":"PT30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of f of x,"},{"Start":"00:04.560 ","End":"00:07.635","Text":"which is x over x squared plus 3."},{"Start":"00:07.635 ","End":"00:12.555","Text":"Notice that it\u0027s defined for all x. x squared plus 3 can never be 0,"},{"Start":"00:12.555 ","End":"00:17.145","Text":"because x squared is at least 0 and plus 3 is certainly positive,"},{"Start":"00:17.145 ","End":"00:20.730","Text":"so there are no problems with f of x and it\u0027s even continuous."},{"Start":"00:20.730 ","End":"00:22.260","Text":"Since it\u0027s so well-behaved,"},{"Start":"00:22.260 ","End":"00:24.825","Text":"it can\u0027t have any vertical asymptotes."},{"Start":"00:24.825 ","End":"00:27.795","Text":"I\u0027ll just write the answer as none."},{"Start":"00:27.795 ","End":"00:31.660","Text":"There are no vertical asymptotes. We\u0027re done."}],"ID":5856},{"Watched":false,"Name":"Exercise 7","Duration":"1m 22s","ChapterTopicVideoID":5859,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5859.jpeg","UploadDate":"2016-05-02T06:49:24.5700000","DurationForVideoObject":"PT1M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.650 ","End":"00:08.055","Text":"which is x over x squared plus x plus 1."},{"Start":"00:08.055 ","End":"00:12.300","Text":"The only place we could possibly find an asymptote is somewhere where"},{"Start":"00:12.300 ","End":"00:16.560","Text":"the denominator is 0 and then the function wouldn\u0027t be defined."},{"Start":"00:16.560 ","End":"00:19.680","Text":"Let\u0027s see if the denominator can be 0."},{"Start":"00:19.680 ","End":"00:27.240","Text":"We try and solve x squared plus x plus 1 equals 0 and see if we can find any solutions."},{"Start":"00:27.240 ","End":"00:30.045","Text":"I can tell you right away that the answer is no."},{"Start":"00:30.045 ","End":"00:32.235","Text":"You could try solving it the usual way."},{"Start":"00:32.235 ","End":"00:34.589","Text":"But if you studied quadratic equations,"},{"Start":"00:34.589 ","End":"00:37.950","Text":"you may remember that there is a concept called Delta,"},{"Start":"00:37.950 ","End":"00:41.230","Text":"which is b squared minus 4ac."},{"Start":"00:41.230 ","End":"00:46.340","Text":"We\u0027re talking about the general case where ax squared plus bx plus c equals 0."},{"Start":"00:46.340 ","End":"00:49.010","Text":"It turns out that if Delta is negative,"},{"Start":"00:49.010 ","End":"00:50.405","Text":"there are no solutions."},{"Start":"00:50.405 ","End":"00:54.365","Text":"If Delta is less than 0, then no solution."},{"Start":"00:54.365 ","End":"00:57.335","Text":"This is what happens in our case because in our case,"},{"Start":"00:57.335 ","End":"00:59.135","Text":"a, b, and c are all 1 here."},{"Start":"00:59.135 ","End":"01:05.645","Text":"So Delta is b squared minus 4ac is 1 squared minus 4 times 1 times 1,"},{"Start":"01:05.645 ","End":"01:08.660","Text":"which is negative 3, which is less than 0."},{"Start":"01:08.660 ","End":"01:10.744","Text":"When this happens, there\u0027s no solution."},{"Start":"01:10.744 ","End":"01:14.460","Text":"Everything is okay with numerator and denominator,"},{"Start":"01:14.460 ","End":"01:16.395","Text":"x is defined everywhere,"},{"Start":"01:16.395 ","End":"01:23.220","Text":"to even continuous, so we have no vertical asymptotes. We\u0027re done."}],"ID":5857},{"Watched":false,"Name":"Exercise 8","Duration":"2m 18s","ChapterTopicVideoID":5860,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5860.jpeg","UploadDate":"2016-05-02T06:49:38.6930000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.599","Text":"In this exercise, we\u0027re asked to find the vertical asymptote"},{"Start":"00:03.599 ","End":"00:07.200","Text":"of the function x minus 1 over x squared."},{"Start":"00:07.200 ","End":"00:08.475","Text":"Now, looking at it,"},{"Start":"00:08.475 ","End":"00:10.469","Text":"I\u0027m looking for trouble, basically,"},{"Start":"00:10.469 ","End":"00:13.340","Text":"somewhat by the functions not defined, for example."},{"Start":"00:13.340 ","End":"00:17.010","Text":"There\u0027s only 1 such place and that\u0027s where the denominator is 0."},{"Start":"00:17.010 ","End":"00:22.320","Text":"In other words, x equals 0 is our only suspect for a vertical asymptote."},{"Start":"00:22.320 ","End":"00:23.895","Text":"So I\u0027ll just write that meanwhile,"},{"Start":"00:23.895 ","End":"00:26.845","Text":"it\u0027s just a suspect and we have to check it."},{"Start":"00:26.845 ","End":"00:30.770","Text":"Now, a vertical asymptote has the property that the limit,"},{"Start":"00:30.770 ","End":"00:35.175","Text":"when x goes to this point, in this case 0,"},{"Start":"00:35.175 ","End":"00:40.100","Text":"from the left or from the right and we\u0027ll take it from the right first of f of x."},{"Start":"00:40.100 ","End":"00:44.105","Text":"If this is equal to plus infinity or minus infinity,"},{"Start":"00:44.105 ","End":"00:47.070","Text":"then we\u0027ll have an asymptote and likewise for the left."},{"Start":"00:47.070 ","End":"00:51.320","Text":"Let\u0027s take this, this is the limit as x goes to 0."},{"Start":"00:51.320 ","End":"00:54.350","Text":"I\u0027m starting off with the right and if we don\u0027t find anything will go to"},{"Start":"00:54.350 ","End":"00:59.000","Text":"the left of x minus 1 over x squared."},{"Start":"00:59.000 ","End":"01:03.650","Text":"Now, essentially what we get here is when x goes to 0 from either direction,"},{"Start":"01:03.650 ","End":"01:06.650","Text":"x minus 1 goes to minus 1,"},{"Start":"01:06.650 ","End":"01:10.460","Text":"and x squared because it\u0027s squared, it\u0027s always positive."},{"Start":"01:10.460 ","End":"01:13.940","Text":"So 0 plus squared is also 0 plus something very,"},{"Start":"01:13.940 ","End":"01:15.200","Text":"very, very, very small."},{"Start":"01:15.200 ","End":"01:16.820","Text":"But if you square it, it\u0027s still very,"},{"Start":"01:16.820 ","End":"01:18.679","Text":"very, very small but positive."},{"Start":"01:18.679 ","End":"01:21.500","Text":"So we get minus 1 over 0 plus,"},{"Start":"01:21.500 ","End":"01:23.300","Text":"and so this is actually equal to"},{"Start":"01:23.300 ","End":"01:27.065","Text":"minus infinity because we have minus or negative over positive."},{"Start":"01:27.065 ","End":"01:28.490","Text":"When this is very, very small,"},{"Start":"01:28.490 ","End":"01:29.735","Text":"the magnitude is large,"},{"Start":"01:29.735 ","End":"01:30.800","Text":"but it\u0027s still negative."},{"Start":"01:30.800 ","End":"01:32.635","Text":"So this is minus infinity."},{"Start":"01:32.635 ","End":"01:34.310","Text":"So the answer is yes,"},{"Start":"01:34.310 ","End":"01:36.350","Text":"x equals 0 is an asymptote,"},{"Start":"01:36.350 ","End":"01:39.785","Text":"but I just want to show you that it\u0027s actually a double-sided asymptote."},{"Start":"01:39.785 ","End":"01:41.510","Text":"I mean, we could stop here and say yes,"},{"Start":"01:41.510 ","End":"01:43.220","Text":"but I want to show you that also,"},{"Start":"01:43.220 ","End":"01:47.150","Text":"if x goes to 0 from the left of the same thing,"},{"Start":"01:47.150 ","End":"01:51.665","Text":"we get exactly the same because even if x is 0 minus,"},{"Start":"01:51.665 ","End":"01:53.630","Text":"x squared is also 0 plus,"},{"Start":"01:53.630 ","End":"01:55.039","Text":"I mean something very tiny,"},{"Start":"01:55.039 ","End":"02:00.005","Text":"minute and negative when you square to something minute and tiny and positive."},{"Start":"02:00.005 ","End":"02:02.640","Text":"We also get minus infinity."},{"Start":"02:02.640 ","End":"02:07.865","Text":"It\u0027s a double-sided asymptote even with minus infinity on both sides."},{"Start":"02:07.865 ","End":"02:12.425","Text":"Anyway, the answer to the question is that x equals 0"},{"Start":"02:12.425 ","End":"02:18.160","Text":"is a vertical asymptote and that\u0027s all. So we\u0027re done."}],"ID":5858},{"Watched":false,"Name":"Exercise 9","Duration":"2m 11s","ChapterTopicVideoID":5861,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5861.jpeg","UploadDate":"2016-05-02T06:49:52.7870000","DurationForVideoObject":"PT2M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this exercise, we have to find the vertical asymptotes of"},{"Start":"00:04.170 ","End":"00:08.370","Text":"the function 2x squared over x plus 1 squared."},{"Start":"00:08.370 ","End":"00:12.030","Text":"Basically, when I look for asymptotes I\u0027m basically looking for trouble."},{"Start":"00:12.030 ","End":"00:13.980","Text":"I\u0027m looking for where something goes wrong,"},{"Start":"00:13.980 ","End":"00:16.140","Text":"mainly where the function\u0027s not defined."},{"Start":"00:16.140 ","End":"00:17.790","Text":"Here if you examine it,"},{"Start":"00:17.790 ","End":"00:21.075","Text":"you\u0027ll see that trouble can happen when the denominator is 0."},{"Start":"00:21.075 ","End":"00:24.644","Text":"Let\u0027s see what happens if we let the denominator equals 0."},{"Start":"00:24.644 ","End":"00:27.060","Text":"This only has 1 solution,"},{"Start":"00:27.060 ","End":"00:29.820","Text":"and that is that x equals minus 1."},{"Start":"00:29.820 ","End":"00:32.835","Text":"This is going to be our suspect for an asymptote."},{"Start":"00:32.835 ","End":"00:34.485","Text":"What we do is check the limit."},{"Start":"00:34.485 ","End":"00:36.389","Text":"Let\u0027s try the limit on the right."},{"Start":"00:36.389 ","End":"00:41.090","Text":"We take the limit as x goes to minus 1,"},{"Start":"00:41.090 ","End":"00:47.835","Text":"but from the right of 2x squared over x plus 1 squared."},{"Start":"00:47.835 ","End":"00:53.880","Text":"What this equals is on the numerator we can substitute minus 1 so twice"},{"Start":"00:53.880 ","End":"01:00.105","Text":"minus 1 squared is just positive 2 and on the denominator,"},{"Start":"01:00.105 ","End":"01:06.570","Text":"minus 1 plus 1 gives us 0 plus 0 squared which is"},{"Start":"01:06.570 ","End":"01:14.540","Text":"positive 0 or the tiny positive number squared so it\u0027s 2 over positive 0,"},{"Start":"01:14.540 ","End":"01:16.265","Text":"so it\u0027s plus infinity."},{"Start":"01:16.265 ","End":"01:18.275","Text":"I\u0027m just writing the plus for emphasis."},{"Start":"01:18.275 ","End":"01:20.940","Text":"It doesn\u0027t matter though whether it was plus or minus infinity,"},{"Start":"01:20.940 ","End":"01:22.175","Text":"we have an asymptote."},{"Start":"01:22.175 ","End":"01:27.110","Text":"This is a confirmed asymptote and we could end the question here,"},{"Start":"01:27.110 ","End":"01:28.610","Text":"but I\u0027d like to be more complete."},{"Start":"01:28.610 ","End":"01:31.905","Text":"Let\u0027s take a look and see what happens at 1 from the left?"},{"Start":"01:31.905 ","End":"01:35.270","Text":"If x goes to minus 1 from the left,"},{"Start":"01:35.270 ","End":"01:36.830","Text":"we get the same thing,"},{"Start":"01:36.830 ","End":"01:39.680","Text":"pretty much of the same thing, ditto,"},{"Start":"01:39.680 ","End":"01:43.340","Text":"the only difference is that we get 0 minus squared,"},{"Start":"01:43.340 ","End":"01:45.815","Text":"but it still comes out to positive 0."},{"Start":"01:45.815 ","End":"01:49.290","Text":"2 over positive 0 will also be plus infinity,"},{"Start":"01:49.290 ","End":"01:52.760","Text":"so we have infinity on both sides and this"},{"Start":"01:52.760 ","End":"01:56.330","Text":"is actually a 2-sided asymptote but we could have ended here because"},{"Start":"01:56.330 ","End":"02:03.245","Text":"the 1-sided asymptote is still fully an asymptote so the answer is that the line x equals"},{"Start":"02:03.245 ","End":"02:07.790","Text":"minus 1 is a vertical asymptote"},{"Start":"02:07.790 ","End":"02:12.750","Text":"and it\u0027s the only 1 for this particular function. We\u0027re done."}],"ID":5859},{"Watched":false,"Name":"Exercise 10","Duration":"2m 41s","ChapterTopicVideoID":5862,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5862.jpeg","UploadDate":"2016-05-02T06:50:09.1470000","DurationForVideoObject":"PT2M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Here, we\u0027re asked to find the vertical asymptotes of"},{"Start":"00:03.210 ","End":"00:07.515","Text":"the function x cubed over x plus 1 squared."},{"Start":"00:07.515 ","End":"00:09.960","Text":"Here we have a polynomial over a polynomial,"},{"Start":"00:09.960 ","End":"00:11.710","Text":"otherwise known as a rational function,"},{"Start":"00:11.710 ","End":"00:16.740","Text":"and the only place we can get an asymptote is where the denominator is 0,"},{"Start":"00:16.740 ","End":"00:18.885","Text":"then the function is not defined."},{"Start":"00:18.885 ","End":"00:23.295","Text":"Basically, we just have to see when the denominator is 0,"},{"Start":"00:23.295 ","End":"00:27.329","Text":"we have the equation x plus 1 squared equals 0,"},{"Start":"00:27.329 ","End":"00:30.330","Text":"and that clearly gives us just 1 possibility,"},{"Start":"00:30.330 ","End":"00:33.420","Text":"and that is that x is equal to minus 1."},{"Start":"00:33.420 ","End":"00:36.060","Text":"This will be our suspect for an asymptote."},{"Start":"00:36.060 ","End":"00:40.205","Text":"The way we check for an asymptote is by considering the limit"},{"Start":"00:40.205 ","End":"00:44.825","Text":"of the function as x goes towards this value on the right, or on the left."},{"Start":"00:44.825 ","End":"00:47.360","Text":"Let\u0027s try the limit on the right first."},{"Start":"00:47.360 ","End":"00:52.535","Text":"We\u0027ll take x goes to minus 1 from the right of f of x,"},{"Start":"00:52.535 ","End":"00:56.750","Text":"which is x cubed over x plus 1 squared,"},{"Start":"00:56.750 ","End":"00:58.895","Text":"and if this is infinity or minus infinity,"},{"Start":"00:58.895 ","End":"01:01.175","Text":"then this confirms it as an asymptote."},{"Start":"01:01.175 ","End":"01:02.600","Text":"If we don\u0027t succeed here,"},{"Start":"01:02.600 ","End":"01:04.700","Text":"we can also try the limit on the left."},{"Start":"01:04.700 ","End":"01:07.690","Text":"It\u0027s enough for 1 of them to be infinity or minus infinity."},{"Start":"01:07.690 ","End":"01:11.330","Text":"Here, by substituting x equals minus 1,"},{"Start":"01:11.330 ","End":"01:14.300","Text":"we get on the numerator minus 1 cubed,"},{"Start":"01:14.300 ","End":"01:17.520","Text":"which is I\u0027ll just leave it as minus 1 cubed,"},{"Start":"01:17.520 ","End":"01:19.040","Text":"and on the denominator,"},{"Start":"01:19.040 ","End":"01:22.320","Text":"x plus 1 is just 0 plus,"},{"Start":"01:22.320 ","End":"01:24.080","Text":"with symbolic for a tiny, tiny,"},{"Start":"01:24.080 ","End":"01:27.380","Text":"tiny quantity, but positive squared."},{"Start":"01:27.380 ","End":"01:28.730","Text":"Now, if we compute this,"},{"Start":"01:28.730 ","End":"01:30.620","Text":"we get this is minus 1,"},{"Start":"01:30.620 ","End":"01:33.560","Text":"and this is positive 0."},{"Start":"01:33.560 ","End":"01:36.050","Text":"Then, though it\u0027s symbolic, there is no such number as 0 plus,"},{"Start":"01:36.050 ","End":"01:37.100","Text":"but we use it symbolically,"},{"Start":"01:37.100 ","End":"01:40.310","Text":"and we know that 1 over 0 plus is infinity,"},{"Start":"01:40.310 ","End":"01:42.920","Text":"but minus 1 would make it minus infinity,"},{"Start":"01:42.920 ","End":"01:44.390","Text":"because it\u0027s minus over plus,"},{"Start":"01:44.390 ","End":"01:45.800","Text":"so we have minus infinity."},{"Start":"01:45.800 ","End":"01:49.010","Text":"This confirms x equals 1 as an asymptote,"},{"Start":"01:49.010 ","End":"01:50.885","Text":"but I\u0027d like to, just for the practice,"},{"Start":"01:50.885 ","End":"01:55.910","Text":"also compute the limit on the other side and see if we have a double-sided asymptote,"},{"Start":"01:55.910 ","End":"01:57.260","Text":"which is also an asymptote."},{"Start":"01:57.260 ","End":"02:01.430","Text":"Let\u0027s try, goes to minus 1 from the left of the same thing."},{"Start":"02:01.430 ","End":"02:04.445","Text":"We just basically get everything the same,"},{"Start":"02:04.445 ","End":"02:09.395","Text":"except that here we get 0 minus a negative 0,"},{"Start":"02:09.395 ","End":"02:13.160","Text":"but a negative 0 squared is also a positive 0,"},{"Start":"02:13.160 ","End":"02:16.835","Text":"so we also get minus 1 over positive 0,"},{"Start":"02:16.835 ","End":"02:18.665","Text":"and again, minus infinity."},{"Start":"02:18.665 ","End":"02:21.065","Text":"The limits on the left and on the right,"},{"Start":"02:21.065 ","End":"02:24.065","Text":"when x goes to minus 1 is minus infinity,"},{"Start":"02:24.065 ","End":"02:26.270","Text":"so we have a 2-sided asymptote."},{"Start":"02:26.270 ","End":"02:27.650","Text":"I don\u0027t know if that\u0027s worth more,"},{"Start":"02:27.650 ","End":"02:29.120","Text":"or if it\u0027s more of an asymptote,"},{"Start":"02:29.120 ","End":"02:30.785","Text":"it\u0027s not just an asymptote."},{"Start":"02:30.785 ","End":"02:38.940","Text":"The answer is that x equals minus 1 is a vertical asymptote,"},{"Start":"02:38.940 ","End":"02:41.770","Text":"and we are done."}],"ID":5860},{"Watched":false,"Name":"Exercise 11","Duration":"2m 40s","ChapterTopicVideoID":5863,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5863.jpeg","UploadDate":"2016-05-02T06:50:26.6230000","DurationForVideoObject":"PT2M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"Here we\u0027re asked to find the vertical asymptotes of"},{"Start":"00:03.870 ","End":"00:08.085","Text":"the function x plus 1 over x minus 1 all cubed."},{"Start":"00:08.085 ","End":"00:09.435","Text":"Copied it over here,"},{"Start":"00:09.435 ","End":"00:12.420","Text":"and I\u0027d like to expand it using the laws of algebra."},{"Start":"00:12.420 ","End":"00:14.235","Text":"Which law am I talking about?"},{"Start":"00:14.235 ","End":"00:19.890","Text":"A over b to the power of n is a to the n over b to the n."},{"Start":"00:19.890 ","End":"00:26.070","Text":"Here, we have x plus 1 cubed over x minus 1 cubed."},{"Start":"00:26.070 ","End":"00:30.075","Text":"What we have here is a rational function, polynomial over polynomial."},{"Start":"00:30.075 ","End":"00:32.250","Text":"I won\u0027t bother expanding it, no need."},{"Start":"00:32.250 ","End":"00:34.970","Text":"For rational functions, the only place to"},{"Start":"00:34.970 ","End":"00:38.125","Text":"look for an asymptote is with a denominator is 0."},{"Start":"00:38.125 ","End":"00:43.240","Text":"I\u0027m going to see if I can find where x minus 1 cubed equals 0."},{"Start":"00:43.240 ","End":"00:47.615","Text":"It\u0027s quite obvious that there\u0027s only one solution and that is that x equals 1."},{"Start":"00:47.615 ","End":"00:52.445","Text":"Now, I\u0027ll check the suspect as to whether this is indeed an asymptote."},{"Start":"00:52.445 ","End":"00:54.650","Text":"I\u0027ll check one of the limits,"},{"Start":"00:54.650 ","End":"00:56.270","Text":"either the right or the left,"},{"Start":"00:56.270 ","End":"01:01.340","Text":"or try the limit on the right of the function and see what we get."},{"Start":"01:01.340 ","End":"01:03.200","Text":"If I look at the function,"},{"Start":"01:03.200 ","End":"01:05.675","Text":"let\u0027s say, I\u0027ll use this form of the function."},{"Start":"01:05.675 ","End":"01:08.504","Text":"Then I get x plus 1 cubed,"},{"Start":"01:08.504 ","End":"01:12.055","Text":"2 cubed over x minus 1,"},{"Start":"01:12.055 ","End":"01:15.710","Text":"1 plus minus 1 gives me 0 plus"},{"Start":"01:15.710 ","End":"01:20.765","Text":"and it\u0027s something very tiny minute infinitesimal but positive cubed."},{"Start":"01:20.765 ","End":"01:26.105","Text":"Altogether, I\u0027ve got positive I\u0027d say, 8 over 0 plus."},{"Start":"01:26.105 ","End":"01:29.930","Text":"This is 8, cube is still positive and that gives us"},{"Start":"01:29.930 ","End":"01:34.160","Text":"plus infinity because it\u0027s positive over positive, so it\u0027s infinity."},{"Start":"01:34.160 ","End":"01:39.255","Text":"That means that x equals 1 is indeed an asymptote."},{"Start":"01:39.255 ","End":"01:40.970","Text":"We could end here and say, yes,"},{"Start":"01:40.970 ","End":"01:43.730","Text":"we have an asymptote at the line x equals 1."},{"Start":"01:43.730 ","End":"01:47.915","Text":"But I want to be more thorough and let\u0027s also check the limit on the left."},{"Start":"01:47.915 ","End":"01:50.270","Text":"If x goes to 1 from the left,"},{"Start":"01:50.270 ","End":"01:52.715","Text":"we get the same thing on the top,"},{"Start":"01:52.715 ","End":"01:55.340","Text":"1 plus 1 cubed, that\u0027s 2 cubed."},{"Start":"01:55.340 ","End":"01:58.340","Text":"But if we take 1 minus,"},{"Start":"01:58.340 ","End":"02:00.680","Text":"which means tiny, tiny bit less than 1,"},{"Start":"02:00.680 ","End":"02:03.200","Text":"and we subtract 1 from it,"},{"Start":"02:03.200 ","End":"02:07.670","Text":"then we get something very close to 0 but negative and that"},{"Start":"02:07.670 ","End":"02:12.409","Text":"equals 8 over minus times minus times minus is minus,"},{"Start":"02:12.409 ","End":"02:14.780","Text":"so we still 0 minus,"},{"Start":"02:14.780 ","End":"02:17.900","Text":"and this time we get negative infinity."},{"Start":"02:17.900 ","End":"02:21.365","Text":"That would also make the line x equals 1 an asymptote,"},{"Start":"02:21.365 ","End":"02:23.825","Text":"so in fact it\u0027s a two-sided asymptote,"},{"Start":"02:23.825 ","End":"02:25.400","Text":"but on one side it\u0027s infinity,"},{"Start":"02:25.400 ","End":"02:27.544","Text":"on the other side it\u0027s minus infinity."},{"Start":"02:27.544 ","End":"02:29.630","Text":"In any event, I want a full answer,"},{"Start":"02:29.630 ","End":"02:34.265","Text":"I would write the line x equals 1 is"},{"Start":"02:34.265 ","End":"02:40.440","Text":"a vertical asymptote and that would be a full answer and we\u0027re done."}],"ID":5861},{"Watched":false,"Name":"Exercise 12","Duration":"1m 57s","ChapterTopicVideoID":5864,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5864.jpeg","UploadDate":"2016-05-02T06:50:39.0470000","DurationForVideoObject":"PT1M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.040","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of the function f of x,"},{"Start":"00:05.040 ","End":"00:07.785","Text":"which is x minus 1 over x cubed."},{"Start":"00:07.785 ","End":"00:11.430","Text":"This is a rational function polynomial over polynomial."},{"Start":"00:11.430 ","End":"00:15.825","Text":"The only place to look for an asymptote is where the denominator is 0."},{"Start":"00:15.825 ","End":"00:20.560","Text":"The denominator is x cubed and we want to see where this is equal to 0."},{"Start":"00:20.560 ","End":"00:24.120","Text":"Well, quite obviously there\u0027s only 1 place and that is x equals 0."},{"Start":"00:24.120 ","End":"00:26.190","Text":"This is our suspect for an asymptote."},{"Start":"00:26.190 ","End":"00:26.670","Text":"So let\u0027s check."},{"Start":"00:26.670 ","End":"00:32.760","Text":"We need to check if 1 of the limits as x goes to 0 of f of x,"},{"Start":"00:32.760 ","End":"00:34.515","Text":"is plus or minus infinity."},{"Start":"00:34.515 ","End":"00:37.180","Text":"Let\u0027s try the limit from the right first,"},{"Start":"00:37.180 ","End":"00:40.130","Text":"x goes to 0 plus of f of x,"},{"Start":"00:40.130 ","End":"00:42.935","Text":"which is x minus 1 over x cubed."},{"Start":"00:42.935 ","End":"00:47.075","Text":"This is equal to 0 minus 1 is minus 1,"},{"Start":"00:47.075 ","End":"00:52.200","Text":"and 0 plus cubed is just 0 plus,"},{"Start":"00:52.200 ","End":"00:54.990","Text":"because positive x positive x positive is"},{"Start":"00:54.990 ","End":"01:00.320","Text":"positive and minus 1 over 0 plus is minus infinity."},{"Start":"01:00.320 ","End":"01:04.850","Text":"Yes, we have plus or minus infinity specifically minus on the right,"},{"Start":"01:04.850 ","End":"01:06.845","Text":"and this already makes it an asymptote,"},{"Start":"01:06.845 ","End":"01:10.760","Text":"but I\u0027d like to just be thorough and see if we also have a 2 sided asymptote."},{"Start":"01:10.760 ","End":"01:16.580","Text":"I\u0027m going to also check the limit as x goes to 0 from the left of the function f of x."},{"Start":"01:16.580 ","End":"01:21.830","Text":"This time, I also get minus 1 here but on the denominator,"},{"Start":"01:21.830 ","End":"01:23.390","Text":"since it\u0027s the power of 3,"},{"Start":"01:23.390 ","End":"01:25.340","Text":"the x cubed we have minus, minus,"},{"Start":"01:25.340 ","End":"01:29.210","Text":"minus, so 0 minus cubed is also 0 minus."},{"Start":"01:29.210 ","End":"01:34.115","Text":"It\u0027s very, very tiny, but still negative because 3 negatives multiplied give a negative."},{"Start":"01:34.115 ","End":"01:35.990","Text":"Negative over negative is positive,"},{"Start":"01:35.990 ","End":"01:37.955","Text":"and here we have plus infinity,"},{"Start":"01:37.955 ","End":"01:40.220","Text":"so again, it\u0027s an asymptote both on the right."},{"Start":"01:40.220 ","End":"01:42.965","Text":"On the left is in fact a 2 sided asymptote."},{"Start":"01:42.965 ","End":"01:45.200","Text":"I won\u0027t bother with the complete sentence."},{"Start":"01:45.200 ","End":"01:49.055","Text":"We can say the line x equals 0,"},{"Start":"01:49.055 ","End":"01:52.100","Text":"this vertical line is an asymptote and"},{"Start":"01:52.100 ","End":"01:55.070","Text":"the only asymptote for the graph of this function."},{"Start":"01:55.070 ","End":"01:57.240","Text":"That\u0027s the answer."}],"ID":5862},{"Watched":false,"Name":"Exercise 13","Duration":"29s","ChapterTopicVideoID":5865,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5865.jpeg","UploadDate":"2016-05-02T06:50:42.1470000","DurationForVideoObject":"PT29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.860","Text":"In this exercise, we\u0027re asked to find"},{"Start":"00:01.860 ","End":"00:06.210","Text":"the vertical asymptotes of the function x minus e_x."},{"Start":"00:06.210 ","End":"00:08.670","Text":"This function is defined everywhere,"},{"Start":"00:08.670 ","End":"00:10.065","Text":"x is defined everywhere,"},{"Start":"00:10.065 ","End":"00:12.075","Text":"e_ x is defined everywhere."},{"Start":"00:12.075 ","End":"00:13.530","Text":"It\u0027s even continuous."},{"Start":"00:13.530 ","End":"00:15.090","Text":"Nothing wrong with this function."},{"Start":"00:15.090 ","End":"00:16.800","Text":"Nowhere to look for an asymptote."},{"Start":"00:16.800 ","End":"00:19.770","Text":"I\u0027ll just write that there are no asymptotes."},{"Start":"00:19.770 ","End":"00:24.000","Text":"There\u0027s no suspect, no denominator 0 or anything like that."},{"Start":"00:24.000 ","End":"00:26.985","Text":"No vertical asymptotes at any rate."},{"Start":"00:26.985 ","End":"00:29.260","Text":"We\u0027re done."}],"ID":5863},{"Watched":false,"Name":"Exercise 14","Duration":"1m 34s","ChapterTopicVideoID":5866,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5866.jpeg","UploadDate":"2016-05-02T06:50:52.4130000","DurationForVideoObject":"PT1M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.065","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes,"},{"Start":"00:04.065 ","End":"00:05.490","Text":"the function f of x,"},{"Start":"00:05.490 ","End":"00:08.370","Text":"which is e to the 1 over x. I\u0027ve copied it here."},{"Start":"00:08.370 ","End":"00:12.870","Text":"If we look at it, the only thing that\u0027s suspicious or that could go wrong to"},{"Start":"00:12.870 ","End":"00:15.375","Text":"make f undefined would be when x is 0"},{"Start":"00:15.375 ","End":"00:18.270","Text":"because then we get a 0 in the denominator."},{"Start":"00:18.270 ","End":"00:25.260","Text":"X equals 0 is our suspect to be an asymptote and now we have to check it out."},{"Start":"00:25.260 ","End":"00:27.509","Text":"Let\u0027s check if it really is an asymptote."},{"Start":"00:27.509 ","End":"00:30.090","Text":"We\u0027ll try 1 of the 1-sided limits."},{"Start":"00:30.090 ","End":"00:32.110","Text":"Let\u0027s try the 1 on the right first,"},{"Start":"00:32.110 ","End":"00:38.360","Text":"we want to see if x goes to 0 from the right of e to the 1 over x to check what this is."},{"Start":"00:38.360 ","End":"00:41.900","Text":"If it comes out to be plus or minus infinity, then we\u0027re okay."},{"Start":"00:41.900 ","End":"00:47.900","Text":"What we get is e to the power of 1 over 0 plus,"},{"Start":"00:47.900 ","End":"00:52.920","Text":"which is e to the power of plus infinity, which is infinity."},{"Start":"00:52.920 ","End":"00:56.570","Text":"Indeed, it is an asymptote on the right."},{"Start":"00:56.570 ","End":"00:58.460","Text":"Let\u0027s also, out of curiosity,"},{"Start":"00:58.460 ","End":"01:03.485","Text":"see what happens when we go to 0 from the left of e to the 1 over x."},{"Start":"01:03.485 ","End":"01:07.565","Text":"Here, we have e to the power of 1 over negative 0,"},{"Start":"01:07.565 ","End":"01:12.665","Text":"which is e to the power of minus infinity and this equals 0"},{"Start":"01:12.665 ","End":"01:18.260","Text":"and so we get that x equals 0 is just a 1-sided asymptote."},{"Start":"01:18.260 ","End":"01:22.595","Text":"It\u0027s still an asymptote, a vertical asymptote."},{"Start":"01:22.595 ","End":"01:24.680","Text":"It\u0027s fully qualified asymptote,"},{"Start":"01:24.680 ","End":"01:31.190","Text":"but I could just add to be more precise on the right or 1-sided, but that\u0027s okay."},{"Start":"01:31.190 ","End":"01:35.070","Text":"Just an additional information anyway, we are done."}],"ID":5864},{"Watched":false,"Name":"Exercise 15","Duration":"30s","ChapterTopicVideoID":5867,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5867.jpeg","UploadDate":"2016-05-02T06:50:56.4870000","DurationForVideoObject":"PT30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of the function f of x,"},{"Start":"00:05.130 ","End":"00:07.740","Text":"which is xe to the minus 2x squared."},{"Start":"00:07.740 ","End":"00:08.955","Text":"I\u0027ve copied it here."},{"Start":"00:08.955 ","End":"00:11.025","Text":"Minus 2x squared is a polynomial,"},{"Start":"00:11.025 ","End":"00:13.725","Text":"no problems with that, defined everywhere,"},{"Start":"00:13.725 ","End":"00:18.950","Text":"e to the power of the exponential function defined for all x and x,"},{"Start":"00:18.950 ","End":"00:22.815","Text":"if you multiply it by this is still everything\u0027s defined."},{"Start":"00:22.815 ","End":"00:24.960","Text":"No problems, no division by 0,"},{"Start":"00:24.960 ","End":"00:27.255","Text":"nothing, nowhere to look for suspects."},{"Start":"00:27.255 ","End":"00:29.475","Text":"There are no vertical asymptotes."},{"Start":"00:29.475 ","End":"00:31.750","Text":"That\u0027s all there is to it."}],"ID":5865},{"Watched":false,"Name":"Exercise 16","Duration":"2m 23s","ChapterTopicVideoID":5868,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5868.jpeg","UploadDate":"2016-05-02T06:51:12.5030000","DurationForVideoObject":"PT2M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.700","Text":"In this exercise, we\u0027re given the function f of x is x plus 2 times e^1/x."},{"Start":"00:05.700 ","End":"00:07.860","Text":"We have to find the vertical asymptotes."},{"Start":"00:07.860 ","End":"00:09.720","Text":"To find the vertical asymptotes,"},{"Start":"00:09.720 ","End":"00:14.280","Text":"we have to look for places where the function\u0027s undefined and if you look at it,"},{"Start":"00:14.280 ","End":"00:16.800","Text":"there\u0027s only 1 source of the problem,"},{"Start":"00:16.800 ","End":"00:19.200","Text":"and that is the denominator being 0."},{"Start":"00:19.200 ","End":"00:23.440","Text":"So all we have here is that x cannot be equal to 0."},{"Start":"00:23.440 ","End":"00:24.735","Text":"That\u0027s in the domain,"},{"Start":"00:24.735 ","End":"00:29.235","Text":"which means that x equals 0 is a suspect for an asymptote."},{"Start":"00:29.235 ","End":"00:31.950","Text":"How do we check if it really is an asymptote?"},{"Start":"00:31.950 ","End":"00:34.335","Text":"We try 1 of the 2-sided limits."},{"Start":"00:34.335 ","End":"00:38.910","Text":"The limit as x goes to either 0 from the right or 0 from the left."},{"Start":"00:38.910 ","End":"00:41.790","Text":"I\u0027ll start off with 0 from the right of f of x"},{"Start":"00:41.790 ","End":"00:44.450","Text":"to see if this is infinity or minus infinity."},{"Start":"00:44.450 ","End":"00:53.245","Text":"So what we have is the limit as x goes to 0 from the right of x plus 2e^1/x,"},{"Start":"00:53.245 ","End":"00:54.955","Text":"which is equal to, now,"},{"Start":"00:54.955 ","End":"01:04.535","Text":"the x plus 2 just substitution is 2 and e^1 over positive 0 is equal to plus infinity."},{"Start":"01:04.535 ","End":"01:09.020","Text":"1 over 0 is infinity and e to the infinity is infinity."},{"Start":"01:09.020 ","End":"01:11.390","Text":"So twice infinity is still infinity."},{"Start":"01:11.390 ","End":"01:13.145","Text":"I\u0027ll write it twice infinity,"},{"Start":"01:13.145 ","End":"01:15.155","Text":"which is infinity and that is good."},{"Start":"01:15.155 ","End":"01:18.665","Text":"That means that we have found a vertical asymptote"},{"Start":"01:18.665 ","End":"01:22.430","Text":"at 0 from the right and that\u0027s enough for it to be an asymptote."},{"Start":"01:22.430 ","End":"01:24.710","Text":"But I\u0027d like to, out of curiosity,"},{"Start":"01:24.710 ","End":"01:26.270","Text":"know what happens on the left."},{"Start":"01:26.270 ","End":"01:31.610","Text":"So I\u0027ll take the limit as x goes to 0 from the left of f of x,"},{"Start":"01:31.610 ","End":"01:34.355","Text":"and I basically get the same thing."},{"Start":"01:34.355 ","End":"01:41.600","Text":"I\u0027ll just skip over this and it\u0027s equal to twice e^1/0 minus tiny,"},{"Start":"01:41.600 ","End":"01:45.080","Text":"tiny bit negative, very, very small negative"},{"Start":"01:45.080 ","End":"01:48.965","Text":"and so 1 over it is minus infinity,"},{"Start":"01:48.965 ","End":"01:53.690","Text":"very large and negative and e to the minus infinity is well known,"},{"Start":"01:53.690 ","End":"01:56.585","Text":"it\u0027s 0, so it\u0027s twice 0."},{"Start":"01:56.585 ","End":"01:59.660","Text":"So it\u0027s 0. Here we don\u0027t have infinity,"},{"Start":"01:59.660 ","End":"02:02.960","Text":"but it\u0027s enough for us to be infinity on 1 side."},{"Start":"02:02.960 ","End":"02:06.500","Text":"So yes, this is a vertical asymptote,"},{"Start":"02:06.500 ","End":"02:12.530","Text":"x equals 0 is a vertical asymptote."},{"Start":"02:12.530 ","End":"02:13.850","Text":"If you want to be precise,"},{"Start":"02:13.850 ","End":"02:15.680","Text":"we can say from the right,"},{"Start":"02:15.680 ","End":"02:20.030","Text":"but enough for either 1 side or both for it to be an asymptote."},{"Start":"02:20.030 ","End":"02:23.130","Text":"So that\u0027s it. We\u0027re done."}],"ID":5866},{"Watched":false,"Name":"Exercise 17","Duration":"2m 34s","ChapterTopicVideoID":5869,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5869.jpeg","UploadDate":"2016-05-02T06:51:29.8070000","DurationForVideoObject":"PT2M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes"},{"Start":"00:03.630 ","End":"00:08.070","Text":"of the function f of x is natural log of x over x,"},{"Start":"00:08.070 ","End":"00:09.480","Text":"and I\u0027ve copied it over here."},{"Start":"00:09.480 ","End":"00:12.045","Text":"Now, if we look at the domain of this function,"},{"Start":"00:12.045 ","End":"00:14.880","Text":"we see that we need for 2 things to happen."},{"Start":"00:14.880 ","End":"00:17.850","Text":"X has to be bigger than 0 because that\u0027s the domain of"},{"Start":"00:17.850 ","End":"00:22.770","Text":"the natural logarithm and x has to be different from 0 because of the denominator."},{"Start":"00:22.770 ","End":"00:28.275","Text":"So bigger than 0 and different from 0 altogether means that x is bigger than 0."},{"Start":"00:28.275 ","End":"00:30.375","Text":"That will satisfy both conditions."},{"Start":"00:30.375 ","End":"00:31.845","Text":"Now, in its domain,"},{"Start":"00:31.845 ","End":"00:33.899","Text":"this function is defined everywhere,"},{"Start":"00:33.899 ","End":"00:36.510","Text":"but this question is unusual."},{"Start":"00:36.510 ","End":"00:38.895","Text":"It\u0027s different than the usual question"},{"Start":"00:38.895 ","End":"00:42.360","Text":"and I don\u0027t usually mention this because there\u0027s no need,"},{"Start":"00:42.360 ","End":"00:44.150","Text":"but when we have an interval,"},{"Start":"00:44.150 ","End":"00:47.420","Text":"the edge of the interval can also be an asymptote."},{"Start":"00:47.420 ","End":"00:49.520","Text":"In other words, x equals 0,"},{"Start":"00:49.520 ","End":"00:51.770","Text":"which is the endpoint of this interval,"},{"Start":"00:51.770 ","End":"00:54.050","Text":"is a suspect, I\u0027m not saying it is an asymptote,"},{"Start":"00:54.050 ","End":"00:55.225","Text":"but it\u0027s a suspect."},{"Start":"00:55.225 ","End":"00:58.730","Text":"At least, we have to check it out what happens at the edge suspect"},{"Start":"00:58.730 ","End":"01:03.725","Text":"for being an asymptote. We have to check it."},{"Start":"01:03.725 ","End":"01:07.130","Text":"Now, we want to check 1 of the two-sided limits."},{"Start":"01:07.130 ","End":"01:11.930","Text":"We have to check the limit as x goes to 0 and we\u0027ll see in a minute whether we"},{"Start":"01:11.930 ","End":"01:16.910","Text":"choose left or right of f of x to see if this is infinity or minus infinity."},{"Start":"01:16.910 ","End":"01:20.195","Text":"Now, since x is only defined for bigger than 0,"},{"Start":"01:20.195 ","End":"01:23.600","Text":"the only one-sided limit we can take is the limit from the right."},{"Start":"01:23.600 ","End":"01:28.790","Text":"I have to put a plus here and if this is plus or minus infinity, then we\u0027re okay."},{"Start":"01:28.790 ","End":"01:30.635","Text":"We got an asymptote otherwise not."},{"Start":"01:30.635 ","End":"01:40.215","Text":"This is equal to the limit as x goes to positive 0 of natural log of x over x."},{"Start":"01:40.215 ","End":"01:44.675","Text":"Now, the natural log of 0 plus is known to be"},{"Start":"01:44.675 ","End":"01:52.245","Text":"minus infinity and here x goes to 0 plus it\u0027s just 0 plus,"},{"Start":"01:52.245 ","End":"01:54.140","Text":"and if I look at it as a product,"},{"Start":"01:54.140 ","End":"01:59.375","Text":"this equals minus infinity times 1 over 0 plus,"},{"Start":"01:59.375 ","End":"02:04.550","Text":"which means that this is minus infinity times plus infinity,"},{"Start":"02:04.550 ","End":"02:07.010","Text":"which makes it minus infinity."},{"Start":"02:07.010 ","End":"02:09.470","Text":"We have a one-sided asymptote,"},{"Start":"02:09.470 ","End":"02:11.480","Text":"which is fully counted as an asymptote,"},{"Start":"02:11.480 ","End":"02:16.655","Text":"so I\u0027ll just write the answer that x equals 0 is an asymptote,"},{"Start":"02:16.655 ","End":"02:21.230","Text":"or more precisely a vertical asymptote."},{"Start":"02:21.230 ","End":"02:23.680","Text":"X equals 0 is a line, a vertical line,"},{"Start":"02:23.680 ","End":"02:27.019","Text":"a vertical asymptote and if we want to be precise,"},{"Start":"02:27.019 ","End":"02:29.090","Text":"we can say on the right,"},{"Start":"02:29.090 ","End":"02:32.750","Text":"but that\u0027s enough for one side for it to be an asymptote."},{"Start":"02:32.750 ","End":"02:34.770","Text":"Okay. We\u0027re done."}],"ID":5867},{"Watched":false,"Name":"Exercise 18","Duration":"3m 39s","ChapterTopicVideoID":5870,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5870.jpeg","UploadDate":"2016-05-02T06:51:53.7300000","DurationForVideoObject":"PT3M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"In this exercise, we\u0027re given the function f of x equals x natural log of x,"},{"Start":"00:05.340 ","End":"00:08.100","Text":"and we have to find its vertical asymptotes."},{"Start":"00:08.100 ","End":"00:09.825","Text":"I\u0027ve copied the exercise here."},{"Start":"00:09.825 ","End":"00:13.725","Text":"You should especially note the domain of definition of the function."},{"Start":"00:13.725 ","End":"00:15.495","Text":"Because of the natural log,"},{"Start":"00:15.495 ","End":"00:18.075","Text":"this is x bigger than 0."},{"Start":"00:18.075 ","End":"00:20.295","Text":"So it\u0027s not defined on the whole line,"},{"Start":"00:20.295 ","End":"00:22.215","Text":"only on the positive numbers."},{"Start":"00:22.215 ","End":"00:23.640","Text":"Now, in this domain,"},{"Start":"00:23.640 ","End":"00:27.165","Text":"the function is defined everywhere, that\u0027s what the domain is."},{"Start":"00:27.165 ","End":"00:30.375","Text":"But when we have a domain which has an endpoint,"},{"Start":"00:30.375 ","End":"00:35.270","Text":"an interval, then we also have to check the endpoint to see if it\u0027s an asymptote."},{"Start":"00:35.270 ","End":"00:36.725","Text":"It\u0027s one of those special cases."},{"Start":"00:36.725 ","End":"00:40.250","Text":"So x equals 0 being the endpoint, is a suspect."},{"Start":"00:40.250 ","End":"00:43.030","Text":"If x equals 0 is suspect,"},{"Start":"00:43.030 ","End":"00:46.700","Text":"when I say suspect, I mean for being a vertical asymptote, of course."},{"Start":"00:46.700 ","End":"00:50.510","Text":"Let\u0027s check out the suspect by taking the limit."},{"Start":"00:50.510 ","End":"00:55.430","Text":"Now, I have to check by letting x go to 0 either from the left or from the right."},{"Start":"00:55.430 ","End":"00:58.610","Text":"But I can\u0027t go to 0 from the left because that would mean that"},{"Start":"00:58.610 ","End":"01:01.940","Text":"x would go through negative values and x has to be positive."},{"Start":"01:01.940 ","End":"01:04.070","Text":"So I can only go to 0 from the right."},{"Start":"01:04.070 ","End":"01:08.880","Text":"My only chance is to see if the limit from the right is plus or minus infinity."},{"Start":"01:08.880 ","End":"01:10.880","Text":"I need the limit of f of x,"},{"Start":"01:10.880 ","End":"01:13.565","Text":"which is x natural log of x,"},{"Start":"01:13.565 ","End":"01:17.119","Text":"and see whether I get infinity or minus infinity."},{"Start":"01:17.119 ","End":"01:20.300","Text":"If I just do it naively by substitution,"},{"Start":"01:20.300 ","End":"01:22.085","Text":"I\u0027ll see that product here,"},{"Start":"01:22.085 ","End":"01:27.155","Text":"the x is 0 and the natural log of x is minus infinity."},{"Start":"01:27.155 ","End":"01:32.315","Text":"As it\u0027s known that the natural log of x goes to minus infinity when x goes to 0."},{"Start":"01:32.315 ","End":"01:34.675","Text":"This is one of those indeterminate forms."},{"Start":"01:34.675 ","End":"01:40.025","Text":"But what I can do is use the same tricks that we use in L\u0027Hopital."},{"Start":"01:40.025 ","End":"01:44.710","Text":"I\u0027m going to erase this and rewrite this in a different form."},{"Start":"01:44.710 ","End":"01:47.960","Text":"0 times minus infinity doesn\u0027t suit me."},{"Start":"01:47.960 ","End":"01:52.280","Text":"What I do is the usual trick of writing it as the limit,"},{"Start":"01:52.280 ","End":"01:58.420","Text":"same x goes to 0 plus of natural log of x over 1 over x."},{"Start":"01:58.420 ","End":"02:01.510","Text":"Otherwise, I put the x into the denominator and invert it."},{"Start":"02:01.510 ","End":"02:08.180","Text":"Because now, what I\u0027ll get is that this thing still goes to minus infinity,"},{"Start":"02:08.180 ","End":"02:13.460","Text":"but this thing goes to 1 over 0 plus, which is infinity."},{"Start":"02:13.460 ","End":"02:15.829","Text":"So I have a form infinity over infinity."},{"Start":"02:15.829 ","End":"02:17.195","Text":"The minus doesn\u0027t matter."},{"Start":"02:17.195 ","End":"02:19.735","Text":"Now, I can use L\u0027Hopital."},{"Start":"02:19.735 ","End":"02:22.160","Text":"This equals, I write it by,"},{"Start":"02:22.160 ","End":"02:26.870","Text":"the infinity over infinity case of L\u0027Hopital\u0027s rule."},{"Start":"02:26.870 ","End":"02:28.220","Text":"What I do is,"},{"Start":"02:28.220 ","End":"02:34.780","Text":"I differentiate the numerator and the denominator. Just a minute."},{"Start":"02:34.780 ","End":"02:43.010","Text":"The limit till x goes to 0 plus the derivative of this is 1 over x,"},{"Start":"02:43.010 ","End":"02:48.080","Text":"and the derivative of the denominator is minus 1 over x squared."},{"Start":"02:48.080 ","End":"02:53.015","Text":"So this is equal to this by L\u0027Hopital\u0027s rule for the infinity over infinity case."},{"Start":"02:53.015 ","End":"02:59.465","Text":"Now, I can continue and say that this is equal to the limit as x goes to 0 plus,"},{"Start":"02:59.465 ","End":"03:03.515","Text":"if you multiply this by minus x squared,"},{"Start":"03:03.515 ","End":"03:05.129","Text":"if I invert it and multiply,"},{"Start":"03:05.129 ","End":"03:08.560","Text":"in short, we get minus x for this thing."},{"Start":"03:08.560 ","End":"03:10.250","Text":"When x goes to 0,"},{"Start":"03:10.250 ","End":"03:13.685","Text":"this is just equal to 0."},{"Start":"03:13.685 ","End":"03:17.645","Text":"So it is not an asymptote."},{"Start":"03:17.645 ","End":"03:23.840","Text":"This is, I want to say, if it\u0027s not equal to plus or minus infinity, the limit is 0,"},{"Start":"03:23.840 ","End":"03:32.070","Text":"so there is no vertical asymptote."},{"Start":"03:32.070 ","End":"03:36.150","Text":"They\u0027re only suspect turned out not [inaudible]"},{"Start":"03:36.150 ","End":"03:39.930","Text":"That\u0027s all I have to say. We\u0027re done."}],"ID":5868},{"Watched":false,"Name":"Exercise 19","Duration":"2m 26s","ChapterTopicVideoID":5871,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5871.jpeg","UploadDate":"2016-05-02T06:52:09.6230000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of the function"},{"Start":"00:05.010 ","End":"00:09.720","Text":"natural log squared of x plus twice natural log of x minus 3."},{"Start":"00:09.720 ","End":"00:10.940","Text":"I\u0027ve copied it here."},{"Start":"00:10.940 ","End":"00:13.260","Text":"Pay special note to the domain,"},{"Start":"00:13.260 ","End":"00:15.450","Text":"the domain here is not all of x,"},{"Start":"00:15.450 ","End":"00:17.595","Text":"but only x positive."},{"Start":"00:17.595 ","End":"00:21.120","Text":"That\u0027s because of the natural logarithm and there\u0027s no other restrictions."},{"Start":"00:21.120 ","End":"00:23.310","Text":"Now, this is one of those unusual cases where we"},{"Start":"00:23.310 ","End":"00:25.815","Text":"have the function defined on an interval,"},{"Start":"00:25.815 ","End":"00:28.260","Text":"and in this case, we have to look at the endpoint or"},{"Start":"00:28.260 ","End":"00:31.605","Text":"endpoints of the interval and they are suspects."},{"Start":"00:31.605 ","End":"00:37.695","Text":"X equals 0, the line x equals 0 will be now a suspect for being a vertical asymptote,"},{"Start":"00:37.695 ","End":"00:39.935","Text":"so I\u0027ll just write down that it\u0027s a suspect."},{"Start":"00:39.935 ","End":"00:42.695","Text":"But of course, we have to check out the suspect by"},{"Start":"00:42.695 ","End":"00:46.160","Text":"figuring out the one-sided limit of f of x."},{"Start":"00:46.160 ","End":"00:47.840","Text":"Now, in this case,"},{"Start":"00:47.840 ","End":"00:50.990","Text":"we can only go to 0 from the right."},{"Start":"00:50.990 ","End":"00:52.700","Text":"We can\u0027t go to 0 from the left,"},{"Start":"00:52.700 ","End":"00:55.415","Text":"because we have to stay in positive numbers."},{"Start":"00:55.415 ","End":"00:58.370","Text":"So we can go to 0 through positive numbers getting"},{"Start":"00:58.370 ","End":"01:02.690","Text":"smaller and smaller and smaller but still positive of f of x,"},{"Start":"01:02.690 ","End":"01:10.555","Text":"which is natural log squared of x plus twice natural log of x minus 3."},{"Start":"01:10.555 ","End":"01:13.155","Text":"I just want to make a note at the side,"},{"Start":"01:13.155 ","End":"01:18.160","Text":"that the natural log of 0 plus is minus infinity."},{"Start":"01:18.160 ","End":"01:20.870","Text":"When x is very, very, very small and positive,"},{"Start":"01:20.870 ","End":"01:23.330","Text":"natural log of x is very large and negative."},{"Start":"01:23.330 ","End":"01:24.730","Text":"This is minus infinity."},{"Start":"01:24.730 ","End":"01:28.395","Text":"Let\u0027s do some arithmetics with this."},{"Start":"01:28.395 ","End":"01:30.120","Text":"I\u0027m going to slightly rewrite it,"},{"Start":"01:30.120 ","End":"01:32.790","Text":"because otherwise I\u0027m going to get an infinity minus infinity."},{"Start":"01:32.790 ","End":"01:37.820","Text":"Let\u0027s rewrite it as the limit of natural log of"},{"Start":"01:37.820 ","End":"01:43.130","Text":"x times natural log of x plus 2 minus 3."},{"Start":"01:43.130 ","End":"01:47.845","Text":"I will take this outside the brackets, by substituting this,"},{"Start":"01:47.845 ","End":"01:51.165","Text":"I can get minus infinity, brackets,"},{"Start":"01:51.165 ","End":"01:55.125","Text":"minus infinity plus 2 minus 3."},{"Start":"01:55.125 ","End":"01:57.570","Text":"Well, minus infinity plus 2,"},{"Start":"01:57.570 ","End":"02:00.530","Text":"first of all is just minus infinity."},{"Start":"02:00.530 ","End":"02:05.245","Text":"Then minus infinity times minus infinity is plus infinity."},{"Start":"02:05.245 ","End":"02:08.000","Text":"Minus 3 infinity minus 3 is infinity,"},{"Start":"02:08.000 ","End":"02:09.440","Text":"so I\u0027ll leave it as infinity."},{"Start":"02:09.440 ","End":"02:11.255","Text":"This is good."},{"Start":"02:11.255 ","End":"02:15.060","Text":"This means that the suspect really is guilty."},{"Start":"02:15.060 ","End":"02:17.265","Text":"In other words, x equals 0,"},{"Start":"02:17.265 ","End":"02:24.905","Text":"the vertical line is a vertical asymptote to this function."},{"Start":"02:24.905 ","End":"02:27.510","Text":"We are done."}],"ID":5869},{"Watched":false,"Name":"Exercise 20","Duration":"38s","ChapterTopicVideoID":5872,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5872.jpeg","UploadDate":"2016-05-02T06:52:13.9470000","DurationForVideoObject":"PT38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.500 ","End":"00:07.830","Text":"which is 1 over the square root of x squared plus 1."},{"Start":"00:07.830 ","End":"00:09.015","Text":"I\u0027ve copied it here."},{"Start":"00:09.015 ","End":"00:10.110","Text":"If we examine this,"},{"Start":"00:10.110 ","End":"00:12.300","Text":"we see that it\u0027s defined for all x,"},{"Start":"00:12.300 ","End":"00:13.365","Text":"there are no problems."},{"Start":"00:13.365 ","End":"00:16.740","Text":"X squared is always positive or non-negative,"},{"Start":"00:16.740 ","End":"00:19.170","Text":"and when we add 1, it\u0027s certainly positive."},{"Start":"00:19.170 ","End":"00:21.170","Text":"We have the square root of a positive number,"},{"Start":"00:21.170 ","End":"00:24.555","Text":"and that\u0027s positive, and 1 over a positive number is defined."},{"Start":"00:24.555 ","End":"00:26.985","Text":"In short, everything is defined for all x,"},{"Start":"00:26.985 ","End":"00:30.525","Text":"and so there\u0027s no suspect for an asymptote."},{"Start":"00:30.525 ","End":"00:36.255","Text":"We\u0027re just going to write that there are no vertical asymptotes."},{"Start":"00:36.255 ","End":"00:39.180","Text":"That\u0027s it, there are none. We\u0027re done."}],"ID":5870},{"Watched":false,"Name":"Exercise 21","Duration":"37s","ChapterTopicVideoID":5873,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5873.jpeg","UploadDate":"2016-05-02T06:52:17.9500000","DurationForVideoObject":"PT37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.050","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.050 ","End":"00:06.930","Text":"which is x over the square root of x squared plus 1."},{"Start":"00:06.930 ","End":"00:09.975","Text":"But this function is defined for all x."},{"Start":"00:09.975 ","End":"00:11.235","Text":"For every x I take,"},{"Start":"00:11.235 ","End":"00:14.250","Text":"x squared is going to be 0 or positive."},{"Start":"00:14.250 ","End":"00:16.335","Text":"If I add 1, it\u0027s strictly positive."},{"Start":"00:16.335 ","End":"00:19.530","Text":"The square root of a positive number is a positive number,"},{"Start":"00:19.530 ","End":"00:21.030","Text":"and on the denominator,"},{"Start":"00:21.030 ","End":"00:22.890","Text":"positive number is no problem."},{"Start":"00:22.890 ","End":"00:25.360","Text":"In other words, there\u0027s absolutely no problem with this function."},{"Start":"00:25.360 ","End":"00:30.990","Text":"It\u0027s defined everywhere and there\u0027s no suspect for an asymptote and no asymptote."},{"Start":"00:30.990 ","End":"00:38.530","Text":"The solution is that there are no vertical asymptotes, and that\u0027s all."}],"ID":5871},{"Watched":false,"Name":"Exercise 22","Duration":"35s","ChapterTopicVideoID":5874,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5874.jpeg","UploadDate":"2016-05-02T06:52:21.9300000","DurationForVideoObject":"PT35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"We\u0027re asked to find the vertical asymptotes of"},{"Start":"00:02.850 ","End":"00:07.320","Text":"the function f of x is x times square root of x squared plus 4."},{"Start":"00:07.320 ","End":"00:09.300","Text":"I claim that there are no asymptotes."},{"Start":"00:09.300 ","End":"00:12.780","Text":"I mean, there\u0027s nothing that can go wrong with this function, it\u0027s defined everywhere."},{"Start":"00:12.780 ","End":"00:15.660","Text":"Look, x squared is 0 or larger,"},{"Start":"00:15.660 ","End":"00:17.820","Text":"when I add 4 it\u0027s strictly positive."},{"Start":"00:17.820 ","End":"00:22.800","Text":"The square root\u0027s defined and I can certainly multiply any number by any other number."},{"Start":"00:22.800 ","End":"00:26.320","Text":"It\u0027s well-behaved, no problem defined everywhere,"},{"Start":"00:26.320 ","End":"00:29.070","Text":"and so there are no vertical asymptotes."},{"Start":"00:29.070 ","End":"00:30.780","Text":"I\u0027ll just write that down in brief,"},{"Start":"00:30.780 ","End":"00:35.380","Text":"no vertical asymptotes, and then we\u0027re done."}],"ID":5872},{"Watched":false,"Name":"Exercise 23","Duration":"36s","ChapterTopicVideoID":5875,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5875.jpeg","UploadDate":"2016-05-02T06:52:25.5470000","DurationForVideoObject":"PT36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we\u0027re asked to find the vertical asymptotes of f of x,"},{"Start":"00:04.410 ","End":"00:08.370","Text":"which is the cube root of x squared times 1 minus x."},{"Start":"00:08.370 ","End":"00:10.605","Text":"Now, this function is well-behaved."},{"Start":"00:10.605 ","End":"00:12.000","Text":"It\u0027s defined everywhere."},{"Start":"00:12.000 ","End":"00:14.565","Text":"1 minus x can be defined for all x,"},{"Start":"00:14.565 ","End":"00:16.590","Text":"x squared, no problem with that."},{"Start":"00:16.590 ","End":"00:18.720","Text":"The cube root of any number is defined,"},{"Start":"00:18.720 ","End":"00:20.340","Text":"especially of a positive number."},{"Start":"00:20.340 ","End":"00:23.865","Text":"But a cube root is always defined and you can always multiply 2 numbers,"},{"Start":"00:23.865 ","End":"00:25.665","Text":"but there\u0027s no problems here,"},{"Start":"00:25.665 ","End":"00:29.475","Text":"defined everywhere, and so here\u0027s no reason for there to be an asymptote,"},{"Start":"00:29.475 ","End":"00:31.635","Text":"no suspects, so I\u0027m just going to write,"},{"Start":"00:31.635 ","End":"00:36.670","Text":"there are no vertical asymptotes, and that\u0027s it."}],"ID":5873},{"Watched":false,"Name":"Exercise 24","Duration":"35s","ChapterTopicVideoID":5876,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5876.jpeg","UploadDate":"2016-05-02T06:52:29.7300000","DurationForVideoObject":"PT35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.800","Text":"In this exercise, we\u0027re asked to find vertical asymptotes of the function,"},{"Start":"00:04.800 ","End":"00:06.960","Text":"cube root of x squared minus 1,"},{"Start":"00:06.960 ","End":"00:08.265","Text":"this whole thing squared."},{"Start":"00:08.265 ","End":"00:10.815","Text":"Now, this function is well-behaved."},{"Start":"00:10.815 ","End":"00:12.870","Text":"It\u0027s defined everywhere, there\u0027s no problems."},{"Start":"00:12.870 ","End":"00:14.895","Text":"Let\u0027s look at it. If I take any x,"},{"Start":"00:14.895 ","End":"00:16.500","Text":"I can certainly square it."},{"Start":"00:16.500 ","End":"00:18.405","Text":"Any number has a cube root,"},{"Start":"00:18.405 ","End":"00:19.860","Text":"I can always subtract 1,"},{"Start":"00:19.860 ","End":"00:21.315","Text":"I can always square it."},{"Start":"00:21.315 ","End":"00:24.750","Text":"There\u0027s no obstacle to it being defined anywhere,"},{"Start":"00:24.750 ","End":"00:28.770","Text":"so there\u0027s no reason for there to be any suspect for a vertical asymptote."},{"Start":"00:28.770 ","End":"00:33.495","Text":"In short, I\u0027m going to write the answer that there are no vertical asymptotes,"},{"Start":"00:33.495 ","End":"00:36.310","Text":"and leave it at that. We\u0027re done."}],"ID":5874},{"Watched":false,"Name":"Exercise 25","Duration":"28s","ChapterTopicVideoID":5877,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5877.jpeg","UploadDate":"2016-05-02T06:52:33.4400000","DurationForVideoObject":"PT28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this exercise, we\u0027re asked to find vertical asymptotes of the function f of x,"},{"Start":"00:04.860 ","End":"00:07.485","Text":"which is the cube root of x squared minus 1."},{"Start":"00:07.485 ","End":"00:10.380","Text":"Now, I can\u0027t find any suspect for asymptotes."},{"Start":"00:10.380 ","End":"00:13.830","Text":"I mean, for any x I can compute x squared minus 1,"},{"Start":"00:13.830 ","End":"00:16.320","Text":"and whatever it is, whether it\u0027s positive or negative,"},{"Start":"00:16.320 ","End":"00:17.655","Text":"it has a cube root."},{"Start":"00:17.655 ","End":"00:19.800","Text":"There\u0027s no problems here at all,"},{"Start":"00:19.800 ","End":"00:23.820","Text":"and no suspects, no vertical asymptotes."},{"Start":"00:23.820 ","End":"00:24.930","Text":"Just leave it at that."},{"Start":"00:24.930 ","End":"00:29.260","Text":"No vertical asymptotes, and we\u0027re done."}],"ID":5875},{"Watched":false,"Name":"Exercise 26","Duration":"3m ","ChapterTopicVideoID":5878,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5878.jpeg","UploadDate":"2016-05-02T06:52:52.7300000","DurationForVideoObject":"PT3M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.575 ","End":"00:09.000","Text":"which is the absolute value of x minus 3 over x minus 2."},{"Start":"00:09.000 ","End":"00:10.140","Text":"I\u0027ve copied it here."},{"Start":"00:10.140 ","End":"00:15.945","Text":"If we look at it, the only problematic x is where x equals 2,"},{"Start":"00:15.945 ","End":"00:19.230","Text":"because then we have 0 on the denominator."},{"Start":"00:19.230 ","End":"00:23.100","Text":"For the domain we have x not equal to 2,"},{"Start":"00:23.100 ","End":"00:25.905","Text":"but that makes x equals 2 as a suspect,"},{"Start":"00:25.905 ","End":"00:30.195","Text":"the line x equals 2 now is a suspect for being a vertical asymptote."},{"Start":"00:30.195 ","End":"00:33.990","Text":"It\u0027s a suspect and then we\u0027ll check it out."},{"Start":"00:33.990 ","End":"00:38.130","Text":"Let\u0027s check the limit as x goes to 2."},{"Start":"00:38.130 ","End":"00:39.690","Text":"We\u0027ll see on the left or the right,"},{"Start":"00:39.690 ","End":"00:43.425","Text":"I think we might be able to get both in 1 shot of f of x,"},{"Start":"00:43.425 ","End":"00:48.905","Text":"which is x minus 3 in absolute value over x minus 2."},{"Start":"00:48.905 ","End":"00:54.780","Text":"Meanwhile, we\u0027ll see if we have the 2 sided limit and better not."},{"Start":"00:54.780 ","End":"00:59.275","Text":"You know what, I\u0027ll go with the right limit first and see what that does."},{"Start":"00:59.275 ","End":"01:05.475","Text":"This is equal to the limit as x goes to 2 from the right."},{"Start":"01:05.475 ","End":"01:11.915","Text":"X minus 3, if x is close to 2 and x is going to be 2 plus a little bit,"},{"Start":"01:11.915 ","End":"01:13.775","Text":"if x is around 2,"},{"Start":"01:13.775 ","End":"01:17.280","Text":"and this is going to be negative inside the absolute value."},{"Start":"01:17.280 ","End":"01:20.525","Text":"The absolute value inverts it and makes it positive."},{"Start":"01:20.525 ","End":"01:23.000","Text":"We have 3 minus x,"},{"Start":"01:23.000 ","End":"01:29.255","Text":"which is the negative of x minus 3 over x minus 2 when x goes to positive 2."},{"Start":"01:29.255 ","End":"01:36.560","Text":"What we\u0027re left with is 3 minus 2 plus over 2 plus minus 2,"},{"Start":"01:36.560 ","End":"01:39.260","Text":"doing arithmetic with these symbols."},{"Start":"01:39.260 ","End":"01:41.915","Text":"Any event, the numerator when x goes to 2,"},{"Start":"01:41.915 ","End":"01:45.905","Text":"goes to 1, which is like 3 minus 2, which is 1."},{"Start":"01:45.905 ","End":"01:48.560","Text":"When x goes to 2 from the right,"},{"Start":"01:48.560 ","End":"01:51.770","Text":"then x minus 2 goes to 0 from the right."},{"Start":"01:51.770 ","End":"01:53.690","Text":"It\u0027s like 1 over 0 plus,"},{"Start":"01:53.690 ","End":"01:55.835","Text":"which is just infinity."},{"Start":"01:55.835 ","End":"01:57.305","Text":"The answer is yes,"},{"Start":"01:57.305 ","End":"02:01.310","Text":"x equals 2 is indeed an asymptote from the right."},{"Start":"02:01.310 ","End":"02:03.200","Text":"But just for completeness,"},{"Start":"02:03.200 ","End":"02:05.540","Text":"I also want to check what happens on the left."},{"Start":"02:05.540 ","End":"02:07.220","Text":"Although we could stop here and say yes,"},{"Start":"02:07.220 ","End":"02:08.419","Text":"we have our asymptote."},{"Start":"02:08.419 ","End":"02:10.610","Text":"Let\u0027s just see what happens on the left."},{"Start":"02:10.610 ","End":"02:14.480","Text":"On the left, we have the same thing over x minus 2,"},{"Start":"02:14.480 ","End":"02:16.520","Text":"which is the limit x is still close to 2,"},{"Start":"02:16.520 ","End":"02:18.200","Text":"whether it\u0027s above or below."},{"Start":"02:18.200 ","End":"02:22.460","Text":"This thing again becomes 3 minus x over x minus 2,"},{"Start":"02:22.460 ","End":"02:25.745","Text":"x goes to 2 from the left so far were the same."},{"Start":"02:25.745 ","End":"02:28.340","Text":"The only thing is that here, instead of 2 plus,"},{"Start":"02:28.340 ","End":"02:32.195","Text":"we have 2 minus and 2 minus, minus 2."},{"Start":"02:32.195 ","End":"02:35.665","Text":"Here we have 1, but here we have negative 0."},{"Start":"02:35.665 ","End":"02:37.220","Text":"This is minus infinity,"},{"Start":"02:37.220 ","End":"02:38.390","Text":"which is also good."},{"Start":"02:38.390 ","End":"02:41.810","Text":"That makes it an asymptote on the other side also."},{"Start":"02:41.810 ","End":"02:43.325","Text":"When I write the answer,"},{"Start":"02:43.325 ","End":"02:46.100","Text":"I can write that x equals 2."},{"Start":"02:46.100 ","End":"02:51.305","Text":"The line is a vertical asymptote,"},{"Start":"02:51.305 ","End":"02:53.465","Text":"and I can even write that it\u0027s 2 sided,"},{"Start":"02:53.465 ","End":"02:54.590","Text":"not that it has to be,"},{"Start":"02:54.590 ","End":"02:56.480","Text":"but you don\u0027t get any bonus points,"},{"Start":"02:56.480 ","End":"03:01.530","Text":"but it is a 2 sided asymptote from the right and from the left, and we\u0027re done."}],"ID":5876},{"Watched":false,"Name":"Exercise 27","Duration":"35s","ChapterTopicVideoID":5879,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5879.jpeg","UploadDate":"2016-05-02T06:52:56.8770000","DurationForVideoObject":"PT35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this exercise, we have to find the vertical asymptotes of the function of x,"},{"Start":"00:04.320 ","End":"00:07.200","Text":"which is x minus twice arctangent of x."},{"Start":"00:07.200 ","End":"00:10.890","Text":"Now, arctangent x is a function which is defined for all x,"},{"Start":"00:10.890 ","End":"00:12.630","Text":"defined and even continuous."},{"Start":"00:12.630 ","End":"00:13.860","Text":"No problems with it."},{"Start":"00:13.860 ","End":"00:16.559","Text":"From minus infinity to infinity it defined,"},{"Start":"00:16.559 ","End":"00:18.150","Text":"x, of course, is defined."},{"Start":"00:18.150 ","End":"00:20.700","Text":"In other words, there\u0027s no limitation on x."},{"Start":"00:20.700 ","End":"00:24.885","Text":"This function, x minus twice arctangent x is defined for all x."},{"Start":"00:24.885 ","End":"00:27.780","Text":"There is no suspect for vertical asymptote."},{"Start":"00:27.780 ","End":"00:29.895","Text":"In fact, there are no vertical asymptotes,"},{"Start":"00:29.895 ","End":"00:31.065","Text":"and I\u0027ll just write that,"},{"Start":"00:31.065 ","End":"00:36.040","Text":"no vertical asymptotes, and we\u0027ll leave it at that."}],"ID":5877},{"Watched":false,"Name":"Exercise 28","Duration":"2m 33s","ChapterTopicVideoID":5880,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5880.jpeg","UploadDate":"2016-05-02T06:53:13.8570000","DurationForVideoObject":"PT2M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.960","Text":"In this exercise, we have to find"},{"Start":"00:01.960 ","End":"00:04.255","Text":"the vertical asymptotes of f of x,"},{"Start":"00:04.255 ","End":"00:07.825","Text":"which is 1 over 1 plus e to the 1 over x."},{"Start":"00:07.825 ","End":"00:09.070","Text":"If we examine this,"},{"Start":"00:09.070 ","End":"00:11.860","Text":"the only source of trouble, so to speak,"},{"Start":"00:11.860 ","End":"00:15.985","Text":"is the x, which is in the denominator of the 1 over x."},{"Start":"00:15.985 ","End":"00:18.310","Text":"Other than that, the denominator here"},{"Start":"00:18.310 ","End":"00:20.080","Text":"is not going to be 0 because"},{"Start":"00:20.080 ","End":"00:21.610","Text":"e to the power of anything where"},{"Start":"00:21.610 ","End":"00:23.800","Text":"it\u0027s defined is positive,"},{"Start":"00:23.800 ","End":"00:25.960","Text":"and 1 plus positive is positive,"},{"Start":"00:25.960 ","End":"00:26.860","Text":"so it\u0027s never 0,"},{"Start":"00:26.860 ","End":"00:28.570","Text":"so really the only source"},{"Start":"00:28.570 ","End":"00:30.910","Text":"of trouble is the x here."},{"Start":"00:30.910 ","End":"00:34.120","Text":"We say that x equals 0 is our suspect."},{"Start":"00:34.120 ","End":"00:35.830","Text":"In fact, it\u0027s the only suspect"},{"Start":"00:35.830 ","End":"00:37.270","Text":"for a vertical asymptote."},{"Start":"00:37.270 ","End":"00:39.730","Text":"What we do is check the 1 sided limits"},{"Start":"00:39.730 ","End":"00:42.435","Text":"as x goes to 0 from the left and from the right."},{"Start":"00:42.435 ","End":"00:43.820","Text":"Let\u0027s try both."},{"Start":"00:43.820 ","End":"00:48.890","Text":"Let\u0027s try x goes to 0 from the right of f of x."},{"Start":"00:48.890 ","End":"00:51.080","Text":"This is equal to the limit"},{"Start":"00:51.080 ","End":"00:53.450","Text":"as x goes to 0 from the right,"},{"Start":"00:53.450 ","End":"00:54.935","Text":"I\u0027ll just copy f of x,"},{"Start":"00:54.935 ","End":"01:00.875","Text":"which is 1 over 1 plus e to the 1 over x."},{"Start":"01:00.875 ","End":"01:09.860","Text":"This is equal to 1 over 1 plus e to the 1 over 0 plus,"},{"Start":"01:09.860 ","End":"01:11.390","Text":"I will do the arithmetic"},{"Start":"01:11.390 ","End":"01:14.285","Text":"with infinities and tiny quantities."},{"Start":"01:14.285 ","End":"01:16.250","Text":"Let\u0027s see, this is equal to 1"},{"Start":"01:16.250 ","End":"01:19.430","Text":"over 1 plus e to the power of,"},{"Start":"01:19.430 ","End":"01:22.880","Text":"now 1 over positive 0 is positive infinity"},{"Start":"01:22.880 ","End":"01:26.945","Text":"and e to the positive infinity is infinity."},{"Start":"01:26.945 ","End":"01:29.285","Text":"If I add 1, it\u0027s still going to be infinity."},{"Start":"01:29.285 ","End":"01:31.040","Text":"I have 1 over infinity,"},{"Start":"01:31.040 ","End":"01:32.330","Text":"so this is 0."},{"Start":"01:32.330 ","End":"01:35.060","Text":"On the right, I don\u0027t have an asymptote."},{"Start":"01:35.060 ","End":"01:37.030","Text":"Checking on the left,"},{"Start":"01:37.030 ","End":"01:40.265","Text":"limit x goes to 0 from the left."},{"Start":"01:40.265 ","End":"01:44.105","Text":"We get same thing but 0 from the left,"},{"Start":"01:44.105 ","End":"01:47.435","Text":"1 over 1 plus e to the 1 over x,"},{"Start":"01:47.435 ","End":"01:51.560","Text":"which equals 1 over 1 plus 1 over,"},{"Start":"01:51.560 ","End":"01:52.850","Text":"and here\u0027s the difference."},{"Start":"01:52.850 ","End":"01:55.400","Text":"We have negative 0,"},{"Start":"01:55.400 ","End":"01:57.335","Text":"so to speak, e to the power of,"},{"Start":"01:57.335 ","End":"02:02.060","Text":"this is equal to 1 over 1 plus e to the 1"},{"Start":"02:02.060 ","End":"02:05.815","Text":"over 0 minus is minus infinity."},{"Start":"02:05.815 ","End":"02:07.050","Text":"Here\u0027s the difference,"},{"Start":"02:07.050 ","End":"02:09.800","Text":"e to the minus infinity is 0."},{"Start":"02:09.800 ","End":"02:13.220","Text":"We have 1 over 1 plus 0,"},{"Start":"02:13.220 ","End":"02:16.820","Text":"and that equals 1."},{"Start":"02:16.820 ","End":"02:18.860","Text":"But again, neither of these,"},{"Start":"02:18.860 ","End":"02:20.240","Text":"we did not get plus"},{"Start":"02:20.240 ","End":"02:22.670","Text":"or minus infinity in either case."},{"Start":"02:22.670 ","End":"02:24.950","Text":"So both of them turn out bad,"},{"Start":"02:24.950 ","End":"02:26.105","Text":"No asymptote."},{"Start":"02:26.105 ","End":"02:27.470","Text":"Well, that\u0027s all I can say,"},{"Start":"02:27.470 ","End":"02:30.680","Text":"is there is no vertical asymptote."},{"Start":"02:30.680 ","End":"02:33.450","Text":"Sorry, and now we\u0027re done."}],"ID":5878},{"Watched":false,"Name":"Exercise 29","Duration":"2m 19s","ChapterTopicVideoID":5881,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5881.jpeg","UploadDate":"2016-05-02T06:53:28.8430000","DurationForVideoObject":"PT2M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.260","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.260 ","End":"00:07.245","Text":"which is x squared minus 1 over x plus 1."},{"Start":"00:07.245 ","End":"00:09.120","Text":"This is a rational function,"},{"Start":"00:09.120 ","End":"00:12.075","Text":"polynomial over polynomial, and we talked about this."},{"Start":"00:12.075 ","End":"00:16.410","Text":"The only place we can find an asymptote is where the denominator is 0."},{"Start":"00:16.410 ","End":"00:17.760","Text":"Just copied it over here."},{"Start":"00:17.760 ","End":"00:21.060","Text":"I was saying, we\u0027re looking for suspects where the"},{"Start":"00:21.060 ","End":"00:24.825","Text":"denominator is 0 and when x plus 1 is 0,"},{"Start":"00:24.825 ","End":"00:30.120","Text":"that means that x is equal to minus 1 and that in fact is outside our domain."},{"Start":"00:30.120 ","End":"00:32.190","Text":"Our domain is all x except minus 1."},{"Start":"00:32.190 ","End":"00:34.830","Text":"Let\u0027s check out what happens at minus 1."},{"Start":"00:34.830 ","End":"00:39.970","Text":"What I need to do is find the limit as x goes to 0 from the right or the left."},{"Start":"00:39.970 ","End":"00:47.025","Text":"We\u0027ll start with the right of f of x and see if we get infinity or minus infinity."},{"Start":"00:47.025 ","End":"00:49.370","Text":"Sorry, it\u0027s not 0, I meant minus 1,"},{"Start":"00:49.370 ","End":"00:52.920","Text":"which equals the limit as x goes to minus 1,"},{"Start":"00:52.920 ","End":"00:58.955","Text":"but from the right of x squared minus 1 over x plus 1."},{"Start":"00:58.955 ","End":"01:01.220","Text":"Now, if you substitute minus 1,"},{"Start":"01:01.220 ","End":"01:03.005","Text":"we get 0 over 0."},{"Start":"01:03.005 ","End":"01:04.865","Text":"We could use L\u0027Hopital,"},{"Start":"01:04.865 ","End":"01:06.320","Text":"but in this case it\u0027s,"},{"Start":"01:06.320 ","End":"01:10.085","Text":"I think easier to actually factorize because,"},{"Start":"01:10.085 ","End":"01:11.300","Text":"I\u0027ll do this at the side,"},{"Start":"01:11.300 ","End":"01:18.800","Text":"x squared minus 1 over x plus 1 is just equal to x minus 1, x plus 1."},{"Start":"01:18.800 ","End":"01:20.700","Text":"That\u0027s difference of squares formula,"},{"Start":"01:20.700 ","End":"01:22.335","Text":"over x plus 1."},{"Start":"01:22.335 ","End":"01:24.020","Text":"X plus 1 cancels,"},{"Start":"01:24.020 ","End":"01:26.120","Text":"which just leaves us with x minus 1."},{"Start":"01:26.120 ","End":"01:28.910","Text":"I think it will be easier doing that than L\u0027Hopital."},{"Start":"01:28.910 ","End":"01:35.645","Text":"This is the limit as x goes to negative 1 from the right of x minus 1."},{"Start":"01:35.645 ","End":"01:42.500","Text":"X minus 1 when x goes to minus 1 from the left or the right is equal to minus 2,"},{"Start":"01:42.500 ","End":"01:46.340","Text":"so this is just equal to minus 2."},{"Start":"01:46.340 ","End":"01:49.790","Text":"As you can easily see, if I put minus 1 from the left everywhere,"},{"Start":"01:49.790 ","End":"01:52.400","Text":"I\u0027d get exactly the same because the limit as x"},{"Start":"01:52.400 ","End":"01:55.310","Text":"goes to 1 from the left or from the right of x,"},{"Start":"01:55.310 ","End":"01:57.310","Text":"minus 1 is just minus 2."},{"Start":"01:57.310 ","End":"02:03.424","Text":"Now, this minus 2 is not equal to infinity or minus infinity,"},{"Start":"02:03.424 ","End":"02:05.960","Text":"so this is not a vertical asymptote."},{"Start":"02:05.960 ","End":"02:09.260","Text":"Our suspect, which was x equals minus 1,"},{"Start":"02:09.260 ","End":"02:10.595","Text":"turned out to be innocent,"},{"Start":"02:10.595 ","End":"02:12.500","Text":"not a vertical asymptote."},{"Start":"02:12.500 ","End":"02:13.955","Text":"There are no other suspects,"},{"Start":"02:13.955 ","End":"02:20.640","Text":"so we have no vertical asymptotes here, and we\u0027re done."}],"ID":5879},{"Watched":false,"Name":"Exercise 30","Duration":"4m 30s","ChapterTopicVideoID":5882,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5882.jpeg","UploadDate":"2016-05-02T06:53:58.2230000","DurationForVideoObject":"PT4M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this exercise, we have to find the vertical asymptotes of f of x,"},{"Start":"00:04.950 ","End":"00:09.045","Text":"which is this rational function polynomial over polynomial."},{"Start":"00:09.045 ","End":"00:11.505","Text":"We already talked about rational functions."},{"Start":"00:11.505 ","End":"00:16.770","Text":"The place to look for suspects for vertical asymptotes is where the denominator 0."},{"Start":"00:16.770 ","End":"00:19.170","Text":"Let\u0027s try solving the equation where"},{"Start":"00:19.170 ","End":"00:24.495","Text":"the denominator x squared minus 3 x plus 2 is equal to 0."},{"Start":"00:24.495 ","End":"00:27.330","Text":"I\u0027m not going to waste time solving a quadratic equation."},{"Start":"00:27.330 ","End":"00:28.525","Text":"You all know to do this."},{"Start":"00:28.525 ","End":"00:30.150","Text":"I\u0027ll just tell you the solutions."},{"Start":"00:30.150 ","End":"00:34.995","Text":"The solutions are x equals 1 and x equals 2."},{"Start":"00:34.995 ","End":"00:39.115","Text":"These are going to be our 2 suspects for vertical asymptote."},{"Start":"00:39.115 ","End":"00:41.345","Text":"Let\u0027s take them 1 at a time."},{"Start":"00:41.345 ","End":"00:42.830","Text":"I\u0027ll take first of all,"},{"Start":"00:42.830 ","End":"00:48.830","Text":"x equals 1 and I\u0027ll try the limit on the right then on the left if need be."},{"Start":"00:48.830 ","End":"00:50.660","Text":"X goes to 1,"},{"Start":"00:50.660 ","End":"00:54.850","Text":"start off on the right of this function and see what we get."},{"Start":"00:54.850 ","End":"00:56.930","Text":"If we get plus or minus infinity,"},{"Start":"00:56.930 ","End":"00:58.685","Text":"then we found an asymptote."},{"Start":"00:58.685 ","End":"01:01.775","Text":"X squared minus 3x plus 2."},{"Start":"01:01.775 ","End":"01:05.165","Text":"Now if I just substitute x equals 1,"},{"Start":"01:05.165 ","End":"01:09.360","Text":"here I get 1 plus 1 minus 2 is 0."},{"Start":"01:09.360 ","End":"01:14.059","Text":"Here I get 1 minus 3 plus 2, which is also 0."},{"Start":"01:14.059 ","End":"01:18.920","Text":"Well, I expected that because these are the solutions that we have, 0 over 0."},{"Start":"01:18.920 ","End":"01:23.265","Text":"In this case, we can go and use L\u0027Hopital."},{"Start":"01:23.265 ","End":"01:26.375","Text":"I\u0027ll just erase this 0 over 0."},{"Start":"01:26.375 ","End":"01:28.235","Text":"Now that we\u0027ve checked that it is."},{"Start":"01:28.235 ","End":"01:37.520","Text":"I\u0027ll put that this is equal to by the 0 over 0 case of L\u0027Hopital to a new limit,"},{"Start":"01:37.520 ","End":"01:42.200","Text":"which is the fraction you get when you differentiate the top and the bottom."},{"Start":"01:42.200 ","End":"01:43.640","Text":"If we differentiate this,"},{"Start":"01:43.640 ","End":"01:46.225","Text":"we get 2x plus 1."},{"Start":"01:46.225 ","End":"01:49.935","Text":"We differentiate this we get 2x minus 3."},{"Start":"01:49.935 ","End":"01:53.655","Text":"This time, if we put x equals 1,"},{"Start":"01:53.655 ","End":"02:02.355","Text":"we\u0027ll get 3 over 2 plus 1 over 2 minus 3 is minus 1."},{"Start":"02:02.355 ","End":"02:06.290","Text":"In any event, it is minus 3 but the point"},{"Start":"02:06.290 ","End":"02:10.775","Text":"is that it\u0027s not equal to infinity or minus infinity."},{"Start":"02:10.775 ","End":"02:15.410","Text":"On the right, we don\u0027t have a vertical asymptote,"},{"Start":"02:15.410 ","End":"02:19.175","Text":"but we still get another chance to try the limit from the left."},{"Start":"02:19.175 ","End":"02:24.125","Text":"When x goes to 1 from the left of the same thing,"},{"Start":"02:24.125 ","End":"02:30.065","Text":"x squared plus x minus 2 over x squared minus 3x plus 2."},{"Start":"02:30.065 ","End":"02:34.730","Text":"What we get this time is if we put in 1 from the left,"},{"Start":"02:34.730 ","End":"02:37.085","Text":"we\u0027ll get exactly the same thing."},{"Start":"02:37.085 ","End":"02:40.900","Text":"There\u0027s no point checking this out because I can see that everywhere along the line,"},{"Start":"02:40.900 ","End":"02:42.725","Text":"if I put 1 from the left,"},{"Start":"02:42.725 ","End":"02:44.995","Text":"I\u0027m going to get exactly the same thing."},{"Start":"02:44.995 ","End":"02:47.060","Text":"Let me just erase that."},{"Start":"02:47.060 ","End":"02:56.210","Text":"Just make a note to myself that x equals 1 is not a vertical asymptote,"},{"Start":"02:56.210 ","End":"02:58.160","Text":"but we have 2 suspects."},{"Start":"02:58.160 ","End":"03:01.225","Text":"We have also the suspect x equals 2."},{"Start":"03:01.225 ","End":"03:03.885","Text":"Let\u0027s try x equals 2."},{"Start":"03:03.885 ","End":"03:06.855","Text":"Again, we\u0027ll try the limits first of all, from the right."},{"Start":"03:06.855 ","End":"03:12.320","Text":"X goes to 2 from the right of x squared plus x"},{"Start":"03:12.320 ","End":"03:18.630","Text":"minus 2 over x squared minus 3x plus 2."},{"Start":"03:18.630 ","End":"03:20.330","Text":"This time we get,"},{"Start":"03:20.330 ","End":"03:23.945","Text":"if we just naively substitute x equals 2,"},{"Start":"03:23.945 ","End":"03:28.845","Text":"here we get 2 squared plus 2 minus 2,"},{"Start":"03:28.845 ","End":"03:30.960","Text":"which comes out to be 4."},{"Start":"03:30.960 ","End":"03:38.270","Text":"On the denominator, we get 2 squared is 4 minus 6 plus 2, which is 0."},{"Start":"03:38.270 ","End":"03:42.680","Text":"Now, I could check if it\u0027s 0 plus or 0 minus,"},{"Start":"03:42.680 ","End":"03:44.375","Text":"but it doesn\u0027t matter."},{"Start":"03:44.375 ","End":"03:48.950","Text":"Either way, it\u0027s going to come out to be plus or minus infinity,"},{"Start":"03:48.950 ","End":"03:51.530","Text":"which means that we do have an asymptote at"},{"Start":"03:51.530 ","End":"03:54.695","Text":"x equals 2 and I don\u0027t even have to check from the left."},{"Start":"03:54.695 ","End":"04:00.260","Text":"I could check if it\u0027s plus or minus 0 because I could factorize this into x minus"},{"Start":"04:00.260 ","End":"04:06.195","Text":"1 and x minus 2 and then I could start computing when x is 2 plus."},{"Start":"04:06.195 ","End":"04:08.250","Text":"We could do it, but we don\u0027t need to know."},{"Start":"04:08.250 ","End":"04:10.910","Text":"It\u0027ll come out to be either plus infinity or minus infinity,"},{"Start":"04:10.910 ","End":"04:13.435","Text":"but I don\u0027t care either 1 is good for me."},{"Start":"04:13.435 ","End":"04:17.225","Text":"I can write that x equals 2, the line,"},{"Start":"04:17.225 ","End":"04:19.715","Text":"the vertical line is a vertical asymptote,"},{"Start":"04:19.715 ","End":"04:22.100","Text":"not even going to check from the 2 from the left,"},{"Start":"04:22.100 ","End":"04:23.330","Text":"this is enough for me,"},{"Start":"04:23.330 ","End":"04:30.970","Text":"is a vertical asymptote of the above function and we\u0027re done."}],"ID":5880},{"Watched":false,"Name":"Exercise 31","Duration":"2m 3s","ChapterTopicVideoID":5883,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5883.jpeg","UploadDate":"2016-05-02T06:54:12.4000000","DurationForVideoObject":"PT2M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this exercise, we have to find the vertical asymptotes of the function f,"},{"Start":"00:04.860 ","End":"00:07.590","Text":"which is natural log of x minus 1."},{"Start":"00:07.590 ","End":"00:10.995","Text":"First thing we need to do is look at the domain."},{"Start":"00:10.995 ","End":"00:13.260","Text":"The domain for the natural logarithm,"},{"Start":"00:13.260 ","End":"00:15.660","Text":"is that its argument must be bigger than 0."},{"Start":"00:15.660 ","End":"00:21.150","Text":"In other words, we must have that x minus 1 has to be bigger than 0,"},{"Start":"00:21.150 ","End":"00:25.410","Text":"which gives us that x is bigger than 1."},{"Start":"00:25.410 ","End":"00:29.540","Text":"This is the domain of the function f. Now,"},{"Start":"00:29.540 ","End":"00:33.830","Text":"when we have a domain with 1 or more endpoints,"},{"Start":"00:33.830 ","End":"00:38.465","Text":"then those endpoints are suspects for vertical asymptote."},{"Start":"00:38.465 ","End":"00:42.005","Text":"In this case, even though x is never equal to 1,"},{"Start":"00:42.005 ","End":"00:44.210","Text":"it\u0027s the edge of a domain."},{"Start":"00:44.210 ","End":"00:47.355","Text":"X equals 1 is a suspect,"},{"Start":"00:47.355 ","End":"00:49.430","Text":"for being a vertical asymptote."},{"Start":"00:49.430 ","End":"00:51.875","Text":"Let\u0027s just check it out."},{"Start":"00:51.875 ","End":"00:55.140","Text":"This time, we only have 1 limit to check,"},{"Start":"00:55.140 ","End":"00:56.610","Text":"and that is the limit from the right,"},{"Start":"00:56.610 ","End":"00:58.575","Text":"because if x is bigger than 1,"},{"Start":"00:58.575 ","End":"01:01.365","Text":"then it has to approach 1 from the right."},{"Start":"01:01.365 ","End":"01:03.855","Text":"It can\u0027t be less than 1, not defined."},{"Start":"01:03.855 ","End":"01:09.620","Text":"We need to check the limit as x goes to 1 from the right."},{"Start":"01:09.620 ","End":"01:13.040","Text":"It\u0027s only defined for bigger than 1 of f of x,"},{"Start":"01:13.040 ","End":"01:16.460","Text":"which is the natural log of x minus 1."},{"Start":"01:16.460 ","End":"01:20.485","Text":"See if we get infinity or minus infinity."},{"Start":"01:20.485 ","End":"01:23.555","Text":"If I put x equals 1 plus here,"},{"Start":"01:23.555 ","End":"01:27.740","Text":"I\u0027m going to get the natural log of 1 plus,"},{"Start":"01:27.740 ","End":"01:33.930","Text":"takeaway 1, which is the natural log of 0 plus."},{"Start":"01:33.930 ","End":"01:37.065","Text":"This is well-known to be minus infinity."},{"Start":"01:37.065 ","End":"01:39.980","Text":"In any event, whether it was plus or minus infinity,"},{"Start":"01:39.980 ","End":"01:41.284","Text":"it\u0027s a kind of infinity,"},{"Start":"01:41.284 ","End":"01:46.880","Text":"which means that x equals 1 is indeed a vertical asymptote or if you like,"},{"Start":"01:46.880 ","End":"01:48.545","Text":"the suspect is guilty."},{"Start":"01:48.545 ","End":"01:54.550","Text":"So x equals 1 is a vertical asymptote."},{"Start":"01:54.550 ","End":"01:56.390","Text":"It\u0027s actually on the right."},{"Start":"01:56.390 ","End":"01:58.115","Text":"It could only have been on the right."},{"Start":"01:58.115 ","End":"02:00.180","Text":"I\u0027ll just add that for more information,"},{"Start":"02:00.180 ","End":"02:01.565","Text":"although we don\u0027t need to."},{"Start":"02:01.565 ","End":"02:04.170","Text":"That\u0027s the answer. We\u0027re done."}],"ID":5881},{"Watched":false,"Name":"Exercise 32","Duration":"1m 52s","ChapterTopicVideoID":5884,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5884.jpeg","UploadDate":"2016-05-02T06:54:25.0370000","DurationForVideoObject":"PT1M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this exercise, we need to find"},{"Start":"00:02.400 ","End":"00:07.860","Text":"the vertical asymptote of the function natural log of 4 minus x."},{"Start":"00:07.860 ","End":"00:09.720","Text":"The important thing is the domain."},{"Start":"00:09.720 ","End":"00:14.520","Text":"A natural logarithm has a domain of positive numbers only."},{"Start":"00:14.520 ","End":"00:18.780","Text":"This means that we have to have 4 minus x positive,"},{"Start":"00:18.780 ","End":"00:23.610","Text":"and this comes out to be the same as x less than 4,"},{"Start":"00:23.610 ","End":"00:25.995","Text":"so this is our domain."},{"Start":"00:25.995 ","End":"00:29.205","Text":"Now, when we have a domain like this with an endpoint,"},{"Start":"00:29.205 ","End":"00:31.020","Text":"we suspect the endpoint,"},{"Start":"00:31.020 ","End":"00:32.835","Text":"in this case, x equals 4,"},{"Start":"00:32.835 ","End":"00:34.410","Text":"even though it\u0027s not part of the domain,"},{"Start":"00:34.410 ","End":"00:36.575","Text":"it\u0027s the endpoint of the domain."},{"Start":"00:36.575 ","End":"00:40.170","Text":"X equals 4 becomes a suspect to be"},{"Start":"00:40.170 ","End":"00:44.540","Text":"a vertical asymptote and what we do is check the one-sided limit."},{"Start":"00:44.540 ","End":"00:49.780","Text":"We can only check the limit as x goes to 4 from the left because x has to be less than 4."},{"Start":"00:49.780 ","End":"00:53.450","Text":"What we need to do is to check the limit as x goes to"},{"Start":"00:53.450 ","End":"00:57.740","Text":"4 from the left because of the less than of the function of x,"},{"Start":"00:57.740 ","End":"01:01.070","Text":"which is natural log of 4 minus x,"},{"Start":"01:01.070 ","End":"01:02.615","Text":"and what are we expecting?"},{"Start":"01:02.615 ","End":"01:04.520","Text":"If we get plus or minus infinity,"},{"Start":"01:04.520 ","End":"01:07.470","Text":"then yes, it\u0027s an asymptote, otherwise not."},{"Start":"01:07.470 ","End":"01:11.820","Text":"Let\u0027s see, if x is 4 teensy-weensy bit less,"},{"Start":"01:11.820 ","End":"01:13.485","Text":"just very close to 4,"},{"Start":"01:13.485 ","End":"01:18.640","Text":"so 4 minus teensy-weensy bit less than 4 is just a bit above 0."},{"Start":"01:18.640 ","End":"01:22.735","Text":"We get the natural log of 0 plus,"},{"Start":"01:22.735 ","End":"01:25.500","Text":"see if x is 3.9999,"},{"Start":"01:25.500 ","End":"01:29.385","Text":"then 4 minus it is 0.0000 something 1,"},{"Start":"01:29.385 ","End":"01:31.335","Text":"very tiny bit above 0."},{"Start":"01:31.335 ","End":"01:34.430","Text":"The natural log of 0 plus is a well-known limit,"},{"Start":"01:34.430 ","End":"01:35.915","Text":"and it\u0027s minus infinity."},{"Start":"01:35.915 ","End":"01:38.989","Text":"Any event we were expecting the plus or minus infinity,"},{"Start":"01:38.989 ","End":"01:42.155","Text":"which confirms that our suspect is guilty, so to speak."},{"Start":"01:42.155 ","End":"01:44.270","Text":"In other words, x equals 4,"},{"Start":"01:44.270 ","End":"01:49.275","Text":"the vertical line is indeed a vertical asymptote,"},{"Start":"01:49.275 ","End":"01:52.570","Text":"and we found it, and we\u0027re done."}],"ID":5882},{"Watched":false,"Name":"Exercise 33","Duration":"2m 12s","ChapterTopicVideoID":5885,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5885.jpeg","UploadDate":"2016-05-02T06:54:39.0770000","DurationForVideoObject":"PT2M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.495","Text":"In this exercise, we have to find the vertical asymptotes"},{"Start":"00:03.495 ","End":"00:07.440","Text":"of f of x which is 1 over the square root of x minus 4."},{"Start":"00:07.440 ","End":"00:10.365","Text":"The important thing here is to find the domain."},{"Start":"00:10.365 ","End":"00:12.570","Text":"Now, there\u0027s 2 things we have to watch out for."},{"Start":"00:12.570 ","End":"00:13.650","Text":"There is a square root,"},{"Start":"00:13.650 ","End":"00:16.710","Text":"so its argument has to be bigger or equal to 0,"},{"Start":"00:16.710 ","End":"00:19.815","Text":"and we also have a denominator which mustn\u0027t be 0."},{"Start":"00:19.815 ","End":"00:20.970","Text":"As for the square root,"},{"Start":"00:20.970 ","End":"00:27.225","Text":"x has to be bigger or equal to 4 in order for x minus 4 to be bigger or equal to 0."},{"Start":"00:27.225 ","End":"00:29.430","Text":"On the other hand, it mustn\u0027t be 0."},{"Start":"00:29.430 ","End":"00:30.950","Text":"Basically, if you think about it,"},{"Start":"00:30.950 ","End":"00:34.694","Text":"we\u0027re left with x bigger than 4 for the domain,"},{"Start":"00:34.694 ","End":"00:37.530","Text":"and then under the square root we\u0027ll have something strictly"},{"Start":"00:37.530 ","End":"00:40.645","Text":"positive and we won\u0027t have a 0 in the denominator."},{"Start":"00:40.645 ","End":"00:43.740","Text":"As I\u0027ve mentioned before when we have an interval"},{"Start":"00:43.740 ","End":"00:46.950","Text":"which has an endpoint as we do in this case because"},{"Start":"00:46.950 ","End":"00:50.210","Text":"the endpoint of the interval is x equals 4 then we"},{"Start":"00:50.210 ","End":"00:53.960","Text":"check that endpoint as a suspect for a vertical asymptote."},{"Start":"00:53.960 ","End":"00:57.910","Text":"X equals 4 is now a suspect,"},{"Start":"00:57.910 ","End":"01:04.190","Text":"and we\u0027ll check it out by testing the limit as x goes to this point of f of x,"},{"Start":"01:04.190 ","End":"01:06.410","Text":"and see if we get plus or minus infinity."},{"Start":"01:06.410 ","End":"01:08.000","Text":"But since x is bigger than 4,"},{"Start":"01:08.000 ","End":"01:10.445","Text":"we can only check the limit from the right."},{"Start":"01:10.445 ","End":"01:12.740","Text":"We\u0027re going to check x goes to 4 from"},{"Start":"01:12.740 ","End":"01:18.170","Text":"the right of 1 over the square root of x minus 4."},{"Start":"01:18.170 ","End":"01:20.180","Text":"Now, this is going to equal,"},{"Start":"01:20.180 ","End":"01:22.670","Text":"x is a tiny bit bigger than 4,"},{"Start":"01:22.670 ","End":"01:29.390","Text":"so what we\u0027re going to get is 1 over the square root of 4 plus minus 4,"},{"Start":"01:29.390 ","End":"01:33.940","Text":"which is equal to 1 over the square root of 0 plus,"},{"Start":"01:33.940 ","End":"01:37.460","Text":"which is the square root of 0 plus something very tiny and"},{"Start":"01:37.460 ","End":"01:40.820","Text":"positive is still tiny and positive 1 over 0 plus."},{"Start":"01:40.820 ","End":"01:44.585","Text":"1 over something very tiny and positive is very large,"},{"Start":"01:44.585 ","End":"01:46.700","Text":"so this basically is infinity."},{"Start":"01:46.700 ","End":"01:50.615","Text":"This is a shorthand way of using the infinity and the 0 plus and so on."},{"Start":"01:50.615 ","End":"01:55.140","Text":"What it means is that we do have a limit of infinity as x goes to 4 from the right,"},{"Start":"01:55.140 ","End":"01:58.010","Text":"and so our suspect is indeed a vertical asymptote."},{"Start":"01:58.010 ","End":"02:07.160","Text":"I can write that the line x equals 4 is indeed a vertical asymptote for this function,"},{"Start":"02:07.160 ","End":"02:10.415","Text":"and it\u0027s from the right, but I\u0027ll leave it at that."},{"Start":"02:10.415 ","End":"02:13.320","Text":"We found our asymptote and we\u0027re done."}],"ID":5883},{"Watched":false,"Name":"Exercise 34","Duration":"1m 25s","ChapterTopicVideoID":5886,"CourseChapterTopicPlaylistID":65383,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5886.jpeg","UploadDate":"2016-05-02T06:54:47.8500000","DurationForVideoObject":"PT1M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.670","Text":"In this exercise, we have to find the vertical asymptotes,"},{"Start":"00:03.670 ","End":"00:05.310","Text":"if any, of f of x,"},{"Start":"00:05.310 ","End":"00:07.410","Text":"which is the square root of x minus 1."},{"Start":"00:07.410 ","End":"00:10.380","Text":"The important thing here is the domain."},{"Start":"00:10.380 ","End":"00:14.790","Text":"The domain is x minus 1 bigger or equal to 0,"},{"Start":"00:14.790 ","End":"00:19.230","Text":"in other words, x bigger or equal to 1, that\u0027s our domain."},{"Start":"00:19.230 ","End":"00:21.660","Text":"When we have a domain which has an end point,"},{"Start":"00:21.660 ","End":"00:25.005","Text":"the only suspect would be the end point of that domain."},{"Start":"00:25.005 ","End":"00:28.110","Text":"So x equals 1 is a suspect."},{"Start":"00:28.110 ","End":"00:31.724","Text":"Let\u0027s just check it out and see if the limit on either side,"},{"Start":"00:31.724 ","End":"00:37.880","Text":"and when the limit as x goes to 1 from the right or from the left later of f of x,"},{"Start":"00:37.880 ","End":"00:40.100","Text":"which is the square root of x minus 1 and see"},{"Start":"00:40.100 ","End":"00:42.530","Text":"if it\u0027s equal to infinity or minus infinity."},{"Start":"00:42.530 ","End":"00:45.200","Text":"Anyway, this limit is equal to,"},{"Start":"00:45.200 ","End":"00:48.595","Text":"there\u0027s no problem here and just substituting x equals 1,"},{"Start":"00:48.595 ","End":"00:52.080","Text":"and in fact, it will work also when we\u0027re going from the right or from the left."},{"Start":"00:52.080 ","End":"00:54.005","Text":"But we can\u0027t go from the left actually,"},{"Start":"00:54.005 ","End":"00:55.730","Text":"because x is bigger or equal to 1."},{"Start":"00:55.730 ","End":"00:57.800","Text":"We only can check the limit from the right."},{"Start":"00:57.800 ","End":"01:00.925","Text":"This is equal to square root of 1 minus 1,"},{"Start":"01:00.925 ","End":"01:02.699","Text":"which is square root of 0."},{"Start":"01:02.699 ","End":"01:03.980","Text":"If you really want it accurate,"},{"Start":"01:03.980 ","End":"01:06.095","Text":"I could say 1 plus minus 1,"},{"Start":"01:06.095 ","End":"01:08.090","Text":"which is the square root of 0 plus,"},{"Start":"01:08.090 ","End":"01:11.245","Text":"which is just 0 plus which is 0 basically."},{"Start":"01:11.245 ","End":"01:15.425","Text":"It\u0027s certainly not equal to infinity or minus infinity."},{"Start":"01:15.425 ","End":"01:18.110","Text":"Our suspect turned out to be innocent."},{"Start":"01:18.110 ","End":"01:23.540","Text":"There is no vertical asymptote."},{"Start":"01:23.540 ","End":"01:26.310","Text":"That\u0027s it. We\u0027re done."}],"ID":5884}],"Thumbnail":null,"ID":65383},{"Name":"Horizontal Asymptotes","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Horizontal Asymptotes","Duration":"4m 20s","ChapterTopicVideoID":8273,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8273.jpeg","UploadDate":"2019-10-29T12:49:53.0530000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"In this clip, I\u0027m going to be talking about horizontal asymptotes."},{"Start":"00:04.035 ","End":"00:06.810","Text":"Asymptotes are first of all, lines."},{"Start":"00:06.810 ","End":"00:09.345","Text":"I just want to remind you the equation of a line."},{"Start":"00:09.345 ","End":"00:12.915","Text":"In general, it\u0027s y equals ax plus b."},{"Start":"00:12.915 ","End":"00:14.970","Text":"But if it\u0027s going to be horizontal,"},{"Start":"00:14.970 ","End":"00:19.845","Text":"then the a will be 0 and we\u0027ll have just y equals b."},{"Start":"00:19.845 ","End":"00:21.330","Text":"Now if I have a function,"},{"Start":"00:21.330 ","End":"00:25.080","Text":"y equals f of x there\u0027s actually"},{"Start":"00:25.080 ","End":"00:29.790","Text":"2 kinds of horizontal asymptotes there\u0027s a right and a left horizontal asymptotes."},{"Start":"00:29.790 ","End":"00:33.010","Text":"Let me start off with the right 1 first."},{"Start":"00:33.010 ","End":"00:38.660","Text":"If I have the limit as x goes to infinity of f of x,"},{"Start":"00:38.660 ","End":"00:43.520","Text":"and if it comes out to be some finite number b,"},{"Start":"00:43.520 ","End":"00:51.710","Text":"then we say that Y equals b is a horizontal asymptote for f from the right,"},{"Start":"00:51.710 ","End":"00:56.840","Text":"there\u0027s a similar definition for a left horizontal asymptote that if we have"},{"Start":"00:56.840 ","End":"01:03.275","Text":"the limit as x goes to minus infinity of f of x is some number b."},{"Start":"01:03.275 ","End":"01:06.410","Text":"Then we say that Y equals b is"},{"Start":"01:06.410 ","End":"01:11.915","Text":"a left horizontal asymptote for f. I want to give you an example."},{"Start":"01:11.915 ","End":"01:21.230","Text":"Let\u0027s take the function y equals x minus 1 over 2x. That\u0027s my f of x."},{"Start":"01:21.230 ","End":"01:29.810","Text":"If I take the limit as x goes to infinity of x minus 1 over 2x,"},{"Start":"01:29.810 ","End":"01:34.130","Text":"what we can do is divide numerator and denominator here by x."},{"Start":"01:34.130 ","End":"01:36.500","Text":"It\u0027s going to infinity, so it\u0027s far away from 0,"},{"Start":"01:36.500 ","End":"01:37.955","Text":"so that\u0027s no problem."},{"Start":"01:37.955 ","End":"01:46.474","Text":"It\u0027s the limit as x goes to infinity of 1 minus 1/x over 2."},{"Start":"01:46.474 ","End":"01:50.960","Text":"Now 1/x goes to 0 at infinity,"},{"Start":"01:50.960 ","End":"01:54.965","Text":"so the limit comes out to be just 1/2."},{"Start":"01:54.965 ","End":"02:02.240","Text":"I can say that y equals 1/2 is the horizontal asymptote on the right."},{"Start":"02:02.240 ","End":"02:05.629","Text":"The thing is that if I took a minus infinity,"},{"Start":"02:05.629 ","End":"02:07.520","Text":"I\u0027d get exactly the same thing."},{"Start":"02:07.520 ","End":"02:10.430","Text":"In other words, if I took minus infinity here,"},{"Start":"02:10.430 ","End":"02:15.320","Text":"minus infinity here, 1 over minus infinity will still be 0."},{"Start":"02:15.320 ","End":"02:20.394","Text":"This is both on the right and on the left."},{"Start":"02:20.394 ","End":"02:23.555","Text":"Here\u0027s a picture of what\u0027s happening here."},{"Start":"02:23.555 ","End":"02:25.370","Text":"There\u0027s also a vertical asymptote,"},{"Start":"02:25.370 ","End":"02:26.720","Text":"but never mind that."},{"Start":"02:26.720 ","End":"02:28.580","Text":"The point is that on the right,"},{"Start":"02:28.580 ","End":"02:30.110","Text":"when x goes to infinity,"},{"Start":"02:30.110 ","End":"02:34.700","Text":"we see the curve approaches a horizontal line, y equals 1/2."},{"Start":"02:34.700 ","End":"02:38.105","Text":"The same thing also happens when x goes to minus infinity."},{"Start":"02:38.105 ","End":"02:41.360","Text":"It also gets near y equals 1/2."},{"Start":"02:41.360 ","End":"02:42.725","Text":"The limit is 1/2."},{"Start":"02:42.725 ","End":"02:48.305","Text":"That\u0027s quite typical for the right asymptote and the left asymptote to be the same,"},{"Start":"02:48.305 ","End":"02:50.510","Text":"there are several possibilities."},{"Start":"02:50.510 ","End":"02:53.495","Text":"The function might have no horizontal asymptotes at all,"},{"Start":"02:53.495 ","End":"02:58.985","Text":"or it might have 2 different ones or it might have 1 on the right and non on the left."},{"Start":"02:58.985 ","End":"03:04.475","Text":"Here\u0027s an example where a function has a horizontal asymptote only on the right,"},{"Start":"03:04.475 ","End":"03:07.970","Text":"the function is 3 minus 2^minus x."},{"Start":"03:07.970 ","End":"03:10.070","Text":"When x goes to infinity,"},{"Start":"03:10.070 ","End":"03:12.080","Text":"2^minus x goes to 0,"},{"Start":"03:12.080 ","End":"03:13.865","Text":"it\u0027s 1 over 2^x."},{"Start":"03:13.865 ","End":"03:16.400","Text":"But when x goes to minus infinity,"},{"Start":"03:16.400 ","End":"03:18.185","Text":"we get 2 to the infinity."},{"Start":"03:18.185 ","End":"03:20.000","Text":"It doesn\u0027t go to anything finite."},{"Start":"03:20.000 ","End":"03:23.815","Text":"We have a horizontal asymptote, y equals 3."},{"Start":"03:23.815 ","End":"03:25.830","Text":"That\u0027s this goes to 0 at infinity,"},{"Start":"03:25.830 ","End":"03:29.360","Text":"we\u0027re left with just 3 and it only has 1 on one side."},{"Start":"03:29.360 ","End":"03:33.410","Text":"Here\u0027s another example of the function it\u0027s not defined everywhere."},{"Start":"03:33.410 ","End":"03:40.880","Text":"It\u0027s only defined for x bigger or equal to 3 or less than or equal to minus 3,"},{"Start":"03:40.880 ","End":"03:43.190","Text":"because x squared has to be bigger than 9."},{"Start":"03:43.190 ","End":"03:45.260","Text":"Because when we take x to infinity,"},{"Start":"03:45.260 ","End":"03:46.895","Text":"x is in the range,"},{"Start":"03:46.895 ","End":"03:48.830","Text":"turns out at infinity,"},{"Start":"03:48.830 ","End":"03:51.589","Text":"the function tends to fall."},{"Start":"03:51.589 ","End":"03:56.240","Text":"The right asymptote would be y equals 4."},{"Start":"03:56.240 ","End":"03:57.635","Text":"But on the other side,"},{"Start":"03:57.635 ","End":"04:03.020","Text":"left asymptote turns out to be y equals minus 4."},{"Start":"04:03.020 ","End":"04:06.080","Text":"It\u0027s possible for 2 asymptotes to be different,"},{"Start":"04:06.080 ","End":"04:11.365","Text":"only 1 asymptote or 2 asymptotes to be the same."},{"Start":"04:11.365 ","End":"04:13.670","Text":"Of course there could be no asymptote at all,"},{"Start":"04:13.670 ","End":"04:18.605","Text":"I\u0027m not getting into too much detail because there are solved exercises after this."},{"Start":"04:18.605 ","End":"04:21.420","Text":"I\u0027ll end the clip here."}],"ID":8448},{"Watched":false,"Name":"Exercise 1","Duration":"1m 54s","ChapterTopicVideoID":5887,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5887.jpeg","UploadDate":"2016-05-02T07:30:52.5570000","DurationForVideoObject":"PT1M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.220","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:05.220 ","End":"00:08.115","Text":"which is x squared minus 2x plus 5."},{"Start":"00:08.115 ","End":"00:12.330","Text":"Now, you could just quote the fact that polynomials,"},{"Start":"00:12.330 ","End":"00:13.710","Text":"and this is a polynomial,"},{"Start":"00:13.710 ","End":"00:16.110","Text":"they have no asymptotes at all of any kind."},{"Start":"00:16.110 ","End":"00:20.565","Text":"You could just write polynomial has no asymptotes and you\u0027d be done with it."},{"Start":"00:20.565 ","End":"00:25.395","Text":"But I\u0027d like to show you in this case why we don\u0027t have a horizontal asymptote."},{"Start":"00:25.395 ","End":"00:30.210","Text":"What I have to do is take the limit of f as x goes to infinity to find"},{"Start":"00:30.210 ","End":"00:35.805","Text":"the right horizontal asymptote and minus infinity for the left horizontal asymptote."},{"Start":"00:35.805 ","End":"00:37.155","Text":"Let\u0027s do that."},{"Start":"00:37.155 ","End":"00:38.310","Text":"But I\u0027ll first of all,"},{"Start":"00:38.310 ","End":"00:45.740","Text":"also rewrite the function as x times x minus 2 plus 5."},{"Start":"00:45.740 ","End":"00:47.975","Text":"Now when x goes to infinity,"},{"Start":"00:47.975 ","End":"00:54.500","Text":"this is equal to x is infinity and infinity minus 2 plus 5."},{"Start":"00:54.500 ","End":"00:56.930","Text":"Let\u0027s do it arithmetic of infinity."},{"Start":"00:56.930 ","End":"01:00.410","Text":"Infinity minus 2 is also infinity."},{"Start":"01:00.410 ","End":"01:04.400","Text":"I get infinity times infinity plus 5."},{"Start":"01:04.400 ","End":"01:06.785","Text":"Infinity times infinity is infinity,"},{"Start":"01:06.785 ","End":"01:09.700","Text":"infinity plus 5 is infinity."},{"Start":"01:09.700 ","End":"01:11.355","Text":"This is not a number."},{"Start":"01:11.355 ","End":"01:12.480","Text":"For it to be an asymptote,"},{"Start":"01:12.480 ","End":"01:15.185","Text":"this has to be an actual number but not infinity."},{"Start":"01:15.185 ","End":"01:20.990","Text":"Similarly, if we do the same thing for x goes to minus infinity,"},{"Start":"01:20.990 ","End":"01:23.265","Text":"the same thing will be true except that here we\u0027ll"},{"Start":"01:23.265 ","End":"01:25.655","Text":"have minus infinity and minus infinity,"},{"Start":"01:25.655 ","End":"01:27.730","Text":"minus infinity times minus infinity,"},{"Start":"01:27.730 ","End":"01:29.450","Text":"we\u0027ll get the same thing from this point."},{"Start":"01:29.450 ","End":"01:31.445","Text":"Infinity plus 5 is infinity,"},{"Start":"01:31.445 ","End":"01:34.910","Text":"so this is also equal to infinity."},{"Start":"01:34.910 ","End":"01:37.130","Text":"We get infinity on the right,"},{"Start":"01:37.130 ","End":"01:38.270","Text":"infinity on the left,"},{"Start":"01:38.270 ","End":"01:40.865","Text":"in no case do we get a constant actual number,"},{"Start":"01:40.865 ","End":"01:42.529","Text":"so it doesn\u0027t have an asymptote."},{"Start":"01:42.529 ","End":"01:48.380","Text":"At the end, we\u0027d write something like no horizontal asymptote."},{"Start":"01:48.380 ","End":"01:54.630","Text":"But you could just quote straight away that a polynomial has no asymptotes. Done."}],"ID":5885},{"Watched":false,"Name":"Exercise 2","Duration":"29s","ChapterTopicVideoID":5888,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5888.jpeg","UploadDate":"2016-05-02T07:30:56.4130000","DurationForVideoObject":"PT29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"In this exercise, we have to find the horizontal asymptotes of the function f of x,"},{"Start":"00:05.280 ","End":"00:07.290","Text":"which is x cubed minus 3x."},{"Start":"00:07.290 ","End":"00:08.520","Text":"I\u0027ve copied it here."},{"Start":"00:08.520 ","End":"00:14.850","Text":"Notice that the function f is a polynomial and it\u0027s well known or should be,"},{"Start":"00:14.850 ","End":"00:19.410","Text":"that a polynomial has no asymptotes of any kind."},{"Start":"00:19.410 ","End":"00:22.260","Text":"Therefore, f has no,"},{"Start":"00:22.260 ","End":"00:26.280","Text":"in particular, horizontal asymptotes."},{"Start":"00:26.280 ","End":"00:29.890","Text":"There\u0027s nothing more to be said. We\u0027re done."}],"ID":5886},{"Watched":false,"Name":"Exercise 3","Duration":"32s","ChapterTopicVideoID":5889,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5889.jpeg","UploadDate":"2016-05-02T07:31:00.8570000","DurationForVideoObject":"PT32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.730","Text":"In this exercise, we need to find the horizontal asymptotes of the following function,"},{"Start":"00:05.730 ","End":"00:09.300","Text":"2x cubed minus 15x squared plus 24x minus 1."},{"Start":"00:09.300 ","End":"00:13.050","Text":"Notice that the function f is a polynomial."},{"Start":"00:13.050 ","End":"00:15.720","Text":"Now, it is known, and you should know,"},{"Start":"00:15.720 ","End":"00:19.680","Text":"that a polynomial has no asymptotes of any kind."},{"Start":"00:19.680 ","End":"00:23.625","Text":"Therefore, in particular, it has no horizontal asymptotes."},{"Start":"00:23.625 ","End":"00:30.315","Text":"F has no horizontal asymptotes and there\u0027s nothing more to be said."},{"Start":"00:30.315 ","End":"00:33.010","Text":"We\u0027re done. There are no asymptotes."}],"ID":5887},{"Watched":false,"Name":"Exercise 4","Duration":"25s","ChapterTopicVideoID":5890,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5890.jpeg","UploadDate":"2016-05-02T07:31:04.1370000","DurationForVideoObject":"PT25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, we are asked to find"},{"Start":"00:03.060 ","End":"00:08.145","Text":"the horizontal asymptotes of this function x to the fourth minus 2x cubed."},{"Start":"00:08.145 ","End":"00:13.260","Text":"Notice that this function is a polynomial and because it\u0027s a polynomial,"},{"Start":"00:13.260 ","End":"00:15.930","Text":"it doesn\u0027t have any asymptotes because it is known"},{"Start":"00:15.930 ","End":"00:18.750","Text":"that polynomials don\u0027t have asymptotes."},{"Start":"00:18.750 ","End":"00:21.225","Text":"We could just write something like that."},{"Start":"00:21.225 ","End":"00:26.020","Text":"Something like f has no horizontal asymptotes and that\u0027s all we\u0027re done."}],"ID":5888},{"Watched":false,"Name":"Exercise 5","Duration":"32s","ChapterTopicVideoID":5891,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5891.jpeg","UploadDate":"2016-05-02T07:31:07.6670000","DurationForVideoObject":"PT32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.300","Text":"In this exercise, we\u0027re given the function f of x equals 3x^5 minus 20x cubed,"},{"Start":"00:06.300 ","End":"00:09.405","Text":"then we have to find its horizontal asymptotes."},{"Start":"00:09.405 ","End":"00:11.955","Text":"Now, if you look at the form of the function f,"},{"Start":"00:11.955 ","End":"00:13.650","Text":"you see it\u0027s a polynomial,"},{"Start":"00:13.650 ","End":"00:17.670","Text":"just the combination of coefficients with powers of x."},{"Start":"00:17.670 ","End":"00:22.635","Text":"It is well known that a polynomial has no asymptotes at all."},{"Start":"00:22.635 ","End":"00:25.980","Text":"In particular, it has no horizontal asymptotes."},{"Start":"00:25.980 ","End":"00:29.760","Text":"You can just quote that a polynomial has no horizontal asymptotes,"},{"Start":"00:29.760 ","End":"00:33.100","Text":"so there\u0027s nothing to find. We\u0027re done."}],"ID":5889},{"Watched":false,"Name":"Exercise 6","Duration":"1m 23s","ChapterTopicVideoID":5892,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5892.jpeg","UploadDate":"2016-05-02T07:31:16.8930000","DurationForVideoObject":"PT1M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this exercise, we have to find"},{"Start":"00:02.070 ","End":"00:03.840","Text":"the horizontal asymptotes"},{"Start":"00:03.840 ","End":"00:05.190","Text":"of the function f of x,"},{"Start":"00:05.190 ","End":"00:07.920","Text":"which is x over x squared plus 3."},{"Start":"00:07.920 ","End":"00:09.720","Text":"Now, the main thing to note here"},{"Start":"00:09.720 ","End":"00:11.933","Text":"is that f is a rational function,"},{"Start":"00:11.933 ","End":"00:13.320","Text":"and in case you\u0027ve forgotten"},{"Start":"00:13.320 ","End":"00:14.640","Text":"what a rational function is,"},{"Start":"00:14.640 ","End":"00:17.325","Text":"it\u0027s a polynomial over a polynomial."},{"Start":"00:17.325 ","End":"00:19.665","Text":"Now, a polynomial has a degree,"},{"Start":"00:19.665 ","End":"00:21.330","Text":"and in this case,"},{"Start":"00:21.330 ","End":"00:23.910","Text":"the polynomial in the numerator"},{"Start":"00:23.910 ","End":"00:26.730","Text":"has degree 1 and the polynomial"},{"Start":"00:26.730 ","End":"00:29.840","Text":"in the denominator has degree 2."},{"Start":"00:29.840 ","End":"00:32.600","Text":"The degree is just the highest power of x that we get."},{"Start":"00:32.600 ","End":"00:34.010","Text":"Here it\u0027s x to the 1,"},{"Start":"00:34.010 ","End":"00:35.555","Text":"and here it\u0027s x to the 2,"},{"Start":"00:35.555 ","End":"00:38.060","Text":"so we have a degree 1 over degree 2."},{"Start":"00:38.060 ","End":"00:41.600","Text":"Also note that 1 is less than 2,"},{"Start":"00:41.600 ","End":"00:42.800","Text":"by which I mean the degree"},{"Start":"00:42.800 ","End":"00:44.240","Text":"in the numerator is less than"},{"Start":"00:44.240 ","End":"00:47.045","Text":"the degree in the denominator."},{"Start":"00:47.045 ","End":"00:48.980","Text":"Now, when this occurs,"},{"Start":"00:48.980 ","End":"00:52.700","Text":"it\u0027s well known that the only asymptote"},{"Start":"00:52.700 ","End":"00:54.620","Text":"that there is the x-axis."},{"Start":"00:54.620 ","End":"00:56.990","Text":"In other words, y equals zero and y"},{"Start":"00:56.990 ","End":"01:01.600","Text":"equals zero is the horizontal asymptote."},{"Start":"01:01.600 ","End":"01:05.240","Text":"By quoting this theorem proposition,"},{"Start":"01:05.240 ","End":"01:08.000","Text":"whatever, we can end the exercise,"},{"Start":"01:08.000 ","End":"01:10.265","Text":"we just say we have a rational function."},{"Start":"01:10.265 ","End":"01:12.020","Text":"The degree in the numerator is less than"},{"Start":"01:12.020 ","End":"01:13.735","Text":"the degree in the denominator,"},{"Start":"01:13.735 ","End":"01:18.050","Text":"so the x-axis is the only horizontal asymptote and by the way,"},{"Start":"01:18.050 ","End":"01:19.640","Text":"it\u0027s the asymptote both at the plus"},{"Start":"01:19.640 ","End":"01:22.927","Text":"infinity side and the minus infinity side."},{"Start":"01:22.927 ","End":"01:24.690","Text":"Done."}],"ID":5890},{"Watched":false,"Name":"Exercise 7","Duration":"1m 4s","ChapterTopicVideoID":5893,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5893.jpeg","UploadDate":"2016-05-02T07:31:23.6600000","DurationForVideoObject":"PT1M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we have to find the horizontal asymptotes for this function,"},{"Start":"00:05.130 ","End":"00:08.175","Text":"x over x^2 plus x plus 1."},{"Start":"00:08.175 ","End":"00:12.960","Text":"The main thing to notice is that f is a rational function."},{"Start":"00:12.960 ","End":"00:15.430","Text":"In case you\u0027ve forgotten what rational function is,"},{"Start":"00:15.430 ","End":"00:17.850","Text":"it\u0027s a polynomial over a polynomial."},{"Start":"00:17.850 ","End":"00:19.676","Text":"Now, a polynomial has a degree,"},{"Start":"00:19.676 ","End":"00:23.400","Text":"and in this case, the numerator has degree 1"},{"Start":"00:23.400 ","End":"00:26.285","Text":"and the denominator has degree 2."},{"Start":"00:26.285 ","End":"00:27.812","Text":"Degree is the highest power of x."},{"Start":"00:27.812 ","End":"00:29.614","Text":"Here, x^2 is the highest power."},{"Start":"00:29.614 ","End":"00:32.150","Text":"Here, x^1 is the highest power."},{"Start":"00:32.150 ","End":"00:34.820","Text":"Also, note that 1 is less than 2,"},{"Start":"00:34.820 ","End":"00:39.320","Text":"by which I mean the degree in the numerator is less than the degree in the denominator."},{"Start":"00:39.320 ","End":"00:40.940","Text":"Now it is known, and you can quote it,"},{"Start":"00:40.940 ","End":"00:42.215","Text":"that when this occurs,"},{"Start":"00:42.215 ","End":"00:46.360","Text":"then the x-axis is the only asymptote."},{"Start":"00:46.360 ","End":"00:48.750","Text":"x-axis is called y equals 0,"},{"Start":"00:48.750 ","End":"00:50.825","Text":"if you want to give the equation of the line,"},{"Start":"00:50.825 ","End":"00:53.937","Text":"is the horizontal asymptote."},{"Start":"00:53.937 ","End":"00:55.035","Text":"We\u0027re done."},{"Start":"00:55.035 ","End":"00:59.078","Text":"But I just might add that this horizontal asymptote is also on"},{"Start":"00:59.078 ","End":"01:02.843","Text":"the plus infinity side as well as on the minus infinity side."},{"Start":"01:02.843 ","End":"01:04.980","Text":"That\u0027s it."}],"ID":5891},{"Watched":false,"Name":"Exercise 8","Duration":"52s","ChapterTopicVideoID":5894,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5894.jpeg","UploadDate":"2016-05-02T07:31:29.2370000","DurationForVideoObject":"PT52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this exercise, we have to find the horizontal asymptotes"},{"Start":"00:03.660 ","End":"00:07.605","Text":"of the function x minus 1 over x squared."},{"Start":"00:07.605 ","End":"00:12.490","Text":"What I\u0027d like you to notice here is that f is a rational function."},{"Start":"00:12.490 ","End":"00:15.860","Text":"Rational function is a polynomial over a polynomial,"},{"Start":"00:15.860 ","End":"00:18.260","Text":"and polynomials have degrees,"},{"Start":"00:18.260 ","End":"00:22.970","Text":"its rational function has a polynomial of degree 1 in the numerator,"},{"Start":"00:22.970 ","End":"00:24.500","Text":"its highest power is 1."},{"Start":"00:24.500 ","End":"00:26.750","Text":"In the denominator the degree is 2,"},{"Start":"00:26.750 ","End":"00:30.890","Text":"the highest power is 2 and 1 is less than 2."},{"Start":"00:30.890 ","End":"00:33.950","Text":"I just mean that the degree in the numerator is less than the degree in"},{"Start":"00:33.950 ","End":"00:36.110","Text":"the denominator and it\u0027s well known in"},{"Start":"00:36.110 ","End":"00:39.365","Text":"this case and you can just quote it that in such a case,"},{"Start":"00:39.365 ","End":"00:42.980","Text":"there is an asymptote and there\u0027s only 1 asymptote,"},{"Start":"00:42.980 ","End":"00:44.870","Text":"and that is the x axis."},{"Start":"00:44.870 ","End":"00:47.180","Text":"In other words, y equals 0,"},{"Start":"00:47.180 ","End":"00:52.920","Text":"which is the x-axis is the horizontal asymptote. We\u0027re done."}],"ID":5892},{"Watched":false,"Name":"Exercise 9","Duration":"1m 44s","ChapterTopicVideoID":5895,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5895.jpeg","UploadDate":"2016-05-02T07:31:41.6670000","DurationForVideoObject":"PT1M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.620","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.620 ","End":"00:08.595","Text":"which is 2x squared over x plus 1 squared."},{"Start":"00:08.595 ","End":"00:11.910","Text":"Note that f of x is a rational function."},{"Start":"00:11.910 ","End":"00:16.290","Text":"In case you\u0027ve forgotten the rational function is a polynomial over a polynomial."},{"Start":"00:16.290 ","End":"00:18.375","Text":"To expand this, you\u0027ll get a polynomial."},{"Start":"00:18.375 ","End":"00:24.765","Text":"Notice also that the degree of the polynomial in the numerator is 2."},{"Start":"00:24.765 ","End":"00:28.005","Text":"In the denominator we also have a degree 2,"},{"Start":"00:28.005 ","End":"00:29.325","Text":"because if we expand it,"},{"Start":"00:29.325 ","End":"00:32.170","Text":"it\u0027s x squared plus 2x plus 1."},{"Start":"00:32.170 ","End":"00:35.260","Text":"The degree at the top is 2 and the bottom is 2,"},{"Start":"00:35.260 ","End":"00:37.245","Text":"and these are equal degrees."},{"Start":"00:37.245 ","End":"00:38.370","Text":"Now in this case,"},{"Start":"00:38.370 ","End":"00:42.160","Text":"it is well known that if we take the leading coefficients,"},{"Start":"00:42.160 ","End":"00:44.660","Text":"we have 2x squared here,"},{"Start":"00:44.660 ","End":"00:46.580","Text":"and here we have x squared."},{"Start":"00:46.580 ","End":"00:51.050","Text":"Let me write it as 1x squared plus 2x plus 1."},{"Start":"00:51.050 ","End":"00:54.395","Text":"I\u0027m writing these where the highest power is at the beginning."},{"Start":"00:54.395 ","End":"00:56.720","Text":"X squared is the leading term,"},{"Start":"00:56.720 ","End":"01:01.715","Text":"and that has degree 2 and 1x squared here is the leading term and it has degree 2."},{"Start":"01:01.715 ","End":"01:04.805","Text":"The theorem or proposition says that in this case,"},{"Start":"01:04.805 ","End":"01:06.365","Text":"where we have equal degrees,"},{"Start":"01:06.365 ","End":"01:09.620","Text":"if I take the leading coefficient,"},{"Start":"01:09.620 ","End":"01:12.340","Text":"which is here 2, and here it\u0027s 1,"},{"Start":"01:12.340 ","End":"01:14.165","Text":"then there is an asymptote,"},{"Start":"01:14.165 ","End":"01:18.140","Text":"horizontal asymptote at y equals this over this."},{"Start":"01:18.140 ","End":"01:22.535","Text":"In other words, if I just compute 2 over 1,"},{"Start":"01:22.535 ","End":"01:27.740","Text":"which equals 2, then y equals 2 is a horizontal asymptote,"},{"Start":"01:27.740 ","End":"01:29.360","Text":"and that\u0027s the only asymptote."},{"Start":"01:29.360 ","End":"01:33.320","Text":"It\u0027s good for both the plus infinity side and the minus infinity side,"},{"Start":"01:33.320 ","End":"01:35.060","Text":"and that\u0027s all there is to it."},{"Start":"01:35.060 ","End":"01:36.710","Text":"Although we did rely heavily on"},{"Start":"01:36.710 ","End":"01:40.760","Text":"this proposition that when we have equal degrees for a rational function,"},{"Start":"01:40.760 ","End":"01:45.420","Text":"then we divide the leading coefficient to get the asymptote."}],"ID":5893},{"Watched":false,"Name":"Exercise 10","Duration":"55s","ChapterTopicVideoID":5896,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5896.jpeg","UploadDate":"2016-05-02T07:31:47.8570000","DurationForVideoObject":"PT55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.590 ","End":"00:07.425","Text":"which is x cubed over x plus 1 squared."},{"Start":"00:07.425 ","End":"00:10.140","Text":"I expanded it just to show you that"},{"Start":"00:10.140 ","End":"00:13.425","Text":"we have a polynomial both on the top and on the bottom,"},{"Start":"00:13.425 ","End":"00:16.259","Text":"which means that f is a rational function."},{"Start":"00:16.259 ","End":"00:17.895","Text":"When we get a rational function,"},{"Start":"00:17.895 ","End":"00:21.105","Text":"we look at the degree on the top and on the bottom."},{"Start":"00:21.105 ","End":"00:23.670","Text":"Here the degree is 3, it\u0027s the highest power."},{"Start":"00:23.670 ","End":"00:29.280","Text":"We have a degree 3 polynomial over a degree 2 polynomial,"},{"Start":"00:29.280 ","End":"00:32.220","Text":"2 is the highest power, and in this case,"},{"Start":"00:32.220 ","End":"00:33.870","Text":"3 is bigger than 2,"},{"Start":"00:33.870 ","End":"00:35.370","Text":"by which I mean that the degree on the"},{"Start":"00:35.370 ","End":"00:38.430","Text":"numerator\u0027s bigger than the degree on the denominator."},{"Start":"00:38.430 ","End":"00:41.400","Text":"In this case, there\u0027s a theorem proposition"},{"Start":"00:41.400 ","End":"00:46.115","Text":"that the rational function has no asymptotes at all,"},{"Start":"00:46.115 ","End":"00:48.170","Text":"also no horizontal asymptote."},{"Start":"00:48.170 ","End":"00:49.910","Text":"So we can just quote that here,"},{"Start":"00:49.910 ","End":"00:52.805","Text":"f has no horizontal asymptote,"},{"Start":"00:52.805 ","End":"00:56.520","Text":"and that\u0027s all. We are done."}],"ID":5894},{"Watched":false,"Name":"Exercise 11","Duration":"1m 40s","ChapterTopicVideoID":5897,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5897.jpeg","UploadDate":"2016-05-02T07:31:59.1870000","DurationForVideoObject":"PT1M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"We have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.995 ","End":"00:06.450","Text":"which is as written here,"},{"Start":"00:06.450 ","End":"00:09.120","Text":"x plus 1 over x minus 1 all cubed."},{"Start":"00:09.120 ","End":"00:10.485","Text":"I like to rewrite it,"},{"Start":"00:10.485 ","End":"00:11.730","Text":"modify it a bit,"},{"Start":"00:11.730 ","End":"00:15.750","Text":"x plus 1 cubed over x minus 1 cubed,"},{"Start":"00:15.750 ","End":"00:18.240","Text":"using algebraic laws of exponents."},{"Start":"00:18.240 ","End":"00:23.580","Text":"I could expand it. I\u0027m just going to say that it begins with x^3 plus other stuff."},{"Start":"00:23.580 ","End":"00:25.758","Text":"It\u0027s a polynomial of degree 3,"},{"Start":"00:25.758 ","End":"00:31.120","Text":"and here we have x^3 plus or minus something and so on."},{"Start":"00:31.120 ","End":"00:34.715","Text":"Basically, we have a polynomial here and a polynomial here,"},{"Start":"00:34.715 ","End":"00:41.705","Text":"and the polynomials are degree 3 at the top and a degree 3 also at the bottom."},{"Start":"00:41.705 ","End":"00:44.090","Text":"This is in fact a rational function."},{"Start":"00:44.090 ","End":"00:49.250","Text":"When we have a rational function with equal degrees, 3 equals 3,"},{"Start":"00:49.250 ","End":"00:51.110","Text":"I\u0027m just writing that to emphasize that we have"},{"Start":"00:51.110 ","End":"00:54.795","Text":"an equal degree polynomial at the top as well as at the bottom,"},{"Start":"00:54.795 ","End":"00:59.795","Text":"in this case, there\u0027s a theorem or proposition that we take the leading term."},{"Start":"00:59.795 ","End":"01:01.040","Text":"In this case, it\u0027s x^3,"},{"Start":"01:01.040 ","End":"01:04.970","Text":"I\u0027d like to write it as 1x^3 and here, 1x^3."},{"Start":"01:04.970 ","End":"01:06.865","Text":"These are the leading coefficients."},{"Start":"01:06.865 ","End":"01:09.890","Text":"It\u0027s the coefficient of the term of the highest power."},{"Start":"01:09.890 ","End":"01:11.750","Text":"To start again, if we have"},{"Start":"01:11.750 ","End":"01:15.395","Text":"a rational function with equal degrees on the top and the bottom,"},{"Start":"01:15.395 ","End":"01:18.710","Text":"then f has a horizontal asymptote,"},{"Start":"01:18.710 ","End":"01:21.900","Text":"where y equals this over this,"},{"Start":"01:21.900 ","End":"01:24.540","Text":"1 over 1 is equal to 1,"},{"Start":"01:24.540 ","End":"01:29.090","Text":"so y equals 1 is the horizontal asymptote."},{"Start":"01:29.090 ","End":"01:32.270","Text":"With rational functions, you never get 2 different ones."},{"Start":"01:32.270 ","End":"01:36.259","Text":"It\u0027s always the same 1 for the plus infinity as for the minus infinity."},{"Start":"01:36.259 ","End":"01:38.660","Text":"The horizontal line y equals 1 is it,"},{"Start":"01:38.660 ","End":"01:41.250","Text":"and we are done."}],"ID":5895},{"Watched":false,"Name":"Exercise 12","Duration":"1m 7s","ChapterTopicVideoID":5898,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5898.jpeg","UploadDate":"2016-05-02T07:32:06.6000000","DurationForVideoObject":"PT1M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.860 ","End":"00:07.650","Text":"which is x minus 1 over x cubed."},{"Start":"00:07.650 ","End":"00:12.720","Text":"The important thing to notice here is that we have a rational function,"},{"Start":"00:12.720 ","End":"00:15.315","Text":"a polynomial over a polynomial."},{"Start":"00:15.315 ","End":"00:19.410","Text":"The degree of the polynomial in the top is 1,"},{"Start":"00:19.410 ","End":"00:21.345","Text":"and on the bottom, it\u0027s 3."},{"Start":"00:21.345 ","End":"00:23.520","Text":"Let me write that down it\u0027s important."},{"Start":"00:23.520 ","End":"00:27.105","Text":"The degree here is 1 because the leading coefficient is x,"},{"Start":"00:27.105 ","End":"00:30.340","Text":"and here it\u0027s 3 because x cubed is the leading coefficient."},{"Start":"00:30.340 ","End":"00:33.080","Text":"We have degree 1 over degree 3,"},{"Start":"00:33.080 ","End":"00:35.540","Text":"and that 3 is bigger than 1."},{"Start":"00:35.540 ","End":"00:38.860","Text":"In other words, we have a higher degree in the denominator."},{"Start":"00:38.860 ","End":"00:43.640","Text":"Now, when you have a rational function with a higher degree in the denominator,"},{"Start":"00:43.640 ","End":"00:48.230","Text":"it is always the case that the asymptote is the x-axis."},{"Start":"00:48.230 ","End":"00:49.970","Text":"In other words, y equals 0,"},{"Start":"00:49.970 ","End":"00:52.055","Text":"which is the equation of the x-axis,"},{"Start":"00:52.055 ","End":"00:54.349","Text":"is the horizontal asymptote."},{"Start":"00:54.349 ","End":"00:57.550","Text":"There is only 1 asymptote at most for a rational function."},{"Start":"00:57.550 ","End":"01:02.015","Text":"It\u0027s always the same at the plus infinity side is the minus infinity side."},{"Start":"01:02.015 ","End":"01:05.060","Text":"Anyway, that\u0027s it. We found the horizontal asymptote,"},{"Start":"01:05.060 ","End":"01:08.490","Text":"the x-axis or y equals 0, and we\u0027re done."}],"ID":5896},{"Watched":false,"Name":"Exercise 13","Duration":"3m 25s","ChapterTopicVideoID":5899,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5899.jpeg","UploadDate":"2016-05-02T07:32:29.4130000","DurationForVideoObject":"PT3M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.890","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.890 ","End":"00:09.015","Text":"which is x minus e to the x. I\u0027ve copied that here."},{"Start":"00:09.015 ","End":"00:11.970","Text":"I want to remind you because we\u0027ll probably need it,"},{"Start":"00:11.970 ","End":"00:16.605","Text":"that e to the power of infinity is infinity,"},{"Start":"00:16.605 ","End":"00:20.414","Text":"but e to the power of minus infinity is 0."},{"Start":"00:20.414 ","End":"00:24.075","Text":"Because of this, when we have exercises involving e to the power of,"},{"Start":"00:24.075 ","End":"00:26.249","Text":"we usually have to check separately"},{"Start":"00:26.249 ","End":"00:30.630","Text":"the asymptote at plus infinity and the asymptote at minus infinity."},{"Start":"00:30.630 ","End":"00:33.480","Text":"Let\u0027s start with the plus infinity first."},{"Start":"00:33.480 ","End":"00:39.000","Text":"In other words, what is the limit as x goes to infinity of f of x,"},{"Start":"00:39.000 ","End":"00:42.025","Text":"which I\u0027ll write as x minus e to the x."},{"Start":"00:42.025 ","End":"00:45.155","Text":"If it comes out to be a constant finite number,"},{"Start":"00:45.155 ","End":"00:48.500","Text":"then we\u0027ll have an asymptote at infinity."},{"Start":"00:48.500 ","End":"00:51.620","Text":"At first sight it\u0027s 1 of those indeterminate,"},{"Start":"00:51.620 ","End":"00:54.425","Text":"undefined forms because x is infinity,"},{"Start":"00:54.425 ","End":"00:56.870","Text":"and e to the x is also infinity,"},{"Start":"00:56.870 ","End":"00:59.435","Text":"so we get infinity minus infinity."},{"Start":"00:59.435 ","End":"01:01.820","Text":"When we have an infinity minus infinity,"},{"Start":"01:01.820 ","End":"01:05.090","Text":"we have to do some algebraic manipulation."},{"Start":"01:05.090 ","End":"01:09.440","Text":"We can get it into either 0 over 0 or infinity over infinity,"},{"Start":"01:09.440 ","End":"01:10.775","Text":"and then use L\u0027Hopital."},{"Start":"01:10.775 ","End":"01:16.840","Text":"What I suggest the following trick, take the x outside the brackets,"},{"Start":"01:16.840 ","End":"01:19.715","Text":"so this is limit as x goes to infinity."},{"Start":"01:19.715 ","End":"01:21.725","Text":"If I take x outside the brackets,"},{"Start":"01:21.725 ","End":"01:28.055","Text":"I\u0027ll get 1 minus e to the power of x over x."},{"Start":"01:28.055 ","End":"01:31.835","Text":"Let me compute the limit of this bit at the side."},{"Start":"01:31.835 ","End":"01:34.100","Text":"What I have is, at the side,"},{"Start":"01:34.100 ","End":"01:35.300","Text":"I\u0027ll do the limit."},{"Start":"01:35.300 ","End":"01:40.250","Text":"As x goes to infinity of e to the x over x,"},{"Start":"01:40.250 ","End":"01:41.690","Text":"and this is equal to,"},{"Start":"01:41.690 ","End":"01:44.360","Text":"now we have infinity over infinity,"},{"Start":"01:44.360 ","End":"01:46.835","Text":"so we can use L\u0027Hopital."},{"Start":"01:46.835 ","End":"01:53.500","Text":"What I\u0027ll do is indicate this as the infinity over infinity, L for L\u0027Hopital,"},{"Start":"01:53.500 ","End":"01:55.520","Text":"which means that we can take a new limit,"},{"Start":"01:55.520 ","End":"02:00.320","Text":"which is gotten from the previous 1 by differentiating top and bottom."},{"Start":"02:00.320 ","End":"02:02.300","Text":"Now we have e to the x,"},{"Start":"02:02.300 ","End":"02:05.380","Text":"a derivative is also e to the x over 1."},{"Start":"02:05.380 ","End":"02:10.040","Text":"Now we have infinity over 1, which is infinity."},{"Start":"02:10.040 ","End":"02:11.960","Text":"Now getting back to here."},{"Start":"02:11.960 ","End":"02:15.625","Text":"What we have now is we can take the limit separately."},{"Start":"02:15.625 ","End":"02:19.315","Text":"We have the limit of x is infinity,"},{"Start":"02:19.315 ","End":"02:22.520","Text":"times 1 minus infinity,"},{"Start":"02:22.520 ","End":"02:25.955","Text":"which is infinity times minus infinity,"},{"Start":"02:25.955 ","End":"02:28.775","Text":"which is minus infinity."},{"Start":"02:28.775 ","End":"02:32.814","Text":"This is infinity and not a finite number."},{"Start":"02:32.814 ","End":"02:36.650","Text":"There is no limit on the right as x goes to infinity,"},{"Start":"02:36.650 ","End":"02:41.440","Text":"but we still have another chance to find an asymptote at the other side on the left."},{"Start":"02:41.440 ","End":"02:47.195","Text":"When x goes to minus infinity of x minus e to the x."},{"Start":"02:47.195 ","End":"02:51.470","Text":"This equals, e to the power of x as x"},{"Start":"02:51.470 ","End":"02:56.465","Text":"goes to minus infinity is e to the minus infinity, and this is equal to 0,"},{"Start":"02:56.465 ","End":"03:00.575","Text":"so we can write here that x is minus infinity,"},{"Start":"03:00.575 ","End":"03:02.885","Text":"and then we have minus 0,"},{"Start":"03:02.885 ","End":"03:05.795","Text":"and this is equal to minus infinity."},{"Start":"03:05.795 ","End":"03:07.640","Text":"Again, we have minus infinity,"},{"Start":"03:07.640 ","End":"03:09.455","Text":"and it\u0027s another finite number,"},{"Start":"03:09.455 ","End":"03:13.730","Text":"so we don\u0027t have a horizontal asymptote at minus infinity either."},{"Start":"03:13.730 ","End":"03:18.735","Text":"In short, we failed twice at infinity and at minus infinity,"},{"Start":"03:18.735 ","End":"03:26.580","Text":"so I can just say that f has no horizontal asymptotes. We are done."}],"ID":5897},{"Watched":false,"Name":"Exercise 14","Duration":"1m 13s","ChapterTopicVideoID":5900,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5900.jpeg","UploadDate":"2016-05-02T07:32:37.7830000","DurationForVideoObject":"PT1M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.680","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.680 ","End":"00:07.934","Text":"which is e^1 over x."},{"Start":"00:07.934 ","End":"00:12.600","Text":"Let\u0027s just go about it and check the limits at infinity and minus infinity."},{"Start":"00:12.600 ","End":"00:22.364","Text":"Limit as x goes to infinity of f of x is just the limit f of x is e^1 over x."},{"Start":"00:22.364 ","End":"00:24.690","Text":"If we let x equal infinity,"},{"Start":"00:24.690 ","End":"00:28.455","Text":"basically we get e^1 over infinity,"},{"Start":"00:28.455 ","End":"00:32.460","Text":"which is e^0, which is equal to 1."},{"Start":"00:32.460 ","End":"00:39.200","Text":"Very similarly, the limit as x goes to minus infinity of the same thing,"},{"Start":"00:39.200 ","End":"00:40.850","Text":"basically you get the same thing here,"},{"Start":"00:40.850 ","End":"00:45.980","Text":"except here we\u0027ll get a minus infinity and 1 over minus infinity is also 0,"},{"Start":"00:45.980 ","End":"00:47.975","Text":"so it\u0027s also equal to 1."},{"Start":"00:47.975 ","End":"00:54.680","Text":"In other words, we have an asymptote at the value of the horizontal line y equals 1,"},{"Start":"00:54.680 ","End":"00:57.470","Text":"both at infinity and at minus infinity."},{"Start":"00:57.470 ","End":"00:58.775","Text":"There I\u0027ve written it out,"},{"Start":"00:58.775 ","End":"01:00.320","Text":"so we have a horizontal asymptote."},{"Start":"01:00.320 ","End":"01:01.655","Text":"If you want to add,"},{"Start":"01:01.655 ","End":"01:07.120","Text":"you can add that it\u0027s good for both infinity and minus infinity."},{"Start":"01:07.120 ","End":"01:09.620","Text":"Both at infinity and minus infinity,"},{"Start":"01:09.620 ","End":"01:10.879","Text":"it\u0027s more specific."},{"Start":"01:10.879 ","End":"01:13.320","Text":"Anyway, we\u0027re done."}],"ID":5898},{"Watched":false,"Name":"Exercise 15","Duration":"2m 59s","ChapterTopicVideoID":5901,"CourseChapterTopicPlaylistID":65384,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/5901.jpeg","UploadDate":"2016-05-02T07:32:56.4930000","DurationForVideoObject":"PT2M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we have to find the horizontal asymptotes of f of x,"},{"Start":"00:04.350 ","End":"00:07.050","Text":"which is xe^minus 2x squared."},{"Start":"00:07.050 ","End":"00:09.840","Text":"Now, when we have the exponential function,"},{"Start":"00:09.840 ","End":"00:15.615","Text":"we usually check both at x goes to infinity and that x goes to minus infinity."},{"Start":"00:15.615 ","End":"00:22.020","Text":"Let\u0027s start out by trying the limit as x goes to plus infinity of our function,"},{"Start":"00:22.020 ","End":"00:25.440","Text":"which I\u0027ll rewrite instead of the minus here,"},{"Start":"00:25.440 ","End":"00:26.910","Text":"I\u0027ll put that in the denominator,"},{"Start":"00:26.910 ","End":"00:31.539","Text":"so I\u0027ll get x/e to the power of plus 2 x squared."},{"Start":"00:31.539 ","End":"00:35.615","Text":"You\u0027ll soon see why I prefer to write it this way because if you look at it,"},{"Start":"00:35.615 ","End":"00:37.565","Text":"when x goes to infinity,"},{"Start":"00:37.565 ","End":"00:41.270","Text":"so does 2x squared, so here we have e to the infinity,"},{"Start":"00:41.270 ","End":"00:46.805","Text":"which is infinity and here we have infinity and then we can use L\u0027Hopital\u0027s rule."},{"Start":"00:46.805 ","End":"00:51.185","Text":"This equals the limit x goes to infinity and it"},{"Start":"00:51.185 ","End":"00:56.510","Text":"equals by L\u0027Hopital according to the infinity over infinity case,"},{"Start":"00:56.510 ","End":"01:01.900","Text":"which sometimes we write like this to indicate L\u0027Hopital for the infinity over infinity."},{"Start":"01:01.900 ","End":"01:03.515","Text":"Now we get a new limit,"},{"Start":"01:03.515 ","End":"01:06.230","Text":"which is what happens when we differentiate both"},{"Start":"01:06.230 ","End":"01:09.680","Text":"the top and the bottom. At the top we have 1 and at"},{"Start":"01:09.680 ","End":"01:12.950","Text":"the bottom we get by the chain rule e^2x"},{"Start":"01:12.950 ","End":"01:17.225","Text":"squared times the derivative of this,