Introduction to Extrema Word Problems
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Geometrical Problems
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Functions and Graphs Problems
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[{"Name":"Introduction to Extrema Word Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro and Example 1","Duration":"7m 13s","ChapterTopicVideoID":8275,"CourseChapterTopicPlaylistID":65388,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8275.jpeg","UploadDate":"2019-11-14T01:29:13.2600000","DurationForVideoObject":"PT7M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.955","Text":"Here I\u0027m going to talk about word problems,"},{"Start":"00:02.955 ","End":"00:05.790","Text":"which many people really dislike,"},{"Start":"00:05.790 ","End":"00:09.750","Text":"but they\u0027re actually closer to real-life problems."},{"Start":"00:09.750 ","End":"00:14.190","Text":"They\u0027re usually expressed in English rather than just mathematical formula"},{"Start":"00:14.190 ","End":"00:19.065","Text":"that isn\u0027t just \"find the maximum of the following function\" and so on and so on."},{"Start":"00:19.065 ","End":"00:21.990","Text":"But you often get a whole story and then you have to"},{"Start":"00:21.990 ","End":"00:24.990","Text":"convert it into mathematical language and solve it."},{"Start":"00:24.990 ","End":"00:26.670","Text":"That\u0027s what we\u0027re going to do here,"},{"Start":"00:26.670 ","End":"00:28.920","Text":"specifically for extremum problems,"},{"Start":"00:28.920 ","End":"00:30.645","Text":"that\u0027s minimum or maximum."},{"Start":"00:30.645 ","End":"00:33.300","Text":"Let me start with example 1."},{"Start":"00:33.300 ","End":"00:40.889","Text":"Here I want you to find 2 numbers whose product is 24 and whose sum is minimal."},{"Start":"00:40.889 ","End":"00:44.930","Text":"Now, 1 thing common to all these problems is that they always tell you"},{"Start":"00:44.930 ","End":"00:49.070","Text":"that something is got to be either minimal or maximal,"},{"Start":"00:49.070 ","End":"00:51.080","Text":"or least or greatest,"},{"Start":"00:51.080 ","End":"00:52.580","Text":"or in other words,"},{"Start":"00:52.580 ","End":"00:54.830","Text":"in this case, it\u0027s the sum."},{"Start":"00:54.830 ","End":"00:57.130","Text":"The sum refers to the 2 numbers."},{"Start":"00:57.130 ","End":"00:59.335","Text":"There are certain conditions imposed,"},{"Start":"00:59.335 ","End":"01:02.810","Text":"like here with the 2 numbers have to have a product of 24 and in general,"},{"Start":"01:02.810 ","End":"01:05.570","Text":"will become constraints on a real-world problem."},{"Start":"01:05.570 ","End":"01:07.805","Text":"But there could be many possibilities,"},{"Start":"01:07.805 ","End":"01:09.590","Text":"like 6 times 4,"},{"Start":"01:09.590 ","End":"01:12.440","Text":"12 times 2, 8 times 3,"},{"Start":"01:12.440 ","End":"01:14.525","Text":"1 times 24, and so on."},{"Start":"01:14.525 ","End":"01:17.000","Text":"We have to find amongst all those possibilities,"},{"Start":"01:17.000 ","End":"01:20.555","Text":"the possibility which gives the minimum or maximum something,"},{"Start":"01:20.555 ","End":"01:22.145","Text":"in this case, the sum."},{"Start":"01:22.145 ","End":"01:26.420","Text":"This is always going to be the dependent variable."},{"Start":"01:26.420 ","End":"01:29.240","Text":"We start with giving it a name or a letter."},{"Start":"01:29.240 ","End":"01:31.895","Text":"This case let\u0027s take S for sum."},{"Start":"01:31.895 ","End":"01:35.030","Text":"The sum will be the sum of the 2 numbers."},{"Start":"01:35.030 ","End":"01:38.660","Text":"So I give them names or call 1 number x,"},{"Start":"01:38.660 ","End":"01:41.035","Text":"and 1 number y."},{"Start":"01:41.035 ","End":"01:44.000","Text":"The idea is to get this in terms of"},{"Start":"01:44.000 ","End":"01:48.440","Text":"1 single variable to choose which is the independent variable."},{"Start":"01:48.440 ","End":"01:52.925","Text":"Now I\u0027m going to go with x to be my independent variable,"},{"Start":"01:52.925 ","End":"01:58.160","Text":"and the condition here on the product will enable me to find y in terms of x,"},{"Start":"01:58.160 ","End":"02:02.240","Text":"and thus I\u0027ll get S all in terms of 1 single variable x."},{"Start":"02:02.240 ","End":"02:03.830","Text":"So here at the side,"},{"Start":"02:03.830 ","End":"02:09.065","Text":"I note that because x times y is 24,"},{"Start":"02:09.065 ","End":"02:14.210","Text":"then I can get y in terms of x is 24 over x."},{"Start":"02:14.210 ","End":"02:16.910","Text":"By the way, I forgot to say with the 2 numbers,"},{"Start":"02:16.910 ","End":"02:19.100","Text":"I want them to be positive."},{"Start":"02:19.100 ","End":"02:21.710","Text":"So now that I have y in terms of x,"},{"Start":"02:21.710 ","End":"02:30.110","Text":"I can say that my sum is 24 over x plus x. I meant it the other way around,"},{"Start":"02:30.110 ","End":"02:31.685","Text":"not that it really matters."},{"Start":"02:31.685 ","End":"02:35.150","Text":"This sort of concludes the first phase of getting"},{"Start":"02:35.150 ","End":"02:39.875","Text":"the word problem into a mathematical problem."},{"Start":"02:39.875 ","End":"02:42.365","Text":"I just have to add that S is this,"},{"Start":"02:42.365 ","End":"02:47.940","Text":"and I want the minimum S. So this gives me a formulation."},{"Start":"02:47.940 ","End":"02:51.165","Text":"At this point, we go over to the calculus,"},{"Start":"02:51.165 ","End":"02:54.620","Text":"and in calculus, we know about minimum and maximum."},{"Start":"02:54.620 ","End":"02:57.560","Text":"We usually differentiate, and then we look for"},{"Start":"02:57.560 ","End":"03:00.530","Text":"suspects amongst the derivative equals 0 and so on."},{"Start":"03:00.530 ","End":"03:02.330","Text":"So let\u0027s start that process."},{"Start":"03:02.330 ","End":"03:03.590","Text":"I could even write this,"},{"Start":"03:03.590 ","End":"03:05.600","Text":"by the way, as S of x,"},{"Start":"03:05.600 ","End":"03:09.770","Text":"just to emphasize that the sum now depends on the variable x,"},{"Start":"03:09.770 ","End":"03:11.330","Text":"which is 1 of the 2 numbers."},{"Start":"03:11.330 ","End":"03:14.740","Text":"So S prime is equal to,"},{"Start":"03:14.740 ","End":"03:17.010","Text":"just differentiate, this is 1,"},{"Start":"03:17.010 ","End":"03:20.000","Text":"derivative of 1 over x is minus 1 over x squared,"},{"Start":"03:20.000 ","End":"03:23.080","Text":"so it\u0027s 1 minus 24 over x squared."},{"Start":"03:23.080 ","End":"03:24.630","Text":"Once I\u0027ve got the derivative,"},{"Start":"03:24.630 ","End":"03:28.565","Text":"I search for suspects amongst derivative equal 0,"},{"Start":"03:28.565 ","End":"03:33.830","Text":"which gives me the 1 minus 24 over x squared equals 0."},{"Start":"03:33.830 ","End":"03:35.480","Text":"This gives me, if you think about it,"},{"Start":"03:35.480 ","End":"03:38.375","Text":"x squared equals 24."},{"Start":"03:38.375 ","End":"03:40.940","Text":"Remember we\u0027re only interested in positive numbers."},{"Start":"03:40.940 ","End":"03:44.330","Text":"So x equals the square root of 24."},{"Start":"03:44.330 ","End":"03:48.050","Text":"This is only a suspect or a candidate for minimum."},{"Start":"03:48.050 ","End":"03:51.200","Text":"The complete answer would involve the 2 numbers."},{"Start":"03:51.200 ","End":"03:52.490","Text":"We have to find the 2 numbers."},{"Start":"03:52.490 ","End":"03:54.425","Text":"So let\u0027s just see what y would be."},{"Start":"03:54.425 ","End":"03:55.925","Text":"If x is this,"},{"Start":"03:55.925 ","End":"04:02.225","Text":"then that would give me that y equals 24 over x,"},{"Start":"04:02.225 ","End":"04:06.005","Text":"which is 24 over the square root of 24,"},{"Start":"04:06.005 ","End":"04:08.990","Text":"which is also the square root of 24."},{"Start":"04:08.990 ","End":"04:11.825","Text":"So it turns out that if it is a minimum,"},{"Start":"04:11.825 ","End":"04:17.210","Text":"that x is square root of 24 and so is y. I would say something like"},{"Start":"04:17.210 ","End":"04:24.440","Text":"the numbers are square root of 24 and square root of 24."},{"Start":"04:24.440 ","End":"04:27.815","Text":"But we still haven\u0027t shown that it is really a minimum."},{"Start":"04:27.815 ","End":"04:33.920","Text":"The 2 main techniques by using the second derivative or by drawing a table of increasing,"},{"Start":"04:33.920 ","End":"04:37.370","Text":"decreasing, and I\u0027m going to go for the table method here."},{"Start":"04:37.370 ","End":"04:41.000","Text":"So what I\u0027m going to do is write a little table,"},{"Start":"04:41.000 ","End":"04:46.340","Text":"where here I have the independent variable x and here I have"},{"Start":"04:46.340 ","End":"04:52.535","Text":"the derivative of whatever it is and here I have the dependent variable itself."},{"Start":"04:52.535 ","End":"04:54.875","Text":"Here I put a list of the suspects,"},{"Start":"04:54.875 ","End":"04:58.655","Text":"and in this particular case I only have 1 suspect for x,"},{"Start":"04:58.655 ","End":"05:01.485","Text":"and that is the square root of 24."},{"Start":"05:01.485 ","End":"05:07.670","Text":"At this point, I know that S prime is 0 and what we want to do is take"},{"Start":"05:07.670 ","End":"05:10.730","Text":"a number to the left and to the right of this and see whether"},{"Start":"05:10.730 ","End":"05:14.375","Text":"this is between increasing and decreasing intervals or what?"},{"Start":"05:14.375 ","End":"05:18.005","Text":"The square root of 24 is a bit less than 5,"},{"Start":"05:18.005 ","End":"05:19.520","Text":"because square root of 25 is 5."},{"Start":"05:19.520 ","End":"05:20.975","Text":"I don\u0027t even need to know what it is."},{"Start":"05:20.975 ","End":"05:26.285","Text":"I\u0027m sure it\u0027s between 4 and 5 because 4 squared is 16 and 5 squared is 25."},{"Start":"05:26.285 ","End":"05:28.115","Text":"But I could do it on the calculator."},{"Start":"05:28.115 ","End":"05:30.160","Text":"I might take 2 numbers, 1 on either side,"},{"Start":"05:30.160 ","End":"05:32.430","Text":"it doesn\u0027t even have to be 4 and 5 and then I"},{"Start":"05:32.430 ","End":"05:35.210","Text":"plug these and figure out what is S prime for these."},{"Start":"05:35.210 ","End":"05:38.550","Text":"But the fact is that I don\u0027t have to know the value for this table,"},{"Start":"05:38.550 ","End":"05:40.489","Text":"I only have to know the sign,"},{"Start":"05:40.489 ","End":"05:42.425","Text":"whether it\u0027s positive or negative."},{"Start":"05:42.425 ","End":"05:46.699","Text":"So as soon as I see that I got 1 minus 24 over 16,"},{"Start":"05:46.699 ","End":"05:49.310","Text":"I know that 24 over 16 is greater than 1."},{"Start":"05:49.310 ","End":"05:51.725","Text":"So immediately I can say this is negative."},{"Start":"05:51.725 ","End":"05:53.905","Text":"Similarly, if I put 5 in here,"},{"Start":"05:53.905 ","End":"05:57.220","Text":"24 over 25 is less than 1,"},{"Start":"05:57.220 ","End":"05:59.730","Text":"so I\u0027m going to get something positive here,"},{"Start":"05:59.730 ","End":"06:01.980","Text":"and then once I know the sign of the derivative,"},{"Start":"06:01.980 ","End":"06:04.385","Text":"I know that the function here is decreasing,"},{"Start":"06:04.385 ","End":"06:08.330","Text":"the function here is increasing to the left of this point."},{"Start":"06:08.330 ","End":"06:10.895","Text":"So I like to draw a little sketch."},{"Start":"06:10.895 ","End":"06:14.915","Text":"The decreasing increasing means that the function looks something like this."},{"Start":"06:14.915 ","End":"06:16.820","Text":"So it is indeed a minimum,"},{"Start":"06:16.820 ","End":"06:19.550","Text":"not a maximum or not something else."},{"Start":"06:19.550 ","End":"06:21.965","Text":"That\u0027s basically the answer."},{"Start":"06:21.965 ","End":"06:24.680","Text":"What we\u0027ve done is we\u0027ve got the 2 numbers."},{"Start":"06:24.680 ","End":"06:27.330","Text":"You\u0027ve proven that it\u0027s a minimum and we are"},{"Start":"06:27.330 ","End":"06:30.200","Text":"done except that I want to complicate the problem a bit."},{"Start":"06:30.200 ","End":"06:34.445","Text":"Sometimes they also ask you to actually find the minimum value."},{"Start":"06:34.445 ","End":"06:37.430","Text":"The minimum value is S,"},{"Start":"06:37.430 ","End":"06:39.260","Text":"which is x plus y,"},{"Start":"06:39.260 ","End":"06:43.385","Text":"which is square root of 24 plus square root of 24."},{"Start":"06:43.385 ","End":"06:47.090","Text":"I mean, I could substitute it in here or in here."},{"Start":"06:47.090 ","End":"06:49.145","Text":"Either way, I\u0027d get the same thing."},{"Start":"06:49.145 ","End":"06:52.520","Text":"This is twice square root of 24."},{"Start":"06:52.520 ","End":"06:54.950","Text":"We could do it on the calculator."},{"Start":"06:54.950 ","End":"06:58.220","Text":"For me, this is perfectly adequate as an answer."},{"Start":"06:58.220 ","End":"07:02.360","Text":"In some questions, they ask you for the minimum or maximum value itself."},{"Start":"07:02.360 ","End":"07:05.300","Text":"For some questions, they ask you to find the variables"},{"Start":"07:05.300 ","End":"07:08.555","Text":"which give the minimum or maximum, or sometimes both."},{"Start":"07:08.555 ","End":"07:10.910","Text":"This is the end of example 1."},{"Start":"07:10.910 ","End":"07:14.010","Text":"Now let\u0027s get on to example 2."}],"ID":8450},{"Watched":false,"Name":"Example 2","Duration":"9m 31s","ChapterTopicVideoID":8276,"CourseChapterTopicPlaylistID":65388,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8276.jpeg","UploadDate":"2019-11-14T01:28:46.2270000","DurationForVideoObject":"PT9M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.895","Text":"This is example 2 from geometry,"},{"Start":"00:02.895 ","End":"00:08.400","Text":"we have to find the largest possible area of an isosceles triangle whose leg is 10."},{"Start":"00:08.400 ","End":"00:10.335","Text":"Isosceles, in case you forgot,"},{"Start":"00:10.335 ","End":"00:12.930","Text":"means triangle with 2 equal sides."},{"Start":"00:12.930 ","End":"00:15.090","Text":"The 2 equal sides are called the legs."},{"Start":"00:15.090 ","End":"00:16.320","Text":"This is a leg, this is a leg,"},{"Start":"00:16.320 ","End":"00:18.374","Text":"and this is actually called a base."},{"Start":"00:18.374 ","End":"00:21.990","Text":"Now we want the 1 with the largest possible area,"},{"Start":"00:21.990 ","End":"00:26.010","Text":"and that already tells you what the dependent variable is."},{"Start":"00:26.010 ","End":"00:29.520","Text":"The one that\u0027s going to be largest to smallest or greatest or least to,"},{"Start":"00:29.520 ","End":"00:32.610","Text":"maximal or minimal, the area is going"},{"Start":"00:32.610 ","End":"00:36.045","Text":"to be the dependent variable and I\u0027m going to call it A."},{"Start":"00:36.045 ","End":"00:38.960","Text":"Question is, what is the independent variable?"},{"Start":"00:38.960 ","End":"00:40.380","Text":"There could be many choices,"},{"Start":"00:40.380 ","End":"00:41.690","Text":"and if you choose well,"},{"Start":"00:41.690 ","End":"00:43.415","Text":"the work will go easier."},{"Start":"00:43.415 ","End":"00:47.090","Text":"It seems like the most natural thing to do would be to take"},{"Start":"00:47.090 ","End":"00:51.365","Text":"the base and let\u0027s call that x and see if we can use that."},{"Start":"00:51.365 ","End":"00:55.009","Text":"Now, how do I express the area in terms of the base?"},{"Start":"00:55.009 ","End":"01:01.070","Text":"Well, the most famous formula for the area of a triangle is a half base times height."},{"Start":"01:01.070 ","End":"01:03.390","Text":"We have the base now we\u0027ve called it x,"},{"Start":"01:03.390 ","End":"01:04.960","Text":"so what is the height?"},{"Start":"01:04.960 ","End":"01:10.160","Text":"I\u0027ve drawn a height in and let\u0027s call it h. We have that"},{"Start":"01:10.160 ","End":"01:15.469","Text":"the area is equal to 1/2 base times height,"},{"Start":"01:15.469 ","End":"01:19.085","Text":"but I only want one independent variable, x."},{"Start":"01:19.085 ","End":"01:25.745","Text":"What I have to do now is use some geometry or trigonometry to find h in terms of x."},{"Start":"01:25.745 ","End":"01:30.095","Text":"The most obvious thing to do is to use Pythagoras\u0027s theorem is to say,"},{"Start":"01:30.095 ","End":"01:32.100","Text":"okay, this is the height,"},{"Start":"01:32.100 ","End":"01:34.310","Text":"it comes here at 90 degrees,"},{"Start":"01:34.310 ","End":"01:36.500","Text":"but the height also is a median,"},{"Start":"01:36.500 ","End":"01:40.670","Text":"so this is x/2 and this is x/22."},{"Start":"01:40.670 ","End":"01:43.060","Text":"Why not use Pythagoras\u0027s theorem?"},{"Start":"01:43.060 ","End":"01:51.020","Text":"X/2^2 plus h^2 equals 10^2."},{"Start":"01:51.020 ","End":"01:59.150","Text":"Then we would get h^2 is 100 minus x^2 over 4."},{"Start":"01:59.150 ","End":"02:01.730","Text":"H is equal to the square root of this,"},{"Start":"02:01.730 ","End":"02:03.545","Text":"the positive square root, of course,"},{"Start":"02:03.545 ","End":"02:09.250","Text":"because all the lengths are positive of 100 minus x^2 over 4."},{"Start":"02:09.250 ","End":"02:13.430","Text":"Then I would put that in here to emphasize that as a function of x,"},{"Start":"02:13.430 ","End":"02:15.140","Text":"we will write it as a of x."},{"Start":"02:15.140 ","End":"02:23.975","Text":"It\u0027s equal to 1.5 times x times the square root of 100 minus x^2 over 4."},{"Start":"02:23.975 ","End":"02:27.440","Text":"Now we\u0027ve got a formula for the dependent variable,"},{"Start":"02:27.440 ","End":"02:30.515","Text":"the one we want to maximize in this case,"},{"Start":"02:30.515 ","End":"02:34.340","Text":"in terms of an independent variable which is the base."},{"Start":"02:34.340 ","End":"02:37.295","Text":"This part is the geometry part."},{"Start":"02:37.295 ","End":"02:40.725","Text":"Let\u0027s continue with calculus techniques."},{"Start":"02:40.725 ","End":"02:44.720","Text":"We get that a prime or a prime of x is equal to,"},{"Start":"02:44.720 ","End":"02:46.685","Text":"I can leave the half as is."},{"Start":"02:46.685 ","End":"02:48.480","Text":"Now I have a product this times this,"},{"Start":"02:48.480 ","End":"02:50.540","Text":"so we know how to derive products."},{"Start":"02:50.540 ","End":"02:54.845","Text":"We take the derivative of the first and multiply by the second,"},{"Start":"02:54.845 ","End":"02:57.035","Text":"and then the other way round."},{"Start":"02:57.035 ","End":"02:59.059","Text":"This time the x is untouched,"},{"Start":"02:59.059 ","End":"03:00.670","Text":"but we differentiate this,"},{"Start":"03:00.670 ","End":"03:03.575","Text":"and whenever you see a square root and differentiated,"},{"Start":"03:03.575 ","End":"03:08.135","Text":"we put the denominator as twice the square root."},{"Start":"03:08.135 ","End":"03:12.860","Text":"In the numerator, the inner derivative of what was under the square root,"},{"Start":"03:12.860 ","End":"03:17.375","Text":"which in this case would be minus 2x/4."},{"Start":"03:17.375 ","End":"03:24.080","Text":"Let\u0027s simplify this by putting everything over a denominator."},{"Start":"03:24.080 ","End":"03:27.995","Text":"Basically, what I\u0027ll get is a 4 in the denominator."},{"Start":"03:27.995 ","End":"03:34.735","Text":"What I want is a common denominator with 4 times the square root in the denominator."},{"Start":"03:34.735 ","End":"03:38.045","Text":"This equals, I\u0027ll go throw in one extra step,"},{"Start":"03:38.045 ","End":"03:39.710","Text":"you know while already there,"},{"Start":"03:39.710 ","End":"03:45.990","Text":"why not take this 2 and combine it with the 4 here and make that an 8?"},{"Start":"03:45.990 ","End":"03:50.120","Text":"In the denominator, what I\u0027m going to get is starting out by"},{"Start":"03:50.120 ","End":"03:56.955","Text":"8 times the square root of a 100 minus x^2 over 4."},{"Start":"03:56.955 ","End":"04:00.210","Text":"This 2 is also going to be under here."},{"Start":"04:00.210 ","End":"04:02.010","Text":"If I put the 2 here,"},{"Start":"04:02.010 ","End":"04:03.810","Text":"and it comes out 8,"},{"Start":"04:03.810 ","End":"04:06.620","Text":"but I still have to put it over 2 here."},{"Start":"04:06.620 ","End":"04:13.130","Text":"The second term here is going to be minus x^2,"},{"Start":"04:13.130 ","End":"04:20.165","Text":"and this one, I have to multiply top and bottom by 4 times the square root."},{"Start":"04:20.165 ","End":"04:25.790","Text":"What I\u0027m left with in the numerator is 4 times the square root of this times itself,"},{"Start":"04:25.790 ","End":"04:33.110","Text":"it is just 4 times 100 minus x^2 over 4 minus x^2."},{"Start":"04:33.110 ","End":"04:40.220","Text":"Now, this is equal to 4 times a 100 is 400,"},{"Start":"04:40.220 ","End":"04:47.325","Text":"4 times x^2 over 4 is just x^2 minus x^2 minus x^2"},{"Start":"04:47.325 ","End":"04:55.395","Text":"over 8 times square root of 100 minus x^2 over 4."},{"Start":"04:55.395 ","End":"04:59.010","Text":"This is actually 400 minus 2x^2,"},{"Start":"04:59.010 ","End":"05:01.725","Text":"and I can take 2 outside the brackets,"},{"Start":"05:01.725 ","End":"05:10.205","Text":"so it\u0027s twice 200 minus x^2 over 8 times the square root"},{"Start":"05:10.205 ","End":"05:13.825","Text":"of 100 minus x^2 over 4."},{"Start":"05:13.825 ","End":"05:18.210","Text":"Again, I can cancel this 2 with this 8 leaves me with 4 here."},{"Start":"05:18.210 ","End":"05:20.895","Text":"This is still A prime."},{"Start":"05:20.895 ","End":"05:22.490","Text":"Now, what do we do with this?"},{"Start":"05:22.490 ","End":"05:24.970","Text":"Well, we\u0027re looking for a maximum."},{"Start":"05:24.970 ","End":"05:27.605","Text":"For a maximum or a minimum, in fact,"},{"Start":"05:27.605 ","End":"05:30.470","Text":"we have to look where a prime is 0."},{"Start":"05:30.470 ","End":"05:32.105","Text":"When we have a fraction,"},{"Start":"05:32.105 ","End":"05:35.345","Text":"I have 0 only the numerator has to be 0,"},{"Start":"05:35.345 ","End":"05:39.394","Text":"so we get 200 minus x^2 equals 0,"},{"Start":"05:39.394 ","End":"05:44.990","Text":"which gives us that x equals the square root of 200,"},{"Start":"05:44.990 ","End":"05:46.040","Text":"the positive square root,"},{"Start":"05:46.040 ","End":"05:49.010","Text":"we only need positive numbers, That\u0027s x,"},{"Start":"05:49.010 ","End":"05:52.404","Text":"but they asked us for the area,"},{"Start":"05:52.404 ","End":"05:54.930","Text":"so let\u0027s see what\u0027s the corresponding area,"},{"Start":"05:54.930 ","End":"05:59.420","Text":"and then after that, we\u0027ll still have to show that it is indeed a maximum."},{"Start":"05:59.420 ","End":"06:03.050","Text":"I\u0027m going to substitute the value of x into it."},{"Start":"06:03.050 ","End":"06:07.640","Text":"By the way, if we\u0027re asking not for the area but for the base length,"},{"Start":"06:07.640 ","End":"06:10.625","Text":"then you could leave the answer like this,"},{"Start":"06:10.625 ","End":"06:14.750","Text":"or you can simplify it by taking either square root of 2 square root of a 100,"},{"Start":"06:14.750 ","End":"06:16.500","Text":"which is 10 square root of 2,"},{"Start":"06:16.500 ","End":"06:19.910","Text":"or you could have done it on the calculator and you would have gotten approximately"},{"Start":"06:19.910 ","End":"06:24.005","Text":"equal to 14.142 or something."},{"Start":"06:24.005 ","End":"06:26.245","Text":"But in this case, they asked for the area."},{"Start":"06:26.245 ","End":"06:27.650","Text":"When we write an answer,"},{"Start":"06:27.650 ","End":"06:29.050","Text":"we also put the units in,"},{"Start":"06:29.050 ","End":"06:30.170","Text":"so if they asked for this,"},{"Start":"06:30.170 ","End":"06:31.940","Text":"it would be centimeters and here we have to"},{"Start":"06:31.940 ","End":"06:34.745","Text":"remember to add centimeter squared at the end."},{"Start":"06:34.745 ","End":"06:40.185","Text":"What we get is that A of square root of 200,"},{"Start":"06:40.185 ","End":"06:42.150","Text":"just put checks as that,"},{"Start":"06:42.150 ","End":"06:49.495","Text":"so it\u0027s 1.5 square root of 200 times the square root of 100."},{"Start":"06:49.495 ","End":"06:53.895","Text":"The square root of 200^2 is just 200 over 4."},{"Start":"06:53.895 ","End":"06:55.859","Text":"200 over 4 is 50,"},{"Start":"06:55.859 ","End":"06:58.170","Text":"a 100 minus 50 is 50,"},{"Start":"06:58.170 ","End":"07:00.860","Text":"so here we have the square root of 50."},{"Start":"07:00.860 ","End":"07:04.415","Text":"Here we have the square root of 200."},{"Start":"07:04.415 ","End":"07:06.580","Text":"Here we have 1/2."},{"Start":"07:06.580 ","End":"07:10.745","Text":"I can combine the square roots and say it\u0027s 1/2"},{"Start":"07:10.745 ","End":"07:16.855","Text":"of the square root of 200 times 50 is 10,000,"},{"Start":"07:16.855 ","End":"07:21.270","Text":"and the square root of 10,000 is a 100."},{"Start":"07:21.270 ","End":"07:27.620","Text":"What we get in the end is just 50 centimeters squared."},{"Start":"07:27.620 ","End":"07:32.345","Text":"The missing bits of the problem is to show that it is actually a maximum,"},{"Start":"07:32.345 ","End":"07:34.250","Text":"and there\u0027s 2 general approaches,"},{"Start":"07:34.250 ","End":"07:40.100","Text":"one is the second derivative and one is the table method for increasing, decreasing."},{"Start":"07:40.100 ","End":"07:42.835","Text":"I prefer the table when possible."},{"Start":"07:42.835 ","End":"07:47.744","Text":"In the table, I\u0027m going to put the independent variable x,"},{"Start":"07:47.744 ","End":"07:50.660","Text":"and here I put the dependent,"},{"Start":"07:50.660 ","End":"07:54.320","Text":"which is A and its derivative here."},{"Start":"07:54.320 ","End":"07:56.720","Text":"Here I put the list of suspects,"},{"Start":"07:56.720 ","End":"07:59.810","Text":"and in our case we only have one suspect,"},{"Start":"07:59.810 ","End":"08:04.265","Text":"and that is the square root of 200."},{"Start":"08:04.265 ","End":"08:06.820","Text":"I\u0027m going to substitute an A prime."},{"Start":"08:06.820 ","End":"08:13.010","Text":"Notice that A prime here is equal to this over this,"},{"Start":"08:13.010 ","End":"08:15.680","Text":"but the denominator is positive,"},{"Start":"08:15.680 ","End":"08:18.985","Text":"the square root is positive and 4 is positive."},{"Start":"08:18.985 ","End":"08:24.785","Text":"All I would have to do would be to substitute in this part here,"},{"Start":"08:24.785 ","End":"08:27.860","Text":"because I don\u0027t actually need A prime,"},{"Start":"08:27.860 ","End":"08:31.025","Text":"I only need its sign plus or minus."},{"Start":"08:31.025 ","End":"08:34.205","Text":"Of course, if I substitute the square root of 200,"},{"Start":"08:34.205 ","End":"08:38.015","Text":"I\u0027m going to get 0 because that\u0027s how I found this where you can check it."},{"Start":"08:38.015 ","End":"08:41.165","Text":"What I want is something to the left and to the right,"},{"Start":"08:41.165 ","End":"08:45.770","Text":"less than and greater than to see where my increase in decrease are."},{"Start":"08:45.770 ","End":"08:48.290","Text":"Let\u0027s put some numbers that are easy to compute,"},{"Start":"08:48.290 ","End":"08:52.440","Text":"why don\u0027t I just put 10 and 20."},{"Start":"08:52.440 ","End":"08:56.930","Text":"Let\u0027s see if I put x equals 10 in the highlighted bit,"},{"Start":"08:56.930 ","End":"09:01.640","Text":"I\u0027ve got 200 minus 10^2 is obviously positive."},{"Start":"09:01.640 ","End":"09:06.125","Text":"If I put 200 minus 20^2, obviously negative."},{"Start":"09:06.125 ","End":"09:10.740","Text":"Here, I\u0027m increasing, here I\u0027m decreasing."},{"Start":"09:10.740 ","End":"09:15.260","Text":"If I imagine the shape is something like this, increasing,"},{"Start":"09:15.260 ","End":"09:19.565","Text":"decreasing, which means that it really is a maximum."},{"Start":"09:19.565 ","End":"09:23.030","Text":"We\u0027ve actually solved the problem."},{"Start":"09:23.030 ","End":"09:26.300","Text":"The maximum area of such a triangle is"},{"Start":"09:26.300 ","End":"09:31.890","Text":"50 centimeters squared and we\u0027ve actually proven that it is maximum."}],"ID":8620},{"Watched":false,"Name":"Example 2 continued","Duration":"6m 34s","ChapterTopicVideoID":8832,"CourseChapterTopicPlaylistID":65388,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8832.jpeg","UploadDate":"2019-11-14T01:29:02.1130000","DurationForVideoObject":"PT6M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"Now, this was quite hard work."},{"Start":"00:01.980 ","End":"00:05.190","Text":"I want to show you some tricks and shortcuts."},{"Start":"00:05.190 ","End":"00:09.495","Text":"I\u0027d like to go back to a certain point in the clip."},{"Start":"00:09.495 ","End":"00:15.960","Text":"I\u0027m going to go back to the point where we had A in terms of x."},{"Start":"00:15.960 ","End":"00:17.190","Text":"This point here."},{"Start":"00:17.190 ","End":"00:20.490","Text":"This trick I\u0027m going to show you will work whenever it\u0027s"},{"Start":"00:20.490 ","End":"00:24.210","Text":"convenient to have the function squared."},{"Start":"00:24.210 ","End":"00:28.230","Text":"Here the square root is cumbersome so if instead of A,"},{"Start":"00:28.230 ","End":"00:30.315","Text":"I take A squared,"},{"Start":"00:30.315 ","End":"00:34.005","Text":"let me say that S is A squared."},{"Start":"00:34.005 ","End":"00:37.340","Text":"So I get that S of x is equal"},{"Start":"00:37.340 ","End":"00:45.710","Text":"to 1/4x^2 times 100 minus x^2/4."},{"Start":"00:45.710 ","End":"00:47.090","Text":"What I claim is,"},{"Start":"00:47.090 ","End":"00:52.435","Text":"is that if we have a function that we need to maximize or minimize and it\u0027s positive,"},{"Start":"00:52.435 ","End":"00:55.440","Text":"notice that this is always positive;"},{"Start":"00:55.440 ","End":"01:01.820","Text":"x is positive in our domain because it\u0027s a length and square root is always positive."},{"Start":"01:01.820 ","End":"01:04.595","Text":"So when we have a function that\u0027s positive,"},{"Start":"01:04.595 ","End":"01:10.280","Text":"we can replace it by its square in order to find the maximum."},{"Start":"01:10.280 ","End":"01:15.255","Text":"We\u0027ll get the same x as we would have with S as with A"},{"Start":"01:15.255 ","End":"01:17.825","Text":"and now we\u0027re going to work only with S and that will"},{"Start":"01:17.825 ","End":"01:20.700","Text":"make things a lot easier on the mathematics."},{"Start":"01:20.700 ","End":"01:27.395","Text":"1/4x^2 times 100 will give me 25x^2 and this with"},{"Start":"01:27.395 ","End":"01:34.795","Text":"this x^2/4 with x^2/4 will give me x^4/4^2 is 16."},{"Start":"01:34.795 ","End":"01:39.860","Text":"Now I do the calculus part that C will go to the derivative."},{"Start":"01:39.860 ","End":"01:49.595","Text":"So S prime is going to be 50x minus 4x^3/16, got x^3/4."},{"Start":"01:49.595 ","End":"01:53.030","Text":"We want to assign S prime equals 0 so we can look for"},{"Start":"01:53.030 ","End":"01:56.765","Text":"suspects for the maximum and this gives us,"},{"Start":"01:56.765 ","End":"02:04.740","Text":"if we multiply it all by 4 that 200x minus x^3 equals 0."},{"Start":"02:04.740 ","End":"02:09.050","Text":"I\u0027ll just multiply this equation by 4 and then I\u0027ll take x outside"},{"Start":"02:09.050 ","End":"02:15.084","Text":"the brackets so I\u0027ve got x times 200 minus x^2 equals 0;"},{"Start":"02:15.084 ","End":"02:20.640","Text":"x is not 0 so I can just divide by it and that gives us that x"},{"Start":"02:20.640 ","End":"02:26.569","Text":"equals the square root of 200 and that\u0027s what we got before."},{"Start":"02:26.569 ","End":"02:29.255","Text":"Once we got the x equals the square root of 200,"},{"Start":"02:29.255 ","End":"02:30.505","Text":"we did it before,"},{"Start":"02:30.505 ","End":"02:32.370","Text":"we just substitute in A."},{"Start":"02:32.370 ","End":"02:38.030","Text":"We got that A was equal to I believe we got the answer was 50,"},{"Start":"02:38.030 ","End":"02:41.105","Text":"same thing here because A hasn\u0027t changed."},{"Start":"02:41.105 ","End":"02:42.860","Text":"Okay, so this is much shorter."},{"Start":"02:42.860 ","End":"02:48.125","Text":"This technique of squaring instead of taking maximum of A,"},{"Start":"02:48.125 ","End":"02:53.285","Text":"we take maximum of A^2 or whatever function it is."},{"Start":"02:53.285 ","End":"02:57.500","Text":"I remind you it\u0027s true as long as we have a positive function."},{"Start":"02:57.500 ","End":"02:59.855","Text":"Okay, now that\u0027s 1 shortcut."},{"Start":"02:59.855 ","End":"03:06.785","Text":"I also want to go back even earlier in the question and show you another possibility."},{"Start":"03:06.785 ","End":"03:09.155","Text":"We have a choice in deciding what"},{"Start":"03:09.155 ","End":"03:13.865","Text":"the independent variable is and x is not the only choice."},{"Start":"03:13.865 ","End":"03:16.610","Text":"I\u0027m going to show you another choice, let me, first of all,"},{"Start":"03:16.610 ","End":"03:19.500","Text":"erase some unnecessary stuff."},{"Start":"03:19.500 ","End":"03:21.469","Text":"We\u0027re back to the original question."},{"Start":"03:21.469 ","End":"03:25.280","Text":"We know that the area is the dependent variable."},{"Start":"03:25.280 ","End":"03:28.760","Text":"We still want to find the largest or the maximal area but"},{"Start":"03:28.760 ","End":"03:33.005","Text":"the independent variable that we took as x last time which was the base,"},{"Start":"03:33.005 ","End":"03:36.440","Text":"there could be other choices and I\u0027m just going to show you from experience,"},{"Start":"03:36.440 ","End":"03:38.120","Text":"what you might try."},{"Start":"03:38.120 ","End":"03:40.265","Text":"How about considering the angle?"},{"Start":"03:40.265 ","End":"03:45.215","Text":"Let\u0027s take this angle Alpha to be the independent variable."},{"Start":"03:45.215 ","End":"03:50.825","Text":"The reason I\u0027m choosing this is because there\u0027s a very nice formula for the area."},{"Start":"03:50.825 ","End":"03:53.870","Text":"Let\u0027s call the area A, just as before,"},{"Start":"03:53.870 ","End":"03:57.500","Text":"we\u0027re going to find what A equals but this time in terms of Alpha."},{"Start":"03:57.500 ","End":"04:00.740","Text":"Now there\u0027s a formula in general with a triangle."},{"Start":"04:00.740 ","End":"04:04.940","Text":"If I have a triangle and let\u0027s say this is a,"},{"Start":"04:04.940 ","End":"04:06.410","Text":"this side is b,"},{"Start":"04:06.410 ","End":"04:10.320","Text":"this side is c, and this angle is Alpha,"},{"Start":"04:10.320 ","End":"04:12.555","Text":"the area is A,"},{"Start":"04:12.555 ","End":"04:16.730","Text":"then the formula from trigonometry says that A"},{"Start":"04:16.730 ","End":"04:23.215","Text":"equals 1/2 b times c times sine Alpha."},{"Start":"04:23.215 ","End":"04:24.545","Text":"So in our case,"},{"Start":"04:24.545 ","End":"04:32.990","Text":"the area is equal to 1/2 times 10 times 10 times sine Alpha,"},{"Start":"04:32.990 ","End":"04:37.770","Text":"which is just equal to 50 sine Alpha."},{"Start":"04:37.770 ","End":"04:42.560","Text":"This time I\u0027m taking A as a function of alpha and this is an aside for those who are"},{"Start":"04:42.560 ","End":"04:47.990","Text":"interested but if I look at the sine function and I look at it between,"},{"Start":"04:47.990 ","End":"04:51.925","Text":"say 0 and 180 degrees or Pi,"},{"Start":"04:51.925 ","End":"04:54.580","Text":"the sine has its maximum."},{"Start":"04:54.580 ","End":"05:02.600","Text":"This point is the maximum and it\u0027s equal to 1 exactly when Alpha is Pi/2."},{"Start":"05:02.600 ","End":"05:05.310","Text":"So I could just say, \"Hey, the maximum,"},{"Start":"05:05.310 ","End":"05:07.150","Text":"the sine can be at most 1,"},{"Start":"05:07.150 ","End":"05:11.095","Text":"it does occur and so the maximum is 50 and we\u0027re done."},{"Start":"05:11.095 ","End":"05:15.970","Text":"But this might be a bit unorthodox so let\u0027s do it with differentiation."},{"Start":"05:15.970 ","End":"05:21.985","Text":"A prime is equal to 50 cosine Alpha and I\u0027m looking for"},{"Start":"05:21.985 ","End":"05:29.599","Text":"A prime equals 0 and that will give me that 50 cosine alpha equals 0."},{"Start":"05:29.599 ","End":"05:32.990","Text":"So cosine Alpha equals 0 and"},{"Start":"05:32.990 ","End":"05:38.220","Text":"the only angle between 0 and 180 degrees which is what it could be,"},{"Start":"05:38.220 ","End":"05:45.499","Text":"but cosine is 0 is Alpha equals 90 degrees or you can do the arc cosine of 0."},{"Start":"05:45.499 ","End":"05:49.540","Text":"Alpha is 90 degrees or Pi/2."},{"Start":"05:49.540 ","End":"05:51.130","Text":"Once I have my Alpha,"},{"Start":"05:51.130 ","End":"05:54.920","Text":"then I substitute in the function A of Alpha."},{"Start":"05:54.920 ","End":"06:00.700","Text":"So A of Pi/2 is"},{"Start":"06:00.700 ","End":"06:07.230","Text":"50 sine Pi/2 or 90 degrees you can think of it and sine of 90 degrees is 1."},{"Start":"06:07.230 ","End":"06:11.085","Text":"That\u0027s like what I was saying before and so this equals 50."},{"Start":"06:11.085 ","End":"06:15.240","Text":"Basically, I\u0027ve shown you a trick and the concept."},{"Start":"06:15.240 ","End":"06:18.860","Text":"The trick is that if you want to square a function which is positive,"},{"Start":"06:18.860 ","End":"06:20.450","Text":"you can go ahead and do it."},{"Start":"06:20.450 ","End":"06:25.160","Text":"The other thing is, you might want to spend a few seconds or minutes thinking about what"},{"Start":"06:25.160 ","End":"06:29.945","Text":"your independent variable is going to be because it can make life a lot simpler."},{"Start":"06:29.945 ","End":"06:35.040","Text":"I\u0027m done with this example and we\u0027re done with the tutorial too."}],"ID":9092}],"Thumbnail":null,"ID":65388},{"Name":"Geometrical Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"9m 29s","ChapterTopicVideoID":8355,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8355.jpeg","UploadDate":"2017-01-28T12:41:48.0330000","DurationForVideoObject":"PT9M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.010","Text":"Here we have a geometric word problem involving an extremum."},{"Start":"00:05.010 ","End":"00:08.520","Text":"The word maximal is what gives it away."},{"Start":"00:08.520 ","End":"00:11.250","Text":"Let\u0027s read through the problem."},{"Start":"00:11.250 ","End":"00:14.700","Text":"In an isosceles trapezoid ABCD,"},{"Start":"00:14.700 ","End":"00:17.295","Text":"the length of the leg is 4."},{"Start":"00:17.295 ","End":"00:19.230","Text":"The leg is this or this,"},{"Start":"00:19.230 ","End":"00:21.180","Text":"so I know that this is 4,"},{"Start":"00:21.180 ","End":"00:23.245","Text":"and this is 4."},{"Start":"00:23.245 ","End":"00:27.480","Text":"The length of the short base is 6 centimeters."},{"Start":"00:27.480 ","End":"00:29.535","Text":"The short base is this 1."},{"Start":"00:29.535 ","End":"00:31.470","Text":"DE is an altitude,"},{"Start":"00:31.470 ","End":"00:33.720","Text":"which means that D is the height."},{"Start":"00:33.720 ","End":"00:38.315","Text":"You can call it h. We have to find the length of the segment AE,"},{"Start":"00:38.315 ","End":"00:40.880","Text":"so let\u0027s call this 1 x."},{"Start":"00:40.880 ","End":"00:44.975","Text":"The thing that we\u0027re trying to maximize or minimize is what I call y."},{"Start":"00:44.975 ","End":"00:48.855","Text":"This segment AE is what I\u0027ll call x,"},{"Start":"00:48.855 ","End":"00:53.315","Text":"and the area of the trapezoid is what I\u0027ll call y."},{"Start":"00:53.315 ","End":"00:57.710","Text":"Then I\u0027ll get y in terms of x and use calculus to differentiate,"},{"Start":"00:57.710 ","End":"00:59.675","Text":"and so on to find the maximum."},{"Start":"00:59.675 ","End":"01:02.090","Text":"I have to remind you of the formula for"},{"Start":"01:02.090 ","End":"01:05.180","Text":"the area of a trapezoid in case you\u0027ve forgotten it."},{"Start":"01:05.180 ","End":"01:08.890","Text":"In general, a trapezoid which has 2 bases."},{"Start":"01:08.890 ","End":"01:10.810","Text":"It has, let\u0027s say this base is a,"},{"Start":"01:10.810 ","End":"01:12.515","Text":"and this base is b,"},{"Start":"01:12.515 ","End":"01:14.970","Text":"and this here is an altitude,"},{"Start":"01:14.970 ","End":"01:18.735","Text":"or height, then the area s,"},{"Start":"01:18.735 ","End":"01:22.680","Text":"is equal a plus b over 2, times the height."},{"Start":"01:22.680 ","End":"01:24.210","Text":"I\u0027ll put that in here."},{"Start":"01:24.210 ","End":"01:28.950","Text":"In our case, y is equal to,"},{"Start":"01:28.950 ","End":"01:31.305","Text":"we don\u0027t have all the pieces yet,"},{"Start":"01:31.305 ","End":"01:32.955","Text":"but we have a good start,"},{"Start":"01:32.955 ","End":"01:35.625","Text":"that we have a, which is 6,"},{"Start":"01:35.625 ","End":"01:38.235","Text":"and now let\u0027s see what the long base is."},{"Start":"01:38.235 ","End":"01:41.930","Text":"Because it\u0027s isosceles, this will also be x."},{"Start":"01:41.930 ","End":"01:43.775","Text":"There is a symmetry to it."},{"Start":"01:43.775 ","End":"01:45.620","Text":"Because if this is a rectangle,"},{"Start":"01:45.620 ","End":"01:48.860","Text":"this bit is 6, so the long base is 6,"},{"Start":"01:48.860 ","End":"01:51.890","Text":"plus 2_x, then times h,"},{"Start":"01:51.890 ","End":"01:55.535","Text":"which we also don\u0027t know yet, over 2."},{"Start":"01:55.535 ","End":"02:00.710","Text":"But certainly, I can simplify it already and see that 6 plus 6 plus 2_x over"},{"Start":"02:00.710 ","End":"02:07.125","Text":"2 is just 6 plus x times h. H, we can figure out."},{"Start":"02:07.125 ","End":"02:09.980","Text":"I\u0027m going to remind you also of Pythagoras\u0027s theorem."},{"Start":"02:09.980 ","End":"02:15.520","Text":"In general, if we have a right triangle where this is a,"},{"Start":"02:15.520 ","End":"02:17.100","Text":"and this is b,"},{"Start":"02:17.100 ","End":"02:19.094","Text":"and this is c,"},{"Start":"02:19.094 ","End":"02:23.795","Text":"then c squared equals a squared plus b squared."},{"Start":"02:23.795 ","End":"02:26.230","Text":"We want h in terms of 4 and x,"},{"Start":"02:26.230 ","End":"02:31.110","Text":"so we just use Pythagoras\u0027s theorem by bringing 1 of these to the other side,"},{"Start":"02:31.110 ","End":"02:36.635","Text":"and we get that h squared is 4 squared minus x squared,"},{"Start":"02:36.635 ","End":"02:39.815","Text":"which is 16 minus x squared."},{"Start":"02:39.815 ","End":"02:49.930","Text":"What I can do now is write y in terms of just x. Y is equal to 6 plus x from here,"},{"Start":"02:49.930 ","End":"02:56.070","Text":"and h is the square root of 16 minus x squared,"},{"Start":"02:56.070 ","End":"02:58.930","Text":"and this is a big part of the problem,"},{"Start":"02:58.930 ","End":"03:05.340","Text":"is just getting the formula for y in terms of x. I\u0027d also like to point out,"},{"Start":"03:05.340 ","End":"03:08.375","Text":"as far as the domain of definition of x, that x,"},{"Start":"03:08.375 ","End":"03:09.440","Text":"because it\u0027s a length,"},{"Start":"03:09.440 ","End":"03:13.130","Text":"has to be bigger than 0, but also,"},{"Start":"03:13.130 ","End":"03:15.680","Text":"I also know that x is less than 4,"},{"Start":"03:15.680 ","End":"03:19.505","Text":"because any side of a right triangle is less than the hypotenuse,"},{"Start":"03:19.505 ","End":"03:21.110","Text":"so it\u0027s less than 4."},{"Start":"03:21.110 ","End":"03:24.470","Text":"What this means is that x squared is less than 16,"},{"Start":"03:24.470 ","End":"03:27.110","Text":"and the square root is defined very well."},{"Start":"03:27.110 ","End":"03:31.280","Text":"Also, when we finally do the calculus differentiate and set to 0,"},{"Start":"03:31.280 ","End":"03:33.665","Text":"we should remember that x is positive."},{"Start":"03:33.665 ","End":"03:36.545","Text":"What we need now is y prime."},{"Start":"03:36.545 ","End":"03:38.395","Text":"This is a product,"},{"Start":"03:38.395 ","End":"03:40.515","Text":"and since I\u0027m reminding you of everything,"},{"Start":"03:40.515 ","End":"03:42.390","Text":"I\u0027ll remind you of the product rule,"},{"Start":"03:42.390 ","End":"03:46.200","Text":"that if I have 2 functions of x, product, derivative."},{"Start":"03:46.200 ","End":"03:48.590","Text":"Is just like a cross-multiplication."},{"Start":"03:48.590 ","End":"03:49.955","Text":"1 of them derived,"},{"Start":"03:49.955 ","End":"03:51.445","Text":"and the other not."},{"Start":"03:51.445 ","End":"03:53.370","Text":"Then 1, as is,"},{"Start":"03:53.370 ","End":"03:55.260","Text":"and the other derived."},{"Start":"03:55.260 ","End":"03:58.680","Text":"In our case, we\u0027ll get that this 1,"},{"Start":"03:58.680 ","End":"04:02.000","Text":"derived is just 1, times the other,"},{"Start":"04:02.000 ","End":"04:06.010","Text":"which is the square root of 16 minus x squared."},{"Start":"04:06.010 ","End":"04:09.210","Text":"Then plus, this 1, as is,"},{"Start":"04:09.210 ","End":"04:14.435","Text":"6 plus x, and the derivative of 16 minus x squared."},{"Start":"04:14.435 ","End":"04:18.350","Text":"The derivative of the square root is 1,"},{"Start":"04:18.350 ","End":"04:22.575","Text":"over twice the square root of the same thing for starters,"},{"Start":"04:22.575 ","End":"04:26.540","Text":"but then we have to also multiply by the inner derivative,"},{"Start":"04:26.540 ","End":"04:28.550","Text":"which is minus 2_x."},{"Start":"04:28.550 ","End":"04:34.160","Text":"At this point, I\u0027d like to already cancel this 2 with this 2."},{"Start":"04:34.160 ","End":"04:38.180","Text":"Now I want to just do some rearranging algebraically."},{"Start":"04:38.180 ","End":"04:42.620","Text":"What we\u0027re going to do is bring everything over as a fraction,"},{"Start":"04:42.620 ","End":"04:47.820","Text":"over square root of 16 minus x squared."},{"Start":"04:47.820 ","End":"04:50.840","Text":"This is already over 16 minus x squared,"},{"Start":"04:50.840 ","End":"04:54.230","Text":"but this bit, I\u0027ll have to multiply top and bottom, so to speak."},{"Start":"04:54.230 ","End":"04:57.335","Text":"I\u0027ll have to multiply by this square root over itself."},{"Start":"04:57.335 ","End":"05:02.570","Text":"The square root of this times the square root of the same thing without the square root,"},{"Start":"05:02.570 ","End":"05:05.450","Text":"it\u0027s just 16 minus x squared."},{"Start":"05:05.450 ","End":"05:07.820","Text":"There\u0027s a minus here, notice,"},{"Start":"05:07.820 ","End":"05:09.670","Text":"so it\u0027s minus 6,"},{"Start":"05:09.670 ","End":"05:13.050","Text":"plus x over this."},{"Start":"05:13.050 ","End":"05:17.960","Text":"What we have to do is let y prime equal 0."},{"Start":"05:17.960 ","End":"05:19.580","Text":"Y prime equals 0."},{"Start":"05:19.580 ","End":"05:21.965","Text":"If I set it, will give me."},{"Start":"05:21.965 ","End":"05:25.390","Text":"Let me just simplify the numerator."},{"Start":"05:25.390 ","End":"05:29.205","Text":"Ops, I forgot the x here, not too late."},{"Start":"05:29.205 ","End":"05:30.975","Text":"There\u0027s another x here."},{"Start":"05:30.975 ","End":"05:33.320","Text":"Now when I simplify,"},{"Start":"05:33.320 ","End":"05:40.080","Text":"what I\u0027ll get is 16 minus 2_x squared and just the x term."},{"Start":"05:40.080 ","End":"05:46.640","Text":"The x term is minus 6_x over the same thing."},{"Start":"05:46.640 ","End":"05:47.900","Text":"I\u0027ll just write the square root."},{"Start":"05:47.900 ","End":"05:49.850","Text":"The reason I\u0027m not bothering to write what\u0027s"},{"Start":"05:49.850 ","End":"05:53.215","Text":"inside is that the square root is also positive."},{"Start":"05:53.215 ","End":"05:57.710","Text":"If this over something positive or non-zero is equal to 0,"},{"Start":"05:57.710 ","End":"05:59.660","Text":"then just the numerator is 0."},{"Start":"05:59.660 ","End":"06:01.550","Text":"I\u0027ll divide by minus 2,"},{"Start":"06:01.550 ","End":"06:03.245","Text":"so I\u0027ll get 1_x squared."},{"Start":"06:03.245 ","End":"06:05.510","Text":"I get, if I divide by minus 2,"},{"Start":"06:05.510 ","End":"06:11.380","Text":"1_x squared plus 3_x minus 8,"},{"Start":"06:11.380 ","End":"06:13.460","Text":"this is equal to 0."},{"Start":"06:13.460 ","End":"06:16.105","Text":"That\u0027s just a quadratic equation."},{"Start":"06:16.105 ","End":"06:19.500","Text":"Just apply the formula that I\u0027m not going to remind you of."},{"Start":"06:19.500 ","End":"06:23.070","Text":"We get x is equal to,"},{"Start":"06:23.070 ","End":"06:25.170","Text":"this is like a is 1,"},{"Start":"06:25.170 ","End":"06:27.040","Text":"b is 3, c is minus 8,"},{"Start":"06:27.040 ","End":"06:32.750","Text":"minus b plus or minus the square root of b squared,"},{"Start":"06:32.750 ","End":"06:36.170","Text":"which is 9, minus 4 a_c,"},{"Start":"06:36.170 ","End":"06:37.610","Text":"minus 4, times 1,"},{"Start":"06:37.610 ","End":"06:40.540","Text":"times minus 8, it\u0027s plus 32,"},{"Start":"06:40.540 ","End":"06:43.500","Text":"all over 2_a which is 2,"},{"Start":"06:43.500 ","End":"06:45.825","Text":"9 plus 32 is 41."},{"Start":"06:45.825 ","End":"06:48.080","Text":"We have 2 possibilities."},{"Start":"06:48.080 ","End":"06:55.365","Text":"Minus 3 minus the square root of 41 over 2,"},{"Start":"06:55.365 ","End":"06:58.140","Text":"and the other possibility is minus 3,"},{"Start":"06:58.140 ","End":"07:01.890","Text":"plus the square root of 41 over 2."},{"Start":"07:01.890 ","End":"07:05.105","Text":"Now, the square root of 41 is bigger than 3,"},{"Start":"07:05.105 ","End":"07:09.509","Text":"so this 1 comes out negative,"},{"Start":"07:09.509 ","End":"07:11.745","Text":"and therefore is inapplicable."},{"Start":"07:11.745 ","End":"07:15.194","Text":"The only 1 that we have is this."},{"Start":"07:15.194 ","End":"07:17.585","Text":"I just did the calculation,"},{"Start":"07:17.585 ","End":"07:24.075","Text":"and this is approximately 1.7 to 1 decimal place."},{"Start":"07:24.075 ","End":"07:26.530","Text":"This is the candidate for extremum,"},{"Start":"07:26.530 ","End":"07:28.690","Text":"but how do I know that it\u0027s a maximum?"},{"Start":"07:28.690 ","End":"07:34.400","Text":"Let\u0027s scroll down a bit first and then make a table,"},{"Start":"07:34.400 ","End":"07:38.280","Text":"only the very simple table. Let\u0027s see."},{"Start":"07:38.280 ","End":"07:40.315","Text":"There are only 2 things I really need."},{"Start":"07:40.315 ","End":"07:42.129","Text":"I need to put in x,"},{"Start":"07:42.129 ","End":"07:44.590","Text":"and I need to put in y prime."},{"Start":"07:44.590 ","End":"07:48.070","Text":"The value I have is 1.7,"},{"Start":"07:48.070 ","End":"07:51.615","Text":"that the derivative is equal to 0."},{"Start":"07:51.615 ","End":"07:55.445","Text":"Question is what happens when it\u0027s a bit less than 1.7?"},{"Start":"07:55.445 ","End":"07:58.855","Text":"Let\u0027s choose a sample value, say 1,"},{"Start":"07:58.855 ","End":"08:01.395","Text":"and a bit bigger than 1.7,"},{"Start":"08:01.395 ","End":"08:04.230","Text":"and I\u0027ll choose the sample value 2."},{"Start":"08:04.230 ","End":"08:08.120","Text":"Now I have to substitute in y prime."},{"Start":"08:08.120 ","End":"08:10.190","Text":"The function is increasing or decreasing"},{"Start":"08:10.190 ","End":"08:12.660","Text":"depending on whether y prime is positive or negative,"},{"Start":"08:12.660 ","End":"08:15.605","Text":"so I only need to know the sign of what\u0027s here."},{"Start":"08:15.605 ","End":"08:18.770","Text":"Y prime is what\u0027s written over here."},{"Start":"08:18.770 ","End":"08:20.915","Text":"The denominator is positive,"},{"Start":"08:20.915 ","End":"08:24.500","Text":"so all I need to know is the sign of the numerator."},{"Start":"08:24.500 ","End":"08:26.435","Text":"I can take it off here."},{"Start":"08:26.435 ","End":"08:28.280","Text":"When x is 1,"},{"Start":"08:28.280 ","End":"08:30.545","Text":"I get 16 minus 6,"},{"Start":"08:30.545 ","End":"08:32.340","Text":"minus 2, is 8."},{"Start":"08:32.340 ","End":"08:38.625","Text":"Anyways, it\u0027s positive, and I\u0027ll put another column in y."},{"Start":"08:38.625 ","End":"08:46.775","Text":"When I put in 2, I get 16 minus 12, minus 8."},{"Start":"08:46.775 ","End":"08:49.715","Text":"It comes out negative,"},{"Start":"08:49.715 ","End":"08:54.904","Text":"which means that the function itself is increasing here,"},{"Start":"08:54.904 ","End":"09:00.005","Text":"decreasing here, and this gives us so that this point,"},{"Start":"09:00.005 ","End":"09:02.200","Text":"this must be a maximum."},{"Start":"09:02.200 ","End":"09:09.555","Text":"The answer is that AE is equal to 1.7,"},{"Start":"09:09.555 ","End":"09:13.730","Text":"or you could even write minus 3,"},{"Start":"09:13.730 ","End":"09:17.450","Text":"plus square root of 41 over 2, to be accurate,"},{"Start":"09:17.450 ","End":"09:20.705","Text":"which is approximately 1.7 for"},{"Start":"09:20.705 ","End":"09:30.600","Text":"maximum area of the trapezoid ABCD. We are done."}],"ID":8557},{"Watched":false,"Name":"Exercise 2","Duration":"4m 33s","ChapterTopicVideoID":8356,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8356.jpeg","UploadDate":"2017-01-28T12:42:30.5730000","DurationForVideoObject":"PT4M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.445","Text":"In this exercise, we\u0027re given a rectangle, A, B, C, D."},{"Start":"00:05.445 ","End":"00:12.734","Text":"Part a asks us to express the area of the rectangle in terms of x."},{"Start":"00:12.734 ","End":"00:18.400","Text":"This is x if the circumference of the rectangle is 60 centimeters."},{"Start":"00:18.590 ","End":"00:24.705","Text":"I\u0027ll let the area be denoted as y."},{"Start":"00:24.705 ","End":"00:28.080","Text":"We have to give y in terms of x."},{"Start":"00:28.080 ","End":"00:32.325","Text":"Now, the area of a rectangle is the length times the width."},{"Start":"00:32.325 ","End":"00:34.155","Text":"We have 1 dimension,"},{"Start":"00:34.155 ","End":"00:36.510","Text":"which is the length, that\u0027s x,"},{"Start":"00:36.510 ","End":"00:39.420","Text":"and this is also x, but we don\u0027t have the width,"},{"Start":"00:39.420 ","End":"00:42.420","Text":"let\u0027s call it w, that would be here and here."},{"Start":"00:42.420 ","End":"00:48.335","Text":"We have to first of all compute w. We do have a hint that the circumference is 60,"},{"Start":"00:48.335 ","End":"00:51.560","Text":"so what we have is that 60,"},{"Start":"00:51.560 ","End":"00:52.685","Text":"which is a circumference,"},{"Start":"00:52.685 ","End":"00:55.520","Text":"is w plus x plus w plus x,"},{"Start":"00:55.520 ","End":"00:58.790","Text":"which is 2w plus 2x,"},{"Start":"00:58.790 ","End":"01:04.355","Text":"which means that 30 is w plus x,"},{"Start":"01:04.355 ","End":"01:08.975","Text":"which means that w is 30 minus x."},{"Start":"01:08.975 ","End":"01:11.000","Text":"Now, we\u0027ve expressed it in terms of x."},{"Start":"01:11.000 ","End":"01:16.295","Text":"Now, getting back to the area y would equal length times width,"},{"Start":"01:16.295 ","End":"01:21.275","Text":"which is x times w. Since w is that,"},{"Start":"01:21.275 ","End":"01:26.860","Text":"it\u0027s equal to x times 30 minus x."},{"Start":"01:26.860 ","End":"01:30.365","Text":"Now, we\u0027ve got equation of"},{"Start":"01:30.365 ","End":"01:34.340","Text":"the thing we need to maximize in terms of the independent variable."},{"Start":"01:34.340 ","End":"01:36.305","Text":"Let me just write that again,"},{"Start":"01:36.305 ","End":"01:41.315","Text":"y equals x times 30 minus x,"},{"Start":"01:41.315 ","End":"01:47.700","Text":"we put it in a box because this is an important first stage."},{"Start":"01:47.700 ","End":"01:55.840","Text":"We have to find out now what is the value of x in order for y to be maximal."},{"Start":"01:55.840 ","End":"02:00.080","Text":"This is now a calculus problem where we\u0027re going to differentiate,"},{"Start":"02:00.080 ","End":"02:01.790","Text":"and that is equal to 0 and so on."},{"Start":"02:01.790 ","End":"02:03.200","Text":"Let\u0027s do that."},{"Start":"02:03.200 ","End":"02:05.450","Text":"We have to find y prime first."},{"Start":"02:05.450 ","End":"02:06.890","Text":"Could use the product rule,"},{"Start":"02:06.890 ","End":"02:15.455","Text":"but I think it\u0027s better to just expand to say that y is equal to 30x minus x squared."},{"Start":"02:15.455 ","End":"02:22.480","Text":"Then we can get that y prime is equal to 30 minus 2x."},{"Start":"02:22.480 ","End":"02:25.850","Text":"For maximum or minimum,"},{"Start":"02:25.850 ","End":"02:31.415","Text":"we need to solve the equation y prime equals 0."},{"Start":"02:31.415 ","End":"02:37.590","Text":"This will give us that 30 minus 2x is equal to 0,"},{"Start":"02:37.590 ","End":"02:42.830","Text":"and so immediately we can see that x is equal to 15."},{"Start":"02:42.830 ","End":"02:45.020","Text":"But how do we know it\u0027s a maximum?"},{"Start":"02:45.020 ","End":"02:47.870","Text":"One of the ways is to make a table,"},{"Start":"02:47.870 ","End":"02:50.380","Text":"so we make a table where we have"},{"Start":"02:50.380 ","End":"02:55.630","Text":"x, y prime, and y."},{"Start":"02:55.630 ","End":"02:59.604","Text":"This is for the increasing decreasing method."},{"Start":"02:59.604 ","End":"03:03.790","Text":"The interesting x is 15."},{"Start":"03:03.790 ","End":"03:07.105","Text":"At this point, y prime is 0."},{"Start":"03:07.105 ","End":"03:10.840","Text":"Then we want to take something smaller than 15, let\u0027s say,"},{"Start":"03:10.840 ","End":"03:13.689","Text":"14, and also something bigger than 15,"},{"Start":"03:13.689 ","End":"03:16.795","Text":"let\u0027s say 16, and see what y prime is there."},{"Start":"03:16.795 ","End":"03:19.180","Text":"Well, if x is 14,"},{"Start":"03:19.180 ","End":"03:23.320","Text":"then here we get 30 minus 28 is 2,"},{"Start":"03:23.320 ","End":"03:26.980","Text":"but we don\u0027t actually care about the value only that it\u0027s positive."},{"Start":"03:26.980 ","End":"03:31.765","Text":"If I put in 16, it\u0027s 30 minus 32 comes out negative."},{"Start":"03:31.765 ","End":"03:34.835","Text":"At this point, the function is increasing."},{"Start":"03:34.835 ","End":"03:42.200","Text":"Here, it\u0027s decreasing and when you pass from increasing to decreasing, it\u0027s a maximum."},{"Start":"03:42.200 ","End":"03:45.050","Text":"But let\u0027s see what the original question was."},{"Start":"03:45.050 ","End":"03:50.750","Text":"It was to find the dimensions of the rectangle with maximal area."},{"Start":"03:50.750 ","End":"03:55.340","Text":"The dimensions means both x and w."},{"Start":"03:55.340 ","End":"04:00.240","Text":"What we want to do is now find x and w,"},{"Start":"04:00.240 ","End":"04:03.720","Text":"where x we have, is 15."},{"Start":"04:03.720 ","End":"04:09.620","Text":"From here, w is equal to 30 minus x,"},{"Start":"04:09.620 ","End":"04:12.530","Text":"which is 30 minus 15,"},{"Start":"04:12.530 ","End":"04:15.770","Text":"which is 15 also."},{"Start":"04:15.770 ","End":"04:25.939","Text":"The dimensions 15 by 15 give maximal area."},{"Start":"04:25.939 ","End":"04:28.955","Text":"It turns out that when it\u0027s a square,"},{"Start":"04:28.955 ","End":"04:31.040","Text":"that\u0027s when the area\u0027s maximal."},{"Start":"04:31.040 ","End":"04:34.050","Text":"We\u0027re done."}],"ID":8558},{"Watched":false,"Name":"Exercise 3","Duration":"5m 57s","ChapterTopicVideoID":8370,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8370.jpeg","UploadDate":"2017-01-28T12:49:37.2070000","DurationForVideoObject":"PT5M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we\u0027re given a word problem in geometry,"},{"Start":"00:03.780 ","End":"00:05.670","Text":"and instead of reading it out loud,"},{"Start":"00:05.670 ","End":"00:08.610","Text":"I\u0027ll just tell you what\u0027s going on here."},{"Start":"00:08.610 ","End":"00:17.085","Text":"What we have is a rectangle and we\u0027re given that the short side is 5 centimeters."},{"Start":"00:17.085 ","End":"00:18.270","Text":"That\u0027s from here to here,"},{"Start":"00:18.270 ","End":"00:20.025","Text":"and also this would be 5."},{"Start":"00:20.025 ","End":"00:23.130","Text":"We\u0027re given that the long side,"},{"Start":"00:23.130 ","End":"00:25.935","Text":"that would be from here to here,"},{"Start":"00:25.935 ","End":"00:31.695","Text":"is 10 centimeters and the same here AB is also 10 centimeters."},{"Start":"00:31.695 ","End":"00:36.590","Text":"We\u0027re also told that this is x centimeters,"},{"Start":"00:36.590 ","End":"00:38.570","Text":"this is x from here to here,"},{"Start":"00:38.570 ","End":"00:41.360","Text":"and this is x, and this is x."},{"Start":"00:41.360 ","End":"00:45.770","Text":"Now we have to look at the parallelogram here, I\u0027ll shade it."},{"Start":"00:45.770 ","End":"00:51.365","Text":"We\u0027re asked to find x such that this area is maximum."},{"Start":"00:51.365 ","End":"00:54.770","Text":"Now since the area is what we have to find out,"},{"Start":"00:54.770 ","End":"00:57.980","Text":"this will be our dependent variable y,"},{"Start":"00:57.980 ","End":"01:01.460","Text":"which is a function of the independent variable x,"},{"Start":"01:01.460 ","End":"01:03.095","Text":"which is what we want to find out."},{"Start":"01:03.095 ","End":"01:05.870","Text":"Which x gives maximum y?"},{"Start":"01:05.870 ","End":"01:10.495","Text":"The first thing to do is to actually find the function of x."},{"Start":"01:10.495 ","End":"01:17.600","Text":"What I\u0027m going to do is take the area of the rectangle as a whole,"},{"Start":"01:17.600 ","End":"01:23.125","Text":"and subtract these 4 triangles that are still unshaded."},{"Start":"01:23.125 ","End":"01:26.120","Text":"What I get is, first of all,"},{"Start":"01:26.120 ","End":"01:29.970","Text":"the area of the rectangle is 5 times 10."},{"Start":"01:30.640 ","End":"01:35.420","Text":"Now I have to subtract the area of this triangle,"},{"Start":"01:35.420 ","End":"01:37.580","Text":"but it appears twice."},{"Start":"01:37.580 ","End":"01:45.200","Text":"Each triangle is 1/2 base times height times x times x,"},{"Start":"01:45.200 ","End":"01:46.900","Text":"but since there\u0027s 2 of them,"},{"Start":"01:46.900 ","End":"01:49.650","Text":"I\u0027ll put a 2 in front here."},{"Start":"01:49.650 ","End":"01:55.670","Text":"The other triangle is also 1/2 base times height."},{"Start":"01:55.670 ","End":"01:58.070","Text":"Now, in this case, or I can look at this 1,"},{"Start":"01:58.070 ","End":"02:01.205","Text":"the height would be this,"},{"Start":"02:01.205 ","End":"02:05.635","Text":"which is 5 minus x, so it\u0027s 1/2."},{"Start":"02:05.635 ","End":"02:07.665","Text":"Well, let\u0027s take the base first,"},{"Start":"02:07.665 ","End":"02:10.110","Text":"base would be 10 minus x,"},{"Start":"02:10.110 ","End":"02:12.580","Text":"that\u0027s from here to here."},{"Start":"02:12.680 ","End":"02:14.740","Text":"The height, as I said,"},{"Start":"02:14.740 ","End":"02:16.720","Text":"is 5 minus x."},{"Start":"02:16.720 ","End":"02:18.660","Text":"But again, there\u0027s 2 of these,"},{"Start":"02:18.660 ","End":"02:21.195","Text":"so I put a 2 in front of here."},{"Start":"02:21.195 ","End":"02:23.200","Text":"This is the area."},{"Start":"02:23.200 ","End":"02:25.195","Text":"Now let\u0027s simplify this a bit."},{"Start":"02:25.195 ","End":"02:27.189","Text":"What we get is 50,"},{"Start":"02:27.189 ","End":"02:29.035","Text":"the 2 with the 1/2 cancels,"},{"Start":"02:29.035 ","End":"02:31.055","Text":"so that\u0027s x squared."},{"Start":"02:31.055 ","End":"02:32.685","Text":"Then here we have,"},{"Start":"02:32.685 ","End":"02:34.650","Text":"just also 2 with 1/2 cancel,"},{"Start":"02:34.650 ","End":"02:38.850","Text":"10 minus x, 5 minus x."},{"Start":"02:38.850 ","End":"02:43.800","Text":"This is equal to 50 minus x squared minus,"},{"Start":"02:43.800 ","End":"02:47.190","Text":"if I multiply this out I get 50,"},{"Start":"02:47.190 ","End":"02:54.915","Text":"minus 5x minus 10x is minus 15x plus x squared."},{"Start":"02:54.915 ","End":"03:02.480","Text":"This is equal to 50 minus x squared minus 50 plus 15x,"},{"Start":"03:02.480 ","End":"03:06.445","Text":"the minus with a minus gives plus, minus x squared."},{"Start":"03:06.445 ","End":"03:12.615","Text":"This equals, the 50 with the 50 cancels,"},{"Start":"03:12.615 ","End":"03:19.355","Text":"and x squared combines with x squared to give 2x squared."},{"Start":"03:19.355 ","End":"03:27.470","Text":"What I end up with is 15x minus 2x squared."},{"Start":"03:27.470 ","End":"03:34.820","Text":"I also want you to notice that x has a certain domain, first of all,"},{"Start":"03:34.820 ","End":"03:39.104","Text":"it\u0027s a positive quantity because all lengths are positive,"},{"Start":"03:39.104 ","End":"03:41.140","Text":"I don\u0027t want to take the case where x is 0,"},{"Start":"03:41.140 ","End":"03:42.595","Text":"it doesn\u0027t really make sense."},{"Start":"03:42.595 ","End":"03:46.540","Text":"Also x has to be less than 5 and it has to be less than 10,"},{"Start":"03:46.540 ","End":"03:49.805","Text":"which means that it\u0027s less than 5."},{"Start":"03:49.805 ","End":"03:58.605","Text":"I have a limitation x has to be between 0 and 5."},{"Start":"03:58.605 ","End":"04:02.290","Text":"We going to now do an extremum problem,"},{"Start":"04:02.290 ","End":"04:08.380","Text":"which means differentiating and setting to 0 and then checking if it\u0027s a maximum."},{"Start":"04:08.380 ","End":"04:18.680","Text":"We get that f prime of x or y prime is equal to 15 minus 4x."},{"Start":"04:18.980 ","End":"04:21.710","Text":"If we want to an extremum,"},{"Start":"04:21.710 ","End":"04:26.225","Text":"we have to set f prime of x to equal 0 for an extremum,"},{"Start":"04:26.225 ","End":"04:32.810","Text":"and that gives me that 15 minus 4x is equal to 0,"},{"Start":"04:32.810 ","End":"04:40.090","Text":"which gives me that x is equal to 15/4 or if you like,"},{"Start":"04:40.090 ","End":"04:44.990","Text":"3 3/4 or 3.75, whichever you prefer."},{"Start":"04:44.990 ","End":"04:48.725","Text":"Now let\u0027s just check that it is indeed a maximum."},{"Start":"04:48.725 ","End":"04:53.045","Text":"One of the ways to show that it\u0027s a maximum is the table method,"},{"Start":"04:53.045 ","End":"04:56.615","Text":"where we make a table with x,"},{"Start":"04:56.615 ","End":"05:00.320","Text":"f prime of x or y prime and f of"},{"Start":"05:00.320 ","End":"05:05.855","Text":"x. I\u0027m going to use this for increasing and decreasing intervals."},{"Start":"05:05.855 ","End":"05:08.260","Text":"The first thing we know is that at,"},{"Start":"05:08.260 ","End":"05:09.750","Text":"let\u0027s take the decimal,"},{"Start":"05:09.750 ","End":"05:14.415","Text":"3.75, the derivative is 0,"},{"Start":"05:14.415 ","End":"05:16.550","Text":"so we check what happens before and after."},{"Start":"05:16.550 ","End":"05:17.720","Text":"Let\u0027s take, for example,"},{"Start":"05:17.720 ","End":"05:21.090","Text":"the value 3 and the value 4,"},{"Start":"05:21.090 ","End":"05:23.505","Text":"now f prime of x, this is here."},{"Start":"05:23.505 ","End":"05:25.260","Text":"If I put it in x equals 3,"},{"Start":"05:25.260 ","End":"05:26.910","Text":"it\u0027s 15 minus 12,"},{"Start":"05:26.910 ","End":"05:29.475","Text":"I don\u0027t care about the answer but it\u0027s positive,"},{"Start":"05:29.475 ","End":"05:30.915","Text":"that\u0027s what I need to know."},{"Start":"05:30.915 ","End":"05:32.400","Text":"If I put x equals 4,"},{"Start":"05:32.400 ","End":"05:36.215","Text":"I get 15 minus 16 in any event it\u0027s negative."},{"Start":"05:36.215 ","End":"05:39.425","Text":"This means that the function here is increasing,"},{"Start":"05:39.425 ","End":"05:44.240","Text":"here it\u0027s decreasing, and if it\u0027s between an increasing and decreasing interval,"},{"Start":"05:44.240 ","End":"05:46.825","Text":"then it must be a maximum."},{"Start":"05:46.825 ","End":"05:55.475","Text":"That means that our answer that x is 3.75 is indeed a maximum,"},{"Start":"05:55.475 ","End":"05:58.079","Text":"and that answers the question."}],"ID":8559},{"Watched":false,"Name":"Exercise 4","Duration":"5m 19s","ChapterTopicVideoID":8357,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8357.jpeg","UploadDate":"2017-01-28T12:43:29.3870000","DurationForVideoObject":"PT5M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"Here we have a word problem and instead of just reading it, I\u0027ll explain it."},{"Start":"00:03.990 ","End":"00:06.900","Text":"We have a right triangle A, B, C,"},{"Start":"00:06.900 ","End":"00:09.810","Text":"and that means that this is 90 degrees."},{"Start":"00:09.810 ","End":"00:13.680","Text":"In a right triangle, these 2 are called legs and this is the hypothenuse."},{"Start":"00:13.680 ","End":"00:16.620","Text":"Now, we build a square on the hypothenuse."},{"Start":"00:16.620 ","End":"00:19.590","Text":"That means this is a square, so these are all 90 degrees"},{"Start":"00:19.590 ","End":"00:22.065","Text":"and this length and this length are equal."},{"Start":"00:22.065 ","End":"00:27.555","Text":"We\u0027re also told that the sum of the lengths is 8 centimeters,"},{"Start":"00:27.555 ","End":"00:35.370","Text":"and we have to find what the legs are in order to make the total area minimal."},{"Start":"00:35.370 ","End":"00:37.080","Text":"The total area, I mean of the"},{"Start":"00:37.080 ","End":"00:41.970","Text":"Pentagon A, E, D, B, C, the whole thing."},{"Start":"00:41.970 ","End":"00:45.030","Text":"Let\u0027s call one of these legs x,"},{"Start":"00:45.030 ","End":"00:47.750","Text":"and that\u0027s going to be our independent variable,"},{"Start":"00:47.750 ","End":"00:50.780","Text":"and the area will be our dependent variable."},{"Start":"00:50.780 ","End":"00:53.630","Text":"Now, I don\u0027t need an extra variable here,"},{"Start":"00:53.630 ","End":"00:55.130","Text":"because straight away,"},{"Start":"00:55.130 ","End":"00:59.155","Text":"I can say that if x plus this is 8,"},{"Start":"00:59.155 ","End":"01:02.855","Text":"so I can immediately say that this is 8 minus x."},{"Start":"01:02.855 ","End":"01:07.385","Text":"All I have to do now is compute the total area."},{"Start":"01:07.385 ","End":"01:09.470","Text":"Well, clearly it breaks up into 2 bits,"},{"Start":"01:09.470 ","End":"01:11.845","Text":"the triangle bit and the square bit."},{"Start":"01:11.845 ","End":"01:16.540","Text":"The hypothenuse squared the length of AB"},{"Start":"01:16.540 ","End":"01:22.880","Text":"squared by Pythagoras is equal to this squared plus this squared."},{"Start":"01:22.880 ","End":"01:27.590","Text":"The other part of the area is the triangle,"},{"Start":"01:27.590 ","End":"01:35.375","Text":"which is 1/2 times the length of AC times the length of CB,"},{"Start":"01:35.375 ","End":"01:41.435","Text":"which in our case is simply 1/2 of x times 8 minus x."},{"Start":"01:41.435 ","End":"01:46.315","Text":"The total area and I call it y or f of x,"},{"Start":"01:46.315 ","End":"01:51.465","Text":"this will equal the triangular bit,"},{"Start":"01:51.465 ","End":"01:57.395","Text":"1/2x times 8 minus x plus the square bit,"},{"Start":"01:57.395 ","End":"02:03.045","Text":"which is x squared plus 8 minus x squared."},{"Start":"02:03.045 ","End":"02:05.850","Text":"I want to find the minimal f."},{"Start":"02:05.850 ","End":"02:11.435","Text":"We do our usual trick of differentiating and then we\u0027ll set it to 0."},{"Start":"02:11.435 ","End":"02:16.055","Text":"Let\u0027s first simplify the expression and see what we get."},{"Start":"02:16.055 ","End":"02:21.890","Text":"We get x times 8 minus x is 8x minus x squared."},{"Start":"02:21.890 ","End":"02:27.760","Text":"If we divide by 2, we get 4x minus x squared over 2."},{"Start":"02:27.760 ","End":"02:30.959","Text":"From here, we get plus x squared."},{"Start":"02:30.959 ","End":"02:34.835","Text":"Then using the formula for A minus B squared,"},{"Start":"02:34.835 ","End":"02:36.080","Text":"we get 8 squared,"},{"Start":"02:36.080 ","End":"02:42.574","Text":"which is 64 minus twice 8 times x plus x squared."},{"Start":"02:42.574 ","End":"02:46.430","Text":"Let\u0027s collect the terms together."},{"Start":"02:46.430 ","End":"02:51.395","Text":"Let\u0027s see, for x squared, I have x squared plus x squared minus x squared over 2."},{"Start":"02:51.395 ","End":"02:52.700","Text":"That\u0027s 2 minus 1/2,"},{"Start":"02:52.700 ","End":"02:54.940","Text":"it\u0027s 1 and 1/2x squared."},{"Start":"02:54.940 ","End":"02:57.945","Text":"That\u0027s 3 over 2x squared."},{"Start":"02:57.945 ","End":"03:05.115","Text":"Let\u0027s see, for Xs I just have the minus 12x and plus 64."},{"Start":"03:05.115 ","End":"03:09.200","Text":"That\u0027s the expression for the total area in terms of x."},{"Start":"03:09.200 ","End":"03:10.600","Text":"Now, I want to find the minimum,"},{"Start":"03:10.600 ","End":"03:12.560","Text":"so I start off by differentiating."},{"Start":"03:12.560 ","End":"03:18.690","Text":"F prime of x is equal to the derivative 2 times 3 over 2 is 3."},{"Start":"03:18.690 ","End":"03:24.395","Text":"It\u0027s 3x minus 12, that\u0027s the derivative."},{"Start":"03:24.395 ","End":"03:27.320","Text":"Now, when we do extrema, maximum or minimum,"},{"Start":"03:27.320 ","End":"03:33.035","Text":"we use the equation f prime of x equals 0 and see what that gives us."},{"Start":"03:33.035 ","End":"03:39.290","Text":"3x minus 12 is 0 and bring the 12 over to the other side,"},{"Start":"03:39.290 ","End":"03:43.550","Text":"divide by 3, x is equal to 4."},{"Start":"03:43.550 ","End":"03:48.740","Text":"This is our suspect for giving the minimum area."},{"Start":"03:48.740 ","End":"03:52.759","Text":"Let\u0027s check that we do indeed get a minimum."},{"Start":"03:52.759 ","End":"03:59.390","Text":"What I\u0027ll do is I\u0027ll use the method of the table for increasing and decreasing."},{"Start":"03:59.390 ","End":"04:03.345","Text":"I get a table with x,"},{"Start":"04:03.345 ","End":"04:07.575","Text":"with f prime of x, and f of x."},{"Start":"04:07.575 ","End":"04:15.845","Text":"The only candidate is 4 which point f prime is 0."},{"Start":"04:15.845 ","End":"04:19.500","Text":"But I want to know what it is to the right and to the left of 4."},{"Start":"04:19.500 ","End":"04:20.855","Text":"Let\u0027s take some values."},{"Start":"04:20.855 ","End":"04:23.485","Text":"Simplest is just to take 3 and 5."},{"Start":"04:23.485 ","End":"04:25.505","Text":"Let\u0027s see what f prime of x is."},{"Start":"04:25.505 ","End":"04:26.750","Text":"F prime of x is here,"},{"Start":"04:26.750 ","End":"04:28.955","Text":"so 3 times 3 is 9 minus 12."},{"Start":"04:28.955 ","End":"04:30.230","Text":"I don\u0027t care the value,"},{"Start":"04:30.230 ","End":"04:32.544","Text":"just if it\u0027s positive or negative."},{"Start":"04:32.544 ","End":"04:37.490","Text":"If I put in 5, I get 15 minus 12 is 3, anyway it\u0027s positive."},{"Start":"04:37.490 ","End":"04:40.160","Text":"Here, we have a decrease,"},{"Start":"04:40.160 ","End":"04:42.570","Text":"and here, we have an increase."},{"Start":"04:42.570 ","End":"04:46.460","Text":"If it\u0027s between a decreasing interval and an increasing interval,"},{"Start":"04:46.460 ","End":"04:48.859","Text":"then it\u0027s a minimum."},{"Start":"04:48.859 ","End":"04:51.020","Text":"Indeed we have a minimum,"},{"Start":"04:51.020 ","End":"04:53.645","Text":"but we haven\u0027t quite answered the question."},{"Start":"04:53.645 ","End":"04:57.605","Text":"The question was, what are the 2 legs equal to?"},{"Start":"04:57.605 ","End":"04:59.960","Text":"I\u0027ll just write it in words."},{"Start":"04:59.960 ","End":"05:11.240","Text":"The legs are x is 4 and 8 minus x, I can do in my head, is also 4,"},{"Start":"05:11.240 ","End":"05:15.410","Text":"so it\u0027s 4 centimeters and 4 centimeters."},{"Start":"05:15.410 ","End":"05:17.330","Text":"They\u0027re both 4 centimeters."},{"Start":"05:17.330 ","End":"05:20.790","Text":"We are done."}],"ID":8560},{"Watched":false,"Name":"Exercise 5","Duration":"6m 48s","ChapterTopicVideoID":8358,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8358.jpeg","UploadDate":"2017-01-28T12:44:31.8870000","DurationForVideoObject":"PT6M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"We have here a word problem and instead of just reading it,"},{"Start":"00:02.430 ","End":"00:04.650","Text":"I\u0027ll explain to you what\u0027s going on."},{"Start":"00:04.650 ","End":"00:08.850","Text":"We have a semicircle and its radius is 8,"},{"Start":"00:08.850 ","End":"00:12.600","Text":"so that would be the center of the semicircle."},{"Start":"00:12.600 ","End":"00:16.740","Text":"We have a rectangle ABCD inscribed in it."},{"Start":"00:16.740 ","End":"00:18.390","Text":"AB is on the diameter,"},{"Start":"00:18.390 ","End":"00:23.280","Text":"as you can see and C and D are on the circumference."},{"Start":"00:23.280 ","End":"00:28.080","Text":"The question is, what should the length of the side AB"},{"Start":"00:28.080 ","End":"00:34.110","Text":"be in order for the rectangle to have the maximum area possible?"},{"Start":"00:34.110 ","End":"00:39.485","Text":"What we need to do is to choose an independent variable."},{"Start":"00:39.485 ","End":"00:45.640","Text":"Normally I would choose AB to be x because that\u0027s what we have to find any way,"},{"Start":"00:45.640 ","End":"00:53.869","Text":"but it turns out that the math is a bit simpler if we let x be just 1/2 of the rectangle."},{"Start":"00:53.869 ","End":"01:03.135","Text":"In other words, just from A to O or from O to B so that the length of AB is actually 2x."},{"Start":"01:03.135 ","End":"01:04.550","Text":"At the end when we find x,"},{"Start":"01:04.550 ","End":"01:10.310","Text":"we must remember that when we answer the question, we have to double it."},{"Start":"01:11.690 ","End":"01:14.345","Text":"How do we find the area?"},{"Start":"01:14.345 ","End":"01:20.675","Text":"Well, Pythagoras comes in handy because the area of a rectangle is the base times height."},{"Start":"01:20.675 ","End":"01:24.170","Text":"We know the base is 2x but we have to figure out the height."},{"Start":"01:24.170 ","End":"01:30.290","Text":"Meanwhile, I can write a function f of x to represent the area of the rectangle."},{"Start":"01:30.290 ","End":"01:33.510","Text":"It\u0027s equal to 2x times,"},{"Start":"01:33.510 ","End":"01:37.140","Text":"let\u0027s call this height h times the height."},{"Start":"01:37.140 ","End":"01:43.730","Text":"We\u0027ll use Pythagoras to find the height so that h is equal"},{"Start":"01:43.730 ","End":"01:50.415","Text":"to the square root of 8 squared minus x squared."},{"Start":"01:50.415 ","End":"01:52.035","Text":"When we put it here,"},{"Start":"01:52.035 ","End":"01:59.825","Text":"we get 2x times the square root of 64 minus x squared."},{"Start":"01:59.825 ","End":"02:05.300","Text":"There\u0027s also a domain of definition of x. I mean,"},{"Start":"02:05.300 ","End":"02:08.435","Text":"x has to be positive because it\u0027s a length."},{"Start":"02:08.435 ","End":"02:11.090","Text":"So x is bigger than 0,"},{"Start":"02:11.090 ","End":"02:14.420","Text":"but x at most can be the radius of"},{"Start":"02:14.420 ","End":"02:18.850","Text":"the circle if x has to be less otherwise the rectangle will be flat."},{"Start":"02:18.850 ","End":"02:21.370","Text":"So x is between 0 and 8."},{"Start":"02:21.370 ","End":"02:25.610","Text":"Now, I\u0027m going to show you a trick."},{"Start":"02:25.610 ","End":"02:28.190","Text":"We want to find the maximum of this."},{"Start":"02:28.190 ","End":"02:33.439","Text":"We want to know what the maximum of f of x equals."},{"Start":"02:33.439 ","End":"02:35.540","Text":"Well, it turns out there\u0027s a trick when you have"},{"Start":"02:35.540 ","End":"02:38.870","Text":"square roots is that if you\u0027re looking for"},{"Start":"02:38.870 ","End":"02:41.960","Text":"a maximum of a positive quantity and this area"},{"Start":"02:41.960 ","End":"02:45.349","Text":"is certainly positive and this trick only works for positive,"},{"Start":"02:45.349 ","End":"02:48.860","Text":"then it\u0027s the same as answering the question,"},{"Start":"02:48.860 ","End":"02:53.900","Text":"what value of x gives the maximum value of f of x squared?"},{"Start":"02:53.900 ","End":"02:57.200","Text":"In other words, the target function to be maximized or"},{"Start":"02:57.200 ","End":"03:01.310","Text":"minimized if you know it\u0027s positive and it is in our case,"},{"Start":"03:01.310 ","End":"03:05.940","Text":"then you can square it and simplify the calculations."},{"Start":"03:05.940 ","End":"03:11.230","Text":"What we have now is another function and I\u0027ll call it g of x."},{"Start":"03:11.230 ","End":"03:13.680","Text":"You don\u0027t have to give it a name, but I\u0027ll call g of x,"},{"Start":"03:13.680 ","End":"03:16.295","Text":"f of x squared is equal to the square of this,"},{"Start":"03:16.295 ","End":"03:22.745","Text":"which gives me 4x squared and the square of the square root is just the thing itself,"},{"Start":"03:22.745 ","End":"03:26.460","Text":"64 minus x squared."},{"Start":"03:28.850 ","End":"03:32.235","Text":"Well, leave the 4 separately,"},{"Start":"03:32.235 ","End":"03:38.495","Text":"x squared times 64 is 64x squared and x squared times x squared is x^4,"},{"Start":"03:38.495 ","End":"03:40.670","Text":"it\u0027s with a minus."},{"Start":"03:40.670 ","End":"03:45.930","Text":"Now I want to differentiate because that\u0027s how we work with maximum or minimum."},{"Start":"03:45.930 ","End":"03:48.270","Text":"Differentiate then we assign to 0."},{"Start":"03:48.270 ","End":"03:52.460","Text":"The 4 stays and we get twice 64 times x,"},{"Start":"03:52.460 ","End":"03:59.385","Text":"which is 128x minus 4x cubed."},{"Start":"03:59.385 ","End":"04:04.700","Text":"I can take another 4 out of here and get 16 times 32."},{"Start":"04:04.700 ","End":"04:07.040","Text":"You know what, I can also take x out."},{"Start":"04:07.040 ","End":"04:12.465","Text":"It\u0027s 32 minus 4x squared times x."},{"Start":"04:12.465 ","End":"04:15.200","Text":"Now I want to solve the equation,"},{"Start":"04:15.200 ","End":"04:19.360","Text":"g prime of x equals 0."},{"Start":"04:19.360 ","End":"04:23.430","Text":"Now remember, I\u0027ll show you that x is not 0,"},{"Start":"04:23.430 ","End":"04:25.875","Text":"x is bigger than 0."},{"Start":"04:25.875 ","End":"04:33.980","Text":"When I\u0027m setting to 0, I can cancel the x and I can ignore the 16 also."},{"Start":"04:33.980 ","End":"04:37.229","Text":"Then just this bit will be 0."},{"Start":"04:37.800 ","End":"04:41.885","Text":"Sorry there\u0027s no 4 here, I\u0027m sorry."},{"Start":"04:41.885 ","End":"04:44.840","Text":"Take the eraser and just erase it because I"},{"Start":"04:44.840 ","End":"04:48.170","Text":"already took the 4 out and put it in with the 16."},{"Start":"04:48.170 ","End":"04:54.165","Text":"32 minus x squared is equal to 0."},{"Start":"04:54.165 ","End":"04:58.850","Text":"So x equals plus or minus the square root of 32,"},{"Start":"04:58.850 ","End":"05:00.320","Text":"but it\u0027s bigger than 0."},{"Start":"05:00.320 ","End":"05:07.415","Text":"So x is the square root of 32 and that\u0027s pretty much our answer for x."},{"Start":"05:07.415 ","End":"05:13.590","Text":"But remember that we have to find the value of AB, which is 2x."},{"Start":"05:13.970 ","End":"05:21.425","Text":"The length of AB for maximum would be twice the square root of 32."},{"Start":"05:21.425 ","End":"05:23.480","Text":"But not so fast,"},{"Start":"05:23.480 ","End":"05:27.345","Text":"I haven\u0027t shown yet that it really is a maximum."},{"Start":"05:27.345 ","End":"05:31.370","Text":"Let\u0027s make our usual table."},{"Start":"05:31.370 ","End":"05:35.840","Text":"What I need for the table is x,"},{"Start":"05:35.840 ","End":"05:39.205","Text":"I need g prime of x,"},{"Start":"05:39.205 ","End":"05:42.510","Text":"and I need g of x."},{"Start":"05:42.510 ","End":"05:44.630","Text":"This is only a suspect,"},{"Start":"05:44.630 ","End":"05:46.940","Text":"so I put in my suspects here."},{"Start":"05:46.940 ","End":"05:49.295","Text":"For the table, I need an approximate value."},{"Start":"05:49.295 ","End":"05:55.830","Text":"Calculator shows me that this is about 5.65 approximately."},{"Start":"05:55.830 ","End":"05:59.465","Text":"Here the derivative is 0 we just found"},{"Start":"05:59.465 ","End":"06:04.040","Text":"and we\u0027ll take something on the left and something on the right."},{"Start":"06:04.040 ","End":"06:08.940","Text":"We\u0027ll take 5 and we\u0027ll take 6."},{"Start":"06:08.940 ","End":"06:12.110","Text":"We have to substitute in g prime."},{"Start":"06:12.110 ","End":"06:14.920","Text":"But we only need to know positive or negative,"},{"Start":"06:14.920 ","End":"06:17.580","Text":"x is positive and 16 is positive."},{"Start":"06:17.580 ","End":"06:20.925","Text":"I can just substitute in this bit."},{"Start":"06:20.925 ","End":"06:23.180","Text":"Let\u0027s see if I put in 5,"},{"Start":"06:23.180 ","End":"06:26.330","Text":"I get 32 minus 25 positive."},{"Start":"06:26.330 ","End":"06:30.230","Text":"If I put in 6, 32 minus 36 negative,"},{"Start":"06:30.230 ","End":"06:32.990","Text":"the function here is increasing."},{"Start":"06:32.990 ","End":"06:35.240","Text":"The function here is decreasing."},{"Start":"06:35.240 ","End":"06:37.325","Text":"Between increasing and decreasing,"},{"Start":"06:37.325 ","End":"06:39.565","Text":"we have a maximum."},{"Start":"06:39.565 ","End":"06:43.920","Text":"Indeed the value of the side AB which gives"},{"Start":"06:43.920 ","End":"06:49.660","Text":"the maximum value is twice the square root of 32 and we are done."}],"ID":8561},{"Watched":false,"Name":"Exercise 6","Duration":"5m 32s","ChapterTopicVideoID":8371,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8371.jpeg","UploadDate":"2017-01-28T12:50:26.8070000","DurationForVideoObject":"PT5M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"Here we have a word problem and I\u0027ll explain it rather than just reading it."},{"Start":"00:04.650 ","End":"00:07.290","Text":"We have here a right angle triangle."},{"Start":"00:07.290 ","End":"00:09.555","Text":"There\u0027s the right angle."},{"Start":"00:09.555 ","End":"00:13.230","Text":"The side opposite the right angle"},{"Start":"00:13.230 ","End":"00:16.590","Text":"is called the hypotenuse and the other 2 are called legs."},{"Start":"00:16.590 ","End":"00:21.840","Text":"We\u0027re told that the sum of the lengths of the legs is 30 centimeters."},{"Start":"00:21.840 ","End":"00:24.405","Text":"That\u0027s AB plus BC."},{"Start":"00:24.405 ","End":"00:28.020","Text":"We also know that AD is the median to the leg BC."},{"Start":"00:28.020 ","End":"00:33.015","Text":"Now, the median means that it divides this leg into 2 equal bits."},{"Start":"00:33.015 ","End":"00:36.075","Text":"This bit is equal to this bit."},{"Start":"00:36.075 ","End":"00:41.450","Text":"What we want to know is what are the lengths of the legs AB and"},{"Start":"00:41.450 ","End":"00:46.280","Text":"BC in order to minimize the square of the length of the median."},{"Start":"00:46.280 ","End":"00:51.965","Text":"In other words, what we want to know is the length of AD squared."},{"Start":"00:51.965 ","End":"00:54.905","Text":"We want it to be minimum."},{"Start":"00:54.905 ","End":"00:58.400","Text":"Now, this is going to be our target function."},{"Start":"00:58.400 ","End":"01:00.610","Text":"That\u0027s what\u0027s going to be our f of x,"},{"Start":"01:00.610 ","End":"01:02.570","Text":"but we haven\u0027t gotten x."},{"Start":"01:02.570 ","End":"01:05.465","Text":"Since we want the length of the legs,"},{"Start":"01:05.465 ","End":"01:08.060","Text":"maybe we should let x be 1 of these legs,"},{"Start":"01:08.060 ","End":"01:09.260","Text":"either this or this."},{"Start":"01:09.260 ","End":"01:14.360","Text":"But since this is divided into 2 pieces and probably we could avoid fractions,"},{"Start":"01:14.360 ","End":"01:19.650","Text":"let\u0027s let x be just half of this length."},{"Start":"01:19.650 ","End":"01:21.575","Text":"But this is x and this is x."},{"Start":"01:21.575 ","End":"01:24.920","Text":"Now, the sum is 30 centimeters,"},{"Start":"01:24.920 ","End":"01:27.440","Text":"so this plus this is 30,"},{"Start":"01:27.440 ","End":"01:29.300","Text":"so 2x plus this is 30."},{"Start":"01:29.300 ","End":"01:33.350","Text":"In our heads, we can say that this is 30 minus 2x."},{"Start":"01:33.350 ","End":"01:40.475","Text":"Now, we\u0027re going to use Pythagoras\u0027s theorem to figure out what is AD squared."},{"Start":"01:40.475 ","End":"01:43.475","Text":"Let\u0027s just say that\u0027s going to be our f of x."},{"Start":"01:43.475 ","End":"01:47.115","Text":"The length of AD squared."},{"Start":"01:47.115 ","End":"01:52.640","Text":"By Pythagoras, it\u0027s going to equal this squared plus this squared."},{"Start":"01:52.640 ","End":"01:55.280","Text":"If I look at the triangle ABD,"},{"Start":"01:55.280 ","End":"02:04.204","Text":"x squared plus 30 minus 2x squared equals this squared."},{"Start":"02:04.204 ","End":"02:05.870","Text":"We don\u0027t want to find AD itself,"},{"Start":"02:05.870 ","End":"02:07.805","Text":"we want to find D square."},{"Start":"02:07.805 ","End":"02:12.925","Text":"Here, we have the formula for f of x and let\u0027s just simplify it."},{"Start":"02:12.925 ","End":"02:19.580","Text":"We get x squared plus 30 squared is 900 minus twice 30 times"},{"Start":"02:19.580 ","End":"02:27.630","Text":"2x is 120x plus 2x all squared is 4x squared,"},{"Start":"02:27.630 ","End":"02:31.595","Text":"which equals the 4 and the x squared and x squared combined,"},{"Start":"02:31.595 ","End":"02:35.060","Text":"that\u0027s 5x squared minus 120x,"},{"Start":"02:35.060 ","End":"02:39.265","Text":"I\u0027m changing the order, plus 900."},{"Start":"02:39.265 ","End":"02:43.280","Text":"Now, since we\u0027re going to do maximum or minimum,"},{"Start":"02:43.280 ","End":"02:46.340","Text":"we use the method of differentiation and setting to 0."},{"Start":"02:46.340 ","End":"02:51.440","Text":"We get that f prime of x is equal to 5 times 2"},{"Start":"02:51.440 ","End":"02:59.845","Text":"is 10x minus 120 this gives nothing."},{"Start":"02:59.845 ","End":"03:03.560","Text":"I can just take 10 outside the brackets"},{"Start":"03:03.560 ","End":"03:07.955","Text":"and say it\u0027s 10 times x minus 12."},{"Start":"03:07.955 ","End":"03:09.695","Text":"I forgot to say by the way,"},{"Start":"03:09.695 ","End":"03:11.495","Text":"that for the function,"},{"Start":"03:11.495 ","End":"03:13.565","Text":"we should check its domain."},{"Start":"03:13.565 ","End":"03:16.350","Text":"X has to be positive that\u0027s for sure,"},{"Start":"03:16.350 ","End":"03:20.510","Text":"2x could most be 30 but it has to be less than 30,"},{"Start":"03:20.510 ","End":"03:22.370","Text":"so x has to be less than 15."},{"Start":"03:22.370 ","End":"03:24.560","Text":"Just in case, we get anything outside this range,"},{"Start":"03:24.560 ","End":"03:26.770","Text":"we can discard it."},{"Start":"03:26.770 ","End":"03:29.480","Text":"F prime of x equals 0,"},{"Start":"03:29.480 ","End":"03:32.990","Text":"will give us suspects for the minimum."},{"Start":"03:32.990 ","End":"03:35.810","Text":"This gives us, but we can ignore the 10,"},{"Start":"03:35.810 ","End":"03:38.495","Text":"that x minus 12 is 0,"},{"Start":"03:38.495 ","End":"03:40.730","Text":"or that x equals 12."},{"Start":"03:40.730 ","End":"03:43.000","Text":"Notice that it is within the range."},{"Start":"03:43.000 ","End":"03:46.020","Text":"Let\u0027s find the length of the legs and then at the end,"},{"Start":"03:46.020 ","End":"03:48.030","Text":"we\u0027ll show that this really is a minimum."},{"Start":"03:48.030 ","End":"03:49.490","Text":"The length of the legs,"},{"Start":"03:49.490 ","End":"03:51.005","Text":"I\u0027ll go back to the picture."},{"Start":"03:51.005 ","End":"03:59.025","Text":"This gives us that BD would be 2x, which is 24,"},{"Start":"03:59.025 ","End":"04:04.130","Text":"and AB, which is 30 minus 2x,"},{"Start":"04:04.130 ","End":"04:09.015","Text":"is just 6 because together they\u0027re 30."},{"Start":"04:09.015 ","End":"04:13.715","Text":"What remains is to check that it is indeed a minimum."},{"Start":"04:13.715 ","End":"04:16.205","Text":"Let\u0027s do it with the table method."},{"Start":"04:16.205 ","End":"04:19.640","Text":"If I draw a table for the increasing,"},{"Start":"04:19.640 ","End":"04:24.415","Text":"decreasing stuff, then what I need is x,"},{"Start":"04:24.415 ","End":"04:26.765","Text":"I need f prime of x,"},{"Start":"04:26.765 ","End":"04:31.925","Text":"and I need f of x at the suspicious point,"},{"Start":"04:31.925 ","End":"04:34.670","Text":"and we only have 1 and that is 12."},{"Start":"04:34.670 ","End":"04:36.230","Text":"Where an x is 12,"},{"Start":"04:36.230 ","End":"04:38.465","Text":"the derivative is 0, we know that."},{"Start":"04:38.465 ","End":"04:41.030","Text":"What we want to know is what is before and after,"},{"Start":"04:41.030 ","End":"04:44.795","Text":"so let\u0027s just take the 11 and 13."},{"Start":"04:44.795 ","End":"04:46.895","Text":"Let\u0027s see what f prime of x is."},{"Start":"04:46.895 ","End":"04:49.130","Text":"We have to substitute in here."},{"Start":"04:49.130 ","End":"04:51.920","Text":"What we get is, if you put 11 in,"},{"Start":"04:51.920 ","End":"04:55.280","Text":"it\u0027s 10 times 11 minus 12, that\u0027s negative."},{"Start":"04:55.280 ","End":"04:56.945","Text":"I only need the sign."},{"Start":"04:56.945 ","End":"05:01.775","Text":"If I put 13 in here,13 minus 12 is positive times 10 is still positive,"},{"Start":"05:01.775 ","End":"05:06.590","Text":"which means that the function here is decreasing and here it\u0027s increasing and of course,"},{"Start":"05:06.590 ","End":"05:08.570","Text":"if this is decreasing and increasing,"},{"Start":"05:08.570 ","End":"05:09.995","Text":"this is a minimum."},{"Start":"05:09.995 ","End":"05:14.810","Text":"We\u0027ve shown that indeed we do have a minimum and the minimum"},{"Start":"05:14.810 ","End":"05:20.450","Text":"occurs when the length of the sides are 24 and 6."},{"Start":"05:20.450 ","End":"05:26.345","Text":"Where the BD here is 24, the long one is 24."},{"Start":"05:26.345 ","End":"05:30.470","Text":"The horizontal 1 and the vertical 1 is 6."},{"Start":"05:30.470 ","End":"05:33.210","Text":"Okay. We are done."}],"ID":8562},{"Watched":false,"Name":"Exercise 7","Duration":"5m 49s","ChapterTopicVideoID":8372,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8372.jpeg","UploadDate":"2017-01-28T12:52:14.7230000","DurationForVideoObject":"PT5M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"Let me explain what\u0027s going on here."},{"Start":"00:02.160 ","End":"00:07.950","Text":"There\u0027s an advertising leaflet or brochure and the area of the page is given."},{"Start":"00:07.950 ","End":"00:11.250","Text":"It has to be 600 square centimeters."},{"Start":"00:11.250 ","End":"00:14.910","Text":"There\u0027s also a stipulation that the margins on"},{"Start":"00:14.910 ","End":"00:17.970","Text":"either side of the printed area is 3 centimeters,"},{"Start":"00:17.970 ","End":"00:20.750","Text":"which means that this is also 3."},{"Start":"00:20.750 ","End":"00:23.090","Text":"Above and below it\u0027s 8 centimeters,"},{"Start":"00:23.090 ","End":"00:25.070","Text":"which means that this is also 8."},{"Start":"00:25.070 ","End":"00:28.370","Text":"It doesn\u0027t look like it, it\u0027s also 8."},{"Start":"00:28.370 ","End":"00:31.410","Text":"What we have to find is the length and width of"},{"Start":"00:31.410 ","End":"00:34.860","Text":"the page so that the printing area is maximized."},{"Start":"00:34.860 ","End":"00:36.765","Text":"That\u0027s the shaded bit."},{"Start":"00:36.765 ","End":"00:40.920","Text":"What I suggest is to let 1 of these dimensions,"},{"Start":"00:40.920 ","End":"00:42.830","Text":"be x or independent variable,"},{"Start":"00:42.830 ","End":"00:44.155","Text":"let\u0027s say it\u0027s this 1."},{"Start":"00:44.155 ","End":"00:49.175","Text":"Now, we don\u0027t need to give a name for this variable because we can mentally compute it."},{"Start":"00:49.175 ","End":"00:52.490","Text":"Since the area of this thing is 600,"},{"Start":"00:52.490 ","End":"00:54.260","Text":"this is x centimeters,"},{"Start":"00:54.260 ","End":"00:59.995","Text":"this is going to be 600 over x centimeters."},{"Start":"00:59.995 ","End":"01:05.090","Text":"What we have to do is, let y or f of x be the printing area,"},{"Start":"01:05.090 ","End":"01:07.625","Text":"the shaded area in the picture."},{"Start":"01:07.625 ","End":"01:11.140","Text":"Let\u0027s see if we can say what f of x is."},{"Start":"01:11.140 ","End":"01:15.380","Text":"It\u0027s also a rectangle and it has a width and a height."},{"Start":"01:15.380 ","End":"01:19.610","Text":"Now, the width of this rectangle is just going to be x,"},{"Start":"01:19.610 ","End":"01:22.295","Text":"do we subtract 3 and the 3."},{"Start":"01:22.295 ","End":"01:24.140","Text":"This is the width,"},{"Start":"01:24.140 ","End":"01:26.284","Text":"is x minus 6,"},{"Start":"01:26.284 ","End":"01:29.870","Text":"and the height is going to be 600 over x."},{"Start":"01:29.870 ","End":"01:33.120","Text":"But after we subtract the 8 and the 8,"},{"Start":"01:33.530 ","End":"01:35.980","Text":"altogether as a product here,"},{"Start":"01:35.980 ","End":"01:38.330","Text":"because it\u0027s this bit times this bit."},{"Start":"01:38.330 ","End":"01:44.430","Text":"This bit we already said and this bit is 600 over x minus 8,"},{"Start":"01:44.430 ","End":"01:47.310","Text":"and minus 8 is minus 16."},{"Start":"01:47.310 ","End":"01:50.390","Text":"This is what we have to maximize."},{"Start":"01:50.390 ","End":"01:57.380","Text":"Now, there are some limitations on x. X can\u0027t go any lower than 6,"},{"Start":"01:57.380 ","End":"02:03.265","Text":"for example, so x has to be certainly bigger than 6."},{"Start":"02:03.265 ","End":"02:09.900","Text":"Similarly, 600 over x has to be at least 16."},{"Start":"02:09.900 ","End":"02:16.450","Text":"I have 600 over x is bigger than 16 and we can switch the places of x and the 16,"},{"Start":"02:16.450 ","End":"02:20.150","Text":"so x has to be less than 600 over 16,"},{"Start":"02:20.150 ","End":"02:22.715","Text":"which is 37 and 1/2."},{"Start":"02:22.715 ","End":"02:29.644","Text":"We have a range of possible values for x in order for the problem to make sense."},{"Start":"02:29.644 ","End":"02:31.970","Text":"Let\u0027s multiply out."},{"Start":"02:31.970 ","End":"02:38.085","Text":"What we get is, x times 600 over x is 600,"},{"Start":"02:38.085 ","End":"02:42.240","Text":"x times minus 16 is minus 16x,"},{"Start":"02:42.240 ","End":"02:51.570","Text":"minus 6 times 600 over x is minus 3600 over x,"},{"Start":"02:51.570 ","End":"02:56.825","Text":"and minus 6 times minus 16 is plus 96."},{"Start":"02:56.825 ","End":"03:00.715","Text":"What we have is, 696"},{"Start":"03:00.715 ","End":"03:08.945","Text":"minus 16x minus 3600 over x."},{"Start":"03:08.945 ","End":"03:13.100","Text":"That\u0027s our function that we have to maximize."},{"Start":"03:13.100 ","End":"03:15.140","Text":"Let\u0027s take the derivative,"},{"Start":"03:15.140 ","End":"03:18.140","Text":"f prime of x is, this gives nothing,"},{"Start":"03:18.140 ","End":"03:20.735","Text":"this gives minus 16,"},{"Start":"03:20.735 ","End":"03:25.790","Text":"the derivative of 1 over x is minus 1 over x squared."},{"Start":"03:25.790 ","End":"03:31.090","Text":"We get plus 3600 over x squared."},{"Start":"03:31.090 ","End":"03:35.210","Text":"Now, in order to get a minimum or any kind of extremum,"},{"Start":"03:35.210 ","End":"03:38.705","Text":"we need f prime of x to equals 0."},{"Start":"03:38.705 ","End":"03:46.595","Text":"This gives us that 3600 over x squared is equal to 16."},{"Start":"03:46.595 ","End":"03:50.800","Text":"X squared, I switched the places is equal to"},{"Start":"03:50.800 ","End":"04:00.255","Text":"3600 over 16, which equals 225."},{"Start":"04:00.255 ","End":"04:05.720","Text":"This is exactly 15 squared and x is positive certainly,"},{"Start":"04:05.720 ","End":"04:09.470","Text":"so x is 15 and that\u0027s fine because it\u0027s in the range."},{"Start":"04:09.470 ","End":"04:11.930","Text":"But that\u0027s not what they asked us,"},{"Start":"04:11.930 ","End":"04:14.840","Text":"they asked us for the dimensions,"},{"Start":"04:14.840 ","End":"04:15.965","Text":"the length and the width."},{"Start":"04:15.965 ","End":"04:18.455","Text":"See here, find the length and the width."},{"Start":"04:18.455 ","End":"04:21.785","Text":"What we need is 600 over x also."},{"Start":"04:21.785 ","End":"04:28.685","Text":"So 600 over x is 600 over 15, which is 40."},{"Start":"04:28.685 ","End":"04:33.695","Text":"This dimension is 15 and this dimension is 40,"},{"Start":"04:33.695 ","End":"04:37.190","Text":"but we still haven\u0027t shown that this is a maximum."},{"Start":"04:37.190 ","End":"04:41.540","Text":"We\u0027ll use the table method to show that it\u0027s a maximum."},{"Start":"04:41.540 ","End":"04:43.115","Text":"What I\u0027m going to do is,"},{"Start":"04:43.115 ","End":"04:45.780","Text":"draw a little table here."},{"Start":"04:45.830 ","End":"04:49.190","Text":"This column will be x,"},{"Start":"04:49.190 ","End":"04:52.800","Text":"this column will be f prime of x."},{"Start":"04:53.140 ","End":"04:58.255","Text":"Suspicious points, there\u0027s only 1, that\u0027s 15."},{"Start":"04:58.255 ","End":"05:00.620","Text":"This is where f prime is 0."},{"Start":"05:00.620 ","End":"05:03.380","Text":"But let\u0027s see what it is a bit before and a bit after."},{"Start":"05:03.380 ","End":"05:06.635","Text":"I don\u0027t know, 10 and 20,"},{"Start":"05:06.635 ","End":"05:08.480","Text":"keep it nice and round."},{"Start":"05:08.480 ","End":"05:12.185","Text":"If x is 10, f prime is,"},{"Start":"05:12.185 ","End":"05:14.470","Text":"10 squared is a 100,"},{"Start":"05:14.470 ","End":"05:18.825","Text":"minus 16 plus 36 is positive,"},{"Start":"05:18.825 ","End":"05:20.775","Text":"I just need to know the sign."},{"Start":"05:20.775 ","End":"05:22.950","Text":"If I take x is 20,"},{"Start":"05:22.950 ","End":"05:25.395","Text":"x squared is 400,"},{"Start":"05:25.395 ","End":"05:29.515","Text":"3600 over 400 is 9,"},{"Start":"05:29.515 ","End":"05:33.030","Text":"minus 16 plus 9 is negative,"},{"Start":"05:33.030 ","End":"05:36.255","Text":"which means that the function here is increasing."},{"Start":"05:36.255 ","End":"05:39.710","Text":"Here it\u0027s decreasing and that means that this point here"},{"Start":"05:39.710 ","End":"05:43.830","Text":"is a maximum and that\u0027s what there remained to show."},{"Start":"05:43.830 ","End":"05:48.270","Text":"The answer is, as we said is 15 centimeters and centimeters,"},{"Start":"05:48.270 ","End":"05:50.320","Text":"and we are done."}],"ID":8563},{"Watched":false,"Name":"Exercise 8","Duration":"5m 31s","ChapterTopicVideoID":8373,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8373.jpeg","UploadDate":"2017-01-28T12:52:43.7270000","DurationForVideoObject":"PT5M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"Here we have another word problem and instead of just reading it,"},{"Start":"00:03.480 ","End":"00:05.775","Text":"I\u0027ll explain what is meant."},{"Start":"00:05.775 ","End":"00:09.000","Text":"We\u0027re given a square ABCD."},{"Start":"00:09.000 ","End":"00:14.085","Text":"A square means that all the sides are equal and all the angles are 90 degrees."},{"Start":"00:14.085 ","End":"00:20.194","Text":"We\u0027re given some extra points, E, F, and G on the sides"},{"Start":"00:20.194 ","End":"00:27.240","Text":"in such a way that this side is equal to this side or this segment"},{"Start":"00:27.240 ","End":"00:30.390","Text":"and this bit is equal to this bit,"},{"Start":"00:30.390 ","End":"00:33.780","Text":"EB equals BF and FC equals CG."},{"Start":"00:33.780 ","End":"00:36.995","Text":"We\u0027re also given that the side of the square,"},{"Start":"00:36.995 ","End":"00:43.440","Text":"say from here to here, is 6 centimeters."},{"Start":"00:43.440 ","End":"00:45.043","Text":"They\u0027re all 6 centimeters."},{"Start":"00:45.043 ","End":"00:47.535","Text":"6, 6, 6 because it\u0027s a square."},{"Start":"00:47.535 ","End":"00:51.680","Text":"In the first part, we have to express in terms of x."},{"Start":"00:51.680 ","End":"00:57.890","Text":"This part is x, so this we\u0027ll call it x and this is also x, they\u0027re equal also."},{"Start":"00:57.890 ","End":"00:59.195","Text":"If one is x, the other is x."},{"Start":"00:59.195 ","End":"01:03.157","Text":"We have to express in terms of x the shaded area."},{"Start":"01:03.157 ","End":"01:06.568","Text":"In other words, the sum of this triangle plus this triangle,"},{"Start":"01:06.568 ","End":"01:09.810","Text":"and, of course, these triangles are 90 degrees,"},{"Start":"01:09.810 ","End":"01:12.740","Text":"just to emphasize that because the square is"},{"Start":"01:12.740 ","End":"01:16.625","Text":"90 degrees everywhere, so here we have 90 degrees."},{"Start":"01:16.625 ","End":"01:22.130","Text":"Let\u0027s do part a first and compute the sum of the areas."},{"Start":"01:22.130 ","End":"01:25.980","Text":"We\u0027ll take the area of this one and the area of this one, and add it."},{"Start":"01:25.990 ","End":"01:29.807","Text":"Let\u0027s call that f of x."},{"Start":"01:29.807 ","End":"01:35.090","Text":"f of x is the area of this triangle first,"},{"Start":"01:35.090 ","End":"01:40.115","Text":"and the area of the triangle is base times height over 2,"},{"Start":"01:40.115 ","End":"01:46.200","Text":"so it\u0027s x times x over 2 plus."},{"Start":"01:46.200 ","End":"01:48.770","Text":"In this triangle, we also know the side"},{"Start":"01:48.770 ","End":"01:52.085","Text":"because if this whole thing is 6 and this is x,"},{"Start":"01:52.085 ","End":"01:56.690","Text":"then this is just 6 minus x and, likewise, this is 6 minus x."},{"Start":"01:56.690 ","End":"01:58.760","Text":"The same thing with this triangle."},{"Start":"01:58.760 ","End":"02:07.820","Text":"Base times height over 2, and that actually answers part a."},{"Start":"02:07.820 ","End":"02:12.240","Text":"It\u0027s in centimeters squared, of course, because it\u0027s area."},{"Start":"02:12.490 ","End":"02:16.530","Text":"Let\u0027s go to part b1."},{"Start":"02:17.360 ","End":"02:26.160","Text":"What we have to find out is for which x is this f of x minimal."},{"Start":"02:26.160 ","End":"02:33.700","Text":"Let\u0027s simplify first and then we\u0027ll do our usual differentiate and set to 0 and so on."},{"Start":"02:33.700 ","End":"02:37.525","Text":"Let\u0027s just write it again."},{"Start":"02:37.525 ","End":"02:40.870","Text":"f of x is equal to,"},{"Start":"02:40.870 ","End":"02:44.285","Text":"let\u0027s put it all over 2."},{"Start":"02:44.285 ","End":"02:46.635","Text":"It seems to naturally go that way."},{"Start":"02:46.635 ","End":"02:48.415","Text":"Here we have x squared,"},{"Start":"02:48.415 ","End":"02:50.977","Text":"and here we have 6 minus x all squared"},{"Start":"02:50.977 ","End":"02:57.565","Text":"so 6 squared is 36 minus twice 6 times x plus x squared."},{"Start":"02:57.565 ","End":"03:03.270","Text":"We get 2x^2 and I\u0027ll divide by 2 as I go along,"},{"Start":"03:03.270 ","End":"03:05.760","Text":"so 2x^2 over 2 is x^2."},{"Start":"03:05.760 ","End":"03:09.839","Text":"Then minus 12x, so minus 6x."},{"Start":"03:09.839 ","End":"03:13.515","Text":"Then plus 36, so plus 18."},{"Start":"03:13.515 ","End":"03:16.940","Text":"This is what f of x equals."},{"Start":"03:16.940 ","End":"03:20.840","Text":"That\u0027s the area of the shaded in terms of x."},{"Start":"03:20.840 ","End":"03:23.075","Text":"Now, let\u0027s differentiate."},{"Start":"03:23.075 ","End":"03:27.890","Text":"f prime of x is 2x minus 6."},{"Start":"03:27.890 ","End":"03:30.320","Text":"To find a possible minimum,"},{"Start":"03:30.320 ","End":"03:34.025","Text":"we should set f prime to 0,"},{"Start":"03:34.025 ","End":"03:36.935","Text":"which gives us, we can do it in our heads."},{"Start":"03:36.935 ","End":"03:38.570","Text":"2x minus 6 is 0,"},{"Start":"03:38.570 ","End":"03:40.188","Text":"so x is 3,"},{"Start":"03:40.188 ","End":"03:43.888","Text":"and if x is 3, that would answer b1"},{"Start":"03:43.888 ","End":"03:47.495","Text":"except that we haven\u0027t really shown that it\u0027s minimal."},{"Start":"03:47.495 ","End":"03:50.395","Text":"Let\u0027s do it with the table method."},{"Start":"03:50.395 ","End":"03:55.650","Text":"I\u0027ll make a table which has 3 rows."},{"Start":"03:55.650 ","End":"04:01.812","Text":"We have x, we have f prime of x, and we have f of x,"},{"Start":"04:01.812 ","End":"04:03.380","Text":"and here we put all the suspects."},{"Start":"04:03.380 ","End":"04:06.455","Text":"We only have one suspect, that\u0027s x equals 3."},{"Start":"04:06.455 ","End":"04:09.380","Text":"At this point, the derivative is 0."},{"Start":"04:09.380 ","End":"04:13.775","Text":"We\u0027d like to know what it is a bit before and a bit after or whatever."},{"Start":"04:13.775 ","End":"04:18.028","Text":"Let\u0027s try 2 and 4 and put that into f prime,"},{"Start":"04:18.028 ","End":"04:20.195","Text":"so f prime is this."},{"Start":"04:20.195 ","End":"04:23.839","Text":"If I put in 2, I get twice 2 minus 6, it\u0027s negative."},{"Start":"04:23.839 ","End":"04:25.173","Text":"All I care about is the sign,"},{"Start":"04:25.173 ","End":"04:28.340","Text":"and if I put 4 to 8 minus 6, it\u0027s positive."},{"Start":"04:28.340 ","End":"04:31.820","Text":"Negative means that the function is decreasing."},{"Start":"04:31.820 ","End":"04:33.965","Text":"Positive means it\u0027s increasing."},{"Start":"04:33.965 ","End":"04:38.509","Text":"So decreasing to the point and then increasing after makes it a minimum,"},{"Start":"04:38.509 ","End":"04:40.410","Text":"so we\u0027re all right."},{"Start":"04:40.410 ","End":"04:46.480","Text":"The answer to part b1 is that x equals 3."},{"Start":"04:46.550 ","End":"04:54.680","Text":"We still have question b2 and we know where the minimum is obtained."},{"Start":"04:54.680 ","End":"04:58.400","Text":"We know that the minimum is obtained when x equals 3."},{"Start":"04:58.400 ","End":"04:59.810","Text":"But what is that minimum?"},{"Start":"04:59.810 ","End":"05:05.726","Text":"All we have to do is find out what is f of 3 equal to"},{"Start":"05:05.726 ","End":"05:08.790","Text":"and that will give us the minimum value."},{"Start":"05:09.080 ","End":"05:17.100","Text":"f of 3 is equal to 3^2 minus 6 times 3 plus 18."},{"Start":"05:17.100 ","End":"05:21.165","Text":"While 6 times 3 is 18, so all I\u0027m left with is 3 squared,"},{"Start":"05:21.165 ","End":"05:28.575","Text":"which equals 9 and the units are centimeters squared and here it\u0027s centimeters."},{"Start":"05:28.575 ","End":"05:32.190","Text":"I think we are done here."}],"ID":8564},{"Watched":false,"Name":"Exercise 9","Duration":"10m 20s","ChapterTopicVideoID":8374,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8374.jpeg","UploadDate":"2017-01-28T12:53:39.6970000","DurationForVideoObject":"PT10M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.000","Text":"Here we have another word problem in geometry."},{"Start":"00:03.000 ","End":"00:05.700","Text":"I\u0027ll just explain instead of reading it out."},{"Start":"00:05.700 ","End":"00:08.550","Text":"Basically what we have here is a square A, B, C, D,"},{"Start":"00:08.550 ","End":"00:13.005","Text":"which means that all 4 sides are equal and all the angles are 90 degrees."},{"Start":"00:13.005 ","End":"00:16.800","Text":"What more is we have a point E outside"},{"Start":"00:16.800 ","End":"00:21.135","Text":"the square in such a way that if we connect it to D and to C,"},{"Start":"00:21.135 ","End":"00:23.700","Text":"then this is an isosceles triangle."},{"Start":"00:23.700 ","End":"00:26.985","Text":"What that means is that this side,"},{"Start":"00:26.985 ","End":"00:30.690","Text":"these are the legs of the isosceles triangle and this are equal in"},{"Start":"00:30.690 ","End":"00:35.430","Text":"length and where they cut is M and N. Now,"},{"Start":"00:35.430 ","End":"00:39.510","Text":"what we\u0027re interested in is what should the length of AM be,"},{"Start":"00:39.510 ","End":"00:42.730","Text":"and let\u0027s call this one x."},{"Start":"00:44.390 ","End":"00:46.770","Text":"Now, the sum of the areas,"},{"Start":"00:46.770 ","End":"00:48.240","Text":"I\u0027ll just shade it."},{"Start":"00:48.240 ","End":"00:53.590","Text":"The sum of all these 3 triangles has got to be minimized."},{"Start":"00:53.590 ","End":"00:58.550","Text":"That\u0027s an indication this is going to be our y or a function of x."},{"Start":"00:58.550 ","End":"01:00.560","Text":"Now, before we get started,"},{"Start":"01:00.560 ","End":"01:03.590","Text":"I just want to say from geometry we can tell."},{"Start":"01:03.590 ","End":"01:07.610","Text":"From geometry, we can conclude that this is also x."},{"Start":"01:07.610 ","End":"01:09.500","Text":"You can see it by symmetry."},{"Start":"01:09.500 ","End":"01:13.535","Text":"But if you really want a more precise reason, well,"},{"Start":"01:13.535 ","End":"01:18.545","Text":"these 2 green triangles are congruent because we have 2 angles."},{"Start":"01:18.545 ","End":"01:21.740","Text":"We have 90 degrees and 90 degrees."},{"Start":"01:21.740 ","End":"01:23.545","Text":"We have this side,"},{"Start":"01:23.545 ","End":"01:25.305","Text":"and that\u0027s what I forgot to say,"},{"Start":"01:25.305 ","End":"01:27.300","Text":"that the side is 10 centimeters."},{"Start":"01:27.300 ","End":"01:30.800","Text":"This is 10 and this is 10 and they\u0027re all 10."},{"Start":"01:30.800 ","End":"01:36.350","Text":"The reason that this angle is also equal to this angle is"},{"Start":"01:36.350 ","End":"01:43.175","Text":"because these angles are equal because they\u0027re the base angles of an isosceles triangle."},{"Start":"01:43.175 ","End":"01:44.450","Text":"Let\u0027s say this is alpha,"},{"Start":"01:44.450 ","End":"01:46.820","Text":"and this is alpha, this alternate with this,"},{"Start":"01:46.820 ","End":"01:48.170","Text":"this alternates with this,"},{"Start":"01:48.170 ","End":"01:50.570","Text":"2 angles in the side, the congruent."},{"Start":"01:50.570 ","End":"01:52.340","Text":"This is also x."},{"Start":"01:52.340 ","End":"01:54.325","Text":"Enough with the geometry."},{"Start":"01:54.325 ","End":"01:58.790","Text":"Now, we want to get an expression for the sum of the 3 triangles,"},{"Start":"01:58.790 ","End":"02:00.515","Text":"which I\u0027ll call f of x,"},{"Start":"02:00.515 ","End":"02:02.975","Text":"where x is our independent variable."},{"Start":"02:02.975 ","End":"02:06.995","Text":"Let\u0027s see. Let\u0027s do this triangle first."},{"Start":"02:06.995 ","End":"02:10.970","Text":"It\u0027s 90 degrees, so it\u0027s just base times height."},{"Start":"02:10.970 ","End":"02:15.455","Text":"It\u0027s x times 10 over 2. x times 10 over 2."},{"Start":"02:15.455 ","End":"02:17.405","Text":"Likewise, this one is congruent,"},{"Start":"02:17.405 ","End":"02:21.150","Text":"so this is also x times 10 over 2."},{"Start":"02:21.150 ","End":"02:22.800","Text":"Now, the hard part,"},{"Start":"02:22.800 ","End":"02:25.745","Text":"what about this triangle, the one at the top?"},{"Start":"02:25.745 ","End":"02:28.055","Text":"How do I find the area of this?"},{"Start":"02:28.055 ","End":"02:30.950","Text":"Well, I need half base times height."},{"Start":"02:30.950 ","End":"02:34.770","Text":"The base is not so bad because this whole thing is 10,"},{"Start":"02:34.770 ","End":"02:36.785","Text":"and here I subtract x and x."},{"Start":"02:36.785 ","End":"02:40.250","Text":"This bit is 10 minus 2x."},{"Start":"02:40.250 ","End":"02:42.290","Text":"I\u0027ve got the base. The question is,"},{"Start":"02:42.290 ","End":"02:43.595","Text":"what about the height?"},{"Start":"02:43.595 ","End":"02:46.010","Text":"Let me draw that in a different color."},{"Start":"02:46.010 ","End":"02:50.330","Text":"What I need to know is the height of this triangle,"},{"Start":"02:50.330 ","End":"02:57.495","Text":"and we\u0027ll call that h. The way I\u0027m going to find h is using similar triangles."},{"Start":"02:57.495 ","End":"03:00.725","Text":"Now, this bit here after the h is 10."},{"Start":"03:00.725 ","End":"03:04.775","Text":"What I claim is that this big isosceles triangle"},{"Start":"03:04.775 ","End":"03:08.134","Text":"is similar to this small isosceles triangle."},{"Start":"03:08.134 ","End":"03:10.490","Text":"Well, I need for similarity of 2 triangles."},{"Start":"03:10.490 ","End":"03:11.600","Text":"I\u0027ve got alpha and alpha,"},{"Start":"03:11.600 ","End":"03:14.150","Text":"and here I\u0027ve got alpha and alpha, so they\u0027re similar."},{"Start":"03:14.150 ","End":"03:15.650","Text":"If there are similar,"},{"Start":"03:15.650 ","End":"03:18.530","Text":"then I can compare corresponding sides."},{"Start":"03:18.530 ","End":"03:22.295","Text":"I can compare the heights and the bases to be in the same ratio."},{"Start":"03:22.295 ","End":"03:25.595","Text":"What I\u0027m saying is that in this triangle,"},{"Start":"03:25.595 ","End":"03:29.089","Text":"the ratio of h,"},{"Start":"03:29.089 ","End":"03:32.090","Text":"the height of the small triangle to the height of the big triangle,"},{"Start":"03:32.090 ","End":"03:33.935","Text":"which is h plus 10,"},{"Start":"03:33.935 ","End":"03:38.840","Text":"is going to be in the same ratio as the base of this to the base of this."},{"Start":"03:38.840 ","End":"03:45.720","Text":"It\u0027s the ratio of 10 minus 2x to 10."},{"Start":"03:45.720 ","End":"03:50.065","Text":"What this means is that this over this equals this over this."},{"Start":"03:50.065 ","End":"03:53.895","Text":"What you do is you just multiply the outsides and the inside."},{"Start":"03:53.895 ","End":"04:00.555","Text":"h times 10 or 10 times h is equal to h plus 10"},{"Start":"04:00.555 ","End":"04:08.280","Text":"times 10 minus 2x.10h is equal."},{"Start":"04:08.280 ","End":"04:10.950","Text":"Now, to multiply out h times both of these,"},{"Start":"04:10.950 ","End":"04:15.355","Text":"I\u0027ve got 10h minus 2hx,"},{"Start":"04:15.355 ","End":"04:18.200","Text":"and then 10 times both of these,"},{"Start":"04:18.200 ","End":"04:22.215","Text":"so plus 100 minus 20x."},{"Start":"04:22.215 ","End":"04:27.425","Text":"I see that 10h and 10h cancel."},{"Start":"04:27.425 ","End":"04:30.995","Text":"Well, let\u0027s bring this to the other side."},{"Start":"04:30.995 ","End":"04:36.020","Text":"I\u0027ve got 2xh equals 100 minus 20x."},{"Start":"04:36.020 ","End":"04:38.285","Text":"I\u0027m trying to isolate h. Finally,"},{"Start":"04:38.285 ","End":"04:45.420","Text":"h equals 100 minus 20x over 2x."},{"Start":"04:45.520 ","End":"04:52.175","Text":"If I divide it out,100 over 2x is 50 over x,"},{"Start":"04:52.175 ","End":"04:57.190","Text":"and 20x over 2x is just 10."},{"Start":"04:57.190 ","End":"05:02.115","Text":"Now, I have a formula for h in terms of x."},{"Start":"05:02.115 ","End":"05:05.060","Text":"Now, I\u0027m going to add the missing piece."},{"Start":"05:05.060 ","End":"05:09.395","Text":"The missing piece here is this triangle,"},{"Start":"05:09.395 ","End":"05:13.565","Text":"and I need half the base times the height, so it\u0027s half."},{"Start":"05:13.565 ","End":"05:18.450","Text":"The base is 10 minus 2x times"},{"Start":"05:18.450 ","End":"05:23.100","Text":"the height h. But I don\u0027t want the height h. I don\u0027t want to copy this line."},{"Start":"05:23.100 ","End":"05:26.180","Text":"Again, I\u0027m just going to put h for what it is equal to here."},{"Start":"05:26.180 ","End":"05:30.260","Text":"Let\u0027s erase that h. Instead,"},{"Start":"05:30.260 ","End":"05:37.085","Text":"I\u0027m going to put 50 over x minus 10."},{"Start":"05:37.085 ","End":"05:39.560","Text":"This is our f of x."},{"Start":"05:39.560 ","End":"05:44.060","Text":"This is the function that\u0027s equal to the area of the green shaded bits."},{"Start":"05:44.060 ","End":"05:48.905","Text":"I\u0027m going to simplify this before we go any further because this looks a mess."},{"Start":"05:48.905 ","End":"05:50.990","Text":"Let\u0027s see what we can get here."},{"Start":"05:50.990 ","End":"05:54.080","Text":"Well, this plus this is just going to be 10x."},{"Start":"05:54.080 ","End":"05:55.730","Text":"It\u0027s over 2 and over 2,"},{"Start":"05:55.730 ","End":"05:59.100","Text":"so it\u0027s just 10x for this bit."},{"Start":"05:59.100 ","End":"06:01.140","Text":"This half can go."},{"Start":"06:01.140 ","End":"06:03.479","Text":"Let\u0027s see, if I get rid of this half,"},{"Start":"06:03.479 ","End":"06:07.110","Text":"I can make this 5 and this just x,"},{"Start":"06:07.110 ","End":"06:09.315","Text":"so it\u0027s 5 minus x."},{"Start":"06:09.315 ","End":"06:14.100","Text":"Let\u0027s multiply out what\u0027s here."},{"Start":"06:14.100 ","End":"06:16.920","Text":"5 minus x times this."},{"Start":"06:16.920 ","End":"06:24.795","Text":"The 5 with these 2 goes as 250 over x minus 50,"},{"Start":"06:24.795 ","End":"06:29.655","Text":"and then the minus x gives me minus 50."},{"Start":"06:29.655 ","End":"06:31.910","Text":"Then minus with minus is plus,"},{"Start":"06:31.910 ","End":"06:34.865","Text":"so it\u0027s plus 10x."},{"Start":"06:34.865 ","End":"06:37.100","Text":"Let\u0027s see what we can collect here."},{"Start":"06:37.100 ","End":"06:41.820","Text":"10x and 10x is 20x."},{"Start":"06:41.820 ","End":"06:46.215","Text":"Minus 50 and minus 50 is minus 100."},{"Start":"06:46.215 ","End":"06:50.595","Text":"Plus 250 over x,"},{"Start":"06:50.595 ","End":"06:53.670","Text":"so this is what f of x equals."},{"Start":"06:53.670 ","End":"06:56.745","Text":"I\u0027ll just highlight that."},{"Start":"06:56.745 ","End":"07:02.410","Text":"That gives the sum of the 3 areas in terms of x."},{"Start":"07:02.570 ","End":"07:06.615","Text":"Okay. This is what we want to minimize."},{"Start":"07:06.615 ","End":"07:16.475","Text":"This is what we\u0027re going to differentiate and then set to 0. f prime of x is equal to 20."},{"Start":"07:16.475 ","End":"07:21.590","Text":"This gives nothing, and the derivative of 1 over x is minus 1 over x squared,"},{"Start":"07:21.590 ","End":"07:25.025","Text":"so we get minus 250 over x squared."},{"Start":"07:25.025 ","End":"07:26.479","Text":"That\u0027s the derivative."},{"Start":"07:26.479 ","End":"07:29.295","Text":"Now, we want to look for suspects for extra mass."},{"Start":"07:29.295 ","End":"07:32.665","Text":"We set f prime of x equals to 0."},{"Start":"07:32.665 ","End":"07:36.750","Text":"That gives us that this minus this equals 0,"},{"Start":"07:36.750 ","End":"07:41.625","Text":"20 minus 250 over x squared equals 0."},{"Start":"07:41.625 ","End":"07:46.590","Text":"250 over x squared is equal to 20."},{"Start":"07:46.590 ","End":"07:55.965","Text":"Then we can switch between these 2 and say that x squared is 250 over 20."},{"Start":"07:55.965 ","End":"08:06.880","Text":"This comes out to be 12-and-a-half I guess, 12.5."},{"Start":"08:06.890 ","End":"08:09.525","Text":"X has to be positive,"},{"Start":"08:09.525 ","End":"08:13.705","Text":"so x is the plus square root of 12.5,"},{"Start":"08:13.705 ","End":"08:17.285","Text":"and that\u0027s approximately equal to 3.5."},{"Start":"08:17.285 ","End":"08:22.165","Text":"But we still haven\u0027t shown that is truly a minimum."},{"Start":"08:22.165 ","End":"08:24.260","Text":"What we need to do is,"},{"Start":"08:24.260 ","End":"08:27.845","Text":"well the several possibilities not going to use the table method."},{"Start":"08:27.845 ","End":"08:33.320","Text":"The table means that I make a table as follows."},{"Start":"08:33.320 ","End":"08:35.520","Text":"I have 3 rows in it."},{"Start":"08:35.520 ","End":"08:40.110","Text":"I have x, I have f prime of x,"},{"Start":"08:40.110 ","End":"08:46.240","Text":"and I have f of x. I put in all the suspicious or candidates for extra math."},{"Start":"08:46.240 ","End":"08:50.165","Text":"I only have 1, and that\u0027s the square root of 12.5."},{"Start":"08:50.165 ","End":"08:55.330","Text":"That\u0027s, as I said, is roughly 3.5."},{"Start":"08:56.030 ","End":"09:00.380","Text":"At this point is where the f prime is 0."},{"Start":"09:00.380 ","End":"09:02.060","Text":"That\u0027s how we got to it."},{"Start":"09:02.060 ","End":"09:06.450","Text":"What I need to know is if it\u0027s increasing or decreasing before or after."},{"Start":"09:06.450 ","End":"09:10.040","Text":"Let\u0027s take 0.3 and 0.4,"},{"Start":"09:10.040 ","End":"09:11.315","Text":"which are around it."},{"Start":"09:11.315 ","End":"09:13.010","Text":"That\u0027s why I need to know approximately."},{"Start":"09:13.010 ","End":"09:15.455","Text":"I need to know something smaller, something bigger."},{"Start":"09:15.455 ","End":"09:19.865","Text":"If I put in x equals 3 in f prime,"},{"Start":"09:19.865 ","End":"09:27.660","Text":"so I get 20 minus 250 over 9."},{"Start":"09:31.580 ","End":"09:36.405","Text":"Well, it\u0027s more than 250 over 10."},{"Start":"09:36.405 ","End":"09:39.500","Text":"It\u0027s more than 25. This comes out negative."},{"Start":"09:39.500 ","End":"09:42.455","Text":"Basically, what I\u0027m saying is this comes out negative."},{"Start":"09:42.455 ","End":"09:46.865","Text":"If I put in x equals 4, then x squared is 16."},{"Start":"09:46.865 ","End":"09:53.840","Text":"250 over 16, well it\u0027s less than 16 because 16 squared is 256,"},{"Start":"09:53.840 ","End":"09:57.785","Text":"and that means that this is positive or you can do it with the calculator also."},{"Start":"09:57.785 ","End":"09:59.490","Text":"Anyway, it\u0027s negative here,"},{"Start":"09:59.490 ","End":"10:01.400","Text":"so the function is decreasing,"},{"Start":"10:01.400 ","End":"10:03.775","Text":"positive here the function is increasing."},{"Start":"10:03.775 ","End":"10:05.990","Text":"Here at the point itself,"},{"Start":"10:05.990 ","End":"10:07.880","Text":"between decreasing and increasing,"},{"Start":"10:07.880 ","End":"10:10.115","Text":"it must be a minimum."},{"Start":"10:10.115 ","End":"10:14.600","Text":"What we have shown is that indeed x,"},{"Start":"10:14.600 ","End":"10:16.415","Text":"which is what we had to find,"},{"Start":"10:16.415 ","End":"10:18.905","Text":"is square root of 12.5,"},{"Start":"10:18.905 ","End":"10:21.330","Text":"and we are done."}],"ID":8565},{"Watched":false,"Name":"Exercise 10","Duration":"13m 54s","ChapterTopicVideoID":8375,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8375.jpeg","UploadDate":"2017-03-07T06:28:25.7030000","DurationForVideoObject":"PT13M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.770","Text":"In this exercise we have a word problem."},{"Start":"00:02.770 ","End":"00:05.520","Text":"The first correction, I apologize,"},{"Start":"00:05.520 ","End":"00:11.795","Text":"I wrote equilateral and I meant isosceles and no harm done."},{"Start":"00:11.795 ","End":"00:15.050","Text":"We have a circle with radius R,"},{"Start":"00:15.050 ","End":"00:17.195","Text":"and that means that,"},{"Start":"00:17.195 ","End":"00:19.680","Text":"say this is R from here to here,"},{"Start":"00:19.680 ","End":"00:27.075","Text":"and this would be R. The isosceles means that this side would equal this side."},{"Start":"00:27.075 ","End":"00:30.120","Text":"The long base is a diameter,"},{"Start":"00:30.120 ","End":"00:32.650","Text":"it goes to the center."},{"Start":"00:32.650 ","End":"00:36.510","Text":"We also have it inscribed in the circle,"},{"Start":"00:36.510 ","End":"00:39.870","Text":"that means that all the points lie on the circle."},{"Start":"00:39.870 ","End":"00:49.215","Text":"What we want to do is we want to express the area of the trapezoid."},{"Start":"00:49.215 ","End":"00:52.870","Text":"The area, let\u0027s see, I\u0027ll shade it."},{"Start":"00:55.030 ","End":"00:58.040","Text":"This is what we\u0027re talking about."},{"Start":"00:58.040 ","End":"01:03.980","Text":"We want to write this in terms of the parameter R and in terms of the short base."},{"Start":"01:03.980 ","End":"01:07.474","Text":"Let\u0027s call the short base x,"},{"Start":"01:07.474 ","End":"01:11.165","Text":"and that will be our independent variable."},{"Start":"01:11.165 ","End":"01:15.400","Text":"This area will be f of x."},{"Start":"01:15.400 ","End":"01:21.525","Text":"We\u0027ll express it as I say in terms of x and in terms of R. Then we want to look"},{"Start":"01:21.525 ","End":"01:27.880","Text":"for the trapezoid as such with maximal area."},{"Start":"01:28.490 ","End":"01:33.830","Text":"First of all, we have to get the formula and then we can start with differentiating,"},{"Start":"01:33.830 ","End":"01:37.100","Text":"setting the derivative equal to 0 and so on."},{"Start":"01:37.100 ","End":"01:42.215","Text":"Let\u0027s see how we can find the area."},{"Start":"01:42.215 ","End":"01:48.695","Text":"We need a bit of construction because we need to figure out the height."},{"Start":"01:48.695 ","End":"01:51.980","Text":"What I mean is that the f of x,"},{"Start":"01:51.980 ","End":"01:53.824","Text":"I can say straight away,"},{"Start":"01:53.824 ","End":"01:59.160","Text":"is by the formula for the area of a trapezoid is sum of the bases."},{"Start":"01:59.160 ","End":"02:01.725","Text":"This base is 2R,"},{"Start":"02:01.725 ","End":"02:07.660","Text":"this base is x times the height over 2."},{"Start":"02:07.660 ","End":"02:10.275","Text":"But the thing is we don\u0027t have the height."},{"Start":"02:10.275 ","End":"02:14.955","Text":"The height, let\u0027s put it in a nice color say red."},{"Start":"02:14.955 ","End":"02:16.635","Text":"This would be the height,"},{"Start":"02:16.635 ","End":"02:18.650","Text":"this is what we\u0027ve got to figure out."},{"Start":"02:18.650 ","End":"02:21.470","Text":"Let\u0027s join this point."},{"Start":"02:21.470 ","End":"02:28.325","Text":"Let me just also draw the height on this side because you see what we have now"},{"Start":"02:28.325 ","End":"02:35.920","Text":"is that from here to here is also x."},{"Start":"02:35.920 ","End":"02:38.690","Text":"This bit in here is x over 2."},{"Start":"02:38.690 ","End":"02:40.370","Text":"You can see it\u0027s symmetrical,"},{"Start":"02:40.370 ","End":"02:43.070","Text":"I\u0027m not going to go into most of the proofs."},{"Start":"02:43.070 ","End":"02:46.345","Text":"This is another radius."},{"Start":"02:46.345 ","End":"02:48.620","Text":"If I just focus on that bit here,"},{"Start":"02:48.620 ","End":"02:51.830","Text":"let me just draw it separately at the side."},{"Start":"02:51.830 ","End":"02:55.910","Text":"That triangle here, we have R,"},{"Start":"02:55.910 ","End":"03:00.935","Text":"we have 90 degrees and we have x over 2,"},{"Start":"03:00.935 ","End":"03:03.455","Text":"and this is the height."},{"Start":"03:03.455 ","End":"03:07.310","Text":"We can figure it out from Pythagoras."},{"Start":"03:07.310 ","End":"03:13.350","Text":"This is equal to 2R plus x,"},{"Start":"03:13.350 ","End":"03:18.690","Text":"and h is just the square root of R squared minus x over"},{"Start":"03:18.690 ","End":"03:24.915","Text":"2 squared over 2."},{"Start":"03:24.915 ","End":"03:31.110","Text":"I\u0027m going to rewrite this with the 1/2 here, it\u0027s more convenient."},{"Start":"03:31.110 ","End":"03:33.710","Text":"Anyway we have a square root here."},{"Start":"03:33.710 ","End":"03:40.325","Text":"So we\u0027re going to use the standard trick that if you have a positive function,"},{"Start":"03:40.325 ","End":"03:42.620","Text":"if it\u0027s all under the square root and you\u0027re okay,"},{"Start":"03:42.620 ","End":"03:44.945","Text":"but here some is under the square root and some is not,"},{"Start":"03:44.945 ","End":"03:50.090","Text":"but this part is positive because we\u0027re only taking positive x."},{"Start":"03:50.090 ","End":"03:52.265","Text":"When you have a positive function,"},{"Start":"03:52.265 ","End":"03:57.375","Text":"you can get the minimum or maximum when you square it."},{"Start":"03:57.375 ","End":"04:01.050","Text":"Let\u0027s say that I don\u0027t know, I\u0027ll call it y,"},{"Start":"04:01.050 ","End":"04:03.570","Text":"but note that is equal f of x,"},{"Start":"04:03.570 ","End":"04:05.910","Text":"but f of x squared."},{"Start":"04:05.910 ","End":"04:14.190","Text":"Then we\u0027ll get that y equals 1/4 times 2R plus x squared."},{"Start":"04:14.190 ","End":"04:17.870","Text":"Then here I get rid of the square root and just have R"},{"Start":"04:17.870 ","End":"04:22.180","Text":"squared minus let say x squared over 4."},{"Start":"04:22.180 ","End":"04:26.800","Text":"Now we want minimum or maximum of y."},{"Start":"04:26.840 ","End":"04:31.200","Text":"The same x will be the minimum or maximum for f,"},{"Start":"04:31.200 ","End":"04:34.100","Text":"because the value of y will be the value of f squared,"},{"Start":"04:34.100 ","End":"04:36.155","Text":"but the x will be the same."},{"Start":"04:36.155 ","End":"04:39.320","Text":"Let\u0027s use the product rule on this,"},{"Start":"04:39.320 ","End":"04:43.355","Text":"where I\u0027ll take this path,"},{"Start":"04:43.355 ","End":"04:48.720","Text":"let\u0027s say to be u,"},{"Start":"04:48.720 ","End":"04:58.275","Text":"and I\u0027ll let this part be v. Remember that uv prime is u prime v plus uv prime,"},{"Start":"04:58.275 ","End":"05:03.635","Text":"but I also have something squared and we\u0027ll use the chain rule for that."},{"Start":"05:03.635 ","End":"05:08.690","Text":"Let\u0027s do it, y prime is equal to,"},{"Start":"05:08.690 ","End":"05:12.095","Text":"now differentiate this, the 1/4 just stays."},{"Start":"05:12.095 ","End":"05:14.465","Text":"Here I have something squared."},{"Start":"05:14.465 ","End":"05:17.075","Text":"I have twice that something,"},{"Start":"05:17.075 ","End":"05:20.790","Text":"which is 2R plus x,"},{"Start":"05:21.380 ","End":"05:24.300","Text":"and then times the inner derivative."},{"Start":"05:24.300 ","End":"05:26.850","Text":"I\u0027m going to write times 1 to show you that I haven\u0027t forgotten,"},{"Start":"05:26.850 ","End":"05:28.920","Text":"but the inner derivative of 2R plus x,"},{"Start":"05:28.920 ","End":"05:32.700","Text":"just 1. That\u0027s u prime."},{"Start":"05:32.700 ","End":"05:39.649","Text":"I still need the v, so I still need R squared minus x squared over 4 plus,"},{"Start":"05:39.649 ","End":"05:41.375","Text":"now I\u0027ve got the second term."},{"Start":"05:41.375 ","End":"05:47.690","Text":"This as is 1/4 2R plus x squared."},{"Start":"05:47.690 ","End":"05:58.130","Text":"Now this is a constant and the derivative of this is minus 2 times x over 4."},{"Start":"05:58.130 ","End":"06:04.890","Text":"This is just minus x over 2."},{"Start":"06:05.000 ","End":"06:07.535","Text":"I\u0027ll just show you this bit again."},{"Start":"06:07.535 ","End":"06:12.170","Text":"The x squared over 4 derivative,"},{"Start":"06:12.170 ","End":"06:15.275","Text":"x squared gives me 2x."},{"Start":"06:15.275 ","End":"06:18.670","Text":"The 4 is a constant so it stays."},{"Start":"06:18.670 ","End":"06:20.970","Text":"Then the 2 with the 4 cancel,"},{"Start":"06:20.970 ","End":"06:25.540","Text":"so I\u0027ve just got 2 here and that gives me that, and there\u0027s the minus."},{"Start":"06:25.540 ","End":"06:28.790","Text":"Now I\u0027m going to simplify this a bit."},{"Start":"06:28.790 ","End":"06:30.320","Text":"Let\u0027s see what I\u0027m going to do."},{"Start":"06:30.320 ","End":"06:31.940","Text":"I got too many fractions."},{"Start":"06:31.940 ","End":"06:36.110","Text":"You know what, I see that 1/4 times 2 is 1/2,"},{"Start":"06:36.110 ","End":"06:37.900","Text":"and here I have 1/4."},{"Start":"06:37.900 ","End":"06:40.280","Text":"If I put it over 8 also here,"},{"Start":"06:40.280 ","End":"06:42.020","Text":"4 and 2 is 8,"},{"Start":"06:42.020 ","End":"06:46.000","Text":"so why don\u0027t I write this as 1/8."},{"Start":"06:46.000 ","End":"06:48.510","Text":"Let\u0027s see what we have left."},{"Start":"06:48.510 ","End":"06:52.425","Text":"This is 1/2, so that disappears."},{"Start":"06:52.425 ","End":"06:55.334","Text":"I\u0027ve got 2R plus x."},{"Start":"06:55.334 ","End":"06:59.415","Text":"Then we\u0027re multiplying this by 2 and this by 4,"},{"Start":"06:59.415 ","End":"07:03.915","Text":"so I have 4R squared minus x squared."},{"Start":"07:03.915 ","End":"07:06.210","Text":"Here I get rid of the 4 and the 2,"},{"Start":"07:06.210 ","End":"07:10.745","Text":"because of the 8, but also the plus with the minus makes it a minus."},{"Start":"07:10.745 ","End":"07:20.670","Text":"I have minus x times 2R plus x all squared."},{"Start":"07:21.260 ","End":"07:27.955","Text":"I\u0027m looking for critical points where y prime is equal to 0,"},{"Start":"07:27.955 ","End":"07:30.145","Text":"and this will give me,"},{"Start":"07:30.145 ","End":"07:32.015","Text":"I can do some canceling."},{"Start":"07:32.015 ","End":"07:36.350","Text":"If this equals 0, then I can multiply both sides by 8,"},{"Start":"07:36.350 ","End":"07:38.629","Text":"so I don\u0027t need this."},{"Start":"07:38.629 ","End":"07:44.370","Text":"Also notice that the 2R plus x here and 2R plus x squared,"},{"Start":"07:44.370 ","End":"07:51.460","Text":"so I can cancel this with 1 of the 2R plus x\u0027s from here."},{"Start":"07:51.710 ","End":"07:55.970","Text":"Then when something minus something is 0, let\u0027s see,"},{"Start":"07:55.970 ","End":"08:01.745","Text":"should I bring this to the other side? I\u0027ll leave it on the left."},{"Start":"08:01.745 ","End":"08:08.909","Text":"We\u0027ve got 4R squared minus x squared"},{"Start":"08:08.909 ","End":"08:14.765","Text":"minus 2Rx from this"},{"Start":"08:14.765 ","End":"08:19.935","Text":"and then minus x squared,"},{"Start":"08:19.935 ","End":"08:23.865","Text":"and this equals 0."},{"Start":"08:23.865 ","End":"08:31.610","Text":"Now look, this with this combines that\u0027s minus 2x squared."},{"Start":"08:31.610 ","End":"08:34.505","Text":"Now everything divides by 2,"},{"Start":"08:34.505 ","End":"08:36.395","Text":"but I\u0027d like the x squared to be positive."},{"Start":"08:36.395 ","End":"08:42.560","Text":"I\u0027m going to divide by minus 2."},{"Start":"08:42.560 ","End":"08:46.114","Text":"Let\u0027s see what we get."},{"Start":"08:46.114 ","End":"08:52.535","Text":"Minus 2x squared over minus 2 is just x squared."},{"Start":"08:52.535 ","End":"08:55.700","Text":"Then I have the minus 2Rx,"},{"Start":"08:55.700 ","End":"08:59.060","Text":"so it becomes plus Rx."},{"Start":"08:59.060 ","End":"09:04.520","Text":"This divided by minus 2"},{"Start":"09:04.520 ","End":"09:12.025","Text":"means minus 2R squared."},{"Start":"09:12.025 ","End":"09:16.205","Text":"Now you could solve this with the quadratic formula."},{"Start":"09:16.205 ","End":"09:18.950","Text":"Let\u0027s do it by factoring."},{"Start":"09:18.950 ","End":"09:26.105","Text":"I mean, it has to be x with something and x with something."},{"Start":"09:26.105 ","End":"09:28.490","Text":"Here I have 2R squared,"},{"Start":"09:28.490 ","End":"09:33.080","Text":"so it\u0027s going to be 2R times R,"},{"Start":"09:33.080 ","End":"09:34.310","Text":"and 1 of them is going to be plus,"},{"Start":"09:34.310 ","End":"09:36.140","Text":"so 1 of them is going to be minus."},{"Start":"09:36.140 ","End":"09:37.865","Text":"But to get the plus here,"},{"Start":"09:37.865 ","End":"09:39.590","Text":"I\u0027ll do it like this,"},{"Start":"09:39.590 ","End":"09:43.310","Text":"and I\u0027ll check x times x is x squared plus 2Rx"},{"Start":"09:43.310 ","End":"09:48.370","Text":"minus Rx is this minus 2R squared, and this equals 0."},{"Start":"09:48.370 ","End":"09:51.530","Text":"When a product is 0, 1 of them has to be 0."},{"Start":"09:51.530 ","End":"09:54.830","Text":"This gives us x equals minus 2R."},{"Start":"09:54.830 ","End":"09:57.475","Text":"This gives us x equals R,"},{"Start":"09:57.475 ","End":"09:59.130","Text":"we\u0027re only working with positive,"},{"Start":"09:59.130 ","End":"10:01.395","Text":"so this is ruled out."},{"Start":"10:01.395 ","End":"10:06.315","Text":"We were looking for x that gives the maximum value,"},{"Start":"10:06.315 ","End":"10:09.050","Text":"and this is the only possibility."},{"Start":"10:09.050 ","End":"10:12.590","Text":"The thing is that we should show that it really is a maximum,"},{"Start":"10:12.590 ","End":"10:15.740","Text":"it\u0027s not a minimum, or something else, it\u0027s a critical point."},{"Start":"10:15.740 ","End":"10:20.300","Text":"Meanwhile, what we\u0027ll do is the 2nd derivative test."},{"Start":"10:20.300 ","End":"10:23.705","Text":"It\u0027s usually good when you have parameters."},{"Start":"10:23.705 ","End":"10:26.160","Text":"Let\u0027s go back to y prime."},{"Start":"10:26.160 ","End":"10:29.990","Text":"Now y prime is this expression before canceling."},{"Start":"10:29.990 ","End":"10:35.160","Text":"It\u0027s really 1/8 and I have to put"},{"Start":"10:35.160 ","End":"10:40.710","Text":"back the 2R plus x that I canceled,"},{"Start":"10:40.710 ","End":"10:43.580","Text":"but the algebra I did here is"},{"Start":"10:43.580 ","End":"10:52.790","Text":"good except that I have to put back that minus 2 that I canceled back,"},{"Start":"10:52.790 ","End":"10:57.755","Text":"because I want the y prime in its original form."},{"Start":"10:57.755 ","End":"11:02.975","Text":"I mean simplified but not divided by all kinds of constants."},{"Start":"11:02.975 ","End":"11:05.930","Text":"Then we have, what is it here, yeah,"},{"Start":"11:05.930 ","End":"11:12.910","Text":"x squared plus Rx minus 2R squared."},{"Start":"11:13.300 ","End":"11:18.535","Text":"Now I\u0027m going to differentiate again to get y double prime."},{"Start":"11:18.535 ","End":"11:26.250","Text":"Now this minus 2 with the 1/8 gives me minus 1/4."},{"Start":"11:26.250 ","End":"11:32.205","Text":"Other than that, I\u0027ll do the product rule on this times this."},{"Start":"11:32.205 ","End":"11:41.675","Text":"Say this part and this part will be my u and the v. The derivative of u is just"},{"Start":"11:41.675 ","End":"11:51.450","Text":"1 and then v as is x squared plus Rx minus 2R squared."},{"Start":"11:51.450 ","End":"11:56.430","Text":"Then the other way round."},{"Start":"11:56.430 ","End":"12:02.255","Text":"I guess I have to put brackets here because the minus 1/4 applies to the whole thing."},{"Start":"12:02.255 ","End":"12:10.270","Text":"The 2R plus x as is and the derivative of this 2x plus"},{"Start":"12:10.270 ","End":"12:17.190","Text":"R. Now what I have to do is"},{"Start":"12:17.190 ","End":"12:25.690","Text":"plug-in y double prime when x is equal to R and see if it comes out positive or negative."},{"Start":"12:25.690 ","End":"12:30.420","Text":"This equals minus 1/4."},{"Start":"12:30.420 ","End":"12:32.700","Text":"Now, if x is R,"},{"Start":"12:32.700 ","End":"12:34.305","Text":"well, I\u0027ll just write it,"},{"Start":"12:34.305 ","End":"12:43.664","Text":"R squared plus R times R is R squared minus 2R squared plus,"},{"Start":"12:43.664 ","End":"12:52.060","Text":"here I have 2R plus R is 3R and 2x plus R is also 3R."},{"Start":"12:52.940 ","End":"12:59.660","Text":"Altogether, what I get is this part is just 0,"},{"Start":"12:59.660 ","End":"13:01.310","Text":"2R squared minus 2R squared,"},{"Start":"13:01.310 ","End":"13:06.405","Text":"and this part is just 9R squared."},{"Start":"13:06.405 ","End":"13:12.060","Text":"What I have is this is negative, this is positive."},{"Start":"13:12.060 ","End":"13:19.880","Text":"We have a negative times a positive and the answer is a negative,"},{"Start":"13:19.880 ","End":"13:23.400","Text":"which means that negative,"},{"Start":"13:23.400 ","End":"13:25.140","Text":"or less than 0."},{"Start":"13:25.140 ","End":"13:31.790","Text":"I\u0027m going to write it as less than 0 and less than 0 means that we"},{"Start":"13:31.790 ","End":"13:39.135","Text":"have a maximum by the 2nd derivative test."},{"Start":"13:39.135 ","End":"13:47.820","Text":"That concludes and the answer is that short base was this x,"},{"Start":"13:47.820 ","End":"13:55.320","Text":"and we found that the answer is that x is equal to the radius. Done."}],"ID":8566},{"Watched":false,"Name":"Exercise 11","Duration":"7m 13s","ChapterTopicVideoID":8376,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8376.jpeg","UploadDate":"2017-01-28T12:55:08.2470000","DurationForVideoObject":"PT7M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.764","Text":"Let me explain this word problem here in geometry."},{"Start":"00:03.764 ","End":"00:06.705","Text":"Also a maximum, minimum problem."},{"Start":"00:06.705 ","End":"00:08.685","Text":"We have a sector,"},{"Start":"00:08.685 ","End":"00:12.225","Text":"O, well this is not labeled this point here."},{"Start":"00:12.225 ","End":"00:14.880","Text":"This point here, this is a quarter of a circle."},{"Start":"00:14.880 ","End":"00:21.135","Text":"Quarter of a circle means that this angle is 90 degrees."},{"Start":"00:21.135 ","End":"00:27.900","Text":"We\u0027re also given a rectangle such that A and B are on the radii,"},{"Start":"00:27.900 ","End":"00:30.390","Text":"A is on this radius and B is on this radius."},{"Start":"00:30.390 ","End":"00:33.855","Text":"Also, DC is tangent to this radius,"},{"Start":"00:33.855 ","End":"00:36.720","Text":"and only tangent at the midpoint."},{"Start":"00:36.720 ","End":"00:40.050","Text":"This bit equals this bit."},{"Start":"00:40.050 ","End":"00:45.980","Text":"Of course, it follows because the tangent is always 90 degrees,"},{"Start":"00:45.980 ","End":"00:48.965","Text":"that also this bit is equal to this bit."},{"Start":"00:48.965 ","End":"00:53.315","Text":"Now, there are many possible such rectangles that we could draw."},{"Start":"00:53.315 ","End":"00:55.940","Text":"Any place here, if I drew AB,"},{"Start":"00:55.940 ","End":"00:59.195","Text":"I could complete it to a rectangle."},{"Start":"00:59.195 ","End":"01:03.575","Text":"What we have to find is among all such possible rectangles,"},{"Start":"01:03.575 ","End":"01:06.605","Text":"which one has the smallest diagonal length?"},{"Start":"01:06.605 ","End":"01:11.375","Text":"The diagonal length means from here to here."},{"Start":"01:11.375 ","End":"01:15.890","Text":"Which one of these rectangles will give the smallest diagonal length."},{"Start":"01:15.890 ","End":"01:23.030","Text":"That will be our dependent variable and I\u0027ll call it l for the moment."},{"Start":"01:23.030 ","End":"01:27.410","Text":"The question is what to take as the independent variable."},{"Start":"01:27.410 ","End":"01:30.440","Text":"There are several possibilities that come to mind."},{"Start":"01:30.440 ","End":"01:31.895","Text":"I could take the length here."},{"Start":"01:31.895 ","End":"01:33.440","Text":"I could take where this bit,"},{"Start":"01:33.440 ","End":"01:37.565","Text":"which I\u0027ve marked, this half of the length of the rectangle."},{"Start":"01:37.565 ","End":"01:44.395","Text":"I thought about it a while and decided to go with this bit and I\u0027ll call this my x."},{"Start":"01:44.395 ","End":"01:47.565","Text":"It\u0027s the height of the rectangle."},{"Start":"01:47.565 ","End":"01:52.805","Text":"If this is x, then because this is a radius,"},{"Start":"01:52.805 ","End":"01:56.435","Text":"all the radii are 10 centimeters. So this is 10."},{"Start":"01:56.435 ","End":"02:00.170","Text":"From O to, oh let\u0027s give it a label."},{"Start":"02:00.170 ","End":"02:03.685","Text":"Let\u0027s call this one M. So OM,"},{"Start":"02:03.685 ","End":"02:06.570","Text":"the length of OM is 10."},{"Start":"02:06.570 ","End":"02:09.050","Text":"Let me give a name for this one,"},{"Start":"02:09.050 ","End":"02:14.210","Text":"E. OE is equal"},{"Start":"02:14.210 ","End":"02:19.355","Text":"to 10 minus x. I\u0027ll just label them in the picture."},{"Start":"02:19.355 ","End":"02:23.390","Text":"This is 10 minus x because this is all 10 and this is x."},{"Start":"02:23.390 ","End":"02:25.735","Text":"This is also 10 minus x,"},{"Start":"02:25.735 ","End":"02:29.085","Text":"and this is also 10 minus x."},{"Start":"02:29.085 ","End":"02:32.900","Text":"Basically, I mean, we have everything we need to know."},{"Start":"02:32.900 ","End":"02:34.820","Text":"We have the rectangle."},{"Start":"02:34.820 ","End":"02:37.130","Text":"I\u0027ll just take out the important points."},{"Start":"02:37.130 ","End":"02:43.095","Text":"Its length is twice 10 minus x and its height is x."},{"Start":"02:43.095 ","End":"02:45.435","Text":"This is the rectangle."},{"Start":"02:45.435 ","End":"02:50.665","Text":"The length of the diagonal is l. We want to find"},{"Start":"02:50.665 ","End":"02:56.845","Text":"x such that the rectangle has the least diagonal."},{"Start":"02:56.845 ","End":"03:03.400","Text":"We can straightaway say that l is equal to using Pythagoras\u0027s theorem to"},{"Start":"03:03.400 ","End":"03:10.345","Text":"the square root of x squared plus,"},{"Start":"03:10.345 ","End":"03:16.510","Text":"let\u0027s see, 4 times 10 minus x squared."},{"Start":"03:16.510 ","End":"03:18.805","Text":"The 4 is 2 squared."},{"Start":"03:18.805 ","End":"03:23.320","Text":"We have to find what is the minimum, the smallest."},{"Start":"03:23.320 ","End":"03:27.180","Text":"We want the minimum for l,"},{"Start":"03:27.180 ","End":"03:30.710","Text":"we have to find which x gives the minimum l. Then we also"},{"Start":"03:30.710 ","End":"03:34.835","Text":"actually have to find the actual l that is the smallest."},{"Start":"03:34.835 ","End":"03:38.810","Text":"I\u0027m going to use a trick and the trick is this."},{"Start":"03:38.810 ","End":"03:45.185","Text":"If we have to minimize the square root of a function or maximize either way,"},{"Start":"03:45.185 ","End":"03:48.620","Text":"we can actually just throw out the square root and"},{"Start":"03:48.620 ","End":"03:52.660","Text":"find x which gives the maximum or minimum without the square root."},{"Start":"03:52.660 ","End":"03:56.310","Text":"Then we can get rid of that and I\u0027ll call it now y,"},{"Start":"03:56.310 ","End":"03:58.359","Text":"which is my l squared,"},{"Start":"03:58.359 ","End":"04:03.440","Text":"which is x squared plus 4 times 10 minus x squared."},{"Start":"04:03.440 ","End":"04:07.639","Text":"As I say, the trick guarantees that the minimum"},{"Start":"04:07.639 ","End":"04:12.990","Text":"for y is also the minimum for l as far as x goes."},{"Start":"04:12.990 ","End":"04:14.810","Text":"When we actually have to find the diagonal,"},{"Start":"04:14.810 ","End":"04:17.015","Text":"of course, we substitute in here."},{"Start":"04:17.015 ","End":"04:25.190","Text":"Let\u0027s differentiate y prime equals 2x plus, let\u0027s see,"},{"Start":"04:25.190 ","End":"04:31.940","Text":"4 times 2 is 8 times 10 minus x times inner derivative,"},{"Start":"04:31.940 ","End":"04:34.235","Text":"which is minus 1."},{"Start":"04:34.235 ","End":"04:37.700","Text":"And let\u0027s see if we can simplify this a bit."},{"Start":"04:37.700 ","End":"04:41.130","Text":"So this is 2x and this is,"},{"Start":"04:41.130 ","End":"04:43.590","Text":"minus 1 and 8 is minus 8,"},{"Start":"04:43.590 ","End":"04:46.579","Text":"so minus 8 times 10 is minus 80,"},{"Start":"04:46.579 ","End":"04:51.620","Text":"minus 8 times minus x is plus 8x."},{"Start":"04:51.620 ","End":"04:56.000","Text":"This is 10x minus 80."},{"Start":"04:56.000 ","End":"05:01.265","Text":"Now suspect for minimum will be when y prime equals 0."},{"Start":"05:01.265 ","End":"05:02.870","Text":"Well, it\u0027s easy to do in our head."},{"Start":"05:02.870 ","End":"05:05.329","Text":"If 10x minus 80 is 0,"},{"Start":"05:05.329 ","End":"05:09.470","Text":"10x is 80, x is equal to 8."},{"Start":"05:09.470 ","End":"05:11.330","Text":"That\u0027s not the answer."},{"Start":"05:11.330 ","End":"05:14.450","Text":"The answer is the diagonal length."},{"Start":"05:14.450 ","End":"05:17.480","Text":"When x is 8, we have l equals,"},{"Start":"05:17.480 ","End":"05:21.485","Text":"let\u0027s call it l of 8 because l is a function of x,"},{"Start":"05:21.485 ","End":"05:26.495","Text":"is equal to the square root of"},{"Start":"05:26.495 ","End":"05:34.255","Text":"8 squared plus 4 times 10 minus 8 squared just substituting here."},{"Start":"05:34.255 ","End":"05:36.170","Text":"Let\u0027s see what it comes out."},{"Start":"05:36.170 ","End":"05:38.180","Text":"10 minus 8 is 2,"},{"Start":"05:38.180 ","End":"05:41.975","Text":"2 squared is 4, 4 times 4 is 16."},{"Start":"05:41.975 ","End":"05:46.710","Text":"16 plus 64 is 80."},{"Start":"05:46.710 ","End":"05:49.710","Text":"The shortest diagonal is 80."},{"Start":"05:49.710 ","End":"05:51.830","Text":"But there is still one thing we haven\u0027t done,"},{"Start":"05:51.830 ","End":"05:53.330","Text":"we haven\u0027t shown that it\u0027s a minimum,"},{"Start":"05:53.330 ","End":"05:54.710","Text":"maybe it\u0027s a maximum."},{"Start":"05:54.710 ","End":"05:58.015","Text":"Let\u0027s use the table method."},{"Start":"05:58.015 ","End":"06:02.060","Text":"Let\u0027s make a little table here with 3 rows."},{"Start":"06:02.060 ","End":"06:04.355","Text":"In this one, I put x, in this one,"},{"Start":"06:04.355 ","End":"06:06.360","Text":"I put y prime,"},{"Start":"06:06.360 ","End":"06:08.105","Text":"and in this one I put y."},{"Start":"06:08.105 ","End":"06:09.905","Text":"Here, I put my suspects."},{"Start":"06:09.905 ","End":"06:14.105","Text":"There is only one suspect and that is x equals 8."},{"Start":"06:14.105 ","End":"06:18.410","Text":"I need to substitute it in y prime."},{"Start":"06:18.410 ","End":"06:21.635","Text":"Actually, the easiest place to substitute,"},{"Start":"06:21.635 ","End":"06:23.300","Text":"this is y prime."},{"Start":"06:23.300 ","End":"06:26.075","Text":"And this would be where I would substitute."},{"Start":"06:26.075 ","End":"06:28.340","Text":"If I put x equals 8, of course,"},{"Start":"06:28.340 ","End":"06:31.475","Text":"I get 0 because that\u0027s how we found the 8."},{"Start":"06:31.475 ","End":"06:33.620","Text":"If I put x equals,"},{"Start":"06:33.620 ","End":"06:36.325","Text":"let\u0027s say 7 and 9,"},{"Start":"06:36.325 ","End":"06:38.745","Text":"at 7, let\u0027s see,"},{"Start":"06:38.745 ","End":"06:42.460","Text":"10 times 7 is 70 minus 80 is,"},{"Start":"06:42.460 ","End":"06:44.300","Text":"it doesn\u0027t matter what the number is."},{"Start":"06:44.300 ","End":"06:46.100","Text":"It\u0027s a negative quantity."},{"Start":"06:46.100 ","End":"06:47.930","Text":"If put x equals 9,"},{"Start":"06:47.930 ","End":"06:50.779","Text":"that\u0027s 90 minus 80, okay, it\u0027s positive."},{"Start":"06:50.779 ","End":"06:52.690","Text":"That\u0027s the important thing."},{"Start":"06:52.690 ","End":"06:54.770","Text":"If y prime is negative,"},{"Start":"06:54.770 ","End":"06:57.080","Text":"that means that y is decreasing."},{"Start":"06:57.080 ","End":"07:02.450","Text":"Here, y prime is increasing and that means that we\u0027ve got something like this,"},{"Start":"07:02.450 ","End":"07:04.685","Text":"which is a minimum."},{"Start":"07:04.685 ","End":"07:06.559","Text":"Indeed we have a minimum."},{"Start":"07:06.559 ","End":"07:08.195","Text":"It\u0027s obtained when x is 8."},{"Start":"07:08.195 ","End":"07:12.080","Text":"But the important thing is that the actual minimum is 80,"},{"Start":"07:12.080 ","End":"07:14.760","Text":"and we are done."}],"ID":8567},{"Watched":false,"Name":"Exercise 12","Duration":"13m 23s","ChapterTopicVideoID":8377,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8377.jpeg","UploadDate":"2017-01-28T13:53:28.6130000","DurationForVideoObject":"PT13M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"What we have in this exercise is a pentagon,"},{"Start":"00:04.830 ","End":"00:08.565","Text":"this house shape, which is made up of 2 pieces."},{"Start":"00:08.565 ","End":"00:10.755","Text":"One is a rectangle here,"},{"Start":"00:10.755 ","End":"00:13.170","Text":"and one is a triangle here."},{"Start":"00:13.170 ","End":"00:15.780","Text":"Now we\u0027re given some details."},{"Start":"00:15.780 ","End":"00:20.670","Text":"That is that the sides of the rectangle,"},{"Start":"00:20.670 ","End":"00:22.860","Text":"this is 2 and this is 2."},{"Start":"00:22.860 ","End":"00:25.380","Text":"We\u0027re working in centimeters, everything."},{"Start":"00:25.380 ","End":"00:29.355","Text":"This triangle, the roof is an isosceles triangle."},{"Start":"00:29.355 ","End":"00:32.635","Text":"Where this is 4 and this is 4."},{"Start":"00:32.635 ","End":"00:38.560","Text":"The question is about the area of the pentagon."},{"Start":"00:38.560 ","End":"00:42.965","Text":"What the greatest area possible?"},{"Start":"00:42.965 ","End":"00:45.804","Text":"Obviously, this is a maximum problem,"},{"Start":"00:45.804 ","End":"00:54.060","Text":"and the dependent variable is going to be the area of the pentagon."},{"Start":"00:54.060 ","End":"00:57.365","Text":"If I let y be the area of the pentagon,"},{"Start":"00:57.365 ","End":"01:02.450","Text":"then we have that y is equal to"},{"Start":"01:02.450 ","End":"01:10.144","Text":"the area of the rectangle plus the area,"},{"Start":"01:10.144 ","End":"01:13.055","Text":"I use that for area, of the triangle."},{"Start":"01:13.055 ","End":"01:14.539","Text":"Let me just be specific."},{"Start":"01:14.539 ","End":"01:17.390","Text":"The rectangle would be E, B,"},{"Start":"01:17.390 ","End":"01:20.940","Text":"C, D, and the triangle will be A,"},{"Start":"01:20.940 ","End":"01:24.260","Text":"B, E. Then I\u0027m going to work out these things bit by"},{"Start":"01:24.260 ","End":"01:27.680","Text":"bit until we get it in terms of the independent variable."},{"Start":"01:27.680 ","End":"01:29.420","Text":"That brings me to the next question."},{"Start":"01:29.420 ","End":"01:31.625","Text":"What is the independent variable?"},{"Start":"01:31.625 ","End":"01:33.770","Text":"Well, many things are possible."},{"Start":"01:33.770 ","End":"01:40.385","Text":"I found it useful to take the angle Alpha as the variable,"},{"Start":"01:40.385 ","End":"01:44.180","Text":"and I can figure out what else I need in terms of Alpha."},{"Start":"01:44.180 ","End":"01:49.579","Text":"For example, in order to compute the area of the rectangle,"},{"Start":"01:49.579 ","End":"01:51.215","Text":"I\u0027d like to know this."},{"Start":"01:51.215 ","End":"01:54.080","Text":"Let me call this x."},{"Start":"01:54.080 ","End":"01:56.270","Text":"That\u0027s this side here,"},{"Start":"01:56.270 ","End":"01:58.100","Text":"and this would be x also."},{"Start":"01:58.100 ","End":"02:02.660","Text":"I\u0027d also like to know what is the height of this triangle,"},{"Start":"02:02.660 ","End":"02:07.055","Text":"because obviously I\u0027m going to use the formula 1/2 base times height."},{"Start":"02:07.055 ","End":"02:10.705","Text":"This is h. Now,"},{"Start":"02:10.705 ","End":"02:12.720","Text":"I can continue with this."},{"Start":"02:12.720 ","End":"02:14.435","Text":"Even though I don\u0027t know everything,"},{"Start":"02:14.435 ","End":"02:19.130","Text":"I can say that this is equal to x times 2,"},{"Start":"02:19.130 ","End":"02:20.870","Text":"because that\u0027s the area of a rectangle,"},{"Start":"02:20.870 ","End":"02:24.510","Text":"base times height, x times 2."},{"Start":"02:24.510 ","End":"02:29.950","Text":"This area of this triangle is going to be a 1/2 times"},{"Start":"02:29.950 ","End":"02:36.335","Text":"x times h. Now what I want to do is get x and h in terms of Alpha."},{"Start":"02:36.335 ","End":"02:42.265","Text":"Let\u0027s start with x. I have 2 separate triangles here."},{"Start":"02:42.265 ","End":"02:44.350","Text":"This is Alpha over 2."},{"Start":"02:44.350 ","End":"02:47.050","Text":"Let me just look at the left triangle,"},{"Start":"02:47.050 ","End":"02:48.940","Text":"and this is x over 2."},{"Start":"02:48.940 ","End":"02:51.650","Text":"It\u0027s isosceles, everything symmetric."},{"Start":"02:51.650 ","End":"02:59.799","Text":"I can say that the sine of Alpha over 2 is the side opposite,"},{"Start":"02:59.799 ","End":"03:05.125","Text":"which is x over 2 over the hypotenuse,"},{"Start":"03:05.125 ","End":"03:07.705","Text":"which is over 4."},{"Start":"03:07.705 ","End":"03:14.865","Text":"So I\u0027ve got that sine Alpha over 2 is x over 8."},{"Start":"03:14.865 ","End":"03:22.395","Text":"So x is equal to 8 sine of Alpha over 2."},{"Start":"03:22.395 ","End":"03:25.200","Text":"That\u0027s 1 result for x."},{"Start":"03:25.200 ","End":"03:28.955","Text":"Now how about h? This time,"},{"Start":"03:28.955 ","End":"03:35.705","Text":"I shall use the cosine of the angle and get that"},{"Start":"03:35.705 ","End":"03:44.000","Text":"the cosine of Alpha over 2 is the side adjacent over the hypotenuse."},{"Start":"03:44.000 ","End":"03:46.805","Text":"This would be h over 4."},{"Start":"03:46.805 ","End":"03:56.595","Text":"So h would equal 4 times cosine Alpha over 2."},{"Start":"03:56.595 ","End":"04:01.070","Text":"Now, I\u0027m all ready to get back here and to write"},{"Start":"04:01.070 ","End":"04:05.915","Text":"this y as a function of Alpha, I\u0027ll emphasize that."},{"Start":"04:05.915 ","End":"04:09.085","Text":"This will equal 2x."},{"Start":"04:09.085 ","End":"04:12.225","Text":"Now, x is 8 sine Alpha over 2."},{"Start":"04:12.225 ","End":"04:17.715","Text":"We have here 2 times x is 16 sine Alpha over 2."},{"Start":"04:17.715 ","End":"04:23.940","Text":"Now plus 1/2, and x is 8 sine Alpha over 2,"},{"Start":"04:23.940 ","End":"04:30.745","Text":"and h is 4 cosine Alpha over 2."},{"Start":"04:30.745 ","End":"04:32.795","Text":"This I\u0027m going to leave as is,"},{"Start":"04:32.795 ","End":"04:36.940","Text":"the 16 sine of Alpha over 2."},{"Start":"04:36.940 ","End":"04:44.345","Text":"But this one, I\u0027m going to simplify using the formula that"},{"Start":"04:44.345 ","End":"04:51.420","Text":"2 sine Theta cosine Theta"},{"Start":"04:51.420 ","End":"04:55.605","Text":"is equal to sine 2 Theta."},{"Start":"04:55.605 ","End":"04:58.800","Text":"If I use it here with Alpha over 2,"},{"Start":"04:58.800 ","End":"05:01.290","Text":"if Theta is Alpha over 2,"},{"Start":"05:01.290 ","End":"05:04.770","Text":"then 2 Theta is Alpha. Now, let\u0027s see."},{"Start":"05:04.770 ","End":"05:07.140","Text":"Just I go to straighten out the constants here."},{"Start":"05:07.140 ","End":"05:10.440","Text":"I have 1/2 times 8 times 4, which is 16."},{"Start":"05:10.440 ","End":"05:15.750","Text":"The 16, I can write as 8 times 2."},{"Start":"05:15.750 ","End":"05:21.555","Text":"So I have 8 times 2 sine Alpha over 2,"},{"Start":"05:21.555 ","End":"05:24.360","Text":"cosine Alpha over 2."},{"Start":"05:24.360 ","End":"05:28.875","Text":"This equals 16 sine Alpha over 2."},{"Start":"05:28.875 ","End":"05:30.740","Text":"Using what I said here,"},{"Start":"05:30.740 ","End":"05:35.595","Text":"I\u0027ve got 8 of sine of Alpha."},{"Start":"05:35.595 ","End":"05:38.685","Text":"Twice Alpha over 2, 8 sine Alpha."},{"Start":"05:38.685 ","End":"05:46.860","Text":"This is my function to express the area of the pentagon in terms of Alpha."},{"Start":"05:47.060 ","End":"05:51.425","Text":"Now, we\u0027ve got a standard extremum problem."},{"Start":"05:51.425 ","End":"05:54.290","Text":"We\u0027re going to differentiate set to 0,"},{"Start":"05:54.290 ","End":"05:58.715","Text":"find Alpha, verify that it\u0027s a maximum,"},{"Start":"05:58.715 ","End":"06:00.980","Text":"and so on. Let\u0027s begin."},{"Start":"06:00.980 ","End":"06:06.470","Text":"Y prime is equal to 16."},{"Start":"06:06.470 ","End":"06:15.485","Text":"Derivative of sine Alpha over 2 is cosine Alpha over 2 times 1/2 in a derivative,"},{"Start":"06:15.485 ","End":"06:19.320","Text":"which will swallow up by the 16,"},{"Start":"06:19.320 ","End":"06:22.545","Text":"it\u0027s just 8 cosine Alpha over 2."},{"Start":"06:22.545 ","End":"06:29.860","Text":"The derivative of this is 8 cosine Alpha."},{"Start":"06:29.860 ","End":"06:34.125","Text":"Now we\u0027re looking for extrema."},{"Start":"06:34.125 ","End":"06:38.100","Text":"We\u0027re going to let y prime equals 0."},{"Start":"06:38.100 ","End":"06:41.600","Text":"By the way, we do have limits for Alpha."},{"Start":"06:41.600 ","End":"06:42.980","Text":"If you look at the picture,"},{"Start":"06:42.980 ","End":"06:47.985","Text":"Alpha has to be between 0 and 190 degree."},{"Start":"06:47.985 ","End":"06:50.689","Text":"If y prime equals 0,"},{"Start":"06:50.689 ","End":"06:53.680","Text":"then this is equal to 0."},{"Start":"06:53.680 ","End":"06:56.805","Text":"I can throw the 8 out."},{"Start":"06:56.805 ","End":"07:04.010","Text":"I get that cosine sine Alpha over 2 plus cosine Alpha equals 0."},{"Start":"07:04.010 ","End":"07:06.605","Text":"Now we\u0027ve got a trigonometric equation."},{"Start":"07:06.605 ","End":"07:08.580","Text":"I\u0027m going to introduce another formula."},{"Start":"07:08.580 ","End":"07:13.995","Text":"Just like we had a formula here for the double angle for sine,"},{"Start":"07:13.995 ","End":"07:15.815","Text":"there\u0027s also one for cosine."},{"Start":"07:15.815 ","End":"07:21.200","Text":"That is that the cosine of 2 Theta is equal"},{"Start":"07:21.200 ","End":"07:27.905","Text":"to 2 cosine squared Theta minus 1."},{"Start":"07:27.905 ","End":"07:31.254","Text":"Theta is going to be Alpha over 2,"},{"Start":"07:31.254 ","End":"07:34.500","Text":"and 2 Theta is going to be Alpha."},{"Start":"07:34.500 ","End":"07:39.550","Text":"If I apply that to what I have here, I have, well,"},{"Start":"07:39.550 ","End":"07:45.730","Text":"cosine Alpha over 2 stays, but this,"},{"Start":"07:45.730 ","End":"07:54.720","Text":"I now write as 2 cosine squared Theta is Alpha over 2 minus 1."},{"Start":"07:54.720 ","End":"07:59.440","Text":"This is equal to 0. I\u0027m going to substitute into here,"},{"Start":"07:59.440 ","End":"08:04.635","Text":"t is equal to cosine Alpha over 2."},{"Start":"08:04.635 ","End":"08:06.925","Text":"If I make this substitution,"},{"Start":"08:06.925 ","End":"08:09.400","Text":"what I have here and then just rearranging the order,"},{"Start":"08:09.400 ","End":"08:17.355","Text":"is 2t squared plus t minus 1 equals 0."},{"Start":"08:17.355 ","End":"08:19.550","Text":"The 2 solutions of this equation,"},{"Start":"08:19.550 ","End":"08:21.695","Text":"and you can check this on your own,"},{"Start":"08:21.695 ","End":"08:30.655","Text":"t equals minus 1 and t equals 1/2."},{"Start":"08:30.655 ","End":"08:33.275","Text":"Those are the possible solutions."},{"Start":"08:33.275 ","End":"08:38.275","Text":"Now, we have to convert back to Alpha."},{"Start":"08:38.275 ","End":"08:42.635","Text":"I either have cosine Alpha over 2 is"},{"Start":"08:42.635 ","End":"08:50.130","Text":"minus 1 or cosine Alpha over 2 equals 1/2."},{"Start":"08:50.130 ","End":"08:55.775","Text":"The cosine is minus 1 when the angle is a multiple of Pi."},{"Start":"08:55.775 ","End":"08:59.665","Text":"I need that Alpha over 2,"},{"Start":"08:59.665 ","End":"09:04.990","Text":"it\u0027s equal to Pi 190 degrees plus multiples of 2Pi."},{"Start":"09:04.990 ","End":"09:08.705","Text":"That means that Alpha is equal to"},{"Start":"09:08.705 ","End":"09:15.890","Text":"2Pi plus 4k Pi,"},{"Start":"09:15.890 ","End":"09:18.485","Text":"which basically is a multiple of 2Pi."},{"Start":"09:18.485 ","End":"09:23.160","Text":"It\u0027s 2Pi times 1 plus 4k."},{"Start":"09:23.200 ","End":"09:25.925","Text":"This is outside the range."},{"Start":"09:25.925 ","End":"09:27.865","Text":"This is ruled out."},{"Start":"09:27.865 ","End":"09:29.820","Text":"I\u0027m going to just say that, \"No,"},{"Start":"09:29.820 ","End":"09:32.175","Text":"this doesn\u0027t lead me to anything.\""},{"Start":"09:32.175 ","End":"09:36.680","Text":"Let\u0027s continue with the other and hopefully we\u0027ll get something from this."},{"Start":"09:36.680 ","End":"09:40.925","Text":"The angle whose cosine is 1/2,"},{"Start":"09:40.925 ","End":"09:44.270","Text":"and if Alpha is between 0 and Pi, of course,"},{"Start":"09:44.270 ","End":"09:48.580","Text":"Alpha over 2 has to be between 0 and Pi over 2."},{"Start":"09:48.580 ","End":"09:52.190","Text":"We\u0027re looking for a first quadrant solution to cosine being 1/2,"},{"Start":"09:52.190 ","End":"09:54.830","Text":"and that happens to be 60 degrees."},{"Start":"09:54.830 ","End":"10:00.335","Text":"Alpha over 2 is 60 degrees."},{"Start":"10:00.335 ","End":"10:02.990","Text":"But here we\u0027re working in radians,"},{"Start":"10:02.990 ","End":"10:07.250","Text":"so it\u0027s Pi over 3 radians."},{"Start":"10:07.250 ","End":"10:11.690","Text":"Alpha is 2Pi over 3,"},{"Start":"10:11.690 ","End":"10:13.640","Text":"in radians, of course."},{"Start":"10:13.640 ","End":"10:15.860","Text":"The thing is that we\u0027re not asked what is Alpha,"},{"Start":"10:15.860 ","End":"10:18.725","Text":"we\u0027re asked what is the area of the pentagon."},{"Start":"10:18.725 ","End":"10:21.410","Text":"That is exactly y."},{"Start":"10:21.410 ","End":"10:24.415","Text":"All I have to do is substitute in y,"},{"Start":"10:24.415 ","End":"10:28.205","Text":"and I have to remember that y was equal to,"},{"Start":"10:28.205 ","End":"10:30.105","Text":"here\u0027s our function."},{"Start":"10:30.105 ","End":"10:31.845","Text":"Here\u0027s our Alpha."},{"Start":"10:31.845 ","End":"10:37.760","Text":"Why don\u0027t we just substitute that in here and we get that y of"},{"Start":"10:37.760 ","End":"10:45.515","Text":"2Pi over 3 is equal to 16 sine."},{"Start":"10:45.515 ","End":"10:53.520","Text":"This will be 1Pi over 3 plus 8 sine of 2Pi over 3."},{"Start":"10:53.520 ","End":"10:55.665","Text":"Let me write this in degree."},{"Start":"10:55.665 ","End":"11:01.185","Text":"Sine of 60, let me just take the 8 outside,"},{"Start":"11:01.185 ","End":"11:06.090","Text":"so it\u0027s 8 outside and then we have 2 here and 1 here."},{"Start":"11:06.090 ","End":"11:11.610","Text":"It\u0027s 2 sine of 60 degrees is root 3 over 2."},{"Start":"11:11.610 ","End":"11:14.895","Text":"Sine of 120 degrees,"},{"Start":"11:14.895 ","End":"11:18.735","Text":"it\u0027s the same as sine of 60 degrees."},{"Start":"11:18.735 ","End":"11:24.225","Text":"This would be root 3 over 2 also."},{"Start":"11:24.225 ","End":"11:28.140","Text":"Altogether I\u0027ve got 3 root 3 over 2."},{"Start":"11:28.140 ","End":"11:37.000","Text":"This comes out to be 24 times root 3 over 2,"},{"Start":"11:37.000 ","End":"11:40.820","Text":"which is 12 times the square root of 3."},{"Start":"11:40.820 ","End":"11:42.650","Text":"The only thing we haven\u0027t done is,"},{"Start":"11:42.650 ","End":"11:45.289","Text":"shown that we have a maximum and not a minimum."},{"Start":"11:45.289 ","End":"11:48.455","Text":"Let me just note that for those who like decimals,"},{"Start":"11:48.455 ","End":"11:58.520","Text":"that this is approximately 20.7846 centimeters squared, of course."},{"Start":"11:58.520 ","End":"12:04.104","Text":"Now, I just have to show that this is indeed the maximum."},{"Start":"12:04.104 ","End":"12:08.395","Text":"This time, I think I\u0027ll go for the second derivative method."},{"Start":"12:08.395 ","End":"12:13.450","Text":"Let me just copy y prime down below."},{"Start":"12:13.450 ","End":"12:16.510","Text":"Using the second derivative method,"},{"Start":"12:16.510 ","End":"12:22.305","Text":"I\u0027ll get y double prime equals,"},{"Start":"12:22.305 ","End":"12:26.365","Text":"let\u0027s see, derivative of cosine is minus sine."},{"Start":"12:26.365 ","End":"12:28.630","Text":"But I also get 1/2,"},{"Start":"12:28.630 ","End":"12:35.445","Text":"so I get minus 4 sine Alpha over 2."},{"Start":"12:35.445 ","End":"12:39.460","Text":"Here I just get minus sine,"},{"Start":"12:39.460 ","End":"12:43.495","Text":"so it\u0027s minus 8 sine Alpha."},{"Start":"12:43.495 ","End":"12:49.410","Text":"Now, when Alpha is 2Pi over 3,"},{"Start":"12:49.410 ","End":"12:56.864","Text":"Alpha over 2 is Pi over 3 and Alpha is 2 Pi over 3."},{"Start":"12:56.864 ","End":"13:02.660","Text":"But the sine is positive all the way from 0 to 180 degrees,"},{"Start":"13:02.660 ","End":"13:07.430","Text":"so the sine is positive and the sine is positive,"},{"Start":"13:07.430 ","End":"13:09.590","Text":"but here we have negative coefficients."},{"Start":"13:09.590 ","End":"13:13.595","Text":"So y double prime is less than 0,"},{"Start":"13:13.595 ","End":"13:19.020","Text":"and that implies that we have a maximum."},{"Start":"13:19.020 ","End":"13:24.910","Text":"This, which is the value is indeed the maximum. We are done."}],"ID":8568},{"Watched":false,"Name":"Exercise 13","Duration":"13m 32s","ChapterTopicVideoID":8388,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8388.jpeg","UploadDate":"2017-01-28T13:54:42.8800000","DurationForVideoObject":"PT13M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.550","Text":"Let me explain this word problem."},{"Start":"00:02.550 ","End":"00:06.750","Text":"We have a right-angled triangle ABC,"},{"Start":"00:06.750 ","End":"00:15.105","Text":"and it bounds a semicircle of radius R. Here\u0027s the semicircle and we\u0027re given its radius."},{"Start":"00:15.105 ","End":"00:20.850","Text":"What we have to find out is amongst all such possible right-angled triangles,"},{"Start":"00:20.850 ","End":"00:22.460","Text":"because there could be many,"},{"Start":"00:22.460 ","End":"00:27.570","Text":"which one has the least sum of legs?"},{"Start":"00:27.570 ","End":"00:31.620","Text":"Now, sum of legs means AB plus AC,"},{"Start":"00:31.620 ","End":"00:33.480","Text":"because in a right triangle,"},{"Start":"00:33.480 ","End":"00:38.220","Text":"this is the hypotenuse and these 2 are legs. What they want to know?"},{"Start":"00:38.220 ","End":"00:40.350","Text":"The angles of this triangle."},{"Start":"00:40.350 ","End":"00:43.040","Text":"Well, we know 1 of them already is 90 degrees,"},{"Start":"00:43.040 ","End":"00:44.855","Text":"so we have to find this and this."},{"Start":"00:44.855 ","End":"00:48.185","Text":"Let me mark some things on the diagram."},{"Start":"00:48.185 ","End":"00:50.110","Text":"This is an important point,"},{"Start":"00:50.110 ","End":"00:52.100","Text":"and this is an important point."},{"Start":"00:52.100 ","End":"00:54.875","Text":"This is a tangent."},{"Start":"00:54.875 ","End":"00:58.720","Text":"In other words, the leg here is tangent to the circle at this point,"},{"Start":"00:58.720 ","End":"01:01.415","Text":"and at this point this leg is tangent."},{"Start":"01:01.415 ","End":"01:04.300","Text":"Let\u0027s call this M, this N,"},{"Start":"01:04.300 ","End":"01:06.775","Text":"and they marked the center for us."},{"Start":"01:06.775 ","End":"01:10.985","Text":"Nice. We have M, N, and O."},{"Start":"01:10.985 ","End":"01:19.339","Text":"Now, notice that if I connect O to N,"},{"Start":"01:19.339 ","End":"01:21.415","Text":"then because this is tangent,"},{"Start":"01:21.415 ","End":"01:23.185","Text":"this is 90 degrees."},{"Start":"01:23.185 ","End":"01:26.550","Text":"Similarly, if I join O to M,"},{"Start":"01:26.550 ","End":"01:29.715","Text":"then I also get 90 degrees."},{"Start":"01:29.715 ","End":"01:32.920","Text":"What\u0027s more, this is a radius because it goes from"},{"Start":"01:32.920 ","End":"01:37.190","Text":"the center to the circumference, so that\u0027s R,"},{"Start":"01:37.190 ","End":"01:44.160","Text":"and this would be R. This thing is a square,"},{"Start":"01:44.160 ","End":"01:46.770","Text":"because all are 90 degrees."},{"Start":"01:46.770 ","End":"01:48.735","Text":"Well, there are 3 90 degrees,"},{"Start":"01:48.735 ","End":"01:52.930","Text":"and that\u0027s enough for the fourth one to be 90 degrees also in fact,"},{"Start":"01:52.930 ","End":"01:56.245","Text":"because the sum of the angles is 360."},{"Start":"01:56.245 ","End":"01:58.390","Text":"This is a square. In particular,"},{"Start":"01:58.390 ","End":"02:01.870","Text":"it\u0027s a parallelogram, so opposite sides are equal."},{"Start":"02:01.870 ","End":"02:05.770","Text":"This is R and this is R. Just from A to N,"},{"Start":"02:05.770 ","End":"02:13.485","Text":"just from A to M. Then there\u0027s 2 other quantities that we don\u0027t know."},{"Start":"02:13.485 ","End":"02:17.355","Text":"That\u0027s this bit here and this bit here."},{"Start":"02:17.355 ","End":"02:23.970","Text":"Let me call them a and b."},{"Start":"02:23.970 ","End":"02:28.020","Text":"As for the angles that we have to find,"},{"Start":"02:32.770 ","End":"02:37.190","Text":"well, this one I\u0027ll call my independent variable."},{"Start":"02:37.190 ","End":"02:39.445","Text":"I\u0027ll call this one x."},{"Start":"02:39.445 ","End":"02:41.370","Text":"Once I\u0027ve found x,"},{"Start":"02:41.370 ","End":"02:45.930","Text":"then I can get this one because this one will be,"},{"Start":"02:45.930 ","End":"02:48.120","Text":"just no room to write it here,"},{"Start":"02:48.120 ","End":"02:49.950","Text":"it\u0027ll be 90 minus x."},{"Start":"02:49.950 ","End":"02:53.010","Text":"Because here I have 90 and these 2 are therefor complementary"},{"Start":"02:53.010 ","End":"02:57.600","Text":"and that\u0027s 90 minus x. X will be the independent variable."},{"Start":"02:57.600 ","End":"03:00.680","Text":"What I want to do is express the sum of"},{"Start":"03:00.680 ","End":"03:04.955","Text":"its legs where it says least or minimum or maximum or greatest,"},{"Start":"03:04.955 ","End":"03:09.080","Text":"that\u0027s the thing I want to take as my dependent variable."},{"Start":"03:09.080 ","End":"03:14.735","Text":"The sum of the legs, I\u0027ll call that y and I\u0027ll find y as a function of x."},{"Start":"03:14.735 ","End":"03:20.345","Text":"Well, the first thing I can do is I can naively write that y,"},{"Start":"03:20.345 ","End":"03:24.580","Text":"the sum of the legs, is a plus R plus R plus b."},{"Start":"03:24.580 ","End":"03:31.605","Text":"Let\u0027s write that as 2R plus a plus b."},{"Start":"03:31.605 ","End":"03:36.370","Text":"See, a, R, R, b."},{"Start":"03:36.740 ","End":"03:40.180","Text":"But I have to figure out what a and b are."},{"Start":"03:40.180 ","End":"03:42.185","Text":"The only things I know are R,"},{"Start":"03:42.185 ","End":"03:47.500","Text":"and I know x in the sense that x is my independent variable."},{"Start":"03:47.500 ","End":"03:52.585","Text":"I want to put a and b in terms of x and R. Let\u0027s see."},{"Start":"03:52.585 ","End":"03:56.470","Text":"I\u0027m going to use trigonometry, of course."},{"Start":"03:56.470 ","End":"04:01.824","Text":"I note that this x is the same."},{"Start":"04:01.824 ","End":"04:04.750","Text":"Let me see, where\u0027s my red? Here it is."},{"Start":"04:04.750 ","End":"04:09.130","Text":"This is also going to be x because"},{"Start":"04:09.130 ","End":"04:16.385","Text":"these 2 lines are parallel and these are therefore corresponding angles."},{"Start":"04:16.385 ","End":"04:18.590","Text":"In each of these triangles,"},{"Start":"04:18.590 ","End":"04:20.330","Text":"I can look at the tangent."},{"Start":"04:20.330 ","End":"04:23.665","Text":"The tangent here will relate R and b,"},{"Start":"04:23.665 ","End":"04:31.775","Text":"and here I\u0027ll relate a and R. What I get is this,"},{"Start":"04:31.775 ","End":"04:40.875","Text":"that tangent x is a over R, opposite over adjacent."},{"Start":"04:40.875 ","End":"04:44.135","Text":"I know also that from this triangle,"},{"Start":"04:44.135 ","End":"04:49.475","Text":"that tangent x is R over b."},{"Start":"04:49.475 ","End":"04:54.044","Text":"If I isolate a and b,"},{"Start":"04:54.044 ","End":"04:59.485","Text":"what we\u0027ll get is that this is equal to 2R,"},{"Start":"04:59.485 ","End":"05:03.739","Text":"now, a is going to be R tangent x,"},{"Start":"05:04.500 ","End":"05:11.120","Text":"and from here, b is going to be R over tangent x."},{"Start":"05:15.870 ","End":"05:19.505","Text":"If take R outside the brackets,"},{"Start":"05:19.505 ","End":"05:24.220","Text":"just to be tidier, my y,"},{"Start":"05:24.220 ","End":"05:28.890","Text":"my dependent variable that I want to minimize,"},{"Start":"05:28.890 ","End":"05:38.969","Text":"is equal to R times 2 plus tangent x plus 1 over tangent x."},{"Start":"05:38.969 ","End":"05:43.610","Text":"Now that I have y in terms of x,"},{"Start":"05:43.610 ","End":"05:47.570","Text":"I\u0027m going to do the usual differentiate, equate to 0,"},{"Start":"05:47.570 ","End":"05:49.940","Text":"find suspects for extra amount,"},{"Start":"05:49.940 ","End":"05:51.350","Text":"show that it\u0027s a minimum,"},{"Start":"05:51.350 ","End":"05:52.655","Text":"and so on and so on."},{"Start":"05:52.655 ","End":"05:54.490","Text":"Let\u0027s get started."},{"Start":"05:54.490 ","End":"05:57.420","Text":"Y prime is equal,"},{"Start":"05:57.420 ","End":"06:00.520","Text":"I\u0027ll keep R here,"},{"Start":"06:01.490 ","End":"06:04.965","Text":"the constant 2 doesn\u0027t contribute anything,"},{"Start":"06:04.965 ","End":"06:07.515","Text":"the derivative of tangent,"},{"Start":"06:07.515 ","End":"06:14.130","Text":"there\u0027s several variations but I know it is 1 over cosine squared x."},{"Start":"06:14.130 ","End":"06:18.810","Text":"This 1 over tangent is actually the cotangent."},{"Start":"06:18.810 ","End":"06:24.930","Text":"Just make a note of that, that is equal to the cotangent, 1 over the tangent."},{"Start":"06:24.930 ","End":"06:30.800","Text":"The derivative of that is minus 1 over sine squared x."},{"Start":"06:30.800 ","End":"06:32.340","Text":"This is one of those immediate,"},{"Start":"06:32.340 ","End":"06:34.380","Text":"well, you can compute it,"},{"Start":"06:34.380 ","End":"06:38.780","Text":"but it\u0027s an immediate one that we remember."},{"Start":"06:38.780 ","End":"06:44.705","Text":"Otherwise you would have to mess with sine and cosine and you\u0027d get the same answer."},{"Start":"06:44.705 ","End":"06:48.710","Text":"Now we have the derivative."},{"Start":"06:48.710 ","End":"06:52.460","Text":"Now, as usual, in extra problems,"},{"Start":"06:52.460 ","End":"06:57.535","Text":"particular minimum problems, we set y prime equal to 0."},{"Start":"06:57.535 ","End":"07:00.400","Text":"What does that give us?"},{"Start":"07:00.920 ","End":"07:05.445","Text":"Well, R is just a constant, it cancels out."},{"Start":"07:05.445 ","End":"07:16.065","Text":"We get that 1 over cosine squared x minus 1 over sine squared x,"},{"Start":"07:16.065 ","End":"07:19.625","Text":"I\u0027m going to say equals 0, but instead,"},{"Start":"07:19.625 ","End":"07:23.300","Text":"why don\u0027t I just put the equals here because I brought it to the other side."},{"Start":"07:23.300 ","End":"07:27.035","Text":"How about that? Let\u0027s cross multiply."},{"Start":"07:27.035 ","End":"07:36.980","Text":"We get that sine squared x times 1 equals cosine squared x times 1."},{"Start":"07:36.980 ","End":"07:42.125","Text":"If I divide by cosine squared x,"},{"Start":"07:42.125 ","End":"07:46.925","Text":"I get tangent squared x equals 1. Follow that?"},{"Start":"07:46.925 ","End":"07:49.415","Text":"Sine over cosine is tangent,"},{"Start":"07:49.415 ","End":"07:53.105","Text":"so sine squared over cosine squared is tangent squared."},{"Start":"07:53.105 ","End":"08:00.425","Text":"Now, the angle x is less than 90 degrees."},{"Start":"08:00.425 ","End":"08:03.920","Text":"I mean, for sure x is less than 90 degrees because"},{"Start":"08:03.920 ","End":"08:07.235","Text":"x and this one together make 90 degrees."},{"Start":"08:07.235 ","End":"08:10.680","Text":"Which means that we\u0027re in the first quadrant,"},{"Start":"08:10.680 ","End":"08:13.425","Text":"so the tangent has to be positive."},{"Start":"08:13.425 ","End":"08:16.200","Text":"Tangent x equals 1."},{"Start":"08:16.200 ","End":"08:17.840","Text":"In the first quadrant,"},{"Start":"08:17.840 ","End":"08:22.095","Text":"the angle whose tangent is 1 is arctangent of 1."},{"Start":"08:22.095 ","End":"08:24.740","Text":"If you look it up in your calculator,"},{"Start":"08:24.740 ","End":"08:26.960","Text":"if you do it in degrees,"},{"Start":"08:26.960 ","End":"08:30.800","Text":"then you\u0027ll get 45-degree but we\u0027re working in radians"},{"Start":"08:30.800 ","End":"08:37.200","Text":"so we get Pi over 4 radians,"},{"Start":"08:37.200 ","End":"08:41.050","Text":"sometimes written with little c for circular measure."},{"Start":"08:41.860 ","End":"08:46.560","Text":"We\u0027ve almost completed the question."},{"Start":"08:46.780 ","End":"08:49.055","Text":"I can write the answer,"},{"Start":"08:49.055 ","End":"08:51.350","Text":"though I haven\u0027t shown yet that it is a minimum,"},{"Start":"08:51.350 ","End":"08:56.750","Text":"because what I want is x here and 90 minus x here."},{"Start":"08:56.750 ","End":"09:00.350","Text":"If I want to be pedantic and it said what are the angles,"},{"Start":"09:00.350 ","End":"09:03.065","Text":"well, I can give the 90 degrees also."},{"Start":"09:03.065 ","End":"09:11.520","Text":"What I can say is that the angles for the answer are: 90 degrees,"},{"Start":"09:11.520 ","End":"09:16.570","Text":"45 degrees, and 45 degrees."},{"Start":"09:16.790 ","End":"09:25.500","Text":"All that remains is to show that this is indeed a minimum."},{"Start":"09:29.030 ","End":"09:32.070","Text":"We have 2 basic methods."},{"Start":"09:32.070 ","End":"09:36.970","Text":"Second derivative method or the table method."},{"Start":"09:36.970 ","End":"09:38.545","Text":"I think I\u0027ll go with the table."},{"Start":"09:38.545 ","End":"09:39.850","Text":"If we run into difficulty,"},{"Start":"09:39.850 ","End":"09:43.330","Text":"we can always fall back on the second derivative."},{"Start":"09:43.330 ","End":"09:46.880","Text":"Let me draw a table."},{"Start":"09:48.780 ","End":"09:55.170","Text":"What I\u0027m going to put here is x,"},{"Start":"09:55.170 ","End":"09:58.875","Text":"y prime, and y."},{"Start":"09:58.875 ","End":"10:04.490","Text":"Our suspect, our independent variable is Pi over 4."},{"Start":"10:04.490 ","End":"10:08.980","Text":"But there\u0027s no harm working in degrees at this point so let me put"},{"Start":"10:08.980 ","End":"10:14.865","Text":"45 degrees because I\u0027m only computing what is y prime."},{"Start":"10:14.865 ","End":"10:19.400","Text":"In fact, I\u0027ll be computing this as the R,"},{"Start":"10:19.400 ","End":"10:20.420","Text":"it doesn\u0027t really matter."},{"Start":"10:20.420 ","End":"10:22.355","Text":"Well, it\u0027s optional."},{"Start":"10:22.355 ","End":"10:27.530","Text":"Positive constant is not going to change the sign of the answer plus or minus."},{"Start":"10:27.530 ","End":"10:31.640","Text":"We want to see something on the left and something on the right."},{"Start":"10:31.640 ","End":"10:33.350","Text":"Let\u0027s take some nice angles,"},{"Start":"10:33.350 ","End":"10:36.800","Text":"the 30 degrees and 60 degrees."},{"Start":"10:36.800 ","End":"10:40.400","Text":"Here the derivative of course is 0 because that\u0027s"},{"Start":"10:40.400 ","End":"10:43.610","Text":"how we got to the 45 degrees in the first place,"},{"Start":"10:43.610 ","End":"10:46.070","Text":"is we let y prime is 0."},{"Start":"10:46.070 ","End":"10:48.845","Text":"Now, what about 30 degrees?"},{"Start":"10:48.845 ","End":"10:51.950","Text":"I\u0027m going to substitute in this expression."},{"Start":"10:51.950 ","End":"10:58.265","Text":"Let me just make a note that some of the familiar angles, I\u0027ll write them here."},{"Start":"10:58.265 ","End":"11:02.120","Text":"The sine of 30 degrees is 1/2,"},{"Start":"11:02.120 ","End":"11:06.050","Text":"and the cosine of 60 degrees is 1/2."},{"Start":"11:06.050 ","End":"11:12.605","Text":"The sine of 60 degrees is the square root of 3 over 2."},{"Start":"11:12.605 ","End":"11:15.290","Text":"The cosine of 30 degrees"},{"Start":"11:15.290 ","End":"11:24.180","Text":"is also"},{"Start":"11:24.180 ","End":"11:25.620","Text":"the square root of 3 over 2."},{"Start":"11:25.620 ","End":"11:30.795","Text":"Yeah, complementary angles; sine of one is the cosine of the other."},{"Start":"11:30.795 ","End":"11:34.520","Text":"Very well, let\u0027s see if we can do this mentally we\u0027re"},{"Start":"11:34.520 ","End":"11:39.890","Text":"substituting 30 degrees in here. Let\u0027s see."},{"Start":"11:39.890 ","End":"11:50.530","Text":"The cosine is root 3 over 2 and the sine is 1/2. I better do it."},{"Start":"11:52.190 ","End":"11:59.880","Text":"Y prime for 30 degrees is,"},{"Start":"11:59.880 ","End":"12:03.315","Text":"well, I\u0027ll leave the R, doesn\u0027t matter,"},{"Start":"12:03.315 ","End":"12:07.680","Text":"1 over cosine of 30,"},{"Start":"12:07.680 ","End":"12:13.845","Text":"root 3 over 2 squared minus,"},{"Start":"12:13.845 ","End":"12:16.875","Text":"sine of 30 degrees is 1/2,"},{"Start":"12:16.875 ","End":"12:22.350","Text":"minus 1 over 1/2 squared,"},{"Start":"12:22.350 ","End":"12:28.740","Text":"which is equal to R times, let\u0027s see."},{"Start":"12:28.740 ","End":"12:31.245","Text":"Root 3 over 2 squared is 3 over 4,"},{"Start":"12:31.245 ","End":"12:34.385","Text":"so this comes out to be 4 over 3."},{"Start":"12:34.385 ","End":"12:36.650","Text":"1/2 squared is 1/4,"},{"Start":"12:36.650 ","End":"12:40.445","Text":"so this comes out to be minus 4."},{"Start":"12:40.445 ","End":"12:45.530","Text":"All I need to know is that this is negative because this minus this is negative,"},{"Start":"12:45.530 ","End":"12:48.334","Text":"R is positive, which is less than 0,"},{"Start":"12:48.334 ","End":"12:52.560","Text":"which means that we have a negative here."},{"Start":"12:53.150 ","End":"12:56.240","Text":"If I put in 60 degrees,"},{"Start":"12:56.240 ","End":"12:58.985","Text":"you\u0027ll get just exactly the opposite."},{"Start":"12:58.985 ","End":"13:01.730","Text":"Everywhere you had root 3 over 2,"},{"Start":"13:01.730 ","End":"13:07.790","Text":"you\u0027ll have 1/2 so we\u0027ll get these 2 in reverse order,"},{"Start":"13:07.790 ","End":"13:10.175","Text":"so it\u0027s going to be positive,"},{"Start":"13:10.175 ","End":"13:17.770","Text":"which means that the function over here is decreasing and here it\u0027s increasing."},{"Start":"13:17.770 ","End":"13:19.400","Text":"Run the 45 degrees,"},{"Start":"13:19.400 ","End":"13:21.065","Text":"we have one of these shapes,"},{"Start":"13:21.065 ","End":"13:26.525","Text":"which means that it is a minimum and that\u0027s what we were required to show."},{"Start":"13:26.525 ","End":"13:30.694","Text":"We have a minimum and the angles are these,"},{"Start":"13:30.694 ","End":"13:33.180","Text":"and we are done."}],"ID":8569},{"Watched":false,"Name":"Exercise 14","Duration":"10m 55s","ChapterTopicVideoID":8389,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8389.jpeg","UploadDate":"2017-01-28T13:56:54.6770000","DurationForVideoObject":"PT10M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.590","Text":"We have another word problem."},{"Start":"00:01.590 ","End":"00:03.645","Text":"I\u0027ll explain what\u0027s going on here."},{"Start":"00:03.645 ","End":"00:09.975","Text":"We have a circle and we know what its radius is, r. Inside it,"},{"Start":"00:09.975 ","End":"00:14.640","Text":"the triangle circumscribed in such a way that 1 of"},{"Start":"00:14.640 ","End":"00:20.325","Text":"the angles is 72 degrees or in radians 2 Pi over 5."},{"Start":"00:20.325 ","End":"00:27.460","Text":"What we have to do is to find the triangle as described with 72 degrees in it,"},{"Start":"00:27.460 ","End":"00:31.985","Text":"such that it has the largest possible circumference from all these triangles."},{"Start":"00:31.985 ","End":"00:34.055","Text":"When we see the word largest,"},{"Start":"00:34.055 ","End":"00:36.395","Text":"then we know it\u0027s a maximum problem."},{"Start":"00:36.395 ","End":"00:38.535","Text":"I see the word circumference,"},{"Start":"00:38.535 ","End":"00:43.120","Text":"I know that\u0027s something that\u0027s going to be my dependent variable."},{"Start":"00:43.120 ","End":"00:46.880","Text":"Let\u0027s mark and label stuff in the picture."},{"Start":"00:46.880 ","End":"00:48.470","Text":"Let\u0027s start with the angles."},{"Start":"00:48.470 ","End":"00:51.800","Text":"Here I have a 72 degree angle,"},{"Start":"00:51.800 ","End":"00:54.200","Text":"and we have 2 other angles,"},{"Start":"00:54.200 ","End":"00:58.995","Text":"I\u0027ll call them Alpha and Beta."},{"Start":"00:58.995 ","End":"01:01.410","Text":"I guess this 1 could be Gamma."},{"Start":"01:01.410 ","End":"01:03.945","Text":"I want to label the sides also."},{"Start":"01:03.945 ","End":"01:06.225","Text":"Opposite Alpha, I have a,"},{"Start":"01:06.225 ","End":"01:08.355","Text":"opposite Beta I have b,"},{"Start":"01:08.355 ","End":"01:17.660","Text":"opposite Gamma I have c. I want the minimum of a plus b"},{"Start":"01:17.660 ","End":"01:27.730","Text":"plus c in such a way that Gamma subject to Gamma being 72 degrees or 2 Pi over 5."},{"Start":"01:27.730 ","End":"01:32.365","Text":"This a plus b plus c will be my circumference and I\u0027ll call it y."},{"Start":"01:32.365 ","End":"01:35.750","Text":"This is going to be a plus b plus c,"},{"Start":"01:35.750 ","End":"01:38.810","Text":"but I want it in terms of an independent variable."},{"Start":"01:38.810 ","End":"01:44.430","Text":"Let\u0027s say that Alpha will be my independent variable."},{"Start":"01:44.430 ","End":"01:47.125","Text":"Although it\u0027s a plus b plus c,"},{"Start":"01:47.125 ","End":"01:50.320","Text":"I want to write each of these in terms of Alpha."},{"Start":"01:50.320 ","End":"01:52.215","Text":"But there\u0027s a formula,"},{"Start":"01:52.215 ","End":"01:56.675","Text":"a over sine Alpha equals b over sine Beta,"},{"Start":"01:56.675 ","End":"01:59.785","Text":"equals c over sine Gamma."},{"Start":"01:59.785 ","End":"02:02.075","Text":"Sometimes it\u0027s just given in this form,"},{"Start":"02:02.075 ","End":"02:03.500","Text":"but in the full form,"},{"Start":"02:03.500 ","End":"02:06.350","Text":"this ratio is known to be 2r,"},{"Start":"02:06.350 ","End":"02:11.944","Text":"where r is the radius of the circumscribing circle."},{"Start":"02:11.944 ","End":"02:18.860","Text":"If you look at it, a is just 2r times sine Alpha."},{"Start":"02:18.860 ","End":"02:24.725","Text":"This is 2r sine Alpha,"},{"Start":"02:24.725 ","End":"02:29.089","Text":"and b is 2r sine Beta,"},{"Start":"02:29.089 ","End":"02:32.140","Text":"and c is 2R sine Gamma."},{"Start":"02:32.140 ","End":"02:35.360","Text":"Now, I know what Beta is,"},{"Start":"02:35.360 ","End":"02:38.495","Text":"because Beta, if I do it in degrees,"},{"Start":"02:38.495 ","End":"02:46.095","Text":"Beta is just 180 minus the sum of the other 2,"},{"Start":"02:46.095 ","End":"02:51.975","Text":"Alpha plus Gamma or Alpha plus 72 degrees."},{"Start":"02:51.975 ","End":"02:54.690","Text":"Gamma, of course, I know what Gamma is,"},{"Start":"02:54.690 ","End":"02:57.210","Text":"it\u0027s in the picture, it\u0027s 72 degrees."},{"Start":"02:57.210 ","End":"03:01.020","Text":"I only need to use Alpha."},{"Start":"03:01.020 ","End":"03:02.750","Text":"I\u0027ll just emphasize this,"},{"Start":"03:02.750 ","End":"03:07.825","Text":"that y is a function of Alpha and r is a parameter like a constant."},{"Start":"03:07.825 ","End":"03:11.610","Text":"I can take the 2r outside the brackets you see,"},{"Start":"03:11.610 ","End":"03:17.480","Text":"so I get sine Alpha plus sine of"},{"Start":"03:17.480 ","End":"03:25.815","Text":"180 degrees minus Alpha plus 72 degrees,"},{"Start":"03:25.815 ","End":"03:33.405","Text":"and then plus sine Gamma is sine of 72 degrees."},{"Start":"03:33.405 ","End":"03:38.080","Text":"I really should write it in terms of radians because when you differentiate,"},{"Start":"03:38.080 ","End":"03:42.090","Text":"trigonometric functions, the formula is only hold for radians."},{"Start":"03:42.090 ","End":"03:44.475","Text":"So y of Alpha is 2r."},{"Start":"03:44.475 ","End":"03:47.790","Text":"This formula was for degrees,"},{"Start":"03:47.790 ","End":"03:50.180","Text":"and it\u0027s not quite what we wanted to know,"},{"Start":"03:50.180 ","End":"03:51.230","Text":"I\u0027ll write it for radians,"},{"Start":"03:51.230 ","End":"03:54.875","Text":"2r sine of Alpha."},{"Start":"03:54.875 ","End":"03:58.190","Text":"Now, the sine of a 180 minus an angle,"},{"Start":"03:58.190 ","End":"03:59.555","Text":"and I\u0027ll write that sine of"},{"Start":"03:59.555 ","End":"04:05.220","Text":"a 180 degrees minus some angle Theta is the same as sine of Theta."},{"Start":"04:05.220 ","End":"04:08.449","Text":"This are supplementary angles and they have the same sign."},{"Start":"04:08.449 ","End":"04:14.810","Text":"This is the same as sine without a 180 minus Alpha plus 2 Pi"},{"Start":"04:14.810 ","End":"04:22.230","Text":"over 5 and sine of 2 Pi over 5."},{"Start":"04:22.230 ","End":"04:25.940","Text":"At this point, we\u0027re now ready for"},{"Start":"04:25.940 ","End":"04:29.750","Text":"the calculus because we have a function y in terms of Alpha,"},{"Start":"04:29.750 ","End":"04:36.824","Text":"and we want the maximum for y."},{"Start":"04:36.824 ","End":"04:39.920","Text":"Well, either way, if it were looking for an extra moment,"},{"Start":"04:39.920 ","End":"04:42.970","Text":"we differentiate and we set to 0."},{"Start":"04:42.970 ","End":"04:46.890","Text":"Y prime of Alpha equals,"},{"Start":"04:46.890 ","End":"04:48.915","Text":"the 2r can stay,"},{"Start":"04:48.915 ","End":"04:55.650","Text":"and what I get is cosine of Alpha plus"},{"Start":"04:55.650 ","End":"05:03.800","Text":"cosine of Alpha plus 2 Pi over 5 times inner derivative,"},{"Start":"05:03.800 ","End":"05:07.460","Text":"which is 1, and this gives me nothing."},{"Start":"05:07.460 ","End":"05:09.299","Text":"I\u0027m going to look for extra-mum,"},{"Start":"05:09.299 ","End":"05:10.875","Text":"specifically for a maximum."},{"Start":"05:10.875 ","End":"05:13.220","Text":"I\u0027m going to set y prime equals 0,"},{"Start":"05:13.220 ","End":"05:14.690","Text":"but this will give me,"},{"Start":"05:14.690 ","End":"05:16.400","Text":"the 2r I can throw out,"},{"Start":"05:16.400 ","End":"05:19.205","Text":"so I get the equation in Alpha,"},{"Start":"05:19.205 ","End":"05:29.015","Text":"cosine Alpha plus cosine of Alpha plus 2 Pi over 5 is equal to 0."},{"Start":"05:29.015 ","End":"05:34.595","Text":"The formula I need is the formula for the sum of 2 cosines."},{"Start":"05:34.595 ","End":"05:38.045","Text":"If I use this formula here,"},{"Start":"05:38.045 ","End":"05:43.055","Text":"what I\u0027ll get is that twice cosine,"},{"Start":"05:43.055 ","End":"05:50.450","Text":"this plus this over 2 will be Alpha plus Alpha plus 2 Pi over 5, all over 2."},{"Start":"05:50.450 ","End":"05:53.945","Text":"If you check it, just get Alpha plus Pi over 5."},{"Start":"05:53.945 ","End":"05:55.460","Text":"In case you\u0027re wondering,"},{"Start":"05:55.460 ","End":"05:57.735","Text":"this is 36 degrees,"},{"Start":"05:57.735 ","End":"06:01.545","Text":"times cosine of Alpha minus Beta over 2,"},{"Start":"06:01.545 ","End":"06:04.350","Text":"so it\u0027s this minus this over 2,"},{"Start":"06:04.350 ","End":"06:11.610","Text":"so it\u0027s cosine of minus Pi over 5,"},{"Start":"06:11.610 ","End":"06:15.675","Text":"and that is equal to 0."},{"Start":"06:15.675 ","End":"06:19.880","Text":"Now, from all of this, this is not 0."},{"Start":"06:19.880 ","End":"06:22.700","Text":"It\u0027s the cosine of minus 36 degrees,"},{"Start":"06:22.700 ","End":"06:24.455","Text":"it\u0027s some number it\u0027s not 0,"},{"Start":"06:24.455 ","End":"06:25.970","Text":"the 2 is not 0."},{"Start":"06:25.970 ","End":"06:28.520","Text":"The only thing that can be 0 is this bit."},{"Start":"06:28.520 ","End":"06:34.295","Text":"We get that the cosine of Alpha plus Pi over 5,"},{"Start":"06:34.295 ","End":"06:37.460","Text":"this is the bit that\u0027s equal to 0."},{"Start":"06:37.460 ","End":"06:44.509","Text":"There\u0027s only 1 angle between 0 and 180 degrees that has a cosine of 0,"},{"Start":"06:44.509 ","End":"06:46.310","Text":"and that is 90 degrees,"},{"Start":"06:46.310 ","End":"06:49.190","Text":"in other words, Pi over 2."},{"Start":"06:49.190 ","End":"06:55.939","Text":"Alpha plus Pi over 5 is the arc cosine of 0,"},{"Start":"06:55.939 ","End":"06:58.980","Text":"which is Pi over 2."},{"Start":"06:59.270 ","End":"07:06.935","Text":"That gives us that Alpha is equal to 3 Pi over 10."},{"Start":"07:06.935 ","End":"07:11.750","Text":"I\u0027ll just write a side that that is 54 degrees."},{"Start":"07:11.750 ","End":"07:14.200","Text":"The other angle is Beta,"},{"Start":"07:14.200 ","End":"07:17.175","Text":"and Beta is equal to,"},{"Start":"07:17.175 ","End":"07:18.885","Text":"I do it in degrees."},{"Start":"07:18.885 ","End":"07:21.735","Text":"If Alpha is 54 degrees,"},{"Start":"07:21.735 ","End":"07:27.900","Text":"54 plus 72 is a 126."},{"Start":"07:27.900 ","End":"07:30.435","Text":"If I subtract this from a 180,"},{"Start":"07:30.435 ","End":"07:34.274","Text":"I get 54 degrees also."},{"Start":"07:34.274 ","End":"07:39.945","Text":"Also 3 Pi over 10, ie, 54 degrees."},{"Start":"07:39.945 ","End":"07:43.320","Text":"We have all the angles, 54, 54,"},{"Start":"07:43.320 ","End":"07:48.770","Text":"72 or 3 Pi over 23 Pi over 10, 2 Pi over 5."},{"Start":"07:48.770 ","End":"07:55.375","Text":"The only thing that remains is to show that we indeed have a maximum."},{"Start":"07:55.375 ","End":"07:59.510","Text":"I\u0027ll do it by usual table that I do."},{"Start":"07:59.510 ","End":"08:03.965","Text":"I have 3 rows in the table,"},{"Start":"08:03.965 ","End":"08:06.200","Text":"I have, normally it\u0027s x,"},{"Start":"08:06.200 ","End":"08:07.600","Text":"but here it\u0027s Alpha,,"},{"Start":"08:07.600 ","End":"08:11.165","Text":"I have y prime and I have y."},{"Start":"08:11.165 ","End":"08:18.205","Text":"The only suspect for extra-mum or minimum is 2 Pi over 5."},{"Start":"08:18.205 ","End":"08:21.650","Text":"Well, I\u0027ll write the degrees,"},{"Start":"08:21.650 ","End":"08:23.510","Text":"I think we\u0027re going to have to just work in degrees,"},{"Start":"08:23.510 ","End":"08:26.570","Text":"I\u0027m not used to working in radians myself."},{"Start":"08:26.570 ","End":"08:29.480","Text":"So 72 degrees at this point,"},{"Start":"08:29.480 ","End":"08:31.570","Text":"y prime is 0,"},{"Start":"08:31.570 ","End":"08:33.455","Text":"that\u0027s how we got to this."},{"Start":"08:33.455 ","End":"08:35.660","Text":"But what is y prime?"},{"Start":"08:35.660 ","End":"08:39.220","Text":"In actual fact, we just dropped the 2r,"},{"Start":"08:39.220 ","End":"08:41.300","Text":"and other than that, we kept it the same."},{"Start":"08:41.300 ","End":"08:49.370","Text":"If I write here that y prime is 2r times this up to here before the equals,"},{"Start":"08:49.370 ","End":"08:51.905","Text":"then this is y prime."},{"Start":"08:51.905 ","End":"08:55.670","Text":"Now, what we need is not y prime,"},{"Start":"08:55.670 ","End":"08:58.565","Text":"but only it\u0027s sign, plus or minus."},{"Start":"08:58.565 ","End":"09:02.640","Text":"This 2r with this 2 is a plus."},{"Start":"09:02.640 ","End":"09:06.780","Text":"The cosine of even a negative angle,"},{"Start":"09:06.780 ","End":"09:11.590","Text":"the cosine is positive between minus Pi over 2 and Pi over 2,"},{"Start":"09:11.590 ","End":"09:13.300","Text":"between minus 90 and 90,"},{"Start":"09:13.300 ","End":"09:14.590","Text":"the cosine is positive,"},{"Start":"09:14.590 ","End":"09:16.114","Text":"so this is positive."},{"Start":"09:16.114 ","End":"09:19.540","Text":"Really the only thing that matters that\u0027s going to give"},{"Start":"09:19.540 ","End":"09:23.050","Text":"us the sine is this middle piece here."},{"Start":"09:23.050 ","End":"09:26.065","Text":"You know what, I\u0027ll just write the whole thing in degrees."},{"Start":"09:26.065 ","End":"09:31.215","Text":"We\u0027re talking about this 54 degree angle."},{"Start":"09:31.215 ","End":"09:34.670","Text":"Let\u0027s take something a bit before and a bit after,"},{"Start":"09:34.670 ","End":"09:39.520","Text":"let\u0027s take 45 degrees and above that we\u0027ll take 60 degrees."},{"Start":"09:39.520 ","End":"09:42.880","Text":"Let\u0027s see, we\u0027re just substituting in this bit here,,"},{"Start":"09:42.880 ","End":"09:44.590","Text":"I\u0027ll even double underline it."},{"Start":"09:44.590 ","End":"09:46.460","Text":"If it\u0027s 54 degrees,"},{"Start":"09:46.460 ","End":"09:51.934","Text":"54 plus 36, just as a check is 90, cosine is 0."},{"Start":"09:51.934 ","End":"09:54.200","Text":"If I plug in 60 degrees,"},{"Start":"09:54.200 ","End":"09:57.950","Text":"I have cosine of 60 plus 36."},{"Start":"09:57.950 ","End":"10:01.630","Text":"It\u0027s cosine of 96 degrees."},{"Start":"10:01.630 ","End":"10:08.705","Text":"The cosine of 96 degrees is negative because in the second quadrant,"},{"Start":"10:08.705 ","End":"10:10.750","Text":"the cosine is negative."},{"Start":"10:10.750 ","End":"10:13.985","Text":"If we put in 45 degrees,"},{"Start":"10:13.985 ","End":"10:23.075","Text":"then we have 45 plus 36 is 81 degrees,"},{"Start":"10:23.075 ","End":"10:29.470","Text":"and cosine of 87 degrees because 81 is in the first quadrant, is positive."},{"Start":"10:29.470 ","End":"10:32.420","Text":"Here the function is increasing,"},{"Start":"10:32.420 ","End":"10:34.550","Text":"here the function is decreasing,"},{"Start":"10:34.550 ","End":"10:36.785","Text":"so we get something like this,"},{"Start":"10:36.785 ","End":"10:39.650","Text":"which is indeed a maximum,"},{"Start":"10:39.650 ","End":"10:44.915","Text":"which is what we wanted and we are done."},{"Start":"10:44.915 ","End":"10:51.320","Text":"We found that there is a maximum and the angles are 54 degrees and 54 degrees,"},{"Start":"10:51.320 ","End":"10:55.740","Text":"those of the other angles besides the original 72 degrees."}],"ID":8570},{"Watched":false,"Name":"Exercise 15","Duration":"7m 17s","ChapterTopicVideoID":8390,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8390.jpeg","UploadDate":"2017-01-28T13:58:30.8470000","DurationForVideoObject":"PT7M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Let me explain what\u0027s going on in this exercise."},{"Start":"00:03.300 ","End":"00:05.325","Text":"There\u0027s a tower so-called,"},{"Start":"00:05.325 ","End":"00:07.890","Text":"built from 2 cubes as you can see in the picture,"},{"Start":"00:07.890 ","End":"00:10.575","Text":"1 placed on top of the other."},{"Start":"00:10.575 ","End":"00:15.225","Text":"What we\u0027re told is that the height of the tower is 8 centimeters."},{"Start":"00:15.225 ","End":"00:17.700","Text":"I\u0027ll explain what that means."},{"Start":"00:17.700 ","End":"00:21.900","Text":"What we\u0027re asked is we have to figure out the length of the side of"},{"Start":"00:21.900 ","End":"00:26.235","Text":"the lower cube in order that the volume of the tower,"},{"Start":"00:26.235 ","End":"00:29.460","Text":"which means the volume of the lower 1 plus the lower of the upper 1,"},{"Start":"00:29.460 ","End":"00:31.485","Text":"together to be minimal."},{"Start":"00:31.485 ","End":"00:35.665","Text":"Let me do a bit of shading to make it look nicer."},{"Start":"00:35.665 ","End":"00:37.850","Text":"Now, these are cubes,"},{"Start":"00:37.850 ","End":"00:39.905","Text":"so all the sides are the same."},{"Start":"00:39.905 ","End":"00:41.660","Text":"Which means that in case of this cube,"},{"Start":"00:41.660 ","End":"00:44.345","Text":"for example, this side which is,"},{"Start":"00:44.345 ","End":"00:48.140","Text":"I don\u0027t know the width, and this side which is the depth,"},{"Start":"00:48.140 ","End":"00:49.880","Text":"and this side which is the height,"},{"Start":"00:49.880 ","End":"00:51.275","Text":"they\u0027re all the same."},{"Start":"00:51.275 ","End":"00:53.255","Text":"Similarly for the other 1,"},{"Start":"00:53.255 ","End":"00:57.410","Text":"this dimension is equal to this dimension,"},{"Start":"00:57.410 ","End":"00:58.610","Text":"is equal to this dimension,"},{"Start":"00:58.610 ","End":"01:01.135","Text":"and everything\u0027s 90 degrees."},{"Start":"01:01.135 ","End":"01:04.070","Text":"What about this height?"},{"Start":"01:04.070 ","End":"01:09.260","Text":"The height of the tower is just the height of this plus the height of this."},{"Start":"01:09.260 ","End":"01:13.415","Text":"Let\u0027s start giving some names for the lengths."},{"Start":"01:13.415 ","End":"01:23.585","Text":"Let\u0027s say that the length of the lower cube and all the edges is going to be x."},{"Start":"01:23.585 ","End":"01:27.230","Text":"The reason I\u0027m choosing for the lower cube is that we\u0027re asked"},{"Start":"01:27.230 ","End":"01:31.340","Text":"what is the length of the side of the lower cube."},{"Start":"01:31.340 ","End":"01:34.370","Text":"I don\u0027t have to use this as my independent variable,"},{"Start":"01:34.370 ","End":"01:36.440","Text":"but I don\u0027t see why I shouldn\u0027t."},{"Start":"01:36.440 ","End":"01:41.295","Text":"I know that the height is 8 centimeters."},{"Start":"01:41.295 ","End":"01:44.060","Text":"The height, which would be,"},{"Start":"01:44.060 ","End":"01:50.830","Text":"say this would be a height and this would be a height."},{"Start":"01:50.830 ","End":"01:55.170","Text":"I know that this plus this is 8."},{"Start":"01:55.170 ","End":"01:57.420","Text":"This I know is x."},{"Start":"01:57.420 ","End":"02:00.240","Text":"If I say that x plus,"},{"Start":"02:00.240 ","End":"02:01.370","Text":"and I don\u0027t know what this is,"},{"Start":"02:01.370 ","End":"02:03.680","Text":"plus something, I don\u0027t want to use letters,"},{"Start":"02:03.680 ","End":"02:06.350","Text":"let\u0027s put question mark, is equal to 8,"},{"Start":"02:06.350 ","End":"02:10.610","Text":"then that question mark is 8 minus x."},{"Start":"02:10.610 ","End":"02:19.835","Text":"In other words, the side of the upper cube is 8 minus x,"},{"Start":"02:19.835 ","End":"02:21.725","Text":"and so is this, and so is this."},{"Start":"02:21.725 ","End":"02:23.000","Text":"All the sides here x,"},{"Start":"02:23.000 ","End":"02:24.560","Text":"all the sides here 8 minus x."},{"Start":"02:24.560 ","End":"02:26.540","Text":"That way, the height will be 8,"},{"Start":"02:26.540 ","End":"02:30.770","Text":"and I\u0027ve only used 1 variable so far, which is x."},{"Start":"02:30.770 ","End":"02:37.220","Text":"Now the thing that\u0027s going to be minimal is the volume of the tower."},{"Start":"02:37.220 ","End":"02:44.030","Text":"I\u0027d like to get the volume expressed in terms of the independent variable x,"},{"Start":"02:44.030 ","End":"02:48.215","Text":"and then I can use calculus to find the minimum."},{"Start":"02:48.215 ","End":"02:51.350","Text":"The volume which I\u0027ll call y,"},{"Start":"02:51.350 ","End":"02:54.390","Text":"just the volume of the lower cube plus the volume of the upper cube."},{"Start":"02:54.390 ","End":"02:57.770","Text":"The volume of the lower cube is x cubed."},{"Start":"02:57.770 ","End":"02:59.510","Text":"It\u0027s x times x times x,"},{"Start":"02:59.510 ","End":"03:01.040","Text":"that\u0027s what it is for a cube,"},{"Start":"03:01.040 ","End":"03:05.990","Text":"and the volume of the upper cube is its side cubed,"},{"Start":"03:05.990 ","End":"03:09.245","Text":"which is 8 minus x cubed."},{"Start":"03:09.245 ","End":"03:15.835","Text":"Now I have a formula for the volume in terms of the side of the lower cube x,"},{"Start":"03:15.835 ","End":"03:23.180","Text":"and then all I need is calculus to find what x is in order that this sum is minimum,"},{"Start":"03:23.180 ","End":"03:26.170","Text":"so we want minimum of y."},{"Start":"03:26.170 ","End":"03:30.500","Text":"We differentiate, we said equal to 0, we find suspects,"},{"Start":"03:30.500 ","End":"03:34.820","Text":"and then we determine whether this gives minimum or maximum,"},{"Start":"03:34.820 ","End":"03:36.260","Text":"and that\u0027s our usual techniques."},{"Start":"03:36.260 ","End":"03:38.180","Text":"Let\u0027s go ahead and do it."},{"Start":"03:38.180 ","End":"03:46.545","Text":"Y prime is equal to 3x squared plus,"},{"Start":"03:46.545 ","End":"03:48.735","Text":"now here I have 3,"},{"Start":"03:48.735 ","End":"03:51.990","Text":"8 minus x squared,"},{"Start":"03:51.990 ","End":"03:56.540","Text":"but there\u0027s an inner derivative which is minus 1."},{"Start":"03:56.540 ","End":"03:59.270","Text":"When we want to find an extremum, in this case,"},{"Start":"03:59.270 ","End":"04:02.960","Text":"a minimum, we set the derivative to be 0."},{"Start":"04:02.960 ","End":"04:06.140","Text":"If y prime is equal to 0,"},{"Start":"04:06.140 ","End":"04:09.560","Text":"then we get that this expression is 0."},{"Start":"04:09.560 ","End":"04:13.445","Text":"Now I\u0027m already going to cancel by 3"},{"Start":"04:13.445 ","End":"04:17.285","Text":"when I said equal to 0 because 3 doesn\u0027t make a difference,"},{"Start":"04:17.285 ","End":"04:19.580","Text":"and then I\u0027m going to get this plus this is 0."},{"Start":"04:19.580 ","End":"04:21.805","Text":"But because of the minus 1,"},{"Start":"04:21.805 ","End":"04:24.855","Text":"I can bring it to the other side."},{"Start":"04:24.855 ","End":"04:31.440","Text":"I get that x squared is equal to 8 minus x squared."},{"Start":"04:31.440 ","End":"04:34.800","Text":"If 2 things squared are equal,"},{"Start":"04:34.800 ","End":"04:39.920","Text":"then 1 is either the same as the other or the negative of the other."},{"Start":"04:39.920 ","End":"04:46.470","Text":"What I\u0027m saying is that x has got to be plus or minus 8 minus x,"},{"Start":"04:46.470 ","End":"04:48.980","Text":"and so I have 2 possibilities."},{"Start":"04:48.980 ","End":"04:52.370","Text":"If I take the plus sign,"},{"Start":"04:52.370 ","End":"04:55.925","Text":"I\u0027ve got x equals 8 minus x,"},{"Start":"04:55.925 ","End":"04:59.435","Text":"which gives me that x is equal to 4."},{"Start":"04:59.435 ","End":"05:01.490","Text":"If I take the minus,"},{"Start":"05:01.490 ","End":"05:05.760","Text":"I get x is equal to x minus 8,"},{"Start":"05:05.760 ","End":"05:07.245","Text":"which is the negative of these things."},{"Start":"05:07.245 ","End":"05:11.030","Text":"That\u0027s impossible, there is no number which is equal to itself minus 8."},{"Start":"05:11.030 ","End":"05:13.190","Text":"I mean, subtract x, we get 0 is minus 8,"},{"Start":"05:13.190 ","End":"05:15.545","Text":"this is no good."},{"Start":"05:15.545 ","End":"05:23.625","Text":"In that case, we already have the value of x and this is equal to 4."},{"Start":"05:23.625 ","End":"05:29.780","Text":"The only question is now whether this really gives a minimum."},{"Start":"05:29.780 ","End":"05:34.270","Text":"Let\u0027s use the table technique."},{"Start":"05:34.270 ","End":"05:37.110","Text":"Here I have x,"},{"Start":"05:37.110 ","End":"05:41.060","Text":"here I have y prime, and here I have y."},{"Start":"05:41.060 ","End":"05:43.925","Text":"Here I put the value of my suspect."},{"Start":"05:43.925 ","End":"05:48.704","Text":"There is only 1 suspect and that is x equals 4,"},{"Start":"05:48.704 ","End":"05:50.045","Text":"and at this point,"},{"Start":"05:50.045 ","End":"05:54.360","Text":"y prime is 0 because that\u0027s how I got it."},{"Start":"05:54.490 ","End":"05:58.489","Text":"What I\u0027m going to do is try another couple of values,"},{"Start":"05:58.489 ","End":"05:59.940","Text":"1 to the left and 1 to the right."},{"Start":"05:59.940 ","End":"06:03.270","Text":"Let\u0027s say, 3 and 5 will do,"},{"Start":"06:03.270 ","End":"06:05.675","Text":"and I plug them into y prime."},{"Start":"06:05.675 ","End":"06:11.330","Text":"Well, I could just rewrite it as 3 times x squared"},{"Start":"06:11.330 ","End":"06:18.600","Text":"minus 8 minus x squared for y prime,"},{"Start":"06:18.600 ","End":"06:22.055","Text":"and the 3 is not going to make any difference, it\u0027s positive."},{"Start":"06:22.055 ","End":"06:23.480","Text":"All I need is the sign."},{"Start":"06:23.480 ","End":"06:25.369","Text":"What I\u0027m saying is that this expression"},{"Start":"06:25.369 ","End":"06:28.610","Text":"underlined will be the expression I\u0027m going to substitute in."},{"Start":"06:28.610 ","End":"06:32.000","Text":"Now just to check that everything\u0027s okay, if I put in 4,"},{"Start":"06:32.000 ","End":"06:33.529","Text":"I get 4 squared,"},{"Start":"06:33.529 ","End":"06:35.930","Text":"minus 8 minus 4 is 4 squared,"},{"Start":"06:35.930 ","End":"06:38.210","Text":"so 4 squared minus 4 squared 0, we\u0027re okay."},{"Start":"06:38.210 ","End":"06:40.115","Text":"Let\u0027s try putting in 3."},{"Start":"06:40.115 ","End":"06:41.915","Text":"We get 3 squared,"},{"Start":"06:41.915 ","End":"06:43.490","Text":"and here we get 5 squared,"},{"Start":"06:43.490 ","End":"06:46.415","Text":"3 squared minus 5 squared, it\u0027s negative."},{"Start":"06:46.415 ","End":"06:49.070","Text":"I don\u0027t need the value if it\u0027s negative."},{"Start":"06:49.070 ","End":"06:50.630","Text":"If I put x is 5,"},{"Start":"06:50.630 ","End":"06:54.790","Text":"I\u0027ve got 5 squared minus 3 squared, so that\u0027s positive."},{"Start":"06:54.790 ","End":"06:58.860","Text":"For less than 4 we are decreasing,"},{"Start":"06:58.860 ","End":"07:01.440","Text":"greater than 4 we are increasing,"},{"Start":"07:01.440 ","End":"07:05.195","Text":"which implies that the shape of the function here is something like this,"},{"Start":"07:05.195 ","End":"07:07.385","Text":"and when it\u0027s like this,"},{"Start":"07:07.385 ","End":"07:10.675","Text":"that means that we have a minimum."},{"Start":"07:10.675 ","End":"07:14.175","Text":"This is indeed a minimum,"},{"Start":"07:14.175 ","End":"07:17.680","Text":"and we are done."}],"ID":8571},{"Watched":false,"Name":"Exercise 16","Duration":"4m 58s","ChapterTopicVideoID":8391,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8391.jpeg","UploadDate":"2017-01-28T14:00:16.3170000","DurationForVideoObject":"PT4M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"Here we have a word problem I\u0027ll explain."},{"Start":"00:02.370 ","End":"00:09.120","Text":"We want to build a box with height y centimeters and the base,"},{"Start":"00:09.120 ","End":"00:12.335","Text":"which is a square with side x centimeters."},{"Start":"00:12.335 ","End":"00:16.025","Text":"Let me just mark those on here."},{"Start":"00:16.025 ","End":"00:21.885","Text":"This height is y and the dimensions of the base are x,"},{"Start":"00:21.885 ","End":"00:24.120","Text":"and here x also and all the way around,"},{"Start":"00:24.120 ","End":"00:26.865","Text":"it\u0027s x throughout its 4 sides."},{"Start":"00:26.865 ","End":"00:30.420","Text":"Now what we\u0027re given is that the circumference of"},{"Start":"00:30.420 ","End":"00:34.995","Text":"each lateral face means that the side face, not the base."},{"Start":"00:34.995 ","End":"00:39.579","Text":"All 4 are identical and the circumference is 12 centimeters."},{"Start":"00:39.579 ","End":"00:44.900","Text":"Perimeter actually, circumference is usually for circles,"},{"Start":"00:44.900 ","End":"00:47.410","Text":"the correct word is perimeter."},{"Start":"00:47.410 ","End":"00:51.140","Text":"Then we have to find what x is going to be with"},{"Start":"00:51.140 ","End":"00:57.395","Text":"these constraints such that the volume is the biggest possible, maximal."},{"Start":"00:57.395 ","End":"01:00.920","Text":"Basically, everything is spelled out for us,"},{"Start":"01:00.920 ","End":"01:03.875","Text":"what the independent and dependent variables are."},{"Start":"01:03.875 ","End":"01:05.960","Text":"Naturally, I would take x to be"},{"Start":"01:05.960 ","End":"01:09.980","Text":"the independent variable unless there\u0027s some good reason to choose something else."},{"Start":"01:09.980 ","End":"01:13.220","Text":"We also know what the dependent variable is,"},{"Start":"01:13.220 ","End":"01:15.810","Text":"that\u0027s going to be the volume."},{"Start":"01:18.110 ","End":"01:20.685","Text":"There are 3 dimensions,"},{"Start":"01:20.685 ","End":"01:23.325","Text":"the product of all 3 is the volume."},{"Start":"01:23.325 ","End":"01:29.824","Text":"The volume is going to be x times x times y."},{"Start":"01:29.824 ","End":"01:32.300","Text":"Now volume is my variable,"},{"Start":"01:32.300 ","End":"01:35.780","Text":"but the independent is x. I want to get y in terms of x."},{"Start":"01:35.780 ","End":"01:39.380","Text":"But I haven\u0027t used the piece of information that the perimeter is 12."},{"Start":"01:39.380 ","End":"01:47.635","Text":"The perimeter is 12 means that 12 is equal to 2x plus 2y,"},{"Start":"01:47.635 ","End":"01:50.290","Text":"or I can write it as twice x plus y."},{"Start":"01:50.290 ","End":"01:53.180","Text":"The reason I did it that way is that where I can divide"},{"Start":"01:53.180 ","End":"01:57.289","Text":"by 2 and say that 6 is equal to x plus y."},{"Start":"01:57.289 ","End":"02:02.285","Text":"Therefore, that y is equal to 6 minus x."},{"Start":"02:02.285 ","End":"02:06.710","Text":"This is good for me because now I can plug that in here and say that"},{"Start":"02:06.710 ","End":"02:11.375","Text":"our volume is equal to x squared from these 2,"},{"Start":"02:11.375 ","End":"02:14.380","Text":"times 6 minus x."},{"Start":"02:14.380 ","End":"02:16.230","Text":"If I open it up,"},{"Start":"02:16.230 ","End":"02:21.555","Text":"it\u0027s 6x squared minus x cubed."},{"Start":"02:21.555 ","End":"02:25.220","Text":"In order to find the maximum,"},{"Start":"02:25.220 ","End":"02:27.540","Text":"we\u0027re going to use calculus methods now."},{"Start":"02:27.540 ","End":"02:30.545","Text":"For extremum, we differentiate set to 0 and so on."},{"Start":"02:30.545 ","End":"02:34.790","Text":"Let\u0027s see, V prime with respect to x is equal"},{"Start":"02:34.790 ","End":"02:42.455","Text":"to 12x minus 3x squared."},{"Start":"02:42.455 ","End":"02:45.710","Text":"I can take 3 out and I can take x out."},{"Start":"02:45.710 ","End":"02:47.660","Text":"I\u0027ve got 3x and what\u0027s left?"},{"Start":"02:47.660 ","End":"02:52.485","Text":"4 minus x."},{"Start":"02:52.485 ","End":"02:54.690","Text":"Now for a derivative,"},{"Start":"02:54.690 ","End":"02:58.389","Text":"we want V prime equals 0."},{"Start":"02:58.460 ","End":"03:02.205","Text":"You can see that we have 2 possibilities,"},{"Start":"03:02.205 ","End":"03:07.445","Text":"either x equals 0 or x equals 4."},{"Start":"03:07.445 ","End":"03:09.860","Text":"But 0 is not a length,"},{"Start":"03:09.860 ","End":"03:12.725","Text":"we have to have strictly positive lengths."},{"Start":"03:12.725 ","End":"03:15.200","Text":"I\u0027m going to rule this 1 out."},{"Start":"03:15.200 ","End":"03:17.905","Text":"We have x equals 4."},{"Start":"03:17.905 ","End":"03:21.590","Text":"Although we weren\u0027t asked to find all the dimensions,"},{"Start":"03:21.590 ","End":"03:23.825","Text":"if we were asked to find the height also,"},{"Start":"03:23.825 ","End":"03:26.990","Text":"I could say that if x equals 4,"},{"Start":"03:26.990 ","End":"03:30.380","Text":"then y equals 2."},{"Start":"03:30.380 ","End":"03:33.110","Text":"Back here, we\u0027ve got the answer,"},{"Start":"03:33.110 ","End":"03:37.990","Text":"but we haven\u0027t verified that it is indeed a maximum."},{"Start":"03:37.990 ","End":"03:40.890","Text":"What we want to do is,"},{"Start":"03:40.890 ","End":"03:42.300","Text":"well there\u0027s 2 main methods,"},{"Start":"03:42.300 ","End":"03:44.815","Text":"there\u0027s the second derivative method and the table method."},{"Start":"03:44.815 ","End":"03:47.425","Text":"I\u0027m going to use the table method."},{"Start":"03:47.425 ","End":"03:49.750","Text":"I\u0027m going to draw a table of,"},{"Start":"03:49.750 ","End":"03:51.890","Text":"I have here x."},{"Start":"03:51.890 ","End":"03:56.790","Text":"Here, I usually have y but here the dependent variable is V,"},{"Start":"03:56.790 ","End":"04:00.630","Text":"V prime and V. I put in all of them."},{"Start":"04:00.630 ","End":"04:03.520","Text":"There\u0027s only 1, which is x equals 4."},{"Start":"04:03.520 ","End":"04:06.635","Text":"At this point, V prime is 0."},{"Start":"04:06.635 ","End":"04:09.865","Text":"I want to know what it is to the left and to the right."},{"Start":"04:09.865 ","End":"04:13.525","Text":"Choose some points, I\u0027m going to choose 3 and 5,"},{"Start":"04:13.525 ","End":"04:17.680","Text":"and we\u0027re going to substitute in V prime."},{"Start":"04:17.680 ","End":"04:20.965","Text":"This will be convenient for me to substitute."},{"Start":"04:20.965 ","End":"04:24.525","Text":"But notice that 3x is positive,"},{"Start":"04:24.525 ","End":"04:28.355","Text":"I only have to look at the 4 minus x."},{"Start":"04:28.355 ","End":"04:30.260","Text":"Clearly, if x is 3,"},{"Start":"04:30.260 ","End":"04:32.345","Text":"4 minus x is positive."},{"Start":"04:32.345 ","End":"04:36.125","Text":"If x is 5, then 4 minus x is negative."},{"Start":"04:36.125 ","End":"04:38.295","Text":"That\u0027s all I needed to know is the sign."},{"Start":"04:38.295 ","End":"04:40.760","Text":"Here it is increasing,"},{"Start":"04:40.760 ","End":"04:43.130","Text":"here it is decreasing,"},{"Start":"04:43.130 ","End":"04:47.495","Text":"which means that the point for the graph looks something like this."},{"Start":"04:47.495 ","End":"04:52.020","Text":"In other words, it\u0027s a maximum indeed."},{"Start":"04:52.020 ","End":"04:56.750","Text":"Our x equals 4 is indeed a maximum,"},{"Start":"04:56.750 ","End":"04:59.040","Text":"and that is our answer."}],"ID":8572},{"Watched":false,"Name":"Exercise 17","Duration":"6m 46s","ChapterTopicVideoID":8392,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8392.jpeg","UploadDate":"2017-01-28T14:01:50.9530000","DurationForVideoObject":"PT6M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.560","Text":"Here we have another word problem,"},{"Start":"00:01.560 ","End":"00:02.970","Text":"I\u0027ll just explain it."},{"Start":"00:02.970 ","End":"00:06.780","Text":"We have a box here which is hollow."},{"Start":"00:06.780 ","End":"00:08.745","Text":"I mean, it\u0027s like a cardboard box,"},{"Start":"00:08.745 ","End":"00:10.080","Text":"but it has a base,"},{"Start":"00:10.080 ","End":"00:11.730","Text":"but it\u0027s an open top."},{"Start":"00:11.730 ","End":"00:15.180","Text":"It has the 4 sides and the base."},{"Start":"00:15.180 ","End":"00:21.960","Text":"We know that the surface area is 75 square centimeters and we"},{"Start":"00:21.960 ","End":"00:28.650","Text":"emphasize the surface area is only 1 base and 4 sides as opposed to the usual 6 sides,"},{"Start":"00:28.650 ","End":"00:30.435","Text":"there\u0027s only 5 sides here."},{"Start":"00:30.435 ","End":"00:33.000","Text":"Now, we have to find,"},{"Start":"00:33.000 ","End":"00:38.135","Text":"under this condition, the dimensions of the box with the maximal volume."},{"Start":"00:38.135 ","End":"00:42.740","Text":"It looks like we\u0027re going to take our dependent variable as the volume,"},{"Start":"00:42.740 ","End":"00:45.650","Text":"that\u0027s the 1 we\u0027re trying to maximize."},{"Start":"00:45.650 ","End":"00:48.220","Text":"Let\u0027s call the volume, V,"},{"Start":"00:48.220 ","End":"00:49.800","Text":"and we haven\u0027t got anything here,"},{"Start":"00:49.800 ","End":"00:51.410","Text":"so let me start labeling stuff."},{"Start":"00:51.410 ","End":"00:53.615","Text":"Let\u0027s say that the base is x."},{"Start":"00:53.615 ","End":"00:57.605","Text":"Now it\u0027s square, that means it\u0027s x here and it\u0027s x here."},{"Start":"00:57.605 ","End":"00:59.390","Text":"The height, I can call it h,"},{"Start":"00:59.390 ","End":"01:05.765","Text":"I can call it y, not to get too used to y being the dependent variable."},{"Start":"01:05.765 ","End":"01:09.860","Text":"Now, V is the product of all 3 dimensions,"},{"Start":"01:09.860 ","End":"01:12.140","Text":"I mean that\u0027s a standard formula for the volume,"},{"Start":"01:12.140 ","End":"01:14.735","Text":"it\u0027s just x times x times y,"},{"Start":"01:14.735 ","End":"01:16.730","Text":"which is x squared y."},{"Start":"01:16.730 ","End":"01:19.160","Text":"Some people say area of base times height,"},{"Start":"01:19.160 ","End":"01:21.784","Text":"but either way you look at it, it\u0027s the product of the 3 dimensions."},{"Start":"01:21.784 ","End":"01:24.650","Text":"Now the thing is I want to get it only in terms of 1 variable."},{"Start":"01:24.650 ","End":"01:27.635","Text":"Let\u0027s say that x is our independent variable,"},{"Start":"01:27.635 ","End":"01:29.480","Text":"so I need to put y in terms of x,"},{"Start":"01:29.480 ","End":"01:32.960","Text":"and I can do that by using this condition. Let\u0027s see."},{"Start":"01:32.960 ","End":"01:36.105","Text":"The surface area, there\u0027s 4 sides,"},{"Start":"01:36.105 ","End":"01:38.490","Text":"so the surface area,"},{"Start":"01:38.490 ","End":"01:41.190","Text":"let\u0027s call that S will be,"},{"Start":"01:41.190 ","End":"01:44.400","Text":"let\u0027s see, x times y is 1 of these things."},{"Start":"01:44.400 ","End":"01:47.250","Text":"Each 1 of these is x times y."},{"Start":"01:47.250 ","End":"01:50.210","Text":"There\u0027s 4 of these because they go all the way"},{"Start":"01:50.210 ","End":"01:54.200","Text":"around and then I have 1 base because it\u0027s open at the top,"},{"Start":"01:54.200 ","End":"01:59.045","Text":"which is x squared and this we\u0027re given is equal to 75."},{"Start":"01:59.045 ","End":"02:02.990","Text":"Now it seems to me like I could isolate y in terms"},{"Start":"02:02.990 ","End":"02:06.725","Text":"of x from here and then I can put it in here and I\u0027ll have just x."},{"Start":"02:06.725 ","End":"02:12.095","Text":"Here goes, I don\u0027t need the name of this S. I just need this equation."},{"Start":"02:12.095 ","End":"02:19.470","Text":"I\u0027ll take x squared to the other side so I\u0027ve got 75 minus x"},{"Start":"02:19.470 ","End":"02:27.830","Text":"squared and then I\u0027ll divide by the 4x and all I\u0027m left with is the y then."},{"Start":"02:27.830 ","End":"02:30.320","Text":"V equals, let\u0027s see,"},{"Start":"02:30.320 ","End":"02:39.560","Text":"x squared times 75 minus x squared over 4x."},{"Start":"02:39.560 ","End":"02:42.440","Text":"Now we can cancel a little bit."},{"Start":"02:42.440 ","End":"02:45.020","Text":"This x can go with 1 of these,"},{"Start":"02:45.020 ","End":"02:46.610","Text":"so I cancel basically the 2,"},{"Start":"02:46.610 ","End":"02:53.365","Text":"so it\u0027s no longer x squared and what I get if I multiply out will be,"},{"Start":"02:53.365 ","End":"02:55.190","Text":"put the 4 as a quarter in front."},{"Start":"02:55.190 ","End":"02:58.250","Text":"I don\u0027t want a big fraction. Let\u0027s see."},{"Start":"02:58.250 ","End":"02:59.435","Text":"Now if I multiply out,"},{"Start":"02:59.435 ","End":"03:05.175","Text":"I get 75x minus x cubed."},{"Start":"03:05.175 ","End":"03:10.085","Text":"This basically, this bit is going to give me"},{"Start":"03:10.085 ","End":"03:15.410","Text":"the equation of the thing I need to maximize in terms of the"},{"Start":"03:15.410 ","End":"03:19.670","Text":"independent variable x. I\u0027m going to use"},{"Start":"03:19.670 ","End":"03:27.885","Text":"calculus extrema methods to find what x gives the maximum V. Let\u0027s see now."},{"Start":"03:27.885 ","End":"03:31.180","Text":"V prime is equal to 1/4,"},{"Start":"03:31.180 ","End":"03:35.150","Text":"that\u0027s a constant so it can stay and differentiate what\u0027s inside."},{"Start":"03:35.150 ","End":"03:40.115","Text":"We have 75 minus 3x squared."},{"Start":"03:40.115 ","End":"03:45.439","Text":"For maximum, I need to have V prime equals 0."},{"Start":"03:45.439 ","End":"03:47.210","Text":"What will that give me?"},{"Start":"03:47.210 ","End":"03:50.375","Text":"The quarter is inconsequential here."},{"Start":"03:50.375 ","End":"03:54.790","Text":"I\u0027ll get 75 minus 3x squared equals 0."},{"Start":"03:54.790 ","End":"04:04.820","Text":"Basically, I\u0027ll get 75 equals 3x squared or if I switch sides and divide by 3,"},{"Start":"04:04.820 ","End":"04:08.105","Text":"I\u0027ll get x squared is 25."},{"Start":"04:08.105 ","End":"04:10.610","Text":"Now, x is a length,"},{"Start":"04:10.610 ","End":"04:11.960","Text":"it\u0027s a positive number,"},{"Start":"04:11.960 ","End":"04:17.930","Text":"it can\u0027t be negative so the only possibility is x is the square root of 25, which is 5."},{"Start":"04:17.930 ","End":"04:21.425","Text":"That\u0027s the only positive number squared that gives 25."},{"Start":"04:21.425 ","End":"04:25.985","Text":"We found what x is that would give the"},{"Start":"04:25.985 ","End":"04:32.195","Text":"maximum not enough to find x. I also have to find what y is."},{"Start":"04:32.195 ","End":"04:34.340","Text":"If x is 5,"},{"Start":"04:34.340 ","End":"04:44.120","Text":"then we have the y equals 75 minus 5 squared over 4 times 5, which is,"},{"Start":"04:44.120 ","End":"04:48.714","Text":"let\u0027s see, 5 squared is 25 from 75 is 50,"},{"Start":"04:48.714 ","End":"04:55.670","Text":"50 over 20 and that\u0027s basically 2 and 1/2 or 5 over 2 or 2.5."},{"Start":"04:55.670 ","End":"04:58.880","Text":"I\u0027ll just write that here as well."},{"Start":"04:58.880 ","End":"05:02.585","Text":"Y equals and I\u0027ll put it as a decimal here,"},{"Start":"05:02.585 ","End":"05:06.965","Text":"2.5 and I\u0027ll highlight that also."},{"Start":"05:06.965 ","End":"05:10.400","Text":"But as I said, we haven\u0027t yet shown that it is indeed"},{"Start":"05:10.400 ","End":"05:13.910","Text":"a maximum and there\u0027s 2 main methods,"},{"Start":"05:13.910 ","End":"05:19.000","Text":"1 is the second derivative method and 1 is the table of increasing, decreasing."},{"Start":"05:19.000 ","End":"05:22.025","Text":"I\u0027m going to use the table method."},{"Start":"05:22.025 ","End":"05:25.155","Text":"I\u0027ll draw a table could be here."},{"Start":"05:25.155 ","End":"05:27.135","Text":"We have 3 rows."},{"Start":"05:27.135 ","End":"05:29.420","Text":"1 of them, I label x,"},{"Start":"05:29.420 ","End":"05:32.465","Text":"the other I label whatever the variable is,"},{"Start":"05:32.465 ","End":"05:36.035","Text":"prime, and then V prime and V in this case."},{"Start":"05:36.035 ","End":"05:38.210","Text":"Here we put the list of suspects,"},{"Start":"05:38.210 ","End":"05:39.500","Text":"not a very long list."},{"Start":"05:39.500 ","End":"05:44.550","Text":"We only have 1 suspect and that is 5 and when x is 5,"},{"Start":"05:44.550 ","End":"05:47.180","Text":"we know that V prime is 0 and our question is,"},{"Start":"05:47.180 ","End":"05:50.195","Text":"what is it to the left of 5 and to the right of 5?"},{"Start":"05:50.195 ","End":"05:53.705","Text":"Let\u0027s take 4 and 6 here."},{"Start":"05:53.705 ","End":"05:57.680","Text":"Now, all I care about is not V prime itself,"},{"Start":"05:57.680 ","End":"06:00.260","Text":"but just the sign whether it\u0027s positive or negative."},{"Start":"06:00.260 ","End":"06:02.255","Text":"I could have written this as"},{"Start":"06:02.255 ","End":"06:08.010","Text":"3/4 times 25 minus x"},{"Start":"06:08.010 ","End":"06:13.115","Text":"squared and then this is positive so we just have to substitute in here."},{"Start":"06:13.115 ","End":"06:15.530","Text":"Let\u0027s just give, for example, if I put in 5,"},{"Start":"06:15.530 ","End":"06:18.305","Text":"I get 25 minus 5 squared it\u0027s zero, that\u0027s correct."},{"Start":"06:18.305 ","End":"06:22.275","Text":"Now let\u0027s try it with 4, 25 minus 16, positive."},{"Start":"06:22.275 ","End":"06:26.355","Text":"It with 6, 25 minus 36 negative."},{"Start":"06:26.355 ","End":"06:28.400","Text":"As far as V itself goes,"},{"Start":"06:28.400 ","End":"06:32.075","Text":"here we have an increasing and here we have a decreasing."},{"Start":"06:32.075 ","End":"06:38.210","Text":"Around 5, the curve is somewhat shaped like this and that\u0027s the shape of a maximum."},{"Start":"06:38.210 ","End":"06:41.495","Text":"This is indeed a maximum."},{"Start":"06:41.495 ","End":"06:47.790","Text":"This pair of dimensions give me the maximum volume and we are done."}],"ID":8573},{"Watched":false,"Name":"Exercise 18","Duration":"7m 9s","ChapterTopicVideoID":8382,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8382.jpeg","UploadDate":"2017-01-28T13:40:37.6570000","DurationForVideoObject":"PT7M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"Let me explain this word problem in geometry."},{"Start":"00:03.660 ","End":"00:06.570","Text":"We have a skeleton of a box,"},{"Start":"00:06.570 ","End":"00:08.685","Text":"sometimes called wire frame,"},{"Start":"00:08.685 ","End":"00:13.035","Text":"means that each of these edges is like a piece of wire."},{"Start":"00:13.035 ","End":"00:18.345","Text":"Actually there\u0027s altogether, let\u0027s see how many these 12 pieces of wire,"},{"Start":"00:18.345 ","End":"00:19.830","Text":"4 for the base,"},{"Start":"00:19.830 ","End":"00:22.560","Text":"4 for the top and 4 supporting,"},{"Start":"00:22.560 ","End":"00:30.015","Text":"and we want the wire-frame box to hold 1,000 cubic centimeters,"},{"Start":"00:30.015 ","End":"00:32.880","Text":"but we want to do it with the least amount of wire,"},{"Start":"00:32.880 ","End":"00:38.605","Text":"basically we\u0027re asking what is the minimum length of wire needed to construct such a box?"},{"Start":"00:38.605 ","End":"00:42.550","Text":"Minimal means an extrema problem and"},{"Start":"00:42.550 ","End":"00:45.770","Text":"the length of wire is going to be our dependent variable."},{"Start":"00:45.770 ","End":"00:48.775","Text":"Let\u0027s call that variable L,"},{"Start":"00:48.775 ","End":"00:51.180","Text":"and it\u0027s going to equal;"},{"Start":"00:51.180 ","End":"00:53.055","Text":"but we haven\u0027t labeled anything yet,"},{"Start":"00:53.055 ","End":"00:58.565","Text":"but let\u0027s say the base is x and this is x also because it\u0027s a square base,"},{"Start":"00:58.565 ","End":"01:01.680","Text":"but the height, we\u0027ll call it y."},{"Start":"01:01.680 ","End":"01:07.185","Text":"Now, the length is going to equal, let\u0027s see now,"},{"Start":"01:07.185 ","End":"01:10.350","Text":"there\u0027s 8 pieces of length x,"},{"Start":"01:10.350 ","End":"01:15.035","Text":"because there\u0027s 4 along the base and then there\u0027s another 4 along here,"},{"Start":"01:15.035 ","End":"01:16.865","Text":"x, x, x and x,"},{"Start":"01:16.865 ","End":"01:18.290","Text":"and there\u0027s 2 more hidden."},{"Start":"01:18.290 ","End":"01:23.450","Text":"Basically, we have 8 x\u0027s and each and 4 of these y\u0027s,"},{"Start":"01:23.450 ","End":"01:27.730","Text":"so 8x plus 4y,"},{"Start":"01:27.730 ","End":"01:30.800","Text":"and this is what we want to minimize,"},{"Start":"01:30.800 ","End":"01:33.020","Text":"but we don\u0027t want 2 independent variables,"},{"Start":"01:33.020 ","End":"01:38.780","Text":"we just want 1, and we can get y in terms of x if we use the other piece of information."},{"Start":"01:38.780 ","End":"01:41.815","Text":"We\u0027re told that the volume is 1,000."},{"Start":"01:41.815 ","End":"01:48.260","Text":"Now, the volume of the box is the product of all 3-dimensions or if you like,"},{"Start":"01:48.260 ","End":"01:49.760","Text":"the area of the base times height,"},{"Start":"01:49.760 ","End":"01:52.490","Text":"either way it\u0027s x squared y,"},{"Start":"01:52.490 ","End":"01:56.315","Text":"and we\u0027re told that is equal to 1,000,"},{"Start":"01:56.315 ","End":"02:00.330","Text":"and I guess I don\u0027t need the letter V, I\u0027ll erase that,"},{"Start":"02:00.330 ","End":"02:02.585","Text":"and in that case,"},{"Start":"02:02.585 ","End":"02:09.245","Text":"we have this equation and we can extract y from it and say it\u0027s 1,000 over x squared."},{"Start":"02:09.245 ","End":"02:13.985","Text":"Now I can put that back in here and say that this is equal to"},{"Start":"02:13.985 ","End":"02:24.165","Text":"8x plus 4 times y is 4000 over x squared,"},{"Start":"02:24.165 ","End":"02:28.845","Text":"and we want the minimum of L,"},{"Start":"02:28.845 ","End":"02:37.005","Text":"that\u0027s what we want to minimize is L. Let\u0027s use calculus, differentiate."},{"Start":"02:37.005 ","End":"02:42.610","Text":"We\u0027ll get L prime is equal to 8x gives 8,"},{"Start":"02:42.610 ","End":"02:45.860","Text":"the derivative of 1 over x squared,"},{"Start":"02:45.860 ","End":"02:48.390","Text":"you can check it\u0027s minus 2 over x cubed,"},{"Start":"02:48.390 ","End":"02:50.160","Text":"if you write it as x^minus 2,"},{"Start":"02:50.160 ","End":"02:52.220","Text":"we get minus 2, x^minus 3,"},{"Start":"02:52.220 ","End":"02:54.934","Text":"and then it\u0027s minus 2 over x cubed,"},{"Start":"02:54.934 ","End":"03:02.130","Text":"so it\u0027s minus 2 times 4,000 over x cubed,"},{"Start":"03:02.130 ","End":"03:05.370","Text":"and we want to minimize,"},{"Start":"03:05.370 ","End":"03:06.770","Text":"so for an extrema problem,"},{"Start":"03:06.770 ","End":"03:09.980","Text":"we want the derivative to be 0 because that\u0027s the only place we"},{"Start":"03:09.980 ","End":"03:13.830","Text":"could find an extremum or suspect anyway."},{"Start":"03:13.830 ","End":"03:19.740","Text":"So that will give us that 8 equals,"},{"Start":"03:19.740 ","End":"03:22.290","Text":"let\u0027s see I bring this to the other side,"},{"Start":"03:22.290 ","End":"03:23.585","Text":"so they\u0027re saying it\u0027s equal to 0,"},{"Start":"03:23.585 ","End":"03:25.850","Text":"8 is equal to 8,000."},{"Start":"03:25.850 ","End":"03:31.745","Text":"That\u0027s the 2 times the 4,000 over x cubed,"},{"Start":"03:31.745 ","End":"03:35.540","Text":"and that will give us, see,"},{"Start":"03:35.540 ","End":"03:37.055","Text":"bring the x cubed over here,"},{"Start":"03:37.055 ","End":"03:38.570","Text":"bring the 8 over here,"},{"Start":"03:38.570 ","End":"03:42.265","Text":"we get x cubed equals 1,000,"},{"Start":"03:42.265 ","End":"03:45.520","Text":"divide by 8, multiply by x cubed."},{"Start":"03:45.520 ","End":"03:47.780","Text":"X is the cube root of 1,000,"},{"Start":"03:47.780 ","End":"03:49.655","Text":"which is a neat 10,"},{"Start":"03:49.655 ","End":"03:52.980","Text":"so that\u0027s going to be an important result,"},{"Start":"03:52.980 ","End":"03:55.930","Text":"and let me just check again what was it that they were asking."},{"Start":"03:55.930 ","End":"03:58.374","Text":"Minimal length of wire,"},{"Start":"03:58.374 ","End":"04:00.260","Text":"so we don\u0027t want the dimensions,"},{"Start":"04:00.260 ","End":"04:04.130","Text":"but we want the actual length of wire, not a problem."},{"Start":"04:04.130 ","End":"04:06.635","Text":"Once we have that x is 10,"},{"Start":"04:06.635 ","End":"04:10.745","Text":"we can then go back to computing y,"},{"Start":"04:10.745 ","End":"04:13.485","Text":"because we have that y is this,"},{"Start":"04:13.485 ","End":"04:15.195","Text":"so if x is 10,"},{"Start":"04:15.195 ","End":"04:21.285","Text":"then y equals 1,000 over 10 squared,"},{"Start":"04:21.285 ","End":"04:24.245","Text":"1,000 over a 100 which is 10."},{"Start":"04:24.245 ","End":"04:31.959","Text":"So that gives us also that x equals 10 and then y equals 10,"},{"Start":"04:31.959 ","End":"04:34.965","Text":"but what we need is L,"},{"Start":"04:34.965 ","End":"04:37.625","Text":"so if I plug it in here,"},{"Start":"04:37.625 ","End":"04:39.995","Text":"I will get that,"},{"Start":"04:39.995 ","End":"04:47.240","Text":"let\u0027s see L is equal to 8x plus 4y,"},{"Start":"04:47.420 ","End":"04:51.810","Text":"which is 8 times 10 plus 4 times,"},{"Start":"04:51.810 ","End":"04:53.895","Text":"so it\u0027s 8 plus 4 times 10,"},{"Start":"04:53.895 ","End":"04:57.830","Text":"so this is 120."},{"Start":"04:57.830 ","End":"04:59.405","Text":"Now the question is,"},{"Start":"04:59.405 ","End":"05:03.350","Text":"how do we know that it\u0027s a minimum?"},{"Start":"05:03.350 ","End":"05:08.510","Text":"2 techniques, second derivative or table,"},{"Start":"05:08.510 ","End":"05:09.810","Text":"those are the 2 main methods."},{"Start":"05:09.810 ","End":"05:12.070","Text":"I\u0027m going to opt for the table method,"},{"Start":"05:12.070 ","End":"05:14.375","Text":"so I\u0027m going to make a table,"},{"Start":"05:14.375 ","End":"05:19.230","Text":"where here I have x my independent variable,"},{"Start":"05:19.230 ","End":"05:21.380","Text":"here I have L prime,"},{"Start":"05:21.380 ","End":"05:23.360","Text":"and here I have L,"},{"Start":"05:23.360 ","End":"05:26.060","Text":"here I put in all the suspects in this case,"},{"Start":"05:26.060 ","End":"05:29.585","Text":"it\u0027s only 1 that happens to be x equals 10."},{"Start":"05:29.585 ","End":"05:33.220","Text":"At this point L prime is 0."},{"Start":"05:33.220 ","End":"05:37.200","Text":"How about I\u0027ll try 8, 20."},{"Start":"05:37.200 ","End":"05:40.080","Text":"Let me just write it again that,"},{"Start":"05:40.080 ","End":"05:49.020","Text":"L prime is 8 minus 8,000 over x cubed,"},{"Start":"05:49.020 ","End":"05:52.110","Text":"that\u0027s where I\u0027m going to be substituting."},{"Start":"05:52.110 ","End":"05:54.284","Text":"If x equals 10,"},{"Start":"05:54.284 ","End":"05:55.950","Text":"10 cubed is 1,000,"},{"Start":"05:55.950 ","End":"05:59.835","Text":"8,000 over 1,000 is 88 minus 8 is 0 that\u0027s fine."},{"Start":"05:59.835 ","End":"06:05.640","Text":"If x is 8, I get 8 cubed is 60,"},{"Start":"06:05.640 ","End":"06:06.810","Text":"I\u0027ll choose a better value,"},{"Start":"06:06.810 ","End":"06:08.490","Text":"let me try 5,"},{"Start":"06:08.490 ","End":"06:10.935","Text":"I think that will work better instead of 8."},{"Start":"06:10.935 ","End":"06:19.799","Text":"I\u0027m going to put 5, 5 cubed is 125, goes into 1,000,"},{"Start":"06:19.799 ","End":"06:24.345","Text":"8 times, so it goes into 8,000,"},{"Start":"06:24.345 ","End":"06:28.815","Text":"64 times, 8 minus 64 it\u0027s negative,"},{"Start":"06:28.815 ","End":"06:30.925","Text":"that\u0027s really all I needed to know."},{"Start":"06:30.925 ","End":"06:33.230","Text":"If I have x equals 20,"},{"Start":"06:33.230 ","End":"06:35.060","Text":"20 cubed is 8,000,"},{"Start":"06:35.060 ","End":"06:36.860","Text":"8,000 over 8,000 is 1,"},{"Start":"06:36.860 ","End":"06:41.450","Text":"8 minus 1, it\u0027s positive is all I need to know,"},{"Start":"06:41.450 ","End":"06:44.570","Text":"and so, here I have decreasing,"},{"Start":"06:44.570 ","End":"06:46.220","Text":"here I have increasing."},{"Start":"06:46.220 ","End":"06:49.235","Text":"The function L around 10"},{"Start":"06:49.235 ","End":"06:53.335","Text":"would be something like this shape in other words, it\u0027s a minimum,"},{"Start":"06:53.335 ","End":"06:57.604","Text":"and so this does give us indeed a minimum,"},{"Start":"06:57.604 ","End":"07:01.105","Text":"and the minimum length is 120,"},{"Start":"07:01.105 ","End":"07:04.640","Text":"and the dimensions that give it are 10,"},{"Start":"07:04.640 ","End":"07:10.140","Text":"everything\u0027s 10, all the pieces of wire are 10. We\u0027re done."}],"ID":8574},{"Watched":false,"Name":"Exercise 19","Duration":"13m 22s","ChapterTopicVideoID":8383,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8383.jpeg","UploadDate":"2017-03-07T06:20:28.7870000","DurationForVideoObject":"PT13M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"Let me explain this word problem in Geometry,"},{"Start":"00:03.915 ","End":"00:09.270","Text":"we have here what is called a right regular pyramid with square base,"},{"Start":"00:09.270 ","End":"00:11.985","Text":"which means that the base is a square,"},{"Start":"00:11.985 ","End":"00:16.395","Text":"all the sides are the same and the angles are 90 degrees."},{"Start":"00:16.395 ","End":"00:22.935","Text":"The word right, means that the apex is directly above the center of the base."},{"Start":"00:22.935 ","End":"00:26.070","Text":"This would be 90 degrees also,"},{"Start":"00:26.070 ","End":"00:30.525","Text":"the height meets this plane of the base and 90 degrees."},{"Start":"00:30.525 ","End":"00:34.215","Text":"What we\u0027re given is that it has a lateral edge,"},{"Start":"00:34.215 ","End":"00:37.055","Text":"lateral means side edge of a centimeter."},{"Start":"00:37.055 ","End":"00:40.700","Text":"This would be a, we are going to just use centimeters,"},{"Start":"00:40.700 ","End":"00:44.195","Text":"and this would be a, and this would be a,"},{"Start":"00:44.195 ","End":"00:46.595","Text":"and I guess the hidden one is also a,"},{"Start":"00:46.595 ","End":"00:48.545","Text":"what our task is,"},{"Start":"00:48.545 ","End":"00:53.675","Text":"is to get the greatest possible volume from all such pyramids."},{"Start":"00:53.675 ","End":"00:59.890","Text":"There are many which fits this description and we want the one with the biggest volume,"},{"Start":"00:59.890 ","End":"01:01.535","Text":"and to find that volume,"},{"Start":"01:01.535 ","End":"01:05.445","Text":"what we need is an independent variable."},{"Start":"01:05.445 ","End":"01:11.119","Text":"I think the side of the base looks fairly natural to me to take,"},{"Start":"01:11.119 ","End":"01:14.660","Text":"let\u0027s call this x, going to be our independent variable,"},{"Start":"01:14.660 ","End":"01:18.170","Text":"this is x and then these two will also be x."},{"Start":"01:18.170 ","End":"01:22.835","Text":"Now we have to express the volume in terms of x."},{"Start":"01:22.835 ","End":"01:28.160","Text":"Normally, the volume of a pyramid is given"},{"Start":"01:28.160 ","End":"01:34.085","Text":"by the formula V is equal to 1/3,"},{"Start":"01:34.085 ","End":"01:36.485","Text":"the area of the base times the height."},{"Start":"01:36.485 ","End":"01:40.310","Text":"Now the area of the base is x squared."},{"Start":"01:40.310 ","End":"01:43.160","Text":"The height would be this,"},{"Start":"01:43.160 ","End":"01:48.185","Text":"I\u0027ll call this h. But the thing is we don\u0027t have h yet,"},{"Start":"01:48.185 ","End":"01:51.035","Text":"we want to express it in terms of x."},{"Start":"01:51.035 ","End":"01:55.760","Text":"This can be done with Pythagoras\u0027s theorem twice."},{"Start":"01:55.760 ","End":"01:59.060","Text":"The first thing is that I want to look at this triangle,"},{"Start":"01:59.060 ","End":"02:07.280","Text":"then I can get h from Pythagoras\u0027s theorem because a is like a known quantity,"},{"Start":"02:07.280 ","End":"02:09.725","Text":"but I don\u0027t have this bit."},{"Start":"02:09.725 ","End":"02:12.035","Text":"I\u0027ll go back again,"},{"Start":"02:12.035 ","End":"02:17.150","Text":"and I\u0027m going to use a different color because we\u0027re going to use a different triangle."},{"Start":"02:17.150 ","End":"02:22.340","Text":"If I call this lets say l. This will"},{"Start":"02:22.340 ","End":"02:28.410","Text":"also be l because the diagonals bisect each other."},{"Start":"02:28.410 ","End":"02:31.675","Text":"Anywhere In a square also this is 90 degrees,"},{"Start":"02:31.675 ","End":"02:33.755","Text":"this angle is 45."},{"Start":"02:33.755 ","End":"02:38.935","Text":"For many reasons, these 2 are the same and this is also 90 degrees."},{"Start":"02:38.935 ","End":"02:41.990","Text":"If I look at the one I\u0027ve shaded in green,"},{"Start":"02:41.990 ","End":"02:50.700","Text":"I can get that l squared plus l squared equals x squared by Pythagoras."},{"Start":"02:51.280 ","End":"02:55.285","Text":"I\u0027ve got that 2l squared is x squared,"},{"Start":"02:55.285 ","End":"03:00.200","Text":"so l squared is 1/2 x squared,"},{"Start":"03:00.200 ","End":"03:03.730","Text":"or x squared over 2 maybe is better."},{"Start":"03:03.730 ","End":"03:07.845","Text":"Now let\u0027s go to the yellow triangle."},{"Start":"03:07.845 ","End":"03:13.385","Text":"I said this isn\u0027t going to be 90 degrees on the yellows got obliterated."},{"Start":"03:13.385 ","End":"03:17.540","Text":"If I apply Pythagoras to the yellow triangle,"},{"Start":"03:17.540 ","End":"03:20.930","Text":"I get that h squared is this squared minus this squared,"},{"Start":"03:20.930 ","End":"03:25.445","Text":"is a squared minus l squared,"},{"Start":"03:25.445 ","End":"03:31.300","Text":"but we know l squared is x squared over 2."},{"Start":"03:33.560 ","End":"03:35.675","Text":"Now I\u0027ve got this,"},{"Start":"03:35.675 ","End":"03:38.195","Text":"I can plug that in here."},{"Start":"03:38.195 ","End":"03:42.030","Text":"I\u0027ve got that v is 1/3 x squared."},{"Start":"03:42.030 ","End":"03:43.430","Text":"Here I have h squared,"},{"Start":"03:43.430 ","End":"03:51.205","Text":"but h is just the square root of a squared minus x squared over 2."},{"Start":"03:51.205 ","End":"03:53.630","Text":"We see that we have a square root,"},{"Start":"03:53.630 ","End":"03:56.225","Text":"which would be awkward if we start differentiating."},{"Start":"03:56.225 ","End":"04:01.625","Text":"We have a standard trick of raising everything squared,"},{"Start":"04:01.625 ","End":"04:03.755","Text":"taking the function squared."},{"Start":"04:03.755 ","End":"04:06.140","Text":"This only works when you have a positive function."},{"Start":"04:06.140 ","End":"04:07.670","Text":"I mean, besides the square root,"},{"Start":"04:07.670 ","End":"04:08.990","Text":"we also have this bit,"},{"Start":"04:08.990 ","End":"04:10.460","Text":"but this is always positive,"},{"Start":"04:10.460 ","End":"04:13.715","Text":"so we\u0027re okay, the positive functions, this trick works."},{"Start":"04:13.715 ","End":"04:17.750","Text":"I\u0027m going to let y be v squared,"},{"Start":"04:17.750 ","End":"04:23.405","Text":"which will be 1/9 x^4,"},{"Start":"04:23.405 ","End":"04:25.610","Text":"and then throughout the square root,"},{"Start":"04:25.610 ","End":"04:30.350","Text":"and I\u0027ve got a squared minus x squared over 2,"},{"Start":"04:30.350 ","End":"04:34.550","Text":"and y is going to have the same maximum or"},{"Start":"04:34.550 ","End":"04:39.530","Text":"minimum x as v. The actual value will be different,"},{"Start":"04:39.530 ","End":"04:41.825","Text":"but the x will be the same."},{"Start":"04:41.825 ","End":"04:47.285","Text":"When we actually find the volume,"},{"Start":"04:47.285 ","End":"04:50.885","Text":"then we have to use v and not y."},{"Start":"04:50.885 ","End":"04:54.815","Text":"Let\u0027s differentiate y prime is,"},{"Start":"04:54.815 ","End":"04:58.040","Text":"now the 1/9 is a constant and the rest I\u0027ll do with"},{"Start":"04:58.040 ","End":"05:04.300","Text":"a product rule like this will be u and this will be v,"},{"Start":"05:04.300 ","End":"05:06.050","Text":"I\u0027m assuming you know the product rule."},{"Start":"05:06.050 ","End":"05:08.749","Text":"We have u prime,"},{"Start":"05:08.749 ","End":"05:11.000","Text":"which is 4x cubed,"},{"Start":"05:11.000 ","End":"05:16.555","Text":"and then v as is a squared minus x squared over 2."},{"Start":"05:16.555 ","End":"05:21.295","Text":"Plus the other way around this one not differentiated,"},{"Start":"05:21.295 ","End":"05:23.465","Text":"and this one differentiated."},{"Start":"05:23.465 ","End":"05:30.450","Text":"Just gives us minus 2x over 2 is just minus x."},{"Start":"05:31.910 ","End":"05:34.860","Text":"We don\u0027t look for critical points first,"},{"Start":"05:34.860 ","End":"05:38.510","Text":"and from those we look for maximum or minimum,"},{"Start":"05:38.510 ","End":"05:42.150","Text":"so y prime is 0 gives us,"},{"Start":"05:42.150 ","End":"05:46.320","Text":"we can ignore this 1/9 is not going to make any difference,"},{"Start":"05:46.320 ","End":"05:49.880","Text":"we multiply both sides by 9,"},{"Start":"05:49.880 ","End":"05:51.905","Text":"it gets still 0."},{"Start":"05:51.905 ","End":"05:57.015","Text":"Not only that, since x is not 0,"},{"Start":"05:57.015 ","End":"05:59.060","Text":"where we\u0027re looking x is positive,"},{"Start":"05:59.060 ","End":"06:00.320","Text":"we can also cancel it,"},{"Start":"06:00.320 ","End":"06:04.220","Text":"lets say this x cubed with x cubed from here,"},{"Start":"06:04.220 ","End":"06:08.360","Text":"so that will get rid of that 4 make it just x."},{"Start":"06:08.360 ","End":"06:14.670","Text":"What do we get? 4a squared minus 4x squared over 2,"},{"Start":"06:14.670 ","End":"06:17.520","Text":"so that\u0027s minus 2x squared."},{"Start":"06:17.520 ","End":"06:24.420","Text":"Here minus x squared because it\u0027s just x minus x."},{"Start":"06:24.460 ","End":"06:29.250","Text":"This is going to equal 0,"},{"Start":"06:30.850 ","End":"06:34.410","Text":"what we get is"},{"Start":"06:37.270 ","End":"06:44.655","Text":"4a squared will equal 3x squared,"},{"Start":"06:44.655 ","End":"06:55.620","Text":"and so x squared is equal to 4 over 3a squared,"},{"Start":"06:55.620 ","End":"06:58.050","Text":"and x has to be positive."},{"Start":"06:58.050 ","End":"07:01.100","Text":"I can only take the positive square root,"},{"Start":"07:01.100 ","End":"07:02.779","Text":"no plus or minus."},{"Start":"07:02.779 ","End":"07:09.060","Text":"Take the square root of 4/3, a square root,"},{"Start":"07:09.060 ","End":"07:15.015","Text":"the positive square root simplify a bit square root of 4 is 2,"},{"Start":"07:15.015 ","End":"07:20.080","Text":"square root of a squared is a if a is positive."},{"Start":"07:20.150 ","End":"07:25.400","Text":"We have the value of x,"},{"Start":"07:25.400 ","End":"07:30.575","Text":"which gives the minimum or maximum or neither its critical point,"},{"Start":"07:30.575 ","End":"07:33.220","Text":"hopefully it\u0027s a maximum."},{"Start":"07:33.220 ","End":"07:36.870","Text":"Let\u0027s do the second derivative test."},{"Start":"07:36.870 ","End":"07:40.610","Text":"For this, we altered y prime a bit,"},{"Start":"07:40.610 ","End":"07:42.080","Text":"we took stuff out."},{"Start":"07:42.080 ","End":"07:45.195","Text":"Let\u0027s get the original y prime back."},{"Start":"07:45.195 ","End":"07:47.515","Text":"I\u0027ll leave the 1.9 then,"},{"Start":"07:47.515 ","End":"07:51.885","Text":"and I\u0027ll also leave the x cubed that I canceled,"},{"Start":"07:51.885 ","End":"07:58.625","Text":"but as for the rest of it we can see from the algebra that what we had was 4a squared,"},{"Start":"07:58.625 ","End":"08:04.170","Text":"I\u0027m reading off here with a simplified minus 3x squared."},{"Start":"08:07.310 ","End":"08:11.290","Text":"Then I need the second derivative,"},{"Start":"08:12.920 ","End":"08:16.605","Text":"and then I\u0027ll substitute the x."},{"Start":"08:16.605 ","End":"08:21.505","Text":"In general, y double prime is equal to,"},{"Start":"08:21.505 ","End":"08:25.170","Text":"I better expand this."},{"Start":"08:27.980 ","End":"08:35.970","Text":"Let\u0027s do it over here we have the 1/9 now we have 4a squared"},{"Start":"08:35.970 ","End":"08:44.450","Text":"x cubed minus 3x^5 because cubed squared fifth."},{"Start":"08:44.450 ","End":"08:49.650","Text":"Now we can differentiate y double prime,"},{"Start":"08:49.650 ","End":"08:51.900","Text":"that 1/9 just sticks,"},{"Start":"08:51.900 ","End":"08:57.565","Text":"and we get 3 times 4 is 12a squared x"},{"Start":"08:57.565 ","End":"09:05.830","Text":"squared,5 times 3 is 15x^4."},{"Start":"09:05.830 ","End":"09:08.570","Text":"It\u0027s just a bit more convenient if I"},{"Start":"09:08.570 ","End":"09:12.960","Text":"take x squared out"},{"Start":"09:12.960 ","End":"09:16.490","Text":"might as well take the 3 out as well you don\u0027t normally expect you to do this,"},{"Start":"09:16.490 ","End":"09:23.435","Text":"but if I take the 3 out I got 1/3,"},{"Start":"09:23.435 ","End":"09:27.890","Text":"the x squared I\u0027m also taking out and then"},{"Start":"09:27.890 ","End":"09:33.035","Text":"I\u0027ve left with 4a squared after I took 3x squared out,"},{"Start":"09:33.035 ","End":"09:37.425","Text":"and from here, if I take 3x squared out,"},{"Start":"09:37.425 ","End":"09:43.000","Text":"I\u0027m left with 5x squared."},{"Start":"09:43.000 ","End":"09:49.895","Text":"Now, I need to know what is y prime at this point where x"},{"Start":"09:49.895 ","End":"09:58.610","Text":"is 2 over root 3a and just to see if it comes up positive or negative."},{"Start":"09:58.610 ","End":"10:01.535","Text":"This thing is obviously positive,"},{"Start":"10:01.535 ","End":"10:03.635","Text":"I\u0027ll just put a plus by that."},{"Start":"10:03.635 ","End":"10:09.360","Text":"The question is whether this is plus or minus,"},{"Start":"10:09.650 ","End":"10:13.845","Text":"I\u0027ll just do this bit at the side."},{"Start":"10:13.845 ","End":"10:23.530","Text":"I\u0027ve got 4a squared minus 5x squared,"},{"Start":"10:23.530 ","End":"10:28.340","Text":"actually I have x squared already over here I could read x squared of this,"},{"Start":"10:28.340 ","End":"10:30.485","Text":"instead of squaring it back again."},{"Start":"10:30.485 ","End":"10:37.800","Text":"It\u0027s times 4/3 a squared from here,"},{"Start":"10:37.800 ","End":"10:40.220","Text":"and this is equal to,"},{"Start":"10:40.220 ","End":"10:50.130","Text":"I can take a squared out of the brackets 4 minus 5 times 4 is 20 over 3."},{"Start":"10:50.130 ","End":"10:54.855","Text":"20 over 3 it\u0027s bigger than 6,"},{"Start":"10:54.855 ","End":"10:58.545","Text":"a squared is positive, so this is negative."},{"Start":"10:58.545 ","End":"11:03.420","Text":"Here we have positive times negative,"},{"Start":"11:03.420 ","End":"11:13.535","Text":"and so this thing is going to be also negative because we got the negative part here,"},{"Start":"11:13.535 ","End":"11:16.380","Text":"and if it\u0027s negative,"},{"Start":"11:17.360 ","End":"11:24.220","Text":"then it means that we have a maximum."},{"Start":"11:24.290 ","End":"11:30.070","Text":"But we\u0027re not done because we were asked for the volume itself,"},{"Start":"11:30.070 ","End":"11:34.970","Text":"so I copy pasted from the beginning what we found to be the volume,"},{"Start":"11:34.970 ","End":"11:41.030","Text":"and now all we have to do is substitute the volume when"},{"Start":"11:41.030 ","End":"11:47.900","Text":"x is 2 over root 3a and see what we get."},{"Start":"11:47.900 ","End":"11:52.414","Text":"We get 1/3 x squared,"},{"Start":"11:52.414 ","End":"11:58.770","Text":"we already said is 4/3 a squared."},{"Start":"12:01.330 ","End":"12:07.460","Text":"Then we have the square root of a squared minus,"},{"Start":"12:07.460 ","End":"12:11.255","Text":"now 1/2 x squared is just a half of this,"},{"Start":"12:11.255 ","End":"12:14.530","Text":"so it\u0027s 2/3 a squared,"},{"Start":"12:14.530 ","End":"12:18.210","Text":"lets\u0027 see a bit more algebra here,"},{"Start":"12:18.210 ","End":"12:25.240","Text":"what we have is 4/9 a squared."},{"Start":"12:25.240 ","End":"12:34.370","Text":"Now, here I can take a squared outside the square root,"},{"Start":"12:34.370 ","End":"12:36.385","Text":"but it comes out as a,"},{"Start":"12:36.385 ","End":"12:37.910","Text":"absolute value of a,"},{"Start":"12:37.910 ","End":"12:42.965","Text":"which is a square root of a squared times 1 minus 2/3,"},{"Start":"12:42.965 ","End":"12:47.730","Text":"so it\u0027s the square root of a third."},{"Start":"12:50.180 ","End":"12:56.480","Text":"What we get, we could slightly simplify this,"},{"Start":"12:56.480 ","End":"13:01.540","Text":"1 over 3 square root is 1 over root 3,"},{"Start":"13:01.540 ","End":"13:07.070","Text":"so I can write it as 4 over 9 root 3,"},{"Start":"13:07.070 ","End":"13:11.670","Text":"and a squared with a is a cubed."},{"Start":"13:12.050 ","End":"13:21.990","Text":"This is the answer for the volume which is maximum, and we\u0027re done."}],"ID":8575},{"Watched":false,"Name":"Exercise 20","Duration":"16m 48s","ChapterTopicVideoID":8384,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8384.jpeg","UploadDate":"2017-01-28T13:46:33.5230000","DurationForVideoObject":"PT16M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"Here I have a word problem involving pyramids."},{"Start":"00:02.910 ","End":"00:08.670","Text":"Let me just explain some of the terms because they\u0027re not always very familiar."},{"Start":"00:08.670 ","End":"00:12.915","Text":"Let\u0027s look at the concept right regular pyramid."},{"Start":"00:12.915 ","End":"00:19.350","Text":"A regular pyramid is 1 where the base is a regular polygon."},{"Start":"00:19.350 ","End":"00:22.260","Text":"Now in our case, we have a square base,"},{"Start":"00:22.260 ","End":"00:25.275","Text":"and a square is certainly a regular polygon."},{"Start":"00:25.275 ","End":"00:29.655","Text":"The word right means that the apex,"},{"Start":"00:29.655 ","End":"00:31.860","Text":"the top of the pyramid,"},{"Start":"00:31.860 ","End":"00:35.645","Text":"is directly above the center of the base."},{"Start":"00:35.645 ","End":"00:37.750","Text":"Which means that if I join these 2,"},{"Start":"00:37.750 ","End":"00:40.135","Text":"that this is a height to the base."},{"Start":"00:40.135 ","End":"00:42.095","Text":"If this is a height,"},{"Start":"00:42.095 ","End":"00:46.010","Text":"I\u0027ll label that height as h. But"},{"Start":"00:46.010 ","End":"00:50.885","Text":"this height is perpendicular to every line that goes through it in the base."},{"Start":"00:50.885 ","End":"00:54.740","Text":"It\u0027s a height. To indicate that we have a square,"},{"Start":"00:54.740 ","End":"00:59.030","Text":"let me just say that this side is x,"},{"Start":"00:59.030 ","End":"01:01.880","Text":"this is x, this is also x,"},{"Start":"01:01.880 ","End":"01:03.290","Text":"this is also x."},{"Start":"01:03.290 ","End":"01:04.640","Text":"Of course, this is a square,"},{"Start":"01:04.640 ","End":"01:07.930","Text":"so all these angles are 90 degrees on the base."},{"Start":"01:07.930 ","End":"01:15.055","Text":"Now we\u0027re given a piece of information that the surface area is 200,"},{"Start":"01:15.055 ","End":"01:19.550","Text":"but there are many such pyramids with"},{"Start":"01:19.550 ","End":"01:25.335","Text":"a surface area of 200 square centimeters which are right regular,"},{"Start":"01:25.335 ","End":"01:29.900","Text":"we want to find the 1 which has maximal volume."},{"Start":"01:29.900 ","End":"01:32.120","Text":"The volume is got to be the greatest."},{"Start":"01:32.120 ","End":"01:35.120","Text":"For 1 thing, this tells us that"},{"Start":"01:35.120 ","End":"01:40.385","Text":"our dependent variable is going to be V because that\u0027s the 1 that we want to maximize."},{"Start":"01:40.385 ","End":"01:44.195","Text":"I\u0027d like to write V in terms of the independent variable, and you know what?"},{"Start":"01:44.195 ","End":"01:46.415","Text":"X is not a bad choice, it seems to me."},{"Start":"01:46.415 ","End":"01:50.300","Text":"We\u0027ll let x be the independent variable and we want to write V in"},{"Start":"01:50.300 ","End":"01:54.260","Text":"terms of x. I can make a start"},{"Start":"01:54.260 ","End":"02:02.320","Text":"by saying that the volume of the pyramid is 1/3 of the area of the base times the height."},{"Start":"02:02.320 ","End":"02:03.890","Text":"The area of the base,"},{"Start":"02:03.890 ","End":"02:05.270","Text":"I can tell straight away,"},{"Start":"02:05.270 ","End":"02:07.560","Text":"it\u0027s just x squared."},{"Start":"02:07.670 ","End":"02:10.100","Text":"The height, I don\u0027t know yet,"},{"Start":"02:10.100 ","End":"02:11.420","Text":"but I\u0027ll write it in there."},{"Start":"02:11.420 ","End":"02:16.315","Text":"My goal now is to get the height in terms of x."},{"Start":"02:16.315 ","End":"02:19.520","Text":"There\u0027s a piece of information I haven\u0027t used in"},{"Start":"02:19.520 ","End":"02:24.800","Text":"the computation that is that the surface area is 200 square centimeters."},{"Start":"02:24.800 ","End":"02:26.930","Text":"To help me with this,"},{"Start":"02:26.930 ","End":"02:29.690","Text":"I\u0027m going to use an intermediate concept,"},{"Start":"02:29.690 ","End":"02:33.020","Text":"and it\u0027s called the slant height."},{"Start":"02:33.020 ","End":"02:35.180","Text":"I\u0027ll show you what the slant height is."},{"Start":"02:35.180 ","End":"02:40.700","Text":"If we take 1 of the faces or the lateral face and I"},{"Start":"02:40.700 ","End":"02:48.230","Text":"join the top to the center of the base because this triangle is isosceles,"},{"Start":"02:48.230 ","End":"02:50.825","Text":"the height and the median are the same."},{"Start":"02:50.825 ","End":"02:54.725","Text":"In other words, this bit is equal to this bit,"},{"Start":"02:54.725 ","End":"02:57.095","Text":"and this is also 90 degrees,"},{"Start":"02:57.095 ","End":"02:59.540","Text":"this is called the slant height."},{"Start":"02:59.540 ","End":"03:03.645","Text":"I\u0027ll call it s for slant."},{"Start":"03:03.645 ","End":"03:07.690","Text":"That\u0027s an intermediate concept I want."},{"Start":"03:07.690 ","End":"03:09.900","Text":"The idea is, first of all,"},{"Start":"03:09.900 ","End":"03:16.350","Text":"find s in terms of x and then I\u0027ll find h in terms of s and x,"},{"Start":"03:16.350 ","End":"03:18.910","Text":"and ultimately, in terms of x."},{"Start":"03:19.280 ","End":"03:26.230","Text":"What I want to say first off is that the surface area is made up of 5 faces;"},{"Start":"03:26.230 ","End":"03:29.120","Text":"there\u0027s 4 side faces and a base."},{"Start":"03:29.120 ","End":"03:35.255","Text":"The surface area, and I\u0027ll call that A for area,"},{"Start":"03:35.255 ","End":"03:39.060","Text":"the surface area is the base,"},{"Start":"03:39.060 ","End":"03:44.220","Text":"I\u0027ll write it first, plus 4 lateral faces."},{"Start":"03:44.220 ","End":"03:46.280","Text":"This 1 is a typical 1."},{"Start":"03:46.280 ","End":"03:47.450","Text":"They\u0027re all going to be the same,"},{"Start":"03:47.450 ","End":"03:48.785","Text":"so it\u0027s 4 times."},{"Start":"03:48.785 ","End":"03:53.005","Text":"Now the area of a triangle is 1/2 base times height."},{"Start":"03:53.005 ","End":"03:56.855","Text":"It\u0027s 1/2x times s,"},{"Start":"03:56.855 ","End":"04:02.020","Text":"which equals x squared plus 2xs."},{"Start":"04:02.020 ","End":"04:04.175","Text":"On the other hand,"},{"Start":"04:04.175 ","End":"04:07.315","Text":"the area is given to be 200."},{"Start":"04:07.315 ","End":"04:09.945","Text":"This is the equation I have."},{"Start":"04:09.945 ","End":"04:13.130","Text":"It\u0027s easy to isolate s because all I would do"},{"Start":"04:13.130 ","End":"04:16.250","Text":"is take the x squared to the other side and get"},{"Start":"04:16.250 ","End":"04:22.985","Text":"200 minus x squared and then divide by 2x."},{"Start":"04:22.985 ","End":"04:25.290","Text":"That gives me s in terms of x,"},{"Start":"04:25.290 ","End":"04:27.360","Text":"but I still haven\u0027t got what I want which is"},{"Start":"04:27.360 ","End":"04:31.735","Text":"h. Now I want to look at a specific triangle."},{"Start":"04:31.735 ","End":"04:39.000","Text":"I\u0027m going to join this to this,"},{"Start":"04:39.000 ","End":"04:46.260","Text":"and I\u0027m going to take this slant and this height."},{"Start":"04:46.260 ","End":"04:48.440","Text":"This is my triangle, and of course,"},{"Start":"04:48.440 ","End":"04:51.650","Text":"this is 90 degrees because h is 90 degrees to"},{"Start":"04:51.650 ","End":"04:56.030","Text":"every line going through the center of the base that\u0027s in the base."},{"Start":"04:56.030 ","End":"05:00.185","Text":"I claim I also know in terms of x what this is."},{"Start":"05:00.185 ","End":"05:03.500","Text":"This is a parallel line to the base,"},{"Start":"05:03.500 ","End":"05:05.270","Text":"so the whole thing is going to be x."},{"Start":"05:05.270 ","End":"05:07.950","Text":"This is just 1/2x, x/2."},{"Start":"05:08.200 ","End":"05:13.180","Text":"If you\u0027re still not sure then let\u0027s look at it from above."},{"Start":"05:13.180 ","End":"05:16.235","Text":"It\u0027s clear that if this is x, this is just 1/2x."},{"Start":"05:16.235 ","End":"05:19.050","Text":"Try and think about it if it\u0027s not."},{"Start":"05:19.390 ","End":"05:24.185","Text":"Now I can use Pythagoras on this yellow."},{"Start":"05:24.185 ","End":"05:32.120","Text":"Then what I will get is that h squared plus this squared is this squared,"},{"Start":"05:32.120 ","End":"05:35.540","Text":"or I can say that it\u0027s equal to this squared minus this squared."},{"Start":"05:35.540 ","End":"05:43.410","Text":"It\u0027s equal to s squared minus x/2 squared."},{"Start":"05:43.410 ","End":"05:48.765","Text":"Now I\u0027m going to substitute for s what it is here."},{"Start":"05:48.765 ","End":"05:58.355","Text":"I\u0027ve got that this is now equal to 200 minus x squared over 2x,"},{"Start":"05:58.355 ","End":"06:04.490","Text":"all this squared minus x/2 squared,"},{"Start":"06:04.490 ","End":"06:08.475","Text":"I\u0027ll just leave it as x/2 squared meanwhile."},{"Start":"06:08.475 ","End":"06:12.965","Text":"I want to do a bit of algebra to simplify this."},{"Start":"06:12.965 ","End":"06:19.785","Text":"Let\u0027s put the whole thing over 2x squared."},{"Start":"06:19.785 ","End":"06:23.315","Text":"Here we have on the numerator,"},{"Start":"06:23.315 ","End":"06:27.230","Text":"200 minus x squared squared,"},{"Start":"06:27.230 ","End":"06:29.975","Text":"and here we have minus,"},{"Start":"06:29.975 ","End":"06:32.900","Text":"if I wanted to have it over 2x squared,"},{"Start":"06:32.900 ","End":"06:36.650","Text":"I would just multiply this by x and this by x."},{"Start":"06:36.650 ","End":"06:39.020","Text":"Then it would be the same as this,"},{"Start":"06:39.020 ","End":"06:41.434","Text":"and so I\u0027d get x squared squared,"},{"Start":"06:41.434 ","End":"06:44.590","Text":"so what I\u0027m going to get is x^4."},{"Start":"06:44.590 ","End":"06:47.870","Text":"Or let\u0027s leave it as x squared squared for the moment."},{"Start":"06:47.870 ","End":"06:50.645","Text":"Now let\u0027s compute further."},{"Start":"06:50.645 ","End":"06:57.125","Text":"Here I have 200 squared is 40,000,"},{"Start":"06:57.125 ","End":"07:06.900","Text":"minus twice this times this is minus 400x plus x^4."},{"Start":"07:06.900 ","End":"07:10.150","Text":"Here I have minus x^4,"},{"Start":"07:10.150 ","End":"07:11.960","Text":"that\u0027s the x squared squared,"},{"Start":"07:11.960 ","End":"07:18.605","Text":"all over 2x, all squared is 4x squared."},{"Start":"07:18.605 ","End":"07:25.635","Text":"Something cancels, at least the x^4 with x^4 cancels."},{"Start":"07:25.635 ","End":"07:34.920","Text":"Let\u0027s see, I can also cancel 4 here and make this just 10,000 and this just 100."},{"Start":"07:35.100 ","End":"07:37.810","Text":"Let\u0027s see any more I can do."},{"Start":"07:37.810 ","End":"07:40.960","Text":"Yes, I could take 100 out here,"},{"Start":"07:40.960 ","End":"07:47.219","Text":"so altogether I\u0027ve got 100 and a 100 into 10,000 goes a 100 times,"},{"Start":"07:47.219 ","End":"07:53.490","Text":"minus x all over x squared."},{"Start":"07:53.490 ","End":"07:55.440","Text":"What I get finally,"},{"Start":"07:55.440 ","End":"08:01.600","Text":"if I take the square root is that h equals take the square root of each piece separately,"},{"Start":"08:01.600 ","End":"08:04.165","Text":"square root of a 100 is 10."},{"Start":"08:04.165 ","End":"08:07.045","Text":"Here I get the square root of"},{"Start":"08:07.045 ","End":"08:14.395","Text":"100 minus x and here I have over x. I\u0027m going to go back up now."},{"Start":"08:14.395 ","End":"08:21.925","Text":"I can get that this is equal to 1/3 x squared times h,"},{"Start":"08:21.925 ","End":"08:31.480","Text":"10 times the square root of a 100 minus x over x."},{"Start":"08:31.480 ","End":"08:33.565","Text":"This is what I wanted."},{"Start":"08:33.565 ","End":"08:38.080","Text":"I wanted what v is in terms of x, and here it is."},{"Start":"08:38.080 ","End":"08:39.910","Text":"Now just simplify a bit."},{"Start":"08:39.910 ","End":"08:42.475","Text":"Let\u0027s see what we can do with this."},{"Start":"08:42.475 ","End":"08:49.060","Text":"Of course, the x here can cancel with the x squared, leaving just x."},{"Start":"08:49.060 ","End":"08:52.720","Text":"What I get is 10/3,"},{"Start":"08:52.720 ","End":"09:01.615","Text":"which is a constant times x times square root of 100 minus x."},{"Start":"09:01.615 ","End":"09:09.025","Text":"That\u0027s my function V in terms of x and the 1 that I need to maximize."},{"Start":"09:09.025 ","End":"09:15.250","Text":"What I want is the maximum of V. I just"},{"Start":"09:15.250 ","End":"09:21.460","Text":"noticed that I made a small error in copying, it\u0027s right here."},{"Start":"09:21.460 ","End":"09:25.825","Text":"When I multiply out 200 minus x squared squared,"},{"Start":"09:25.825 ","End":"09:30.205","Text":"the middle term is twice this times this is 400x squared."},{"Start":"09:30.205 ","End":"09:34.990","Text":"I apologize about that and I have to carry this all the way through."},{"Start":"09:34.990 ","End":"09:41.020","Text":"That\u0027s x squared and this is a 100 minus x squared and over"},{"Start":"09:41.020 ","End":"09:47.290","Text":"here it\u0027s x squared and over here it\u0027s x squared."},{"Start":"09:47.290 ","End":"09:50.140","Text":"Caught it in time, no harm done."},{"Start":"09:50.140 ","End":"09:52.884","Text":"V prime of x,"},{"Start":"09:52.884 ","End":"09:55.645","Text":"let\u0027s see what this equals."},{"Start":"09:55.645 ","End":"09:59.110","Text":"The 10/3 will stay."},{"Start":"09:59.110 ","End":"10:02.290","Text":"It\u0027s just a multiplicative constant."},{"Start":"10:02.290 ","End":"10:04.120","Text":"Here I have a product,"},{"Start":"10:04.120 ","End":"10:07.855","Text":"so I\u0027m going to use the product rule and take,"},{"Start":"10:07.855 ","End":"10:10.090","Text":"first of all, derive this 1,"},{"Start":"10:10.090 ","End":"10:14.050","Text":"so I\u0027ve got 1 and leave the other 1 as is square root of"},{"Start":"10:14.050 ","End":"10:20.875","Text":"100 minus x squared plus x as is and the derivative of this."},{"Start":"10:20.875 ","End":"10:23.530","Text":"The derivative of square root you\u0027ve had this many times,"},{"Start":"10:23.530 ","End":"10:30.055","Text":"is 1 over twice the square root and then times the derivative of what\u0027s inside,"},{"Start":"10:30.055 ","End":"10:32.980","Text":"which is minus 2x."},{"Start":"10:32.980 ","End":"10:40.100","Text":"What\u0027s under the square root is a 100 minus x squared and close the brackets."},{"Start":"10:40.470 ","End":"10:43.960","Text":"Let\u0027s see now what we can do with this."},{"Start":"10:43.960 ","End":"10:50.590","Text":"The only thing that seems to cancel is this 2, with this 2."},{"Start":"10:50.590 ","End":"10:54.505","Text":"What I suggest is we put it all over"},{"Start":"10:54.505 ","End":"11:02.000","Text":"this square root and so we\u0027ll get that this is equal to 10/3."},{"Start":"11:02.520 ","End":"11:05.560","Text":"Now on the denominator here,"},{"Start":"11:05.560 ","End":"11:12.520","Text":"the square root of 100 minus x squared. Let\u0027s see."},{"Start":"11:12.520 ","End":"11:18.340","Text":"The first bit is just this thing times itself,"},{"Start":"11:18.340 ","End":"11:20.365","Text":"which means without the square root,"},{"Start":"11:20.365 ","End":"11:22.900","Text":"we have a 100 minus x squared."},{"Start":"11:22.900 ","End":"11:26.305","Text":"That\u0027s for this bit. For this bit we already have the denominator,"},{"Start":"11:26.305 ","End":"11:32.125","Text":"so I\u0027ll take the minus x with the x and get minus x squared."},{"Start":"11:32.125 ","End":"11:35.470","Text":"Now to find the extremum,"},{"Start":"11:35.470 ","End":"11:39.340","Text":"I assign V prime to be 0."},{"Start":"11:39.340 ","End":"11:42.310","Text":"If V prime is equal to 0,"},{"Start":"11:42.310 ","End":"11:46.300","Text":"because that\u0027s where I look for possible maxima and minima,"},{"Start":"11:46.300 ","End":"11:48.520","Text":"then this whole thing is 0."},{"Start":"11:48.520 ","End":"11:50.050","Text":"This fraction must be 0,"},{"Start":"11:50.050 ","End":"11:52.315","Text":"its numerator must be 0."},{"Start":"11:52.315 ","End":"11:59.665","Text":"So I have 100 minus x squared minus x squared."},{"Start":"11:59.665 ","End":"12:02.454","Text":"This has got to be equal to 0."},{"Start":"12:02.454 ","End":"12:07.030","Text":"If you think about this 100 minus 2x squared is 0,"},{"Start":"12:07.030 ","End":"12:15.460","Text":"so x squared is equal to 50 and I can take the square root of this,"},{"Start":"12:15.460 ","End":"12:17.770","Text":"x equals the square root of 50."},{"Start":"12:17.770 ","End":"12:20.680","Text":"But if I write the 50, you don\u0027t have to do this."},{"Start":"12:20.680 ","End":"12:23.770","Text":"But I just noticed that it\u0027s 25 times 2,"},{"Start":"12:23.770 ","End":"12:29.605","Text":"then it might be nicer to write x to the square root of 50 as 5 times square root of 2."},{"Start":"12:29.605 ","End":"12:31.630","Text":"Square root of 25 is this,"},{"Start":"12:31.630 ","End":"12:33.235","Text":"times square root of 2."},{"Start":"12:33.235 ","End":"12:36.290","Text":"That gives us x."},{"Start":"12:36.810 ","End":"12:39.760","Text":"Well, there\u0027s 2 things that we have still to do."},{"Start":"12:39.760 ","End":"12:42.190","Text":"1 is to find the volume and the second is to"},{"Start":"12:42.190 ","End":"12:45.055","Text":"prove that this really is a maximum and not a minimum."},{"Start":"12:45.055 ","End":"12:48.145","Text":"Let me show that it\u0027s a maximum first."},{"Start":"12:48.145 ","End":"12:51.370","Text":"I\u0027m going to use the table method of showing that it\u0027s a maximum,"},{"Start":"12:51.370 ","End":"12:55.690","Text":"but I also for convenience would like to rewrite V prime."},{"Start":"12:55.690 ","End":"13:02.845","Text":"Now, the point I want to make is I\u0027m going to be substituting various values in V prime,"},{"Start":"13:02.845 ","End":"13:04.480","Text":"not to get the answer,"},{"Start":"13:04.480 ","End":"13:09.145","Text":"but just to get whether it comes out positive or negative."},{"Start":"13:09.145 ","End":"13:14.320","Text":"The only thing that really matters is this bit here,"},{"Start":"13:14.320 ","End":"13:18.520","Text":"because this is positive and this fraction is positive."},{"Start":"13:18.520 ","End":"13:22.000","Text":"If I substitute here and it comes out positive or negative,"},{"Start":"13:22.000 ","End":"13:24.700","Text":"that\u0027s applies to the whole of V prime."},{"Start":"13:24.700 ","End":"13:29.830","Text":"My table has 3 rows to it."},{"Start":"13:29.830 ","End":"13:37.510","Text":"It has x, it has the dependent variable and its derivative."},{"Start":"13:37.510 ","End":"13:40.075","Text":"In here we put all our suspects,"},{"Start":"13:40.075 ","End":"13:46.600","Text":"but we only have 1 suspect and that is 5 square root of 2."},{"Start":"13:46.600 ","End":"13:52.300","Text":"If I put in 5 square root of 2 I\u0027d better get an approximate value for that."},{"Start":"13:52.300 ","End":"13:54.745","Text":"Let me look at the calculator,"},{"Start":"13:54.745 ","End":"14:03.670","Text":"I make it approximately 7.071, something like that."},{"Start":"14:03.670 ","End":"14:06.280","Text":"Anyway, it\u0027s over 7."},{"Start":"14:06.280 ","End":"14:08.695","Text":"If I put this here,"},{"Start":"14:08.695 ","End":"14:11.500","Text":"5 square root of 2,"},{"Start":"14:11.500 ","End":"14:13.900","Text":"I put it in V prime,"},{"Start":"14:13.900 ","End":"14:17.860","Text":"then this comes out to be 0, which I expected."},{"Start":"14:17.860 ","End":"14:20.350","Text":"But I want to know what happens before and after if"},{"Start":"14:20.350 ","End":"14:23.530","Text":"it\u0027s increasing or decreasing the function."},{"Start":"14:23.530 ","End":"14:27.370","Text":"Take something bigger, I\u0027ll take 8,"},{"Start":"14:27.370 ","End":"14:30.610","Text":"and I\u0027ll also take something smaller."},{"Start":"14:30.610 ","End":"14:32.905","Text":"Could just be 7,"},{"Start":"14:32.905 ","End":"14:36.010","Text":"6, whatever, unless it\u0027s smaller."},{"Start":"14:36.010 ","End":"14:38.035","Text":"Now, as I said,"},{"Start":"14:38.035 ","End":"14:44.530","Text":"I don\u0027t want V prime itself only its sign so I can substitute in this highlighted bit."},{"Start":"14:44.530 ","End":"14:48.175","Text":"If I put x equals 8,"},{"Start":"14:48.175 ","End":"14:51.535","Text":"8 squared is 64 times 2 is 128."},{"Start":"14:51.535 ","End":"14:54.730","Text":"This comes out to be negative."},{"Start":"14:54.730 ","End":"14:57.294","Text":"If I put x equals 7,"},{"Start":"14:57.294 ","End":"15:00.370","Text":"7 squared is 49 times 2 is 98,"},{"Start":"15:00.370 ","End":"15:03.700","Text":"100 minus 98 is positive."},{"Start":"15:03.700 ","End":"15:08.920","Text":"The function is here increasing and here decreasing,"},{"Start":"15:08.920 ","End":"15:12.320","Text":"and just the look of it means that the shape is like this."},{"Start":"15:12.320 ","End":"15:15.930","Text":"This is a maximum indeed."},{"Start":"15:16.210 ","End":"15:23.300","Text":"What I highlighted in green is indeed the x that gives the maximum,"},{"Start":"15:23.300 ","End":"15:25.640","Text":"except that what they asked for was"},{"Start":"15:25.640 ","End":"15:29.045","Text":"the maximum itself so we have to know what V itself is."},{"Start":"15:29.045 ","End":"15:32.735","Text":"Let me rewrite what V was."},{"Start":"15:32.735 ","End":"15:35.450","Text":"Here it is, I just copied the formula from above."},{"Start":"15:35.450 ","End":"15:41.270","Text":"What I want is to put this x into here."},{"Start":"15:41.270 ","End":"15:46.960","Text":"I want V of 5 square root of 2 and see what this is equal to."},{"Start":"15:46.960 ","End":"15:51.800","Text":"This is 10/3, x is 5 root 2,"},{"Start":"15:51.800 ","End":"15:57.235","Text":"and the square root of 100 minus,"},{"Start":"15:57.235 ","End":"16:02.385","Text":"now if I square this thing gets 25 times 2, which is 50."},{"Start":"16:02.385 ","End":"16:04.450","Text":"What I get is,"},{"Start":"16:04.450 ","End":"16:08.335","Text":"let\u0027s see, 10 times 5 is 50."},{"Start":"16:08.335 ","End":"16:15.375","Text":"I get 50/3 square root of 2,"},{"Start":"16:15.375 ","End":"16:18.245","Text":"and here square root of 50."},{"Start":"16:18.245 ","End":"16:22.690","Text":"But the square root of 2 times the square root of 50,"},{"Start":"16:22.690 ","End":"16:26.680","Text":"this I can combine together as a square root of 100."},{"Start":"16:26.680 ","End":"16:31.315","Text":"If this is the square root of 100 we\u0027ll do this in our heads, it\u0027s 10."},{"Start":"16:31.315 ","End":"16:38.930","Text":"This is equal to 50 times 10/3, which is 500/3."},{"Start":"16:38.930 ","End":"16:44.855","Text":"This is now going to be our answer for the volume,"},{"Start":"16:44.855 ","End":"16:46.580","Text":"and that answers the question."},{"Start":"16:46.580 ","End":"16:49.380","Text":"So we are done."}],"ID":8576},{"Watched":false,"Name":"Exercise 21","Duration":"11m 42s","ChapterTopicVideoID":8385,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8385.jpeg","UploadDate":"2017-03-07T06:24:00.3870000","DurationForVideoObject":"PT11M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.280","Text":"Here we have another word problem in geometry,"},{"Start":"00:03.280 ","End":"00:05.340","Text":"let me explain what\u0027s going on."},{"Start":"00:05.340 ","End":"00:08.580","Text":"We have a cylinder as in the picture,"},{"Start":"00:08.580 ","End":"00:09.810","Text":"and it\u0027s a right cylinder."},{"Start":"00:09.810 ","End":"00:13.545","Text":"That just means that if I take its axis,"},{"Start":"00:13.545 ","End":"00:20.500","Text":"then the axis is perpendicular to the base, to the basis."},{"Start":"00:20.660 ","End":"00:27.315","Text":"This rectangle here is called the axial cross-section."},{"Start":"00:27.315 ","End":"00:30.870","Text":"What we\u0027re given is its diagonal length of 12."},{"Start":"00:30.870 ","End":"00:32.835","Text":"Everything is in centimeters."},{"Start":"00:32.835 ","End":"00:38.600","Text":"What we have to find is from all such cylinders which"},{"Start":"00:38.600 ","End":"00:44.344","Text":"have this 12 in the diagonal of the axial cross-section,"},{"Start":"00:44.344 ","End":"00:49.055","Text":"which has the largest volume, volume is maximal?"},{"Start":"00:49.055 ","End":"00:54.589","Text":"This tells us already that volume is going to be the dependent variable."},{"Start":"00:54.589 ","End":"00:58.730","Text":"That\u0027s the variable we\u0027re going to have to find the maximum of."},{"Start":"00:58.730 ","End":"01:01.685","Text":"Let me just write something in general."},{"Start":"01:01.685 ","End":"01:05.480","Text":"That in general for a cylinder,"},{"Start":"01:05.480 ","End":"01:07.640","Text":"we have a formula from geometry,"},{"Start":"01:07.640 ","End":"01:12.620","Text":"V equals Pi r squared h,"},{"Start":"01:12.620 ","End":"01:16.875","Text":"where r is the radius of the base and h is the height."},{"Start":"01:16.875 ","End":"01:19.260","Text":"Now there\u0027s nothing marked here."},{"Start":"01:19.260 ","End":"01:24.510","Text":"Let\u0027s take x as the radius."},{"Start":"01:24.510 ","End":"01:27.765","Text":"This is x and this is x,"},{"Start":"01:27.765 ","End":"01:32.440","Text":"and let\u0027s take the height as y."},{"Start":"01:32.990 ","End":"01:37.565","Text":"This gives us that in terms of our variables,"},{"Start":"01:37.565 ","End":"01:47.715","Text":"that V is equal to Pi x squared y."},{"Start":"01:47.715 ","End":"01:51.454","Text":"Now I only want 1 independent variable,"},{"Start":"01:51.454 ","End":"01:54.910","Text":"and I\u0027m going to use x as the independent variable."},{"Start":"01:54.910 ","End":"01:57.935","Text":"What I have to do is get y in terms of x,"},{"Start":"01:57.935 ","End":"02:00.515","Text":"and then I\u0027ll have V just in terms of x."},{"Start":"02:00.515 ","End":"02:03.950","Text":"What I\u0027m going to do is that I\u0027m going to use the piece of"},{"Start":"02:03.950 ","End":"02:07.495","Text":"information that we have, that this is 12,"},{"Start":"02:07.495 ","End":"02:11.420","Text":"in order to find a relationship between y and x. I"},{"Start":"02:11.420 ","End":"02:15.320","Text":"want to use Pythagoras\u0027s theorem and I want to show you which triangle."},{"Start":"02:15.320 ","End":"02:21.440","Text":"I\u0027m going to take the triangle as this and this, and this."},{"Start":"02:21.440 ","End":"02:23.765","Text":"This is a right triangle;"},{"Start":"02:23.765 ","End":"02:27.500","Text":"I can say that y squared"},{"Start":"02:27.500 ","End":"02:34.610","Text":"plus 2x squared is equal to 12 squared."},{"Start":"02:34.610 ","End":"02:38.525","Text":"That\u0027s Pythagoras\u0027s theorem on the yellow highlighted triangle."},{"Start":"02:38.525 ","End":"02:44.840","Text":"This gives me that y squared equals, let\u0027s see,"},{"Start":"02:44.840 ","End":"02:47.780","Text":"the 12 squared is a 144 and I\u0027m going to bring"},{"Start":"02:47.780 ","End":"02:50.975","Text":"this to the other side and also at the same time,"},{"Start":"02:50.975 ","End":"02:53.585","Text":"open it up as 4x squared,"},{"Start":"02:53.585 ","End":"02:56.785","Text":"which gives me that y."},{"Start":"02:56.785 ","End":"03:00.200","Text":"Let me also take the 4 outside."},{"Start":"03:00.200 ","End":"03:01.910","Text":"I like to cancel where possible,"},{"Start":"03:01.910 ","End":"03:03.920","Text":"especially since 4 is a square number,"},{"Start":"03:03.920 ","End":"03:07.280","Text":"and get 36 minus x squared."},{"Start":"03:07.280 ","End":"03:11.285","Text":"Y is equal to the square root of this,"},{"Start":"03:11.285 ","End":"03:13.220","Text":"which is the square root of 4 is 2."},{"Start":"03:13.220 ","End":"03:18.365","Text":"The square root of this is just the square root of 36 minus x squared."},{"Start":"03:18.365 ","End":"03:20.360","Text":"Now that I have that,"},{"Start":"03:20.360 ","End":"03:25.935","Text":"then I can go back here and say that this is equal to"},{"Start":"03:25.935 ","End":"03:32.310","Text":"Pi times x squared times y which is twice this."},{"Start":"03:32.310 ","End":"03:38.460","Text":"I\u0027ll put the 2 in front of the Pi and here 36 minus x squared."},{"Start":"03:38.460 ","End":"03:48.370","Text":"Basically, this is my equation for V in terms of x. I\u0027ll highlight that."},{"Start":"03:50.900 ","End":"03:54.490","Text":"This is okay."},{"Start":"03:55.370 ","End":"03:57.890","Text":"That\u0027s the geometry part."},{"Start":"03:57.890 ","End":"04:02.660","Text":"Now let\u0027s get onto the calculus because we have an extremum problems specifically,"},{"Start":"04:02.660 ","End":"04:05.410","Text":"you want to find the maximal volume."},{"Start":"04:05.410 ","End":"04:08.030","Text":"Now normally what we would do would be to"},{"Start":"04:08.030 ","End":"04:12.065","Text":"differentiate and then find critical points by setting to 0,"},{"Start":"04:12.065 ","End":"04:14.255","Text":"but because of the square root here,"},{"Start":"04:14.255 ","End":"04:16.325","Text":"differentiation will be messy."},{"Start":"04:16.325 ","End":"04:21.210","Text":"So we\u0027ll use our standard trick that in general,"},{"Start":"04:21.210 ","End":"04:24.755","Text":"when we have a positive function which this 1 is,"},{"Start":"04:24.755 ","End":"04:29.420","Text":"we can square it and then the maximum or minimum will occur for the same"},{"Start":"04:29.420 ","End":"04:34.530","Text":"x. I\u0027m going to pick a letter I can\u0027t use x or y."},{"Start":"04:34.530 ","End":"04:40.140","Text":"See, after V is make it W. Let\u0027s take W to be V squared"},{"Start":"04:40.140 ","End":"04:50.020","Text":"which is 4Pi squared x^4,"},{"Start":"04:50.020 ","End":"04:55.525","Text":"and then just 36 minus x squared."},{"Start":"04:55.525 ","End":"04:58.305","Text":"Let me open the brackets here,"},{"Start":"04:58.305 ","End":"05:00.420","Text":"W is 4Pi squared."},{"Start":"05:00.420 ","End":"05:01.990","Text":"At least I\u0027ll leave out the brackets,"},{"Start":"05:01.990 ","End":"05:04.975","Text":"but I\u0027ll put the x^4 inside."},{"Start":"05:04.975 ","End":"05:09.475","Text":"I\u0027ve got 36x^4 minus x^6."},{"Start":"05:09.475 ","End":"05:19.965","Text":"Now we\u0027ll find the maximum or minimum or whatever for w. W prime is 4Pi squared sticks,"},{"Start":"05:19.965 ","End":"05:29.050","Text":"4 times 36 is a 144x cubed minus 6x^5."},{"Start":"05:31.540 ","End":"05:35.240","Text":"I can take stuff outside the brackets."},{"Start":"05:35.240 ","End":"05:40.475","Text":"I see that 6 goes into this and x cubed goes into this."},{"Start":"05:40.475 ","End":"05:48.540","Text":"I\u0027ll take the 6x cubed out of the brackets to the front."},{"Start":"05:48.540 ","End":"05:59.805","Text":"Let\u0027s see 6 with the 4 is 24Pi squared x cubed."},{"Start":"05:59.805 ","End":"06:09.510","Text":"Then what we\u0027re left with after we take 6 out is just 24 minus,"},{"Start":"06:09.510 ","End":"06:14.085","Text":"I have taken out 6, 24 minus x squared."},{"Start":"06:14.085 ","End":"06:17.100","Text":"If W prime equals 0,"},{"Start":"06:17.100 ","End":"06:18.405","Text":"which is for critical point,"},{"Start":"06:18.405 ","End":"06:21.510","Text":"this part is positive,"},{"Start":"06:21.510 ","End":"06:23.060","Text":"you don\u0027t have to mess with that."},{"Start":"06:23.060 ","End":"06:25.735","Text":"We need this to be 0."},{"Start":"06:25.735 ","End":"06:30.335","Text":"We get x squared equals 24,"},{"Start":"06:30.335 ","End":"06:32.975","Text":"but x is positive,"},{"Start":"06:32.975 ","End":"06:38.630","Text":"so x is the positive square root of 24."},{"Start":"06:38.630 ","End":"06:42.610","Text":"That\u0027s an important intermediate result."},{"Start":"06:42.610 ","End":"06:47.095","Text":"In fact, that\u0027s 1 of the things we wanted, I believe."},{"Start":"06:47.095 ","End":"06:49.365","Text":"Yeah, we need to find x and y."},{"Start":"06:49.365 ","End":"06:53.970","Text":"But we haven\u0027t shown that this is a maximum yet."},{"Start":"06:54.100 ","End":"06:56.525","Text":"The several ways of doing it,"},{"Start":"06:56.525 ","End":"07:00.660","Text":"I\u0027ll use the second derivative test."},{"Start":"07:01.850 ","End":"07:05.840","Text":"First of all, we\u0027ll take W double prime,"},{"Start":"07:05.840 ","End":"07:10.430","Text":"the second derivative, and then we substitute it to our given x."},{"Start":"07:10.430 ","End":"07:13.910","Text":"In general, second derivative is,"},{"Start":"07:13.910 ","End":"07:15.845","Text":"I\u0027m reading from here,"},{"Start":"07:15.845 ","End":"07:18.950","Text":"the 4Pi squared is still there,"},{"Start":"07:18.950 ","End":"07:25.260","Text":"the derivative of this 3 times 144, now those are big numbers."},{"Start":"07:25.260 ","End":"07:26.760","Text":"You know what, let\u0027s just take the 6 out,"},{"Start":"07:26.760 ","End":"07:30.120","Text":"not the 6x cubed, that\u0027s 24 here."},{"Start":"07:30.120 ","End":"07:36.165","Text":"Now this would be,"},{"Start":"07:36.165 ","End":"07:38.310","Text":"back to W prime,"},{"Start":"07:38.310 ","End":"07:41.370","Text":"just retaking the 6 out to work with smaller numbers."},{"Start":"07:41.370 ","End":"07:45.780","Text":"It\u0027s 24 x cubed minus x^5."},{"Start":"07:45.780 ","End":"07:53.200","Text":"Now I\u0027ll do the second derivative which is 24Pi squared."},{"Start":"07:56.450 ","End":"07:58.830","Text":"That\u0027s the constant part."},{"Start":"07:58.830 ","End":"08:00.615","Text":"Here is 3 times 24,"},{"Start":"08:00.615 ","End":"08:03.900","Text":"72x squared and"},{"Start":"08:03.900 ","End":"08:12.930","Text":"then minus 5x^4."},{"Start":"08:12.930 ","End":"08:17.040","Text":"Now I can take the x squared out;"},{"Start":"08:17.040 ","End":"08:22.365","Text":"24Pi squared x squared"},{"Start":"08:22.365 ","End":"08:28.815","Text":"times 72 minus 5x squared."},{"Start":"08:28.815 ","End":"08:30.785","Text":"Now the question is,"},{"Start":"08:30.785 ","End":"08:37.310","Text":"what happens when I plug in the square root of 24 for x?"},{"Start":"08:37.310 ","End":"08:42.780","Text":"Because this is W prime double prime as a function of x."},{"Start":"08:43.000 ","End":"08:45.365","Text":"What we get, well,"},{"Start":"08:45.365 ","End":"08:52.085","Text":"this part is positive and all I need to know is whether it\u0027s positive or negative."},{"Start":"08:52.085 ","End":"08:56.285","Text":"If I just plug in this part,"},{"Start":"08:56.285 ","End":"08:59.870","Text":"I\u0027ve got 72 minus"},{"Start":"08:59.870 ","End":"09:07.820","Text":"5 times root 24 squared,"},{"Start":"09:07.820 ","End":"09:10.760","Text":"which is let\u0027s see,"},{"Start":"09:10.760 ","End":"09:18.120","Text":"72 minus 24 square root squared is just 24,"},{"Start":"09:18.120 ","End":"09:24.075","Text":"5 times 24 is a 120, well, it\u0027s negative."},{"Start":"09:24.075 ","End":"09:26.025","Text":"This part is positive,"},{"Start":"09:26.025 ","End":"09:27.870","Text":"this part is negative,"},{"Start":"09:27.870 ","End":"09:30.229","Text":"so this is negative."},{"Start":"09:30.229 ","End":"09:32.795","Text":"We don\u0027t need the value just the sign."},{"Start":"09:32.795 ","End":"09:34.400","Text":"If it\u0027s a negative,"},{"Start":"09:34.400 ","End":"09:39.660","Text":"then it means that we do have a maximum."},{"Start":"09:40.480 ","End":"09:45.050","Text":"But we\u0027re not done because we had to find x and y. X was"},{"Start":"09:45.050 ","End":"09:48.905","Text":"the radius of the base of the cylinder and y was the height of the cylinder."},{"Start":"09:48.905 ","End":"09:53.940","Text":"Let\u0027s see, let me go back to where I got y in terms of x."},{"Start":"09:54.680 ","End":"09:58.060","Text":"We have it somewhere."},{"Start":"09:59.470 ","End":"10:06.900","Text":"There it is. I want to see them both."},{"Start":"10:06.900 ","End":"10:12.010","Text":"I\u0027m going to take this and plug it into here."},{"Start":"10:12.830 ","End":"10:16.680","Text":"We\u0027ll call this x, well, it\u0027s not x max,"},{"Start":"10:16.680 ","End":"10:22.060","Text":"it\u0027s x that gives the maximum volume."},{"Start":"10:22.100 ","End":"10:27.980","Text":"I\u0027ll get the y max which is the y that gives the maximum volume."},{"Start":"10:27.980 ","End":"10:29.810","Text":"Was it volume that they wanted?"},{"Start":"10:29.810 ","End":"10:33.480","Text":"Yeah, volume maximal."},{"Start":"10:34.850 ","End":"10:45.595","Text":"We get twice the square root of 36 minus x squared is just 24,"},{"Start":"10:45.595 ","End":"10:52.590","Text":"this is equal to 2 root 12."},{"Start":"10:52.590 ","End":"10:59.400","Text":"I\u0027ll write again y max and highlight that."},{"Start":"10:59.400 ","End":"11:06.305","Text":"This and this together give us the x and the y."},{"Start":"11:06.305 ","End":"11:10.090","Text":"You might be able to simplify this a little bit."},{"Start":"11:10.090 ","End":"11:15.840","Text":"Actually this is square root of 4 times 6."},{"Start":"11:15.840 ","End":"11:20.435","Text":"This is actually twice square root of 6."},{"Start":"11:20.435 ","End":"11:25.790","Text":"This is the square root of 4 times 3."},{"Start":"11:25.790 ","End":"11:33.335","Text":"It\u0027s twice square root of 3 times the 2,so this is 4 times square root of 3."},{"Start":"11:33.335 ","End":"11:39.590","Text":"Anyway, I don\u0027t want to do a numerical evaluation on the calculator."},{"Start":"11:39.590 ","End":"11:42.870","Text":"We\u0027ll just leave it at that and we\u0027re done."}],"ID":8577},{"Watched":false,"Name":"Exercise 22","Duration":"8m 32s","ChapterTopicVideoID":8386,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8386.jpeg","UploadDate":"2017-01-28T13:49:46.3170000","DurationForVideoObject":"PT8M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"Here we have another geometrical problem."},{"Start":"00:03.525 ","End":"00:07.545","Text":"What\u0027s going on here is that we have a cylindrical container,"},{"Start":"00:07.545 ","End":"00:11.610","Text":"meaning it\u0027s not a solid cylinder, it\u0027s hollow."},{"Start":"00:11.610 ","End":"00:14.115","Text":"It\u0027s open on top, means it has a base."},{"Start":"00:14.115 ","End":"00:16.875","Text":"It\u0027s like a tin can that\u0027s been opened."},{"Start":"00:16.875 ","End":"00:21.750","Text":"There is a base but there\u0027s no top and we\u0027re given that its volume is"},{"Start":"00:21.750 ","End":"00:26.895","Text":"64 meters cubed and it\u0027s made entirely of tin."},{"Start":"00:26.895 ","End":"00:29.910","Text":"Of course, you want to save on the cost,"},{"Start":"00:29.910 ","End":"00:32.729","Text":"so we want the area of tin to be minimal."},{"Start":"00:32.729 ","End":"00:35.850","Text":"The question asked us to show that this is"},{"Start":"00:35.850 ","End":"00:40.050","Text":"true when the base radius is 4 over cube root of Pi."},{"Start":"00:40.050 ","End":"00:43.740","Text":"Let\u0027s mark some variables."},{"Start":"00:43.740 ","End":"00:46.710","Text":"Let the base radius be this,"},{"Start":"00:46.710 ","End":"00:52.935","Text":"I\u0027ll call it x and let the height be y."},{"Start":"00:52.935 ","End":"00:57.150","Text":"In general, there\u0027s 2 formulas we need from geometry,"},{"Start":"00:57.150 ","End":"01:00.625","Text":"1 is for volume and 1 is for surface area."},{"Start":"01:00.625 ","End":"01:04.250","Text":"The area means the surface area of the tin here."},{"Start":"01:04.250 ","End":"01:08.075","Text":"Now, the volume is given by, in general,"},{"Start":"01:08.075 ","End":"01:15.525","Text":"it\u0027s Pir squared H. But here radius is x and the H is y."},{"Start":"01:15.525 ","End":"01:18.710","Text":"The volume is Pix squared y."},{"Start":"01:18.710 ","End":"01:24.335","Text":"The surface area is given by the area of the base,"},{"Start":"01:24.335 ","End":"01:25.530","Text":"there\u0027s only 1 base,"},{"Start":"01:25.530 ","End":"01:31.545","Text":"so that\u0027s Pix squared and we have the sides."},{"Start":"01:31.545 ","End":"01:34.980","Text":"This part is given by, well, you don\u0027t have to remember it,"},{"Start":"01:34.980 ","End":"01:38.660","Text":"you can just think of it as the length of the circumference times the height."},{"Start":"01:38.660 ","End":"01:45.135","Text":"The length of the circumference is 2Pix and here times the y."},{"Start":"01:45.135 ","End":"01:48.840","Text":"This is the formula for this amount of tin."},{"Start":"01:48.840 ","End":"01:53.905","Text":"Now, we\u0027re given that S is 64."},{"Start":"01:53.905 ","End":"01:55.925","Text":"Now here\u0027s my plan."},{"Start":"01:55.925 ","End":"01:58.760","Text":"Using the fact that S is 64,"},{"Start":"01:58.760 ","End":"02:04.145","Text":"I get a relationship between x and y. I can get y in terms of x."},{"Start":"02:04.145 ","End":"02:06.395","Text":"The plan is this,"},{"Start":"02:06.395 ","End":"02:09.620","Text":"the dependent variable is the thing we want to minimize,"},{"Start":"02:09.620 ","End":"02:15.740","Text":"and that\u0027s the area I would have called S. S is equal to,"},{"Start":"02:15.740 ","End":"02:21.420","Text":"I\u0027ll write it again, Pix squared plus 2Pixy."},{"Start":"02:21.560 ","End":"02:25.100","Text":"But I want 1 independent variable,"},{"Start":"02:25.100 ","End":"02:29.120","Text":"and I want that variable to be x. I don\u0027t want the y here."},{"Start":"02:29.120 ","End":"02:32.735","Text":"What I\u0027m going to do is use the fact that I have V,"},{"Start":"02:32.735 ","End":"02:36.760","Text":"that V is equal to in fact 64."},{"Start":"02:36.760 ","End":"02:42.650","Text":"Then I\u0027ll get y in terms of x and substitute that here,"},{"Start":"02:42.650 ","End":"02:46.415","Text":"and then I\u0027ll get S in terms of all x. Let\u0027s see."},{"Start":"02:46.415 ","End":"02:47.930","Text":"If this is true,"},{"Start":"02:47.930 ","End":"02:49.860","Text":"then I can see it right away."},{"Start":"02:49.860 ","End":"02:58.029","Text":"From this and this y is equal to 64 over Pix squared."},{"Start":"02:58.029 ","End":"03:07.105","Text":"Now, I can put that in here and get that this is equal to Pix squared plus 2Pix,"},{"Start":"03:07.105 ","End":"03:15.080","Text":"and instead of y, 64 over Pix squared,"},{"Start":"03:15.080 ","End":"03:19.220","Text":"Let\u0027s start simplifying and see what we can do."},{"Start":"03:19.220 ","End":"03:24.020","Text":"Certainly, this x will cancel with 1 of these xs,"},{"Start":"03:24.020 ","End":"03:26.990","Text":"so I\u0027ll just cancel the twos, so to speak."},{"Start":"03:26.990 ","End":"03:31.350","Text":"This Pi will cancel with this Pi."},{"Start":"03:31.420 ","End":"03:37.880","Text":"What we are left with is that S is equal"},{"Start":"03:37.880 ","End":"03:48.979","Text":"to Pix squared plus what are we left with twice 64 is 128 over x."},{"Start":"03:48.979 ","End":"03:57.650","Text":"Because this is our main equation relating the area to the radius of the base."},{"Start":"03:57.650 ","End":"04:02.135","Text":"We\u0027re hoping to show that x is going to equal this when S is minimum."},{"Start":"04:02.135 ","End":"04:06.030","Text":"Now, this is the end of the geometry part."},{"Start":"04:06.030 ","End":"04:08.525","Text":"Now, we come to the calculus part."},{"Start":"04:08.525 ","End":"04:11.905","Text":"We want to find minimum S,"},{"Start":"04:11.905 ","End":"04:14.765","Text":"minimum or maximum we differentiate."},{"Start":"04:14.765 ","End":"04:19.100","Text":"I\u0027m going to get that S prime is equal to"},{"Start":"04:19.100 ","End":"04:25.010","Text":"2Pix and the derivative of 1 over x is minus 1 over x squared,"},{"Start":"04:25.010 ","End":"04:29.230","Text":"so minus 128 over x squared."},{"Start":"04:29.230 ","End":"04:32.375","Text":"Now, for an extremum maximum or minimum,"},{"Start":"04:32.375 ","End":"04:36.725","Text":"I check where S prime is 0 and get my suspects."},{"Start":"04:36.725 ","End":"04:41.295","Text":"If this is 0, then I get that"},{"Start":"04:41.295 ","End":"04:49.475","Text":"2Pix minus 128 over x squared equals 0."},{"Start":"04:49.475 ","End":"04:54.640","Text":"I can multiply by x squared."},{"Start":"04:54.640 ","End":"04:59.940","Text":"I\u0027ve got that 2Pix cubed,"},{"Start":"04:59.940 ","End":"05:03.315","Text":"and I can bring the 128 to the other side,"},{"Start":"05:03.315 ","End":"05:06.880","Text":"so Pix cubed is 64."},{"Start":"05:06.880 ","End":"05:10.040","Text":"If I divide by 2, that\u0027s a 3 there."},{"Start":"05:10.040 ","End":"05:12.525","Text":"We shouldn\u0027t divide it by Pi."},{"Start":"05:12.525 ","End":"05:17.835","Text":"Let\u0027s just erase the Pi and put it on the other side."},{"Start":"05:17.835 ","End":"05:23.020","Text":"Now, I can take the cube root of both sides and get x equals."},{"Start":"05:23.020 ","End":"05:25.210","Text":"Now, the cube root of 64 is just"},{"Start":"05:25.210 ","End":"05:30.010","Text":"4 and the cube root of Pi that\u0027s just the cube root of Pi."},{"Start":"05:30.010 ","End":"05:31.540","Text":"I can do it on the calculator,"},{"Start":"05:31.540 ","End":"05:32.770","Text":"but that\u0027s just the expression."},{"Start":"05:32.770 ","End":"05:35.530","Text":"I\u0027m going to leave it like that because if we look up,"},{"Start":"05:35.530 ","End":"05:39.040","Text":"then this is exactly what we were asked to find."},{"Start":"05:39.040 ","End":"05:43.490","Text":"But we\u0027re not done because here talks"},{"Start":"05:43.490 ","End":"05:48.050","Text":"about minimal and we haven\u0027t shown whether this suspect for extremum is a minimum."},{"Start":"05:48.050 ","End":"05:50.060","Text":"It could be a maximum or something else."},{"Start":"05:50.060 ","End":"05:53.495","Text":"Let\u0027s use the table method."},{"Start":"05:53.495 ","End":"05:55.490","Text":"This also the second derivative method."},{"Start":"05:55.490 ","End":"05:58.190","Text":"I prefer the table when I can."},{"Start":"05:58.190 ","End":"06:01.220","Text":"Let\u0027s make a table of values."},{"Start":"06:01.220 ","End":"06:04.775","Text":"The table is as follows."},{"Start":"06:04.775 ","End":"06:09.200","Text":"Here I put the independent variable x here,"},{"Start":"06:09.200 ","End":"06:11.690","Text":"the dependent variable prime."},{"Start":"06:11.690 ","End":"06:13.670","Text":"Here, the dependent variable,"},{"Start":"06:13.670 ","End":"06:16.070","Text":"here the list of suspects."},{"Start":"06:16.070 ","End":"06:24.214","Text":"There is only 1 suspect and that is 4 over cube root of Pi."},{"Start":"06:24.214 ","End":"06:26.750","Text":"To make the computation a bit easier,"},{"Start":"06:26.750 ","End":"06:31.130","Text":"let me just rewrite S prime to be just like I did"},{"Start":"06:31.130 ","End":"06:35.810","Text":"here when I took the 2 out and I put the x cubed multiplied by x squared."},{"Start":"06:35.810 ","End":"06:43.040","Text":"Let me just say here that I can take 2 out and I can also put it over x squared."},{"Start":"06:43.040 ","End":"06:49.685","Text":"What I get is Pix cubed minus 64."},{"Start":"06:49.685 ","End":"06:51.650","Text":"Just to check that we\u0027re on track."},{"Start":"06:51.650 ","End":"06:54.320","Text":"If I put in this value of x, I should get 0."},{"Start":"06:54.320 ","End":"06:56.090","Text":"Let\u0027s see that we do."},{"Start":"06:56.090 ","End":"06:59.495","Text":"If x is 4 over cube root of Pi,"},{"Start":"06:59.495 ","End":"07:07.905","Text":"x cubed is just 64 over Pi multiplied by Pi or 64 minus 64 is 0."},{"Start":"07:07.905 ","End":"07:09.560","Text":"That looks okay."},{"Start":"07:09.560 ","End":"07:13.040","Text":"Let\u0027s take a value smaller and a value larger."},{"Start":"07:13.040 ","End":"07:15.770","Text":"It\u0027s suits me to keep the cube root of Pi,"},{"Start":"07:15.770 ","End":"07:17.570","Text":"I don\u0027t have to take whole numbers."},{"Start":"07:17.570 ","End":"07:25.175","Text":"I could take 5 over cube root of Pi and I could take 3 over cube root of Pi."},{"Start":"07:25.175 ","End":"07:29.715","Text":"Then if I put this value in here,"},{"Start":"07:29.715 ","End":"07:39.260","Text":"then here I get x cubed is equal to 27 over Pi."},{"Start":"07:39.260 ","End":"07:47.540","Text":"In here, it\u0027s 27 and 27 minus 64 is negative and the rest of it\u0027s positive,"},{"Start":"07:47.540 ","End":"07:49.250","Text":"so this is negative."},{"Start":"07:49.250 ","End":"07:51.770","Text":"Similarly, if I put 5 in,"},{"Start":"07:51.770 ","End":"07:58.425","Text":"I\u0027ll just get inside the brackets 125 minus 64, which is positive."},{"Start":"07:58.425 ","End":"08:00.870","Text":"Here the function is decreasing,"},{"Start":"08:00.870 ","End":"08:02.930","Text":"here the function is increasing,"},{"Start":"08:02.930 ","End":"08:05.270","Text":"which means that the shape is like this,"},{"Start":"08:05.270 ","End":"08:07.990","Text":"which is the shape of a minimum."},{"Start":"08:07.990 ","End":"08:15.890","Text":"We\u0027ve verified that it\u0027s a minimum and I\u0027ll just highlight the answers."},{"Start":"08:15.890 ","End":"08:20.250","Text":"The answer is that x is this,"},{"Start":"08:20.250 ","End":"08:23.385","Text":"which is the radius of, maybe write in words,"},{"Start":"08:23.385 ","End":"08:28.010","Text":"the base is this and the height is this."},{"Start":"08:28.010 ","End":"08:31.234","Text":"This does give us a minimum."},{"Start":"08:31.234 ","End":"08:33.630","Text":"We are done."}],"ID":8578},{"Watched":false,"Name":"Exercise 23","Duration":"9m 1s","ChapterTopicVideoID":8387,"CourseChapterTopicPlaylistID":65389,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8387.jpeg","UploadDate":"2017-03-07T06:25:12.7270000","DurationForVideoObject":"PT9M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"In this exercise, we have a right cone,"},{"Start":"00:03.360 ","End":"00:08.775","Text":"which means that this is 90 degrees."},{"Start":"00:08.775 ","End":"00:13.320","Text":"This is the center of the circle,"},{"Start":"00:13.320 ","End":"00:15.330","Text":"which is the base."},{"Start":"00:15.330 ","End":"00:18.270","Text":"We\u0027re given that the generator is 10,"},{"Start":"00:18.270 ","End":"00:20.800","Text":"everything is in centimeters."},{"Start":"00:20.870 ","End":"00:24.930","Text":"We\u0027ve got to figure out the volume of the cone,"},{"Start":"00:24.930 ","End":"00:29.829","Text":"which is maximal, the greatest possible volume of such a cone."},{"Start":"00:30.200 ","End":"00:34.740","Text":"Let\u0027s mark some variables here."},{"Start":"00:34.740 ","End":"00:40.110","Text":"I\u0027d like to take the radius as x,"},{"Start":"00:40.110 ","End":"00:42.120","Text":"so that\u0027s x and that\u0027s x,"},{"Start":"00:42.120 ","End":"00:47.040","Text":"and let the height be y."},{"Start":"00:47.040 ","End":"00:50.100","Text":"The thing to be maximal is the volume,"},{"Start":"00:50.100 ","End":"00:53.880","Text":"so the volume is going to be our dependent variable."},{"Start":"00:55.340 ","End":"00:58.265","Text":"We\u0027ll have everything in terms of x."},{"Start":"00:58.265 ","End":"01:01.880","Text":"Now initially I\u0027ll write the volume as the standard formula,"},{"Start":"01:01.880 ","End":"01:07.925","Text":"which is 1/3 times area of base times height,"},{"Start":"01:07.925 ","End":"01:13.850","Text":"which is 1/3 times Pi x squared times y."},{"Start":"01:13.850 ","End":"01:20.045","Text":"In general, the volume is 1/3 Pi r squared h,"},{"Start":"01:20.045 ","End":"01:21.530","Text":"but in our case,"},{"Start":"01:21.530 ","End":"01:28.230","Text":"r is x and h is y. I want it all in terms of 1 variable."},{"Start":"01:28.230 ","End":"01:33.000","Text":"I\u0027d like x to be my independent variable so I"},{"Start":"01:33.000 ","End":"01:38.225","Text":"want to get y in terms of x and I haven\u0027t used the fact that the generator is 10,"},{"Start":"01:38.225 ","End":"01:40.040","Text":"and you\u0027re probably thinking what I\u0027m thinking,"},{"Start":"01:40.040 ","End":"01:41.990","Text":"why not use Pythagoras."},{"Start":"01:41.990 ","End":"01:46.140","Text":"Let me just highlight the triangle,"},{"Start":"01:46.540 ","End":"01:49.990","Text":"this, this, and this."},{"Start":"01:49.990 ","End":"01:59.150","Text":"We have that 10 squared equals x squared plus y squared."},{"Start":"01:59.150 ","End":"02:06.530","Text":"It\u0027s easy to see that y is just the square root of 10 squared minus x squared,"},{"Start":"02:06.530 ","End":"02:12.290","Text":"which means that our volume as a function of x is"},{"Start":"02:12.290 ","End":"02:17.030","Text":"1/3 Pi x squared and times"},{"Start":"02:17.030 ","End":"02:23.135","Text":"this square root of 100 minus x squared,"},{"Start":"02:23.135 ","End":"02:29.690","Text":"we want the maximal V. Now we come to the calculus part."},{"Start":"02:29.690 ","End":"02:33.590","Text":"Normally, we would differentiate and then find critical points,"},{"Start":"02:33.590 ","End":"02:36.155","Text":"but because of the square root,"},{"Start":"02:36.155 ","End":"02:38.735","Text":"the derivative is awkward,"},{"Start":"02:38.735 ","End":"02:44.060","Text":"so we use the famous trick of squaring both sides."},{"Start":"02:44.060 ","End":"02:47.810","Text":"You can do this when the function here is always positive."},{"Start":"02:47.810 ","End":"02:50.945","Text":"This is positive x squared, positive square roots."},{"Start":"02:50.945 ","End":"02:54.215","Text":"Let\u0027s take a new letter."},{"Start":"02:54.215 ","End":"02:56.315","Text":"After v comes w,"},{"Start":"02:56.315 ","End":"03:00.010","Text":"let me take W to equal V squared,"},{"Start":"03:00.010 ","End":"03:02.660","Text":"and so W also a function of x,"},{"Start":"03:02.660 ","End":"03:07.550","Text":"is going to equal 1/9 Pi"},{"Start":"03:07.550 ","End":"03:14.945","Text":"squared x^4 100 minus x squared,"},{"Start":"03:14.945 ","End":"03:24.795","Text":"and the maximum x for W will also be the same maximum x for V. Let\u0027s see."},{"Start":"03:24.795 ","End":"03:27.315","Text":"We want to find critical points."},{"Start":"03:27.315 ","End":"03:31.709","Text":"I think I\u0027ll multiply the x^4,"},{"Start":"03:31.709 ","End":"03:37.125","Text":"so we get 1/9 Pi squared"},{"Start":"03:37.125 ","End":"03:42.885","Text":"and then 100x_4 minus x^6,"},{"Start":"03:42.885 ","End":"03:44.795","Text":"4 plus 2 is 6."},{"Start":"03:44.795 ","End":"03:47.900","Text":"That\u0027s going to be easier for me to differentiate."},{"Start":"03:47.900 ","End":"03:51.905","Text":"The constant just stays 1/9 Pi squared."},{"Start":"03:51.905 ","End":"03:54.980","Text":"Here we differentiate, and 4 times 100 is"},{"Start":"03:54.980 ","End":"04:04.205","Text":"400 and it\u0027s x cubed minus 6x^5."},{"Start":"04:04.205 ","End":"04:09.270","Text":"I can also take x cubed out of the brackets."},{"Start":"04:09.270 ","End":"04:13.365","Text":"Let me just say this is Pi squared, x cubed,"},{"Start":"04:13.365 ","End":"04:15.689","Text":"and I\u0027ll write the 9 over here,"},{"Start":"04:15.689 ","End":"04:17.475","Text":"and what are we left with?"},{"Start":"04:17.475 ","End":"04:25.090","Text":"400 minus 6x squared."},{"Start":"04:25.220 ","End":"04:27.745","Text":"We want critical points."},{"Start":"04:27.745 ","End":"04:31.930","Text":"In other words, W prime equals 0,"},{"Start":"04:31.930 ","End":"04:36.530","Text":"which gives us, well, this is not 0 because x is positive."},{"Start":"04:36.530 ","End":"04:39.585","Text":"This would have to be 0,"},{"Start":"04:39.585 ","End":"04:49.545","Text":"so we\u0027d get that 400 minus 6x squared is 0."},{"Start":"04:49.545 ","End":"04:54.245","Text":"That will give us that, let\u0027s see,"},{"Start":"04:54.245 ","End":"04:57.745","Text":"that x squared is equal to,"},{"Start":"04:57.745 ","End":"05:01.310","Text":"it will be 400 over 6."},{"Start":"05:03.800 ","End":"05:09.730","Text":"X has to be positive so I\u0027ll take the square root of top and bottom,"},{"Start":"05:09.730 ","End":"05:13.250","Text":"square root of 400 is 20,"},{"Start":"05:13.250 ","End":"05:17.740","Text":"square root of 6, I\u0027ll just leave it as square root of 6."},{"Start":"05:17.800 ","End":"05:26.030","Text":"We found x, but we don\u0027t know that it really is a maximum."},{"Start":"05:26.030 ","End":"05:27.860","Text":"There are various tests,"},{"Start":"05:27.860 ","End":"05:30.865","Text":"I\u0027ll use the second derivative test."},{"Start":"05:30.865 ","End":"05:33.300","Text":"The second derivative test,"},{"Start":"05:33.300 ","End":"05:38.445","Text":"I\u0027ll take W prime from here and say that"},{"Start":"05:38.445 ","End":"05:47.340","Text":"W double-prime is equal to 1/9 Pi squared,"},{"Start":"05:47.340 ","End":"05:55.095","Text":"and then 3 times 400 is 1200x squared,"},{"Start":"05:55.095 ","End":"05:59.110","Text":"and here 5 times 6 is 30x^4."},{"Start":"06:01.990 ","End":"06:08.435","Text":"Let\u0027s see, I can take out x squared and I can also take out 30,"},{"Start":"06:08.435 ","End":"06:17.075","Text":"so it\u0027s going to be 30 over 9 Pi squared,"},{"Start":"06:17.075 ","End":"06:20.185","Text":"and then the x squared."},{"Start":"06:20.185 ","End":"06:25.220","Text":"What we\u0027re left with after that is just 40."},{"Start":"06:25.220 ","End":"06:32.250","Text":"Divide this by 30, you get 40 minus x squared."},{"Start":"06:32.420 ","End":"06:44.400","Text":"What we need to know is W double-prime at this point 20 over root 6,"},{"Start":"06:44.400 ","End":"06:47.520","Text":"if it\u0027s bigger than 0 or less than 0."},{"Start":"06:47.520 ","End":"06:50.765","Text":"I just need the sign of it."},{"Start":"06:50.765 ","End":"06:54.790","Text":"Obviously, this bit here is positive."},{"Start":"06:54.790 ","End":"06:57.945","Text":"The question is, what is this bit here?"},{"Start":"06:57.945 ","End":"07:05.880","Text":"This side I\u0027ll do 40 minus x squared, this thing squared."},{"Start":"07:05.880 ","End":"07:07.905","Text":"I haven\u0027t squared here already,"},{"Start":"07:07.905 ","End":"07:11.445","Text":"so it\u0027s 400 over 6."},{"Start":"07:11.445 ","End":"07:16.415","Text":"400 over 6, obviously bigger than 40,"},{"Start":"07:16.415 ","End":"07:19.460","Text":"comes out to about 67 or something,"},{"Start":"07:19.460 ","End":"07:21.095","Text":"but anyway, it\u0027s bigger than 40,"},{"Start":"07:21.095 ","End":"07:25.250","Text":"so this is less than 0 and this is bigger than 0,"},{"Start":"07:25.250 ","End":"07:26.885","Text":"this is less than 0."},{"Start":"07:26.885 ","End":"07:29.450","Text":"Positive times negative is negative."},{"Start":"07:29.450 ","End":"07:31.040","Text":"That\u0027s all we need, the sign."},{"Start":"07:31.040 ","End":"07:32.405","Text":"We don\u0027t need the value."},{"Start":"07:32.405 ","End":"07:35.885","Text":"When we have a second derivative negative,"},{"Start":"07:35.885 ","End":"07:39.395","Text":"then it means that we really do have a maximum."},{"Start":"07:39.395 ","End":"07:43.100","Text":"But the question we wanted x and y,"},{"Start":"07:43.100 ","End":"07:45.035","Text":"we still have y up here."},{"Start":"07:45.035 ","End":"07:50.000","Text":"Let\u0027s call this the x max gives the maximum volume,"},{"Start":"07:50.000 ","End":"07:54.950","Text":"and so we now have that y max is equal to"},{"Start":"07:54.950 ","End":"08:03.560","Text":"the square root of 100 minus I\u0027m just substituting x in here."},{"Start":"08:03.560 ","End":"08:05.960","Text":"Before, we have x squared already,"},{"Start":"08:05.960 ","End":"08:10.890","Text":"which is 400 over 6."},{"Start":"08:10.890 ","End":"08:13.739","Text":"See if we can simplify it a bit,"},{"Start":"08:13.739 ","End":"08:15.090","Text":"so it\u0027s a square root,"},{"Start":"08:15.090 ","End":"08:18.585","Text":"let\u0027s put it all over 6,"},{"Start":"08:18.585 ","End":"08:26.970","Text":"so it\u0027s 600 minus 400 over 6."},{"Start":"08:26.970 ","End":"08:29.970","Text":"That\u0027s 200 over 6,"},{"Start":"08:29.970 ","End":"08:33.550","Text":"which is 100 over 3,"},{"Start":"08:33.550 ","End":"08:40.460","Text":"and so square root of that would be square root of the top is 10 over square root of 3."},{"Start":"08:40.460 ","End":"08:48.160","Text":"Let me repeat the y max and highlight it."},{"Start":"08:48.160 ","End":"08:50.955","Text":"These are the quantities we wanted."},{"Start":"08:50.955 ","End":"08:54.455","Text":"I think this was the base radius and this was the height."},{"Start":"08:54.455 ","End":"09:01.050","Text":"Anyway, this is the answer and we\u0027ve shown that it\u0027s a maximum and we\u0027re done."}],"ID":8579}],"Thumbnail":null,"ID":65389},{"Name":"Functions and Graphs Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"4m 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58s","ChapterTopicVideoID":8421,"CourseChapterTopicPlaylistID":65390,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8421.jpeg","UploadDate":"2017-01-28T14:13:35.2500000","DurationForVideoObject":"PT12M58S","Description":null,"VideoComments":[],"Subtitles":[],"ID":8556}],"Thumbnail":null,"ID":65390},{"Name":"Business Applications Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Economic Functions 1","Duration":"10m 40s","ChapterTopicVideoID":8393,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8393.jpeg","UploadDate":"2019-11-14T01:26:53.6630000","DurationForVideoObject":"PT10M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.490","Text":"In this clip, I\u0027ll be introducing some of"},{"Start":"00:02.490 ","End":"00:07.590","Text":"the basic and important functions used in economics,"},{"Start":"00:07.590 ","End":"00:10.155","Text":"functions in the mathematical sense."},{"Start":"00:10.155 ","End":"00:15.450","Text":"The first one will be the demand function."},{"Start":"00:15.450 ","End":"00:17.985","Text":"It will be written as,"},{"Start":"00:17.985 ","End":"00:22.515","Text":"usually, P as a function of Q."},{"Start":"00:22.515 ","End":"00:25.035","Text":"Now, what are P and Q?"},{"Start":"00:25.035 ","End":"00:33.380","Text":"P is the price, and Q is the quantity,"},{"Start":"00:33.380 ","End":"00:36.635","Text":"meaning the number of items that are sold."},{"Start":"00:36.635 ","End":"00:42.785","Text":"So Q items are sold at the price of P each."},{"Start":"00:42.785 ","End":"00:47.435","Text":"Typically, this is a decreasing function"},{"Start":"00:47.435 ","End":"00:50.690","Text":"because, as the quantity you want to sell goes up,"},{"Start":"00:50.690 ","End":"00:52.220","Text":"the price goes down,"},{"Start":"00:52.220 ","End":"00:53.600","Text":"or if you look at it the other way,"},{"Start":"00:53.600 ","End":"00:54.680","Text":"the lower the price,"},{"Start":"00:54.680 ","End":"00:56.555","Text":"the greater the quantity sold."},{"Start":"00:56.555 ","End":"00:58.070","Text":"Let\u0027s give an example."},{"Start":"00:58.070 ","End":"01:06.990","Text":"The example is P of Q equals minus 2Q plus 40."},{"Start":"01:06.990 ","End":"01:10.590","Text":"In this example, if Q equals 1,"},{"Start":"01:10.590 ","End":"01:13.279","Text":"in other words, 1 item sold,"},{"Start":"01:13.279 ","End":"01:20.390","Text":"then the price will equal minus 2 times 1 plus 40 is 38."},{"Start":"01:20.390 ","End":"01:24.815","Text":"Let\u0027s says it\u0027s in dollars, so $38."},{"Start":"01:24.815 ","End":"01:27.364","Text":"If Q is 2,"},{"Start":"01:27.364 ","End":"01:30.815","Text":"we\u0027d expect the answer to be less, and let\u0027s see,"},{"Start":"01:30.815 ","End":"01:33.350","Text":"minus 2 times 2 is minus 4,"},{"Start":"01:33.350 ","End":"01:39.475","Text":"plus 40 is $36, and so on."},{"Start":"01:39.475 ","End":"01:41.360","Text":"You substitute the value of Q,"},{"Start":"01:41.360 ","End":"01:47.320","Text":"you get value of P. That was example 1."},{"Start":"01:47.320 ","End":"01:50.175","Text":"I\u0027ll underline this too."},{"Start":"01:50.175 ","End":"01:58.460","Text":"Now I want to give another example of a special kind of a demand function."},{"Start":"01:58.460 ","End":"02:04.490","Text":"Sometimes a demand function can be constant or fixed,"},{"Start":"02:04.490 ","End":"02:06.740","Text":"and it never changes."},{"Start":"02:06.740 ","End":"02:13.355","Text":"An example would be P of Q equals 5."},{"Start":"02:13.355 ","End":"02:16.370","Text":"Whatever Q is, P is 5."},{"Start":"02:16.370 ","End":"02:17.780","Text":"It could be many reasons for this."},{"Start":"02:17.780 ","End":"02:20.119","Text":"It could be government-controlled price,"},{"Start":"02:20.119 ","End":"02:26.660","Text":"or it could be the manufacturer suggests that this box of cereal should be sold for $5."},{"Start":"02:26.660 ","End":"02:30.815","Text":"So if the quantity sold was 7 units,"},{"Start":"02:30.815 ","End":"02:33.565","Text":"the price per unit is 5."},{"Start":"02:33.565 ","End":"02:36.710","Text":"If you sold 11 units,"},{"Start":"02:36.710 ","End":"02:39.640","Text":"still the price per unit is 5."},{"Start":"02:39.640 ","End":"02:44.015","Text":"I think you get the idea. It\u0027s a constant demand function."},{"Start":"02:44.015 ","End":"02:48.605","Text":"Now, an important note, and that is,"},{"Start":"02:48.605 ","End":"02:54.500","Text":"that not always is the demand function given as P in terms of Q,"},{"Start":"02:54.500 ","End":"02:59.675","Text":"in some books or some professors would do it the other way around,"},{"Start":"02:59.675 ","End":"03:05.810","Text":"and you would have Q as a function of P. This"},{"Start":"03:05.810 ","End":"03:12.275","Text":"would give you the quantity sold as a function of the price per unit."},{"Start":"03:12.275 ","End":"03:14.375","Text":"Again, it would be decreasing."},{"Start":"03:14.375 ","End":"03:18.175","Text":"The lower the price, the more you expect to sell."},{"Start":"03:18.175 ","End":"03:20.639","Text":"I\u0027ll just say, this is sometimes."},{"Start":"03:20.639 ","End":"03:23.060","Text":"I don\u0027t know how frequently that occurs,"},{"Start":"03:23.060 ","End":"03:24.350","Text":"but you should be aware of it."},{"Start":"03:24.350 ","End":"03:30.365","Text":"For example, it might be given that Q is equal to"},{"Start":"03:30.365 ","End":"03:36.990","Text":"100 minus P over 7."},{"Start":"03:36.990 ","End":"03:39.330","Text":"Let\u0027s say P is the price in dollars."},{"Start":"03:39.330 ","End":"03:42.744","Text":"If P equals 2,"},{"Start":"03:42.744 ","End":"03:44.405","Text":"the price is $2 each,"},{"Start":"03:44.405 ","End":"03:51.570","Text":"the quantity sold would be Q would equal 100 minus 2 over 7,"},{"Start":"03:51.570 ","End":"03:55.485","Text":"98 over 7 would be 14."},{"Start":"03:55.485 ","End":"03:58.470","Text":"You should be aware that it could be the other way."},{"Start":"03:58.470 ","End":"04:02.990","Text":"On another note, and this applies not just to demand functions,"},{"Start":"04:02.990 ","End":"04:08.015","Text":"but to pretty much most of the functions in this clip,"},{"Start":"04:08.015 ","End":"04:10.880","Text":"is that, often, instead of the letter Q,"},{"Start":"04:10.880 ","End":"04:13.520","Text":"because it\u0027s the independent variable,"},{"Start":"04:13.520 ","End":"04:18.560","Text":"it\u0027s often replaced by X to make it more mathematical."},{"Start":"04:18.560 ","End":"04:20.930","Text":"So in exercises or elsewhere,"},{"Start":"04:20.930 ","End":"04:23.560","Text":"you would see, for example,"},{"Start":"04:23.560 ","End":"04:26.545","Text":"P as a function of X."},{"Start":"04:26.545 ","End":"04:27.920","Text":"So not to be alarmed,"},{"Start":"04:27.920 ","End":"04:30.305","Text":"not to get used to just the letter Q,"},{"Start":"04:30.305 ","End":"04:33.305","Text":"it could be X or another letter."},{"Start":"04:33.305 ","End":"04:37.120","Text":"Let\u0027s move on to the next function."},{"Start":"04:37.120 ","End":"04:45.940","Text":"The next function we will consider will be the total revenue."},{"Start":"04:46.060 ","End":"04:50.960","Text":"It\u0027s abbreviated, usually, TR,"},{"Start":"04:50.960 ","End":"04:53.105","Text":"and it\u0027s a function of Q,"},{"Start":"04:53.105 ","End":"04:55.745","Text":"sometimes just R for revenue,"},{"Start":"04:55.745 ","End":"04:57.970","Text":"but we\u0027ll call it TR."},{"Start":"04:57.970 ","End":"05:01.790","Text":"The total revenue, another way of saying it, is the total receipts,"},{"Start":"05:01.790 ","End":"05:04.100","Text":"R also stands for receipts."},{"Start":"05:04.100 ","End":"05:08.540","Text":"Let\u0027s say, for sure, that this is receipts."},{"Start":"05:08.540 ","End":"05:16.175","Text":"Now, what this would be is the price per unit times the number of units sold."},{"Start":"05:16.175 ","End":"05:19.580","Text":"It\u0027s going to be equal to the price,"},{"Start":"05:19.580 ","End":"05:23.165","Text":"I\u0027m leaving a space here deliberately,"},{"Start":"05:23.165 ","End":"05:26.570","Text":"times the number of items sold,"},{"Start":"05:26.570 ","End":"05:28.980","Text":"which is the quantity."},{"Start":"05:29.060 ","End":"05:36.635","Text":"Now, the price is actually a function of Q."},{"Start":"05:36.635 ","End":"05:42.330","Text":"In fact, this is just equal to the demand function."},{"Start":"05:42.890 ","End":"05:48.475","Text":"Q is the quantity,"},{"Start":"05:48.475 ","End":"05:51.755","Text":"the number of items sold."},{"Start":"05:51.755 ","End":"05:54.410","Text":"Price times quantity, the total receipts,"},{"Start":"05:54.410 ","End":"05:56.120","Text":"and we call that the total revenue."},{"Start":"05:56.120 ","End":"05:58.070","Text":"Let\u0027s move on to the next one."},{"Start":"05:58.070 ","End":"06:00.625","Text":"The next one\u0027s going to be the total cost function."},{"Start":"06:00.625 ","End":"06:02.535","Text":"I\u0027ll copy it from here,"},{"Start":"06:02.535 ","End":"06:12.585","Text":"and I\u0027ll just erase revenue and erase the R and total cost function, TC of Q."},{"Start":"06:12.585 ","End":"06:14.310","Text":"It also depends on the quantity."},{"Start":"06:14.310 ","End":"06:16.460","Text":"Sometimes, instead of TC,"},{"Start":"06:16.460 ","End":"06:18.590","Text":"we write just C for cost,"},{"Start":"06:18.590 ","End":"06:21.835","Text":"but we\u0027ll write TC, total cost."},{"Start":"06:21.835 ","End":"06:28.250","Text":"The total cost, basically, it\u0027s all the costs that you"},{"Start":"06:28.250 ","End":"06:35.610","Text":"incurred in producing these Q items."},{"Start":"06:35.610 ","End":"06:37.965","Text":"Now, let\u0027s give an example."},{"Start":"06:37.965 ","End":"06:41.130","Text":"We had example 3 before,"},{"Start":"06:41.130 ","End":"06:44.680","Text":"I think, so this makes this one number 4."},{"Start":"06:45.530 ","End":"06:56.450","Text":"We\u0027ll take TC of Q to equal Q squared plus Q plus 1."},{"Start":"06:56.450 ","End":"06:57.620","Text":"It could be anything,"},{"Start":"06:57.620 ","End":"06:58.790","Text":"and this is an example."},{"Start":"06:58.790 ","End":"07:02.030","Text":"So in this case, if we were told that the number of"},{"Start":"07:02.030 ","End":"07:06.095","Text":"items to be produced or that were produced is 4,"},{"Start":"07:06.095 ","End":"07:10.895","Text":"then we would get the total cost of producing those 4 items, TC of 4,"},{"Start":"07:10.895 ","End":"07:13.070","Text":"and just plug it in here,"},{"Start":"07:13.070 ","End":"07:16.850","Text":"4 squared plus 4 plus 1,"},{"Start":"07:16.850 ","End":"07:19.265","Text":"that comes out to 21,"},{"Start":"07:19.265 ","End":"07:22.050","Text":"and presumably, this is in dollars."},{"Start":"07:22.100 ","End":"07:25.020","Text":"Let\u0027s move on to the next function."},{"Start":"07:25.020 ","End":"07:27.970","Text":"It\u0027s going to be the average cost function."},{"Start":"07:27.970 ","End":"07:29.915","Text":"I\u0027ll just copy this."},{"Start":"07:29.915 ","End":"07:36.370","Text":"I\u0027ll erase total, and I\u0027ll erase the T, and write here"},{"Start":"07:36.370 ","End":"07:45.820","Text":"the average, and make this average cost of Q."},{"Start":"07:45.820 ","End":"07:49.760","Text":"What it is is the average cost of producing,"},{"Start":"07:49.760 ","End":"07:51.970","Text":"not the whole Q units,"},{"Start":"07:51.970 ","End":"07:55.130","Text":"but just 1 unit."},{"Start":"07:55.130 ","End":"07:57.000","Text":"There\u0027s a formula for this."},{"Start":"07:57.000 ","End":"08:01.035","Text":"If we know the total cost,"},{"Start":"08:01.035 ","End":"08:05.770","Text":"so that would be TC of Q,"},{"Start":"08:07.400 ","End":"08:10.790","Text":"and we divide it by the number of units,"},{"Start":"08:10.790 ","End":"08:16.015","Text":"which is Q, then that will give us the average for each unit."},{"Start":"08:16.015 ","End":"08:20.815","Text":"Of course, we can\u0027t do the average cost if the quantity was"},{"Start":"08:20.815 ","End":"08:27.939","Text":"0, so let me just note that it has to be Q bigger than 0."},{"Start":"08:27.939 ","End":"08:30.100","Text":"I should have mentioned earlier."},{"Start":"08:30.100 ","End":"08:32.615","Text":"It almost goes without saying in economics that"},{"Start":"08:32.615 ","End":"08:36.670","Text":"the quantity is always bigger or equal to 0."},{"Start":"08:36.670 ","End":"08:38.950","Text":"You can produce 0 items,"},{"Start":"08:38.950 ","End":"08:41.620","Text":"but you can\u0027t produce a negative number of items."},{"Start":"08:41.620 ","End":"08:43.450","Text":"But if you\u0027re going to do an average,"},{"Start":"08:43.450 ","End":"08:45.640","Text":"you can\u0027t do an average of 0 items."},{"Start":"08:45.640 ","End":"08:48.140","Text":"It has to be at least 1."},{"Start":"08:48.530 ","End":"08:52.820","Text":"The next economics function I want to"},{"Start":"08:52.820 ","End":"08:58.325","Text":"introduce is a very important one, the profit function."},{"Start":"08:58.325 ","End":"08:59.900","Text":"What are we going to call it?"},{"Start":"08:59.900 ","End":"09:02.030","Text":"Are we going to call it P?"},{"Start":"09:02.030 ","End":"09:08.585","Text":"No, we can\u0027t because P is already taken for price for the demand function."},{"Start":"09:08.585 ","End":"09:13.340","Text":"So instead of P, we\u0027ll use the Greek letter Pi."},{"Start":"09:13.340 ","End":"09:18.555","Text":"So it\u0027s going to be Pi also depends on Q."},{"Start":"09:18.555 ","End":"09:25.460","Text":"We can define the profit in terms of the revenue and the cost functions."},{"Start":"09:25.460 ","End":"09:29.270","Text":"The profit is what we get if we take the revenue,"},{"Start":"09:29.270 ","End":"09:32.705","Text":"which is how much we took in in receipts,"},{"Start":"09:32.705 ","End":"09:36.055","Text":"and we subtract from that costs,"},{"Start":"09:36.055 ","End":"09:38.420","Text":"what\u0027s leftover is the profit."},{"Start":"09:38.420 ","End":"09:42.650","Text":"So this is equal to TR, that\u0027s"},{"Start":"09:42.650 ","End":"09:48.825","Text":"the income, minus the expenses, the cost."},{"Start":"09:48.825 ","End":"09:54.545","Text":"Total revenue minus total cost is going to be the profit."},{"Start":"09:54.545 ","End":"09:56.875","Text":"Let me just label this."},{"Start":"09:56.875 ","End":"10:07.770","Text":"So this is the profit, and this is the total receipts, so revenue."},{"Start":"10:08.570 ","End":"10:14.330","Text":"We subtract from that total costs,"},{"Start":"10:14.330 ","End":"10:20.209","Text":"but note that it could come out negative or 0,"},{"Start":"10:20.209 ","End":"10:25.339","Text":"if the revenue is bigger than the cost than profit."},{"Start":"10:25.339 ","End":"10:27.320","Text":"If revenue is less than cost,"},{"Start":"10:27.320 ","End":"10:29.035","Text":"it\u0027s a negative profit."},{"Start":"10:29.035 ","End":"10:34.100","Text":"You can call a negative profit a loss, or you could just leave it as a negative number."},{"Start":"10:34.100 ","End":"10:37.130","Text":"That\u0027s something to bear in mind."},{"Start":"10:37.130 ","End":"10:40.260","Text":"That\u0027s the end of this clip."}],"ID":8528},{"Watched":false,"Name":"Economic Functions 2","Duration":"5m 30s","ChapterTopicVideoID":8394,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8394.jpeg","UploadDate":"2019-11-14T01:27:01.9000000","DurationForVideoObject":"PT5M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.485","Text":"This is a continuation of the previous clip on functions in economics."},{"Start":"00:04.485 ","End":"00:10.560","Text":"This time, we\u0027re going to talk about a new concept: marginal."},{"Start":"00:10.560 ","End":"00:15.000","Text":"Most of the economics functions have a marginal version."},{"Start":"00:15.000 ","End":"00:18.970","Text":"In a moment, I\u0027ll explain what this marginal means."},{"Start":"00:18.970 ","End":"00:22.280","Text":"Corresponding to the total revenue function,"},{"Start":"00:22.280 ","End":"00:29.565","Text":"there is also something called the marginal revenue function,"},{"Start":"00:29.565 ","End":"00:32.490","Text":"written MR of Q, often."},{"Start":"00:32.490 ","End":"00:36.480","Text":"Corresponding to the total cost function,"},{"Start":"00:36.480 ","End":"00:44.250","Text":"there will be another function called the marginal cost function, MC."},{"Start":"00:44.250 ","End":"00:46.920","Text":"Similarly, for the others,"},{"Start":"00:46.920 ","End":"00:51.335","Text":"there\u0027s an average cost function and a marginal average cost function,"},{"Start":"00:51.335 ","End":"00:54.565","Text":"a profit function and a marginal profit function."},{"Start":"00:54.565 ","End":"00:56.624","Text":"They all work similarly,"},{"Start":"00:56.624 ","End":"01:02.030","Text":"so we\u0027ll just focus on the total cost function and the marginal cost function."},{"Start":"01:02.030 ","End":"01:06.575","Text":"In general, let\u0027s start with an economic function."},{"Start":"01:06.575 ","End":"01:08.915","Text":"It doesn\u0027t have to be of Q."},{"Start":"01:08.915 ","End":"01:14.660","Text":"Let\u0027s just call it a function of x, and we\u0027ll call the function f."},{"Start":"01:14.660 ","End":"01:23.915","Text":"Then the marginal corresponding function is f-prime of x,"},{"Start":"01:23.915 ","End":"01:29.555","Text":"whereby this, I mean, the derivative function as in calculus."},{"Start":"01:29.555 ","End":"01:31.670","Text":"I\u0027m presuming you know some calculus,"},{"Start":"01:31.670 ","End":"01:33.935","Text":"at least, you know what a derivative is."},{"Start":"01:33.935 ","End":"01:40.350","Text":"In this case, the marginal cost function, it\u0027s not of x,"},{"Start":"01:40.350 ","End":"01:47.755","Text":"it\u0027s of Q, is just the derivative TC-prime of Q."},{"Start":"01:47.755 ","End":"01:54.710","Text":"Earlier in the clip, we had an example where the total cost was equal,"},{"Start":"01:54.710 ","End":"02:00.569","Text":"if I recall, to Q squared plus Q plus 1."},{"Start":"02:01.100 ","End":"02:06.300","Text":"In this case, the marginal cost function"},{"Start":"02:06.300 ","End":"02:11.870","Text":"of quantity Q would equal the derivative of this."},{"Start":"02:11.870 ","End":"02:15.060","Text":"The derivative of Q squared is 2Q,"},{"Start":"02:15.060 ","End":"02:17.640","Text":"the derivative of Q is 1,"},{"Start":"02:17.640 ","End":"02:20.159","Text":"and the derivative of 1 is nothing."},{"Start":"02:20.159 ","End":"02:21.795","Text":"We also, I remember,"},{"Start":"02:21.795 ","End":"02:29.785","Text":"had an example where we computed TC when the quantity was 4, and we got 21."},{"Start":"02:29.785 ","End":"02:33.685","Text":"I remember it was 4 squared plus 4 plus 1, it came out to be 21."},{"Start":"02:33.685 ","End":"02:39.625","Text":"Now, a common question is what happens if I increase the quantity by 1 unit?"},{"Start":"02:39.625 ","End":"02:43.600","Text":"In other words, I want to know what TC of 5 equals."},{"Start":"02:43.600 ","End":"02:47.080","Text":"The marginal function answers this kind of question of what"},{"Start":"02:47.080 ","End":"02:51.150","Text":"happens when we increase our variable by 1."},{"Start":"02:51.150 ","End":"02:56.510","Text":"Note, by the way, that the marginal cost when Q is"},{"Start":"02:56.510 ","End":"03:01.955","Text":"4 is equal to twice 4 plus 1 is 9."},{"Start":"03:01.955 ","End":"03:05.315","Text":"If we have a general economic function, f,"},{"Start":"03:05.315 ","End":"03:14.525","Text":"and you can think of f as being TC, and we know that f-prime at a is equal to b."},{"Start":"03:14.525 ","End":"03:23.430","Text":"What it means is that if we increase x from a to a plus 1, i.e.,"},{"Start":"03:23.430 ","End":"03:25.820","Text":"from 4-5 in our case,"},{"Start":"03:25.820 ","End":"03:31.100","Text":"then f increases by"},{"Start":"03:31.100 ","End":"03:36.120","Text":"b units, approximately, not exact."},{"Start":"03:36.120 ","End":"03:37.440","Text":"I\u0027ll repeat what I\u0027m saying."},{"Start":"03:37.440 ","End":"03:40.964","Text":"We have a certain economic function,"},{"Start":"03:40.964 ","End":"03:45.900","Text":"it could be total cost, and then,"},{"Start":"03:45.900 ","End":"03:50.239","Text":"in our case, x would be Q, the independent variable."},{"Start":"03:50.239 ","End":"03:55.760","Text":"If we increase the quantity from a to a plus 1, say, from 4-5,"},{"Start":"03:55.760 ","End":"04:01.460","Text":"then we\u0027re saying that f would increase by b units,"},{"Start":"04:01.460 ","End":"04:04.755","Text":"in our case, 9 units, but approximately."},{"Start":"04:04.755 ","End":"04:07.530","Text":"That\u0027s one thing I wanted to stress, it\u0027s approximately."},{"Start":"04:07.530 ","End":"04:10.515","Text":"The other point I want to make is that, actually,"},{"Start":"04:10.515 ","End":"04:14.510","Text":"b could be negative. If b is negative,"},{"Start":"04:14.510 ","End":"04:18.730","Text":"then that means that f decreases."},{"Start":"04:18.730 ","End":"04:22.235","Text":"Now let\u0027s take a look at what happens in our example."},{"Start":"04:22.235 ","End":"04:26.750","Text":"If we increase the quantity from 4-5,"},{"Start":"04:26.750 ","End":"04:31.670","Text":"the total cost for 5 units is approximately equal to,"},{"Start":"04:31.670 ","End":"04:36.660","Text":"if we take the 21 and add the marginal cost,"},{"Start":"04:36.660 ","End":"04:40.085","Text":"what we would get would be 30."},{"Start":"04:40.085 ","End":"04:43.595","Text":"Let\u0027s actually compare with the real cost."},{"Start":"04:43.595 ","End":"04:50.375","Text":"The actual cost, TC, of 5 is, we would plug in 5 here,"},{"Start":"04:50.375 ","End":"04:57.095","Text":"5 squared plus 5 plus 1, which is 31."},{"Start":"04:57.095 ","End":"05:02.180","Text":"This is the approximate, and the exact is the 31."},{"Start":"05:02.180 ","End":"05:03.785","Text":"Just be aware of that,"},{"Start":"05:03.785 ","End":"05:06.005","Text":"that that\u0027s only an approximation."},{"Start":"05:06.005 ","End":"05:13.880","Text":"The derivative of an economic function is the marginal function and is approximately"},{"Start":"05:13.880 ","End":"05:22.660","Text":"equal to the increase in the function if we increase the x or the Q by 1 unit."},{"Start":"05:23.300 ","End":"05:28.145","Text":"The others are similar, and they will appear in the exercises."},{"Start":"05:28.145 ","End":"05:31.110","Text":"That\u0027s it for this clip."}],"ID":8529},{"Watched":false,"Name":"Exercise 1","Duration":"8m 43s","ChapterTopicVideoID":8395,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8395.jpeg","UploadDate":"2017-01-28T13:34:33.5130000","DurationForVideoObject":"PT8M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.640","Text":"In this exercise, we\u0027re talking about a dairy company."},{"Start":"00:03.640 ","End":"00:07.125","Text":"When it sells x gallons of chocolate milk a day,"},{"Start":"00:07.125 ","End":"00:15.780","Text":"it can get a price of p of x given by this formula, dollars per gallon."},{"Start":"00:15.780 ","End":"00:19.949","Text":"By the way, sometimes we use the letter q for quantity,"},{"Start":"00:19.949 ","End":"00:22.885","Text":"x is just the quantity number of gallons."},{"Start":"00:22.885 ","End":"00:24.470","Text":"Then there\u0027s various questions,"},{"Start":"00:24.470 ","End":"00:26.075","Text":"we\u0027ll deal with them one at a time."},{"Start":"00:26.075 ","End":"00:35.505","Text":"A, we want to know what the price is per gallon if it sells 4 gallons a day."},{"Start":"00:35.505 ","End":"00:38.415","Text":"Now 4 is the x,"},{"Start":"00:38.415 ","End":"00:42.555","Text":"so all we have to do is plug in 4 in the formula."},{"Start":"00:42.555 ","End":"00:44.090","Text":"In general, just write it again."},{"Start":"00:44.090 ","End":"00:49.070","Text":"P of x is minus 1/4 x plus 10."},{"Start":"00:49.070 ","End":"00:54.485","Text":"But here we have to substitute x equals 4."},{"Start":"00:54.485 ","End":"01:05.305","Text":"We get p of 4 is minus 1/4 times 4 plus 10."},{"Start":"01:05.305 ","End":"01:11.970","Text":"Minus a 1/4 times 4 is minus 1 plus the 10,"},{"Start":"01:11.970 ","End":"01:18.735","Text":"so the answer is 9 dollars for part a."},{"Start":"01:18.735 ","End":"01:22.850","Text":"Now part b is very similar."},{"Start":"01:22.850 ","End":"01:26.615","Text":"It\u0027s just the same thing but with 12 instead of 4."},{"Start":"01:26.615 ","End":"01:28.760","Text":"What we do is say,"},{"Start":"01:28.760 ","End":"01:30.980","Text":"what is p of 12?"},{"Start":"01:30.980 ","End":"01:36.830","Text":"Well, that\u0027s minus a 1/4 times 12 instead of the 4,"},{"Start":"01:36.830 ","End":"01:40.624","Text":"instead of x quantity plus 10,"},{"Start":"01:40.624 ","End":"01:48.420","Text":"and 12 over 4 is 3 minus 3 plus 10,"},{"Start":"01:48.420 ","End":"01:53.710","Text":"so it\u0027s 7 dollars per gallon."},{"Start":"01:54.440 ","End":"01:57.390","Text":"I\u0027ll do part c over here."},{"Start":"01:57.390 ","End":"02:03.650","Text":"What it asked is, how many gallons a day are sold if the price is 6 dollars per gallon?"},{"Start":"02:03.650 ","End":"02:05.810","Text":"This is like a reverse question."},{"Start":"02:05.810 ","End":"02:09.675","Text":"We\u0027re given p, we want to find x."},{"Start":"02:09.675 ","End":"02:12.245","Text":"All we do is to solve an equation."},{"Start":"02:12.245 ","End":"02:20.280","Text":"We know that p of x is equal to 6,"},{"Start":"02:20.280 ","End":"02:25.065","Text":"but p of x is just this formula."},{"Start":"02:25.065 ","End":"02:26.745","Text":"We have all even from here."},{"Start":"02:26.745 ","End":"02:33.180","Text":"I have minus 1/4 x plus 10 is equal to 6,"},{"Start":"02:33.180 ","End":"02:35.115","Text":"and that\u0027s an algebraic equation."},{"Start":"02:35.115 ","End":"02:37.309","Text":"Let\u0027s bring the 10 to the other side,"},{"Start":"02:37.309 ","End":"02:39.900","Text":"minus 1/4 x equal to,"},{"Start":"02:39.900 ","End":"02:45.910","Text":"bring it over, 6 minus 10 is minus 4."},{"Start":"02:48.290 ","End":"02:56.485","Text":"If I divide both sides by minus 1/4,"},{"Start":"02:56.485 ","End":"03:01.730","Text":"which is the same as multiplying by minus 4,"},{"Start":"03:01.730 ","End":"03:07.075","Text":"then we get that x is equal to 16."},{"Start":"03:07.075 ","End":"03:11.980","Text":"Really we should indicate the unit, it\u0027s 16 gallons."},{"Start":"03:11.980 ","End":"03:17.760","Text":"That\u0027s the answer to part c. Now I\u0027ll move on to a new page."},{"Start":"03:17.760 ","End":"03:20.330","Text":"In part d, we want to sketch a graph of"},{"Start":"03:20.330 ","End":"03:24.815","Text":"the demand function and to say what its domain is."},{"Start":"03:24.815 ","End":"03:28.100","Text":"Now if we look at the function and I\u0027ll just highlight it,"},{"Start":"03:28.100 ","End":"03:32.065","Text":"you can see it\u0027s a linear function."},{"Start":"03:32.065 ","End":"03:35.090","Text":"In general, if we have a function of x,"},{"Start":"03:35.090 ","End":"03:40.310","Text":"which is of the form some constant times x plus another constant,"},{"Start":"03:40.310 ","End":"03:42.890","Text":"that\u0027s a and b, which is what we have in this case,"},{"Start":"03:42.890 ","End":"03:45.390","Text":"then we know it\u0027s a straight line."},{"Start":"03:45.440 ","End":"03:49.400","Text":"In our case, let\u0027s just make a little table."},{"Start":"03:49.400 ","End":"03:53.959","Text":"We really only need 2 points to draw a line."},{"Start":"03:53.959 ","End":"03:58.295","Text":"Let\u0027s label the columns x and p of x."},{"Start":"03:58.295 ","End":"04:02.090","Text":"As I said, we had already 2 points because in part a,"},{"Start":"04:02.090 ","End":"04:05.374","Text":"we know that if 4 gallons were sold,"},{"Start":"04:05.374 ","End":"04:08.450","Text":"then the price was 9."},{"Start":"04:08.450 ","End":"04:12.635","Text":"We also had that when 12 gallons were sold,"},{"Start":"04:12.635 ","End":"04:15.470","Text":"the price was 7."},{"Start":"04:15.470 ","End":"04:17.900","Text":"Just for extra accuracy,"},{"Start":"04:17.900 ","End":"04:19.909","Text":"let\u0027s throw in another point,"},{"Start":"04:19.909 ","End":"04:21.440","Text":"which is easy to compute."},{"Start":"04:21.440 ","End":"04:24.440","Text":"Let\u0027s say the quantity is 0."},{"Start":"04:24.440 ","End":"04:27.860","Text":"If x is 0, are we going to do it in our heads?"},{"Start":"04:27.860 ","End":"04:30.410","Text":"This is 0 and it\u0027ll just be 10."},{"Start":"04:30.410 ","End":"04:33.605","Text":"Let me draw some coordinate axis."},{"Start":"04:33.605 ","End":"04:38.360","Text":"Let\u0027s see. Horizontal axis will be x,"},{"Start":"04:38.360 ","End":"04:40.160","Text":"which is the quantity,"},{"Start":"04:40.160 ","End":"04:42.530","Text":"the vertical axis p,"},{"Start":"04:42.530 ","End":"04:44.850","Text":"which is the price."},{"Start":"04:45.140 ","End":"04:51.890","Text":"In general, we have to have both of these being positive or non-negative."},{"Start":"04:51.890 ","End":"04:54.470","Text":"I wouldn\u0027t have a negative price,"},{"Start":"04:54.470 ","End":"04:57.060","Text":"there\u0027s not someone who is not going to pay you to take the product."},{"Start":"04:57.060 ","End":"05:00.020","Text":"Also number of gallons is going to be at least 0,"},{"Start":"05:00.020 ","End":"05:01.745","Text":"we can\u0027t have negative quantity."},{"Start":"05:01.745 ","End":"05:07.400","Text":"In fact, we already answered the last part of d. The domain is"},{"Start":"05:07.400 ","End":"05:13.655","Text":"x bigger or equal to 0, but not quite."},{"Start":"05:13.655 ","End":"05:19.370","Text":"Later, I\u0027ll put in a condition that the price also has to be bigger or equal to 0."},{"Start":"05:19.370 ","End":"05:20.930","Text":"But for the moment there,"},{"Start":"05:20.930 ","End":"05:22.325","Text":"let\u0027s put the 3 points in."},{"Start":"05:22.325 ","End":"05:24.810","Text":"So I have to choose a scale."},{"Start":"05:25.610 ","End":"05:34.230","Text":"Let\u0027s say that this is 10 because I think that\u0027s the highest and then we keep decreasing."},{"Start":"05:34.230 ","End":"05:37.570","Text":"Let me put some scale here."},{"Start":"05:37.640 ","End":"05:41.385","Text":"Let\u0027s see. The 0s here,"},{"Start":"05:41.385 ","End":"05:43.500","Text":"and then I put 4 here,"},{"Start":"05:43.500 ","End":"05:46.559","Text":"and then 12 would be here,"},{"Start":"05:46.559 ","End":"05:55.125","Text":"and 0,10 would be this point here, and then 4,9."},{"Start":"05:55.125 ","End":"06:00.105","Text":"Well, let\u0027s say that this is 9 and I see we\u0027re also going to need a 7,"},{"Start":"06:00.105 ","End":"06:04.410","Text":"and here\u0027s a 7."},{"Start":"06:04.410 ","End":"06:08.070","Text":"Sketch in 4,9 and"},{"Start":"06:08.070 ","End":"06:16.095","Text":"12,7 and I draw a line through them."},{"Start":"06:16.095 ","End":"06:18.560","Text":"That\u0027s what the graph looks like."},{"Start":"06:18.560 ","End":"06:23.100","Text":"Now I said something about the domain because actually at some point,"},{"Start":"06:23.100 ","End":"06:29.905","Text":"this is going to hit the x-axis where the price is 0 and then goes negative."},{"Start":"06:29.905 ","End":"06:32.620","Text":"You could leave the answer like this,"},{"Start":"06:32.620 ","End":"06:33.769","Text":"but to be precise,"},{"Start":"06:33.769 ","End":"06:41.450","Text":"we also want to say that the price is bigger or equal to 0 here,"},{"Start":"06:41.450 ","End":"06:46.805","Text":"minus 1/4x plus 10 is bigger or equal to 0."},{"Start":"06:46.805 ","End":"06:49.775","Text":"Then we could add 1/4 x to both sides,"},{"Start":"06:49.775 ","End":"06:54.350","Text":"so 10 is bigger or equal to 1/4x."},{"Start":"06:54.350 ","End":"06:59.374","Text":"Then you can multiply both sides by 4 because 4 is positive and we get 40"},{"Start":"06:59.374 ","End":"07:05.459","Text":"bigger or equal to x. I guess, it hits at 40."},{"Start":"07:05.459 ","End":"07:07.765","Text":"Really, if we combine these 2,"},{"Start":"07:07.765 ","End":"07:13.355","Text":"the proper answer for the domain would be that"},{"Start":"07:13.355 ","End":"07:19.950","Text":"x is at least 0 and not greater than 40."},{"Start":"07:19.950 ","End":"07:24.910","Text":"Now let\u0027s go on to part e. In part e,"},{"Start":"07:24.910 ","End":"07:32.420","Text":"we notice that the demand function is given as P in terms of x."},{"Start":"07:32.420 ","End":"07:38.675","Text":"In other words, the price is given as a function of the quantity,"},{"Start":"07:38.675 ","End":"07:40.820","Text":"but we want to now reverse it,"},{"Start":"07:40.820 ","End":"07:44.740","Text":"so the quantity is a function of the price."},{"Start":"07:44.740 ","End":"07:52.830","Text":"Let\u0027s start with saying that P equals minus 1/4 x plus 10."},{"Start":"07:52.830 ","End":"07:56.540","Text":"The idea is to extract x from the equation."},{"Start":"07:56.540 ","End":"08:01.070","Text":"What I can do is throw this over to the left and this on the right,"},{"Start":"08:01.070 ","End":"08:05.880","Text":"so I get 1/4x is equal to,"},{"Start":"08:05.880 ","End":"08:07.500","Text":"and I subtract P,"},{"Start":"08:07.500 ","End":"08:09.365","Text":"I could put it after the 10,"},{"Start":"08:09.365 ","End":"08:15.669","Text":"10 minus P. That\u0027s just by flipping sides."},{"Start":"08:15.669 ","End":"08:19.565","Text":"Now we can multiply both sides by 4,"},{"Start":"08:19.565 ","End":"08:26.165","Text":"so we\u0027ll get that x is equal to 4 times 10 minus"},{"Start":"08:26.165 ","End":"08:29.210","Text":"P. They\u0027ll put the answer in the form"},{"Start":"08:29.210 ","End":"08:33.760","Text":"x and emphasize that it\u0027s a function of P and is equal to,"},{"Start":"08:33.760 ","End":"08:35.075","Text":"I\u0027ll either just copy this,"},{"Start":"08:35.075 ","End":"08:37.295","Text":"but maybe I\u0027ll open the brackets."},{"Start":"08:37.295 ","End":"08:43.940","Text":"Both ways would do and that would be our answer. That\u0027s it."}],"ID":8530},{"Watched":false,"Name":"Exercise 2","Duration":"6m 58s","ChapterTopicVideoID":8396,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8396.jpeg","UploadDate":"2017-01-28T13:36:25.5970000","DurationForVideoObject":"PT6M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.815","Text":"In this exercise, we\u0027re given the demand function of a certain product."},{"Start":"00:04.815 ","End":"00:10.170","Text":"It means that x is the quantity or the number of units"},{"Start":"00:10.170 ","End":"00:16.365","Text":"sold and P is the price that depends on the quantity sold according to this formula,"},{"Start":"00:16.365 ","End":"00:18.585","Text":"and we\u0027re working in dollars."},{"Start":"00:18.585 ","End":"00:25.455","Text":"In part a, we have to find the total revenue function and its domain."},{"Start":"00:25.455 ","End":"00:29.490","Text":"Now, total revenue, the function name TR,"},{"Start":"00:29.490 ","End":"00:34.080","Text":"and sometimes we just use R. What it\u0027s"},{"Start":"00:34.080 ","End":"00:39.540","Text":"equal to is the number of units times the price per unit."},{"Start":"00:39.540 ","End":"00:43.770","Text":"The number of units is x,"},{"Start":"00:43.770 ","End":"00:48.215","Text":"that\u0027s the quantity, sometimes called Q for quantity."},{"Start":"00:48.215 ","End":"00:51.040","Text":"P of x is the price per unit,"},{"Start":"00:51.040 ","End":"00:55.530","Text":"so multiply them and that\u0027ll give us our revenue or total revenue."},{"Start":"00:55.700 ","End":"01:00.045","Text":"If I expand, p of x is,"},{"Start":"01:00.045 ","End":"01:01.500","Text":"I\u0027m copying it from here,"},{"Start":"01:01.500 ","End":"01:06.315","Text":"minus 0.6x plus 120."},{"Start":"01:06.315 ","End":"01:14.360","Text":"Then I can open the brackets and say minus 0.6x squared plus 120x."},{"Start":"01:14.950 ","End":"01:21.470","Text":"I\u0027ll just indicate that R is a function of x. I could have put this brackets all along,"},{"Start":"01:21.470 ","End":"01:23.240","Text":"it doesn\u0027t really matter."},{"Start":"01:23.240 ","End":"01:27.170","Text":"But R, revenue function of x is this."},{"Start":"01:27.170 ","End":"01:30.770","Text":"All I have to do now is give you its domain."},{"Start":"01:30.770 ","End":"01:33.960","Text":"The domain doesn\u0027t come from here,"},{"Start":"01:33.960 ","End":"01:42.140","Text":"just comes from real life where the domain x or quantity can\u0027t be negative."},{"Start":"01:42.140 ","End":"01:44.585","Text":"You can\u0027t produce a negative number of units,"},{"Start":"01:44.585 ","End":"01:49.100","Text":"so the domain is just x bigger or equal to 0,"},{"Start":"01:49.100 ","End":"01:53.329","Text":"could be 0, you might not manufacture anything."},{"Start":"01:53.329 ","End":"01:55.280","Text":"Well, then that\u0027s part a."},{"Start":"01:55.280 ","End":"01:58.324","Text":"Now in part b,"},{"Start":"01:58.324 ","End":"02:01.100","Text":"we\u0027re given that x is 20,"},{"Start":"02:01.100 ","End":"02:03.514","Text":"meaning there were 20 units produced,"},{"Start":"02:03.514 ","End":"02:05.900","Text":"and we want to find what the price is,"},{"Start":"02:05.900 ","End":"02:07.700","Text":"and what the revenue is."},{"Start":"02:07.700 ","End":"02:10.190","Text":"But we have both functions, the price function,"},{"Start":"02:10.190 ","End":"02:12.710","Text":"which is the demand function, is here."},{"Start":"02:12.710 ","End":"02:16.325","Text":"What I have to say is that x equals 20,"},{"Start":"02:16.325 ","End":"02:21.270","Text":"I need to find what is P of 20."},{"Start":"02:21.370 ","End":"02:30.780","Text":"I need to know then what is the revenue or total revenue for 20 equals,"},{"Start":"02:30.780 ","End":"02:32.760","Text":"that\u0027s like 2 questions in 1."},{"Start":"02:32.760 ","End":"02:34.835","Text":"As for P of 20,"},{"Start":"02:34.835 ","End":"02:39.570","Text":"we substitute x equals 20 in the demand function."},{"Start":"02:39.570 ","End":"02:47.385","Text":"We get P of 20 equals minus 0.6 times,"},{"Start":"02:47.385 ","End":"02:49.580","Text":"I\u0027ll write little times like this,"},{"Start":"02:49.580 ","End":"02:51.860","Text":"not to confuse with the decimal point,"},{"Start":"02:51.860 ","End":"02:56.305","Text":"times 20 plus 120."},{"Start":"02:56.305 ","End":"03:05.850","Text":"That is equal to 0.6 times 20 is minus 12 plus 120,"},{"Start":"03:05.850 ","End":"03:11.895","Text":"which is 120 minus 12 is 108."},{"Start":"03:11.895 ","End":"03:14.360","Text":"Remember, we\u0027re working in dollars,"},{"Start":"03:14.360 ","End":"03:18.860","Text":"so the price of a unit is a $108."},{"Start":"03:18.860 ","End":"03:22.650","Text":"The idea is to use the revenue function,"},{"Start":"03:23.470 ","End":"03:31.200","Text":"I just have to substitute this time in this function."},{"Start":"03:31.490 ","End":"03:40.805","Text":"R of 20 is equal to minus 0.6 times"},{"Start":"03:40.805 ","End":"03:51.040","Text":"20 squared plus 120 times 20."},{"Start":"03:51.040 ","End":"03:54.330","Text":"R of 20 equals,"},{"Start":"03:54.330 ","End":"03:56.040","Text":"I don\u0027t think we need to calculate it."},{"Start":"03:56.040 ","End":"03:58.005","Text":"20 squared is 400."},{"Start":"03:58.005 ","End":"04:01.400","Text":"400 times 6 is 2400,"},{"Start":"04:01.400 ","End":"04:03.635","Text":"but it\u0027s 0.6, so it\u0027s 240."},{"Start":"04:03.635 ","End":"04:06.835","Text":"We have minus 240 plus,"},{"Start":"04:06.835 ","End":"04:12.090","Text":"12 times 2 is 24, so it\u0027s 2400."},{"Start":"04:12.090 ","End":"04:15.325","Text":"If we subtract, that gives us what?"},{"Start":"04:15.325 ","End":"04:22.340","Text":"2160, and that would be in dollars."},{"Start":"04:22.340 ","End":"04:27.085","Text":"So this is the prices of the total revenue that does part b."},{"Start":"04:27.085 ","End":"04:28.270","Text":"Just as a check,"},{"Start":"04:28.270 ","End":"04:33.984","Text":"you could take this $108 and multiply it by 20 units,"},{"Start":"04:33.984 ","End":"04:37.895","Text":"and 108 times 20 is 2160,"},{"Start":"04:37.895 ","End":"04:39.650","Text":"so that\u0027s an extra check."},{"Start":"04:39.650 ","End":"04:44.405","Text":"Now part c, this time we\u0027re not given the quantity,"},{"Start":"04:44.405 ","End":"04:46.085","Text":"we are given the price,"},{"Start":"04:46.085 ","End":"04:48.605","Text":"and we have to find the revenue."},{"Start":"04:48.605 ","End":"04:52.835","Text":"Now what we have to first do is get x,"},{"Start":"04:52.835 ","End":"04:58.100","Text":"the quantity, and we\u0027ll do that from the demand function."},{"Start":"04:58.100 ","End":"05:07.015","Text":"Part c, p of x equals minus 0.6x plus 120."},{"Start":"05:07.015 ","End":"05:09.410","Text":"But in our case, we\u0027re given the price,"},{"Start":"05:09.410 ","End":"05:12.220","Text":"we\u0027re given that it\u0027s $12."},{"Start":"05:12.220 ","End":"05:18.215","Text":"I\u0027m just going to erase this and write 12 in its place."},{"Start":"05:18.215 ","End":"05:22.040","Text":"Now this is an equation in x and we want to find out what x is."},{"Start":"05:22.040 ","End":"05:23.660","Text":"That\u0027s just your basic algebra,"},{"Start":"05:23.660 ","End":"05:25.555","Text":"we\u0027ll move this over to that side,"},{"Start":"05:25.555 ","End":"05:27.015","Text":"this over to that side."},{"Start":"05:27.015 ","End":"05:34.740","Text":"I\u0027ve got 0.6x equals 120 minus 12."},{"Start":"05:34.740 ","End":"05:39.070","Text":"Well this we can see already is 108."},{"Start":"05:39.170 ","End":"05:41.600","Text":"X is going to equal,"},{"Start":"05:41.600 ","End":"05:48.340","Text":"if we divide both sides by 0.6, 108 over 0.6."},{"Start":"05:48.340 ","End":"05:52.725","Text":"I don\u0027t think we need a calculator, well, you could."},{"Start":"05:52.725 ","End":"05:56.490","Text":"Anyway, it comes out to 180."},{"Start":"05:56.490 ","End":"05:59.280","Text":"That\u0027s the x, but that\u0027s not what we\u0027re asked for,"},{"Start":"05:59.280 ","End":"06:01.540","Text":"we\u0027re asked for the revenue."},{"Start":"06:01.540 ","End":"06:05.360","Text":"What we do is we plug in this 180,"},{"Start":"06:05.360 ","End":"06:06.740","Text":"which is now the quantity,"},{"Start":"06:06.740 ","End":"06:11.575","Text":"the number of units into the revenue function here,"},{"Start":"06:11.575 ","End":"06:13.770","Text":"let\u0027s plug it right into here."},{"Start":"06:13.770 ","End":"06:16.380","Text":"Then we get that R of x,"},{"Start":"06:16.380 ","End":"06:19.455","Text":"in our case, R of 180,"},{"Start":"06:19.455 ","End":"06:24.180","Text":"will equal minus 0.6 times"},{"Start":"06:24.180 ","End":"06:32.260","Text":"180 squared plus 120 times 180."},{"Start":"06:32.320 ","End":"06:36.020","Text":"I\u0027ll spare you the details of the computation."},{"Start":"06:36.020 ","End":"06:39.755","Text":"That comes out to be 2160,"},{"Start":"06:39.755 ","End":"06:41.905","Text":"and that\u0027s also in dollars."},{"Start":"06:41.905 ","End":"06:47.990","Text":"It\u0027s just a coincidence that when we produce 20 units or a 180 units,"},{"Start":"06:47.990 ","End":"06:52.170","Text":"it comes out the same, just fluke."},{"Start":"06:52.170 ","End":"06:55.100","Text":"We\u0027ve answered a, we\u0027ve answered"},{"Start":"06:55.100 ","End":"06:59.650","Text":"the b both parts and we\u0027ve answered c, and so we\u0027re done."}],"ID":8531},{"Watched":false,"Name":"Exercise 3","Duration":"11m 7s","ChapterTopicVideoID":8397,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8397.jpeg","UploadDate":"2017-01-28T13:39:35.4000000","DurationForVideoObject":"PT11M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.035","Text":"In this exercise, we\u0027re given the total revenue function of a certain product."},{"Start":"00:07.035 ","End":"00:14.700","Text":"We call it R, but sometimes it\u0027s called TR for total revenue."},{"Start":"00:14.700 ","End":"00:18.105","Text":"Also remember that x is really Q,"},{"Start":"00:18.105 ","End":"00:21.735","Text":"the quantity and this is the formula."},{"Start":"00:21.735 ","End":"00:24.390","Text":"Notice that it\u0027s quadratic,"},{"Start":"00:24.390 ","End":"00:26.475","Text":"I\u0027ll say more in a moment."},{"Start":"00:26.475 ","End":"00:28.080","Text":"The first part is,"},{"Start":"00:28.080 ","End":"00:31.900","Text":"what is the domain of the revenue function?"},{"Start":"00:32.210 ","End":"00:34.800","Text":"If you just look at it mathematically,"},{"Start":"00:34.800 ","End":"00:36.239","Text":"x could be anything."},{"Start":"00:36.239 ","End":"00:39.120","Text":"But since it\u0027s a real life problem, x,"},{"Start":"00:39.120 ","End":"00:43.785","Text":"which is the quantity produced, can\u0027t be negative."},{"Start":"00:43.785 ","End":"00:49.105","Text":"The domain is x bigger or equal to 0."},{"Start":"00:49.105 ","End":"00:55.935","Text":"Now on to part b. I mentioned the word quadratic and if we think of R,"},{"Start":"00:55.935 ","End":"01:00.065","Text":"the revenue as like y in mathematics, if we think of x and y,"},{"Start":"01:00.065 ","End":"01:10.115","Text":"then y is of the form ax squared plus bx plus c. That\u0027s a quadratic and in fact,"},{"Start":"01:10.115 ","End":"01:17.630","Text":"we can say that a is minus 0.08, b is 40."},{"Start":"01:17.630 ","End":"01:19.460","Text":"There is no constant term,"},{"Start":"01:19.460 ","End":"01:22.410","Text":"which means that it\u0027s there but it\u0027s 0."},{"Start":"01:22.490 ","End":"01:25.790","Text":"I\u0027ll give you a reminder of the properties,"},{"Start":"01:25.790 ","End":"01:27.545","Text":"some of them, of the parabola."},{"Start":"01:27.545 ","End":"01:29.375","Text":"There\u0027s 2 kinds of parabola."},{"Start":"01:29.375 ","End":"01:36.530","Text":"There\u0027s ones that face up or open up and there\u0027s 1 that face down."},{"Start":"01:36.530 ","End":"01:38.855","Text":"Some people say open down."},{"Start":"01:38.855 ","End":"01:48.005","Text":"It\u0027s upward facing when a is bigger than 0 and downward facing when a is less than 0."},{"Start":"01:48.005 ","End":"01:56.570","Text":"The other thing I wanted to say about parabolas is they have a vertex here or here."},{"Start":"01:56.570 ","End":"01:58.265","Text":"Actually in our case,"},{"Start":"01:58.265 ","End":"02:01.820","Text":"we are going to be a is negative."},{"Start":"02:01.820 ","End":"02:04.430","Text":"We\u0027re actually going to be in this case."},{"Start":"02:04.430 ","End":"02:09.100","Text":"Maybe I\u0027ll just highlight it to say this is what we have in our case."},{"Start":"02:09.100 ","End":"02:12.230","Text":"I also want to label this vertex,"},{"Start":"02:12.230 ","End":"02:15.140","Text":"just give names to the x and the y coordinates."},{"Start":"02:15.140 ","End":"02:17.540","Text":"I\u0027ll call it x sub v,"},{"Start":"02:17.540 ","End":"02:18.935","Text":"x of the vertex,"},{"Start":"02:18.935 ","End":"02:21.155","Text":"and y of the vertex."},{"Start":"02:21.155 ","End":"02:24.450","Text":"There\u0027s a formula for x of the vertex."},{"Start":"02:24.450 ","End":"02:26.550","Text":"I\u0027ll give you in a moment."},{"Start":"02:26.550 ","End":"02:31.730","Text":"Back here, there\u0027s 3 things I usually want to know about a parabola before sketching it."},{"Start":"02:31.730 ","End":"02:35.240","Text":"The first is whether it\u0027s facing upwards or downwards,"},{"Start":"02:35.240 ","End":"02:40.910","Text":"and I can say that this parabola faces downwards."},{"Start":"02:40.910 ","End":"02:44.630","Text":"The other thing is the vertex."},{"Start":"02:44.630 ","End":"02:50.065","Text":"I want to know what the coordinates are so I used the formula,"},{"Start":"02:50.065 ","End":"02:52.040","Text":"maybe I\u0027ll write it over here."},{"Start":"02:52.040 ","End":"02:59.160","Text":"In general, the x of the vertex is equal to minus b over 2a,"},{"Start":"02:59.160 ","End":"03:01.880","Text":"where b and a we have,"},{"Start":"03:01.880 ","End":"03:05.540","Text":"and we get the y by substituting in the equation."},{"Start":"03:05.540 ","End":"03:10.684","Text":"So x_v, from this formula,"},{"Start":"03:10.684 ","End":"03:16.490","Text":"minus b is minus 40 over 2a,"},{"Start":"03:16.490 ","End":"03:21.310","Text":"which is minus 0.08."},{"Start":"03:21.310 ","End":"03:23.930","Text":"I\u0027ll do the computation for you."},{"Start":"03:23.930 ","End":"03:29.150","Text":"It comes out to 250 positive because there\u0027s a minus over a minus."},{"Start":"03:29.150 ","End":"03:32.705","Text":"Then y of the vertex,"},{"Start":"03:32.705 ","End":"03:38.445","Text":"we get by substituting 250 in our formula."},{"Start":"03:38.445 ","End":"03:44.105","Text":"This will be minus 0.08 times"},{"Start":"03:44.105 ","End":"03:51.860","Text":"250 squared plus 40 times 250."},{"Start":"03:51.860 ","End":"03:54.755","Text":"Again, I\u0027ll do the computation for you."},{"Start":"03:54.755 ","End":"03:58.745","Text":"It comes out to be 5,000,"},{"Start":"03:58.745 ","End":"04:04.250","Text":"so now we know that the vertex is at"},{"Start":"04:04.250 ","End":"04:10.665","Text":"the point 250, 5,000."},{"Start":"04:10.665 ","End":"04:13.865","Text":"I want to know the intersection with the axes."},{"Start":"04:13.865 ","End":"04:21.260","Text":"Let\u0027s see, the intersection with the x-axis is what we get when we set y equals 0,"},{"Start":"04:21.260 ","End":"04:29.050","Text":"we get that minus 0.08x squared plus 40x,"},{"Start":"04:29.050 ","End":"04:30.965","Text":"just copied it from here,"},{"Start":"04:30.965 ","End":"04:33.545","Text":"is equal to 0."},{"Start":"04:33.545 ","End":"04:37.010","Text":"We can factor this because x is in both of them."},{"Start":"04:37.010 ","End":"04:38.180","Text":"There\u0027s no constant term,"},{"Start":"04:38.180 ","End":"04:46.890","Text":"so it\u0027s x times minus 0.08x plus 40 is equal to 0."},{"Start":"04:46.890 ","End":"04:49.320","Text":"Just took x out of the brackets."},{"Start":"04:49.320 ","End":"04:52.080","Text":"Then we\u0027ll get 2 solutions."},{"Start":"04:52.080 ","End":"04:59.000","Text":"1 of them will be when x equals 0 and the other will"},{"Start":"04:59.000 ","End":"05:07.265","Text":"be when minus 0.08x plus 40 equals 0."},{"Start":"05:07.265 ","End":"05:08.960","Text":"Because of course, when a product is 0,"},{"Start":"05:08.960 ","End":"05:11.075","Text":"1 or the other must be 0."},{"Start":"05:11.075 ","End":"05:14.855","Text":"When x is 0 is already a value of x."},{"Start":"05:14.855 ","End":"05:18.305","Text":"But here we have to solve a simple linear equation."},{"Start":"05:18.305 ","End":"05:20.285","Text":"What x is equal to,"},{"Start":"05:20.285 ","End":"05:23.780","Text":"I\u0027ll take the 40 to the other side,"},{"Start":"05:23.780 ","End":"05:33.560","Text":"that\u0027ll be minus 40 and divide by the coefficient of x minus 0.08, the minuses cancel."},{"Start":"05:33.560 ","End":"05:39.265","Text":"This just comes out to be 500."},{"Start":"05:39.265 ","End":"05:44.975","Text":"Notice that there\u0027s no need to find the y-intercept,"},{"Start":"05:44.975 ","End":"05:48.200","Text":"meaning we\u0027ve already found it because x equals 0,"},{"Start":"05:48.200 ","End":"05:49.520","Text":"then y is 0."},{"Start":"05:49.520 ","End":"05:58.975","Text":"The 2 intercepts are 0,0 and 500,0."},{"Start":"05:58.975 ","End":"06:02.465","Text":"We\u0027ve got all the information we need for a sketch."},{"Start":"06:02.465 ","End":"06:09.750","Text":"The parabola faces down and its vertex is this point."},{"Start":"06:09.750 ","End":"06:15.580","Text":"These are the intersections with the axes."},{"Start":"06:16.700 ","End":"06:19.235","Text":"We\u0027ll do the sketch."},{"Start":"06:19.235 ","End":"06:22.070","Text":"The horizontal will be x,"},{"Start":"06:22.070 ","End":"06:23.495","Text":"which is the quantity,"},{"Start":"06:23.495 ","End":"06:28.390","Text":"the vertical, just call it y or I could call it R, the revenue."},{"Start":"06:28.390 ","End":"06:35.900","Text":"The intersection with the axes is 0,0 and then I need 500,0."},{"Start":"06:35.900 ","End":"06:38.870","Text":"Let\u0027s just say that this is here,"},{"Start":"06:38.870 ","End":"06:42.470","Text":"500 and this is 0."},{"Start":"06:42.470 ","End":"06:46.580","Text":"The vertex is 250,"},{"Start":"06:46.580 ","End":"06:51.170","Text":"exactly halfway between the 0 and the 500."},{"Start":"06:51.170 ","End":"06:59.255","Text":"Let\u0027s just say this is 5,000 for y or R. We want this point somewhere here."},{"Start":"06:59.255 ","End":"07:02.990","Text":"You get the general idea of the shape. It\u0027s not perfect."},{"Start":"07:02.990 ","End":"07:07.310","Text":"That\u0027s part b. I\u0027ll just clear some space."},{"Start":"07:07.310 ","End":"07:11.810","Text":"I\u0027ll continue over here with part c. We want"},{"Start":"07:11.810 ","End":"07:17.540","Text":"the demand function and we can get it from the revenue function."},{"Start":"07:17.540 ","End":"07:22.250","Text":"Because the formula is that the revenue function,"},{"Start":"07:22.250 ","End":"07:25.370","Text":"let\u0027s say R is in terms of x,"},{"Start":"07:25.370 ","End":"07:31.425","Text":"is equal to x times P of x."},{"Start":"07:31.425 ","End":"07:34.485","Text":"We\u0027re given R of x,"},{"Start":"07:34.485 ","End":"07:37.080","Text":"is it still here?"},{"Start":"07:37.080 ","End":"07:39.285","Text":"I can copy it from here."},{"Start":"07:39.285 ","End":"07:45.570","Text":"I have the minus 0.08x squared plus"},{"Start":"07:45.570 ","End":"07:51.650","Text":"40x is x times the demand function,"},{"Start":"07:51.650 ","End":"07:53.330","Text":"which is P of x."},{"Start":"07:53.330 ","End":"07:56.555","Text":"That gives the price in terms of the quantity."},{"Start":"07:56.555 ","End":"07:59.915","Text":"All you have to do is divide both sides by x,"},{"Start":"07:59.915 ","End":"08:02.000","Text":"x comes out of the brackets."},{"Start":"08:02.000 ","End":"08:12.140","Text":"We get 0.08x plus"},{"Start":"08:12.140 ","End":"08:18.775","Text":"40 is equal to P of x."},{"Start":"08:18.775 ","End":"08:23.490","Text":"Should have written that backwards."},{"Start":"08:23.490 ","End":"08:27.140","Text":"This is the answer for the first part of c,"},{"Start":"08:27.140 ","End":"08:30.030","Text":"that\u0027s the demand function."},{"Start":"08:30.290 ","End":"08:37.255","Text":"Might as well say that it\u0027s domain also is x bigger or equal to 0."},{"Start":"08:37.255 ","End":"08:39.700","Text":"X is still the quantity."},{"Start":"08:39.700 ","End":"08:42.960","Text":"For the sketch, we see it\u0027s linear,"},{"Start":"08:42.960 ","End":"08:44.990","Text":"so we really only need 2 points."},{"Start":"08:44.990 ","End":"08:47.450","Text":"Let me just get some axes in."},{"Start":"08:47.450 ","End":"08:49.610","Text":"I borrowed the earlier sketches,"},{"Start":"08:49.610 ","End":"08:52.060","Text":"just fix it up a bit."},{"Start":"08:52.060 ","End":"08:54.560","Text":"We can make a little table."},{"Start":"08:54.560 ","End":"08:56.390","Text":"We only need 2 values."},{"Start":"08:56.390 ","End":"08:59.495","Text":"Sometimes I do a third if they\u0027re too close."},{"Start":"08:59.495 ","End":"09:04.530","Text":"We want x, this is now P. We need x"},{"Start":"09:04.530 ","End":"09:09.560","Text":"and P. The easiest point to compute always is when x is naught,"},{"Start":"09:09.560 ","End":"09:12.905","Text":"you don\u0027t produce anything."},{"Start":"09:12.905 ","End":"09:14.990","Text":"Put x equals naught here,"},{"Start":"09:14.990 ","End":"09:18.275","Text":"this bit is naught and we just get 40."},{"Start":"09:18.275 ","End":"09:22.560","Text":"X is naught, P is 40."},{"Start":"09:22.560 ","End":"09:28.040","Text":"Why don\u0027t I go for the 500?"},{"Start":"09:28.040 ","End":"09:31.940","Text":"Because I noticed earlier that the revenue was 0 there."},{"Start":"09:31.940 ","End":"09:36.120","Text":"Let\u0027s try x is 500,"},{"Start":"09:36.120 ","End":"09:38.560","Text":"and then I plug it in."},{"Start":"09:38.560 ","End":"09:48.995","Text":"Minus 0.08 times 500 just happens to be minus 40 plus 40, it\u0027s 0."},{"Start":"09:48.995 ","End":"09:51.620","Text":"Why is this not a surprise to me?"},{"Start":"09:51.620 ","End":"09:58.220","Text":"Because earlier we had the parabola and the revenue is 0 here."},{"Start":"09:58.220 ","End":"10:03.245","Text":"The quantity was 500, but the revenue is 0,"},{"Start":"10:03.245 ","End":"10:06.940","Text":"which means that the price was 0."},{"Start":"10:06.940 ","End":"10:08.984","Text":"That makes sense."},{"Start":"10:08.984 ","End":"10:12.900","Text":"Zero, 40, 40 is the largest value."},{"Start":"10:12.900 ","End":"10:16.095","Text":"Let\u0027s say here we have 40,"},{"Start":"10:16.095 ","End":"10:20.415","Text":"the point 0,40 is then here."},{"Start":"10:20.415 ","End":"10:25.155","Text":"The point 500,0 is here."},{"Start":"10:25.155 ","End":"10:28.160","Text":"You know what? Let\u0027s just throw in another 1."},{"Start":"10:28.160 ","End":"10:31.250","Text":"Let\u0027s say we took a quantity of 200,"},{"Start":"10:31.250 ","End":"10:37.850","Text":"200 times point 0,8 would be 16,"},{"Start":"10:37.850 ","End":"10:40.220","Text":"I think but with a minus."},{"Start":"10:40.220 ","End":"10:46.750","Text":"Minus 16 plus 40 would be 24."},{"Start":"10:46.750 ","End":"10:52.110","Text":"Here we have 200 and roughly,"},{"Start":"10:52.110 ","End":"11:00.270","Text":"let\u0027s say 24 means more than halfway somewhere around here."},{"Start":"11:00.270 ","End":"11:04.150","Text":"Hopefully, these are a straight line."},{"Start":"11:05.030 ","End":"11:08.560","Text":"Done with this exercise."}],"ID":8532},{"Watched":false,"Name":"Exercise 4","Duration":"6m 46s","ChapterTopicVideoID":8398,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8398.jpeg","UploadDate":"2017-01-28T13:42:17.5170000","DurationForVideoObject":"PT6M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"In this exercise, we\u0027re given the demand function of a certain product."},{"Start":"00:04.980 ","End":"00:08.955","Text":"P stands for price and x is the quantity."},{"Start":"00:08.955 ","End":"00:13.515","Text":"For each quantity we have the price as this formula."},{"Start":"00:13.515 ","End":"00:16.725","Text":"Let\u0027s take each of these 4 parts separately."},{"Start":"00:16.725 ","End":"00:22.335","Text":"Part a, we\u0027re just asked to find the domain of the function."},{"Start":"00:22.335 ","End":"00:26.520","Text":"Mathematically, there\u0027s no reason you can\u0027t substitute any value of x,"},{"Start":"00:26.520 ","End":"00:32.460","Text":"but it\u0027s a real life question and x being the quantity can\u0027t be negative."},{"Start":"00:32.460 ","End":"00:38.355","Text":"We just write x is not negative means x bigger or equal to 0."},{"Start":"00:38.355 ","End":"00:46.920","Text":"Now part b, the total revenue function and the average revenue function is,"},{"Start":"00:46.920 ","End":"00:49.275","Text":"first we\u0027ll do the total revenue function."},{"Start":"00:49.275 ","End":"00:51.000","Text":"I\u0027ll call it TR of x,"},{"Start":"00:51.000 ","End":"00:54.410","Text":"it\u0027s sometimes abbreviated to just R of x."},{"Start":"00:54.410 ","End":"01:05.450","Text":"What it is, is the number of units sold times the price of each unit."},{"Start":"01:05.450 ","End":"01:09.800","Text":"But we have P of x. P of x is just we have to copy it from there,"},{"Start":"01:09.800 ","End":"01:14.185","Text":"it\u0027s minus 0.4 x plus 100."},{"Start":"01:14.185 ","End":"01:21.910","Text":"We multiply that, we get minus 0.4x squared plus 100x."},{"Start":"01:22.930 ","End":"01:25.760","Text":"That\u0027s the total revenue."},{"Start":"01:25.760 ","End":"01:31.740","Text":"Now, the average revenue"},{"Start":"01:34.540 ","End":"01:42.560","Text":"of x is equal to the total revenue,"},{"Start":"01:42.560 ","End":"01:44.750","Text":"but the average means per unit."},{"Start":"01:44.750 ","End":"01:48.120","Text":"So we divide by the number of units."},{"Start":"01:52.460 ","End":"01:55.725","Text":"TR of x, total revenue,"},{"Start":"01:55.725 ","End":"02:00.015","Text":"minus 0.4x squared plus 100x,"},{"Start":"02:00.015 ","End":"02:04.860","Text":"and then divide that by x."},{"Start":"02:04.860 ","End":"02:10.880","Text":"What we would get, since each of these divides by x,"},{"Start":"02:10.880 ","End":"02:16.370","Text":"we\u0027ll divide everything by x and we get minus 0.4,"},{"Start":"02:16.370 ","End":"02:21.170","Text":"just x plus 100."},{"Start":"02:21.170 ","End":"02:26.330","Text":"Now of course, all this works when x is bigger than 0."},{"Start":"02:26.330 ","End":"02:28.220","Text":"I mean the whole average concept."},{"Start":"02:28.220 ","End":"02:30.320","Text":"You can\u0027t take the average of 0 items,"},{"Start":"02:30.320 ","End":"02:34.025","Text":"so it\u0027s not enough for x to be bigger or equal to,"},{"Start":"02:34.025 ","End":"02:36.475","Text":"has to be strictly positive."},{"Start":"02:36.475 ","End":"02:39.530","Text":"Part c, that\u0027s something else."},{"Start":"02:39.530 ","End":"02:43.075","Text":"That\u0027s the marginal revenue function."},{"Start":"02:43.075 ","End":"02:45.050","Text":"When anything is marginal,"},{"Start":"02:45.050 ","End":"02:46.955","Text":"we take its derivative."},{"Start":"02:46.955 ","End":"02:48.770","Text":"Take a derivative of the revenue."},{"Start":"02:48.770 ","End":"02:50.780","Text":"Well, here I called it TR."},{"Start":"02:50.780 ","End":"02:57.400","Text":"It\u0027s just the derivative of TR of x."},{"Start":"03:03.800 ","End":"03:08.520","Text":"From here, I just take the derivative,"},{"Start":"03:08.520 ","End":"03:13.665","Text":"2 times minus 0.4 is minus 0.8,"},{"Start":"03:13.665 ","End":"03:15.840","Text":"and we lower the power by 1,"},{"Start":"03:15.840 ","End":"03:20.135","Text":"100x becomes just 100,"},{"Start":"03:20.135 ","End":"03:23.870","Text":"so this is the marginal."},{"Start":"03:23.870 ","End":"03:26.570","Text":"Now in part d,"},{"Start":"03:26.570 ","End":"03:30.200","Text":"we have to find the maximal total revenue."},{"Start":"03:30.200 ","End":"03:38.080","Text":"We need to ask which x maximizes,"},{"Start":"03:38.080 ","End":"03:42.240","Text":"and we\u0027re just going to copy the function here,"},{"Start":"03:42.240 ","End":"03:49.930","Text":"maximizes minus 0.4x squared plus 100x."},{"Start":"03:49.940 ","End":"03:52.980","Text":"That\u0027s another way of putting it."},{"Start":"03:52.980 ","End":"03:57.050","Text":"Now we\u0027re going to use our knowledge of parabolas and quadratics."},{"Start":"03:57.050 ","End":"04:02.415","Text":"After all, this is of the general form ax squared"},{"Start":"04:02.415 ","End":"04:08.220","Text":"plus bx plus c. Our marginal revenue,"},{"Start":"04:08.220 ","End":"04:10.605","Text":"well, we\u0027ll just call that y."},{"Start":"04:10.605 ","End":"04:14.615","Text":"Simplicity in maths, we like x and y."},{"Start":"04:14.615 ","End":"04:20.960","Text":"This is the quadratic and we\u0027re going to use the property of quadratic curves,"},{"Start":"04:20.960 ","End":"04:24.560","Text":"which are parabolas instead of using calculus to"},{"Start":"04:24.560 ","End":"04:29.180","Text":"differentiate and set the derivative to 0 and so on."},{"Start":"04:29.180 ","End":"04:32.555","Text":"If a is negative,"},{"Start":"04:32.555 ","End":"04:38.340","Text":"then it faces down, which is what we have."},{"Start":"04:38.340 ","End":"04:40.505","Text":"In the other way, if a was positive,"},{"Start":"04:40.505 ","End":"04:42.230","Text":"it would face up,"},{"Start":"04:42.230 ","End":"04:45.605","Text":"or we say it opens upwards, opens downwards."},{"Start":"04:45.605 ","End":"04:51.410","Text":"This is a, this is b, and c is 0."},{"Start":"04:51.410 ","End":"04:55.855","Text":"The maximum would occur at the vertex,"},{"Start":"04:55.855 ","End":"04:57.070","Text":"and if it\u0027s the other way around,"},{"Start":"04:57.070 ","End":"04:59.365","Text":"then this is the vertex."},{"Start":"04:59.365 ","End":"05:03.810","Text":"There\u0027s a formula, the x of the vertex,"},{"Start":"05:03.810 ","End":"05:10.335","Text":"I\u0027ll call it x_v, is given by minus b over 2a, in general."},{"Start":"05:10.335 ","End":"05:15.645","Text":"In our case, minus b will be minus"},{"Start":"05:15.645 ","End":"05:22.965","Text":"100 over twice a is minus 0.4."},{"Start":"05:22.965 ","End":"05:30.780","Text":"This is going to equal minus 100 over minus 0.8."},{"Start":"05:30.780 ","End":"05:33.605","Text":"Of course, the minuses cancel,"},{"Start":"05:33.605 ","End":"05:39.115","Text":"and this gives us 125."},{"Start":"05:39.115 ","End":"05:41.450","Text":"This will be the x that will give"},{"Start":"05:41.450 ","End":"05:46.400","Text":"the maximal total revenue because where we have an upside down parabola,"},{"Start":"05:46.400 ","End":"05:49.085","Text":"there is a maximum, otherwise there\u0027d be a minimum."},{"Start":"05:49.085 ","End":"05:55.310","Text":"The last thing we have to find out is what is the maximal total revenue."},{"Start":"05:55.310 ","End":"06:03.140","Text":"What we have to do is substitute this x in the revenue function, this 1 here."},{"Start":"06:03.140 ","End":"06:11.655","Text":"What we need is TR of 125 and see what that equals."},{"Start":"06:11.655 ","End":"06:13.965","Text":"I\u0027ll do it over here."},{"Start":"06:13.965 ","End":"06:15.920","Text":"It\u0027s just a computation,"},{"Start":"06:15.920 ","End":"06:21.095","Text":"minus 0.4 times 125 squared"},{"Start":"06:21.095 ","End":"06:27.330","Text":"plus 100 times 125."},{"Start":"06:27.330 ","End":"06:29.865","Text":"This is equal to,"},{"Start":"06:29.865 ","End":"06:33.120","Text":"I make it 625,"},{"Start":"06:33.120 ","End":"06:35.335","Text":"and that would be dollars."},{"Start":"06:35.335 ","End":"06:37.820","Text":"Whoops, I missed a 0."},{"Start":"06:37.820 ","End":"06:41.580","Text":"It\u0027s actually $6,250."},{"Start":"06:42.650 ","End":"06:47.410","Text":"That\u0027s the answer to d, and we\u0027re done."}],"ID":8533},{"Watched":false,"Name":"Exercise 5","Duration":"9m 40s","ChapterTopicVideoID":8399,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8399.jpeg","UploadDate":"2017-01-28T13:45:51.8500000","DurationForVideoObject":"PT9M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.580","Text":"In this exercise, we\u0027re given here the demand function of a product,"},{"Start":"00:05.580 ","End":"00:08.850","Text":"x is the quantity,"},{"Start":"00:08.850 ","End":"00:11.700","Text":"the number of units of the product produced,"},{"Start":"00:11.700 ","End":"00:13.740","Text":"sometimes we use letter q,"},{"Start":"00:13.740 ","End":"00:17.835","Text":"here we use x. P is the price per unit,"},{"Start":"00:17.835 ","End":"00:20.925","Text":"and there are 3 questions,"},{"Start":"00:20.925 ","End":"00:23.385","Text":"we\u0027ll start 1 at a time,"},{"Start":"00:23.385 ","End":"00:25.560","Text":"a asks for 2 things,"},{"Start":"00:25.560 ","End":"00:28.815","Text":"the total revenue and the marginal revenue."},{"Start":"00:28.815 ","End":"00:33.240","Text":"The total revenue, sometimes you write it as TR,"},{"Start":"00:33.240 ","End":"00:35.564","Text":"or sometimes just R for revenue,"},{"Start":"00:35.564 ","End":"00:40.200","Text":"is the total receipt of money from the sale,"},{"Start":"00:40.200 ","End":"00:44.490","Text":"it means the number of units sold,"},{"Start":"00:44.490 ","End":"00:48.120","Text":"which is x, times the price per unit,"},{"Start":"00:48.120 ","End":"00:50.025","Text":"which is p of x."},{"Start":"00:50.025 ","End":"00:52.350","Text":"Now, we\u0027re given p of x over here,"},{"Start":"00:52.350 ","End":"00:53.730","Text":"so I just have to copy it,"},{"Start":"00:53.730 ","End":"00:57.420","Text":"minus 6x squared plus 240x,"},{"Start":"00:57.420 ","End":"01:02.040","Text":"plus 1800, and then expound brackets,"},{"Start":"01:02.040 ","End":"01:04.890","Text":"so you get minus 6x cubed,"},{"Start":"01:04.890 ","End":"01:11.230","Text":"plus 240x squared, plus 1800x."},{"Start":"01:12.410 ","End":"01:17.265","Text":"This is the total revenue,"},{"Start":"01:17.265 ","End":"01:23.010","Text":"and I guess I should write it as R of x to remind you that it\u0027s a function of x."},{"Start":"01:23.010 ","End":"01:25.170","Text":"Next bit is the marginal revenue,"},{"Start":"01:25.170 ","End":"01:29.775","Text":"now when you see the word marginal, it means derivative."},{"Start":"01:29.775 ","End":"01:32.235","Text":"Let me just continue over here,"},{"Start":"01:32.235 ","End":"01:35.715","Text":"marginal revenue written MR,"},{"Start":"01:35.715 ","End":"01:39.675","Text":"is defined to be just the derivative of the total revenue."},{"Start":"01:39.675 ","End":"01:42.480","Text":"So all we have to do is just differentiate this,"},{"Start":"01:42.480 ","End":"01:46.965","Text":"we get minus 18x squared,"},{"Start":"01:46.965 ","End":"01:51.780","Text":"plus 240 times 2 is 480,"},{"Start":"01:51.780 ","End":"01:57.795","Text":"that\u0027s x, and here plus 1800."},{"Start":"01:57.795 ","End":"01:59.940","Text":"This also is in dollars,"},{"Start":"01:59.940 ","End":"02:04.510","Text":"this all is in, write it everywhere but dollars."},{"Start":"02:06.530 ","End":"02:14.940","Text":"In Part b, we assume that our current production level quantity is 40,"},{"Start":"02:14.940 ","End":"02:18.855","Text":"and we want to know whether or not to increase it."},{"Start":"02:18.855 ","End":"02:25.320","Text":"So let\u0027s say we just talk about increasing it by 1 unit,"},{"Start":"02:25.320 ","End":"02:30.435","Text":"the reason I\u0027m just saying by 1 unit is then we can then use it the marginal,"},{"Start":"02:30.435 ","End":"02:40.065","Text":"so what we do is we figure out what is the marginal revenue at 40,"},{"Start":"02:40.065 ","End":"02:42.795","Text":"and if it comes out positive,"},{"Start":"02:42.795 ","End":"02:45.870","Text":"then we should increase production because this"},{"Start":"02:45.870 ","End":"02:49.050","Text":"approximates the additional additional revenue"},{"Start":"02:49.050 ","End":"02:55.590","Text":"we get when we increase the quantity say from 40-41 in this case."},{"Start":"02:55.590 ","End":"02:58.289","Text":"The marginal revenue at 40,"},{"Start":"02:58.289 ","End":"03:03.135","Text":"we just get by substituting 40 in here,"},{"Start":"03:03.135 ","End":"03:05.340","Text":"we just say x equals 40,"},{"Start":"03:05.340 ","End":"03:11.100","Text":"so we get minus 18 times 40 squared,"},{"Start":"03:11.100 ","End":"03:16.710","Text":"plus 4800 times 40,"},{"Start":"03:16.710 ","End":"03:21.420","Text":"plus 1,800, and this is equal"},{"Start":"03:21.420 ","End":"03:29.560","Text":"to minus 7,800, that\u0027s dollars."},{"Start":"03:29.600 ","End":"03:33.645","Text":"So if we increase the production by 1 unit,"},{"Start":"03:33.645 ","End":"03:38.865","Text":"we lose roughly 7800 in revenue,"},{"Start":"03:38.865 ","End":"03:44.115","Text":"so I guess the answer to b is no."},{"Start":"03:44.115 ","End":"03:46.169","Text":"Just another point on notation,"},{"Start":"03:46.169 ","End":"03:50.040","Text":"if you prefer, don\u0027t have to write marginal, MR,"},{"Start":"03:50.040 ","End":"03:57.600","Text":"we could also write it as R prime and just leave it as that, let\u0027s see,"},{"Start":"03:57.600 ","End":"04:06.780","Text":"number c. We want to know which x gives the maximal total revenue,"},{"Start":"04:06.780 ","End":"04:11.835","Text":"and then we have to find out what that maximum value is."},{"Start":"04:11.835 ","End":"04:13.500","Text":"R of x we\u0027re trying to maximum,"},{"Start":"04:13.500 ","End":"04:15.810","Text":"we\u0027ll just call it y for convenience,"},{"Start":"04:15.810 ","End":"04:19.725","Text":"y equals and I\u0027ll copy that, minus 600x cubed,"},{"Start":"04:19.725 ","End":"04:27.780","Text":"plus 240x squared, plus 1800x."},{"Start":"04:27.780 ","End":"04:30.240","Text":"Now, this is a cubic equation,"},{"Start":"04:30.240 ","End":"04:33.525","Text":"unlike in previous exercises we had"},{"Start":"04:33.525 ","End":"04:37.845","Text":"quadratic and then we could use the properties of the parabola, but here,"},{"Start":"04:37.845 ","End":"04:41.130","Text":"we\u0027ll have to use calculus and the derivative,"},{"Start":"04:41.130 ","End":"04:43.980","Text":"so we\u0027re going to look for critical points,"},{"Start":"04:43.980 ","End":"04:45.975","Text":"points where the derivative is 0."},{"Start":"04:45.975 ","End":"04:49.485","Text":"The derivative y prime is just R prime,"},{"Start":"04:49.485 ","End":"04:53.010","Text":"and I just copied it from here to here."},{"Start":"04:53.010 ","End":"04:56.970","Text":"We want to set the derivative to 0,"},{"Start":"04:56.970 ","End":"04:59.400","Text":"because I\u0027m looking for maximum or minimum,"},{"Start":"04:59.400 ","End":"05:01.980","Text":"we set the derivative to 0,"},{"Start":"05:01.980 ","End":"05:05.235","Text":"so that gives us that this,"},{"Start":"05:05.235 ","End":"05:08.085","Text":"is equal to 0."},{"Start":"05:08.085 ","End":"05:11.310","Text":"I see everything to this by 6, and you know what,"},{"Start":"05:11.310 ","End":"05:14.415","Text":"I\u0027ll divide everything by minus 6,"},{"Start":"05:14.415 ","End":"05:17.250","Text":"I\u0027d like the a to be positive,"},{"Start":"05:17.250 ","End":"05:20.155","Text":"so we get 3x"},{"Start":"05:20.155 ","End":"05:27.645","Text":"square minus 80x because 480 over 6 is 80x,"},{"Start":"05:27.645 ","End":"05:34.830","Text":"minus, and from here minus 300 equals 0."},{"Start":"05:34.830 ","End":"05:37.815","Text":"So let\u0027s use the quadratic formula,"},{"Start":"05:37.815 ","End":"05:40.155","Text":"there\u0027s 2 values of x;"},{"Start":"05:40.155 ","End":"05:45.060","Text":"minus b, which is 80, plus or minus,"},{"Start":"05:45.060 ","End":"05:46.995","Text":"that\u0027s what gives us the 2 values,"},{"Start":"05:46.995 ","End":"05:50.520","Text":"the square root of b squared,"},{"Start":"05:50.520 ","End":"05:54.225","Text":"6,400, it\u0027s 80 times 80,"},{"Start":"05:54.225 ","End":"05:57.495","Text":"minus 4ac, it\u0027s going to be minus minus,"},{"Start":"05:57.495 ","End":"05:59.010","Text":"it\u0027s going to be plus,"},{"Start":"05:59.010 ","End":"06:04.425","Text":"4 times 3 times 300."},{"Start":"06:04.425 ","End":"06:07.770","Text":"This is a minus, this is a little to the minus so it\u0027s a plus,"},{"Start":"06:07.770 ","End":"06:13.170","Text":"and all of this over 2a which is 6."},{"Start":"06:13.170 ","End":"06:17.655","Text":"Now, under the square root sign we have 6400,"},{"Start":"06:17.655 ","End":"06:21.060","Text":"and this thing comes out to be 3600, because look,"},{"Start":"06:21.060 ","End":"06:23.955","Text":"4 times 3 times 3 is 4 times 9 is 36,"},{"Start":"06:23.955 ","End":"06:25.980","Text":"and the 2 0s,"},{"Start":"06:25.980 ","End":"06:32.505","Text":"and this plus this is 10,000 and the square root of 10,000 is 100."},{"Start":"06:32.505 ","End":"06:40.935","Text":"So what we get is 80 plus or minus 100 over 6."},{"Start":"06:40.935 ","End":"06:44.865","Text":"Now we take the plus and 1s to the minus."},{"Start":"06:44.865 ","End":"06:47.115","Text":"If we take the plus,"},{"Start":"06:47.115 ","End":"06:53.910","Text":"then we have 180 over 6,"},{"Start":"06:53.910 ","End":"06:55.860","Text":"if we take the minus,"},{"Start":"06:55.860 ","End":"07:01.690","Text":"we\u0027ll get minus 20 over 6,"},{"Start":"07:01.700 ","End":"07:04.800","Text":"but x is a quantity,"},{"Start":"07:04.800 ","End":"07:08.550","Text":"the number of units produced has to be non-negative,"},{"Start":"07:08.550 ","End":"07:10.230","Text":"so we can\u0027t have this,"},{"Start":"07:10.230 ","End":"07:12.795","Text":"so we\u0027re left with just this,"},{"Start":"07:12.795 ","End":"07:19.710","Text":"180 over 6 is 30 so I can say, x equals 30."},{"Start":"07:19.710 ","End":"07:25.290","Text":"Now we don\u0027t know if x equal 30 is a maximum point,"},{"Start":"07:25.290 ","End":"07:26.790","Text":"we want the maximal,"},{"Start":"07:26.790 ","End":"07:29.580","Text":"it could be the minimum or it could be something else,"},{"Start":"07:29.580 ","End":"07:32.955","Text":"what we\u0027re going to do is use the second derivative test."},{"Start":"07:32.955 ","End":"07:35.715","Text":"I hope you remember your calculus,"},{"Start":"07:35.715 ","End":"07:38.295","Text":"that with the second derivative test,"},{"Start":"07:38.295 ","End":"07:41.550","Text":"what I will do is, I\u0027ll take y-double prime,"},{"Start":"07:41.550 ","End":"07:42.945","Text":"I\u0027ll move back over here."},{"Start":"07:42.945 ","End":"07:46.110","Text":"Y-double prime is equal to,"},{"Start":"07:46.110 ","End":"07:48.615","Text":"it\u0027s the derivative of the derivative so that would be"},{"Start":"07:48.615 ","End":"07:51.690","Text":"minus 36x"},{"Start":"07:51.690 ","End":"07:59.010","Text":"plus 4800."},{"Start":"07:59.010 ","End":"08:02.160","Text":"What I have to do is now plug in,"},{"Start":"08:02.160 ","End":"08:08.235","Text":"I have to figure out what is y-double prime when x is 30,"},{"Start":"08:08.235 ","End":"08:09.975","Text":"I mean this is a function of x,"},{"Start":"08:09.975 ","End":"08:17.025","Text":"this is equal to minus 36 times 30,"},{"Start":"08:17.025 ","End":"08:22.080","Text":"plus 4800, which is,"},{"Start":"08:22.080 ","End":"08:28.830","Text":"well you can see it\u0027s going to be negative but this anyway is minus 1080 plus 4800,"},{"Start":"08:28.830 ","End":"08:32.040","Text":"it doesn\u0027t matter the actual answer, it\u0027s negative."},{"Start":"08:32.040 ","End":"08:33.630","Text":"That\u0027s the main thing we want to know."},{"Start":"08:33.630 ","End":"08:40.960","Text":"If it\u0027s negative, then it means that it is a maximum."},{"Start":"08:41.360 ","End":"08:46.680","Text":"Bigger than 0 is minimum and less than 0 is maximum."},{"Start":"08:46.680 ","End":"08:49.890","Text":"We know that we have the maximum at x equals 30,"},{"Start":"08:49.890 ","End":"08:54.170","Text":"but we want to know what is the maximal value,"},{"Start":"08:54.170 ","End":"08:59.180","Text":"so there is 1 more thing we have to do is substitute this in the revenue function,"},{"Start":"08:59.180 ","End":"09:04.820","Text":"R of 30 equals minus"},{"Start":"09:04.820 ","End":"09:09.650","Text":"6 times 30 cubed plus"},{"Start":"09:09.650 ","End":"09:15.885","Text":"240 times 30 squared,"},{"Start":"09:15.885 ","End":"09:21.720","Text":"plus 1800 times 30,"},{"Start":"09:21.720 ","End":"09:25.170","Text":"and I\u0027ll save you the tedious work,"},{"Start":"09:25.170 ","End":"09:33.655","Text":"it comes out to be 108,000 and remember we\u0027re working in dollars,"},{"Start":"09:33.655 ","End":"09:38.475","Text":"and that\u0027s the answer to the second 1/2 of part c,"},{"Start":"09:38.475 ","End":"09:41.020","Text":"and we are done."}],"ID":8534},{"Watched":false,"Name":"Exercise 6","Duration":"9m 14s","ChapterTopicVideoID":8400,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8400.jpeg","UploadDate":"2017-01-28T13:48:24.8470000","DurationForVideoObject":"PT9M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.815","Text":"In this exercise, we\u0027re given the demand function of a certain product,"},{"Start":"00:04.815 ","End":"00:07.995","Text":"but note that it\u0027s given the other way around."},{"Start":"00:07.995 ","End":"00:12.164","Text":"In other words, the independent variable x is p,"},{"Start":"00:12.164 ","End":"00:16.470","Text":"meaning it\u0027s the price per unit, and q,"},{"Start":"00:16.470 ","End":"00:19.575","Text":"the dependent variable is the quantity,"},{"Start":"00:19.575 ","End":"00:22.589","Text":"number of items sold at that price."},{"Start":"00:22.589 ","End":"00:26.220","Text":"More customarily, we have price as a function of quantity,"},{"Start":"00:26.220 ","End":"00:29.085","Text":"but it can be the other way around too."},{"Start":"00:29.085 ","End":"00:33.120","Text":"There\u0027s 3 parts let\u0027s do each 1 at a time."},{"Start":"00:33.120 ","End":"00:39.060","Text":"In part A, we have to find which unit price gives the maximal total revenue."},{"Start":"00:39.060 ","End":"00:42.295","Text":"Let\u0027s first, remember what is total revenue?"},{"Start":"00:42.295 ","End":"00:45.805","Text":"We write it as R or TR."},{"Start":"00:45.805 ","End":"00:48.544","Text":"I\u0027ll choose TR, total revenue."},{"Start":"00:48.544 ","End":"00:52.160","Text":"It\u0027s the number of items that are sold,"},{"Start":"00:52.160 ","End":"00:57.840","Text":"which is q times the price per item."},{"Start":"00:57.840 ","End":"01:00.390","Text":"I better write it as Q of x, like this."},{"Start":"01:00.390 ","End":"01:04.280","Text":"The price here is x and if we expand this,"},{"Start":"01:04.280 ","End":"01:06.470","Text":"we get, well, I\u0027ll write it again."},{"Start":"01:06.470 ","End":"01:14.070","Text":"It\u0027s x times 10x minus x squared over 5 and"},{"Start":"01:14.070 ","End":"01:22.440","Text":"that is equal to 10x squared minus x cubed over 5,"},{"Start":"01:22.440 ","End":"01:25.530","Text":"and this is a function of x, of course."},{"Start":"01:25.530 ","End":"01:26.960","Text":"We want to find which x,"},{"Start":"01:26.960 ","End":"01:29.780","Text":"which price gives the maximum total revenue."},{"Start":"01:29.780 ","End":"01:33.600","Text":"This is a cubic equation,"},{"Start":"01:33.600 ","End":"01:36.365","Text":"there\u0027s a power of 3 here so we can\u0027t use"},{"Start":"01:36.365 ","End":"01:40.445","Text":"the knowledge of parabolas to help us as we did previously,"},{"Start":"01:40.445 ","End":"01:43.040","Text":"we\u0027re going to have to use differentiation."},{"Start":"01:43.040 ","End":"01:46.850","Text":"What we do is we first of all differentiate"},{"Start":"01:46.850 ","End":"01:51.810","Text":"the total revenue function and let\u0027s see what that gives us,"},{"Start":"01:51.810 ","End":"02:01.700","Text":"20x minus 3/5 x squared and when we\u0027re looking for a maximum,"},{"Start":"02:01.700 ","End":"02:05.870","Text":"we\u0027re looking for a critical point it\u0027s where the derivative is 0."},{"Start":"02:05.870 ","End":"02:08.975","Text":"This is going to be a suspect for a maximum."},{"Start":"02:08.975 ","End":"02:19.175","Text":"We set 20x minus 3/5 x squared equals 0 and x is not going to be 0,"},{"Start":"02:19.175 ","End":"02:21.590","Text":"because if x was 0,"},{"Start":"02:21.590 ","End":"02:24.875","Text":"then the revenue would be 0."},{"Start":"02:24.875 ","End":"02:35.650","Text":"We can divide by x and say 20 minus 3/5 x equals 0."},{"Start":"02:35.740 ","End":"02:40.975","Text":"Then bring x to the other side and divide by 3/5."},{"Start":"02:40.975 ","End":"02:47.705","Text":"We get x is 20 divided by 3/5 is like times 5/3."},{"Start":"02:47.705 ","End":"02:50.450","Text":"20 times 5 is 100, 100 over 3,"},{"Start":"02:50.450 ","End":"02:54.880","Text":"$33 and a 1/3."},{"Start":"02:54.880 ","End":"03:00.140","Text":"How do we know if this 33 and a 1/3 is minimum or maximum or neither?"},{"Start":"03:00.140 ","End":"03:02.450","Text":"Well, we use the second derivative test."},{"Start":"03:02.450 ","End":"03:07.505","Text":"Let\u0027s do the second derivative of the revenue,"},{"Start":"03:07.505 ","End":"03:10.475","Text":"which is the derivative of the derivative."},{"Start":"03:10.475 ","End":"03:13.490","Text":"This is going to equal,"},{"Start":"03:13.490 ","End":"03:16.820","Text":"I\u0027m taking this and differentiating it."},{"Start":"03:16.820 ","End":"03:27.190","Text":"We get 20 minus 6/5 x, we know what x is."},{"Start":"03:27.190 ","End":"03:32.050","Text":"We want the second derivative at the point 33 and"},{"Start":"03:32.050 ","End":"03:38.255","Text":"a 1/3 and this is 20 minus,"},{"Start":"03:38.255 ","End":"03:44.895","Text":"let\u0027s see 6/5, 33 and a 1/3 is a 100 over 3,"},{"Start":"03:44.895 ","End":"03:52.560","Text":"3 goes into 6 twice,"},{"Start":"03:52.560 ","End":"03:56.970","Text":"and 5 goes into 100, 20 times."},{"Start":"03:56.970 ","End":"04:01.490","Text":"We\u0027ve got 20 minus 40."},{"Start":"04:01.490 ","End":"04:04.629","Text":"It\u0027s minus 20, but that doesn\u0027t matter."},{"Start":"04:04.629 ","End":"04:08.240","Text":"All we care about is that it\u0027s negative."},{"Start":"04:08.450 ","End":"04:12.140","Text":"When the second derivative is negative,"},{"Start":"04:12.140 ","End":"04:18.150","Text":"then it means that we have a maximum."},{"Start":"04:18.150 ","End":"04:20.385","Text":"That was part A."},{"Start":"04:20.385 ","End":"04:21.685","Text":"Now on to part B,"},{"Start":"04:21.685 ","End":"04:30.090","Text":"we want to know what is the demand that is Q quantity in this case."},{"Start":"04:30.200 ","End":"04:39.170","Text":"What we need to do is to just substitute this value of x in the demand function."},{"Start":"04:39.170 ","End":"04:46.040","Text":"What we need is Q of 33 and a 1/3."},{"Start":"04:46.040 ","End":"04:54.515","Text":"This is equal to just by substituting 10 times 33 and a 1/3."},{"Start":"04:54.515 ","End":"05:05.570","Text":"Then minus x squared over 5 so 33 and a 1/3 squared over 5."},{"Start":"05:05.570 ","End":"05:07.699","Text":"I\u0027ll do this as a side exercise,"},{"Start":"05:07.699 ","End":"05:09.860","Text":"maybe over here somewhere."},{"Start":"05:09.860 ","End":"05:15.425","Text":"What we can do is take 33 and a 1/3 outside the brackets,"},{"Start":"05:15.425 ","End":"05:20.750","Text":"and then we\u0027re left with 10 minus 33 and a 1/3,"},{"Start":"05:20.750 ","End":"05:28.110","Text":"only over 5 is 100 over 3 over 5,"},{"Start":"05:28.110 ","End":"05:31.170","Text":"which is 100 over 15."},{"Start":"05:31.170 ","End":"05:36.525","Text":"5 cancels here 3, here 20."},{"Start":"05:36.525 ","End":"05:40.040","Text":"If I put this bit over 3,"},{"Start":"05:40.040 ","End":"05:47.260","Text":"this is 30 over 3 minus 20 over 3 so this is 10 over 3."},{"Start":"05:47.260 ","End":"05:49.650","Text":"This part is 100 over 3,"},{"Start":"05:49.650 ","End":"05:56.980","Text":"100 over 3 times 10 over 3 is 1,000 over 9,"},{"Start":"05:57.740 ","End":"06:07.530","Text":"which is 111 and 1/9."},{"Start":"06:07.530 ","End":"06:09.225","Text":"I\u0027ll just write it over here."},{"Start":"06:09.225 ","End":"06:14.610","Text":"111 and 1/9, the answer to part B."},{"Start":"06:14.610 ","End":"06:22.415","Text":"Now for part C, when you see the word marginal, it means derivative."},{"Start":"06:22.415 ","End":"06:27.640","Text":"From Q of x, we need to derive and get Q prime of"},{"Start":"06:27.640 ","End":"06:34.795","Text":"x and this is equal to,"},{"Start":"06:34.795 ","End":"06:37.750","Text":"let\u0027s see, derivative of 10x is just 10."},{"Start":"06:37.750 ","End":"06:40.780","Text":"Derivative of x squared is 2x,"},{"Start":"06:40.780 ","End":"06:42.580","Text":"and the rest of it\u0027s constant,"},{"Start":"06:42.580 ","End":"06:46.730","Text":"so it\u0027s minus 2x over 5."},{"Start":"06:46.730 ","End":"06:56.295","Text":"We need Q prime when x is 33 and a 1/3,"},{"Start":"06:56.295 ","End":"06:58.125","Text":"which is equal to,"},{"Start":"06:58.125 ","End":"07:01.470","Text":"let\u0027s see, 10 minus,"},{"Start":"07:01.470 ","End":"07:04.770","Text":"again I\u0027ll do a little side computation,"},{"Start":"07:04.770 ","End":"07:10.065","Text":"2/5 of x is 2/5 and x is 33 and a 1/3,"},{"Start":"07:10.065 ","End":"07:13.230","Text":"which we said was 100 over 3."},{"Start":"07:13.230 ","End":"07:17.250","Text":"Now 5 goes into 100 20 times,"},{"Start":"07:17.250 ","End":"07:18.960","Text":"20 times 2 is 40."},{"Start":"07:18.960 ","End":"07:21.600","Text":"This is 40 over 3,"},{"Start":"07:21.600 ","End":"07:28.410","Text":"or 13 and a 1/3 and 10 minus 13 and"},{"Start":"07:28.410 ","End":"07:35.880","Text":"a 1/3 is going to equal minus 3 and a 1/3,"},{"Start":"07:35.880 ","End":"07:41.850","Text":"so that answers the numerical part of C. But there\u0027s another last bit,"},{"Start":"07:41.850 ","End":"07:43.564","Text":"what does it mean?"},{"Start":"07:43.564 ","End":"07:49.925","Text":"The marginal means that if we increase the independent variable by 1, in this case,"},{"Start":"07:49.925 ","End":"07:57.650","Text":"we increase the price from $33 and a 1/3 and we add a dollar,"},{"Start":"07:57.650 ","End":"08:02.470","Text":"that would make it $34 and a 1/3,"},{"Start":"08:02.470 ","End":"08:11.690","Text":"then the demand Q will drop by 3 and a 1/3 units."},{"Start":"08:11.690 ","End":"08:19.605","Text":"The old Q that we had was 111 and 1/9."},{"Start":"08:19.605 ","End":"08:21.630","Text":"It will drop minus 3 and a 1/3,"},{"Start":"08:21.630 ","End":"08:23.045","Text":"and we\u0027ll get the new Q,"},{"Start":"08:23.045 ","End":"08:31.485","Text":"that\u0027s approximately equal to 111 and a 1/9 minus 3 and a 1/3."},{"Start":"08:31.485 ","End":"08:35.075","Text":"The demand will drop by approximately this."},{"Start":"08:35.075 ","End":"08:42.935","Text":"It\u0027s 107 and 7/9 so I make it the new quantity."},{"Start":"08:42.935 ","End":"08:44.410","Text":"The numbers don\u0027t matter,"},{"Start":"08:44.410 ","End":"08:51.680","Text":"it\u0027s the idea that this marginal demand and it\u0027s minus 3 and a 1/3,"},{"Start":"08:51.680 ","End":"08:55.759","Text":"means that if we increase the price by a dollar,"},{"Start":"08:55.759 ","End":"09:04.250","Text":"the quantity will drop by 3 and a 1/3 and it will drop from this to this approximately."},{"Start":"09:04.250 ","End":"09:10.545","Text":"You could actually compute Q of 34 and a 1/3,"},{"Start":"09:10.545 ","End":"09:15.330","Text":"you\u0027ll get something close. That\u0027s it."}],"ID":8535},{"Watched":false,"Name":"Exercise 7","Duration":"7m 50s","ChapterTopicVideoID":8401,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8401.jpeg","UploadDate":"2017-01-28T13:50:27.7400000","DurationForVideoObject":"PT7M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.255","Text":"In this exercise, we\u0027re told that a manufacturer produces x pounds of coffee a day."},{"Start":"00:06.255 ","End":"00:12.015","Text":"X is the quantity and the total dollar cost of C of x,"},{"Start":"00:12.015 ","End":"00:15.885","Text":"cost function, 5x plus 150."},{"Start":"00:15.885 ","End":"00:19.680","Text":"Sometimes instead of C, we use TC for total cost."},{"Start":"00:19.680 ","End":"00:21.480","Text":"4 parts, part a,"},{"Start":"00:21.480 ","End":"00:24.705","Text":"we have to sketch your graph of the total cost function."},{"Start":"00:24.705 ","End":"00:27.660","Text":"Also, to answer the question,"},{"Start":"00:27.660 ","End":"00:29.280","Text":"what are the fixed costs?"},{"Start":"00:29.280 ","End":"00:31.500","Text":"Let\u0027s just do them 1 at a time."},{"Start":"00:31.500 ","End":"00:34.290","Text":"A, the graph, now,"},{"Start":"00:34.290 ","End":"00:35.790","Text":"it\u0027s going to be a straight line."},{"Start":"00:35.790 ","End":"00:40.890","Text":"It\u0027s a linear function at some number x plus a number, Ax plus B."},{"Start":"00:40.890 ","End":"00:42.980","Text":"If it\u0027s a linear graph, a straight line,"},{"Start":"00:42.980 ","End":"00:46.785","Text":"we only need 2 points. Let\u0027s see."},{"Start":"00:46.785 ","End":"00:49.590","Text":"We need 2 columns,"},{"Start":"00:49.590 ","End":"00:52.035","Text":"1 column for x,"},{"Start":"00:52.035 ","End":"00:53.520","Text":"and 1 for the cost,"},{"Start":"00:53.520 ","End":"00:55.610","Text":"C, or C of x."},{"Start":"00:55.610 ","End":"00:59.930","Text":"The obvious value of x to take would be 0,"},{"Start":"00:59.930 ","End":"01:03.260","Text":"no pounds of coffee. Plug that in."},{"Start":"01:03.260 ","End":"01:05.285","Text":"We\u0027ve got C of 0,"},{"Start":"01:05.285 ","End":"01:10.530","Text":"5 times 0 plus 150 is 150. I\u0027m just remarking."},{"Start":"01:10.530 ","End":"01:14.525","Text":"That\u0027s what you get when you plug in 0 into the cost function."},{"Start":"01:14.525 ","End":"01:17.015","Text":"Take another number, I don\u0027t know."},{"Start":"01:17.015 ","End":"01:19.670","Text":"I\u0027ll take 10 pounds of coffee."},{"Start":"01:19.670 ","End":"01:22.895","Text":"What we want is C of 10."},{"Start":"01:22.895 ","End":"01:25.640","Text":"Plug in 10 into this formula,"},{"Start":"01:25.640 ","End":"01:27.860","Text":"and 10 times 5 is 50,"},{"Start":"01:27.860 ","End":"01:30.575","Text":"plus 150 is 200."},{"Start":"01:30.575 ","End":"01:33.030","Text":"Now we need to sketch some axis."},{"Start":"01:35.930 ","End":"01:40.140","Text":"This 1 is x and this 1 is the cost,"},{"Start":"01:40.140 ","End":"01:43.815","Text":"C. We have an idea of scale now."},{"Start":"01:43.815 ","End":"01:46.140","Text":"We want to, on the vertical axis,"},{"Start":"01:46.140 ","End":"01:48.605","Text":"put in 150 and 200."},{"Start":"01:48.605 ","End":"01:52.070","Text":"I don\u0027t know. Let\u0027s just say this is 150."},{"Start":"01:52.070 ","End":"01:53.390","Text":"Then that would be,"},{"Start":"01:53.390 ","End":"01:55.580","Text":"then if we break it up into 50s,"},{"Start":"01:55.580 ","End":"01:57.680","Text":"that would be 200."},{"Start":"01:57.680 ","End":"02:03.585","Text":"If 0s here, let\u0027s say that 10 is here,"},{"Start":"02:03.585 ","End":"02:09.300","Text":"0,150 would be this point here, and 10,"},{"Start":"02:09.300 ","End":"02:13.200","Text":"200 and have to go up from the 10 and right from the 200,"},{"Start":"02:13.200 ","End":"02:15.520","Text":"could put dotted lines,"},{"Start":"02:15.520 ","End":"02:18.680","Text":"sometimes helps, anyway, somewhere here."},{"Start":"02:18.680 ","End":"02:21.415","Text":"Then just take a ruler,"},{"Start":"02:21.415 ","End":"02:25.370","Text":"something like this, and notice the number of pounds of coffee,"},{"Start":"02:25.370 ","End":"02:28.475","Text":"the quantity is going to be bigger or equal to 0."},{"Start":"02:28.475 ","End":"02:32.245","Text":"That was the first of part a."},{"Start":"02:32.245 ","End":"02:35.090","Text":"Next question is, what are the fixed costs?"},{"Start":"02:35.090 ","End":"02:38.000","Text":"How is it related to this sketching the graph?"},{"Start":"02:38.000 ","End":"02:41.510","Text":"Well, turns out the fixed costs are just where the graph cuts"},{"Start":"02:41.510 ","End":"02:45.505","Text":"the y-axis or in our case the C-axis."},{"Start":"02:45.505 ","End":"02:47.495","Text":"Do it in a different color."},{"Start":"02:47.495 ","End":"02:51.935","Text":"This point here is the fixed cost, 150."},{"Start":"02:51.935 ","End":"02:56.420","Text":"The reason is, that\u0027s it\u0027s fixed without producing anything,"},{"Start":"02:56.420 ","End":"02:59.660","Text":"producing 0 pounds of coffee,"},{"Start":"02:59.660 ","End":"03:02.060","Text":"there are still expenses."},{"Start":"03:02.060 ","End":"03:07.085","Text":"The $150 is the fixed cost."},{"Start":"03:07.085 ","End":"03:12.635","Text":"Now we come to part b and it\u0027s a reverse question."},{"Start":"03:12.635 ","End":"03:18.125","Text":"We want to know how many pounds of coffee does the manufacturer produce."},{"Start":"03:18.125 ","End":"03:20.915","Text":"This will be the x and it will be the unknown,"},{"Start":"03:20.915 ","End":"03:22.940","Text":"but the cost is known,"},{"Start":"03:22.940 ","End":"03:24.440","Text":"it\u0027s going to be 1,000."},{"Start":"03:24.440 ","End":"03:31.420","Text":"In other words, we know that the cost of x pounds is 1,000."},{"Start":"03:31.420 ","End":"03:34.190","Text":"What is x equal to?"},{"Start":"03:34.190 ","End":"03:35.765","Text":"It\u0027s an equation."},{"Start":"03:35.765 ","End":"03:42.175","Text":"We know that the cost of producing x pounds is 5x plus 150,"},{"Start":"03:42.175 ","End":"03:46.180","Text":"and that\u0027s got to equal 1,000,"},{"Start":"03:46.180 ","End":"03:48.405","Text":"and then we just solve the equation."},{"Start":"03:48.405 ","End":"03:52.715","Text":"We bring the 150 over to the other side."},{"Start":"03:52.715 ","End":"03:59.020","Text":"We\u0027ve got 5x equals 850."},{"Start":"03:59.020 ","End":"04:01.810","Text":"After dividing by 5,"},{"Start":"04:01.810 ","End":"04:06.410","Text":"we get that x is equal to 170,"},{"Start":"04:06.410 ","End":"04:11.275","Text":"but that\u0027s not dollars, that\u0027s pounds."},{"Start":"04:11.275 ","End":"04:13.310","Text":"Now, we could have done it on the graph,"},{"Start":"04:13.310 ","End":"04:16.010","Text":"but I didn\u0027t choose the scale right."},{"Start":"04:16.010 ","End":"04:18.455","Text":"If it would have gone up to 1,000,"},{"Start":"04:18.455 ","End":"04:25.685","Text":"let\u0027s just say it was cut and then we went up to 1,000 and then we went across here."},{"Start":"04:25.685 ","End":"04:29.300","Text":"This was also to scale,"},{"Start":"04:29.300 ","End":"04:32.690","Text":"and the 1,000 would hit somewhere here."},{"Start":"04:32.690 ","End":"04:34.460","Text":"You\u0027d go down here,"},{"Start":"04:34.460 ","End":"04:36.800","Text":"and then you\u0027d get to 170."},{"Start":"04:36.800 ","End":"04:41.420","Text":"But algebraically is the best method anyway rather than the graph,"},{"Start":"04:41.420 ","End":"04:43.170","Text":"but I was just mentioning."},{"Start":"04:43.170 ","End":"04:47.090","Text":"That\u0027s the answer to part b."},{"Start":"04:47.630 ","End":"04:53.335","Text":"Now, part c is similar to part a."},{"Start":"04:53.335 ","End":"04:57.980","Text":"We want to know the cost of producing 20 pounds of coffee."},{"Start":"04:57.980 ","End":"05:00.560","Text":"Well, we already did 0 and 10."},{"Start":"05:00.560 ","End":"05:03.235","Text":"I\u0027ll just move things around a bit."},{"Start":"05:03.235 ","End":"05:05.970","Text":"We just need a third row,"},{"Start":"05:05.970 ","End":"05:08.850","Text":"0,10, now we want 20."},{"Start":"05:08.850 ","End":"05:16.255","Text":"What we put here is what we get when we plug 20 into the cost function."},{"Start":"05:16.255 ","End":"05:21.840","Text":"Here it is, 5 times 20 is"},{"Start":"05:21.840 ","End":"05:28.815","Text":"100.100 plus 150 that is 250."},{"Start":"05:28.815 ","End":"05:33.630","Text":"This 250, which is $250,"},{"Start":"05:33.630 ","End":"05:41.390","Text":"is the answer to part c. Now we have 1 more part which is d,"},{"Start":"05:41.390 ","End":"05:43.670","Text":"the marginal cost function,"},{"Start":"05:43.670 ","End":"05:46.760","Text":"the word marginal to indicate derivative."},{"Start":"05:46.760 ","End":"05:49.760","Text":"Cost function of x is equal"},{"Start":"05:49.760 ","End":"05:57.570","Text":"to 5x plus 150."},{"Start":"05:57.570 ","End":"06:01.520","Text":"The marginal cost is the derivative C prime of x."},{"Start":"06:01.520 ","End":"06:05.705","Text":"Actually, sometimes you could even label it like marginal cost might be"},{"Start":"06:05.705 ","End":"06:14.720","Text":"MC of x is equal to same thing as the derivative of the cost function."},{"Start":"06:14.720 ","End":"06:17.390","Text":"In this case, it\u0027s just a constant 5,"},{"Start":"06:17.390 ","End":"06:19.740","Text":"a linear function, 150 gives nothing,"},{"Start":"06:19.740 ","End":"06:21.250","Text":"and 5x is 5."},{"Start":"06:21.250 ","End":"06:23.360","Text":"This is also in dollars."},{"Start":"06:23.360 ","End":"06:24.920","Text":"If the cost is in dollars,"},{"Start":"06:24.920 ","End":"06:27.770","Text":"marginal is also in dollars."},{"Start":"06:27.770 ","End":"06:31.340","Text":"I would like to tell you what it means."},{"Start":"06:31.340 ","End":"06:37.225","Text":"It means that if we increase the production by 1 pound,"},{"Start":"06:37.225 ","End":"06:40.425","Text":"then we get an extra $5."},{"Start":"06:40.425 ","End":"06:50.320","Text":"For example, if 20 pounds gave us $250,"},{"Start":"06:50.320 ","End":"06:54.490","Text":"then if I raise this to 21 pounds,"},{"Start":"06:54.490 ","End":"06:59.470","Text":"so you add 1 unit to the quantity in pounds,"},{"Start":"06:59.470 ","End":"07:07.240","Text":"then what we would expect to get is an increase in $5,"},{"Start":"07:07.240 ","End":"07:12.460","Text":"and we would expect it to be $255."},{"Start":"07:12.460 ","End":"07:16.285","Text":"Now, it just so happens that this turns out exactly,"},{"Start":"07:16.285 ","End":"07:19.645","Text":"because if I put 5 times 21 here,"},{"Start":"07:19.645 ","End":"07:26.935","Text":"that\u0027s 105 plus 150 is 255."},{"Start":"07:26.935 ","End":"07:34.699","Text":"Normally, the marginal, it\u0027s only the approximate increase in cost,"},{"Start":"07:34.699 ","End":"07:41.185","Text":"the dependent variable if you increase x by 1 unit."},{"Start":"07:41.185 ","End":"07:43.235","Text":"That\u0027s how it is in general."},{"Start":"07:43.235 ","End":"07:46.370","Text":"In a case of a linear function, it\u0027s not approximately,"},{"Start":"07:46.370 ","End":"07:48.450","Text":"it\u0027s exact."},{"Start":"07:48.550 ","End":"07:51.840","Text":"That\u0027s all then"}],"ID":8536},{"Watched":false,"Name":"Exercise 8","Duration":"12m 50s","ChapterTopicVideoID":8402,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8402.jpeg","UploadDate":"2017-01-28T13:55:49.0700000","DurationForVideoObject":"PT12M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this exercise, we are given that the total cost function"},{"Start":"00:04.530 ","End":"00:08.969","Text":"of a hat manufacturer is given by TC."},{"Start":"00:08.969 ","End":"00:14.579","Text":"That\u0027s total cost as a function of x. X is the quantity,"},{"Start":"00:14.579 ","End":"00:21.630","Text":"the number of hats as given in this formula and it\u0027s in dollars per day."},{"Start":"00:21.630 ","End":"00:23.970","Text":"Then there are 3 questions,"},{"Start":"00:23.970 ","End":"00:27.015","Text":"I\u0027ll just take them 1 at a time."},{"Start":"00:27.015 ","End":"00:33.690","Text":"A, compute the average cost for producing 40 hats per day."},{"Start":"00:33.690 ","End":"00:36.620","Text":"So we\u0027ll do this in 2 steps."},{"Start":"00:36.620 ","End":"00:42.170","Text":"First of all, I\u0027ll compute the average cost in general for x,"},{"Start":"00:42.170 ","End":"00:44.855","Text":"and then we\u0027ll substitute x equals 40."},{"Start":"00:44.855 ","End":"00:47.705","Text":"Now, the average cost,"},{"Start":"00:47.705 ","End":"00:50.585","Text":"often written as AC,"},{"Start":"00:50.585 ","End":"00:54.410","Text":"is just how you would figure of computing an average."},{"Start":"00:54.410 ","End":"00:57.545","Text":"I mean, you take the total cost for"},{"Start":"00:57.545 ","End":"01:02.255","Text":"x hats and you divide by the number of hats, which is x."},{"Start":"01:02.255 ","End":"01:03.875","Text":"That will give you the average."},{"Start":"01:03.875 ","End":"01:07.850","Text":"Now, we spell this out."},{"Start":"01:07.850 ","End":"01:16.115","Text":"I\u0027ll copy the total cost from here and now I just have to divide this by x."},{"Start":"01:16.115 ","End":"01:18.800","Text":"So x on the denominator."},{"Start":"01:18.800 ","End":"01:22.505","Text":"It\u0027s easier if we split it up into 3 pieces."},{"Start":"01:22.505 ","End":"01:24.050","Text":"So this over this,"},{"Start":"01:24.050 ","End":"01:29.755","Text":"the x squared over x cancels to just becoming x and then"},{"Start":"01:29.755 ","End":"01:37.600","Text":"10x over x is just 10 and then 400 over x as is."},{"Start":"01:37.600 ","End":"01:40.325","Text":"Of course, when we talk about average,"},{"Start":"01:40.325 ","End":"01:44.164","Text":"we\u0027re always assuming that x is bigger than 0."},{"Start":"01:44.164 ","End":"01:47.945","Text":"You can\u0027t have an average of 0 hats."},{"Start":"01:47.945 ","End":"01:52.500","Text":"So there\u0027s no problem with dividing by x."},{"Start":"01:52.910 ","End":"01:59.420","Text":"Now that\u0027s AC average cost of x in general,"},{"Start":"01:59.420 ","End":"02:03.785","Text":"where it\u0027s x hats per day and we want for 40 hats per day."},{"Start":"02:03.785 ","End":"02:08.555","Text":"So all we do is substitute 40 in place of x here."},{"Start":"02:08.555 ","End":"02:14.435","Text":"So 0.04 times 40"},{"Start":"02:14.435 ","End":"02:20.755","Text":"plus 10 plus 400 over 40."},{"Start":"02:20.755 ","End":"02:31.575","Text":"Now let\u0027s see, 0.04 times 40 is 1.6 this is 10 400 over 40 is 10."},{"Start":"02:31.575 ","End":"02:39.290","Text":"So what we have is that the average cost for making"},{"Start":"02:39.290 ","End":"02:48.205","Text":"40 hats is 21.6 I\u0027ll write it 0.60 put a dollar sign 21 dollars and 60 cents per hat."},{"Start":"02:48.205 ","End":"02:50.630","Text":"So that was part a."},{"Start":"02:50.630 ","End":"02:55.670","Text":"In part b, we want to minimize"},{"Start":"02:55.670 ","End":"03:02.464","Text":"the average cost function of x, which I\u0027ll copy."},{"Start":"03:02.464 ","End":"03:05.780","Text":"I prefer it in this form and by the way,"},{"Start":"03:05.780 ","End":"03:14.435","Text":"I expect the minimum average cost to be less than or most,"},{"Start":"03:14.435 ","End":"03:17.090","Text":"21 dollars 60 cents probably less."},{"Start":"03:17.090 ","End":"03:19.060","Text":"We\u0027re going to use calculus."},{"Start":"03:19.060 ","End":"03:21.160","Text":"We\u0027re going to take"},{"Start":"03:21.160 ","End":"03:27.280","Text":"this average cost function and differentiate it and set it equal to 0,"},{"Start":"03:27.280 ","End":"03:29.410","Text":"find critical points and so on."},{"Start":"03:29.410 ","End":"03:35.785","Text":"So we start by differentiating the derivative of AC of x. I\u0027ll have 3 terms,"},{"Start":"03:35.785 ","End":"03:37.610","Text":"take them 1 at a time."},{"Start":"03:37.610 ","End":"03:39.840","Text":"Constant times x."},{"Start":"03:39.840 ","End":"03:45.309","Text":"So it\u0027s just that constant 0.04 the 10 is a constant,"},{"Start":"03:45.309 ","End":"03:50.140","Text":"so nothing but I\u0027ll just put a 0 here just to show that we haven\u0027t forgotten"},{"Start":"03:50.140 ","End":"03:56.455","Text":"and the 400 over x comes out minus 400 over x squared."},{"Start":"03:56.455 ","End":"03:59.530","Text":"I\u0027ll give you a little reminder why this is so."},{"Start":"03:59.530 ","End":"04:04.330","Text":"The derivative of 1 over x is minus 1 over x"},{"Start":"04:04.330 ","End":"04:09.590","Text":"squared 1 way to show this is by writing it as an exponent."},{"Start":"04:09.590 ","End":"04:16.055","Text":"This is x to the minus 1 and if I take the derivative of a power,"},{"Start":"04:16.055 ","End":"04:24.065","Text":"then I put the exponent in front minus 1 and then we take the exponent and lower it by 1,"},{"Start":"04:24.065 ","End":"04:28.730","Text":"minus 1 lowered by 1 is minus 2 and minus 1x to"},{"Start":"04:28.730 ","End":"04:34.185","Text":"the minus 2 is just minus 1 over x squared."},{"Start":"04:34.185 ","End":"04:39.845","Text":"So as the way of seeing it and here it\u0027s 400 and the 400 just stays."},{"Start":"04:39.845 ","End":"04:43.495","Text":"So actually I won\u0027t put the minus 400 like that,"},{"Start":"04:43.495 ","End":"04:46.839","Text":"I\u0027ll just put the minus in front."},{"Start":"04:46.839 ","End":"04:52.429","Text":"Next, we want to see where the derivative is equal to 0."},{"Start":"04:52.429 ","End":"04:54.529","Text":"So if we set it equal to 0,"},{"Start":"04:54.529 ","End":"05:00.380","Text":"we get an equation that 0.04 forget that"},{"Start":"05:00.380 ","End":"05:07.280","Text":"0 there minus 400 over x squared equals 0."},{"Start":"05:07.280 ","End":"05:09.750","Text":"Start solving it."},{"Start":"05:09.820 ","End":"05:13.519","Text":"Bring this to the other side."},{"Start":"05:13.519 ","End":"05:21.970","Text":"So 0.04 is equal to 400 over"},{"Start":"05:21.970 ","End":"05:25.320","Text":"x squared and then I"},{"Start":"05:25.320 ","End":"05:32.200","Text":"can switch the positions of these to like multiply by x squared and divide by 0.04."},{"Start":"05:32.200 ","End":"05:37.660","Text":"So I get x squared equals 400 over"},{"Start":"05:37.660 ","End":"05:42.880","Text":"0.04 and this is"},{"Start":"05:42.880 ","End":"05:49.290","Text":"equal to 10,000."},{"Start":"05:49.290 ","End":"05:51.750","Text":"So if x squared is 10,000,"},{"Start":"05:51.750 ","End":"05:56.110","Text":"x is the square root of 10,000 which is 100."},{"Start":"05:56.110 ","End":"05:57.540","Text":"I took the square root."},{"Start":"05:57.540 ","End":"06:02.560","Text":"I only took the positive square root because we know that the quantity,"},{"Start":"06:02.560 ","End":"06:07.400","Text":"the number of hats is positive can\u0027t be minus 100. What do we do with this?"},{"Start":"06:07.400 ","End":"06:09.740","Text":"This is a critical point where the derivative is 0."},{"Start":"06:09.740 ","End":"06:10.925","Text":"Now we have to test,"},{"Start":"06:10.925 ","End":"06:12.860","Text":"is it really a minimum?"},{"Start":"06:12.860 ","End":"06:15.440","Text":"So there several ways of doing this,"},{"Start":"06:15.440 ","End":"06:19.790","Text":"but the easiest is the second order derivative test."},{"Start":"06:19.790 ","End":"06:23.600","Text":"We compute the second derivative of AC double prime of"},{"Start":"06:23.600 ","End":"06:28.115","Text":"x and according to the sign of the answer we\u0027ll know maximum or minimum."},{"Start":"06:28.115 ","End":"06:34.550","Text":"Here\u0027s AC prime and I need to differentiate this."},{"Start":"06:34.550 ","End":"06:38.700","Text":"So the 0.04 is a constant,"},{"Start":"06:38.700 ","End":"06:44.520","Text":"so it gives me nothing and minus 400 over x squared."},{"Start":"06:44.520 ","End":"06:49.610","Text":"I\u0027ll copy it. Minus 400 over x squared"},{"Start":"06:49.610 ","End":"06:57.380","Text":"derivative is equal to 800 over x cubed."},{"Start":"06:57.380 ","End":"07:01.970","Text":"Again, if you need an explanation the derivative of 1"},{"Start":"07:01.970 ","End":"07:06.770","Text":"over x squared is minus 2 over x cubed."},{"Start":"07:06.770 ","End":"07:09.440","Text":"For similar reasons, if you add it to the exponent,"},{"Start":"07:09.440 ","End":"07:12.800","Text":"this is x to the minus 2 and its derivative,"},{"Start":"07:12.800 ","End":"07:17.540","Text":"the minus 2 comes out in front and the exponent goes down by 1."},{"Start":"07:17.540 ","End":"07:20.330","Text":"So it\u0027s minus 3, which is this."},{"Start":"07:20.330 ","End":"07:25.490","Text":"But in our case we had a minus 400 also this is what we get."},{"Start":"07:25.490 ","End":"07:32.720","Text":"Now we need to substitute AC double prime at 100,"},{"Start":"07:32.720 ","End":"07:35.990","Text":"which is the suspected minimum that the"},{"Start":"07:35.990 ","End":"07:40.940","Text":"100 but you don\u0027t have to do any actual computations."},{"Start":"07:40.940 ","End":"07:45.380","Text":"I can start writing it out for you it\u0027s 800 over 100 cubed."},{"Start":"07:45.380 ","End":"07:51.980","Text":"But all I need to know is that this thing is a positive number."},{"Start":"07:51.980 ","End":"07:54.890","Text":"When the second derivative is positive,"},{"Start":"07:54.890 ","End":"08:00.020","Text":"then it means that the 100 is"},{"Start":"08:00.020 ","End":"08:05.390","Text":"a minimum and if it\u0027s negative, it\u0027s a maximum."},{"Start":"08:05.390 ","End":"08:12.245","Text":"I\u0027ll just write that, that the answer is 100 hats and highlight it."},{"Start":"08:12.245 ","End":"08:16.430","Text":"So here we are back at part c. I\u0027ll just clean the board."},{"Start":"08:16.430 ","End":"08:24.055","Text":"In part c, we want the marginal cost for producing 100 hats per day."},{"Start":"08:24.055 ","End":"08:28.030","Text":"Then there\u0027s a second part 2 can we conclude?"},{"Start":"08:28.030 ","End":"08:33.290","Text":"It\u0027s just a coincidence that this 100 hats per day was"},{"Start":"08:33.290 ","End":"08:38.705","Text":"the value that gave the minimum average cost."},{"Start":"08:38.705 ","End":"08:43.630","Text":"But I\u0027m not going to use that fact coincidence."},{"Start":"08:44.160 ","End":"08:47.080","Text":"Now, when you see marginal,"},{"Start":"08:47.080 ","End":"08:50.200","Text":"that should ring a bell and say derivative."},{"Start":"08:50.200 ","End":"08:59.140","Text":"So the marginal cost to marginal total cost is the derivative of the cost,"},{"Start":"08:59.140 ","End":"09:01.360","Text":"sometimes written C, sometimes TC."},{"Start":"09:01.360 ","End":"09:03.580","Text":"It\u0027s TC prime."},{"Start":"09:03.580 ","End":"09:08.890","Text":"I should write of x here of x, the derivative function."},{"Start":"09:08.890 ","End":"09:15.445","Text":"So we have the total cost function over here just differentiate it."},{"Start":"09:15.445 ","End":"09:22.479","Text":"2 times 0.04 is 0.08x"},{"Start":"09:22.479 ","End":"09:29.960","Text":"derivative of 10x is 10 and the derivative of 400 is nothing."},{"Start":"09:29.960 ","End":"09:32.850","Text":"So that\u0027s just this."},{"Start":"09:33.110 ","End":"09:38.890","Text":"Now we need the marginal cost for when we\u0027re producing 100."},{"Start":"09:38.890 ","End":"09:40.945","Text":"So x is 100."},{"Start":"09:40.945 ","End":"09:44.545","Text":"So just plug in 100,"},{"Start":"09:44.545 ","End":"09:50.810","Text":"we get 0.08 times"},{"Start":"09:50.810 ","End":"09:58.170","Text":"100 plus 10 and that is equal to 0.08 of 100 is 8,"},{"Start":"09:58.170 ","End":"10:01.870","Text":"8 plus 10 is 18 dollars."},{"Start":"10:01.870 ","End":"10:10.080","Text":"That\u0027s the marginal cost when the production level is at 100 hats per day."},{"Start":"10:10.080 ","End":"10:11.850","Text":"Now what does this mean?"},{"Start":"10:11.850 ","End":"10:13.455","Text":"What does this 18 dollars?"},{"Start":"10:13.455 ","End":"10:15.955","Text":"Let\u0027s see what can we conclude?"},{"Start":"10:15.955 ","End":"10:23.690","Text":"Well, in general, it\u0027s an approximation that if we produce 1 more unit,"},{"Start":"10:23.690 ","End":"10:27.665","Text":"in other words, if we go from 100 to 101."},{"Start":"10:27.665 ","End":"10:35.145","Text":"So we go from 100 to 101 in x,"},{"Start":"10:35.145 ","End":"10:39.855","Text":"then the total cost"},{"Start":"10:39.855 ","End":"10:45.825","Text":"will increase by let\u0027s write it."},{"Start":"10:45.825 ","End":"10:55.510","Text":"Increases by approximately, this is not an exact 18 dollars."},{"Start":"10:55.510 ","End":"11:00.440","Text":"I don\u0027t just want to leave it at that I want to show you that it really is"},{"Start":"11:00.440 ","End":"11:05.630","Text":"a very good approximation by actually seeing how much the total cost for 100,"},{"Start":"11:05.630 ","End":"11:10.290","Text":"how much for 101 and seeing what the difference is."},{"Start":"11:10.640 ","End":"11:12.665","Text":"I\u0027ll do it over here."},{"Start":"11:12.665 ","End":"11:16.415","Text":"Let\u0027s see the total cost for 100."},{"Start":"11:16.415 ","End":"11:18.559","Text":"We already did this for 40."},{"Start":"11:18.559 ","End":"11:19.894","Text":"The same idea."},{"Start":"11:19.894 ","End":"11:24.050","Text":"It\u0027s just by plugging in the value here."},{"Start":"11:24.050 ","End":"11:33.755","Text":"So we get 0.04 times 100 squared"},{"Start":"11:33.755 ","End":"11:41.685","Text":"plus 10 times 100 plus"},{"Start":"11:41.685 ","End":"11:46.230","Text":"400 and this is equal to,"},{"Start":"11:46.230 ","End":"11:50.009","Text":"this comes out to 1,800,"},{"Start":"11:50.009 ","End":"11:51.630","Text":"I\u0027ll leave you to check that,"},{"Start":"11:51.630 ","End":"11:53.770","Text":"and that\u0027s in dollars."},{"Start":"11:53.770 ","End":"11:59.385","Text":"So total cost of producing 100 hats is this 1,800 dollars."},{"Start":"11:59.385 ","End":"12:05.050","Text":"The total cost for 101 hats."},{"Start":"12:05.240 ","End":"12:09.220","Text":"Similar expression, just 101 instead of 100."},{"Start":"12:09.220 ","End":"12:10.895","Text":"I\u0027ll leave you the details."},{"Start":"12:10.895 ","End":"12:14.100","Text":"I\u0027ll just give you the answer, it comes out to"},{"Start":"12:14.500 ","End":"12:21.910","Text":"1,818 dollars and 4 cents."},{"Start":"12:21.910 ","End":"12:24.875","Text":"Now if you take the difference between these 2,"},{"Start":"12:24.875 ","End":"12:27.604","Text":"you don\u0027t get 18 dollars,"},{"Start":"12:27.604 ","End":"12:31.100","Text":"you get 18 dollars and 4 cents."},{"Start":"12:31.100 ","End":"12:38.400","Text":"But like we said, this is an approximate value of the marginal cost."},{"Start":"12:38.400 ","End":"12:41.380","Text":"We see it\u0027s pretty close."},{"Start":"12:41.810 ","End":"12:44.445","Text":"That\u0027s what we can conclude."},{"Start":"12:44.445 ","End":"12:50.160","Text":"We even went further and we\u0027ve done with this exercise."}],"ID":8537},{"Watched":false,"Name":"Exercise 9","Duration":"9m 56s","ChapterTopicVideoID":8403,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8403.jpeg","UploadDate":"2017-01-28T13:59:49.3970000","DurationForVideoObject":"PT9M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.555","Text":"In this exercise, we\u0027re given the total cost function of a certain product, C of x."},{"Start":"00:06.555 ","End":"00:08.520","Text":"Now, C is the total cost,"},{"Start":"00:08.520 ","End":"00:11.040","Text":"sometimes it\u0027s written as TC."},{"Start":"00:11.040 ","End":"00:13.845","Text":"X is the quantity,"},{"Start":"00:13.845 ","End":"00:17.130","Text":"sometimes written as Q."},{"Start":"00:17.130 ","End":"00:27.900","Text":"Anyway, the cost of producing x units of the product is given by this expression."},{"Start":"00:27.900 ","End":"00:30.240","Text":"I think I\u0027d like to highlight it."},{"Start":"00:30.240 ","End":"00:32.520","Text":"There are 4 parts."},{"Start":"00:32.520 ","End":"00:34.365","Text":"We\u0027ll just take them 1 at a time."},{"Start":"00:34.365 ","End":"00:39.660","Text":"In part a, we have to compute the cost when x is 100 and when x is 101."},{"Start":"00:39.660 ","End":"00:42.160","Text":"Let\u0027s start off with x equals 100."},{"Start":"00:42.160 ","End":"00:48.870","Text":"Really, all we have to do is compute the cost of 100 instead of x, put 100."},{"Start":"00:48.870 ","End":"00:56.340","Text":"So what we get is 0.004 and we need 100"},{"Start":"00:56.340 ","End":"01:05.190","Text":"squared plus 10 times 100 plus 200."},{"Start":"01:05.190 ","End":"01:09.735","Text":"Let\u0027s see, this is equal too."},{"Start":"01:09.735 ","End":"01:13.755","Text":"Here we have 10,000 times 0.04 is 40,"},{"Start":"01:13.755 ","End":"01:15.930","Text":"10 times the 1000, sorry,"},{"Start":"01:15.930 ","End":"01:20.145","Text":"10 times 100 is 1000 and another 200,"},{"Start":"01:20.145 ","End":"01:24.940","Text":"and so what we have is 1,240."},{"Start":"01:24.940 ","End":"01:28.580","Text":"Let\u0027s just say we\u0027re working, everything\u0027s in dollars."},{"Start":"01:28.580 ","End":"01:33.515","Text":"This is the cost of producing 100 units is this,"},{"Start":"01:33.515 ","End":"01:36.695","Text":"and now let\u0027s do it for 101."},{"Start":"01:36.695 ","End":"01:43.835","Text":"The cost of producing 101 units would be similar expression to this."},{"Start":"01:43.835 ","End":"01:51.120","Text":"I just copied and I\u0027m going to replace the 100 with 101,"},{"Start":"01:51.120 ","End":"01:53.480","Text":"and of pay the details,"},{"Start":"01:53.480 ","End":"01:54.770","Text":"the answer comes out to"},{"Start":"01:54.770 ","End":"02:05.290","Text":"be 1,250.804."},{"Start":"02:05.650 ","End":"02:14.290","Text":"That\u0027s part a, 2 halves of it. Now part b."},{"Start":"02:14.290 ","End":"02:19.095","Text":"Part b, we want the marginal cost when x is 100."},{"Start":"02:19.095 ","End":"02:23.175","Text":"When you see marginal, you think derivative."},{"Start":"02:23.175 ","End":"02:28.735","Text":"Marginal cost is the derivative of the cost function."},{"Start":"02:28.735 ","End":"02:31.895","Text":"What we do is just differentiate."},{"Start":"02:31.895 ","End":"02:34.805","Text":"There\u0027s the cost function differentiate it."},{"Start":"02:34.805 ","End":"02:40.620","Text":"2 times 0.004 is 0.008."},{"Start":"02:41.360 ","End":"02:46.440","Text":"X derivative of 10x is just 10,"},{"Start":"02:46.440 ","End":"02:48.570","Text":"derivative of 200 is nothing,"},{"Start":"02:48.570 ","End":"02:51.420","Text":"this is all we have."},{"Start":"02:51.420 ","End":"02:58.280","Text":"The marginal cost, well,"},{"Start":"02:58.280 ","End":"03:05.210","Text":"it\u0027s really a function of x. I should have put in here of x, MC,"},{"Start":"03:05.210 ","End":"03:08.470","Text":"marginal cost at 100 is just,"},{"Start":"03:08.470 ","End":"03:10.124","Text":"you plug in 100,"},{"Start":"03:10.124 ","End":"03:16.319","Text":"0.008 times 100 plus 10,"},{"Start":"03:16.319 ","End":"03:19.540","Text":"and this comes out to be,"},{"Start":"03:19.670 ","End":"03:25.515","Text":"well, 100 times 0.008 is 0.8 plus 10."},{"Start":"03:25.515 ","End":"03:30.255","Text":"It\u0027s 10.8 and it\u0027s in dollars,"},{"Start":"03:30.255 ","End":"03:36.000","Text":"and that the end Part b."},{"Start":"03:36.000 ","End":"03:39.710","Text":"Maybe I\u0027ll highlight, this is the answer to part b,"},{"Start":"03:39.710 ","End":"03:41.269","Text":"and there were 2 parts,"},{"Start":"03:41.269 ","End":"03:44.780","Text":"of a was this and there was this."},{"Start":"03:44.780 ","End":"03:52.790","Text":"We\u0027re going to use all of this when we do part c. In part c, first of all,"},{"Start":"03:52.790 ","End":"03:55.009","Text":"we have to use part a,"},{"Start":"03:55.009 ","End":"03:58.400","Text":"these 2 figures to compute the cost increase when we"},{"Start":"03:58.400 ","End":"04:04.215","Text":"increase the quantity from 100 to 101."},{"Start":"04:04.215 ","End":"04:06.420","Text":"All we have to do for part c,"},{"Start":"04:06.420 ","End":"04:08.475","Text":"first of all is to subtract."},{"Start":"04:08.475 ","End":"04:14.280","Text":"We want the cost for"},{"Start":"04:14.280 ","End":"04:23.655","Text":"101 units minus the cost for 100 units,"},{"Start":"04:23.655 ","End":"04:27.450","Text":"which is just this,"},{"Start":"04:27.450 ","End":"04:38.315","Text":"1,250.804 minus 1,240,"},{"Start":"04:38.315 ","End":"04:40.130","Text":"and when it comes out"},{"Start":"04:40.130 ","End":"04:49.370","Text":"is $10.804."},{"Start":"04:49.370 ","End":"04:51.365","Text":"I\u0027ll highlight it."},{"Start":"04:51.365 ","End":"04:54.380","Text":"Now look if you compare these 2,"},{"Start":"04:54.380 ","End":"05:00.030","Text":"well, they\u0027re not exactly the same, but very close."},{"Start":"05:00.040 ","End":"05:03.065","Text":"What can we conclude,"},{"Start":"05:03.065 ","End":"05:07.339","Text":"in this case, we can say that they\u0027re roughly equal."},{"Start":"05:07.339 ","End":"05:08.869","Text":"This is the idea,"},{"Start":"05:08.869 ","End":"05:10.249","Text":"is that in general,"},{"Start":"05:10.249 ","End":"05:17.780","Text":"the marginal cost is roughly equal to the cost of that 1 extra unit."},{"Start":"05:17.780 ","End":"05:19.430","Text":"Maybe I\u0027ll write something down."},{"Start":"05:19.430 ","End":"05:23.430","Text":"I\u0027ll say that the marginal cost when,"},{"Start":"05:23.430 ","End":"05:26.885","Text":"in this case it\u0027s when x equals 100,"},{"Start":"05:26.885 ","End":"05:29.735","Text":"that\u0027s our MC of 100,"},{"Start":"05:29.735 ","End":"05:35.195","Text":"is approximately or roughly equal,"},{"Start":"05:35.195 ","End":"05:46.240","Text":"say roughly equal to the cost of producing that extra item."},{"Start":"05:46.240 ","End":"05:52.115","Text":"That would be the 101st item or the next item."},{"Start":"05:52.115 ","End":"05:56.725","Text":"Say the next 1 more item."},{"Start":"05:56.725 ","End":"05:59.890","Text":"It works in general, not just with 100."},{"Start":"05:59.890 ","End":"06:02.465","Text":"Here we see that it really is close."},{"Start":"06:02.465 ","End":"06:06.065","Text":"Actually increase 10.804."},{"Start":"06:06.065 ","End":"06:08.635","Text":"The marginal cost."},{"Start":"06:08.635 ","End":"06:15.610","Text":"At that point 100 is exactly 10.8 or $10.80."},{"Start":"06:15.610 ","End":"06:19.195","Text":"So it\u0027s like a fraction of a cent there."},{"Start":"06:19.195 ","End":"06:22.880","Text":"Now part d is a bit strange."},{"Start":"06:22.880 ","End":"06:24.440","Text":"I\u0027ll clear the board."},{"Start":"06:24.440 ","End":"06:28.175","Text":"Well, actually I\u0027m going to keep this bit and erase the rest."},{"Start":"06:28.175 ","End":"06:31.770","Text":"Yeah, this is the bit I wanted to keep."},{"Start":"06:33.440 ","End":"06:41.585","Text":"Now let\u0027s read d. Is the rate of change of the cost function increasing or decreasing?"},{"Start":"06:41.585 ","End":"06:45.425","Text":"Well, first of all, what does the rate of change mean?"},{"Start":"06:45.425 ","End":"06:51.185","Text":"Whenever you see rate of change, it means derivative."},{"Start":"06:51.185 ","End":"06:55.110","Text":"That\u0027s the calculus concept."},{"Start":"06:55.110 ","End":"06:56.825","Text":"I mean in the day-to-day life,"},{"Start":"06:56.825 ","End":"06:59.930","Text":"derivative is interpreted as rate of change."},{"Start":"06:59.930 ","End":"07:04.460","Text":"Now, this is the cost function and the rate of change is the derivative,"},{"Start":"07:04.460 ","End":"07:07.110","Text":"and this is what we have."},{"Start":"07:07.280 ","End":"07:11.069","Text":"Here\u0027s the cost function,"},{"Start":"07:11.069 ","End":"07:14.195","Text":"and this is its rate of change,"},{"Start":"07:14.195 ","End":"07:16.265","Text":"which is also a function."},{"Start":"07:16.265 ","End":"07:18.620","Text":"Now this new function,"},{"Start":"07:18.620 ","End":"07:22.330","Text":"we can ask, is it increasing or decreasing?"},{"Start":"07:22.330 ","End":"07:25.610","Text":"How do we tell if a function\u0027s increasing or decreasing?"},{"Start":"07:25.610 ","End":"07:27.470","Text":"Well, there\u0027s several ways."},{"Start":"07:27.470 ","End":"07:29.765","Text":"Now, we\u0027re going to use calculus."},{"Start":"07:29.765 ","End":"07:31.310","Text":"In this particular instance,"},{"Start":"07:31.310 ","End":"07:32.945","Text":"we wouldn\u0027t have to."},{"Start":"07:32.945 ","End":"07:41.815","Text":"I can show you a shortcut way because this is a linear function of the form ax plus b,"},{"Start":"07:41.815 ","End":"07:47.285","Text":"then we know that when we have a function like this,"},{"Start":"07:47.285 ","End":"07:52.175","Text":"that if a is bigger than 0, it\u0027s increasing,"},{"Start":"07:52.175 ","End":"07:58.070","Text":"and if a is less than 0, it\u0027s decreasing."},{"Start":"07:58.070 ","End":"08:03.450","Text":"In our case, our a is 0.08,"},{"Start":"08:03.450 ","End":"08:10.039","Text":"which is positive, so this is increasing for all values of x, it\u0027s increasing everywhere."},{"Start":"08:10.039 ","End":"08:14.225","Text":"But this is not using calculus."},{"Start":"08:14.225 ","End":"08:18.560","Text":"I want to use a more general method that increasing"},{"Start":"08:18.560 ","End":"08:24.390","Text":"is positive derivative and decreasing is negative derivative."},{"Start":"08:24.950 ","End":"08:28.145","Text":"I want the derivative of c prime."},{"Start":"08:28.145 ","End":"08:31.790","Text":"The derivative of c prime is just c double prime."},{"Start":"08:31.790 ","End":"08:35.810","Text":"It\u0027s like the second derivative of the original cost function."},{"Start":"08:35.810 ","End":"08:38.225","Text":"If we differentiate this,"},{"Start":"08:38.225 ","End":"08:42.825","Text":"we get just 0.008,"},{"Start":"08:42.825 ","End":"08:44.690","Text":"x drops out altogether."},{"Start":"08:44.690 ","End":"08:46.475","Text":"It doesn\u0027t depend on x."},{"Start":"08:46.475 ","End":"08:51.570","Text":"This is always positive for every x."},{"Start":"08:51.570 ","End":"08:56.315","Text":"Since it\u0027s positive, the derivative of this function is positive,"},{"Start":"08:56.315 ","End":"09:02.860","Text":"that means that this is increasing."},{"Start":"09:06.320 ","End":"09:09.685","Text":"That\u0027s the same answer as we got"},{"Start":"09:09.685 ","End":"09:14.555","Text":"here without calculus using the properties of the straight line."},{"Start":"09:14.555 ","End":"09:18.065","Text":"But in general, you would do this."},{"Start":"09:18.065 ","End":"09:19.640","Text":"Once again, if you\u0027re asked,"},{"Start":"09:19.640 ","End":"09:24.050","Text":"rate of change of a function is increasing or decreasing,"},{"Start":"09:24.050 ","End":"09:27.035","Text":"the rate of change is the first derivative,"},{"Start":"09:27.035 ","End":"09:29.194","Text":"and then you differentiate again,"},{"Start":"09:29.194 ","End":"09:30.670","Text":"get the second derivative,"},{"Start":"09:30.670 ","End":"09:35.255","Text":"and if it\u0027s positive, increasing, negative, decreasing."},{"Start":"09:35.255 ","End":"09:37.940","Text":"Of course in general, it could be sometimes increasing and"},{"Start":"09:37.940 ","End":"09:41.170","Text":"sometimes decreasing because it could depend on x,"},{"Start":"09:41.170 ","End":"09:44.780","Text":"and then you would deal with it accordingly."},{"Start":"09:44.780 ","End":"09:46.820","Text":"If it said for a certain x,"},{"Start":"09:46.820 ","End":"09:49.325","Text":"you\u0027d see if it\u0027s increasing or decreasing."},{"Start":"09:49.325 ","End":"09:52.925","Text":"Anyway here it\u0027s increasing everywhere."},{"Start":"09:52.925 ","End":"09:57.810","Text":"That answer is d, and so we\u0027re done."}],"ID":8538},{"Watched":false,"Name":"Exercise 10","Duration":"6m 31s","ChapterTopicVideoID":8404,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8404.jpeg","UploadDate":"2017-01-28T14:02:09.2670000","DurationForVideoObject":"PT6M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.870","Text":"There\u0027s a manufacturer who makes a product and it has"},{"Start":"00:03.870 ","End":"00:10.770","Text":"a demand function P of x. P is price and x is quantity,"},{"Start":"00:10.770 ","End":"00:14.220","Text":"sometimes we use the letter Q,"},{"Start":"00:14.220 ","End":"00:19.650","Text":"of 100 minus 0.06x and it\u0027s in dollars per unit."},{"Start":"00:19.650 ","End":"00:24.345","Text":"He also has a total cost function 200 plus 4x."},{"Start":"00:24.345 ","End":"00:28.005","Text":"Let\u0027s do a first and then we\u0027ll do B."},{"Start":"00:28.005 ","End":"00:32.100","Text":"In a, we\u0027re asked what the amount X is,"},{"Start":"00:32.100 ","End":"00:33.715","Text":"that he needs to produce,"},{"Start":"00:33.715 ","End":"00:38.090","Text":"how many units in order to get a maximum profit."},{"Start":"00:38.090 ","End":"00:44.990","Text":"Now, in general, profit is equal to the revenue minus the cost."},{"Start":"00:44.990 ","End":"00:47.240","Text":"How much you take in minus how much you spend."},{"Start":"00:47.240 ","End":"00:52.200","Text":"I\u0027ll just write it as in words as revenue or total revenue,"},{"Start":"00:52.200 ","End":"00:56.310","Text":"you could say, minus cost or total cost."},{"Start":"00:56.310 ","End":"00:58.905","Text":"In terms of letters,"},{"Start":"00:58.905 ","End":"01:01.020","Text":"there we\u0027ve used letter P for price,"},{"Start":"01:01.020 ","End":"01:04.385","Text":"so we use the letter Pi for profit."},{"Start":"01:04.385 ","End":"01:08.150","Text":"The profit of the function of x is the revenue,"},{"Start":"01:08.150 ","End":"01:10.220","Text":"which I\u0027ll call TR,"},{"Start":"01:10.220 ","End":"01:13.610","Text":"total revenue, sometimes just R,"},{"Start":"01:13.610 ","End":"01:17.675","Text":"minus the cost or total cost of x."},{"Start":"01:17.675 ","End":"01:22.215","Text":"Now, TC, we are given."},{"Start":"01:22.215 ","End":"01:25.445","Text":"We have to figure out what TR is."},{"Start":"01:25.445 ","End":"01:29.690","Text":"Now, TR as a function of x,"},{"Start":"01:29.690 ","End":"01:36.485","Text":"the revenue is just how many items he produced times how many dollars he got per item."},{"Start":"01:36.485 ","End":"01:42.405","Text":"It\u0027s x items and the price of each is P of X."},{"Start":"01:42.405 ","End":"01:44.580","Text":"But we have what P of X is."},{"Start":"01:44.580 ","End":"01:46.110","Text":"It\u0027s over here."},{"Start":"01:46.110 ","End":"01:51.010","Text":"Why can\u0027t I just highlight it what P of x is."},{"Start":"01:52.190 ","End":"01:58.860","Text":"I\u0027m just copying this now, 100 minus 0.06x."},{"Start":"01:58.860 ","End":"02:01.710","Text":"Then expand the brackets."},{"Start":"02:01.710 ","End":"02:08.090","Text":"100x minus 0.06x squared."},{"Start":"02:08.090 ","End":"02:12.290","Text":"Now this TR, which is what I need here, is given by this."},{"Start":"02:12.290 ","End":"02:13.715","Text":"I\u0027m going to highlight it."},{"Start":"02:13.715 ","End":"02:16.075","Text":"What we\u0027re going to have to subtract,"},{"Start":"02:16.075 ","End":"02:20.700","Text":"the total cost is given here."},{"Start":"02:20.700 ","End":"02:25.710","Text":"Perhaps I\u0027ll highlight this color is this color."},{"Start":"02:25.710 ","End":"02:29.035","Text":"This is what\u0027s over here."},{"Start":"02:29.035 ","End":"02:36.260","Text":"We get, now, I\u0027m copying from here 100x minus"},{"Start":"02:36.260 ","End":"02:45.200","Text":"0.06x^2 minus TC of x. I\u0027ll just put them both with a minus,"},{"Start":"02:45.200 ","End":"02:49.315","Text":"minus 200 minus 4x."},{"Start":"02:49.315 ","End":"02:52.504","Text":"Now, if I collect like terms,"},{"Start":"02:52.504 ","End":"02:56.430","Text":"we get minus 0.06x^2."},{"Start":"02:56.680 ","End":"02:59.270","Text":"Then x, is I have 2 of them."},{"Start":"02:59.270 ","End":"03:01.295","Text":"I have a 100 minus 4."},{"Start":"03:01.295 ","End":"03:09.135","Text":"That\u0027s plus 96x and constants minus 200."},{"Start":"03:09.135 ","End":"03:12.785","Text":"Then I\u0027ll highlight the profit function in a different color."},{"Start":"03:12.785 ","End":"03:15.650","Text":"This is the 1 we want to maximize."},{"Start":"03:15.650 ","End":"03:20.510","Text":"At least you want to find out what x should be in order for this to be maximum."},{"Start":"03:20.510 ","End":"03:24.694","Text":"Notice that this is a quadratic expression polynomial"},{"Start":"03:24.694 ","End":"03:28.985","Text":"in x. I want to remind you about quadratics and parabolas."},{"Start":"03:28.985 ","End":"03:34.280","Text":"In algebra, you probably studied parabolas y equals ax"},{"Start":"03:34.280 ","End":"03:39.795","Text":"squared plus bx plus c in general and in our case,"},{"Start":"03:39.795 ","End":"03:43.710","Text":"this would be a which is minus 0.06,"},{"Start":"03:43.710 ","End":"03:48.195","Text":"this is b, 96 and c is minus 200."},{"Start":"03:48.195 ","End":"03:51.785","Text":"If a is positive,"},{"Start":"03:51.785 ","End":"03:58.065","Text":"then the parabola opens upwards."},{"Start":"03:58.065 ","End":"04:00.650","Text":"If a is negative,"},{"Start":"04:00.650 ","End":"04:04.490","Text":"it faces downwards like this."},{"Start":"04:04.490 ","End":"04:08.164","Text":"Here, we have a situation with a minimum,"},{"Start":"04:08.164 ","End":"04:11.015","Text":"and here, we have a situation with a maximum."},{"Start":"04:11.015 ","End":"04:15.390","Text":"Now, in our case, a is less than 0,"},{"Start":"04:15.390 ","End":"04:17.780","Text":"we\u0027re okay for maximum."},{"Start":"04:17.780 ","End":"04:21.470","Text":"The other thing that we know about these parabolas,"},{"Start":"04:21.470 ","End":"04:26.555","Text":"they have a vertex where the minimum is and the maximum is where the maximum is."},{"Start":"04:26.555 ","End":"04:31.520","Text":"It\u0027s the tip and the x of the vertex,"},{"Start":"04:31.520 ","End":"04:33.410","Text":"which is label it V,"},{"Start":"04:33.410 ","End":"04:34.654","Text":"this is the vertex,"},{"Start":"04:34.654 ","End":"04:39.755","Text":"the x of the vertex is minus b over 2a."},{"Start":"04:39.755 ","End":"04:46.060","Text":"Back here, the x of the vertex is going to equal minus b,"},{"Start":"04:46.060 ","End":"05:00.050","Text":"which is minus 96 over 2a, is minus 0.12, and the calculator is 800."},{"Start":"05:00.050 ","End":"05:03.320","Text":"That\u0027s the answer to part a."},{"Start":"05:03.320 ","End":"05:12.175","Text":"He has to produce 800 units per day to maximize profits."},{"Start":"05:12.175 ","End":"05:14.490","Text":"Now, let\u0027s look at part b."},{"Start":"05:14.490 ","End":"05:16.940","Text":"Part b is the natural question that arises,"},{"Start":"05:16.940 ","End":"05:19.160","Text":"this is how many he has to produce,"},{"Start":"05:19.160 ","End":"05:21.910","Text":"but what is the maximum profit?"},{"Start":"05:21.910 ","End":"05:26.900","Text":"Let\u0027s call the maximum profit Pi max,"},{"Start":"05:26.900 ","End":"05:31.895","Text":"which is Pi when you substitute 800."},{"Start":"05:31.895 ","End":"05:38.045","Text":"Now, all we have to do then is look at this profit function which is here,"},{"Start":"05:38.045 ","End":"05:41.025","Text":"and I just substitute."},{"Start":"05:41.025 ","End":"05:46.915","Text":"I\u0027ve got minus 0.06,"},{"Start":"05:46.915 ","End":"05:58.870","Text":"800^2 plus 96 times 800 minus 200."},{"Start":"05:59.390 ","End":"06:01.800","Text":"I won\u0027t make you do the work,"},{"Start":"06:01.800 ","End":"06:04.200","Text":"I\u0027ll just give you the answer."},{"Start":"06:04.200 ","End":"06:14.050","Text":"I make it 38,200 and we\u0027re working in dollars."},{"Start":"06:14.120 ","End":"06:17.725","Text":"This is the answer to part b,"},{"Start":"06:17.725 ","End":"06:25.535","Text":"and it\u0027ll produces 800 units to get a maximum profit of $38,200."},{"Start":"06:25.535 ","End":"06:29.370","Text":"I just forgot to write part b here."},{"Start":"06:29.370 ","End":"06:31.990","Text":"Okay. Now, we\u0027re done."}],"ID":8539},{"Watched":false,"Name":"Exercise 11","Duration":"7m 3s","ChapterTopicVideoID":8405,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8405.jpeg","UploadDate":"2017-01-28T14:03:47.1370000","DurationForVideoObject":"PT7M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, it concerns a manufacturer"},{"Start":"00:04.560 ","End":"00:08.820","Text":"who has a product and it has a constant demand function."},{"Start":"00:08.820 ","End":"00:11.955","Text":"The price is always 20,"},{"Start":"00:11.955 ","End":"00:14.775","Text":"sorry we are working in dollars,"},{"Start":"00:14.775 ","End":"00:17.910","Text":"no matter how much the quantity is,"},{"Start":"00:17.910 ","End":"00:21.680","Text":"how much the demand is x is they\u0027re sometimes called Q."},{"Start":"00:21.680 ","End":"00:28.140","Text":"The total cost function TC is given by the following formula."},{"Start":"00:28.140 ","End":"00:33.870","Text":"In a, we need to find which quantity he needs to produce,"},{"Start":"00:33.870 ","End":"00:37.905","Text":"how many units in order to maximize his profit?"},{"Start":"00:37.905 ","End":"00:40.525","Text":"Then after we found out the x,"},{"Start":"00:40.525 ","End":"00:45.090","Text":"we want to know what actually is the maximum profit."},{"Start":"00:45.380 ","End":"00:51.600","Text":"Let\u0027s start with a word formula for profit."},{"Start":"00:52.930 ","End":"00:56.390","Text":"It\u0027s the total revenue,"},{"Start":"00:56.390 ","End":"01:00.560","Text":"how much you receive from selling the product,"},{"Start":"01:00.560 ","End":"01:06.925","Text":"minus what your expenses are, the total cost."},{"Start":"01:06.925 ","End":"01:10.340","Text":"If we write that in symbols,"},{"Start":"01:10.340 ","End":"01:13.160","Text":"profit, at least in this course,"},{"Start":"01:13.160 ","End":"01:16.925","Text":"we use the letter pi because P is reserved for price."},{"Start":"01:16.925 ","End":"01:22.625","Text":"Pi of quantity x is the total revenue,"},{"Start":"01:22.625 ","End":"01:29.590","Text":"TR of x minus the total cost for x units."},{"Start":"01:29.590 ","End":"01:31.350","Text":"Now as it stands,"},{"Start":"01:31.350 ","End":"01:35.070","Text":"we have TC exactly over here."},{"Start":"01:35.070 ","End":"01:38.205","Text":"I think I\u0027ll highlight it."},{"Start":"01:38.205 ","End":"01:46.350","Text":"That\u0027s our TC and I highlighted here also."},{"Start":"01:46.350 ","End":"01:49.460","Text":"Total revenue is not given,"},{"Start":"01:49.460 ","End":"01:51.620","Text":"but we can compute it."},{"Start":"01:51.620 ","End":"01:57.919","Text":"Now, total revenue is the number of units x,"},{"Start":"01:57.919 ","End":"02:01.850","Text":"and they\u0027re sold at P of x. P of x is 20,"},{"Start":"02:01.850 ","End":"02:04.850","Text":"so I don\u0027t even have to put times P of x,"},{"Start":"02:04.850 ","End":"02:06.050","Text":"it\u0027s x times 20."},{"Start":"02:06.050 ","End":"02:13.140","Text":"In other words, 20 bucks a piece x pieces 20x dollars."},{"Start":"02:13.140 ","End":"02:16.960","Text":"That\u0027s total revenue. I should\u0027ve said of x."},{"Start":"02:16.960 ","End":"02:24.870","Text":"That\u0027s better. I\u0027ll do that in this color and there\u0027s our total revenue."},{"Start":"02:25.300 ","End":"02:31.220","Text":"Profit pi is going to equal total revenue from here,"},{"Start":"02:31.220 ","End":"02:36.530","Text":"that\u0027s 20x dollars minus the cost."},{"Start":"02:36.530 ","End":"02:39.030","Text":"Take that from here."},{"Start":"02:39.430 ","End":"02:45.400","Text":"That\u0027s 300 plus 2x squared."},{"Start":"02:45.770 ","End":"02:49.320","Text":"Just a little bit of simplification."},{"Start":"02:49.320 ","End":"02:50.900","Text":"They still put them in order."},{"Start":"02:50.900 ","End":"02:54.485","Text":"I\u0027ll put the x squared first,"},{"Start":"02:54.485 ","End":"02:59.130","Text":"and then I\u0027ll put the xs and then the numbers."},{"Start":"02:59.130 ","End":"03:03.890","Text":"I\u0027ll highlight the profit in a different color."},{"Start":"03:03.890 ","End":"03:07.790","Text":"Now to find minimum or maximum, in this case,"},{"Start":"03:07.790 ","End":"03:12.335","Text":"maximum, we normally would use calculus."},{"Start":"03:12.335 ","End":"03:15.310","Text":"But some of you may not have learned calculus."},{"Start":"03:15.310 ","End":"03:19.820","Text":"If you haven\u0027t, I hope you\u0027ve learned about the quadratic function and parabolas."},{"Start":"03:19.820 ","End":"03:26.300","Text":"Because this is like a quadratic function where y equals"},{"Start":"03:26.300 ","End":"03:34.095","Text":"ax squared plus bx plus c, I\u0027ve changed color."},{"Start":"03:34.095 ","End":"03:39.665","Text":"This is of this form where this would be our a negative 2,"},{"Start":"03:39.665 ","End":"03:41.720","Text":"b would be 20,"},{"Start":"03:41.720 ","End":"03:44.825","Text":"and c would be minus 300."},{"Start":"03:44.825 ","End":"03:55.730","Text":"What we know about such curves is that if a is bigger than 0,"},{"Start":"03:55.730 ","End":"04:06.250","Text":"then the parabola faces upwards or opens upwards."},{"Start":"04:06.250 ","End":"04:09.090","Text":"If a is negative,"},{"Start":"04:09.090 ","End":"04:15.080","Text":"then it opens the faces downwards."},{"Start":"04:15.080 ","End":"04:19.410","Text":"In our case, our a is negative."},{"Start":"04:20.650 ","End":"04:24.095","Text":"We\u0027re in this case, which is good,"},{"Start":"04:24.095 ","End":"04:27.760","Text":"because in this case, we have a maximum."},{"Start":"04:27.760 ","End":"04:30.540","Text":"It occurs that this vertex, here"},{"Start":"04:30.540 ","End":"04:32.090","Text":"also, there\u0027s a vertex,"},{"Start":"04:32.090 ","End":"04:33.500","Text":"but it\u0027s a minimum."},{"Start":"04:33.500 ","End":"04:37.170","Text":"Anyway, we\u0027re only concerned now with this 1."},{"Start":"04:37.170 ","End":"04:40.065","Text":"The vertex, so call it v for this point."},{"Start":"04:40.065 ","End":"04:45.590","Text":"There is a formula that the x-value of"},{"Start":"04:45.590 ","End":"04:52.260","Text":"the vertex is given by minus b over 2a."},{"Start":"04:52.260 ","End":"04:57.065","Text":"That will do for the maximum or at least will have the x of it."},{"Start":"04:57.065 ","End":"05:05.360","Text":"Here, we will do that and we get that x_v is minus b,"},{"Start":"05:05.360 ","End":"05:12.765","Text":"which is minus 20 over 2a is twice minus 2."},{"Start":"05:12.765 ","End":"05:14.580","Text":"We can do that in our heads,"},{"Start":"05:14.580 ","End":"05:19.560","Text":"minus 20 over minus 4 is 5."},{"Start":"05:19.560 ","End":"05:25.859","Text":"What that means is that the answer to part a"},{"Start":"05:25.859 ","End":"05:33.415","Text":"is that 5 units are needed for the minimum."},{"Start":"05:33.415 ","End":"05:35.995","Text":"Now, I have another color left,"},{"Start":"05:35.995 ","End":"05:41.150","Text":"say this color is for the answer for part a."},{"Start":"05:41.150 ","End":"05:47.060","Text":"Part b is easy because all we have to do is plug"},{"Start":"05:47.060 ","End":"05:54.110","Text":"in this value of x 5 into the profit function, which is this."},{"Start":"05:54.110 ","End":"06:01.370","Text":"In other words, I want pi of 5 and see what this is equal to."},{"Start":"06:01.370 ","End":"06:06.665","Text":"This will be my, let\u0027s call it pi_max,"},{"Start":"06:06.665 ","End":"06:12.690","Text":"pi profit, max for maximum is 5."},{"Start":"06:13.120 ","End":"06:16.655","Text":"It\u0027s equal to, I\u0027m plugging it here,"},{"Start":"06:16.655 ","End":"06:21.650","Text":"minus 2 times 5 squared plus 20"},{"Start":"06:21.650 ","End":"06:28.255","Text":"times 5 minus 300."},{"Start":"06:28.255 ","End":"06:31.440","Text":"Let\u0027s see, 5 squared is 25,"},{"Start":"06:31.440 ","End":"06:33.615","Text":"so that\u0027s minus 50."},{"Start":"06:33.615 ","End":"06:38.820","Text":"Another negative, so that\u0027s altogether minus 350 plus a 100."},{"Start":"06:38.820 ","End":"06:45.490","Text":"What we end up with is minus 250 dollars,"},{"Start":"06:45.490 ","End":"06:49.335","Text":"the way you put the dollars,"},{"Start":"06:49.335 ","End":"06:52.350","Text":"left to the minus so to the right of the minus."},{"Start":"06:52.350 ","End":"06:55.020","Text":"That you ask is the maximum,"},{"Start":"06:55.020 ","End":"06:57.145","Text":"it\u0027s a loss not a profit."},{"Start":"06:57.145 ","End":"06:58.715","Text":"But it\u0027s the best you could do."},{"Start":"06:58.715 ","End":"07:03.510","Text":"Anyway, I\u0027ll just highlight it and declare that we are done."}],"ID":8540},{"Watched":false,"Name":"Exercise 12","Duration":"11m 36s","ChapterTopicVideoID":8406,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8406.jpeg","UploadDate":"2017-01-28T14:06:30.8370000","DurationForVideoObject":"PT11M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.860","Text":"Manufacturer has this product and it has a demand"},{"Start":"00:04.860 ","End":"00:09.300","Text":"function P of x. X is the quantity or number of"},{"Start":"00:09.300 ","End":"00:13.485","Text":"units of the product he sells and"},{"Start":"00:13.485 ","End":"00:19.390","Text":"that quantity he can get price P for it and it\u0027s given by this formula."},{"Start":"00:21.380 ","End":"00:24.460","Text":"He also has a marginal cost function."},{"Start":"00:24.460 ","End":"00:27.950","Text":"We\u0027re given MC of x is this formula."},{"Start":"00:27.950 ","End":"00:30.655","Text":"Everything\u0027s in dollars."},{"Start":"00:30.655 ","End":"00:38.365","Text":"Just note that sometimes the letter Q is used instead of x for quantity but we use x."},{"Start":"00:38.365 ","End":"00:42.140","Text":"The question is, what is the quantity?"},{"Start":"00:42.140 ","End":"00:49.860","Text":"In other words, what x does he need to produce in order to get the maximum profit?"},{"Start":"00:50.150 ","End":"00:53.464","Text":"We need to get some equations, formulas."},{"Start":"00:53.464 ","End":"00:55.714","Text":"We will start off with something simple,"},{"Start":"00:55.714 ","End":"00:58.970","Text":"that the profit and we\u0027ve seen this before,"},{"Start":"00:58.970 ","End":"01:01.100","Text":"is equal to the,"},{"Start":"01:01.100 ","End":"01:03.335","Text":"I\u0027ll just say revenue for short,"},{"Start":"01:03.335 ","End":"01:08.840","Text":"not total revenue and instead of total cost I\u0027ll just write cost."},{"Start":"01:08.840 ","End":"01:13.655","Text":"If we make it specific as functions of x,"},{"Start":"01:13.655 ","End":"01:19.040","Text":"then we\u0027ve got the profit we use the letter pi of x is equal to revenue,"},{"Start":"01:19.040 ","End":"01:21.690","Text":"it\u0027s either R or TR,"},{"Start":"01:21.690 ","End":"01:28.924","Text":"revenue or total revenue of x minus cost or total cost we\u0027ll go with TC of x."},{"Start":"01:28.924 ","End":"01:31.760","Text":"We don\u0027t have the total cost,"},{"Start":"01:31.760 ","End":"01:33.665","Text":"we have the marginal cost."},{"Start":"01:33.665 ","End":"01:35.630","Text":"We don\u0027t even have a total revenue,"},{"Start":"01:35.630 ","End":"01:36.800","Text":"but we do have the price,"},{"Start":"01:36.800 ","End":"01:38.840","Text":"so we\u0027re going to have to build these."},{"Start":"01:38.840 ","End":"01:47.450","Text":"Let\u0027s see. Let\u0027s start off and see if we can get what TR of x is equal to."},{"Start":"01:47.450 ","End":"01:53.250","Text":"Total revenue is the number of items times the price per each."},{"Start":"01:53.250 ","End":"02:01.555","Text":"We know that we sold x items and we know the price for each was P of x."},{"Start":"02:01.555 ","End":"02:04.805","Text":"We\u0027re even given P of x is this,"},{"Start":"02:04.805 ","End":"02:12.930","Text":"so we have that this is equal to x times minus 0.15x plus 50."},{"Start":"02:12.930 ","End":"02:21.600","Text":"If we expand we\u0027ve got minus 0.15x squared plus 50x."},{"Start":"02:22.480 ","End":"02:30.080","Text":"Let me highlight that we already have the total revenue function and it\u0027s over here."},{"Start":"02:30.080 ","End":"02:33.695","Text":"Now, next thing we need is TC."},{"Start":"02:33.695 ","End":"02:36.300","Text":"Let\u0027s see how we\u0027ll get that."},{"Start":"02:36.640 ","End":"02:43.565","Text":"We are given the marginal cost as a function."},{"Start":"02:43.565 ","End":"02:45.740","Text":"I\u0027ll just copy it from there."},{"Start":"02:45.740 ","End":"02:50.850","Text":"It\u0027s 0.06x squared plus 20."},{"Start":"02:51.110 ","End":"02:59.360","Text":"What we do know is that the marginal cost is the derivative of the total cost."},{"Start":"02:59.360 ","End":"03:06.635","Text":"I\u0027ve got TC prime of x is equal to this same thing,"},{"Start":"03:06.635 ","End":"03:10.370","Text":"0.06 x squared plus 20."},{"Start":"03:10.370 ","End":"03:16.655","Text":"I\u0027m thinking, I have TC prime and I can\u0027t get TC."},{"Start":"03:16.655 ","End":"03:19.655","Text":"You might say, well, why not integrate it?"},{"Start":"03:19.655 ","End":"03:23.030","Text":"The reason I can\u0027t integrate this is because"},{"Start":"03:23.030 ","End":"03:26.330","Text":"I\u0027ll have a constant of integration and I won\u0027t know what that is."},{"Start":"03:26.330 ","End":"03:28.850","Text":"You would need an initial condition or something."},{"Start":"03:28.850 ","End":"03:35.135","Text":"What we can do is if we can\u0027t integrate this at least we can differentiate that."},{"Start":"03:35.135 ","End":"03:42.335","Text":"How about we differentiate everything here and get that the derivative of the profit is"},{"Start":"03:42.335 ","End":"03:50.435","Text":"the derivative of the total revenue minus the derivative of the total cost."},{"Start":"03:50.435 ","End":"03:53.630","Text":"Well, I have got this."},{"Start":"03:53.630 ","End":"03:55.820","Text":"I\u0027ve got TC prime."},{"Start":"03:55.820 ","End":"04:01.035","Text":"This is equal to this,"},{"Start":"04:01.035 ","End":"04:08.620","Text":"but what I should be caring about is not TR and I remove the highlighting."},{"Start":"04:08.630 ","End":"04:12.975","Text":"What we\u0027ll need is TR prime."},{"Start":"04:12.975 ","End":"04:18.610","Text":"Now, I don\u0027t have TR prime but if I just differentiate this then I will."},{"Start":"04:18.610 ","End":"04:24.255","Text":"Let me take this and differentiate it and put the answer here."},{"Start":"04:24.255 ","End":"04:29.245","Text":"TR prime of x is equal to,"},{"Start":"04:29.245 ","End":"04:31.820","Text":"well, maybe not all the way from here."},{"Start":"04:31.820 ","End":"04:35.490","Text":"Maybe we can take it from here."},{"Start":"04:35.540 ","End":"04:38.670","Text":"I\u0027m going to differentiate this."},{"Start":"04:38.670 ","End":"04:48.420","Text":"Let\u0027s see, twice minus 0.5 is minus 0.3x plus 50."},{"Start":"04:48.420 ","End":"04:50.670","Text":"Now I can shade it."},{"Start":"04:50.670 ","End":"04:52.815","Text":"Now I\u0027ve got these 2 pieces,"},{"Start":"04:52.815 ","End":"05:00.680","Text":"so I can get the derivative of the profit is equal to this green minus yellow."},{"Start":"05:00.680 ","End":"05:08.585","Text":"It\u0027s minus 0.3x plus 50 minus,"},{"Start":"05:08.585 ","End":"05:16.250","Text":"minus in front of each of the minus 0.06x squared minus 20."},{"Start":"05:16.250 ","End":"05:24.845","Text":"Collecting like terms I get minus 0.06x squared"},{"Start":"05:24.845 ","End":"05:32.390","Text":"minus 0.3x plus 50 minus"},{"Start":"05:32.390 ","End":"05:36.695","Text":"20 is plus 30."},{"Start":"05:36.695 ","End":"05:40.570","Text":"That it\u0027s not the profit,"},{"Start":"05:40.570 ","End":"05:46.295","Text":"it\u0027s the derivative of the profit that actually is even better."},{"Start":"05:46.295 ","End":"05:50.329","Text":"This by the way is also called the marginal profit."},{"Start":"05:50.329 ","End":"05:53.375","Text":"Anything with derivative is marginal."},{"Start":"05:53.375 ","End":"05:56.465","Text":"I prefer to leave it in this form as a derivative."},{"Start":"05:56.465 ","End":"05:59.180","Text":"Now, why do I want a derivative?"},{"Start":"05:59.180 ","End":"06:02.510","Text":"The question doesn\u0027t say to find the maximum profit,"},{"Start":"06:02.510 ","End":"06:10.285","Text":"it just wants to know what quantity I need in order to guarantee that it\u0027s maximum."},{"Start":"06:10.285 ","End":"06:16.740","Text":"I can solve the equation pi prime of x equals 0 and"},{"Start":"06:16.740 ","End":"06:20.210","Text":"then look for critical points and then check if"},{"Start":"06:20.210 ","End":"06:24.870","Text":"they give me a maximum or minimum or neither."},{"Start":"06:25.820 ","End":"06:34.580","Text":"We\u0027ll just get an equation by setting this and I just copied it here to equal 0."},{"Start":"06:34.580 ","End":"06:39.710","Text":"Now this equation we can multiply everything by"},{"Start":"06:39.710 ","End":"06:45.065","Text":"100 or even make it minus 100 and then we\u0027ll get something simpler."},{"Start":"06:45.065 ","End":"06:48.945","Text":"We\u0027ll get 6x squared."},{"Start":"06:48.945 ","End":"06:51.645","Text":"The minus will become plus,"},{"Start":"06:51.645 ","End":"07:01.560","Text":"times 100 is 30x minus 3,000 equals 0."},{"Start":"07:01.560 ","End":"07:08.370","Text":"Let\u0027s divide now by 6 and then we\u0027ve got x squared plus"},{"Start":"07:08.370 ","End":"07:18.195","Text":"5x minus 500 is equal to 0."},{"Start":"07:18.195 ","End":"07:20.130","Text":"Continue over here."},{"Start":"07:20.130 ","End":"07:25.430","Text":"We\u0027ll do it using the formula for quadratic and we get that x equals,"},{"Start":"07:25.430 ","End":"07:27.695","Text":"I\u0027m assuming you know the formula,"},{"Start":"07:27.695 ","End":"07:33.725","Text":"minus B plus or minus the square root of"},{"Start":"07:33.725 ","End":"07:40.245","Text":"B squared is 25 minus 4 times,"},{"Start":"07:40.245 ","End":"07:41.985","Text":"well, A is just 1."},{"Start":"07:41.985 ","End":"07:46.935","Text":"I could put a 1x squared there for A times C,"},{"Start":"07:46.935 ","End":"07:50.230","Text":"C is minus 500."},{"Start":"07:51.740 ","End":"07:56.490","Text":"All this is over 2A,"},{"Start":"07:56.490 ","End":"07:59.050","Text":"which is just 2."},{"Start":"07:59.050 ","End":"08:02.765","Text":"Now what\u0027s under the square root sign is going to be,"},{"Start":"08:02.765 ","End":"08:04.610","Text":"this is plus with minus, minus."},{"Start":"08:04.610 ","End":"08:07.865","Text":"This is 20,000 plus 25."},{"Start":"08:07.865 ","End":"08:12.275","Text":"I need the square root of 2025."},{"Start":"08:12.275 ","End":"08:17.255","Text":"You could do it on the calculator and it comes out to be 45."},{"Start":"08:17.255 ","End":"08:21.230","Text":"What we get is that x is equal to minus 5 plus"},{"Start":"08:21.230 ","End":"08:28.915","Text":"or minus 45 over 2."},{"Start":"08:28.915 ","End":"08:35.080","Text":"Now if we take the plus then we will get minus 5"},{"Start":"08:35.080 ","End":"08:41.575","Text":"plus 45 is 40 over 2 is 20."},{"Start":"08:41.575 ","End":"08:48.930","Text":"If we take the minus then we\u0027ve got minus 50 over 2 which is minus"},{"Start":"08:48.930 ","End":"08:52.660","Text":"25 which is going to have to be"},{"Start":"08:52.660 ","End":"08:58.790","Text":"excluded because it\u0027s negative and our x as a quantity it has to be positive."},{"Start":"08:59.090 ","End":"09:06.370","Text":"We know that the x which makes the derivative of"},{"Start":"09:06.370 ","End":"09:14.010","Text":"the profit 0 is just x equals 20."},{"Start":"09:14.140 ","End":"09:21.400","Text":"We could say yes, the answer is 20 but we still have to show that this really is"},{"Start":"09:21.400 ","End":"09:28.330","Text":"a maximum because derivative 0 can happen for other reasons."},{"Start":"09:28.330 ","End":"09:32.800","Text":"The simplest thing to do is to use the second derivative test."},{"Start":"09:32.800 ","End":"09:36.580","Text":"If we figure out the second derivative and it\u0027s negative,"},{"Start":"09:36.580 ","End":"09:38.880","Text":"then we\u0027re guaranteed a maximum."},{"Start":"09:38.880 ","End":"09:43.870","Text":"Let\u0027s continue and go for the second derivative."},{"Start":"09:43.870 ","End":"09:48.110","Text":"Pi double-prime of x."},{"Start":"09:48.110 ","End":"09:51.165","Text":"Now this is Pi prime,"},{"Start":"09:51.165 ","End":"09:54.430","Text":"so double prime is just the derivative of that."},{"Start":"09:54.430 ","End":"10:02.395","Text":"We have minus 0.12x,"},{"Start":"10:02.395 ","End":"10:08.620","Text":"from here we have minus 0.3."},{"Start":"10:08.990 ","End":"10:17.775","Text":"What we have to check is whether it\u0027s positive or negative or what but when x is 20."},{"Start":"10:17.775 ","End":"10:22.000","Text":"I still need 1 more calculation,"},{"Start":"10:22.000 ","End":"10:33.695","Text":"Pi double-prime at 20 is minus 0.12 times 20 minus 0.3."},{"Start":"10:33.695 ","End":"10:36.545","Text":"Well, now it\u0027s obviously negative."},{"Start":"10:36.545 ","End":"10:39.410","Text":"I mean, everything here, it is negative."},{"Start":"10:39.410 ","End":"10:42.375","Text":"You don\u0027t need the actual answer."},{"Start":"10:42.375 ","End":"10:44.070","Text":"It\u0027s negative."},{"Start":"10:44.070 ","End":"10:51.620","Text":"If it\u0027s negative then this thing here is a maximum."},{"Start":"10:51.620 ","End":"11:01.550","Text":"The thing is that we know what quantity to produce to"},{"Start":"11:01.550 ","End":"11:05.420","Text":"get the maximum but we can\u0027t actually find"},{"Start":"11:05.420 ","End":"11:10.969","Text":"the maximum because somewhere along the line we were given only a derivative."},{"Start":"11:10.969 ","End":"11:15.680","Text":"We were given the marginal cost and we weren\u0027t given the total cost."},{"Start":"11:15.680 ","End":"11:20.850","Text":"We weren\u0027t given some extra information so that\u0027s the best we can do."},{"Start":"11:20.850 ","End":"11:23.330","Text":"I can say to the manufacturer, \"Yeah,"},{"Start":"11:23.330 ","End":"11:26.180","Text":"make 20 items and then you\u0027ll get"},{"Start":"11:26.180 ","End":"11:30.780","Text":"maximum profit,\" but I can\u0027t tell him what that profit is."},{"Start":"11:32.570 ","End":"11:37.890","Text":"I\u0027ll just say that this is the answer and we\u0027re done."}],"ID":8541},{"Watched":false,"Name":"Exercise 13","Duration":"7m 56s","ChapterTopicVideoID":8407,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8407.jpeg","UploadDate":"2017-01-28T14:08:19.7000000","DurationForVideoObject":"PT7M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.760","Text":"A manufacturer has a product with demand function as follows;"},{"Start":"00:05.760 ","End":"00:10.185","Text":"x is equal to 5000 minus 50P over 3,"},{"Start":"00:10.185 ","End":"00:16.005","Text":"and a total cost function TC of x is equal to 200 plus 4x."},{"Start":"00:16.005 ","End":"00:18.540","Text":"Everything is in dollars here."},{"Start":"00:18.540 ","End":"00:23.480","Text":"But notice that the total cost is given in terms of x,"},{"Start":"00:23.480 ","End":"00:31.895","Text":"which is the quantity number of items or units produced."},{"Start":"00:31.895 ","End":"00:35.640","Text":"But this one is the other way round which we sometimes get,"},{"Start":"00:35.640 ","End":"00:40.885","Text":"it\u0027s quantity in terms of price and not price in terms of quantity."},{"Start":"00:40.885 ","End":"00:44.450","Text":"Notice that there\u0027s something reversed here."},{"Start":"00:44.450 ","End":"00:46.400","Text":"Now let\u0027s get on to part (a),"},{"Start":"00:46.400 ","End":"00:47.960","Text":"we\u0027ll do them one at a time."},{"Start":"00:47.960 ","End":"00:51.950","Text":"In (a) we want to find the quantity"},{"Start":"00:51.950 ","End":"00:56.800","Text":"x that the manufacturer needs to produce to maximize his profit."},{"Start":"00:56.800 ","End":"01:03.600","Text":"Then later in (b) we\u0027ll actually find out what is that maximum profit."},{"Start":"01:04.130 ","End":"01:08.605","Text":"We have to remember how to find profit."},{"Start":"01:08.605 ","End":"01:11.990","Text":"Profit is the difference of two things."},{"Start":"01:11.990 ","End":"01:17.260","Text":"It\u0027s the total income or revenue,"},{"Start":"01:17.260 ","End":"01:19.995","Text":"minus what is spent,"},{"Start":"01:19.995 ","End":"01:24.540","Text":"which is the cost or total cost."},{"Start":"01:24.540 ","End":"01:26.505","Text":"If we put this in symbols,"},{"Start":"01:26.505 ","End":"01:29.430","Text":"profit, we use the Greek letter pi,"},{"Start":"01:29.430 ","End":"01:32.280","Text":"because p stands for price."},{"Start":"01:32.280 ","End":"01:35.615","Text":"Pi of x is equal to"},{"Start":"01:35.615 ","End":"01:45.305","Text":"the total revenue as a function of x minus the total cost as a function of x."},{"Start":"01:45.305 ","End":"01:47.810","Text":"Now we have total cost here,"},{"Start":"01:47.810 ","End":"01:50.480","Text":"but we don\u0027t have total revenue."},{"Start":"01:50.480 ","End":"01:55.580","Text":"However, total revenue, there is a formula for it, the total revenue,"},{"Start":"01:55.580 ","End":"02:02.369","Text":"how much he takes in is the number of items or products"},{"Start":"02:02.369 ","End":"02:10.340","Text":"times the price per each item, the unit price."},{"Start":"02:10.340 ","End":"02:13.280","Text":"But we don\u0027t have p in terms of x,"},{"Start":"02:13.280 ","End":"02:16.490","Text":"we have x in terms of p. What we want to do"},{"Start":"02:16.490 ","End":"02:20.815","Text":"now is a bit of algebra to reverse this equation."},{"Start":"02:20.815 ","End":"02:24.540","Text":"This one I want to write the other way round."},{"Start":"02:24.540 ","End":"02:26.180","Text":"Let\u0027s do the algebra at"},{"Start":"02:26.180 ","End":"02:30.590","Text":"the side and I just move that formula here so we\u0027ll have room here."},{"Start":"02:30.590 ","End":"02:37.000","Text":"I\u0027m going to copy x equals 5000 minus 50P over 3."},{"Start":"02:37.000 ","End":"02:42.210","Text":"I want to extract p. Multiply both sides by 3,"},{"Start":"02:42.210 ","End":"02:48.150","Text":"so I\u0027ve got 3x equals 5000 minus 50P."},{"Start":"02:48.150 ","End":"02:51.795","Text":"Now let me put the p\u0027s on the left and the x\u0027s on the right,"},{"Start":"02:51.795 ","End":"02:58.740","Text":"so I have 50P equals 5000 minus 3x."},{"Start":"02:58.740 ","End":"03:02.730","Text":"Then all I have left to do is divide by 50,"},{"Start":"03:04.270 ","End":"03:08.845","Text":"5000 divided by 50 is 100,"},{"Start":"03:08.845 ","End":"03:12.425","Text":"and 3 over 50,"},{"Start":"03:12.425 ","End":"03:17.795","Text":"I\u0027ll just write it as a decimal, it\u0027s 0.06x."},{"Start":"03:17.795 ","End":"03:24.504","Text":"Now that I have this and I can write this as p in terms of x,"},{"Start":"03:24.504 ","End":"03:31.940","Text":"I could just substitute that in here and continue and get"},{"Start":"03:31.940 ","End":"03:41.610","Text":"the total revenue is x times 100 minus 0.06x."},{"Start":"03:42.310 ","End":"03:45.110","Text":"Now I can continue with this formula."},{"Start":"03:45.110 ","End":"03:47.240","Text":"Let me just highlight to make things clearer."},{"Start":"03:47.240 ","End":"03:53.605","Text":"The total revenue I\u0027m going to get from here,"},{"Start":"03:53.605 ","End":"04:00.760","Text":"and the total cost I can get from here."},{"Start":"04:00.860 ","End":"04:03.485","Text":"If I substitute these,"},{"Start":"04:03.485 ","End":"04:06.654","Text":"I can now figure out what is the profit."},{"Start":"04:06.654 ","End":"04:15.920","Text":"I\u0027ve got pi profit of x is equal to this green color minus the magenta color."},{"Start":"04:15.920 ","End":"04:18.230","Text":"From here I multiply out,"},{"Start":"04:18.230 ","End":"04:26.840","Text":"I\u0027ve got 100x minus 0.06x squared."},{"Start":"04:26.840 ","End":"04:29.515","Text":"Then I have to subtract this,"},{"Start":"04:29.515 ","End":"04:31.679","Text":"so I\u0027ll just put them both with a minus;"},{"Start":"04:31.679 ","End":"04:35.340","Text":"minus 200, minus 4x,"},{"Start":"04:35.340 ","End":"04:39.725","Text":"and I just collect like terms and order them."},{"Start":"04:39.725 ","End":"04:49.490","Text":"I\u0027ve got minus 0.06x squared plus 100x minus 4x,"},{"Start":"04:49.490 ","End":"04:55.090","Text":"that\u0027s plus 96x minus 200."},{"Start":"04:55.090 ","End":"04:56.940","Text":"Then I think I\u0027ll highlight this too,"},{"Start":"04:56.940 ","End":"04:59.340","Text":"this is the profit function."},{"Start":"04:59.340 ","End":"05:03.710","Text":"Now we have the function that we need to maximize."},{"Start":"05:03.710 ","End":"05:10.150","Text":"We want to find the x that gives the largest value of the profit and what the profit is."},{"Start":"05:10.150 ","End":"05:13.880","Text":"Now we could do this with calculus and differentiation."},{"Start":"05:13.880 ","End":"05:16.600","Text":"We don\u0027t need to because this is a quadratic function."},{"Start":"05:16.600 ","End":"05:24.829","Text":"I want to remind you when we have y equals ax squared plus bx plus c,"},{"Start":"05:24.829 ","End":"05:28.115","Text":"then depending on the sign of a,"},{"Start":"05:28.115 ","End":"05:30.560","Text":"if a is negative,"},{"Start":"05:30.560 ","End":"05:32.840","Text":"we have a parabola like this,"},{"Start":"05:32.840 ","End":"05:35.855","Text":"facing down or opening downwards."},{"Start":"05:35.855 ","End":"05:37.520","Text":"If a is positive,"},{"Start":"05:37.520 ","End":"05:40.250","Text":"which it isn\u0027t, then it would be this way around."},{"Start":"05:40.250 ","End":"05:42.860","Text":"In each case, there\u0027s a vertex,"},{"Start":"05:42.860 ","End":"05:45.965","Text":"which is either the minimum or the maximum."},{"Start":"05:45.965 ","End":"05:50.675","Text":"In our case, we have that this is our a which is negative,"},{"Start":"05:50.675 ","End":"05:55.225","Text":"so we have a maximum here."},{"Start":"05:55.225 ","End":"05:57.740","Text":"If we label the vertex v,"},{"Start":"05:57.740 ","End":"06:04.280","Text":"there\u0027s a formula that the x of the vertex is minus b over 2a."},{"Start":"06:04.280 ","End":"06:06.110","Text":"It\u0027s the same formula here,"},{"Start":"06:06.110 ","End":"06:08.515","Text":"but we don\u0027t need this one."},{"Start":"06:08.515 ","End":"06:13.680","Text":"Back here, well, I\u0027ll just continue labeling,"},{"Start":"06:13.680 ","End":"06:19.120","Text":"b is 96, we don\u0027t even need c but it\u0027s minus 200."},{"Start":"06:19.670 ","End":"06:24.920","Text":"We get the x of the vertex is minus b over 2a,"},{"Start":"06:24.920 ","End":"06:34.050","Text":"is minus 96 over twice, minus 0.06."},{"Start":"06:35.210 ","End":"06:38.435","Text":"Minus will cancel with the minus,"},{"Start":"06:38.435 ","End":"06:42.090","Text":"and we\u0027ll get 96 over 0.12,"},{"Start":"06:43.130 ","End":"06:47.550","Text":"and this comes out to be 800."},{"Start":"06:47.550 ","End":"06:52.820","Text":"Eight hundred is the answer to part (a) and that is"},{"Start":"06:52.820 ","End":"06:58.700","Text":"the quantity or the number of units that have to be sold for maximum profit."},{"Start":"06:58.700 ","End":"07:04.845","Text":"All we have to do now in part (b) is find out what"},{"Start":"07:04.845 ","End":"07:11.610","Text":"the profit is for 800 units."},{"Start":"07:11.610 ","End":"07:14.375","Text":"I still have the formula written up here."},{"Start":"07:14.375 ","End":"07:18.965","Text":"What I want is pi maximum,"},{"Start":"07:18.965 ","End":"07:21.080","Text":"which is the pi,"},{"Start":"07:21.080 ","End":"07:29.010","Text":"the profit for 800 units and that is equal to from here,"},{"Start":"07:29.010 ","End":"07:35.415","Text":"minus 0.06 times 800 squared,"},{"Start":"07:35.415 ","End":"07:42.880","Text":"plus 96 times 800 minus 200."},{"Start":"07:43.250 ","End":"07:47.430","Text":"Comes out to be 38,200,"},{"Start":"07:47.430 ","End":"07:51.320","Text":"I did the calculation for you and it\u0027s in dollars."},{"Start":"07:51.320 ","End":"07:57.390","Text":"That concludes part (b) and so we\u0027ve done with this exercise."}],"ID":8542},{"Watched":false,"Name":"Exercise 14","Duration":"4m 42s","ChapterTopicVideoID":8408,"CourseChapterTopicPlaylistID":65391,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8408.jpeg","UploadDate":"2017-01-28T14:09:23.0570000","DurationForVideoObject":"PT4M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.020","Text":"In this exercise, we will be using integrals from calculus."},{"Start":"00:04.020 ","End":"00:05.925","Text":"If you\u0027ve completely forgotten them,"},{"Start":"00:05.925 ","End":"00:09.795","Text":"just go and review your basic integrals."},{"Start":"00:09.795 ","End":"00:14.895","Text":"As usual, we have a manufacturer who has a product,"},{"Start":"00:14.895 ","End":"00:19.440","Text":"and this time we\u0027re given the marginal cost function MC"},{"Start":"00:19.440 ","End":"00:25.125","Text":"of x. X is the quantity sometimes called q,"},{"Start":"00:25.125 ","End":"00:30.760","Text":"and it\u0027s equal to 0.06x squared plus 20 and we\u0027re talking in US dollars."},{"Start":"00:30.760 ","End":"00:34.865","Text":"Now, we have to figure out from the marginal cost function,"},{"Start":"00:34.865 ","End":"00:41.555","Text":"what is the total cost function and we\u0027re given an extra piece of information,"},{"Start":"00:41.555 ","End":"00:44.035","Text":"and I\u0027ll return to that later."},{"Start":"00:44.035 ","End":"00:46.605","Text":"The idea is this,"},{"Start":"00:46.605 ","End":"00:57.220","Text":"is that the marginal cost MC is just the derivative of the total cost function."},{"Start":"00:57.220 ","End":"01:01.295","Text":"We learn this, the question is,"},{"Start":"01:01.295 ","End":"01:04.220","Text":"if we\u0027re given the marginal cost,"},{"Start":"01:04.220 ","End":"01:06.440","Text":"how can we get back to the total cost?"},{"Start":"01:06.440 ","End":"01:07.820","Text":"We know the derivative,"},{"Start":"01:07.820 ","End":"01:09.645","Text":"how do we find the function?"},{"Start":"01:09.645 ","End":"01:12.860","Text":"We\u0027re going to do an anti-differentiation."},{"Start":"01:12.860 ","End":"01:16.970","Text":"An anti-differentiation is what we call a primitive,"},{"Start":"01:16.970 ","End":"01:20.495","Text":"or more commonly the indefinite integral."},{"Start":"01:20.495 ","End":"01:26.360","Text":"If I know that TC prime of x is equal"},{"Start":"01:26.360 ","End":"01:33.225","Text":"to 0.06x squared plus 20,"},{"Start":"01:33.225 ","End":"01:38.025","Text":"then I can say what TC of x is,"},{"Start":"01:38.025 ","End":"01:41.735","Text":"it\u0027s equal to just the integral of this thing,"},{"Start":"01:41.735 ","End":"01:48.860","Text":"0.06x squared plus 20dx, indefinite."},{"Start":"01:48.860 ","End":"01:52.310","Text":"But in the end, we\u0027ll get a constant and we\u0027ll see what to do with that."},{"Start":"01:52.310 ","End":"01:55.380","Text":"What is the integral?"},{"Start":"01:56.800 ","End":"01:59.325","Text":"I\u0027ll give you the formula."},{"Start":"01:59.325 ","End":"02:06.440","Text":"In general, when you have a monomial ax^ndx,"},{"Start":"02:07.200 ","End":"02:13.260","Text":"then the integral, what you do is you raise the power of x by 1,"},{"Start":"02:13.260 ","End":"02:14.930","Text":"from n you go to n plus 1,"},{"Start":"02:14.930 ","End":"02:17.525","Text":"then you divide by the new exponent."},{"Start":"02:17.525 ","End":"02:21.425","Text":"You take the a and divide by that new exponent,"},{"Start":"02:21.425 ","End":"02:25.480","Text":"and then there\u0027s a plus a constant of integration."},{"Start":"02:25.480 ","End":"02:29.695","Text":"Here, we get from this part,"},{"Start":"02:29.695 ","End":"02:35.850","Text":"raise the power by 1 so it\u0027s x^3 or x cubed,"},{"Start":"02:35.850 ","End":"02:40.680","Text":"and then divide by 3, so it\u0027s 0.02."},{"Start":"02:40.680 ","End":"02:45.290","Text":"For 20, it\u0027s like x^0 where you"},{"Start":"02:45.290 ","End":"02:49.550","Text":"might just remember that the integral of a constant is that constant times x^1,"},{"Start":"02:49.550 ","End":"02:57.905","Text":"which is just x, and you only need 1 constant and that\u0027s plus C. Now,"},{"Start":"02:57.905 ","End":"03:01.385","Text":"the question is, how do we find this constant?"},{"Start":"03:01.385 ","End":"03:02.930","Text":"I\u0027ll just repeat that,"},{"Start":"03:02.930 ","End":"03:05.060","Text":"TC of x is this."},{"Start":"03:05.060 ","End":"03:08.120","Text":"That\u0027s where this extra hint comes in,"},{"Start":"03:08.120 ","End":"03:12.049","Text":"we\u0027re told that when x is 10,"},{"Start":"03:12.049 ","End":"03:16.365","Text":"the total cost is 225."},{"Start":"03:16.365 ","End":"03:27.830","Text":"If I plugin that x equals 10 and then TC is 225,"},{"Start":"03:27.830 ","End":"03:37.340","Text":"then I get that 225 is equal to 0.02"},{"Start":"03:37.340 ","End":"03:42.210","Text":"times 10 which is x cubed plus"},{"Start":"03:42.210 ","End":"03:48.810","Text":"20 times 10 plus C. Let\u0027s see,"},{"Start":"03:48.810 ","End":"03:51.120","Text":"10 cubed is 1,000,"},{"Start":"03:51.120 ","End":"03:55.950","Text":"1,000 times 0.2 is 20,"},{"Start":"03:55.950 ","End":"04:05.100","Text":"20 times 10 is 200 plus the C. Clearly this together is 220,"},{"Start":"04:05.100 ","End":"04:10.725","Text":"this is 225, so C equals 5."},{"Start":"04:10.725 ","End":"04:13.380","Text":"Now that we have C,"},{"Start":"04:13.380 ","End":"04:19.535","Text":"we can plug this C back into here,"},{"Start":"04:19.535 ","End":"04:24.845","Text":"which is our total cost and then we can finally get without any constants,"},{"Start":"04:24.845 ","End":"04:32.925","Text":"that the total cost function of x is 0.02x cubed plus 20,"},{"Start":"04:32.925 ","End":"04:42.160","Text":"x plus 5, and this is our answer. We\u0027re done."}],"ID":8543}],"Thumbnail":null,"ID":65391},{"Name":"Pricing Problems","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"12m 57s","ChapterTopicVideoID":8434,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8434.jpeg","UploadDate":"2017-01-30T04:52:15.5730000","DurationForVideoObject":"PT12M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Let\u0027s read this exercise."},{"Start":"00:03.060 ","End":"00:05.895","Text":"We have a washing machine manufacturer,"},{"Start":"00:05.895 ","End":"00:10.620","Text":"and he sells 500 washing machines a week at a price of"},{"Start":"00:10.620 ","End":"00:18.675","Text":"225 dollars for each machine and each machine has a production cost of 125 dollars."},{"Start":"00:18.675 ","End":"00:25.230","Text":"Now, a market survey shows that if he would lower the price,"},{"Start":"00:25.230 ","End":"00:27.915","Text":"he\u0027d sell more and let\u0027s be more precise."},{"Start":"00:27.915 ","End":"00:31.365","Text":"For each 5 dollar, reduction in price,"},{"Start":"00:31.365 ","End":"00:38.030","Text":"the number of machines sold per week would increase by 50 and so for example,"},{"Start":"00:38.030 ","End":"00:41.210","Text":"if he reduces the price by 10 dollars,"},{"Start":"00:41.210 ","End":"00:45.035","Text":"it would increase by 100 and so on proportionally."},{"Start":"00:45.035 ","End":"00:49.660","Text":"That\u0027s 2 questions, A, what should he charged per machine to maximize his profit?"},{"Start":"00:49.660 ","End":"00:54.500","Text":"In part b, we want to know what his production costs are"},{"Start":"00:54.500 ","End":"01:00.120","Text":"in the case of a and are they necessarily minimal?"},{"Start":"01:00.120 ","End":"01:02.155","Text":"Second half of B."},{"Start":"01:02.155 ","End":"01:08.405","Text":"Let\u0027s start with A. I\u0027ll start by writing something very general,"},{"Start":"01:08.405 ","End":"01:11.375","Text":"the formula for profit in words,"},{"Start":"01:11.375 ","End":"01:14.145","Text":"it\u0027s the total revenues,"},{"Start":"01:14.145 ","End":"01:17.910","Text":"or just revenue minus the costs in this case,"},{"Start":"01:17.910 ","End":"01:23.030","Text":"it\u0027s the production costs and then we have to figure out what is revenue,"},{"Start":"01:23.030 ","End":"01:27.830","Text":"what is cost and we even don\u0027t have a name of"},{"Start":"01:27.830 ","End":"01:33.560","Text":"a variable yet so let\u0027s judiciously choose what our variable is going to be."},{"Start":"01:33.560 ","End":"01:35.210","Text":"After thinking about it,"},{"Start":"01:35.210 ","End":"01:38.840","Text":"it looks like this word each holds the key,"},{"Start":"01:38.840 ","End":"01:43.265","Text":"each 5 dollar reduction in price could be"},{"Start":"01:43.265 ","End":"01:48.690","Text":"a unit we\u0027ll just count how many price reductions if it\u0027s 1,"},{"Start":"01:48.690 ","End":"01:49.875","Text":"it\u0027s 5 dollar reduction,"},{"Start":"01:49.875 ","End":"01:52.440","Text":"2 is 10 dollars ,3 is 15 dollars,"},{"Start":"01:52.440 ","End":"01:55.155","Text":"and so on, and I\u0027ll call that x."},{"Start":"01:55.155 ","End":"02:04.170","Text":"Let x equal the number of 5 dollar price reductions."},{"Start":"02:04.170 ","End":"02:12.109","Text":"Now, let\u0027s see if we can build up the revenue and the cost and then the profit."},{"Start":"02:12.109 ","End":"02:15.035","Text":"Let\u0027s start with the revenue,"},{"Start":"02:15.035 ","End":"02:17.475","Text":"revenue is sometimes written,"},{"Start":"02:17.475 ","End":"02:20.520","Text":"R, sometimes TR, I\u0027ll just write it as R,"},{"Start":"02:20.520 ","End":"02:27.845","Text":"R for revenue as a function of this x is equal to,"},{"Start":"02:27.845 ","End":"02:29.735","Text":"well, product of 2 things."},{"Start":"02:29.735 ","End":"02:36.645","Text":"It\u0027s the number of machines sold and let me just write that number of machines sold"},{"Start":"02:36.645 ","End":"02:45.579","Text":"times the price per machine and I can write each of these in terms of x."},{"Start":"02:45.579 ","End":"02:49.640","Text":"The price per machine I\u0027ll start with this factor is"},{"Start":"02:49.640 ","End":"02:58.630","Text":"the 225 dollars minus x jumps of 5 dollars so it\u0027s 5x."},{"Start":"02:59.810 ","End":"03:04.160","Text":"Then the number of machines sold is the original"},{"Start":"03:04.160 ","End":"03:12.665","Text":"500 plus x times an increase of 50 so it\u0027s plus 50x."},{"Start":"03:12.665 ","End":"03:14.510","Text":"This is the number of machines,"},{"Start":"03:14.510 ","End":"03:17.060","Text":"this is the price per machine in dollars and this is"},{"Start":"03:17.060 ","End":"03:22.670","Text":"the product and this is equal to this times this,"},{"Start":"03:22.670 ","End":"03:23.780","Text":"and then this times this,"},{"Start":"03:23.780 ","End":"03:26.590","Text":"this times this, this times this."},{"Start":"03:26.590 ","End":"03:32.110","Text":"Let\u0027s see this times this is 112,500."},{"Start":"03:34.240 ","End":"03:38.090","Text":"I\u0027ll take this with this next because it\u0027s similarities,"},{"Start":"03:38.090 ","End":"03:44.850","Text":"it\u0027s just 50 instead of 500 so it\u0027ll be 1, 1, 2, 5,"},{"Start":"03:44.850 ","End":"03:48.495","Text":"0, 1, 0 less, but with an x,"},{"Start":"03:48.495 ","End":"03:55.840","Text":"then this with this minus 5 times 500 is 2500x."},{"Start":"03:56.510 ","End":"04:01.895","Text":"Then finally, this with this will give us"},{"Start":"04:01.895 ","End":"04:08.850","Text":"minus 250x squared so that\u0027s the revenue part."},{"Start":"04:08.850 ","End":"04:12.015","Text":"Now, let\u0027s do the cost part."},{"Start":"04:12.015 ","End":"04:16.565","Text":"The cost, which also depends on x, well,"},{"Start":"04:16.565 ","End":"04:21.950","Text":"it\u0027s just a 125 dollars for each machine and we\u0027ve got the number of machines,"},{"Start":"04:21.950 ","End":"04:30.890","Text":"which is this bit here so it\u0027s 500 plus 50x number of machines"},{"Start":"04:30.890 ","End":"04:40.840","Text":"times 125 and let\u0027s see what this equals, 500 times 125."},{"Start":"04:40.840 ","End":"04:46.805","Text":"Let\u0027s see 62,500 for this times this,"},{"Start":"04:46.805 ","End":"04:48.230","Text":"then times 50, it\u0027ll be 1,"},{"Start":"04:48.230 ","End":"04:53.580","Text":"0 less, it\u0027ll be 6250x."},{"Start":"04:56.180 ","End":"05:04.880","Text":"The profit that we usually use the letter Pi because p is reserved for price."},{"Start":"05:04.880 ","End":"05:09.650","Text":"Now I can write an expression for the profit and say"},{"Start":"05:09.650 ","End":"05:14.940","Text":"that the profit as a function of x is revenue minus cost."},{"Start":"05:14.940 ","End":"05:19.570","Text":"I need to take this minus this,"},{"Start":"05:19.570 ","End":"05:23.540","Text":"do a subtraction, perhaps if I highlight them, it\u0027ll be easier."},{"Start":"05:23.540 ","End":"05:28.364","Text":"This the revenue and that\u0027s the cost,"},{"Start":"05:28.364 ","End":"05:31.580","Text":"I\u0027ll subtract then collect like terms at the same time."},{"Start":"05:31.580 ","End":"05:34.145","Text":"Let\u0027s go for the constants first."},{"Start":"05:34.145 ","End":"05:37.920","Text":"Here on this 1 I have just this,"},{"Start":"05:37.920 ","End":"05:43.140","Text":"112,500 less 62,500 so that"},{"Start":"05:43.140 ","End":"05:49.350","Text":"just gives me 50,000."},{"Start":"05:49.350 ","End":"05:55.500","Text":"Now this minus this is 5000x minus 2500x,"},{"Start":"05:55.500 ","End":"05:57.870","Text":"should just leave me with"},{"Start":"05:57.870 ","End":"06:08.225","Text":"2500x and we still have another minus 250x squared."},{"Start":"06:08.225 ","End":"06:11.210","Text":"That\u0027s the profit function."},{"Start":"06:11.210 ","End":"06:20.460","Text":"What we want to do to maximize it is, well, there\u0027s 2 ways."},{"Start":"06:20.460 ","End":"06:22.515","Text":"You can either use calculus,"},{"Start":"06:22.515 ","End":"06:23.860","Text":"but I\u0027d rather not,"},{"Start":"06:23.860 ","End":"06:25.840","Text":"because some people haven\u0027t studied it or we could use"},{"Start":"06:25.840 ","End":"06:28.445","Text":"the properties of a quadratic equation."},{"Start":"06:28.445 ","End":"06:31.300","Text":"I\u0027d like to give you a quick reminder of"},{"Start":"06:31.300 ","End":"06:37.225","Text":"quadratic functions or polynomials and parabolas."},{"Start":"06:37.225 ","End":"06:44.350","Text":"In algebra, when you have y equals ax squared plus bx plus c, where a, b,"},{"Start":"06:44.350 ","End":"06:49.920","Text":"and c are constants then depending on the sign of a,"},{"Start":"06:49.920 ","End":"06:51.740","Text":"we have 1 of 2 cases."},{"Start":"06:51.740 ","End":"06:56.375","Text":"We have that if a is bigger than 0,"},{"Start":"06:56.375 ","End":"07:02.270","Text":"then the parabola faces upwards or opens upwards and"},{"Start":"07:02.270 ","End":"07:09.575","Text":"there\u0027s also a vertex here which is the minimum point."},{"Start":"07:09.575 ","End":"07:12.875","Text":"If a is negative,"},{"Start":"07:12.875 ","End":"07:20.575","Text":"then the parabola faces down or opens down and again it has a vertex."},{"Start":"07:20.575 ","End":"07:22.010","Text":"I\u0027ll call the vertex v,"},{"Start":"07:22.010 ","End":"07:24.620","Text":"but this case it\u0027s a maximum."},{"Start":"07:24.620 ","End":"07:26.465","Text":"Now, what we have,"},{"Start":"07:26.465 ","End":"07:27.830","Text":"we\u0027ve written it in reverse,"},{"Start":"07:27.830 ","End":"07:30.080","Text":"but if you look at it, this is a,"},{"Start":"07:30.080 ","End":"07:32.380","Text":"which is the minus 250,"},{"Start":"07:32.380 ","End":"07:36.090","Text":"this is b, we don\u0027t actually need c,"},{"Start":"07:36.090 ","End":"07:37.545","Text":"but there it is."},{"Start":"07:37.545 ","End":"07:45.539","Text":"In our case, a is negative so we are in this case,"},{"Start":"07:45.539 ","End":"07:47.730","Text":"but just to make a note,"},{"Start":"07:47.730 ","End":"07:50.125","Text":"this is the case where at, a is negative,"},{"Start":"07:50.125 ","End":"07:54.860","Text":"so we know that there\u0027s a maximum and what is more,"},{"Start":"07:54.860 ","End":"08:02.255","Text":"there\u0027s even a formula that tells us the x coordinate of the vertex."},{"Start":"08:02.255 ","End":"08:04.250","Text":"It works in both cases,"},{"Start":"08:04.250 ","End":"08:12.015","Text":"but we just needed to hear that x is equal to minus b over 2a,"},{"Start":"08:12.015 ","End":"08:14.565","Text":"the x of the vertex."},{"Start":"08:14.565 ","End":"08:16.785","Text":"Back in our case,"},{"Start":"08:16.785 ","End":"08:22.400","Text":"our x is going to equal the minus b over"},{"Start":"08:22.400 ","End":"08:29.090","Text":"2a is minus 2500 over twice,"},{"Start":"08:29.090 ","End":"08:32.240","Text":"a is minus 250"},{"Start":"08:32.240 ","End":"08:41.419","Text":"is 5 so this is the value for x,"},{"Start":"08:41.419 ","End":"08:44.510","Text":"which gives us the maximum profit."},{"Start":"08:44.510 ","End":"08:48.800","Text":"However, the question asks what the new prices,"},{"Start":"08:48.800 ","End":"08:50.510","Text":"I\u0027ll go back and show you."},{"Start":"08:50.510 ","End":"08:54.860","Text":"The question says, how much should he charged per machine?"},{"Start":"08:54.860 ","End":"09:02.420","Text":"The price is given by this expression, 225 minus 5x."},{"Start":"09:02.420 ","End":"09:10.010","Text":"So what I have is that the new price for maximum profit is going to"},{"Start":"09:10.010 ","End":"09:18.720","Text":"be 225 minus 5x or 5x_v,"},{"Start":"09:18.720 ","End":"09:21.960","Text":"which in this case, well, just there,"},{"Start":"09:21.960 ","End":"09:23.980","Text":"write instead of x_v,"},{"Start":"09:24.200 ","End":"09:29.760","Text":"I\u0027ll just write 5 and perhaps I\u0027ll just highlight."},{"Start":"09:29.760 ","End":"09:33.680","Text":"This is the expression I\u0027m using this is the price,"},{"Start":"09:33.680 ","End":"09:39.185","Text":"I\u0027m going to write a p over it and this is the x,"},{"Start":"09:39.185 ","End":"09:44.705","Text":"which here is equal to 5 that\u0027s this x here."},{"Start":"09:44.705 ","End":"09:46.700","Text":"Now what does this equal?"},{"Start":"09:46.700 ","End":"09:51.980","Text":"200 and remember that it is in dollars."},{"Start":"09:51.980 ","End":"09:54.500","Text":"Let\u0027s see what else was asked."},{"Start":"09:54.500 ","End":"09:57.110","Text":"What are the production costs?"},{"Start":"09:57.110 ","End":"10:05.555","Text":"The production cost is given here as this c of x of what we want,"},{"Start":"10:05.555 ","End":"10:08.215","Text":"but we know that x is 5,"},{"Start":"10:08.215 ","End":"10:10.555","Text":"we can take it from here,"},{"Start":"10:10.555 ","End":"10:20.835","Text":"I just have to put in x equals 5 here so it\u0027s 62,500 plus"},{"Start":"10:20.835 ","End":"10:30.060","Text":"6,250 times 5 and that is"},{"Start":"10:30.060 ","End":"10:32.290","Text":"equal to"},{"Start":"10:35.660 ","End":"10:44.675","Text":"93,750 dollars."},{"Start":"10:44.675 ","End":"10:50.825","Text":"I guess I should have said that this is the beginning of Part B here."},{"Start":"10:50.825 ","End":"10:54.110","Text":"The answer to Part A was 200 dollars,"},{"Start":"10:54.110 ","End":"10:58.550","Text":"the price per machine at a given maximum profit,"},{"Start":"10:58.550 ","End":"11:00.185","Text":"and at this point,"},{"Start":"11:00.185 ","End":"11:04.820","Text":"the production costs, so it\u0027s costs this."},{"Start":"11:04.820 ","End":"11:06.695","Text":"There is another way of getting to this,"},{"Start":"11:06.695 ","End":"11:10.040","Text":"I\u0027ll show you how. It wouldn\u0027t hurt."},{"Start":"11:10.040 ","End":"11:15.305","Text":"If we look over here, the number of machines sold is 500 plus 50x."},{"Start":"11:15.305 ","End":"11:18.020","Text":"So if x is 5,"},{"Start":"11:18.020 ","End":"11:23.105","Text":"then the number of machines sold would"},{"Start":"11:23.105 ","End":"11:29.010","Text":"be 500 plus 50 times 5,"},{"Start":"11:29.010 ","End":"11:35.010","Text":"50 times 5 is 250 plus 500 is 750 machines."},{"Start":"11:35.010 ","End":"11:43.250","Text":"Then I could have multiplied this by the 125 dollars production cost per machine."},{"Start":"11:43.250 ","End":"11:52.390","Text":"If you do this, you will get also 93,750."},{"Start":"11:52.390 ","End":"11:57.470","Text":"Just sometimes I like to show an alternative way and we\u0027re not quite done."},{"Start":"11:57.470 ","End":"12:01.295","Text":"There was 1 extra question there let\u0027s see,"},{"Start":"12:01.295 ","End":"12:05.005","Text":"production costs are they necessarily minimal?"},{"Start":"12:05.005 ","End":"12:08.644","Text":"And the answer is a resounding no."},{"Start":"12:08.644 ","End":"12:12.320","Text":"Just because this maximizes profit,"},{"Start":"12:12.320 ","End":"12:15.440","Text":"doesn\u0027t mean that the costs are least."},{"Start":"12:15.440 ","End":"12:20.440","Text":"For example since the production costs"},{"Start":"12:20.440 ","End":"12:27.615","Text":"basically is 125 times the number of machines,"},{"Start":"12:27.615 ","End":"12:33.875","Text":"all I have to do is produce less machines and I\u0027ll get lower production costs."},{"Start":"12:33.875 ","End":"12:37.475","Text":"So not minimal."},{"Start":"12:37.475 ","End":"12:39.320","Text":"For example,"},{"Start":"12:39.320 ","End":"12:42.290","Text":"if we took x equals 0,"},{"Start":"12:42.290 ","End":"12:47.925","Text":"i.e to produce 500 machines,"},{"Start":"12:47.925 ","End":"12:54.604","Text":"certainly, 500 machines have less production cost than 750 machines."},{"Start":"12:54.604 ","End":"12:58.380","Text":"That answers b and we\u0027re finally done."}],"ID":8621},{"Watched":false,"Name":"Exercise 2","Duration":"9m 35s","ChapterTopicVideoID":8426,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8426.jpeg","UploadDate":"2017-01-30T04:40:57.9570000","DurationForVideoObject":"PT9M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.890","Text":"In this exercise, we have a cell phone company, international."},{"Start":"00:07.890 ","End":"00:18.975","Text":"They offer packages where you talk 200 minutes international overseas a month for $100."},{"Start":"00:18.975 ","End":"00:20.955","Text":"But it\u0027s pro-rated."},{"Start":"00:20.955 ","End":"00:23.100","Text":"If you go over the 200,"},{"Start":"00:23.100 ","End":"00:25.800","Text":"they take it at $0.50 a minute."},{"Start":"00:25.800 ","End":"00:32.475","Text":"The market survey showed that for each $2 reduction in rate,"},{"Start":"00:32.475 ","End":"00:34.680","Text":"meaning instead of $100,"},{"Start":"00:34.680 ","End":"00:38.355","Text":"it would be 98 or 96 or 94 and so on,"},{"Start":"00:38.355 ","End":"00:40.650","Text":"for each $2 reduction,"},{"Start":"00:40.650 ","End":"00:45.705","Text":"the customers will talk 10 more minutes a month."},{"Start":"00:45.705 ","End":"00:48.620","Text":"In the first example, you reduce it by $2,"},{"Start":"00:48.620 ","End":"00:51.605","Text":"you increase the number of minutes by 10."},{"Start":"00:51.605 ","End":"00:54.275","Text":"What we\u0027re asked to do is, a,"},{"Start":"00:54.275 ","End":"00:59.050","Text":"to find the package price which will maximize the revenue."},{"Start":"00:59.050 ","End":"01:02.795","Text":"We\u0027re not talking about profit, talking about revenue."},{"Start":"01:02.795 ","End":"01:06.755","Text":"The second part is once we found this price,"},{"Start":"01:06.755 ","End":"01:14.630","Text":"then to also find the corresponding number of minutes that the customers will talk."},{"Start":"01:14.630 ","End":"01:17.420","Text":"If it was 98, it would be 210."},{"Start":"01:17.420 ","End":"01:21.660","Text":"If it was 96, it would be 220 minutes and so on."},{"Start":"01:21.670 ","End":"01:25.220","Text":"What I\u0027ll do first is choose a variable."},{"Start":"01:25.220 ","End":"01:27.845","Text":"It\u0027s not clear what the variable should be."},{"Start":"01:27.845 ","End":"01:29.900","Text":"But if you think about it,"},{"Start":"01:29.900 ","End":"01:37.730","Text":"each unit is a $2 reduction in the rate and then the corresponding increase in minutes."},{"Start":"01:37.730 ","End":"01:39.440","Text":"Each one of those,"},{"Start":"01:39.440 ","End":"01:41.255","Text":"like a pulse or a tick."},{"Start":"01:41.255 ","End":"01:43.730","Text":"Let\u0027s take that as our unit."},{"Start":"01:43.730 ","End":"01:46.850","Text":"Let x be, how should I describe it?"},{"Start":"01:46.850 ","End":"01:54.915","Text":"The number of $2 discounts or reductions."},{"Start":"01:54.915 ","End":"01:56.850","Text":"We are talking about revenue."},{"Start":"01:56.850 ","End":"01:59.975","Text":"Let\u0027s get an expression for revenue."},{"Start":"01:59.975 ","End":"02:02.705","Text":"Let\u0027s figure out the package price,"},{"Start":"02:02.705 ","End":"02:06.185","Text":"I\u0027ll call that p, which depends on x."},{"Start":"02:06.185 ","End":"02:15.425","Text":"It\u0027s just the $100 minus x times the $2 reduction."},{"Start":"02:15.425 ","End":"02:18.830","Text":"Do not bother with the dollar symbol."},{"Start":"02:18.830 ","End":"02:22.785","Text":"Let\u0027s just write this as 2x."},{"Start":"02:22.785 ","End":"02:24.375","Text":"We know it\u0027s in dollars."},{"Start":"02:24.375 ","End":"02:27.810","Text":"Now, what I want is the price per minute."},{"Start":"02:27.810 ","End":"02:30.605","Text":"The packages are still for 200 minutes."},{"Start":"02:30.605 ","End":"02:36.075","Text":"The price per minute would be 100 minus 2x,"},{"Start":"02:36.075 ","End":"02:41.220","Text":"which is the package price, divided by 200."},{"Start":"02:41.220 ","End":"02:43.445","Text":"That\u0027s what it\u0027s going to be pro-rated at."},{"Start":"02:43.445 ","End":"02:45.410","Text":"Let\u0027s just simplify this,"},{"Start":"02:45.410 ","End":"02:49.715","Text":"100 over 200 is 1.5."},{"Start":"02:49.715 ","End":"02:55.395","Text":"Then we have the 2 over 2 cancels, x over 100."},{"Start":"02:55.395 ","End":"02:58.180","Text":"That\u0027s the price per minute."},{"Start":"02:59.180 ","End":"03:06.410","Text":"I want to figure out the number of minutes that the customers will talk."},{"Start":"03:06.410 ","End":"03:10.100","Text":"Number of minutes, this will equal,"},{"Start":"03:10.100 ","End":"03:20.319","Text":"we have the 200 minutes original and 10 extra minutes reduction"},{"Start":"03:20.319 ","End":"03:21.670","Text":"for each x."},{"Start":"03:21.670 ","End":"03:26.690","Text":"What it is, is plus 10 times x."},{"Start":"03:26.690 ","End":"03:31.810","Text":"Now we can multiply price per minute times the number of minutes."},{"Start":"03:31.810 ","End":"03:35.515","Text":"If I take this and this and multiply them,"},{"Start":"03:35.515 ","End":"03:40.710","Text":"I\u0027ll get the revenue for that customer."},{"Start":"03:40.710 ","End":"03:42.865","Text":"We\u0027ll call it R,"},{"Start":"03:42.865 ","End":"03:46.510","Text":"revenue, which also depends on x,"},{"Start":"03:46.510 ","End":"03:52.560","Text":"is going to be 1.5 minus x over 100,"},{"Start":"03:52.560 ","End":"03:54.390","Text":"the price per minute,"},{"Start":"03:54.390 ","End":"03:57.134","Text":"times number of minutes,"},{"Start":"03:57.134 ","End":"04:02.220","Text":"is 200 plus 10x from here."},{"Start":"04:02.220 ","End":"04:06.000","Text":"This is the thing that we want to maximize."},{"Start":"04:06.000 ","End":"04:08.850","Text":"Let\u0027s multiply this out."},{"Start":"04:08.850 ","End":"04:13.080","Text":"I\u0027ll take the 1.5 first with each of these,"},{"Start":"04:13.080 ","End":"04:16.365","Text":"1/2 times 200 is 100,"},{"Start":"04:16.365 ","End":"04:19.695","Text":"1/2 times 10x is 5x."},{"Start":"04:19.695 ","End":"04:23.940","Text":"Now, I\u0027ll take this minus x over 100 with each of these."},{"Start":"04:23.940 ","End":"04:30.135","Text":"We\u0027re going to get minus x over 100 times 200 is 2x,"},{"Start":"04:30.135 ","End":"04:35.700","Text":"and x over 100 times 10x,"},{"Start":"04:35.700 ","End":"04:38.730","Text":"it\u0027ll be with a minus, 10 over 100,"},{"Start":"04:38.730 ","End":"04:40.660","Text":"it will be 1/10,"},{"Start":"04:40.660 ","End":"04:45.000","Text":"and the x with x will be x squared."},{"Start":"04:45.000 ","End":"04:48.590","Text":"If I rewrite it and also change the order,"},{"Start":"04:48.590 ","End":"04:53.850","Text":"it\u0027s minus 1/10x squared plus 5x minus"},{"Start":"04:53.850 ","End":"04:59.924","Text":"2x is plus 3x and then plus 100."},{"Start":"04:59.924 ","End":"05:05.250","Text":"I\u0027ll repeat the R of x is equal to this."},{"Start":"05:05.250 ","End":"05:07.890","Text":"This is a quadratic."},{"Start":"05:07.890 ","End":"05:12.470","Text":"I\u0027m going to remind you about quadratics in case you\u0027ve forgotten."},{"Start":"05:12.470 ","End":"05:17.885","Text":"We talked about y equals what we have as R,"},{"Start":"05:17.885 ","End":"05:22.850","Text":"ax squared plus bx plus c quadratic function."},{"Start":"05:22.850 ","End":"05:25.100","Text":"We know the shape is a parabola."},{"Start":"05:25.100 ","End":"05:29.315","Text":"If a is bigger than 0,"},{"Start":"05:29.315 ","End":"05:35.795","Text":"then the parabola faces up or opens upwards,"},{"Start":"05:35.795 ","End":"05:39.465","Text":"and there\u0027s a vertex at the tip,"},{"Start":"05:39.465 ","End":"05:41.940","Text":"which is a minimum, I\u0027ll call that point v,"},{"Start":"05:41.940 ","End":"05:44.640","Text":"and this is a minimum point."},{"Start":"05:44.640 ","End":"05:48.389","Text":"If a is less than 0,"},{"Start":"05:48.389 ","End":"05:55.445","Text":"then we have an upside-down parabola or facing down, opening downwards."},{"Start":"05:55.445 ","End":"05:59.000","Text":"It also has a vertex v, also."},{"Start":"05:59.000 ","End":"06:03.130","Text":"In this case, this is a maximum."},{"Start":"06:03.130 ","End":"06:05.130","Text":"In each of these cases,"},{"Start":"06:05.130 ","End":"06:08.625","Text":"it doesn\u0027t matter the x of the vertex,"},{"Start":"06:08.625 ","End":"06:11.865","Text":"I\u0027ll call it x_v,"},{"Start":"06:11.865 ","End":"06:17.160","Text":"is equal to minus b over 2a."},{"Start":"06:17.160 ","End":"06:20.690","Text":"In our case, this is the a,"},{"Start":"06:20.690 ","End":"06:22.685","Text":"notice that it\u0027s negative."},{"Start":"06:22.685 ","End":"06:27.930","Text":"We really do have a maximum for the revenue."},{"Start":"06:27.930 ","End":"06:32.885","Text":"That\u0027s a, and this plus 3 is"},{"Start":"06:32.885 ","End":"06:39.620","Text":"b plus 100 is c. A,"},{"Start":"06:39.620 ","End":"06:45.170","Text":"b, and c. What we can do is we can compute the x for"},{"Start":"06:45.170 ","End":"06:53.195","Text":"the vertex and I\u0027ll call it x max because the vertex is the x that gives us the maximum,"},{"Start":"06:53.195 ","End":"06:56.230","Text":"is equal to minus b over 2a,"},{"Start":"06:56.230 ","End":"07:01.320","Text":"which is minus 3 over twice a,"},{"Start":"07:01.320 ","End":"07:07.725","Text":"is minus 1/10, and the minuses cancel."},{"Start":"07:07.725 ","End":"07:10.395","Text":"This just gives us,"},{"Start":"07:10.395 ","End":"07:13.380","Text":"It\u0027s like 3 times 10 over 2,"},{"Start":"07:13.380 ","End":"07:20.655","Text":"which is 15, the number of $2 reductions."},{"Start":"07:20.655 ","End":"07:25.485","Text":"We don\u0027t want x, we want the package price."},{"Start":"07:25.485 ","End":"07:30.260","Text":"The package price is given by p. I want p,"},{"Start":"07:30.260 ","End":"07:33.305","Text":"not of x, but p of 15."},{"Start":"07:33.305 ","End":"07:38.610","Text":"From this formula, it\u0027s 100 minus twice 15,"},{"Start":"07:38.610 ","End":"07:43.455","Text":"and so this is equal to 100 minus 30"},{"Start":"07:43.455 ","End":"07:50.230","Text":"is $70 for the package."},{"Start":"07:52.580 ","End":"07:55.850","Text":"We also need the number of minutes,"},{"Start":"07:55.850 ","End":"07:57.904","Text":"which in this case,"},{"Start":"07:57.904 ","End":"08:03.545","Text":"number of minutes is equal to 200 plus"},{"Start":"08:03.545 ","End":"08:11.670","Text":"10x is 200 plus 10 times 15."},{"Start":"08:12.320 ","End":"08:15.420","Text":"That equals, let\u0027s see,"},{"Start":"08:15.420 ","End":"08:22.200","Text":"10 times 15 is 150 plus 200, that is 350."},{"Start":"08:22.200 ","End":"08:25.575","Text":"I\u0027ll just write it as minutes."},{"Start":"08:25.575 ","End":"08:29.810","Text":"I can highlight the solution that a,"},{"Start":"08:29.810 ","End":"08:33.350","Text":"the package price is $70,"},{"Start":"08:33.350 ","End":"08:39.060","Text":"and the number of minutes is 350."},{"Start":"08:39.070 ","End":"08:45.635","Text":"Now we\u0027re done, but I\u0027m just curious as to how much the revenue did increase by."},{"Start":"08:45.635 ","End":"08:52.080","Text":"Originally, revenue was $100 because that\u0027s the package price,"},{"Start":"08:52.080 ","End":"08:54.549","Text":"and it was 200 minutes."},{"Start":"08:54.560 ","End":"08:59.430","Text":"But now if we have $70 package,"},{"Start":"08:59.430 ","End":"09:06.795","Text":"the price per minute is only 35 cents, 0.35."},{"Start":"09:06.795 ","End":"09:10.030","Text":"On the other hand, the customer speaks longer,"},{"Start":"09:10.030 ","End":"09:24.395","Text":"350 times $0.35 is equal to $122.50,"},{"Start":"09:24.395 ","End":"09:27.515","Text":"which is certainly more than $100."},{"Start":"09:27.515 ","End":"09:29.480","Text":"The revenue is definitely increasing."},{"Start":"09:29.480 ","End":"09:31.850","Text":"This is the maximum revenue,"},{"Start":"09:31.850 ","End":"09:36.810","Text":"but we weren\u0027t asked for that. Now I\u0027m done."}],"ID":8622},{"Watched":false,"Name":"Exercise 3","Duration":"13m 4s","ChapterTopicVideoID":8427,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8427.jpeg","UploadDate":"2017-01-30T04:42:41.0830000","DurationForVideoObject":"PT13M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.970","Text":"In this exercise, we have an artist and he creates pieces of jewelry,"},{"Start":"00:05.970 ","End":"00:10.810","Text":"and it costs him 30 dollars each to produce."},{"Start":"00:11.930 ","End":"00:15.585","Text":"We found that when he charges 40 dollars each,"},{"Start":"00:15.585 ","End":"00:18.435","Text":"then he can sell 100 pieces a day."},{"Start":"00:18.435 ","End":"00:23.490","Text":"But for each 2 dollar price hike, he sells less."},{"Start":"00:23.490 ","End":"00:25.200","Text":"Of course, if you charge more, you\u0027ll sell less,"},{"Start":"00:25.200 ","End":"00:27.945","Text":"he sells 4 a day less."},{"Start":"00:27.945 ","End":"00:31.875","Text":"Then there are 3 questions."},{"Start":"00:31.875 ","End":"00:36.444","Text":"How many a day should he make to get maximum profit?"},{"Start":"00:36.444 ","End":"00:40.130","Text":"Then under this number of pieces,"},{"Start":"00:40.130 ","End":"00:43.510","Text":"what price would he get each?"},{"Start":"00:43.510 ","End":"00:46.770","Text":"Again, under the same conditions,"},{"Start":"00:46.770 ","End":"00:51.455","Text":"what\u0027s his total production cost per day for the jewelry?"},{"Start":"00:51.455 ","End":"00:56.340","Text":"What I\u0027d like to do here is think of a good variable."},{"Start":"00:56.340 ","End":"00:59.850","Text":"If you choose the variable right things go easier."},{"Start":"00:59.850 ","End":"01:02.500","Text":"Is the variable going to be the price of the piece?"},{"Start":"01:02.500 ","End":"01:05.915","Text":"Or how many pieces he makes a day?"},{"Start":"01:05.915 ","End":"01:08.335","Text":"I suggest something simpler."},{"Start":"01:08.335 ","End":"01:10.980","Text":"Notice this word each."},{"Start":"01:10.980 ","End":"01:13.370","Text":"We have discrete steps,"},{"Start":"01:13.370 ","End":"01:17.705","Text":"like quantum steps of 2 dollars price increase."},{"Start":"01:17.705 ","End":"01:21.095","Text":"Let\u0027s just count the number of price increases."},{"Start":"01:21.095 ","End":"01:30.500","Text":"I\u0027m going to let x be the number of 2 dollar price increases."},{"Start":"01:30.500 ","End":"01:34.925","Text":"That\u0027s there\u0027ll be my x, and everything else will fall from that."},{"Start":"01:34.925 ","End":"01:37.770","Text":"Let me also say something general."},{"Start":"01:39.710 ","End":"01:45.830","Text":"You\u0027ve seen this before is basically what you take in minus how much you spend."},{"Start":"01:45.830 ","End":"01:53.210","Text":"It\u0027s the revenue from the sale of the jewelry minus its costs,"},{"Start":"01:53.210 ","End":"01:56.660","Text":"in this case, that\u0027s the production costs."},{"Start":"01:56.660 ","End":"02:00.560","Text":"Now, we\u0027re going to have to figure out each 1 of these."},{"Start":"02:00.560 ","End":"02:09.470","Text":"Then the revenue is going to be the number of pieces times the price per piece."},{"Start":"02:09.470 ","End":"02:15.770","Text":"We\u0027ll need the number of pieces and the price per piece."},{"Start":"02:16.900 ","End":"02:20.850","Text":"It starts off with 40 dollars,"},{"Start":"02:20.850 ","End":"02:24.345","Text":"and then he has x price hikes of 2 dollars each."},{"Start":"02:24.345 ","End":"02:27.255","Text":"The price is now 40 plus 2x."},{"Start":"02:27.255 ","End":"02:32.180","Text":"On the other hand, the quantity that goes"},{"Start":"02:32.180 ","End":"02:38.765","Text":"by the 100 pieces minus 4 each time,"},{"Start":"02:38.765 ","End":"02:41.270","Text":"and there\u0027s x times x jumps."},{"Start":"02:41.270 ","End":"02:46.150","Text":"It\u0027s 100 minus 4x."},{"Start":"02:46.380 ","End":"02:50.170","Text":"Now the revenue is this times this."},{"Start":"02:50.170 ","End":"02:54.250","Text":"If I multiply, I get that"},{"Start":"02:54.250 ","End":"03:01.920","Text":"the revenue is equal to just this times this."},{"Start":"03:01.920 ","End":"03:04.230","Text":"It doesn\u0027t matter in which order,"},{"Start":"03:04.230 ","End":"03:09.015","Text":"40 plus 2x each,"},{"Start":"03:09.015 ","End":"03:12.405","Text":"and he sells 100 minus 4x,"},{"Start":"03:12.405 ","End":"03:15.840","Text":"so the product is the revenue in dollars."},{"Start":"03:15.840 ","End":"03:18.390","Text":"That takes care of revenue."},{"Start":"03:18.390 ","End":"03:21.160","Text":"Now, what about the cost?"},{"Start":"03:21.870 ","End":"03:26.710","Text":"I should have maybe said total revenue and total cost."},{"Start":"03:26.710 ","End":"03:29.710","Text":"This is the total revenue."},{"Start":"03:29.710 ","End":"03:32.380","Text":"These are all total costs."},{"Start":"03:32.380 ","End":"03:35.030","Text":"Total revenue, total cost."},{"Start":"03:35.100 ","End":"03:37.975","Text":"That\u0027s his production costs."},{"Start":"03:37.975 ","End":"03:40.960","Text":"Where\u0027s that? That\u0027s 30 dollars each."},{"Start":"03:40.960 ","End":"03:49.345","Text":"It\u0027s 30 dollars times the number of pieces itself which is the quantity,"},{"Start":"03:49.345 ","End":"03:53.570","Text":"which is 100 minus 4x."},{"Start":"03:53.690 ","End":"04:00.010","Text":"Now that we have the cost and we have the revenue,"},{"Start":"04:00.010 ","End":"04:02.545","Text":"we can substitute them in here."},{"Start":"04:02.545 ","End":"04:05.330","Text":"Perhaps I should use a bit of color."},{"Start":"04:05.330 ","End":"04:10.500","Text":"The revenue is over here."},{"Start":"04:10.500 ","End":"04:14.190","Text":"Its revenue is this,"},{"Start":"04:14.190 ","End":"04:22.600","Text":"and the cost which I need over here is this."},{"Start":"04:22.820 ","End":"04:26.825","Text":"Then what l want is the profit."},{"Start":"04:26.825 ","End":"04:29.510","Text":"I\u0027ll just highlight it for now."},{"Start":"04:29.510 ","End":"04:35.035","Text":"The quick copy-paste, will equal this,"},{"Start":"04:35.035 ","End":"04:37.605","Text":"which I\u0027ll copy from here,"},{"Start":"04:37.605 ","End":"04:39.360","Text":"and then I\u0027ll subtract the other."},{"Start":"04:39.360 ","End":"04:44.610","Text":"I\u0027ve got 40 plus 2x times"},{"Start":"04:44.610 ","End":"04:50.925","Text":"100 minus 4x minus the cost which is from here,"},{"Start":"04:50.925 ","End":"04:57.970","Text":"30 times 100 minus 4x."},{"Start":"04:58.120 ","End":"05:04.020","Text":"Now notice that these share a common factor."},{"Start":"05:05.060 ","End":"05:12.920","Text":"Therefore I can take outside the brackets the 100 minus"},{"Start":"05:12.920 ","End":"05:18.030","Text":"4x and say that it\u0027s"},{"Start":"05:18.030 ","End":"05:24.760","Text":"equal to 40 plus 2x minus 30."},{"Start":"05:28.360 ","End":"05:32.015","Text":"There is a shortcut way we could have got here."},{"Start":"05:32.015 ","End":"05:33.950","Text":"We could have said, okay,"},{"Start":"05:33.950 ","End":"05:36.185","Text":"this is the quantity,"},{"Start":"05:36.185 ","End":"05:43.700","Text":"and what he takes in the profit on each piece is what he sells it for,"},{"Start":"05:43.700 ","End":"05:50.075","Text":"which is 40 plus 2x minus his production cost of 30."},{"Start":"05:50.075 ","End":"05:57.110","Text":"This is like what he sells it for minus the production cost."},{"Start":"05:57.110 ","End":"06:00.110","Text":"This is the number of units."},{"Start":"06:00.110 ","End":"06:02.780","Text":"This is like quantity Q,"},{"Start":"06:02.780 ","End":"06:07.115","Text":"this is the price per unit,"},{"Start":"06:07.115 ","End":"06:13.610","Text":"and this is the production cost per unit of the total cost."},{"Start":"06:13.610 ","End":"06:16.790","Text":"We could have gotten to this right away."},{"Start":"06:16.790 ","End":"06:19.535","Text":"It doesn\u0027t matter as long as we\u0027re getting the same thing."},{"Start":"06:19.535 ","End":"06:21.965","Text":"Now I\u0027ll do the simplification,"},{"Start":"06:21.965 ","End":"06:24.155","Text":"and I\u0027ll give it a name."},{"Start":"06:24.155 ","End":"06:29.670","Text":"I use the letter Pi for profit, and a Greek letter."},{"Start":"06:29.670 ","End":"06:31.230","Text":"It\u0027s the Greek P because usually,"},{"Start":"06:31.230 ","End":"06:35.110","Text":"P is price, so Pi is profit."},{"Start":"06:35.300 ","End":"06:42.045","Text":"That is going to equal 100 minus 4x."},{"Start":"06:42.045 ","End":"06:47.980","Text":"Then this is 10 plus 2x."},{"Start":"06:48.590 ","End":"06:51.330","Text":"Let\u0027s just do some algebra now."},{"Start":"06:51.330 ","End":"06:56.210","Text":"Expand and see what we get. You know what?"},{"Start":"06:56.210 ","End":"07:00.950","Text":"I\u0027m going to choose them in the order that I want them in decreasing powers"},{"Start":"07:00.950 ","End":"07:06.125","Text":"of x. X squared times comes from this times this."},{"Start":"07:06.125 ","End":"07:09.750","Text":"I\u0027ve got minus 8x squared."},{"Start":"07:09.750 ","End":"07:15.470","Text":"Then xs can come from this times this is minus 40x,"},{"Start":"07:15.470 ","End":"07:19.630","Text":"and this times this is plus 200x,"},{"Start":"07:19.630 ","End":"07:23.290","Text":"so altogether I\u0027m plus 106x."},{"Start":"07:23.420 ","End":"07:29.430","Text":"Then this with this gives me 100 times 10 is 1,000."},{"Start":"07:29.430 ","End":"07:33.695","Text":"Now I\u0027ve got the profit function in terms of x,"},{"Start":"07:33.695 ","End":"07:37.695","Text":"which is the number of 2 dollar hikes,"},{"Start":"07:37.695 ","End":"07:44.560","Text":"and just find which x makes this maximum."},{"Start":"07:44.600 ","End":"07:50.179","Text":"It\u0027s possible to solve this problem with calculus and derivatives,"},{"Start":"07:50.179 ","End":"07:52.070","Text":"but I\u0027m not going to."},{"Start":"07:52.070 ","End":"07:54.110","Text":"I\u0027m going to use quadratics and parabolas."},{"Start":"07:54.110 ","End":"07:55.640","Text":"I\u0027ll give you a quick reminder."},{"Start":"07:55.640 ","End":"08:00.060","Text":"If I have y is some quadratic function of x,"},{"Start":"08:00.060 ","End":"08:04.215","Text":"ax squared plus bx plus c,"},{"Start":"08:04.215 ","End":"08:13.055","Text":"just like here, we do if this minus 8 is a and this 160 is b,"},{"Start":"08:13.055 ","End":"08:14.780","Text":"and this 1,000 is c,"},{"Start":"08:14.780 ","End":"08:16.710","Text":"and this is y."},{"Start":"08:16.710 ","End":"08:23.140","Text":"Then there\u0027s 2 parabola depending on whether a is positive or negative."},{"Start":"08:23.140 ","End":"08:27.485","Text":"A is not 0 because then it wouldn\u0027t be quadratic, it would be linear."},{"Start":"08:27.485 ","End":"08:31.805","Text":"If a is bigger than 0 positive,"},{"Start":"08:31.805 ","End":"08:36.150","Text":"then you get a parabola facing upwards,"},{"Start":"08:36.150 ","End":"08:42.070","Text":"and it has a vertex which is in this case the minimum."},{"Start":"08:42.070 ","End":"08:45.210","Text":"This is V, the vertex."},{"Start":"08:45.210 ","End":"08:48.715","Text":"If a is negative,"},{"Start":"08:48.715 ","End":"08:56.270","Text":"then we have an upside-down parabola opens downwards."},{"Start":"08:56.270 ","End":"08:58.310","Text":"Also, has a vertex,"},{"Start":"08:58.310 ","End":"09:03.530","Text":"in this case, the vertex is the maximum."},{"Start":"09:04.980 ","End":"09:10.210","Text":"Now we are in the case where a is negative,"},{"Start":"09:10.210 ","End":"09:13.735","Text":"so that\u0027s good, we\u0027re in the maximum case."},{"Start":"09:13.735 ","End":"09:16.165","Text":"But in both of them,"},{"Start":"09:16.165 ","End":"09:21.610","Text":"the x-value of the vertex,"},{"Start":"09:21.610 ","End":"09:29.610","Text":"let\u0027s call it x sub v is equal to minus b over 2a."},{"Start":"09:30.730 ","End":"09:35.940","Text":"This is in both cases."},{"Start":"09:36.740 ","End":"09:39.434","Text":"Returning to our case,"},{"Start":"09:39.434 ","End":"09:45.810","Text":"our x_v, but I\u0027d prefer to call it x max."},{"Start":"09:45.810 ","End":"09:51.540","Text":"Here the better name because it remind me that it\u0027s x for the maximum."},{"Start":"09:51.540 ","End":"09:55.815","Text":"It\u0027s equal to minus b over 2a."},{"Start":"09:55.815 ","End":"10:02.310","Text":"It\u0027s minus 160, that\u0027s y minus b over twice a,"},{"Start":"10:02.310 ","End":"10:05.050","Text":"which is minus 8."},{"Start":"10:05.630 ","End":"10:12.765","Text":"This is 160 over 16 is just 10,"},{"Start":"10:12.765 ","End":"10:14.460","Text":"the minus is canceled."},{"Start":"10:14.460 ","End":"10:16.755","Text":"We found our x."},{"Start":"10:16.755 ","End":"10:19.055","Text":"But what is this x?"},{"Start":"10:19.055 ","End":"10:21.860","Text":"It\u0027s the number of jumps."},{"Start":"10:21.860 ","End":"10:24.200","Text":"I think it\u0027s just off the board."},{"Start":"10:24.200 ","End":"10:25.355","Text":"There it is."},{"Start":"10:25.355 ","End":"10:29.090","Text":"It\u0027s the number of 2 dollar increases."},{"Start":"10:29.090 ","End":"10:31.880","Text":"Once I have x, I can get the price and"},{"Start":"10:31.880 ","End":"10:37.195","Text":"the quantity which is what I need for parts a and b."},{"Start":"10:37.195 ","End":"10:44.400","Text":"Actually, I\u0027m working on part a and b together. I\u0027ll just erase that."},{"Start":"10:44.400 ","End":"10:47.370","Text":"We\u0027ll work on the whole problem as a whole a, b,"},{"Start":"10:47.370 ","End":"10:54.890","Text":"and c. Let\u0027s see now the price for each,"},{"Start":"10:54.890 ","End":"10:58.045","Text":"which will be the answer to part a,"},{"Start":"10:58.045 ","End":"11:00.010","Text":"and I think I got that backward."},{"Start":"11:00.010 ","End":"11:03.380","Text":"I think the quantity is part a and the price is part b."},{"Start":"11:03.380 ","End":"11:11.280","Text":"The quantity would be 100 minus 4x."},{"Start":"11:11.290 ","End":"11:16.400","Text":"In our case when x is 10 is equal to"},{"Start":"11:16.400 ","End":"11:23.620","Text":"100 minus 4 times 10,"},{"Start":"11:23.620 ","End":"11:27.300","Text":"and 4 times 10 is 40,"},{"Start":"11:27.300 ","End":"11:29.670","Text":"100 minus 40 is 60."},{"Start":"11:29.670 ","End":"11:36.340","Text":"It\u0027s 60 pieces, and that\u0027s the answer to part a."},{"Start":"11:37.220 ","End":"11:45.160","Text":"It was part b that are for the price which is 40 plus 2x."},{"Start":"11:45.160 ","End":"11:49.514","Text":"In our case where x is 10,"},{"Start":"11:49.514 ","End":"11:56.820","Text":"for the maximum is 40 plus 2 times 10."},{"Start":"11:56.820 ","End":"12:02.235","Text":"This is just the 40 dollars and 10 price hikes of 2 dollars,"},{"Start":"12:02.235 ","End":"12:06.910","Text":"10 times 2 is 20 plus 40 is 60."},{"Start":"12:08.390 ","End":"12:12.510","Text":"Here I put dollars and here units."},{"Start":"12:12.510 ","End":"12:16.325","Text":"Both 60. This highlights the answers."},{"Start":"12:16.325 ","End":"12:18.595","Text":"This is the answer to a,"},{"Start":"12:18.595 ","End":"12:23.580","Text":"and 60 pieces sold and part b at 60 dollars each,"},{"Start":"12:23.580 ","End":"12:31.210","Text":"and part c if I remember was the production costs."},{"Start":"12:31.880 ","End":"12:36.995","Text":"If I recall, the production cost was 30 dollars each,"},{"Start":"12:36.995 ","End":"12:42.455","Text":"so I take 60 pieces and I multiply that"},{"Start":"12:42.455 ","End":"12:49.805","Text":"by 30 dollars cost production per piece."},{"Start":"12:49.805 ","End":"12:59.680","Text":"I get that to equal 1,800 dollars."},{"Start":"12:59.680 ","End":"13:05.050","Text":"I\u0027ll just highlight part c and say that we are done."}],"ID":8623},{"Watched":false,"Name":"Exercise 4","Duration":"7m 32s","ChapterTopicVideoID":8428,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8428.jpeg","UploadDate":"2017-01-30T04:43:41.2870000","DurationForVideoObject":"PT7M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.980","Text":"This is an exercise of the kind where we have to maximize profit."},{"Start":"00:04.980 ","End":"00:07.560","Text":"As a company that rents buses,"},{"Start":"00:07.560 ","End":"00:12.070","Text":"coaches we just gave it a fictitious name, Good Trip."},{"Start":"00:12.230 ","End":"00:15.840","Text":"It rents to groups of 30 tourists,"},{"Start":"00:15.840 ","End":"00:18.434","Text":"and it\u0027s a 100 dollars a person."},{"Start":"00:18.434 ","End":"00:22.110","Text":"But for each additional tourist,"},{"Start":"00:22.110 ","End":"00:27.035","Text":"the group gets a price discount of 2 dollars each."},{"Start":"00:27.035 ","End":"00:30.680","Text":"I mean, all the tourists on the bus get a 2 dollar discount,"},{"Start":"00:30.680 ","End":"00:33.200","Text":"not just the additional ones."},{"Start":"00:33.200 ","End":"00:40.055","Text":"The question is, what number of tourists will give the maximum profit?"},{"Start":"00:40.055 ","End":"00:44.450","Text":"Now for that, we also need the costs, not just revenues."},{"Start":"00:44.450 ","End":"00:46.265","Text":"But we assume that costs are fixed,"},{"Start":"00:46.265 ","End":"00:49.865","Text":"and it actually doesn\u0027t matter what those fixed costs are."},{"Start":"00:49.865 ","End":"00:54.230","Text":"But they\u0027re fixed. So how do we go about this?"},{"Start":"00:54.230 ","End":"00:57.680","Text":"What I suggest is choosing a variable"},{"Start":"00:57.680 ","End":"01:01.385","Text":"and a good choice of variable will make the problem easier."},{"Start":"01:01.385 ","End":"01:08.015","Text":"So the natural choice would either be the number of tourists,"},{"Start":"01:08.015 ","End":"01:10.760","Text":"but I think it\u0027ll be easier if we just take"},{"Start":"01:10.760 ","End":"01:13.730","Text":"the number of additional tourists beyond the 30,"},{"Start":"01:13.730 ","End":"01:14.840","Text":"it will come out simpler."},{"Start":"01:14.840 ","End":"01:22.770","Text":"So let me declare that x is the number of additional tourists."},{"Start":"01:22.770 ","End":"01:24.060","Text":"So when I say additional,"},{"Start":"01:24.060 ","End":"01:29.640","Text":"I mean over the 30 original ones."},{"Start":"01:29.640 ","End":"01:32.689","Text":"Now we can compute everything."},{"Start":"01:32.689 ","End":"01:39.510","Text":"In general, profit is the total revenue,"},{"Start":"01:39.510 ","End":"01:41.730","Text":"I\u0027ll just say revenue."},{"Start":"01:41.730 ","End":"01:47.265","Text":"From this, we have to subtract the total costs so let\u0027s say revenue minus cost."},{"Start":"01:47.265 ","End":"01:51.655","Text":"Now, the cost will just be"},{"Start":"01:51.655 ","End":"01:57.640","Text":"a fixed quantity so let\u0027s just call it C. C for constant or C for cost and"},{"Start":"01:57.640 ","End":"02:02.335","Text":"the revenue will equal the number of"},{"Start":"02:02.335 ","End":"02:11.370","Text":"tourists multiplied by the price each."},{"Start":"02:11.370 ","End":"02:15.465","Text":"Now, these 2, I\u0027ll express in terms of x."},{"Start":"02:15.465 ","End":"02:19.060","Text":"The number of tourists is going to be"},{"Start":"02:19.060 ","End":"02:24.430","Text":"just 30 plus x. I think that\u0027s pretty clear by the way we set x up,"},{"Start":"02:24.430 ","End":"02:27.335","Text":"it\u0027s the number of tourists beyond the 30."},{"Start":"02:27.335 ","End":"02:34.725","Text":"Now the price each comes from this discount of 2 dollars each."},{"Start":"02:34.725 ","End":"02:37.530","Text":"We started from 100 dollars."},{"Start":"02:37.530 ","End":"02:43.650","Text":"So it\u0027s a 100 dollars minus x times 2 dollars,"},{"Start":"02:43.650 ","End":"02:46.105","Text":"x times 2 is 2x."},{"Start":"02:46.105 ","End":"02:48.755","Text":"So we have to do is multiply these 2,"},{"Start":"02:48.755 ","End":"02:53.525","Text":"and I\u0027ll call this revenue R, R of x."},{"Start":"02:53.525 ","End":"03:01.745","Text":"The profit will use the letter Pi because the letter p is already reserved for price."},{"Start":"03:01.745 ","End":"03:05.210","Text":"So what I can write is that Pi of x,"},{"Start":"03:05.210 ","End":"03:07.940","Text":"this is Pi of x,"},{"Start":"03:07.940 ","End":"03:10.250","Text":"revenue R of x."},{"Start":"03:10.250 ","End":"03:13.630","Text":"Cost in general, it depends on x,"},{"Start":"03:13.630 ","End":"03:16.060","Text":"but in this case, it\u0027s a constant."},{"Start":"03:16.060 ","End":"03:19.675","Text":"So the revenue, which is"},{"Start":"03:19.675 ","End":"03:27.580","Text":"30 plus x times a 100 minus 2x minus the cost,"},{"Start":"03:27.580 ","End":"03:30.400","Text":"which is that constant fixed cost."},{"Start":"03:30.400 ","End":"03:34.480","Text":"This is our profit function and this is what we have to maximize."},{"Start":"03:34.480 ","End":"03:37.840","Text":"So let\u0027s start by expanding the brackets."},{"Start":"03:37.840 ","End":"03:42.385","Text":"I\u0027m going to choose the powers of x as I want them."},{"Start":"03:42.385 ","End":"03:44.230","Text":"I want the x squared first."},{"Start":"03:44.230 ","End":"03:50.309","Text":"So I can get from x times minus 2x minus 2x squared."},{"Start":"03:50.309 ","End":"03:54.565","Text":"Let\u0027s collect x\u0027s from this and this we have plus a 100x,"},{"Start":"03:54.565 ","End":"03:58.840","Text":"and from this and this minus 60x."},{"Start":"03:58.840 ","End":"04:02.930","Text":"So a 100 minus 60 is 40."},{"Start":"04:02.930 ","End":"04:12.385","Text":"So plus 40x, and then 30 times a 100 is 3,000 minus"},{"Start":"04:12.385 ","End":"04:22.815","Text":"C. What we want to find is the maximum of profit."},{"Start":"04:22.815 ","End":"04:27.265","Text":"Actually, we want the x that gives the maximum."},{"Start":"04:27.265 ","End":"04:33.385","Text":"Now, there\u0027s 2 ways of doing this 1 is with calculus and differentiation,"},{"Start":"04:33.385 ","End":"04:37.525","Text":"and 1 is using the properties of the quadratic function and parabolas."},{"Start":"04:37.525 ","End":"04:40.645","Text":"But I\u0027m not sure if you\u0027ve studied calculus,"},{"Start":"04:40.645 ","End":"04:43.510","Text":"and so I\u0027ll do it without calculus."},{"Start":"04:43.510 ","End":"04:46.990","Text":"I\u0027ll remind you of the quadratic function."},{"Start":"04:46.990 ","End":"04:50.310","Text":"In general, quick algebra review,"},{"Start":"04:50.310 ","End":"04:56.309","Text":"when you have a function y equals ax squared plus"},{"Start":"04:56.309 ","End":"05:04.680","Text":"bx plus C. Assuming it\u0027s not 0,"},{"Start":"05:04.680 ","End":"05:06.880","Text":"otherwise, it\u0027s not quadratic, it\u0027s linear."},{"Start":"05:06.880 ","End":"05:09.640","Text":"Then there\u0027s 2 general cases,"},{"Start":"05:09.640 ","End":"05:12.700","Text":"if a is bigger than 0,"},{"Start":"05:12.700 ","End":"05:19.964","Text":"then the graph looks like a parabola facing upwards, opening upwards."},{"Start":"05:19.964 ","End":"05:23.480","Text":"That\u0027s when a is bigger than 0 and there\u0027s a vertex here,"},{"Start":"05:23.480 ","End":"05:27.705","Text":"and this vertex is the minimum."},{"Start":"05:27.705 ","End":"05:31.205","Text":"When a is negative,"},{"Start":"05:31.205 ","End":"05:35.540","Text":"the parabola opens downwards or faces downwards,"},{"Start":"05:35.540 ","End":"05:38.960","Text":"and it still has a vertex v,"},{"Start":"05:38.960 ","End":"05:42.170","Text":"but this time it\u0027s a maximum,"},{"Start":"05:42.170 ","End":"05:44.790","Text":"and in each case,"},{"Start":"05:48.830 ","End":"05:53.345","Text":"the x value of the vertex,"},{"Start":"05:53.345 ","End":"05:55.295","Text":"just the x coordinate,"},{"Start":"05:55.295 ","End":"05:59.495","Text":"is equal to minus b over 2a."},{"Start":"05:59.495 ","End":"06:01.835","Text":"Now, of course, in our case,"},{"Start":"06:01.835 ","End":"06:08.400","Text":"we have a less than 0 because this is our parabola."},{"Start":"06:08.400 ","End":"06:11.460","Text":"Y is, this is like the y."},{"Start":"06:11.460 ","End":"06:17.470","Text":"This would be a which is negative, 40 is b,"},{"Start":"06:17.470 ","End":"06:26.515","Text":"and this whole thing is little c. So let\u0027s compute the x for the vertex,"},{"Start":"06:26.515 ","End":"06:30.280","Text":"which will actually be the x that gives us the maximum."},{"Start":"06:30.280 ","End":"06:32.340","Text":"I\u0027ll call it x_max."},{"Start":"06:32.340 ","End":"06:34.120","Text":"Like I said a is negative."},{"Start":"06:34.120 ","End":"06:42.910","Text":"So we do have a maximum it\u0027s equal to minus b over 2a in our case is minus 40 over 2,"},{"Start":"06:42.910 ","End":"06:45.780","Text":"a is minus 2,"},{"Start":"06:45.780 ","End":"06:52.815","Text":"and minus 40 over minus 4 is just equal to 10."},{"Start":"06:52.815 ","End":"06:55.110","Text":"But in the question it doesn\u0027t say what is x,"},{"Start":"06:55.110 ","End":"06:57.510","Text":"it said what is the number of passengers,"},{"Start":"06:57.510 ","End":"07:03.870","Text":"10 plus 30 or rather 30 plus 10,"},{"Start":"07:03.870 ","End":"07:07.290","Text":"straightly speaking, which is equal to 40."},{"Start":"07:07.290 ","End":"07:12.410","Text":"That\u0027s the number of passengers for which there is maximum profit."},{"Start":"07:12.410 ","End":"07:17.825","Text":"Because remember we had 30 plus x, here it is,"},{"Start":"07:17.825 ","End":"07:26.010","Text":"30 plus x is the number of tourists, passengers tourists."},{"Start":"07:26.010 ","End":"07:28.925","Text":"Yeah, tourists not passengers."},{"Start":"07:28.925 ","End":"07:33.210","Text":"Anyway, this is the answer and we\u0027re done."}],"ID":8624},{"Watched":false,"Name":"Exercise 5","Duration":"5m 38s","ChapterTopicVideoID":8429,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8429.jpeg","UploadDate":"2017-01-30T04:44:26.0130000","DurationForVideoObject":"PT5M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"In this exercise, we have a cell phone company, cellular services,"},{"Start":"00:06.090 ","End":"00:10.395","Text":"and they charge 10 cents for each SMS sent,"},{"Start":"00:10.395 ","End":"00:13.980","Text":"and the average customer sends 200 a month."},{"Start":"00:13.980 ","End":"00:16.890","Text":"Now, they did some market research and it turns out that"},{"Start":"00:16.890 ","End":"00:20.265","Text":"for every penny raise in the price,"},{"Start":"00:20.265 ","End":"00:25.350","Text":"the customer will send 10 less SMSs."},{"Start":"00:25.350 ","End":"00:32.905","Text":"The question is, what should they charge for an SMS to maximize the revenue?"},{"Start":"00:32.905 ","End":"00:36.810","Text":"Now, what I\u0027d like to start with in the solution is deciding what"},{"Start":"00:36.810 ","End":"00:40.980","Text":"the variable is going to be for an equation,"},{"Start":"00:40.980 ","End":"00:46.665","Text":"and I could let the variable be the new price,"},{"Start":"00:46.665 ","End":"00:52.370","Text":"but I think it\u0027s easier if we just take the number of raises."},{"Start":"00:52.370 ","End":"00:56.660","Text":"I\u0027m going to let x be"},{"Start":"00:56.660 ","End":"01:04.160","Text":"the number of 1 cent raises."},{"Start":"01:04.160 ","End":"01:10.190","Text":"Now, the revenue is going to be equal, first step,"},{"Start":"01:10.190 ","End":"01:14.690","Text":"to the quantity number of SMS,"},{"Start":"01:14.690 ","End":"01:22.035","Text":"I\u0027ll call it quantity times the price."},{"Start":"01:22.035 ","End":"01:26.360","Text":"Now, I just put it a quantity means number of SMSs monthly for"},{"Start":"01:26.360 ","End":"01:31.220","Text":"the average customer and the price is per SMS."},{"Start":"01:31.220 ","End":"01:34.950","Text":"Now, I\u0027ll call this R, well,"},{"Start":"01:34.950 ","End":"01:39.930","Text":"it depends on x so it\u0027s a function R of x equals the quantity"},{"Start":"01:39.930 ","End":"01:46.980","Text":"of SMSs sent by the average customer is going to be not 200,"},{"Start":"01:46.980 ","End":"01:52.815","Text":"but it\u0027s going to drop by 10 for every penny raise."},{"Start":"01:52.815 ","End":"02:03.395","Text":"It\u0027s minus 10x and then times the price starts out at 10 cents,"},{"Start":"02:03.395 ","End":"02:10.065","Text":"and this x raises so it goes up to 10 plus x cents."},{"Start":"02:10.065 ","End":"02:15.495","Text":"I\u0027m working in cents now not in dollars."},{"Start":"02:15.495 ","End":"02:18.990","Text":"Let\u0027s see what this is equal to."},{"Start":"02:18.990 ","End":"02:24.824","Text":"I\u0027ll start with the x squared, minus 10x squared,"},{"Start":"02:24.824 ","End":"02:31.155","Text":"and then x\u0027s I\u0027ll get minus 100 plus 200 is 100x,"},{"Start":"02:31.155 ","End":"02:33.450","Text":"and like we said,"},{"Start":"02:33.450 ","End":"02:39.360","Text":"200 times 10 is 2,000."},{"Start":"02:39.360 ","End":"02:45.055","Text":"The reason I put it in this order is because I want to use the properties of"},{"Start":"02:45.055 ","End":"02:52.970","Text":"the parabola and the quadratic ax squared plus bx plus c,"},{"Start":"02:52.970 ","End":"02:55.050","Text":"instead of using calculus."},{"Start":"02:55.050 ","End":"02:59.200","Text":"It\u0027s possible to do these maximization questions with differentiation in calculus,"},{"Start":"02:59.200 ","End":"03:01.765","Text":"and I\u0027ll assume you maybe"},{"Start":"03:01.765 ","End":"03:07.405","Text":"haven\u0027t studied calculus so we\u0027ll do it using properties of the quadratic."},{"Start":"03:07.405 ","End":"03:10.420","Text":"What we know about such quadratics,"},{"Start":"03:10.420 ","End":"03:12.444","Text":"a, b, and c are constants,"},{"Start":"03:12.444 ","End":"03:16.880","Text":"is that when a is bigger than 0,"},{"Start":"03:16.880 ","End":"03:25.610","Text":"we have a facing up parabola for the graph and it has a vertex which is the minimum."},{"Start":"03:26.300 ","End":"03:30.020","Text":"When a is negative,"},{"Start":"03:30.020 ","End":"03:35.465","Text":"the parabola faces down or opens downwards and it has a vertex,"},{"Start":"03:35.465 ","End":"03:39.350","Text":"but this time the vertex is a maximum."},{"Start":"03:39.350 ","End":"03:43.430","Text":"We don\u0027t have the case a equals 0 because then it\u0027s not a quadratic."},{"Start":"03:43.430 ","End":"03:45.215","Text":"Now, in our case,"},{"Start":"03:45.215 ","End":"03:50.630","Text":"a is minus 10 so already we see that we\u0027re"},{"Start":"03:50.630 ","End":"03:56.414","Text":"in this case and b is 100,"},{"Start":"03:56.414 ","End":"03:59.025","Text":"and c is 2,000."},{"Start":"03:59.025 ","End":"04:06.230","Text":"Now, what I didn\u0027t tell you yet and we need this is that the x coordinate of"},{"Start":"04:06.230 ","End":"04:15.160","Text":"the vertex x_v in both cases is minus b/2a."},{"Start":"04:19.330 ","End":"04:25.400","Text":"In our exercise, this is like the y here, the R of x."},{"Start":"04:25.400 ","End":"04:30.020","Text":"Let\u0027s see which x gives the maximum."},{"Start":"04:30.020 ","End":"04:35.180","Text":"Won\u0027t call it x_v, I\u0027ll call it x_max."},{"Start":"04:35.180 ","End":"04:44.870","Text":"Now, it\u0027s minus b/2a which is minus 100 is my b over twice minus 10."},{"Start":"04:44.870 ","End":"04:47.645","Text":"It\u0027s 100/20,"},{"Start":"04:47.645 ","End":"04:56.085","Text":"the minuses cancel and this is equal to 5."},{"Start":"04:56.085 ","End":"04:57.720","Text":"Now, this is not the answer,"},{"Start":"04:57.720 ","End":"04:58.820","Text":"we have to look what did they ask."},{"Start":"04:58.820 ","End":"05:02.450","Text":"What should the company charge for an SMS?"},{"Start":"05:02.450 ","End":"05:07.160","Text":"That\u0027s the price and that\u0027s the 10 plus x,"},{"Start":"05:07.160 ","End":"05:13.375","Text":"so all I need to do now is to replace x by 5."},{"Start":"05:13.375 ","End":"05:19.070","Text":"If we take this expression and let x equals 5,"},{"Start":"05:19.070 ","End":"05:26.305","Text":"then I have that 10 plus x is equal to 15."},{"Start":"05:26.305 ","End":"05:34.019","Text":"This 15 is already 15 cents and this is the answer to the question,"},{"Start":"05:34.019 ","End":"05:39.670","Text":"that\u0027s the new price for an SMS. We\u0027re done."}],"ID":8625},{"Watched":false,"Name":"Exercise 6","Duration":"7m 35s","ChapterTopicVideoID":8430,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8430.jpeg","UploadDate":"2017-01-30T04:45:26.7170000","DurationForVideoObject":"PT7M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.995","Text":"In this exercise, we have a theater,"},{"Start":"00:03.995 ","End":"00:09.690","Text":"the Rialto, that sells tickets to movies."},{"Start":"00:09.690 ","End":"00:16.710","Text":"They are 9 dollars a ticket and they sell 60 tickets a week at this price."},{"Start":"00:16.710 ","End":"00:20.505","Text":"They researched that if they reduced the price,"},{"Start":"00:20.505 ","End":"00:22.650","Text":"they\u0027ll obviously sell more tickets,"},{"Start":"00:22.650 ","End":"00:28.200","Text":"but specifically for each 10 cents lowering of the price, in other words,"},{"Start":"00:28.200 ","End":"00:29.880","Text":"from 9 down to 8.90,"},{"Start":"00:29.880 ","End":"00:32.430","Text":"$8.80, $8.70, and so on,"},{"Start":"00:32.430 ","End":"00:37.860","Text":"each such reduction results in 2 more ticket sales a week."},{"Start":"00:37.860 ","End":"00:41.840","Text":"The question is, what should the price of a ticket"},{"Start":"00:41.840 ","End":"00:46.255","Text":"be in order to maximize the theater\u0027s weekly revenue,"},{"Start":"00:46.255 ","End":"00:52.110","Text":"and what is this weekly revenue that is maximal?"},{"Start":"00:52.300 ","End":"00:58.224","Text":"I would like to start by assigning a variable x,"},{"Start":"00:58.224 ","End":"01:02.510","Text":"but judiciously, what is my variable going to be?"},{"Start":"01:02.510 ","End":"01:05.690","Text":"You might say it\u0027s the price of a ticket,"},{"Start":"01:05.690 ","End":"01:09.785","Text":"but I think it\u0027s easier if we take x to be"},{"Start":"01:09.785 ","End":"01:17.610","Text":"the number of price reductions."},{"Start":"01:17.610 ","End":"01:22.960","Text":"That\u0027s going to help us because from this we can find out what we need."},{"Start":"01:22.960 ","End":"01:30.735","Text":"For example, the ticket price will be 9 dollars."},{"Start":"01:30.735 ","End":"01:32.175","Text":"We\u0027ll work in dollars now,"},{"Start":"01:32.175 ","End":"01:33.495","Text":"so everything is in dollars;"},{"Start":"01:33.495 ","End":"01:40.650","Text":"9 minus 10 cents is 0.1 dollars."},{"Start":"01:40.650 ","End":"01:44.540","Text":"So we have 0.1x,"},{"Start":"01:44.540 ","End":"01:47.350","Text":"and that\u0027s going to be the ticket price."},{"Start":"01:47.350 ","End":"01:52.330","Text":"I can also figure out the number of tickets sold."},{"Start":"01:52.330 ","End":"01:57.385","Text":"The number of tickets sold is more than 60,"},{"Start":"01:57.385 ","End":"02:02.215","Text":"each reduction causes an increase in 2 more,"},{"Start":"02:02.215 ","End":"02:06.060","Text":"so I get 2x more."},{"Start":"02:06.060 ","End":"02:07.830","Text":"This should be also weekly."},{"Start":"02:07.830 ","End":"02:11.805","Text":"Tickets sold weekly, is this."},{"Start":"02:11.805 ","End":"02:16.935","Text":"Now, the revenue is just equal, again,"},{"Start":"02:16.935 ","End":"02:23.400","Text":"weekly, is the number of tickets sold, which is this."},{"Start":"02:23.840 ","End":"02:26.850","Text":"I\u0027ll write it in shorthand, numbers sold,"},{"Start":"02:26.850 ","End":"02:29.335","Text":"and we understand it\u0027s in tickets per week,"},{"Start":"02:29.335 ","End":"02:34.950","Text":"multiplied by the price of a ticket."},{"Start":"02:35.090 ","End":"02:37.680","Text":"If I write this in symbols,"},{"Start":"02:37.680 ","End":"02:39.915","Text":"let the revenue be R,"},{"Start":"02:39.915 ","End":"02:42.420","Text":"and actually, R is a function of x,"},{"Start":"02:42.420 ","End":"02:44.999","Text":"it depends on x, is the number sold,"},{"Start":"02:44.999 ","End":"02:48.810","Text":"which is this, 60 plus 2x,"},{"Start":"02:48.810 ","End":"02:56.200","Text":"times the price, which is 9 minus 0.1x."},{"Start":"02:57.350 ","End":"03:02.130","Text":"That\u0027s the function that I want to maximize."},{"Start":"03:02.130 ","End":"03:08.345","Text":"Let\u0027s multiply out, and I want to arrange it so that x squared\u0027s first;"},{"Start":"03:08.345 ","End":"03:16.075","Text":"2x times minus 0.1x is minus 0.2x^2."},{"Start":"03:16.075 ","End":"03:21.255","Text":"Now, the x\u0027s, I\u0027ve got 9 times 2 is 18,"},{"Start":"03:21.255 ","End":"03:25.140","Text":"minus 60 times 0.1 is 6."},{"Start":"03:25.140 ","End":"03:29.505","Text":"So 18 minus 6 is 12,"},{"Start":"03:29.505 ","End":"03:34.425","Text":"plus 12x, and then 60 times 9,"},{"Start":"03:34.425 ","End":"03:39.570","Text":"well, 6 times 9 is 54, so it\u0027s 540."},{"Start":"03:39.570 ","End":"03:47.435","Text":"This is the weekly revenue in terms of x which is the number of price reductions."},{"Start":"03:47.435 ","End":"03:49.460","Text":"Now to maximize it,"},{"Start":"03:49.460 ","End":"03:53.825","Text":"there\u0027s 2 strategies: you could use calculus and differentiate,"},{"Start":"03:53.825 ","End":"03:57.815","Text":"but I\u0027m not going to assume you\u0027ve learned calculus,"},{"Start":"03:57.815 ","End":"04:03.810","Text":"so we\u0027ll do it using quadratic functions and parabolas."},{"Start":"04:04.250 ","End":"04:06.900","Text":"Do a quick review over here."},{"Start":"04:06.900 ","End":"04:12.800","Text":"When you have a parabola y equals ax^2 plus bx plus c,"},{"Start":"04:12.800 ","End":"04:17.540","Text":"or should I say a quadratic polynomial function,"},{"Start":"04:17.540 ","End":"04:20.555","Text":"then there are 2 cases, generally."},{"Start":"04:20.555 ","End":"04:23.195","Text":"When a is bigger than 0,"},{"Start":"04:23.195 ","End":"04:25.550","Text":"the parabola faces up,"},{"Start":"04:25.550 ","End":"04:31.560","Text":"opens upwards, and it has a minimum point which is also the vertex."},{"Start":"04:32.600 ","End":"04:36.170","Text":"When a is negative,"},{"Start":"04:36.170 ","End":"04:40.885","Text":"the parabola faces downwards or opens downwards,"},{"Start":"04:40.885 ","End":"04:44.855","Text":"and this point is also called the vertex of the parabola."},{"Start":"04:44.855 ","End":"04:47.390","Text":"Only this time, it\u0027s a maximum."},{"Start":"04:47.390 ","End":"04:49.370","Text":"Now, in our case,"},{"Start":"04:49.370 ","End":"04:53.340","Text":"this is a, and this is b,"},{"Start":"04:53.340 ","End":"05:02.135","Text":"and this is c. We are in the case where a is negative,"},{"Start":"05:02.135 ","End":"05:04.775","Text":"because this is a minus here,"},{"Start":"05:04.775 ","End":"05:06.395","Text":"so it does face down,"},{"Start":"05:06.395 ","End":"05:07.955","Text":"and we do have a maximum."},{"Start":"05:07.955 ","End":"05:09.500","Text":"Now there is a formula,"},{"Start":"05:09.500 ","End":"05:12.710","Text":"it doesn\u0027t give us the vertex,"},{"Start":"05:12.710 ","End":"05:15.020","Text":"but the x of the vertex."},{"Start":"05:15.020 ","End":"05:22.975","Text":"The x of the vertex is given by minus b over 2a,"},{"Start":"05:22.975 ","End":"05:27.665","Text":"R of x takes the place of y over here."},{"Start":"05:27.665 ","End":"05:30.980","Text":"Anyway, let\u0027s do the computation."},{"Start":"05:30.980 ","End":"05:33.590","Text":"The x we need of the vertex,"},{"Start":"05:33.590 ","End":"05:37.410","Text":"I\u0027ll call it x max because specifically,"},{"Start":"05:37.410 ","End":"05:39.745","Text":"I know the x gives me the maximum,"},{"Start":"05:39.745 ","End":"05:43.340","Text":"is equal to minus b over 2a,"},{"Start":"05:43.340 ","End":"05:46.750","Text":"and it\u0027s important enough I\u0027ll highlight it."},{"Start":"05:46.750 ","End":"05:50.925","Text":"In our case, minus 12, that\u0027s the b,"},{"Start":"05:50.925 ","End":"05:59.040","Text":"over twice minus 0.2 comes out to be 30."},{"Start":"05:59.040 ","End":"06:00.840","Text":"There were 2 questions."},{"Start":"06:00.840 ","End":"06:03.930","Text":"What should the ticket price be,"},{"Start":"06:03.930 ","End":"06:07.210","Text":"and what is the revenue weekly?"},{"Start":"06:07.210 ","End":"06:11.869","Text":"If I let x equals 30 in each of these,"},{"Start":"06:11.869 ","End":"06:17.220","Text":"then the ticket price becomes"},{"Start":"06:17.220 ","End":"06:26.460","Text":"9 minus 0.1 times 30 is 9 minus 3 is 6."},{"Start":"06:26.460 ","End":"06:29.670","Text":"This, I know the units is dollars."},{"Start":"06:29.670 ","End":"06:38.160","Text":"When x is 30, I get 60 plus twice 30 is another 60, it\u0027s 120."},{"Start":"06:38.160 ","End":"06:41.204","Text":"So 6 dollars a ticket,"},{"Start":"06:41.204 ","End":"06:43.755","Text":"they now sell 120."},{"Start":"06:43.755 ","End":"06:47.940","Text":"This part was 120,"},{"Start":"06:47.940 ","End":"06:50.655","Text":"this part was 6,"},{"Start":"06:50.655 ","End":"06:56.190","Text":"and 120 times 6 is 720,"},{"Start":"06:56.190 ","End":"06:57.990","Text":"and it\u0027s in dollars,"},{"Start":"06:57.990 ","End":"06:59.735","Text":"so we found everything."},{"Start":"06:59.735 ","End":"07:02.839","Text":"This was the price of a ticket,"},{"Start":"07:02.839 ","End":"07:10.460","Text":"and this is the maximal revenue, and we\u0027re done."},{"Start":"07:10.460 ","End":"07:15.800","Text":"But I just wanted to say something that the second part,"},{"Start":"07:15.800 ","End":"07:21.110","Text":"this we didn\u0027t have to do by figuring out ticket price and number of tickets sold."},{"Start":"07:21.110 ","End":"07:26.420","Text":"You could have just plugged it in blindly into this formula here,"},{"Start":"07:26.420 ","End":"07:28.805","Text":"letting x equals 30,"},{"Start":"07:28.805 ","End":"07:31.429","Text":"and it would have come out the same."},{"Start":"07:31.429 ","End":"07:33.440","Text":"That\u0027s just another option."},{"Start":"07:33.440 ","End":"07:35.820","Text":"Okay. We\u0027re done."}],"ID":8626},{"Watched":false,"Name":"Exercise 7","Duration":"9m 15s","ChapterTopicVideoID":8431,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8431.jpeg","UploadDate":"2017-01-30T04:46:41.3900000","DurationForVideoObject":"PT9M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"Here we have an exercise in maximizing profit."},{"Start":"00:04.080 ","End":"00:12.420","Text":"Production of a SpongeBob SquarePants doll costs the Nickelodeon company $5 to produce."},{"Start":"00:12.420 ","End":"00:20.220","Text":"Now, the company sells the doll for $9 and it can sell 200 units a day at that price."},{"Start":"00:20.220 ","End":"00:22.920","Text":"I guess they did some market research that"},{"Start":"00:22.920 ","End":"00:26.400","Text":"showed that for each 10 cent reduction in price,"},{"Start":"00:26.400 ","End":"00:29.940","Text":"it can sell an additional $10 a day."},{"Start":"00:29.940 ","End":"00:33.930","Text":"The question is, what\u0027s the maximum daily profit?"},{"Start":"00:33.930 ","End":"00:35.595","Text":"I guess it should say,"},{"Start":"00:35.595 ","End":"00:41.590","Text":"that the company can make on the sale of these dolls."},{"Start":"00:42.320 ","End":"00:47.590","Text":"Something general that we know is going to be useful to us here that"},{"Start":"00:47.590 ","End":"00:52.330","Text":"the profit is the revenue minus the cost."},{"Start":"00:52.330 ","End":"00:56.460","Text":"What you take in minus what you spend."},{"Start":"00:56.460 ","End":"00:59.830","Text":"I\u0027m going to be talking everything daily."},{"Start":"00:59.830 ","End":"01:05.240","Text":"The daily profit will be the daily revenue minus the daily cost."},{"Start":"01:05.250 ","End":"01:08.500","Text":"The question is how to figure these out."},{"Start":"01:08.500 ","End":"01:11.740","Text":"If we choose the variable appropriately,"},{"Start":"01:11.740 ","End":"01:21.130","Text":"it\u0027ll make work easier and I suggest that let x be the number of reductions in price."},{"Start":"01:21.130 ","End":"01:27.880","Text":"Let x be the number 10 cents reductions."},{"Start":"01:27.880 ","End":"01:38.990","Text":"Only it\u0027s best if we work in the same unit dollars,10 cents is 0.1 or $0.10,"},{"Start":"01:38.990 ","End":"01:41.470","Text":"I guess we write the dollars on the left."},{"Start":"01:41.470 ","End":"01:45.170","Text":"Let\u0027s see then from this we can figure out"},{"Start":"01:45.170 ","End":"01:50.345","Text":"the price and we can tell how many units a day. Let\u0027s see."},{"Start":"01:50.345 ","End":"01:55.530","Text":"The price of a doll,"},{"Start":"01:55.530 ","End":"01:59.625","Text":"it will equal, now it\u0027s going to be less than $9."},{"Start":"01:59.625 ","End":"02:02.370","Text":"It\u0027s going to be 9."},{"Start":"02:02.370 ","End":"02:04.365","Text":"Let\u0027s work in dollars from now on."},{"Start":"02:04.365 ","End":"02:11.715","Text":"Will be 9 minus x reductions each 1 0.10."},{"Start":"02:11.715 ","End":"02:18.225","Text":"So minus 0.10x, that will be the price of a doll."},{"Start":"02:18.225 ","End":"02:23.960","Text":"Then the quantity, say quantity but I mean the number"},{"Start":"02:23.960 ","End":"02:29.780","Text":"of dolls that it can sell per day will be not the 200,"},{"Start":"02:29.780 ","End":"02:33.320","Text":"but now it\u0027s got extra x times."},{"Start":"02:33.320 ","End":"02:36.885","Text":"We have to increase 10 dolls a day."},{"Start":"02:36.885 ","End":"02:41.230","Text":"So 200 plus 10x."},{"Start":"02:41.230 ","End":"02:46.175","Text":"The revenue is going to be,"},{"Start":"02:46.175 ","End":"02:51.815","Text":"how much you take in is the price for each doll times the number of dolls you sell."},{"Start":"02:51.815 ","End":"02:54.020","Text":"It\u0027s going to be, well,"},{"Start":"02:54.020 ","End":"02:58.355","Text":"this times this, then you get the revenue."},{"Start":"02:58.355 ","End":"03:08.030","Text":"It\u0027s going to be 9 minus 0.1x times 200 plus 10x."},{"Start":"03:08.030 ","End":"03:12.900","Text":"Now the daily costs."},{"Start":"03:14.960 ","End":"03:20.580","Text":"Because it cost them $5 to produce this number of"},{"Start":"03:20.580 ","End":"03:27.315","Text":"quantity 200 plus 10x times,"},{"Start":"03:27.315 ","End":"03:29.595","Text":"the cost is $5,"},{"Start":"03:29.595 ","End":"03:31.635","Text":"I\u0027ll put the 5 in front."},{"Start":"03:31.635 ","End":"03:38.255","Text":"If I subtract revenue minus production costs,"},{"Start":"03:38.255 ","End":"03:40.070","Text":"then I\u0027ll get the profit."},{"Start":"03:40.070 ","End":"03:47.965","Text":"Finally, I can get the profit and I\u0027ll use the letter Pi depending on x,"},{"Start":"03:47.965 ","End":"03:54.200","Text":"daily profit, is equal to this minus this."},{"Start":"03:54.200 ","End":"04:02.450","Text":"So it\u0027s 9 minus 0.1x times 200"},{"Start":"04:02.450 ","End":"04:12.065","Text":"plus 10x minus 5 times 200 plus 10x."},{"Start":"04:12.065 ","End":"04:19.790","Text":"All this expression is the profit and this is what we want to maximize."},{"Start":"04:19.790 ","End":"04:23.435","Text":"In general, there\u0027s 2 main approaches to doing this."},{"Start":"04:23.435 ","End":"04:28.595","Text":"1 uses calculus, but I\u0027m not sure if you\u0027ve"},{"Start":"04:28.595 ","End":"04:35.140","Text":"done calculus and finding maximum from differentiation and so on."},{"Start":"04:35.140 ","End":"04:36.815","Text":"So I\u0027ll use the other method,"},{"Start":"04:36.815 ","End":"04:41.435","Text":"which uses the properties of quadratic functions and parabolas."},{"Start":"04:41.435 ","End":"04:45.959","Text":"Let me continue and I will just use Algebra."},{"Start":"04:46.330 ","End":"04:48.770","Text":"I can open the brackets,"},{"Start":"04:48.770 ","End":"04:50.720","Text":"but look, I have a common factor,"},{"Start":"04:50.720 ","End":"04:54.275","Text":"so I think it\u0027ll be easier if I take the 200 plus"},{"Start":"04:54.275 ","End":"04:59.390","Text":"10x outside the brackets and then I\u0027ve got this minus this."},{"Start":"04:59.390 ","End":"05:05.730","Text":"This will just give me 9 minus 5 is 4 minus 0.1x."},{"Start":"05:05.730 ","End":"05:09.675","Text":"I just took this minus the 5 there."},{"Start":"05:09.675 ","End":"05:11.930","Text":"Let\u0027s multiply out."},{"Start":"05:11.930 ","End":"05:15.770","Text":"I want to choose the x squared first."},{"Start":"05:15.770 ","End":"05:18.650","Text":"10x times 0.1x."},{"Start":"05:18.650 ","End":"05:24.870","Text":"It\u0027s a minus, is just 1x squared."},{"Start":"05:25.030 ","End":"05:29.630","Text":"But there\u0027s a minus plus with minus."},{"Start":"05:29.630 ","End":"05:32.180","Text":"Next the xs."},{"Start":"05:32.180 ","End":"05:36.980","Text":"This with this is 40x."},{"Start":"05:36.980 ","End":"05:42.095","Text":"This with this will give me minus 20x."},{"Start":"05:42.095 ","End":"05:47.020","Text":"Altogether plus 20x, 40 minus 20."},{"Start":"05:47.020 ","End":"05:54.190","Text":"Then lastly, this with this will give me plus 800."},{"Start":"05:54.190 ","End":"06:01.010","Text":"On right again that this is the profit function and this is what we have to maximize."},{"Start":"06:01.010 ","End":"06:03.850","Text":"I think I\u0027ll highlight it."},{"Start":"06:03.850 ","End":"06:07.980","Text":"That\u0027s the important thing. As I said,"},{"Start":"06:07.980 ","End":"06:10.145","Text":"we\u0027re not going to use calculus so we\u0027ll use"},{"Start":"06:10.145 ","End":"06:14.080","Text":"properties of quadratics and I\u0027ll remind you."},{"Start":"06:14.080 ","End":"06:18.754","Text":"This function of x is like the dependent variable y,"},{"Start":"06:18.754 ","End":"06:20.690","Text":"y in terms of x, and in general,"},{"Start":"06:20.690 ","End":"06:28.760","Text":"this looks of the form y equals ax squared plus bx plus c,"},{"Start":"06:28.760 ","End":"06:33.050","Text":"where a is minus 1,"},{"Start":"06:33.050 ","End":"06:43.230","Text":"b is 20, c is 800,"},{"Start":"06:43.230 ","End":"06:46.335","Text":"and I\u0027ll return to that in the moment."},{"Start":"06:46.335 ","End":"06:49.775","Text":"The theory is that a parabola,"},{"Start":"06:49.775 ","End":"06:53.150","Text":"depending on whether a is positive or negative,"},{"Start":"06:53.150 ","End":"06:55.730","Text":"will look 1 of 2 shapes."},{"Start":"06:55.730 ","End":"06:58.910","Text":"If the coefficient of x squared,"},{"Start":"06:58.910 ","End":"07:01.190","Text":"if the a is positive,"},{"Start":"07:01.190 ","End":"07:05.260","Text":"then it\u0027s a facing up parabola or opens upwards."},{"Start":"07:05.260 ","End":"07:13.745","Text":"It has a point called the vertex and that vertex is actually the minimum point."},{"Start":"07:13.745 ","End":"07:18.515","Text":"This is not the case here because in our case, a is negative."},{"Start":"07:18.515 ","End":"07:21.410","Text":"When a is negative,"},{"Start":"07:21.410 ","End":"07:27.815","Text":"the parabola\u0027s an upside down parabola and it also has a vertex."},{"Start":"07:27.815 ","End":"07:36.150","Text":"It faces down and this vertex is in this case a maximum point."},{"Start":"07:36.150 ","End":"07:43.130","Text":"The other piece of the puzzle is that I do know the formula for the x."},{"Start":"07:43.130 ","End":"07:48.520","Text":"The x coordinate of the vertex doesn\u0027t matter in either case."},{"Start":"07:48.520 ","End":"07:54.755","Text":"The x of the vertex is given by minus b over 2a."},{"Start":"07:54.755 ","End":"07:57.440","Text":"We can get the x coordinate from the x coordinate,"},{"Start":"07:57.440 ","End":"07:59.090","Text":"we can get the y coordinate."},{"Start":"07:59.090 ","End":"08:01.980","Text":"Let\u0027s apply this back here."},{"Start":"08:03.460 ","End":"08:07.850","Text":"The x from the graph that gives me the maximum,"},{"Start":"08:07.850 ","End":"08:11.885","Text":"and I\u0027ll call it x max other than xv,"},{"Start":"08:11.885 ","End":"08:18.965","Text":"is equal to minus b over 2a applied here is"},{"Start":"08:18.965 ","End":"08:26.615","Text":"minus 20 over twice minus 1 is a. Minuses cancel,"},{"Start":"08:26.615 ","End":"08:29.605","Text":"this clearly just comes out to be 10."},{"Start":"08:29.605 ","End":"08:36.150","Text":"X is 10, but that\u0027s not what we were asked for."},{"Start":"08:36.150 ","End":"08:39.720","Text":"We want the maximum daily profit."},{"Start":"08:39.720 ","End":"08:42.915","Text":"All we have to do really is substitute"},{"Start":"08:42.915 ","End":"08:47.580","Text":"this answer 10 in here and that will give me the maximum."},{"Start":"08:47.580 ","End":"08:50.880","Text":"I need Pi of 10, this 10 here,"},{"Start":"08:50.880 ","End":"09:00.560","Text":"and this is equal minus 10 squared plus 20 times 10 plus 800."},{"Start":"09:00.560 ","End":"09:04.145","Text":"It\u0027s minus 100 plus 200 plus 800."},{"Start":"09:04.145 ","End":"09:12.610","Text":"Answer comes out to be $900."},{"Start":"09:12.610 ","End":"09:15.440","Text":"I\u0027ll highlight this answer."}],"ID":8627},{"Watched":false,"Name":"Exercise 8","Duration":"12m 8s","ChapterTopicVideoID":8432,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8432.jpeg","UploadDate":"2017-01-30T04:48:15.3700000","DurationForVideoObject":"PT12M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.560 ","End":"00:05.819","Text":"We often have a profit-maximizing type of question."},{"Start":"00:05.819 ","End":"00:10.125","Text":"Here we have a proof question about minimum profit."},{"Start":"00:10.125 ","End":"00:12.225","Text":"Let\u0027s see how we tackle this."},{"Start":"00:12.225 ","End":"00:14.025","Text":"Let me read it first."},{"Start":"00:14.025 ","End":"00:20.365","Text":"Office Depot company purchases a number of equally priced items for a total of $800."},{"Start":"00:20.365 ","End":"00:23.870","Text":"We don\u0027t know how many, but they\u0027re all priced the same."},{"Start":"00:23.870 ","End":"00:30.079","Text":"Now, it sells 5 of the items at a 20 percent profit,"},{"Start":"00:30.079 ","End":"00:36.835","Text":"and the remaining items at a markup of 2 dollars each."},{"Start":"00:36.835 ","End":"00:45.285","Text":"I think the word profit in this sentence means a markup of 20 percent."},{"Start":"00:45.285 ","End":"00:47.450","Text":"In both cases, it\u0027s a markup."},{"Start":"00:47.450 ","End":"00:50.055","Text":"In 1 case it\u0027s a fixed markup, 2 dollars,"},{"Start":"00:50.055 ","End":"00:53.465","Text":"on the other case it\u0027s a 20 percent markup."},{"Start":"00:53.465 ","End":"01:00.230","Text":"We have to prove that the profit the company makes on the deal is at least $70."},{"Start":"01:00.230 ","End":"01:02.945","Text":"How do we go about this?"},{"Start":"01:02.945 ","End":"01:08.420","Text":"1 of the things I would do would be to choose a variable."},{"Start":"01:08.420 ","End":"01:13.370","Text":"For example, the number of items to me seems the most"},{"Start":"01:13.370 ","End":"01:18.950","Text":"obvious thing to do because then you can figure out how much we paid for each item,"},{"Start":"01:18.950 ","End":"01:22.865","Text":"and then we can add either 2 dollars or 20 percent and so on and so on."},{"Start":"01:22.865 ","End":"01:30.450","Text":"Let me just declare that x equal the number of items."},{"Start":"01:30.450 ","End":"01:34.020","Text":"We\u0027re talking about the number of items here that were equally priced,"},{"Start":"01:34.020 ","End":"01:36.910","Text":"that cost $800 together."},{"Start":"01:36.950 ","End":"01:41.375","Text":"Now, in general, when we compute profit,"},{"Start":"01:41.375 ","End":"01:46.610","Text":"profit is the revenue minus the cost."},{"Start":"01:46.610 ","End":"01:48.440","Text":"That\u0027s a general thing."},{"Start":"01:48.440 ","End":"01:53.195","Text":"Total revenue or just revenue minus cost or total cost."},{"Start":"01:53.195 ","End":"01:58.280","Text":"Now, the cost is easy because the cost,"},{"Start":"01:58.280 ","End":"02:03.060","Text":"we know it\u0027s a fixed cost of $800."},{"Start":"02:03.100 ","End":"02:10.949","Text":"But the revenue, it splits up into 2 parts."},{"Start":"02:11.740 ","End":"02:20.430","Text":"5 items at 20 percent profit or markup,"},{"Start":"02:20.430 ","End":"02:22.080","Text":"I\u0027ll just put an upward arrow,"},{"Start":"02:22.080 ","End":"02:23.620","Text":"and the remaining, well,"},{"Start":"02:23.620 ","End":"02:26.450","Text":"how many remain if there were x altogether,"},{"Start":"02:26.450 ","End":"02:31.540","Text":"that would be x minus 5 remaining items"},{"Start":"02:31.540 ","End":"02:38.735","Text":"at 2 dollars markup or profit, whatever."},{"Start":"02:38.735 ","End":"02:40.565","Text":"What else do we know?"},{"Start":"02:40.565 ","End":"02:45.620","Text":"We can figure out how much the cost per item was."},{"Start":"02:45.620 ","End":"02:48.600","Text":"The total cost is 800."},{"Start":"02:48.770 ","End":"02:58.350","Text":"Then I can say the cost per item is going to be, well,"},{"Start":"02:58.350 ","End":"03:08.240","Text":"I just have to divide the $800 divided by x items is the number of dollars per item."},{"Start":"03:08.240 ","End":"03:10.625","Text":"I think it\u0027ll highlight this, it\u0027s important."},{"Start":"03:10.625 ","End":"03:12.830","Text":"I\u0027m going to do 2 side calculations."},{"Start":"03:12.830 ","End":"03:20.300","Text":"I want to take this 800 over x and I want to mark it up, add 20 percent."},{"Start":"03:20.300 ","End":"03:29.825","Text":"Adding 20 percent in mathematics means multiplying by 120 percent or by 1.2."},{"Start":"03:29.825 ","End":"03:37.630","Text":"So it\u0027s 1.2 times 800 over x."},{"Start":"03:38.180 ","End":"03:45.590","Text":"The other kind of adding 2 dollars is just 800 over x plus 2."},{"Start":"03:45.590 ","End":"03:48.910","Text":"There\u0027s no further computation to be done."},{"Start":"03:48.910 ","End":"03:54.080","Text":"Going back here, we can now make a calculation of what the revenue is."},{"Start":"03:54.080 ","End":"03:59.765","Text":"The revenue from here was 5 items and each 1"},{"Start":"03:59.765 ","End":"04:09.585","Text":"costs 1.2 times 800 over x."},{"Start":"04:09.585 ","End":"04:12.960","Text":"That part 1 of the revenue from the 5 items."},{"Start":"04:12.960 ","End":"04:18.015","Text":"Part 2 of the revenue is where I have x minus 5 items."},{"Start":"04:18.015 ","End":"04:27.210","Text":"These cost each 800 over x plus 2."},{"Start":"04:27.210 ","End":"04:32.750","Text":"I have to add this plus this and that will just give me my revenue."},{"Start":"04:32.750 ","End":"04:35.480","Text":"If I add them together, these 2,"},{"Start":"04:35.480 ","End":"04:37.490","Text":"that will give me the revenue,"},{"Start":"04:37.490 ","End":"04:40.010","Text":"and then I\u0027ll have to subtract the total cost,"},{"Start":"04:40.010 ","End":"04:42.280","Text":"which will be the 800."},{"Start":"04:42.280 ","End":"04:48.000","Text":"The total revenue is,"},{"Start":"04:48.000 ","End":"04:50.400","Text":"let\u0027s see I can simplify it a bit,"},{"Start":"04:50.400 ","End":"04:53.430","Text":"5 times 1.2 is 6,"},{"Start":"04:53.430 ","End":"04:55.815","Text":"we\u0027ll simplify more later, just a little bit,"},{"Start":"04:55.815 ","End":"04:59.970","Text":"6 times 800 over x plus,"},{"Start":"04:59.970 ","End":"05:02.730","Text":"you know what, I\u0027m going to go back to the way it was."},{"Start":"05:02.730 ","End":"05:06.940","Text":"I also figure I can copy-paste. Here we are."},{"Start":"05:06.940 ","End":"05:13.040","Text":"The total cost we said was 800."},{"Start":"05:13.210 ","End":"05:16.250","Text":"In principle, both of these depend on x,"},{"Start":"05:16.250 ","End":"05:18.919","Text":"so I should have rewritten of x."},{"Start":"05:18.919 ","End":"05:22.085","Text":"Of x is supposed to be a constant function."},{"Start":"05:22.085 ","End":"05:25.530","Text":"If I subtract, I get the profit."},{"Start":"05:25.530 ","End":"05:27.405","Text":"I use the letter Pi."},{"Start":"05:27.405 ","End":"05:33.870","Text":"The profit would be this plus this minus this."},{"Start":"05:36.740 ","End":"05:40.765","Text":"I\u0027ll just start multiplying out."},{"Start":"05:40.765 ","End":"05:45.270","Text":"This is, 5 times 1.2 is 6."},{"Start":"05:45.270 ","End":"05:55.345","Text":"6 times 800 is 4,800 over x."},{"Start":"05:55.345 ","End":"06:04.315","Text":"Then the next bit of the revenue from the remaining non 5 items. Let\u0027s see."},{"Start":"06:04.315 ","End":"06:10.650","Text":"X times 800 over x is just 800,"},{"Start":"06:10.650 ","End":"06:16.180","Text":"x times 2 is 2x."},{"Start":"06:18.200 ","End":"06:24.970","Text":"5 times 800 is"},{"Start":"06:25.580 ","End":"06:32.555","Text":"4,000 over x with a minus and minus 5 times 2 is minus 10."},{"Start":"06:32.555 ","End":"06:37.060","Text":"Now, I also have to subtract the 800."},{"Start":"06:37.060 ","End":"06:44.705","Text":"That\u0027s the cost in the revenue minus cost part."},{"Start":"06:44.705 ","End":"06:52.129","Text":"Notice we have a bit of luck because this 800 and this 800 cancel each other out."},{"Start":"06:52.129 ","End":"06:55.720","Text":"If I collect like terms,"},{"Start":"06:55.720 ","End":"06:58.545","Text":"we get that Pi of x,"},{"Start":"06:58.545 ","End":"07:08.325","Text":"our profit function, is equal to this 4,800 minus 4,000 they\u0027re both over x."},{"Start":"07:08.325 ","End":"07:10.095","Text":"What I get is"},{"Start":"07:10.095 ","End":"07:19.695","Text":"800 over x and then plus 2x minus 10."},{"Start":"07:19.695 ","End":"07:23.610","Text":"This is the profit function."},{"Start":"07:23.610 ","End":"07:25.460","Text":"What I have to prove,"},{"Start":"07:25.460 ","End":"07:27.710","Text":"is that this is at least,"},{"Start":"07:27.710 ","End":"07:33.075","Text":"but we don\u0027t know that so I\u0027ll put a question mark, 70."},{"Start":"07:33.075 ","End":"07:34.800","Text":"That\u0027s what we have to show."},{"Start":"07:34.800 ","End":"07:39.530","Text":"What I would say is take this function, find its minimum."},{"Start":"07:39.530 ","End":"07:44.635","Text":"If the minimum is bigger or equal to 70, then we\u0027re done."},{"Start":"07:44.635 ","End":"07:48.530","Text":"We have a minimum question and we\u0027re going"},{"Start":"07:48.530 ","End":"07:54.880","Text":"to do it with calculus by differentiating and setting to 0 and so on."},{"Start":"07:54.880 ","End":"08:00.260","Text":"I think I\u0027ll change color now that we\u0027re going to go into calculus."},{"Start":"08:00.260 ","End":"08:03.770","Text":"We want to differentiate the profit function."},{"Start":"08:03.770 ","End":"08:07.920","Text":"When you think we want to find maximum or minimum, we differentiate."},{"Start":"08:07.920 ","End":"08:13.580","Text":"What we get is the derivative of 1 over x is minus 1 over x squared."},{"Start":"08:13.580 ","End":"08:17.480","Text":"Here we have minus 800 over x squared."},{"Start":"08:17.480 ","End":"08:26.880","Text":"I\u0027ll trust you to remember that 1 over x derivative is minus 1 over x squared."},{"Start":"08:27.170 ","End":"08:31.885","Text":"The derivative of 2x is just 2,"},{"Start":"08:31.885 ","End":"08:35.330","Text":"and the derivative of a constant is 0."},{"Start":"08:35.330 ","End":"08:37.040","Text":"This is the derivative."},{"Start":"08:37.040 ","End":"08:45.850","Text":"Then we look for critical points by setting the derivative equal to 0."},{"Start":"08:45.850 ","End":"08:49.340","Text":"We get that this plus this is 0,"},{"Start":"08:49.340 ","End":"08:53.585","Text":"which means that if I bring the minus to the other side,"},{"Start":"08:53.585 ","End":"08:59.395","Text":"that 800 over x squared equals 2."},{"Start":"08:59.395 ","End":"09:03.050","Text":"Let\u0027s see what we can do with that."},{"Start":"09:03.160 ","End":"09:07.730","Text":"I can multiply this by x squared then divide by 2."},{"Start":"09:07.730 ","End":"09:12.890","Text":"I have x squared equals 400."},{"Start":"09:12.890 ","End":"09:20.495","Text":"Now, normally I would say that x equals plus or minus 20."},{"Start":"09:20.495 ","End":"09:26.550","Text":"But x can\u0027t be negative because it\u0027s a number of items."},{"Start":"09:26.570 ","End":"09:31.180","Text":"Take the eraser and just erase that."},{"Start":"09:31.250 ","End":"09:35.670","Text":"X equals 20 is a critical point."},{"Start":"09:35.670 ","End":"09:39.390","Text":"To see that this is a minimum,"},{"Start":"09:39.390 ","End":"09:44.595","Text":"we\u0027ll use the second derivative test. Let\u0027s see."},{"Start":"09:44.595 ","End":"09:49.360","Text":"Pi double-prime second derivative,"},{"Start":"09:49.360 ","End":"09:55.340","Text":"I get that from differentiating this here."},{"Start":"09:55.340 ","End":"10:03.630","Text":"The derivative of 1 over x squared is minus 2 over x cubed,"},{"Start":"10:03.630 ","End":"10:05.100","Text":"I\u0027ll leave you to verify,"},{"Start":"10:05.100 ","End":"10:07.035","Text":"or you might remember."},{"Start":"10:07.035 ","End":"10:17.300","Text":"What we get from the differentiating is minus 800 together with minus"},{"Start":"10:17.300 ","End":"10:23.250","Text":"2 would give me"},{"Start":"10:23.250 ","End":"10:29.455","Text":"plus 1,600 over x cubed."},{"Start":"10:29.455 ","End":"10:33.160","Text":"The derivative of 2 is nothing."},{"Start":"10:36.250 ","End":"10:40.740","Text":"We just have to plug in x equals 20."},{"Start":"10:41.900 ","End":"10:46.405","Text":"Obviously, Pi double-prime of 20,"},{"Start":"10:46.405 ","End":"10:48.835","Text":"I don\u0027t even have to do the computation."},{"Start":"10:48.835 ","End":"10:51.880","Text":"No need to, because everything\u0027s positive here."},{"Start":"10:51.880 ","End":"10:55.165","Text":"If it\u0027s a positive second derivative,"},{"Start":"10:55.165 ","End":"11:00.310","Text":"that means that we have a minimum when x is 20."},{"Start":"11:00.310 ","End":"11:04.030","Text":"Now we know that 20 is a minimum point."},{"Start":"11:04.030 ","End":"11:08.440","Text":"We need to know the value of the function Pi,"},{"Start":"11:08.440 ","End":"11:10.840","Text":"the profit function, that 20."},{"Start":"11:10.840 ","End":"11:14.545","Text":"I\u0027ll continue from here."},{"Start":"11:14.545 ","End":"11:18.895","Text":"I want to figure out what it is Pi, the profit,"},{"Start":"11:18.895 ","End":"11:25.510","Text":"where the number of items being 20 is equal to 800"},{"Start":"11:25.510 ","End":"11:35.100","Text":"over 20 plus twice 20 minus 10."},{"Start":"11:35.100 ","End":"11:41.685","Text":"800 over 20 is 40 plus"},{"Start":"11:41.685 ","End":"11:48.615","Text":"40 minus 10 which is 70."},{"Start":"11:48.615 ","End":"11:50.865","Text":"The minimum is 70,"},{"Start":"11:50.865 ","End":"11:54.055","Text":"so the profit is at least 70."},{"Start":"11:54.055 ","End":"11:57.350","Text":"By the way, if this came out higher than 70,"},{"Start":"11:57.350 ","End":"11:58.610","Text":"that would also be good."},{"Start":"11:58.610 ","End":"12:02.110","Text":"If I got 75 or even 100 here,"},{"Start":"12:02.110 ","End":"12:04.985","Text":"that would still prove that the profit is at least 70."},{"Start":"12:04.985 ","End":"12:09.300","Text":"Anyway, we are finally done."}],"ID":8628},{"Watched":false,"Name":"Exercise 9","Duration":"6m 50s","ChapterTopicVideoID":8433,"CourseChapterTopicPlaylistID":65392,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8433.jpeg","UploadDate":"2017-01-30T04:49:37.3900000","DurationForVideoObject":"PT6M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.665","Text":"In this exercise there\u0027s a bicycle company that sells"},{"Start":"00:04.665 ","End":"00:10.320","Text":"300 of a certain kind of bicycle weekly at a $100 each."},{"Start":"00:10.320 ","End":"00:15.689","Text":"Then it starts with this program that if it sells any additional bicycles,"},{"Start":"00:15.689 ","End":"00:17.850","Text":"it gives a discount."},{"Start":"00:17.850 ","End":"00:21.870","Text":"If there\u0027s x additional bicycles,"},{"Start":"00:21.870 ","End":"00:28.185","Text":"I\u0027ll highlight that, then it lowers the price on a 2-tier system."},{"Start":"00:28.185 ","End":"00:30.975","Text":"Each of the bicycles beyond the 300,"},{"Start":"00:30.975 ","End":"00:35.744","Text":"it lowers by this formula 0.4x each,"},{"Start":"00:35.744 ","End":"00:44.595","Text":"and the price on the first 300 is lowered also but only by 0.2x each."},{"Start":"00:44.595 ","End":"00:48.200","Text":"Just to make sense, it\u0027s a retroactive discount,"},{"Start":"00:48.200 ","End":"00:53.015","Text":"meaning they mail them a rebate check after the weekly sales are in."},{"Start":"00:53.015 ","End":"00:57.320","Text":"The question is, how many bicycles does"},{"Start":"00:57.320 ","End":"01:02.245","Text":"the company need to sell to get the maximum revenue?"},{"Start":"01:02.245 ","End":"01:05.480","Text":"Now, 1 thing that\u0027s easy here is that we don\u0027t have"},{"Start":"01:05.480 ","End":"01:08.480","Text":"to figure out what the variable is going to be."},{"Start":"01:08.480 ","End":"01:13.010","Text":"We\u0027ll take x to be the variable which is the number of additional bicycles."},{"Start":"01:13.010 ","End":"01:15.990","Text":"Now, it\u0027s a 2-tier system."},{"Start":"01:15.990 ","End":"01:18.075","Text":"There\u0027s 2 kinds of bicycles;"},{"Start":"01:18.075 ","End":"01:22.805","Text":"there\u0027s the 300 original bicycles,"},{"Start":"01:22.805 ","End":"01:28.000","Text":"and it sells those at the rate of,"},{"Start":"01:28.000 ","End":"01:32.685","Text":"let\u0027s see how much does it sell them for, it\u0027s $100."},{"Start":"01:32.685 ","End":"01:40.830","Text":"We\u0027ll work in dollars. I won\u0027t write the dollars each time;100 minus the discount,"},{"Start":"01:40.830 ","End":"01:47.375","Text":"which is 0.2x each,"},{"Start":"01:47.375 ","End":"01:49.460","Text":"and the additional bicycles,"},{"Start":"01:49.460 ","End":"01:55.490","Text":"x bikes, at the rate of 100"},{"Start":"01:55.490 ","End":"02:02.310","Text":"minus a bigger discount 0.4x each."},{"Start":"02:02.310 ","End":"02:05.225","Text":"If I want to figure out the total revenue,"},{"Start":"02:05.225 ","End":"02:06.830","Text":"which is a function of x,"},{"Start":"02:06.830 ","End":"02:08.620","Text":"I\u0027ll call it R of x,"},{"Start":"02:08.620 ","End":"02:13.580","Text":"it\u0027s equal to the revenue from the first 300,"},{"Start":"02:13.580 ","End":"02:22.190","Text":"which is 300 bicycles times 100 minus 0.2x a piece,"},{"Start":"02:22.190 ","End":"02:25.535","Text":"plus the x bicycles,"},{"Start":"02:25.535 ","End":"02:28.310","Text":"number of bicycles times the price each,"},{"Start":"02:28.310 ","End":"02:33.110","Text":"which is 100 minus 0.4x."},{"Start":"02:33.110 ","End":"02:36.300","Text":"This is what we have to minimize."},{"Start":"02:36.470 ","End":"02:42.245","Text":"Let\u0027s just start by expanding brackets and simplifying."},{"Start":"02:42.245 ","End":"02:48.395","Text":"300 times 100 is 30,000,"},{"Start":"02:48.395 ","End":"02:55.170","Text":"minus 300 times 0.2 is 60."},{"Start":"02:55.960 ","End":"02:59.590","Text":"Then x times 100 is a 100x,"},{"Start":"02:59.590 ","End":"03:07.710","Text":"and then here we have minus 0.4x squared."},{"Start":"03:07.710 ","End":"03:13.940","Text":"I\u0027d like to collect like terms and write it in descending order of powers of x."},{"Start":"03:13.940 ","End":"03:16.400","Text":"We\u0027re going to get a quadratic function."},{"Start":"03:16.400 ","End":"03:22.770","Text":"We have minus 0.4x squared,"},{"Start":"03:22.770 ","End":"03:28.510","Text":"and then these 2 together give me plus 40x,"},{"Start":"03:28.610 ","End":"03:35.745","Text":"and then we also have a plus 30,000."},{"Start":"03:35.745 ","End":"03:41.690","Text":"This is the function that we have to maximize."},{"Start":"03:41.690 ","End":"03:48.185","Text":"Now as usual, I mentioned there\u0027s more than 1 way to do it."},{"Start":"03:48.185 ","End":"03:52.760","Text":"You can do these with calculus and differentiation,"},{"Start":"03:52.760 ","End":"03:56.705","Text":"but I\u0027m not going to assume that you know calculus."},{"Start":"03:56.705 ","End":"03:59.885","Text":"We\u0027ll do it with the properties of the quadratic function,"},{"Start":"03:59.885 ","End":"04:02.365","Text":"and I\u0027ll give you a reminder."},{"Start":"04:02.365 ","End":"04:05.330","Text":"I\u0027ll think of R of x as my y,"},{"Start":"04:05.330 ","End":"04:08.570","Text":"my dependent variable and x is the independent variable."},{"Start":"04:08.570 ","End":"04:12.815","Text":"Then in general, when you have a formula,"},{"Start":"04:12.815 ","End":"04:15.200","Text":"y equals some constant x"},{"Start":"04:15.200 ","End":"04:20.285","Text":"squared plus another constant x plus another constant x squared plus bx plus c,"},{"Start":"04:20.285 ","End":"04:22.850","Text":"then we know that it\u0027s shape is a parabola,"},{"Start":"04:22.850 ","End":"04:24.965","Text":"and there are 2 different kinds."},{"Start":"04:24.965 ","End":"04:27.480","Text":"If a is bigger than 0, it\u0027s 1 thing,"},{"Start":"04:27.480 ","End":"04:29.975","Text":"if a is less than 0, it\u0027s something else."},{"Start":"04:29.975 ","End":"04:33.440","Text":"A is not 0 because it wouldn\u0027t be a parabola then."},{"Start":"04:33.440 ","End":"04:41.465","Text":"In our case, we know that we\u0027re in this case here because this is a, which is negative,"},{"Start":"04:41.465 ","End":"04:48.200","Text":"and this 40 is b and 30,000 is c. When a is negative,"},{"Start":"04:48.200 ","End":"04:53.404","Text":"it\u0027s an upside down parabola facing down opens downwards,"},{"Start":"04:53.404 ","End":"04:54.960","Text":"different names for it,"},{"Start":"04:54.960 ","End":"04:58.055","Text":"and the parabola has a vertex."},{"Start":"04:58.055 ","End":"05:01.735","Text":"The vertex is v, let\u0027s say,"},{"Start":"05:01.735 ","End":"05:04.880","Text":"and by the shape of it,"},{"Start":"05:04.880 ","End":"05:06.620","Text":"it has a maximum."},{"Start":"05:06.620 ","End":"05:11.855","Text":"Just the completeness, I\u0027ll mention that if a was the other way around, was positive,"},{"Start":"05:11.855 ","End":"05:15.424","Text":"then we would still have a vertex v,"},{"Start":"05:15.424 ","End":"05:19.085","Text":"and this time it would be a minimum."},{"Start":"05:19.085 ","End":"05:21.019","Text":"But in any event,"},{"Start":"05:21.019 ","End":"05:25.070","Text":"we have a formula not for the coordinates of v,"},{"Start":"05:25.070 ","End":"05:29.285","Text":"but just the x component of the vertex,"},{"Start":"05:29.285 ","End":"05:30.875","Text":"which I care about here,"},{"Start":"05:30.875 ","End":"05:33.395","Text":"because it\u0027s going to be the x for the maximum,"},{"Start":"05:33.395 ","End":"05:39.710","Text":"is equal to minus b over 2a where the b and the a are from here."},{"Start":"05:39.710 ","End":"05:43.910","Text":"Let\u0027s do this in our particular case."},{"Start":"05:43.910 ","End":"05:48.320","Text":"We have that our x_v,"},{"Start":"05:48.320 ","End":"05:49.920","Text":"but I\u0027ll change the name,"},{"Start":"05:49.920 ","End":"05:55.340","Text":"I\u0027ll call it x_max because we know in the case of a negative that it\u0027s a maximum,"},{"Start":"05:55.340 ","End":"05:57.275","Text":"is equal to, now,"},{"Start":"05:57.275 ","End":"06:04.205","Text":"minus b over 2a in our case is minus 40,"},{"Start":"06:04.205 ","End":"06:10.535","Text":"which is minus b over 2a is minus 0.4,"},{"Start":"06:10.535 ","End":"06:17.520","Text":"which is just equal to 50."},{"Start":"06:17.520 ","End":"06:22.085","Text":"The minuses cancel and 40 over 0.8 is 50."},{"Start":"06:22.085 ","End":"06:27.930","Text":"That tells us how many extra bicycles."},{"Start":"06:28.220 ","End":"06:31.970","Text":"I guess when it says how many bicycles that it need to sell,"},{"Start":"06:31.970 ","End":"06:34.340","Text":"it includes the original 300."},{"Start":"06:34.340 ","End":"06:43.920","Text":"I would just say 300 plus the 50 equals 350 bikes"},{"Start":"06:44.150 ","End":"06:50.770","Text":"will give the maximum revenue, and we\u0027re done."}],"ID":8629}],"Thumbnail":null,"ID":65392}]

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