Definite Integrals
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Inequalities
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Riemann Sum and Integrability
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- Riemann sum
- Riemann Sum and Integrability
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10
- Exercise 11
- Exercise 12
- Exercise 13
- Exercise 14 part 1
- Exercise 14 part 2
- Exercise 15 part 1
- Exercise 15 part 2
- Exercise 16
- Exercise 17
- Exercise 18

Fundamental Theorm of Calculus
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- Fundamental Theorem of Calculus
- First Fundamental Theorem I
- First Fundamental Theorem II
- Second Fundamental Theorem
- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9 part 1
- Exercise 9 part 2
- Exercise 9 part 3
- Exercise 9 part 4
- Exercise 10 part 1
- Exercise 10 part 2
- Exercise 10 part 3
- Exercise 11

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[{"Name":"Definite Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"6m 40s","ChapterTopicVideoID":6358,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/6358.jpeg","UploadDate":"2019-12-10T12:45:10.8030000","DurationForVideoObject":"PT6M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this clip, I\u0027m going to talk about definite integral,"},{"Start":"00:02.700 ","End":"00:04.530","Text":"first of all how to write it."},{"Start":"00:04.530 ","End":"00:06.990","Text":"We also use the integral sign,"},{"Start":"00:06.990 ","End":"00:10.230","Text":"but we put 2 numbers, 1 here,"},{"Start":"00:10.230 ","End":"00:12.810","Text":"let\u0027s call it a, and 1 here b in practice,"},{"Start":"00:12.810 ","End":"00:14.430","Text":"they could be actual numbers."},{"Start":"00:14.430 ","End":"00:17.085","Text":"We have a function f. Let\u0027s assume f is continuous,"},{"Start":"00:17.085 ","End":"00:19.320","Text":"not always necessary, but just to be safe,"},{"Start":"00:19.320 ","End":"00:21.300","Text":"we\u0027ll deal with continuous functions."},{"Start":"00:21.300 ","End":"00:23.190","Text":"I have the integral,"},{"Start":"00:23.190 ","End":"00:25.860","Text":"let me say from a to b of f of x dx,"},{"Start":"00:25.860 ","End":"00:27.645","Text":"and this is how it\u0027s written."},{"Start":"00:27.645 ","End":"00:29.730","Text":"Now how do we define it as"},{"Start":"00:29.730 ","End":"00:32.160","Text":"actually 2 definitions under"},{"Start":"00:32.160 ","End":"00:35.295","Text":"the fundamental theorem of calculus that says they\u0027re the same."},{"Start":"00:35.295 ","End":"00:36.900","Text":"1 of them is geometric,"},{"Start":"00:36.900 ","End":"00:39.150","Text":"1 relates to the indefinite integral."},{"Start":"00:39.150 ","End":"00:42.000","Text":"I\u0027ll go with the geometric definition first."},{"Start":"00:42.000 ","End":"00:46.145","Text":"I brought in a picture I found on the Internet to illustrate this."},{"Start":"00:46.145 ","End":"00:49.430","Text":"Here we have a pair of axis, x and y."},{"Start":"00:49.430 ","End":"00:54.675","Text":"Here is the graph of the function f of x here with the points a and b here."},{"Start":"00:54.675 ","End":"00:58.050","Text":"Usually, a is on the left b is on the right, but not necessarily."},{"Start":"00:58.050 ","End":"01:00.800","Text":"For the moment we\u0027ll assume a is on the left b is on the right."},{"Start":"01:00.800 ","End":"01:03.650","Text":"If not, you\u0027ll have to reverse the answer."},{"Start":"01:03.650 ","End":"01:08.240","Text":"The integral from a to b of f of x dx as written here,"},{"Start":"01:08.240 ","End":"01:13.265","Text":"is just the area that\u0027s between the curve and the x-axis."},{"Start":"01:13.265 ","End":"01:16.985","Text":"Also going to assume that here we have a positive function."},{"Start":"01:16.985 ","End":"01:19.040","Text":"If a is bigger than b,"},{"Start":"01:19.040 ","End":"01:21.455","Text":"if a is on the right and b is on the left,"},{"Start":"01:21.455 ","End":"01:24.920","Text":"then we just take minus of the answer."},{"Start":"01:24.920 ","End":"01:31.340","Text":"I would say that the integral from b to a of f of"},{"Start":"01:31.340 ","End":"01:39.305","Text":"x dx is minus the integral from a to b of f of x dx."},{"Start":"01:39.305 ","End":"01:40.985","Text":"That\u0027s 1 thing."},{"Start":"01:40.985 ","End":"01:44.390","Text":"We can always assume that b is on the right and if not,"},{"Start":"01:44.390 ","End":"01:46.955","Text":"we just reverse and take the negative answer."},{"Start":"01:46.955 ","End":"01:48.829","Text":"The other thing is in this picture,"},{"Start":"01:48.829 ","End":"01:51.050","Text":"I assumed that the function was above."},{"Start":"01:51.050 ","End":"01:53.300","Text":"In general, that may not happen,"},{"Start":"01:53.300 ","End":"01:57.050","Text":"so I\u0027ll bring another picture where f goes from a to b,"},{"Start":"01:57.050 ","End":"02:00.320","Text":"but it\u0027s sometimes positive and sometimes negative,"},{"Start":"02:00.320 ","End":"02:01.535","Text":"and in this case,"},{"Start":"02:01.535 ","End":"02:05.180","Text":"it\u0027s not exactly the area under the curve."},{"Start":"02:05.180 ","End":"02:09.369","Text":"It is in essence if the curve is above the x-axis,"},{"Start":"02:09.369 ","End":"02:14.960","Text":"we take this area with a plus sign and the bits that the curve is below the x-axis,"},{"Start":"02:14.960 ","End":"02:17.380","Text":"we take those areas with a negative sign."},{"Start":"02:17.380 ","End":"02:21.230","Text":"This is like a negative area and then a bit of positive as well."},{"Start":"02:21.230 ","End":"02:23.705","Text":"We have this minus this plus this,"},{"Start":"02:23.705 ","End":"02:26.900","Text":"that will be what we call the definite integral from a to"},{"Start":"02:26.900 ","End":"02:30.800","Text":"b. I\u0027m not going to go into much more detail than that."},{"Start":"02:30.800 ","End":"02:34.070","Text":"There are exercises later on areas and things."},{"Start":"02:34.070 ","End":"02:37.850","Text":"But what I do want is the other definition of the definite integral,"},{"Start":"02:37.850 ","End":"02:39.890","Text":"which ties in with the indefinite."},{"Start":"02:39.890 ","End":"02:45.770","Text":"Suppose I have a function capital F of x,"},{"Start":"02:45.770 ","End":"02:48.470","Text":"which is the indefinite integral,"},{"Start":"02:48.470 ","End":"02:51.050","Text":"which is written also with this symbol,"},{"Start":"02:51.050 ","End":"02:55.415","Text":"but without anything written here and here of f of x dx."},{"Start":"02:55.415 ","End":"02:59.675","Text":"That means that F is an antiderivative of f,"},{"Start":"02:59.675 ","End":"03:02.300","Text":"usually become determinant precisely,"},{"Start":"03:02.300 ","End":"03:05.240","Text":"so we put this constant of integration."},{"Start":"03:05.240 ","End":"03:06.710","Text":"But in any event,"},{"Start":"03:06.710 ","End":"03:10.640","Text":"it means that F prime of x equals f."},{"Start":"03:10.640 ","End":"03:15.470","Text":"Let me just say that F will be 1 specific anti-derivative."},{"Start":"03:15.470 ","End":"03:16.610","Text":"Choose a particular C,"},{"Start":"03:16.610 ","End":"03:22.715","Text":"just fix some F. Then the other way of defining the integral from a to"},{"Start":"03:22.715 ","End":"03:29.150","Text":"b of f of x dx is just the antiderivative,"},{"Start":"03:29.150 ","End":"03:37.085","Text":"the indefinite integral applied to the point b minus f applied to the point a."},{"Start":"03:37.085 ","End":"03:40.220","Text":"There is a mathematical notation when you take a function,"},{"Start":"03:40.220 ","End":"03:42.395","Text":"substitute 2 values and subtract,"},{"Start":"03:42.395 ","End":"03:45.725","Text":"that\u0027s just f of x,"},{"Start":"03:45.725 ","End":"03:51.000","Text":"then we put a vertical line and we put here a and b,"},{"Start":"03:51.000 ","End":"03:54.750","Text":"which means substitute bx equals a and subtract,"},{"Start":"03:54.750 ","End":"03:57.395","Text":"sometimes we emphasize it by putting x equals,"},{"Start":"03:57.395 ","End":"04:01.400","Text":"there\u0027s a more old-fashioned notation which puts it in"},{"Start":"04:01.400 ","End":"04:06.050","Text":"square brackets and then you put the a here and the b here."},{"Start":"04:06.050 ","End":"04:08.900","Text":"Basically, if you see an expression not an integral with"},{"Start":"04:08.900 ","End":"04:11.990","Text":"a number on top and bottom usually means substitute this,"},{"Start":"04:11.990 ","End":"04:14.015","Text":"substitute this and subtract."},{"Start":"04:14.015 ","End":"04:18.185","Text":"I want to point out that the C would not make any difference."},{"Start":"04:18.185 ","End":"04:23.270","Text":"Suppose I took g of x is equal to f of x plus 5,"},{"Start":"04:23.270 ","End":"04:27.725","Text":"just some constant, and then if instead of F I use g,"},{"Start":"04:27.725 ","End":"04:33.260","Text":"I mean g of b minus g of a is going to come out the same because g"},{"Start":"04:33.260 ","End":"04:42.740","Text":"of b is going to be f of b plus 5 minus f of a plus 5."},{"Start":"04:42.740 ","End":"04:45.830","Text":"The 5s are going to cancel and it\u0027s just going to be"},{"Start":"04:45.830 ","End":"04:48.740","Text":"the same as this wherever constant I put,"},{"Start":"04:48.740 ","End":"04:50.660","Text":"it\u0027s going to appear plus and minus,"},{"Start":"04:50.660 ","End":"04:53.420","Text":"so it really doesn\u0027t depend on which"},{"Start":"04:53.420 ","End":"04:56.720","Text":"primitive or anti-derivative you choose here, though,"},{"Start":"04:56.720 ","End":"04:59.240","Text":"we don\u0027t really need the C. Let me just show you in"},{"Start":"04:59.240 ","End":"05:03.335","Text":"practice using this thing how we go about writing it."},{"Start":"05:03.335 ","End":"05:05.630","Text":"I\u0027m not going to relate to the area at the moment."},{"Start":"05:05.630 ","End":"05:10.160","Text":"But if I gave you an example let say I want the"},{"Start":"05:10.160 ","End":"05:17.715","Text":"integral from 1-2 of 3x squared dx,"},{"Start":"05:17.715 ","End":"05:20.020","Text":"then using this method,"},{"Start":"05:20.020 ","End":"05:23.470","Text":"we take F, which is an antiderivative."},{"Start":"05:23.470 ","End":"05:27.955","Text":"So what we do is we take the integral and irregular in the old sense,"},{"Start":"05:27.955 ","End":"05:32.230","Text":"which is the function x cubed right at the side."},{"Start":"05:32.230 ","End":"05:36.835","Text":"If I just say the integral of 3x squared dx,"},{"Start":"05:36.835 ","End":"05:42.850","Text":"then it\u0027s x cubed plus C. But here we just take any particular primitive."},{"Start":"05:42.850 ","End":"05:44.560","Text":"I\u0027ll take C equals 0 here,"},{"Start":"05:44.560 ","End":"05:48.730","Text":"and then we just put the numbers 1 and 2 here,"},{"Start":"05:48.730 ","End":"05:51.745","Text":"and then it means substitute 2,"},{"Start":"05:51.745 ","End":"05:56.080","Text":"so I\u0027ve got 2 cubed and then subtract substitute 1,"},{"Start":"05:56.080 ","End":"06:01.155","Text":"I get 1 cubed and that\u0027s 8 minus 1 is 7."},{"Start":"06:01.155 ","End":"06:03.650","Text":"Here\u0027s an example of an actual computation,"},{"Start":"06:03.650 ","End":"06:05.435","Text":"and this is how it typically goes."},{"Start":"06:05.435 ","End":"06:06.830","Text":"You\u0027re given numbers here,"},{"Start":"06:06.830 ","End":"06:10.520","Text":"you do the indefinite integral and you plug in the top limit,"},{"Start":"06:10.520 ","End":"06:13.595","Text":"plug in the bottom limit and subtract and you get the answer."},{"Start":"06:13.595 ","End":"06:17.570","Text":"But we also know that if you drew the graph of 3x squared,"},{"Start":"06:17.570 ","End":"06:20.905","Text":"and then I took 1 and I took 2,"},{"Start":"06:20.905 ","End":"06:26.775","Text":"then this area between 1 and 2 of the function 3x squared will also come out 7,"},{"Start":"06:26.775 ","End":"06:28.980","Text":"so that there are these 2 definitions,"},{"Start":"06:28.980 ","End":"06:34.265","Text":"1 involving area, 1 involving antiderivatives or indefinite integrals."},{"Start":"06:34.265 ","End":"06:35.630","Text":"That\u0027s all I have to say."},{"Start":"06:35.630 ","End":"06:41.250","Text":"The rest will be covered in the solved exercises. That\u0027s all for now."}],"ID":6370},{"Watched":false,"Name":"Exercise 1","Duration":"2m 10s","ChapterTopicVideoID":4509,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4509.jpeg","UploadDate":"2017-01-26T07:14:24.9030000","DurationForVideoObject":"PT2M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.155","Text":"In this exercise, we have to compute a definite integral."},{"Start":"00:04.155 ","End":"00:05.370","Text":"How do I know it\u0027s definite?"},{"Start":"00:05.370 ","End":"00:07.260","Text":"Because it\u0027s got numbers here and here."},{"Start":"00:07.260 ","End":"00:09.840","Text":"The way we approach this, is,"},{"Start":"00:09.840 ","End":"00:16.260","Text":"we start off with the indefinite integral 2x squared gives us raise the power,"},{"Start":"00:16.260 ","End":"00:18.465","Text":"it\u0027s x cubed and divide by 3."},{"Start":"00:18.465 ","End":"00:21.480","Text":"It\u0027s 2/3 x cubed."},{"Start":"00:21.480 ","End":"00:24.360","Text":"Then the 4x raise the power by 1,"},{"Start":"00:24.360 ","End":"00:26.940","Text":"I get x squared and divide by 2."},{"Start":"00:26.940 ","End":"00:30.794","Text":"That gives me minus 2x squared."},{"Start":"00:30.794 ","End":"00:36.299","Text":"The 1 is just x. I don\u0027t need a constant in a definite integral."},{"Start":"00:36.299 ","End":"00:43.505","Text":"All I have to do is indicate that this is going to be from 1 to 4."},{"Start":"00:43.505 ","End":"00:49.290","Text":"Now, the 1 to 4 I colored for me like in the red means minus,"},{"Start":"00:49.290 ","End":"00:50.980","Text":"and then the black means plus,"},{"Start":"00:50.980 ","End":"00:53.690","Text":"is I substitute x equals 4 first,"},{"Start":"00:53.690 ","End":"00:58.340","Text":"and then I subtract from it what I get when I substitute x equals 1."},{"Start":"00:58.340 ","End":"01:09.275","Text":"This is going to equal 2-thirds times 4 cubed minus 2 times 4 squared plus 4."},{"Start":"01:09.275 ","End":"01:12.209","Text":"That\u0027s the 4 part."},{"Start":"01:12.209 ","End":"01:15.795","Text":"Then I\u0027m going to get minus."},{"Start":"01:15.795 ","End":"01:18.600","Text":"We\u0027re going to do the red part."},{"Start":"01:18.600 ","End":"01:21.435","Text":"2/3 times 1 cubed,"},{"Start":"01:21.435 ","End":"01:25.100","Text":"minus 2 times 1 squared, plus 1."},{"Start":"01:25.100 ","End":"01:27.980","Text":"In each case, I just copied this and instead of x,"},{"Start":"01:27.980 ","End":"01:29.450","Text":"I put 4, instead of x,"},{"Start":"01:29.450 ","End":"01:31.895","Text":"I took 1 and I subtract the 2."},{"Start":"01:31.895 ","End":"01:34.595","Text":"Now it\u0027s just computations."},{"Start":"01:34.595 ","End":"01:38.270","Text":"4 cubed is 64,"},{"Start":"01:38.270 ","End":"01:43.950","Text":"2/3 of 64 is 42 and 2/3."},{"Start":"01:43.950 ","End":"01:45.905","Text":"Here we have at C,"},{"Start":"01:45.905 ","End":"01:48.890","Text":"4 squared is 16 minus 32,"},{"Start":"01:48.890 ","End":"01:53.090","Text":"plus 4, is like minus 28."},{"Start":"01:53.090 ","End":"01:55.850","Text":"Then we\u0027re going to have here"},{"Start":"01:55.850 ","End":"02:04.935","Text":"minus 2/3 powers of 1 or just 1 minus 2, and plus 1."},{"Start":"02:04.935 ","End":"02:06.555","Text":"What do we get?"},{"Start":"02:06.555 ","End":"02:11.170","Text":"13? That\u0027s the answer. Done."}],"ID":4518},{"Watched":false,"Name":"Exercise 2","Duration":"1m 37s","ChapterTopicVideoID":4510,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4510.jpeg","UploadDate":"2017-01-26T07:15:08.9070000","DurationForVideoObject":"PT1M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we have to compute the definite end to"},{"Start":"00:03.930 ","End":"00:08.085","Text":"grow it\u0027s definite because it has numbers here of this function."},{"Start":"00:08.085 ","End":"00:11.130","Text":"We happen to be lucky here because I noticed"},{"Start":"00:11.130 ","End":"00:14.400","Text":"immediately and perhaps if you have sharp eyes,"},{"Start":"00:14.400 ","End":"00:15.840","Text":"you would notice it too,"},{"Start":"00:15.840 ","End":"00:19.410","Text":"that the derivative of x squared plus x plus"},{"Start":"00:19.410 ","End":"00:24.240","Text":"1 is exactly 2x plus 1 because from here I get 2x, from here I get 1."},{"Start":"00:24.240 ","End":"00:27.180","Text":"It\u0027s worth noting this thing because then we\u0027re lucky,"},{"Start":"00:27.180 ","End":"00:33.855","Text":"we can use the formula that the integral of f prime over f,"},{"Start":"00:33.855 ","End":"00:35.790","Text":"where f is a function of x,"},{"Start":"00:35.790 ","End":"00:38.845","Text":"is exactly the natural log of"},{"Start":"00:38.845 ","End":"00:43.250","Text":"f. Here this is exactly the case where f is the denominator,"},{"Start":"00:43.250 ","End":"00:50.510","Text":"f prime is the numerator so we get the natural log of x squared"},{"Start":"00:50.510 ","End":"00:58.710","Text":"plus x plus 1 and this we have to take between 0 and 2,"},{"Start":"00:58.710 ","End":"01:00.520","Text":"which means that we plugin 2,"},{"Start":"01:00.520 ","End":"01:02.855","Text":"we plugin 0 and we subtract."},{"Start":"01:02.855 ","End":"01:04.700","Text":"What we get if we put in 2?"},{"Start":"01:04.700 ","End":"01:11.760","Text":"2 squared is 4 plus 2 plus 1 is 7."},{"Start":"01:13.760 ","End":"01:16.520","Text":"Don\u0027t need the absolute value of the 7,"},{"Start":"01:16.520 ","End":"01:20.180","Text":"that\u0027s also 7 less plugin 0,"},{"Start":"01:20.180 ","End":"01:25.835","Text":"we get natural log of 1 because 0 squared plus 0 plus 1 is 1."},{"Start":"01:25.835 ","End":"01:33.080","Text":"Now, natural log of 1 is 0 so we\u0027re just left with the natural log of 7."},{"Start":"01:33.080 ","End":"01:38.010","Text":"No need to compute it this is precise. We\u0027re done."}],"ID":4519},{"Watched":false,"Name":"Exercise 3","Duration":"4m 2s","ChapterTopicVideoID":4511,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4511.jpeg","UploadDate":"2017-01-26T07:16:18.8270000","DurationForVideoObject":"PT4M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"Here we have a definite integral to compute and in this case,"},{"Start":"00:04.770 ","End":"00:10.800","Text":"it looks like integration by parts and I\u0027m going to remind you of the formula that\u0027s the"},{"Start":"00:10.800 ","End":"00:19.455","Text":"integral of udv is equal to uv minus the integral of vdu."},{"Start":"00:19.455 ","End":"00:22.665","Text":"Now, when we have integration by parts,"},{"Start":"00:22.665 ","End":"00:26.985","Text":"there\u0027s 2 ways of solving it and I\u0027m going to do it both ways."},{"Start":"00:26.985 ","End":"00:33.030","Text":"One way, which I\u0027m going to do over here is just computing the indefinite integral"},{"Start":"00:33.030 ","End":"00:39.555","Text":"first so actually what I\u0027m going to do is just erase this and this and at the end,"},{"Start":"00:39.555 ","End":"00:42.195","Text":"I\u0027m going to substitute the 2 and the 3."},{"Start":"00:42.195 ","End":"00:45.890","Text":"This 1 is better to take as u and this 1 as"},{"Start":"00:45.890 ","End":"00:51.130","Text":"dv because we don\u0027t want to integrate the x we want to differentiate it."},{"Start":"00:51.130 ","End":"00:54.660","Text":"We get the u times v,"},{"Start":"00:54.660 ","End":"01:00.200","Text":"so here\u0027s u, we need to know du and v because we have u and dv,"},{"Start":"01:00.200 ","End":"01:03.410","Text":"du is just 1 dx,"},{"Start":"01:03.410 ","End":"01:06.890","Text":"which means dx and v is the integral of this,"},{"Start":"01:06.890 ","End":"01:09.830","Text":"so it\u0027s minus e to the minus x."},{"Start":"01:09.830 ","End":"01:13.160","Text":"Now if I substitute here, I get u,"},{"Start":"01:13.160 ","End":"01:14.930","Text":"which is x, v,"},{"Start":"01:14.930 ","End":"01:17.510","Text":"which is minus e to the minus x,"},{"Start":"01:17.510 ","End":"01:23.675","Text":"minus e to the minus x minus the integral of v,"},{"Start":"01:23.675 ","End":"01:32.040","Text":"which is minus e to the minus x and du, which is dx."},{"Start":"01:32.040 ","End":"01:35.345","Text":"This becomes minus x,"},{"Start":"01:35.345 ","End":"01:39.350","Text":"e to the minus x and then the integral of minus e to"},{"Start":"01:39.350 ","End":"01:43.625","Text":"the minus x is just plus e to the minus x."},{"Start":"01:43.625 ","End":"01:47.585","Text":"We still end up with a minus plus constant,"},{"Start":"01:47.585 ","End":"01:52.670","Text":"that\u0027s the indefinite and I want to go now to the definite,"},{"Start":"01:52.670 ","End":"01:55.235","Text":"so I just plug in the 2 and the 3."},{"Start":"01:55.235 ","End":"01:59.030","Text":"What I need to do here is to take minus x,"},{"Start":"01:59.030 ","End":"02:00.770","Text":"e to the minus x,"},{"Start":"02:00.770 ","End":"02:06.270","Text":"minus e to the minus x between 2 and 3."},{"Start":"02:06.270 ","End":"02:10.460","Text":"Then I\u0027ll take minus e to the minus x outside the brackets,"},{"Start":"02:10.460 ","End":"02:12.815","Text":"and I\u0027m left with x plus 1."},{"Start":"02:12.815 ","End":"02:16.530","Text":"That might be easier to substitute and then I\u0027ll take it from 2-3,"},{"Start":"02:16.530 ","End":"02:21.900","Text":"the whole thing and so we will get, if put in 3,"},{"Start":"02:21.900 ","End":"02:24.539","Text":"I\u0027ll get x plus 1 is 4,"},{"Start":"02:24.539 ","End":"02:26.130","Text":"so it\u0027s minus 4,"},{"Start":"02:26.130 ","End":"02:27.690","Text":"e to the minus 4,"},{"Start":"02:27.690 ","End":"02:31.559","Text":"and if I put in 2, that becomes 3,"},{"Start":"02:31.559 ","End":"02:33.950","Text":"so it\u0027s minus, minus,"},{"Start":"02:33.950 ","End":"02:35.500","Text":"that\u0027s a plus here,"},{"Start":"02:35.500 ","End":"02:39.240","Text":"3, e to the minus 2,"},{"Start":"02:39.240 ","End":"02:41.655","Text":"this should be a minus 3, sorry."},{"Start":"02:41.655 ","End":"02:46.445","Text":"In summary, this method is to find the indefinite integral"},{"Start":"02:46.445 ","End":"02:51.350","Text":"and when we found it then to plug in the 3 and the 2 and make it definite."},{"Start":"02:51.350 ","End":"02:57.200","Text":"The other method is to constantly stay with the definite with the 2 and the 3."},{"Start":"02:57.200 ","End":"03:00.950","Text":"Copying from here minus x,"},{"Start":"03:00.950 ","End":"03:06.680","Text":"e to the minus x and all this from 2-3, so not an integral,"},{"Start":"03:06.680 ","End":"03:09.290","Text":"just substitute because this is not an integral already,"},{"Start":"03:09.290 ","End":"03:17.875","Text":"minus the integral from 2-3 of minus e to the minus x, dx."},{"Start":"03:17.875 ","End":"03:19.670","Text":"I get minus x,"},{"Start":"03:19.670 ","End":"03:23.945","Text":"e to the minus x from 2-3,"},{"Start":"03:23.945 ","End":"03:32.945","Text":"minus this integral becomes e to the minus x also from 2-3."},{"Start":"03:32.945 ","End":"03:38.810","Text":"I can just combine them and have 1 thing going from 2-3, so let\u0027s see what we have."},{"Start":"03:38.810 ","End":"03:41.090","Text":"We have minus x,"},{"Start":"03:41.090 ","End":"03:43.384","Text":"e to the minus x,"},{"Start":"03:43.384 ","End":"03:51.620","Text":"minus e to the minus x from 2-3 and at this point,"},{"Start":"03:51.620 ","End":"03:54.935","Text":"we just finish it the same way as here."},{"Start":"03:54.935 ","End":"03:59.105","Text":"I mean, you can go back to here because this is the same as this."},{"Start":"03:59.105 ","End":"04:02.550","Text":"We\u0027re done, I mean, this is the answer."}],"ID":4520},{"Watched":false,"Name":"Exercise 4","Duration":"3m 36s","ChapterTopicVideoID":4512,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4512.jpeg","UploadDate":"2017-01-26T07:17:27.9800000","DurationForVideoObject":"PT3M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Here, we have to compute this definite integral,"},{"Start":"00:03.570 ","End":"00:05.970","Text":"has numbers here, so it\u0027s a definite integral,"},{"Start":"00:05.970 ","End":"00:09.960","Text":"and it looks like a case for integration by substitution,"},{"Start":"00:09.960 ","End":"00:12.360","Text":"where I\u0027m going to substitute the natural log of x."},{"Start":"00:12.360 ","End":"00:13.800","Text":"Now when we have a substitution,"},{"Start":"00:13.800 ","End":"00:16.500","Text":"there are 2 methods, and I\u0027m going to demonstrate both method."},{"Start":"00:16.500 ","End":"00:18.540","Text":"I\u0027ll demonstrate 1 method over here."},{"Start":"00:18.540 ","End":"00:21.690","Text":"Just copy the exercise from 1 to 4,"},{"Start":"00:21.690 ","End":"00:28.545","Text":"the integral of natural log^4 of x over x dx."},{"Start":"00:28.545 ","End":"00:33.800","Text":"I\u0027m going to make the substitution t equals natural log of x."},{"Start":"00:33.800 ","End":"00:37.340","Text":"Now in the first method, by substitution,"},{"Start":"00:37.340 ","End":"00:42.500","Text":"I\u0027m going to go into the world of t and I\u0027m never going to come back to the world of x."},{"Start":"00:42.500 ","End":"00:46.820","Text":"In the other method, we just do the indefinite integral first,"},{"Start":"00:46.820 ","End":"00:50.555","Text":"and then at the end we substitute the limits 1 and 4 and subtract."},{"Start":"00:50.555 ","End":"00:54.500","Text":"Let\u0027s go with this way where we just go totally over to t. What"},{"Start":"00:54.500 ","End":"00:59.970","Text":"we get is dt is 1 over x dx."},{"Start":"00:59.970 ","End":"01:04.045","Text":"I see here that I already have 1 over x dx,"},{"Start":"01:04.045 ","End":"01:07.010","Text":"so I don\u0027t need to play with this and"},{"Start":"01:07.010 ","End":"01:10.130","Text":"get what dx\u0027s separately because I only need dx over x,"},{"Start":"01:10.130 ","End":"01:11.780","Text":"so I get the integral."},{"Start":"01:11.780 ","End":"01:13.415","Text":"Before I write the limits,"},{"Start":"01:13.415 ","End":"01:16.175","Text":"these are limits for x, or let me just leave it blank for a moment."},{"Start":"01:16.175 ","End":"01:21.175","Text":"What I have is t^4 and dx over x is dt."},{"Start":"01:21.175 ","End":"01:26.130","Text":"But the thing is, that the limits have to be substituted also."},{"Start":"01:26.130 ","End":"01:28.460","Text":"If t is natural log of x,"},{"Start":"01:28.460 ","End":"01:30.665","Text":"when x is 1,"},{"Start":"01:30.665 ","End":"01:34.130","Text":"then t is natural log of 1."},{"Start":"01:34.130 ","End":"01:39.810","Text":"When x is 4, then t is natural log of 4."},{"Start":"01:39.810 ","End":"01:43.830","Text":"What I get is natural log of 1 is 0."},{"Start":"01:43.830 ","End":"01:49.170","Text":"Natural log of 4 is just natural log of t^4 dt."},{"Start":"01:49.170 ","End":"01:54.750","Text":"I\u0027m continuing here, so we get t^5 over"},{"Start":"01:54.750 ","End":"02:01.525","Text":"5 between the limits of 0 and natural log of 4."},{"Start":"02:01.525 ","End":"02:04.110","Text":"When t is 0,"},{"Start":"02:04.110 ","End":"02:05.520","Text":"this thing is 0."},{"Start":"02:05.520 ","End":"02:13.680","Text":"Basically, what I get is 1/5 natural log of 4^5 like the 5 here."},{"Start":"02:13.680 ","End":"02:15.570","Text":"That would be my answer."},{"Start":"02:15.570 ","End":"02:17.390","Text":"That\u0027s 1 method."},{"Start":"02:17.390 ","End":"02:23.375","Text":"Now, the other method is to just do the indefinite integral."},{"Start":"02:23.375 ","End":"02:26.360","Text":"Actually, I\u0027ve already done that in a way,"},{"Start":"02:26.360 ","End":"02:29.390","Text":"the indefinite integral of this."},{"Start":"02:29.390 ","End":"02:36.550","Text":"Then I make the substitution where t is natural log of x, the same substitution."},{"Start":"02:36.550 ","End":"02:38.240","Text":"I\u0027ll use it over there,"},{"Start":"02:38.240 ","End":"02:43.015","Text":"and then I get the integral of t^4 dt,"},{"Start":"02:43.015 ","End":"02:44.950","Text":"but indefinite, there\u0027s no limit."},{"Start":"02:44.950 ","End":"02:52.040","Text":"Then I get the same thing which is t^5 over 5 plus a constant,"},{"Start":"02:52.040 ","End":"02:54.110","Text":"and then I substitute back."},{"Start":"02:54.110 ","End":"02:55.730","Text":"Now here\u0027s the difference; here,"},{"Start":"02:55.730 ","End":"02:58.190","Text":"we didn\u0027t go back from t to x."},{"Start":"02:58.190 ","End":"03:02.720","Text":"Here, we go back and we say that t is natural log of x,"},{"Start":"03:02.720 ","End":"03:07.875","Text":"so I get natural log^5 of x over 5,"},{"Start":"03:07.875 ","End":"03:13.500","Text":"and then I put in the original limits 4 and 1."},{"Start":"03:13.500 ","End":"03:16.615","Text":"Then when x is equal to 4,"},{"Start":"03:16.615 ","End":"03:22.880","Text":"I get natural log^5 of 4 over 5."},{"Start":"03:22.880 ","End":"03:26.240","Text":"When x is 1, the natural log of x is 0,"},{"Start":"03:26.240 ","End":"03:31.035","Text":"so minus 0, and we basically got the same answer,"},{"Start":"03:31.035 ","End":"03:33.380","Text":"this, just erase the 0."},{"Start":"03:33.380 ","End":"03:36.570","Text":"These are the 2 ways and we are done."}],"ID":4521},{"Watched":false,"Name":"Exercise 5","Duration":"2m 18s","ChapterTopicVideoID":4513,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4513.jpeg","UploadDate":"2017-01-26T07:18:09.3230000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"Here we have to compute this definite integral,"},{"Start":"00:03.075 ","End":"00:04.650","Text":"just copied it over here."},{"Start":"00:04.650 ","End":"00:07.700","Text":"I\u0027m going to use a trigonometric identity to help us,"},{"Start":"00:07.700 ","End":"00:10.950","Text":"and that is that the cosine squared"},{"Start":"00:10.950 ","End":"00:15.660","Text":"of an angle Alpha is just 1/2 of 1"},{"Start":"00:15.660 ","End":"00:19.950","Text":"plus cosine of twice the angle, 2 Alpha."},{"Start":"00:19.950 ","End":"00:23.295","Text":"In our case, the Alpha is 4x."},{"Start":"00:23.295 ","End":"00:25.620","Text":"What I\u0027m going to get"},{"Start":"00:25.620 ","End":"00:29.370","Text":"is the integral from 1 to Pi."},{"Start":"00:29.370 ","End":"00:31.800","Text":"Now, the half I can take in front,"},{"Start":"00:31.800 ","End":"00:33.360","Text":"that\u0027s the 1/2,"},{"Start":"00:33.360 ","End":"00:35.180","Text":"and I\u0027ve got 1 plus cosine,"},{"Start":"00:35.180 ","End":"00:39.810","Text":"2 Alpha is just 8x dx."},{"Start":"00:39.810 ","End":"00:42.940","Text":"This gives me 1/2."},{"Start":"00:42.940 ","End":"00:44.645","Text":"Now, the integral of this,"},{"Start":"00:44.645 ","End":"00:46.370","Text":"the integral of 1 is x"},{"Start":"00:46.370 ","End":"00:49.040","Text":"and the integral of cosine 8x."},{"Start":"00:49.040 ","End":"00:50.824","Text":"Well, I\u0027ll give you another formula."},{"Start":"00:50.824 ","End":"00:57.095","Text":"The integral of cosine of ax dx in general,"},{"Start":"00:57.095 ","End":"00:59.645","Text":"is equal to 1 over a,"},{"Start":"00:59.645 ","End":"01:03.245","Text":"the sine of ax plus constant."},{"Start":"01:03.245 ","End":"01:07.849","Text":"What I get here is 1/2 x plus,"},{"Start":"01:07.849 ","End":"01:13.655","Text":"1 over a is 1 over 8 sine 8x."},{"Start":"01:13.655 ","End":"01:16.715","Text":"Now, I have to also put in the limits,"},{"Start":"01:16.715 ","End":"01:19.850","Text":"so that\u0027s from 1 to Pi,"},{"Start":"01:19.850 ","End":"01:21.635","Text":"and let\u0027s see what we get."},{"Start":"01:21.635 ","End":"01:23.450","Text":"We get 1/2."},{"Start":"01:23.450 ","End":"01:26.130","Text":"Now, if I put in Pi,"},{"Start":"01:26.130 ","End":"01:30.795","Text":"I get Pi, and what is sine of 8 Pi?"},{"Start":"01:30.795 ","End":"01:34.815","Text":"Sine of 8 Pi is like 4 times 2 Pi."},{"Start":"01:34.815 ","End":"01:36.860","Text":"It\u0027s 4 complete circles,"},{"Start":"01:36.860 ","End":"01:38.780","Text":"it\u0027s like the sine of 0."},{"Start":"01:38.780 ","End":"01:41.209","Text":"So the sine of 0 is 0."},{"Start":"01:41.209 ","End":"01:44.800","Text":"I\u0027m saying that this thing here is 0,"},{"Start":"01:44.800 ","End":"01:46.655","Text":"sine of 8 Pi,"},{"Start":"01:46.655 ","End":"01:49.580","Text":"and that just gives me nothing."},{"Start":"01:49.580 ","End":"01:51.410","Text":"Now, we subtract the lower bit."},{"Start":"01:51.410 ","End":"01:53.750","Text":"So it\u0027s minus 1,"},{"Start":"01:53.750 ","End":"01:58.815","Text":"and minus 1/8 of sine of 8x,"},{"Start":"01:58.815 ","End":"02:00.850","Text":"which is just 8."},{"Start":"02:00.850 ","End":"02:03.290","Text":"If I combine this, what do I get?"},{"Start":"02:03.290 ","End":"02:08.690","Text":"This disappears and I get 1/2 of Pi"},{"Start":"02:08.690 ","End":"02:13.690","Text":"minus 1 minus 1/8 sine 8,"},{"Start":"02:13.690 ","End":"02:16.350","Text":"and that\u0027s the answer."},{"Start":"02:16.350 ","End":"02:18.820","Text":"We\u0027re done."}],"ID":4522},{"Watched":false,"Name":"Exercise 6","Duration":"3m 10s","ChapterTopicVideoID":4514,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4514.jpeg","UploadDate":"2017-01-26T07:19:12.0270000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.810","Text":"In this exercise, we have to compute a definite integral which is defined piece-wise."},{"Start":"00:06.810 ","End":"00:12.390","Text":"I mean the function is defined piece-wise or to split function from 0 to 1, it\u0027s 1 way,"},{"Start":"00:12.390 ","End":"00:16.050","Text":"and from bigger than or equal to 1 it\u0027s another way."},{"Start":"00:16.050 ","End":"00:18.120","Text":"I\u0027ve copied the exercise."},{"Start":"00:18.120 ","End":"00:22.935","Text":"The way we handle this is we just split up the range from 0 to 4,"},{"Start":"00:22.935 ","End":"00:25.230","Text":"2 sub ranges from 0 to 1,"},{"Start":"00:25.230 ","End":"00:26.880","Text":"and from 1 to 4."},{"Start":"00:26.880 ","End":"00:32.100","Text":"What we get is the integral from 0 to 1 of f of"},{"Start":"00:32.100 ","End":"00:40.835","Text":"x dx plus the integral from 1 to 4 of f of x dx."},{"Start":"00:40.835 ","End":"00:42.515","Text":"By which I mean,"},{"Start":"00:42.515 ","End":"00:47.390","Text":"we can just rewrite this because we know what f of x is between 0 and 1."},{"Start":"00:47.390 ","End":"00:53.970","Text":"I\u0027ve got the integral of the square roots of x from 0 to 1 dx."},{"Start":"00:53.970 ","End":"00:55.265","Text":"Then the second bit,"},{"Start":"00:55.265 ","End":"00:57.650","Text":"f of x here is 1 over x squared."},{"Start":"00:57.650 ","End":"01:02.720","Text":"I have the integral from 1 to 4 of 1 over x squared dx."},{"Start":"01:02.720 ","End":"01:09.920","Text":"Now I\u0027d like to write this in exponential notation because we going to use the formula,"},{"Start":"01:09.920 ","End":"01:15.530","Text":"standard formula that the integral of x^n dx"},{"Start":"01:15.530 ","End":"01:22.925","Text":"is x^n plus 1 over n plus 1 plus the constant in the indefinite case."},{"Start":"01:22.925 ","End":"01:25.040","Text":"Coming back here, what we get,"},{"Start":"01:25.040 ","End":"01:27.020","Text":"first of all put into an exponential."},{"Start":"01:27.020 ","End":"01:29.815","Text":"This is x to the power of 1 half."},{"Start":"01:29.815 ","End":"01:35.925","Text":"This 1 is from 1 to 4 x to the minus 2 dx."},{"Start":"01:35.925 ","End":"01:38.030","Text":"At this point I apply this formula,"},{"Start":"01:38.030 ","End":"01:42.720","Text":"x to the 1 half plus 1 is x to the power of 3"},{"Start":"01:42.720 ","End":"01:48.020","Text":"over 2 and divide it by 3 over 2, like this."},{"Start":"01:48.020 ","End":"01:55.295","Text":"Then plus x to the minus 1 over minus 1."},{"Start":"01:55.295 ","End":"02:00.390","Text":"This 1 is taken from 0 to 1,"},{"Start":"02:00.390 ","End":"02:05.205","Text":"and this 1 I\u0027m taking from 1 to 4."},{"Start":"02:05.205 ","End":"02:07.425","Text":"Let\u0027s see, what do we get for the first 1?"},{"Start":"02:07.425 ","End":"02:12.275","Text":"When x is 1, it\u0027s 1 to the power of 3 over 2,"},{"Start":"02:12.275 ","End":"02:14.090","Text":"which is just 1,"},{"Start":"02:14.090 ","End":"02:17.735","Text":"1 over 3 over 2 is 2/3."},{"Start":"02:17.735 ","End":"02:21.560","Text":"When x is 0, it\u0027s just 0."},{"Start":"02:21.560 ","End":"02:24.980","Text":"This is 2/3 minus 0 plus."},{"Start":"02:24.980 ","End":"02:30.215","Text":"Now here I have to subtract what I get when x is 1 from what I get when x is 4."},{"Start":"02:30.215 ","End":"02:35.420","Text":"When x is 4, x to the minus 1 is 1 over 4,"},{"Start":"02:35.420 ","End":"02:40.795","Text":"so it\u0027s 1/4 over minus 1."},{"Start":"02:40.795 ","End":"02:45.075","Text":"When x is 1, it\u0027s 1 over 1,"},{"Start":"02:45.075 ","End":"02:48.915","Text":"which is 1 over minus 1."},{"Start":"02:48.915 ","End":"02:50.385","Text":"In other words, let\u0027s see,"},{"Start":"02:50.385 ","End":"02:57.965","Text":"this becomes minus 1/4 and this bit becomes plus 1,"},{"Start":"02:57.965 ","End":"03:00.395","Text":"so altogether what do I get;"},{"Start":"03:00.395 ","End":"03:04.160","Text":"2/3 minus 1/4 plus 1."},{"Start":"03:04.160 ","End":"03:08.010","Text":"I think it\u0027s 1 and 5/12 if I haven\u0027t made a mistake."},{"Start":"03:08.010 ","End":"03:10.990","Text":"Anyway, this should be our answer."}],"ID":4523},{"Watched":false,"Name":"Exercise 7","Duration":"6m 27s","ChapterTopicVideoID":4515,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4515.jpeg","UploadDate":"2017-03-12T00:28:52.4330000","DurationForVideoObject":"PT6M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.160","Text":"In this exercise, we have to compute the definite integral of this function,"},{"Start":"00:05.160 ","End":"00:07.635","Text":"which contains an absolute value."},{"Start":"00:07.635 ","End":"00:09.720","Text":"I don\u0027t like the absolute value,"},{"Start":"00:09.720 ","End":"00:11.850","Text":"so I\u0027m going to rewrite it as a split function,"},{"Start":"00:11.850 ","End":"00:13.575","Text":"or piece-wise defined function."},{"Start":"00:13.575 ","End":"00:18.060","Text":"So what we\u0027re integrating is some function f of x,"},{"Start":"00:18.060 ","End":"00:26.680","Text":"which is equal to the square root of 4 plus x minus 1 in absolute value."},{"Start":"00:26.960 ","End":"00:30.400","Text":"Well, let\u0027s remember what the absolute value is."},{"Start":"00:30.400 ","End":"00:38.000","Text":"The absolute value is defined as absolute value of sum number a is equal to piece-wise,"},{"Start":"00:38.000 ","End":"00:39.695","Text":"is either equal to a itself,"},{"Start":"00:39.695 ","End":"00:41.990","Text":"if a is already non-negative,"},{"Start":"00:41.990 ","End":"00:43.550","Text":"but if a is negative,"},{"Start":"00:43.550 ","End":"00:46.700","Text":"then we have to make it minus a to make it positive,"},{"Start":"00:46.700 ","End":"00:49.115","Text":"so this is how it\u0027s defined."},{"Start":"00:49.115 ","End":"00:50.860","Text":"What we get here,"},{"Start":"00:50.860 ","End":"00:54.200","Text":"if x minus 1 is bigger, or equal to 0,"},{"Start":"00:54.200 ","End":"00:57.110","Text":"then we can just drop the absolute value and we say it\u0027s"},{"Start":"00:57.110 ","End":"01:01.550","Text":"4 plus just x minus 1 under the square root,"},{"Start":"01:01.550 ","End":"01:05.695","Text":"provided that x minus 1 is bigger or equal to 0,"},{"Start":"01:05.695 ","End":"01:09.170","Text":"and if x minus 1 is less than 0,"},{"Start":"01:09.170 ","End":"01:15.425","Text":"then we get the square root of 4 minus from this minus here, x minus 1."},{"Start":"01:15.425 ","End":"01:19.165","Text":"Let me simplify this a bit, this will equal,"},{"Start":"01:19.165 ","End":"01:24.450","Text":"let\u0027s see, 4 plus x minus 1 is 3 plus x,"},{"Start":"01:24.450 ","End":"01:30.440","Text":"so let\u0027s write this as the square root of 3 plus x or x plus 3,"},{"Start":"01:30.440 ","End":"01:33.320","Text":"this will be when x minus 1 is bigger equal to 0,"},{"Start":"01:33.320 ","End":"01:35.630","Text":"which means x is bigger or equal to 1,"},{"Start":"01:35.630 ","End":"01:41.845","Text":"and it will equal 4 minus x minus 1 is 5 minus x,"},{"Start":"01:41.845 ","End":"01:44.700","Text":"that\u0027s when x minus 1 less than 0,"},{"Start":"01:44.700 ","End":"01:47.565","Text":"or x less than 1."},{"Start":"01:47.565 ","End":"01:55.260","Text":"What I have here is actually the integral of f of x dx,"},{"Start":"01:55.260 ","End":"01:57.075","Text":"because this is what my f be,"},{"Start":"01:57.075 ","End":"01:58.735","Text":"this was from minus 1-4."},{"Start":"01:58.735 ","End":"02:01.235","Text":"Now, what\u0027s something happens at x equals 1?"},{"Start":"02:01.235 ","End":"02:05.430","Text":"So I have to break the integral up before and after x equals 1,"},{"Start":"02:05.430 ","End":"02:12.180","Text":"so a first bit is from minus 1 to 1 of f of x dx,"},{"Start":"02:12.180 ","End":"02:18.780","Text":"and the second bit is from 1 up to 4 of f of x dx,"},{"Start":"02:18.780 ","End":"02:21.275","Text":"but I need to replace f with what it is,"},{"Start":"02:21.275 ","End":"02:31.080","Text":"so what it is is the integral from minus 1 to 1 of f of x. X is on the less than 1 bit,"},{"Start":"02:31.080 ","End":"02:35.880","Text":"so it\u0027s square root of 5 minus x dx,"},{"Start":"02:35.880 ","End":"02:40.440","Text":"and from 1-4, it\u0027s only bigger than 1 bit,"},{"Start":"02:40.440 ","End":"02:45.060","Text":"so it\u0027s the square root of x plus 3 dx."},{"Start":"02:45.060 ","End":"02:47.510","Text":"Now, we have to compute these 2 integrals separately,"},{"Start":"02:47.510 ","End":"02:49.430","Text":"and then add the results."},{"Start":"02:49.430 ","End":"02:51.740","Text":"I\u0027m going to write a formula here,"},{"Start":"02:51.740 ","End":"02:53.410","Text":"which will help me here,"},{"Start":"02:53.410 ","End":"02:55.325","Text":"and the formula is this,"},{"Start":"02:55.325 ","End":"03:00.155","Text":"which basically tells me how to integrate an exponent when it\u0027s not x,"},{"Start":"03:00.155 ","End":"03:01.820","Text":"but a linear function of x,"},{"Start":"03:01.820 ","End":"03:03.860","Text":"and it\u0027s very similar to what we do for x,"},{"Start":"03:03.860 ","End":"03:05.540","Text":"just raise the power by 1,"},{"Start":"03:05.540 ","End":"03:06.980","Text":"and divide by the new power,"},{"Start":"03:06.980 ","End":"03:12.065","Text":"except that we have to also divide by the coefficient of x."},{"Start":"03:12.065 ","End":"03:13.910","Text":"Now in our case, we don\u0027t have an exponent,"},{"Start":"03:13.910 ","End":"03:15.410","Text":"but yes we do really,"},{"Start":"03:15.410 ","End":"03:18.050","Text":"because the square root is to the power of 1/2,"},{"Start":"03:18.050 ","End":"03:22.820","Text":"so this is equal to the integral of minus 1 to 1,"},{"Start":"03:22.820 ","End":"03:28.230","Text":"of 5 minus x to the power of 1/2 dx,"},{"Start":"03:28.230 ","End":"03:34.370","Text":"plus the integral from 1 to 4 x plus 3 to the power of 1/2 dx."},{"Start":"03:34.370 ","End":"03:39.755","Text":"Now, if I interpret this formula for the case where n is 1/2,"},{"Start":"03:39.755 ","End":"03:43.310","Text":"what I get is that the integral of the square root of ax,"},{"Start":"03:43.310 ","End":"03:49.370","Text":"plus b dx is going to equal 1 over a,"},{"Start":"03:49.370 ","End":"03:57.980","Text":"ax plus b to the power of 1/2 plus 1 is 3 over 2 divided by 3 over 2,"},{"Start":"03:57.980 ","End":"04:00.125","Text":"so in this case,"},{"Start":"04:00.125 ","End":"04:07.980","Text":"we get 5 minus x to the power of 3 over 2 over 3 over 2,"},{"Start":"04:07.980 ","End":"04:11.270","Text":"but the a here is minus 1 is the coefficient of x."},{"Start":"04:11.270 ","End":"04:13.730","Text":"So I have to put a minus here,"},{"Start":"04:13.730 ","End":"04:18.490","Text":"and this will be taken between minus 1 and 1,"},{"Start":"04:18.490 ","End":"04:25.310","Text":"and the next bit will be x plus 3 to the power of 3 over 2,"},{"Start":"04:25.310 ","End":"04:27.995","Text":"also over 3 over 2,"},{"Start":"04:27.995 ","End":"04:32.305","Text":"this time between 1 and 4."},{"Start":"04:32.305 ","End":"04:36.230","Text":"This divided by 3 over 2 is a bit of a nuisance."},{"Start":"04:36.230 ","End":"04:40.645","Text":"I\u0027m going to take that divided by 3 over 2 is like multiplying by 2/3,"},{"Start":"04:40.645 ","End":"04:45.135","Text":"so I\u0027m just going to write this as 2/3 of"},{"Start":"04:45.135 ","End":"04:52.730","Text":"minus 5 minus x to the 3 over 2 from minus 1 to 1,"},{"Start":"04:52.730 ","End":"04:57.725","Text":"plus x plus 3 to the power of 3 over 2,"},{"Start":"04:57.725 ","End":"05:01.550","Text":"from 1 to 4, so let\u0027s see what we get, 2/3."},{"Start":"05:01.550 ","End":"05:05.265","Text":"Now, here we have to plug in 1 and minus 1 and subtract."},{"Start":"05:05.265 ","End":"05:07.395","Text":"When x is 1,"},{"Start":"05:07.395 ","End":"05:10.994","Text":"we get 5 minus 1 is 4,"},{"Start":"05:10.994 ","End":"05:14.290","Text":"4 to the power of 3 over 2 is 8,"},{"Start":"05:14.290 ","End":"05:17.045","Text":"and this would give me minus 8."},{"Start":"05:17.045 ","End":"05:19.550","Text":"When x is minus 1,"},{"Start":"05:19.550 ","End":"05:23.240","Text":"I\u0027ve got 5 minus minus 1, which is 6,"},{"Start":"05:23.240 ","End":"05:26.275","Text":"so it\u0027s 6 to the power of 3 over 2,"},{"Start":"05:26.275 ","End":"05:28.650","Text":"minus 6 to the power of 3 over 2,"},{"Start":"05:28.650 ","End":"05:30.690","Text":"which makes it plus, because I\u0027m subtracting,"},{"Start":"05:30.690 ","End":"05:35.010","Text":"it\u0027s minus minus 6 to the power of 3 over 2."},{"Start":"05:35.010 ","End":"05:40.265","Text":"Then the next bit is where I put x equals 4,"},{"Start":"05:40.265 ","End":"05:46.370","Text":"so I\u0027ve got 4 plus 3 is 7 to the power of 3 over 2,"},{"Start":"05:46.370 ","End":"05:48.620","Text":"and when x is 1,"},{"Start":"05:48.620 ","End":"05:52.235","Text":"I get 4 to the power of 3 over 2,"},{"Start":"05:52.235 ","End":"05:57.399","Text":"which is 8, but that\u0027s in a minus sign."},{"Start":"05:57.399 ","End":"06:00.060","Text":"We could leave it like this."},{"Start":"06:00.060 ","End":"06:04.410","Text":"I just like to make 1 simplification or maybe 2,"},{"Start":"06:04.410 ","End":"06:10.845","Text":"and what I want to say is that the minus 8 with the minus 8 become minus 16,"},{"Start":"06:10.845 ","End":"06:13.380","Text":"and if you\u0027re going to do it on the calculator,"},{"Start":"06:13.380 ","End":"06:19.460","Text":"it maybe easier to write it as decimal 6 to the 1.5,"},{"Start":"06:19.460 ","End":"06:25.205","Text":"and here 7 to the 1.5, and still with the 2/3 in front,"},{"Start":"06:25.205 ","End":"06:28.050","Text":"and that\u0027s the answer."}],"ID":4524},{"Watched":false,"Name":"Exercise 8","Duration":"7m 2s","ChapterTopicVideoID":4516,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4516.jpeg","UploadDate":"2017-03-12T23:23:45.5130000","DurationForVideoObject":"PT7M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this exercise, you have to compute the"},{"Start":"00:02.430 ","End":"00:05.160","Text":"following definite integral which I have copied over here."},{"Start":"00:05.160 ","End":"00:07.830","Text":"I want to warn you this exercise is going to be a bit"},{"Start":"00:07.830 ","End":"00:11.685","Text":"tedious and also we\u0027re going to use a trick that is not intuitive."},{"Start":"00:11.685 ","End":"00:13.019","Text":"Let\u0027s get started."},{"Start":"00:13.019 ","End":"00:19.740","Text":"The trick is to let x equals pi minus t. This trip won\u0027t work in general,"},{"Start":"00:19.740 ","End":"00:21.540","Text":"specifically with 0 and pi,"},{"Start":"00:21.540 ","End":"00:23.310","Text":"it\u0027s going to work and maybe with other numbers,"},{"Start":"00:23.310 ","End":"00:24.765","Text":"but in general not."},{"Start":"00:24.765 ","End":"00:31.485","Text":"Dx is equal to just minus 1 dt. It\u0027s minus dt."},{"Start":"00:31.485 ","End":"00:41.480","Text":"After the substitution, what we get is the integral x is pi minus t. Sine x is"},{"Start":"00:41.480 ","End":"00:45.005","Text":"sine of pi minus t"},{"Start":"00:45.005 ","End":"00:51.890","Text":"over 1 plus cosine x is cosine of pi minus t,"},{"Start":"00:51.890 ","End":"00:55.715","Text":"but it\u0027s squared, and dx is minus dt."},{"Start":"00:55.715 ","End":"01:00.065","Text":"I can put the minus here and the dt here."},{"Start":"01:00.065 ","End":"01:02.915","Text":"Finally, we have to also switch the limits."},{"Start":"01:02.915 ","End":"01:06.035","Text":"You see when x equals 0,"},{"Start":"01:06.035 ","End":"01:11.060","Text":"then t is equal to pi minus x."},{"Start":"01:11.060 ","End":"01:15.335","Text":"When x is 0, t is pi and when x is pi,"},{"Start":"01:15.335 ","End":"01:17.285","Text":"then t is 0."},{"Start":"01:17.285 ","End":"01:24.160","Text":"What we end up getting if we continue down here is minus the integral."},{"Start":"01:24.160 ","End":"01:27.739","Text":"Now, this was from pi to 0."},{"Start":"01:27.739 ","End":"01:32.260","Text":"Now, we don\u0027t usually like the upper limit to be less than the lower limit,"},{"Start":"01:32.260 ","End":"01:36.770","Text":"but there is a well-known rule that if you switch the top and the bottom,"},{"Start":"01:36.770 ","End":"01:38.875","Text":"then you can get rid of the minus."},{"Start":"01:38.875 ","End":"01:42.740","Text":"In fact, I\u0027m just going to erase this minus and write"},{"Start":"01:42.740 ","End":"01:47.400","Text":"it as the integral from 0 to pi of the same thing."},{"Start":"01:47.400 ","End":"01:51.695","Text":"Now I\u0027d like to remind you of some trigonometrical identities."},{"Start":"01:51.695 ","End":"01:57.880","Text":"1 of them is that the sine of pi minus t,"},{"Start":"01:57.880 ","End":"02:00.190","Text":"pi is 180 degrees, remember,"},{"Start":"02:00.190 ","End":"02:02.840","Text":"is the same as the sine of the angle."},{"Start":"02:02.840 ","End":"02:05.970","Text":"But with the cosine, it\u0027s a minus."},{"Start":"02:05.970 ","End":"02:13.880","Text":"Cosine of pi minus t is minus cosine t. We have the integral from 0 to pi of"},{"Start":"02:13.880 ","End":"02:22.145","Text":"pi minus t and sine of pi minus t is just sine t. Now cosine of pi t is minus cosine t,"},{"Start":"02:22.145 ","End":"02:28.235","Text":"but it\u0027s squared, so it\u0027s still cosine squared t because the minus squared is a plus."},{"Start":"02:28.235 ","End":"02:31.115","Text":"I\u0027m going to break it up into 2 bits."},{"Start":"02:31.115 ","End":"02:36.830","Text":"I\u0027m going to break up according to the pi minus t. There\u0027s a minus here,"},{"Start":"02:36.830 ","End":"02:40.760","Text":"and this minus enables me to break it up into"},{"Start":"02:40.760 ","End":"02:44.915","Text":"2 integrals pi with this thing and t with this thing."},{"Start":"02:44.915 ","End":"02:49.565","Text":"I\u0027m going to write it now as the integral from 0"},{"Start":"02:49.565 ","End":"02:54.350","Text":"to pi of pi times sine t. Now even write the pi"},{"Start":"02:54.350 ","End":"03:03.620","Text":"outside of sine t over 1 plus cosine squared t dt minus the"},{"Start":"03:03.620 ","End":"03:08.420","Text":"integral from 0 to pi t sine t"},{"Start":"03:08.420 ","End":"03:15.050","Text":"over 1 plus cosine squared of t. Now let\u0027s see where this gets us."},{"Start":"03:15.050 ","End":"03:17.530","Text":"Now here\u0027s the other trick that we\u0027re going to use."},{"Start":"03:17.530 ","End":"03:20.570","Text":"The original integral from 0 to pi."},{"Start":"03:20.570 ","End":"03:23.120","Text":"Let\u0027s give it a letter I for integrals,"},{"Start":"03:23.120 ","End":"03:32.095","Text":"so I\u0027ll call it letter I. I claim that this integral here is also equal to i."},{"Start":"03:32.095 ","End":"03:33.695","Text":"This is exactly the same as this."},{"Start":"03:33.695 ","End":"03:36.215","Text":"The t or the x is a dummy variable."},{"Start":"03:36.215 ","End":"03:39.290","Text":"I get 2i is equal to this thing,"},{"Start":"03:39.290 ","End":"03:41.690","Text":"so i is 1.5 of this thing."},{"Start":"03:41.690 ","End":"03:48.080","Text":"Basically what I get is i equals pi over 2 times the"},{"Start":"03:48.080 ","End":"03:55.300","Text":"integral of sine t over 1 plus cosine squared t dt."},{"Start":"03:55.300 ","End":"03:57.335","Text":"I\u0027ll explain that again."},{"Start":"03:57.335 ","End":"03:58.520","Text":"If I have that,"},{"Start":"03:58.520 ","End":"04:04.559","Text":"let\u0027s say i is equal to something minus i,"},{"Start":"04:04.559 ","End":"04:12.015","Text":"then 2i is equal to that something and i is equal to 1.5 of that something."},{"Start":"04:12.015 ","End":"04:14.550","Text":"That\u0027s just what I did here basically."},{"Start":"04:14.550 ","End":"04:18.420","Text":"I forgot the limits from 0 to pi."},{"Start":"04:18.420 ","End":"04:22.710","Text":"What remains is to compute this integral."},{"Start":"04:22.710 ","End":"04:26.445","Text":"This is equal to pi over 2."},{"Start":"04:26.445 ","End":"04:29.720","Text":"This yellow thing is actually equal to pi over 2,"},{"Start":"04:29.720 ","End":"04:31.370","Text":"and that\u0027s what we\u0027re going to leave to the end."},{"Start":"04:31.370 ","End":"04:33.470","Text":"I\u0027ll also highlight that."},{"Start":"04:33.470 ","End":"04:39.040","Text":"The final answer is pi squared over 4,"},{"Start":"04:39.040 ","End":"04:44.515","Text":"but I have the debt of showing you that this thing is equal to this,"},{"Start":"04:44.515 ","End":"04:46.660","Text":"and that\u0027s what I\u0027ll do now."},{"Start":"04:46.660 ","End":"04:52.325","Text":"Now I\u0027m going to show you that the integral of sine t"},{"Start":"04:52.325 ","End":"04:59.820","Text":"over 1 plus cosine squared t dt from 0 to pi is equal to this."},{"Start":"04:59.820 ","End":"05:01.685","Text":"I\u0027m going to do it by substitution."},{"Start":"05:01.685 ","End":"05:08.585","Text":"I\u0027m going to say that let z equal cosine t,"},{"Start":"05:08.585 ","End":"05:15.145","Text":"and then dz is equal to minus sine t dt."},{"Start":"05:15.145 ","End":"05:20.725","Text":"What I want to do is put a minus here and a minus here,"},{"Start":"05:20.725 ","End":"05:24.200","Text":"and then I will get minus sine t dt."},{"Start":"05:24.200 ","End":"05:32.385","Text":"I\u0027ll get the integral of 1 over 1 plus z squared dz."},{"Start":"05:32.385 ","End":"05:37.900","Text":"We have to substitute the limit when t is 0,"},{"Start":"05:37.900 ","End":"05:43.395","Text":"that means that z is cosine of 0 which is 1,"},{"Start":"05:43.395 ","End":"05:51.295","Text":"and when t is pi then z is cosine pi which is minus 1."},{"Start":"05:51.295 ","End":"06:01.115","Text":"This is equal to minus the integral from minus 1 to 1 of 1 over 1 plus z squared dz."},{"Start":"06:01.115 ","End":"06:05.775","Text":"This is an immediate integral and it\u0027s the arctangent."},{"Start":"06:05.775 ","End":"06:14.490","Text":"We get minus the arctangent of z from minus 1 to 1."},{"Start":"06:14.490 ","End":"06:17.240","Text":"I made a small mistake."},{"Start":"06:17.240 ","End":"06:20.240","Text":"I forgot to carry the minus over."},{"Start":"06:20.240 ","End":"06:22.310","Text":"If this is a minus,"},{"Start":"06:22.310 ","End":"06:24.505","Text":"then this becomes a plus."},{"Start":"06:24.505 ","End":"06:29.675","Text":"What we get is the arctangent of"},{"Start":"06:29.675 ","End":"06:35.510","Text":"1 minus the arctangent of minus 1."},{"Start":"06:35.510 ","End":"06:37.490","Text":"Now I know that the angle whose tangent is 1 is"},{"Start":"06:37.490 ","End":"06:42.365","Text":"45 degrees and here it\u0027s minus 45 degrees, but it\u0027s pi over 4."},{"Start":"06:42.365 ","End":"06:45.170","Text":"Basically, this is pi over 4 minus,"},{"Start":"06:45.170 ","End":"06:47.225","Text":"minus pi over 4,"},{"Start":"06:47.225 ","End":"06:50.445","Text":"which equals pi over 2."},{"Start":"06:50.445 ","End":"06:54.155","Text":"This is what we have to show that this yellow equals this yellow,"},{"Start":"06:54.155 ","End":"06:57.260","Text":"and indeed this is equal to pi over 2."},{"Start":"06:57.260 ","End":"07:03.510","Text":"That\u0027s the debt we owed and we are done that this is the answer."}],"ID":4525},{"Watched":false,"Name":"Exercise 9","Duration":"5m 2s","ChapterTopicVideoID":4517,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4517.jpeg","UploadDate":"2017-01-26T07:21:39.4330000","DurationForVideoObject":"PT5M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this exercise, we have to compute a definite"},{"Start":"00:02.940 ","End":"00:07.620","Text":"integral from 0 to pi over 2 of this stuff dx."},{"Start":"00:07.620 ","End":"00:09.510","Text":"I\u0027ve copied it over here."},{"Start":"00:09.510 ","End":"00:13.035","Text":"We\u0027re actually going to use a trick here."},{"Start":"00:13.035 ","End":"00:17.440","Text":"We\u0027re going to rely on the fact that specifically it\u0027s from 0 to pi over 2."},{"Start":"00:17.440 ","End":"00:20.345","Text":"Let me just call this integral I,"},{"Start":"00:20.345 ","End":"00:21.950","Text":"because I\u0027ll need that later."},{"Start":"00:21.950 ","End":"00:23.510","Text":"Now, here\u0027s the trick,"},{"Start":"00:23.510 ","End":"00:26.000","Text":"not exactly a trick, but something that\u0027s not intuitive,"},{"Start":"00:26.000 ","End":"00:34.664","Text":"is to make a substitution where x is equal to pi over 2 minus t. Therefore,"},{"Start":"00:34.664 ","End":"00:40.420","Text":"dx will equal minus 1 dt or just minus dt."},{"Start":"00:40.420 ","End":"00:42.050","Text":"But when we substitute,"},{"Start":"00:42.050 ","End":"00:44.620","Text":"we also have to substitute the limits."},{"Start":"00:44.620 ","End":"00:47.475","Text":"When x is 0,"},{"Start":"00:47.475 ","End":"00:53.535","Text":"then t is pi over 2 minus 0 is pi over 2."},{"Start":"00:53.535 ","End":"00:56.429","Text":"If x is pi over 2,"},{"Start":"00:56.429 ","End":"00:59.580","Text":"then t is equal to 0."},{"Start":"00:59.580 ","End":"01:02.145","Text":"We get the integral,"},{"Start":"01:02.145 ","End":"01:12.690","Text":"but from pi over 2 to 0 of the 4th root of sine of pi over 2 minus t,"},{"Start":"01:12.690 ","End":"01:19.439","Text":"over the 4th root of sine pi over 2 minus t,"},{"Start":"01:19.439 ","End":"01:27.285","Text":"plus the 4th root of cosine of pi over 2 minus t,"},{"Start":"01:27.285 ","End":"01:31.440","Text":"and all this dx is minus dt."},{"Start":"01:31.440 ","End":"01:35.960","Text":"I\u0027m going to do several things now to simplify this."},{"Start":"01:35.960 ","End":"01:38.540","Text":"The first thing I\u0027m going to do is that if we have the"},{"Start":"01:38.540 ","End":"01:41.990","Text":"integral of some function from a to b,"},{"Start":"01:41.990 ","End":"01:47.420","Text":"then this is the same thing as minus the integral from b to a of"},{"Start":"01:47.420 ","End":"01:53.030","Text":"that same f. I\u0027m going to basically switch these 2 and get rid of this minus."},{"Start":"01:53.030 ","End":"01:58.010","Text":"The second thing is that pi over 2 minus an angle is a complimentary angle."},{"Start":"01:58.010 ","End":"02:02.930","Text":"We know that the sine of an angle is equal to the cosine of the complimentary angle."},{"Start":"02:02.930 ","End":"02:09.575","Text":"Basically, what I need to know is that sine of pi over 2 minus t is cosine t."},{"Start":"02:09.575 ","End":"02:16.915","Text":"Cosine of pi over 2 minus t is sine t. Eventually I get,"},{"Start":"02:16.915 ","End":"02:21.015","Text":"I\u0027ve switched the order 0 to pi over 2."},{"Start":"02:21.015 ","End":"02:26.990","Text":"Here I have the 4th root of cosine t from what I wrote here,"},{"Start":"02:26.990 ","End":"02:32.220","Text":"and here the 4th root of sine t and plus"},{"Start":"02:32.220 ","End":"02:40.594","Text":"the 4th root of cosine t. What I\u0027d like to do is replace t by x."},{"Start":"02:40.594 ","End":"02:42.990","Text":"The actual letter is not important,"},{"Start":"02:42.990 ","End":"02:45.065","Text":"it\u0027s still going to be the same thing."},{"Start":"02:45.065 ","End":"02:50.015","Text":"This is equal to the integral from 0 to pi over 2,"},{"Start":"02:50.015 ","End":"02:52.430","Text":"just a letter replacement or if you want to think of"},{"Start":"02:52.430 ","End":"02:54.920","Text":"it as a substitution, that\u0027s okay too,"},{"Start":"02:54.920 ","End":"03:00.930","Text":"of the 4th root of cosine x over the 4th root of"},{"Start":"03:00.930 ","End":"03:07.810","Text":"sine x plus the 4th root of cosine x dx."},{"Start":"03:07.810 ","End":"03:12.455","Text":"Now this has got to also equal I because I started from I,"},{"Start":"03:12.455 ","End":"03:14.890","Text":"so this is also equal to I."},{"Start":"03:14.890 ","End":"03:17.555","Text":"But now you\u0027ll see where the trick comes in."},{"Start":"03:17.555 ","End":"03:20.240","Text":"If I say what is I plus I,"},{"Start":"03:20.240 ","End":"03:23.915","Text":"I can get that 2I is equal to,"},{"Start":"03:23.915 ","End":"03:27.730","Text":"so I plus I I can take this plus this."},{"Start":"03:27.730 ","End":"03:29.600","Text":"I don\u0027t have to take twice this or twice this,"},{"Start":"03:29.600 ","End":"03:32.705","Text":"I can take 1 of these and 1 of these each of them is equal to I."},{"Start":"03:32.705 ","End":"03:36.315","Text":"It\u0027s the integral from 0 to pi over 2."},{"Start":"03:36.315 ","End":"03:37.880","Text":"You know what? I\u0027ll use copy paste."},{"Start":"03:37.880 ","End":"03:40.040","Text":"The first one I copied here."},{"Start":"03:40.040 ","End":"03:42.230","Text":"This form I copied here,"},{"Start":"03:42.230 ","End":"03:44.065","Text":"and that\u0027s equal to 2I."},{"Start":"03:44.065 ","End":"03:48.090","Text":"Now look, these 2 integrals have the same upper and lower limit,"},{"Start":"03:48.090 ","End":"03:50.285","Text":"so I can add the 2 functions."},{"Start":"03:50.285 ","End":"03:51.695","Text":"I can add this to this."},{"Start":"03:51.695 ","End":"03:53.740","Text":"Now they have the same denominator."},{"Start":"03:53.740 ","End":"04:00.980","Text":"What I\u0027m going to get is the integral from 0 to pi over 2 of the sum of these 2 things."},{"Start":"04:00.980 ","End":"04:09.860","Text":"Now the denominator is just the 4th root of sine x plus the fourth root of cosine x."},{"Start":"04:09.860 ","End":"04:12.320","Text":"Then I add the numerators from here,"},{"Start":"04:12.320 ","End":"04:15.455","Text":"the 4th root of sine x,"},{"Start":"04:15.455 ","End":"04:20.825","Text":"from here, the 4th root of cosine x and dx."},{"Start":"04:20.825 ","End":"04:24.935","Text":"But look, this fraction has the same numerator as the denominator."},{"Start":"04:24.935 ","End":"04:26.975","Text":"That\u0027s got to be equal to 1."},{"Start":"04:26.975 ","End":"04:32.565","Text":"It\u0027s equal to the integral from 0 to pi over 2 of 1 dx."},{"Start":"04:32.565 ","End":"04:38.850","Text":"The integral of 1 is just x taken from 0 to pi over 2."},{"Start":"04:38.850 ","End":"04:40.500","Text":"If I put pi over 2,"},{"Start":"04:40.500 ","End":"04:41.910","Text":"it\u0027s pi over 2."},{"Start":"04:41.910 ","End":"04:44.160","Text":"If x is 0, then x is 0."},{"Start":"04:44.160 ","End":"04:47.690","Text":"All together, this is equal to pi over 2, but this is 2I."},{"Start":"04:47.690 ","End":"04:49.010","Text":"I\u0027ll just copy it over here."},{"Start":"04:49.010 ","End":"04:50.950","Text":"This is 2I."},{"Start":"04:50.950 ","End":"04:54.889","Text":"Finally, I which is the integral itself,"},{"Start":"04:54.889 ","End":"05:03.480","Text":"is equal to pi over 4 because I divide this by 2 and this will be my answer. We are done."}],"ID":4526},{"Watched":false,"Name":"Exercise 10 part a","Duration":"3m 40s","ChapterTopicVideoID":4518,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4518.jpeg","UploadDate":"2017-01-26T07:22:49.6170000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.795","Text":"In this exercise, f is a continuous function and we have to prove 2 things."},{"Start":"00:06.795 ","End":"00:09.119","Text":"A, if f is even,"},{"Start":"00:09.119 ","End":"00:15.750","Text":"then the integral from minus a to a of f is twice the integral from 0 to a."},{"Start":"00:15.750 ","End":"00:17.430","Text":"Let\u0027s leave b for later,"},{"Start":"00:17.430 ","End":"00:22.360","Text":"and let\u0027s start off with a. I\u0027d like to remind you what even means."},{"Start":"00:22.360 ","End":"00:31.144","Text":"Means in general that f of minus x is the same as f of x for all x."},{"Start":"00:31.144 ","End":"00:33.079","Text":"Okay. If that\u0027s the case,"},{"Start":"00:33.079 ","End":"00:38.680","Text":"then what we have to do is split this up into 2 integrals."},{"Start":"00:38.680 ","End":"00:46.940","Text":"The integral from minus a to a of f of x dx x equals 0 is a mirror image, it\u0027s an axis."},{"Start":"00:46.940 ","End":"00:49.820","Text":"What we\u0027re going to do is take the interval from minus a to"},{"Start":"00:49.820 ","End":"00:53.755","Text":"a and break it up into 2 intervals from minus a to 0,"},{"Start":"00:53.755 ","End":"00:57.045","Text":"and then from 0 to a."},{"Start":"00:57.045 ","End":"00:59.950","Text":"Now here we have f of x dx,"},{"Start":"00:59.950 ","End":"01:04.720","Text":"and here we have also f of x dx."},{"Start":"01:04.720 ","End":"01:09.890","Text":"Now, what I\u0027m going to do is use the fact that in the negative interval,"},{"Start":"01:09.890 ","End":"01:11.930","Text":"the even property holds."},{"Start":"01:11.930 ","End":"01:16.280","Text":"This is equal to the integral from minus a to"},{"Start":"01:16.280 ","End":"01:22.980","Text":"0 of f of minus x dx plus same thing,"},{"Start":"01:22.980 ","End":"01:26.995","Text":"integral 0 to a f of x dx."},{"Start":"01:26.995 ","End":"01:29.870","Text":"Now here I\u0027d like to make a substitution."},{"Start":"01:29.870 ","End":"01:38.780","Text":"I\u0027d like to say that t is going to be minus x. T is minus x,"},{"Start":"01:38.780 ","End":"01:43.640","Text":"and so dt is minus dx,"},{"Start":"01:43.640 ","End":"01:53.795","Text":"and the limits 0 for x goes to 0 for t. If I put x equals minus a,"},{"Start":"01:53.795 ","End":"01:56.090","Text":"then t becomes a."},{"Start":"01:56.090 ","End":"01:59.660","Text":"In other words, these are the x values and these are the t values."},{"Start":"01:59.660 ","End":"02:07.815","Text":"We have to substitute everything and what I get is this equals the integral."},{"Start":"02:07.815 ","End":"02:14.855","Text":"Now, minus a to 0 becomes a to 0 minus x is t,"},{"Start":"02:14.855 ","End":"02:16.940","Text":"so that\u0027s f of t,"},{"Start":"02:16.940 ","End":"02:20.165","Text":"and if dt is minus dx,"},{"Start":"02:20.165 ","End":"02:27.335","Text":"I could put the dx on the other side and say this is minus dt plus other side the same."},{"Start":"02:27.335 ","End":"02:30.920","Text":"Now we have our usual trick that if we"},{"Start":"02:30.920 ","End":"02:35.360","Text":"reverse the top and bottom limits and make it from 0 to a,"},{"Start":"02:35.360 ","End":"02:37.390","Text":"we have to throw it a minus somewhere,"},{"Start":"02:37.390 ","End":"02:41.255","Text":"so I can just get rid of this minus and say this is f of t,"},{"Start":"02:41.255 ","End":"02:44.590","Text":"dt plus the same."},{"Start":"02:44.590 ","End":"02:46.785","Text":"Now here\u0027s the thing,"},{"Start":"02:46.785 ","End":"02:52.055","Text":"there\u0027s no significance to the particular letter t. If I put here f of u du,"},{"Start":"02:52.055 ","End":"02:54.110","Text":"or f of z dz,"},{"Start":"02:54.110 ","End":"02:55.430","Text":"or f of x dx,"},{"Start":"02:55.430 ","End":"02:56.810","Text":"even it would be the same thing,"},{"Start":"02:56.810 ","End":"02:58.120","Text":"the letter doesn\u0027t matter."},{"Start":"02:58.120 ","End":"02:59.990","Text":"I\u0027m going to go back to x."},{"Start":"02:59.990 ","End":"03:05.750","Text":"This is equal to the integral from 0 to a of f of x dx plus,"},{"Start":"03:05.750 ","End":"03:11.500","Text":"and now I will copy it out in full 0 to a f of x dx,"},{"Start":"03:11.500 ","End":"03:14.150","Text":"but look, this expression is the same as this expression,"},{"Start":"03:14.150 ","End":"03:15.665","Text":"so we have 2 of these."},{"Start":"03:15.665 ","End":"03:22.525","Text":"This is twice the integral from 0 to a of f of x dx."},{"Start":"03:22.525 ","End":"03:25.205","Text":"Let\u0027s see what we were asked to prove."},{"Start":"03:25.205 ","End":"03:30.365","Text":"You were asked to prove that this integral is twice the integral from 0 to a,"},{"Start":"03:30.365 ","End":"03:32.420","Text":"and I think this is what we have here,"},{"Start":"03:32.420 ","End":"03:37.760","Text":"so this is it what we had to prove for part a."},{"Start":"03:37.760 ","End":"03:41.010","Text":"Now let\u0027s go to part b."}],"ID":4527},{"Watched":false,"Name":"Exercise 10 part b","Duration":"2m 50s","ChapterTopicVideoID":4519,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4519.jpeg","UploadDate":"2017-03-12T23:32:01.1670000","DurationForVideoObject":"PT2M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"We just finished Part A,"},{"Start":"00:01.890 ","End":"00:03.480","Text":"now onto Part B."},{"Start":"00:03.480 ","End":"00:06.600","Text":"Previously, it was where f was even and now,"},{"Start":"00:06.600 ","End":"00:08.250","Text":"we have the case where f is odd,"},{"Start":"00:08.250 ","End":"00:11.685","Text":"and I\u0027ll remind you that odd means that f of minus x,"},{"Start":"00:11.685 ","End":"00:14.430","Text":"in general, is equal to minus f of x."},{"Start":"00:14.430 ","End":"00:15.830","Text":"What I\u0027m going to do is just,"},{"Start":"00:15.830 ","End":"00:17.535","Text":"first of all, copy this,"},{"Start":"00:17.535 ","End":"00:21.020","Text":"minus a to a of f of x dx,"},{"Start":"00:21.020 ","End":"00:24.555","Text":"and just like before, we split it up into 2 integrals."},{"Start":"00:24.555 ","End":"00:31.515","Text":"We first go from minus a to 0 and then from 0 to a."},{"Start":"00:31.515 ","End":"00:35.760","Text":"This here, I\u0027m going to do by a substitution."},{"Start":"00:35.760 ","End":"00:37.909","Text":"The first thing before the substitution,"},{"Start":"00:37.909 ","End":"00:42.445","Text":"I\u0027m going to say that this is the integral from minus a to 0,"},{"Start":"00:42.445 ","End":"00:49.580","Text":"and f of x is equal to minus f of minus x. I could put this minus on"},{"Start":"00:49.580 ","End":"00:57.855","Text":"the other side and say that this is minus f of minus x dx plus the same bit."},{"Start":"00:57.855 ","End":"01:00.450","Text":"You know what? I\u0027ll just put it like ditto sign."},{"Start":"01:00.450 ","End":"01:05.855","Text":"At this point, I\u0027m going to make a substitution for t to be equal to minus x,"},{"Start":"01:05.855 ","End":"01:09.530","Text":"so if t is equal to minus x,"},{"Start":"01:09.530 ","End":"01:13.140","Text":"then dt is minus 1dx."},{"Start":"01:13.140 ","End":"01:18.720","Text":"The limits. When x is equal to minus a,"},{"Start":"01:18.720 ","End":"01:22.560","Text":"then the t, which is minus x, is plus a,"},{"Start":"01:22.560 ","End":"01:26.960","Text":"and when x is 0 and t is minus 0,"},{"Start":"01:26.960 ","End":"01:34.915","Text":"which is also 0, so what I get is the integral from minus a to 0."},{"Start":"01:34.915 ","End":"01:41.970","Text":"The minus with the dx could be dt and minus x is t,"},{"Start":"01:41.970 ","End":"01:45.670","Text":"so I get f of t dt."},{"Start":"01:45.830 ","End":"01:50.580","Text":"I\u0027m going to switch upper and lower limits,"},{"Start":"01:50.580 ","End":"01:55.205","Text":"and we can do that provided that we introduce a minus sign."},{"Start":"01:55.205 ","End":"01:57.095","Text":"Why did I write minus a?"},{"Start":"01:57.095 ","End":"01:59.210","Text":"This is a. I fixed that,"},{"Start":"01:59.210 ","End":"02:00.815","Text":"and I completed the line."},{"Start":"02:00.815 ","End":"02:05.995","Text":"What I did is switch the 0 and the a and introduce an extra minus sign,"},{"Start":"02:05.995 ","End":"02:07.755","Text":"and this thing, all along,"},{"Start":"02:07.755 ","End":"02:12.475","Text":"have been dragging this thing and this thing from here."},{"Start":"02:12.475 ","End":"02:15.320","Text":"I\u0027m going to write it out in full again,"},{"Start":"02:15.320 ","End":"02:20.705","Text":"integral from 0 to a of f of x dx,"},{"Start":"02:20.705 ","End":"02:22.890","Text":"and this part, just like in Part A,"},{"Start":"02:22.890 ","End":"02:28.295","Text":"there\u0027s no special meaning to a letter t. I could make it back to x again."},{"Start":"02:28.295 ","End":"02:30.920","Text":"The letter itself has no particular significance,"},{"Start":"02:30.920 ","End":"02:35.030","Text":"so I get minus the integral of 0 to a of f"},{"Start":"02:35.030 ","End":"02:39.385","Text":"of x dx by just simply switching the name of the letter."},{"Start":"02:39.385 ","End":"02:43.775","Text":"Examine this. Here, I have minus something plus the same something."},{"Start":"02:43.775 ","End":"02:46.130","Text":"That is equal to 0,"},{"Start":"02:46.130 ","End":"02:48.725","Text":"and that\u0027s what we have to prove in Part B,"},{"Start":"02:48.725 ","End":"02:50.970","Text":"and now we\u0027re done."}],"ID":4528},{"Watched":false,"Name":"Exercise 11","Duration":"3m 27s","ChapterTopicVideoID":4520,"CourseChapterTopicPlaylistID":65395,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/4520.jpeg","UploadDate":"2017-03-12T23:48:56.4170000","DurationForVideoObject":"PT3M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this exercise, which is really 2 exercises,"},{"Start":"00:03.060 ","End":"00:04.904","Text":"we have to compute the integrals."},{"Start":"00:04.904 ","End":"00:08.910","Text":"Now the previous exercise was all about odd and even functions."},{"Start":"00:08.910 ","End":"00:11.430","Text":"We\u0027re going to use the results of that here."},{"Start":"00:11.430 ","End":"00:17.640","Text":"In part 1, we have the integral from minus 4^4 of"},{"Start":"00:17.640 ","End":"00:24.135","Text":"cosine x over x cubed plus x^5 dx."},{"Start":"00:24.135 ","End":"00:27.145","Text":"Now let\u0027s look at this function, the integrant."},{"Start":"00:27.145 ","End":"00:31.355","Text":"I\u0027m claiming that it\u0027s an odd function because the numerator,"},{"Start":"00:31.355 ","End":"00:34.384","Text":"this cosine x is even."},{"Start":"00:34.384 ","End":"00:38.780","Text":"X cubed is odd and x^5 is odd."},{"Start":"00:38.780 ","End":"00:40.250","Text":"That\u0027s easy to see."},{"Start":"00:40.250 ","End":"00:41.990","Text":"If I replace x by minus x,"},{"Start":"00:41.990 ","End":"00:44.865","Text":"I get minus^5 or minus^3,"},{"Start":"00:44.865 ","End":"00:46.610","Text":"because these are odd numbers."},{"Start":"00:46.610 ","End":"00:51.710","Text":"This is odd, this is odd and the sum of odd is odd and even over odd is odd."},{"Start":"00:51.710 ","End":"00:56.135","Text":"Altogether, I get that this function is odd."},{"Start":"00:56.135 ","End":"01:01.220","Text":"Now we have the result that if f is odd,"},{"Start":"01:01.220 ","End":"01:09.515","Text":"then the integral from minus a^a of f of x dx is equal to 0."},{"Start":"01:09.515 ","End":"01:15.430","Text":"This is what we exactly have here because this whole thing is an odd function."},{"Start":"01:15.430 ","End":"01:18.655","Text":"Using this theorem, where a is 4,"},{"Start":"01:18.655 ","End":"01:25.595","Text":"we just get that the answer is that this thing is equal to 0, and we\u0027re done."},{"Start":"01:25.595 ","End":"01:28.095","Text":"Now, let\u0027s see part 2."},{"Start":"01:28.095 ","End":"01:38.615","Text":"We have the integral from minus 1^1 of sine x plus 1 over x squared plus 1 dx."},{"Start":"01:38.615 ","End":"01:41.015","Text":"Now, here it\u0027s a little bit trickier."},{"Start":"01:41.015 ","End":"01:45.395","Text":"The denominator is certainly an even function."},{"Start":"01:45.395 ","End":"01:51.760","Text":"The thing is about the numerator is that this is odd and this is even."},{"Start":"01:51.760 ","End":"01:56.315","Text":"I don\u0027t know what an odd plus an even is but if I break it up into 2,"},{"Start":"01:56.315 ","End":"01:57.710","Text":"that will help me more."},{"Start":"01:57.710 ","End":"02:01.685","Text":"Let\u0027s break it up into 2 separate integrals."},{"Start":"02:01.685 ","End":"02:08.480","Text":"Then we have minus 1^1 of sine x over x squared plus"},{"Start":"02:08.480 ","End":"02:12.290","Text":"1 dx plus the integral from"},{"Start":"02:12.290 ","End":"02:19.550","Text":"minus 1^1 of 1 over x squared plus 1 dx."},{"Start":"02:19.550 ","End":"02:26.835","Text":"In this case, I have that this is still even and this is still odd."},{"Start":"02:26.835 ","End":"02:32.985","Text":"That makes this whole thing odd a and minus a is 1, and minus 1."},{"Start":"02:32.985 ","End":"02:36.860","Text":"Then we get this using the same theorem that this is equal to 0."},{"Start":"02:36.860 ","End":"02:39.154","Text":"I still have this part."},{"Start":"02:39.154 ","End":"02:40.680","Text":"Now this is an even function,"},{"Start":"02:40.680 ","End":"02:43.910","Text":"and there is a similar rule for even functions,"},{"Start":"02:43.910 ","End":"02:45.590","Text":"but it\u0027s not really going to help me."},{"Start":"02:45.590 ","End":"02:47.945","Text":"I\u0027m just going to carry on with this."},{"Start":"02:47.945 ","End":"02:51.440","Text":"Now this equals and I\u0027m just continuing with this part."},{"Start":"02:51.440 ","End":"02:59.000","Text":"Immediate integral is arctangent of x and I have to take this between minus 1 and 1."},{"Start":"02:59.000 ","End":"03:06.620","Text":"Now, the arctangent of 1 minus the arctangent of minus 1."},{"Start":"03:06.620 ","End":"03:08.810","Text":"Now if you remember your trigonometry,"},{"Start":"03:08.810 ","End":"03:14.570","Text":"then you\u0027d remember that the arctangent of 1 is 45 degrees or Pi over"},{"Start":"03:14.570 ","End":"03:21.945","Text":"4 and the arctangent of minus 1 is minus Pi over 4 minus 45 degrees."},{"Start":"03:21.945 ","End":"03:25.710","Text":"All together we get Pi over 2."},{"Start":"03:25.710 ","End":"03:28.120","Text":"That\u0027s the answer"}],"ID":4529}],"Thumbnail":null,"ID":65395},{"Name":"Inequalities","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Inequalities - Part 1","Duration":"4m 40s","ChapterTopicVideoID":8330,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8330.jpeg","UploadDate":"2019-12-10T12:44:23.6600000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this clip, I\u0027ll be talking about some"},{"Start":"00:02.910 ","End":"00:06.300","Text":"inequalities involving the indefinite integral."},{"Start":"00:06.300 ","End":"00:08.070","Text":"These are particularly useful when"},{"Start":"00:08.070 ","End":"00:11.115","Text":"we can evaluate the indefinite integral."},{"Start":"00:11.115 ","End":"00:12.360","Text":"At least not exactly,"},{"Start":"00:12.360 ","End":"00:14.190","Text":"but we can give an estimate from above"},{"Start":"00:14.190 ","End":"00:16.170","Text":"or below as to what it might be."},{"Start":"00:16.170 ","End":"00:19.800","Text":"I\u0027m going to be giving 2 inequalities."},{"Start":"00:19.800 ","End":"00:21.800","Text":"I\u0027ll write them each 1,"},{"Start":"00:21.800 ","End":"00:23.180","Text":"and then I\u0027ll explain."},{"Start":"00:23.180 ","End":"00:25.820","Text":"Here\u0027s the first formula which"},{"Start":"00:25.820 ","End":"00:28.280","Text":"gives us an estimate of an integral."},{"Start":"00:28.280 ","End":"00:30.380","Text":"It gives an upper bound and a lower bound."},{"Start":"00:30.380 ","End":"00:32.030","Text":"It says that if we have the integral"},{"Start":"00:32.030 ","End":"00:33.995","Text":"from a to b of a function,"},{"Start":"00:33.995 ","End":"00:35.960","Text":"then it\u0027s less than or equal to"},{"Start":"00:35.960 ","End":"00:38.435","Text":"big M times b minus a."},{"Start":"00:38.435 ","End":"00:39.620","Text":"This is b and this is a."},{"Start":"00:39.620 ","End":"00:41.510","Text":"In a moment I\u0027ll say what M is,"},{"Start":"00:41.510 ","End":"00:44.480","Text":"and it\u0027s bigger or equal to little m times b minus a."},{"Start":"00:44.480 ","End":"00:47.540","Text":"Big M is the absolute maximum"},{"Start":"00:47.540 ","End":"00:50.087","Text":"of the function on the interval a,b,"},{"Start":"00:50.087 ","End":"00:52.820","Text":"means when x is between a and b inclusive,"},{"Start":"00:52.820 ","End":"00:55.390","Text":"and m is the absolute minimum."},{"Start":"00:55.390 ","End":"00:57.570","Text":"Here is a couple of axes."},{"Start":"00:57.570 ","End":"01:00.420","Text":"Let\u0027s take a and b along the x-axis."},{"Start":"01:00.420 ","End":"01:02.690","Text":"Let\u0027s say that this here is a,"},{"Start":"01:02.690 ","End":"01:05.805","Text":"and this here is b."},{"Start":"01:05.805 ","End":"01:08.510","Text":"Let\u0027s take some function f of x,"},{"Start":"01:08.510 ","End":"01:11.195","Text":"y equals f of x."},{"Start":"01:11.195 ","End":"01:12.220","Text":"Here\u0027s a."},{"Start":"01:12.220 ","End":"01:15.395","Text":"I want to take a vertical line from a,"},{"Start":"01:15.395 ","End":"01:18.530","Text":"and I want to take a vertical line from b."},{"Start":"01:18.530 ","End":"01:22.280","Text":"I want to shade the area under the curve."},{"Start":"01:22.280 ","End":"01:24.770","Text":"This green area actually represents"},{"Start":"01:24.770 ","End":"01:27.920","Text":"the integral from a to b of f of x/dx."},{"Start":"01:27.920 ","End":"01:32.390","Text":"Let me draw the absolute maximum in the interval a,b."},{"Start":"01:32.390 ","End":"01:34.850","Text":"I can see that the maximum is here,"},{"Start":"01:34.850 ","End":"01:38.914","Text":"and the absolute minimum would be here."},{"Start":"01:38.914 ","End":"01:44.780","Text":"This will be the absolute maximum of f of x"},{"Start":"01:44.780 ","End":"01:48.455","Text":"from a to b on the interval a,b"},{"Start":"01:48.455 ","End":"01:52.730","Text":"and this here will be the absolute minimum"},{"Start":"01:52.730 ","End":"01:56.330","Text":"on the interval a,b of the function f."},{"Start":"01:56.330 ","End":"02:00.125","Text":"I want to estimate the area under the curve."},{"Start":"02:00.125 ","End":"02:02.510","Text":"I want to estimate it from above and from below."},{"Start":"02:02.510 ","End":"02:05.524","Text":"If I draw a line here,"},{"Start":"02:05.524 ","End":"02:08.395","Text":"the area of this rectangle,"},{"Start":"02:08.395 ","End":"02:10.035","Text":"that I\u0027ve outlined in blue,"},{"Start":"02:10.035 ","End":"02:13.340","Text":"certainly its area is less than the area"},{"Start":"02:13.340 ","End":"02:16.820","Text":"of the shaded green because it\u0027s part of it."},{"Start":"02:16.820 ","End":"02:21.410","Text":"If I abbreviate this as little m"},{"Start":"02:21.410 ","End":"02:24.230","Text":"and this 1 big M,"},{"Start":"02:24.230 ","End":"02:27.950","Text":"then the area of this rectangle here"},{"Start":"02:27.950 ","End":"02:30.320","Text":"is going to be base times height"},{"Start":"02:30.320 ","End":"02:32.015","Text":"or height times base."},{"Start":"02:32.015 ","End":"02:33.919","Text":"The height is m,"},{"Start":"02:33.919 ","End":"02:37.685","Text":"and the base is b minus a."},{"Start":"02:37.685 ","End":"02:39.500","Text":"The area under the curve"},{"Start":"02:39.500 ","End":"02:41.000","Text":"which is shaded in green as we said,"},{"Start":"02:41.000 ","End":"02:48.920","Text":"is the integral from a to b of f of x/dx."},{"Start":"02:48.920 ","End":"02:51.485","Text":"This is less than or equal to this,"},{"Start":"02:51.485 ","End":"02:52.850","Text":"and before I write it,"},{"Start":"02:52.850 ","End":"02:54.470","Text":"I want to go for the third thing."},{"Start":"02:54.470 ","End":"02:56.390","Text":"This third thing I\u0027ll sketch in a moment"},{"Start":"02:56.390 ","End":"02:58.460","Text":"will be the rectangle that goes up"},{"Start":"02:58.460 ","End":"03:01.090","Text":"to the maximum which is in red."},{"Start":"03:01.090 ","End":"03:02.420","Text":"Here\u0027s the rectangle I mean,"},{"Start":"03:02.420 ","End":"03:05.660","Text":"and this rectangle certainly has an area"},{"Start":"03:05.660 ","End":"03:08.210","Text":"bigger or equal to the green area,"},{"Start":"03:08.210 ","End":"03:10.490","Text":"because the green area is included in it."},{"Start":"03:10.490 ","End":"03:12.050","Text":"What I can say is that"},{"Start":"03:12.050 ","End":"03:16.445","Text":"the red rectangle has an area of,"},{"Start":"03:16.445 ","End":"03:17.870","Text":"again, base times height"},{"Start":"03:17.870 ","End":"03:18.980","Text":"or height times base."},{"Start":"03:18.980 ","End":"03:21.020","Text":"In this case, the height is big M,"},{"Start":"03:21.020 ","End":"03:26.720","Text":"and the base is also b minus a."},{"Start":"03:26.720 ","End":"03:28.010","Text":"Now, let\u0027s put together"},{"Start":"03:28.010 ","End":"03:29.585","Text":"what I\u0027ve said so far."},{"Start":"03:29.585 ","End":"03:30.800","Text":"Then we have that"},{"Start":"03:30.800 ","End":"03:33.665","Text":"the smallest is the blue rectangle,"},{"Start":"03:33.665 ","End":"03:38.004","Text":"little m times b minus a,"},{"Start":"03:38.004 ","End":"03:40.035","Text":"less than or equal to."},{"Start":"03:40.035 ","End":"03:42.260","Text":"In this diagram, it\u0027s clearly less than."},{"Start":"03:42.260 ","End":"03:43.790","Text":"But it could be that it\u0027s less than"},{"Start":"03:43.790 ","End":"03:45.934","Text":"or equal to the green,"},{"Start":"03:45.934 ","End":"03:48.260","Text":"which is the area under the curve,"},{"Start":"03:48.260 ","End":"03:49.550","Text":"and that\u0027s the integral"},{"Start":"03:49.550 ","End":"03:54.535","Text":"from a to b of f of x/dx."},{"Start":"03:54.535 ","End":"03:57.410","Text":"The biggest of all is the red 1,"},{"Start":"03:57.410 ","End":"04:02.660","Text":"which is big M times b minus a."},{"Start":"04:02.660 ","End":"04:08.120","Text":"Little m, is the minimum of f of x."},{"Start":"04:08.120 ","End":"04:11.120","Text":"When x goes on the interval a,b,"},{"Start":"04:11.120 ","End":"04:14.990","Text":"and big M is the maximum"},{"Start":"04:14.990 ","End":"04:19.800","Text":"on the interval a,b of f of x."},{"Start":"04:19.800 ","End":"04:22.500","Text":"Of course, I mean the absolute minimum"},{"Start":"04:22.500 ","End":"04:24.075","Text":"and absolute maximum."},{"Start":"04:24.075 ","End":"04:26.040","Text":"Basically, we\u0027ve proven"},{"Start":"04:26.040 ","End":"04:27.990","Text":"what we set out to do,"},{"Start":"04:27.990 ","End":"04:30.320","Text":"and all that I would add here"},{"Start":"04:30.320 ","End":"04:31.820","Text":"would be an example."},{"Start":"04:31.820 ","End":"04:33.440","Text":"But there are examples"},{"Start":"04:33.440 ","End":"04:36.005","Text":"right after this theoretical clip."},{"Start":"04:36.005 ","End":"04:40.350","Text":"Let\u0027s get to the second inequality."}],"ID":8607},{"Watched":false,"Name":"Inequalities - Part 2","Duration":"3m 26s","ChapterTopicVideoID":8331,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8331.jpeg","UploadDate":"2019-12-10T12:45:42.5730000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.195","Text":"Here\u0027s our second inequality."},{"Start":"00:03.195 ","End":"00:06.420","Text":"I\u0027ve just written it and now explain it."},{"Start":"00:06.420 ","End":"00:10.290","Text":"It says that if we have 2 functions, f and g,"},{"Start":"00:10.290 ","End":"00:16.335","Text":"and f of x is less than or equal to g of x for all x between a and b inclusive,"},{"Start":"00:16.335 ","End":"00:19.830","Text":"then the integral and that same interval from a to b of"},{"Start":"00:19.830 ","End":"00:23.790","Text":"f is less than or equal to the integral of g. Other words,"},{"Start":"00:23.790 ","End":"00:25.515","Text":"if f is less than or equal to g,"},{"Start":"00:25.515 ","End":"00:27.960","Text":"then the integral of f is also less than or equal to the"},{"Start":"00:27.960 ","End":"00:30.615","Text":"integral of g. But of course, they have to match up."},{"Start":"00:30.615 ","End":"00:32.550","Text":"This has to be true for the interval a b,"},{"Start":"00:32.550 ","End":"00:34.650","Text":"and this is the integral from a to b."},{"Start":"00:34.650 ","End":"00:38.025","Text":"Here it is, and now I\u0027ll explain why this is so."},{"Start":"00:38.025 ","End":"00:41.225","Text":"Let\u0027s start with 1 of the functions f of x,"},{"Start":"00:41.225 ","End":"00:42.825","Text":"and I\u0027ve sketched it here actually,"},{"Start":"00:42.825 ","End":"00:45.275","Text":"I borrowed the sketch from the previous 1,"},{"Start":"00:45.275 ","End":"00:48.775","Text":"and now let\u0027s add g of x into the picture."},{"Start":"00:48.775 ","End":"00:50.300","Text":"Here is our second function,"},{"Start":"00:50.300 ","End":"00:51.550","Text":"y equals g of x,"},{"Start":"00:51.550 ","End":"00:54.080","Text":"and now I want to do a bit more shading because you"},{"Start":"00:54.080 ","End":"00:56.930","Text":"see the green represents the integral of"},{"Start":"00:56.930 ","End":"01:02.340","Text":"f of x. I\u0027ve shaded the area under the curve g of x between a and b."},{"Start":"01:02.340 ","End":"01:07.340","Text":"It\u0027s clear that the area that\u0027s highlighted in green is"},{"Start":"01:07.340 ","End":"01:13.135","Text":"less than or equal to the area that is marked with diagonal stripes."},{"Start":"01:13.135 ","End":"01:14.735","Text":"Here\u0027s what I mean."},{"Start":"01:14.735 ","End":"01:21.185","Text":"If I look at the diagonal purple lines that represents the integral of g of x dx."},{"Start":"01:21.185 ","End":"01:24.470","Text":"The green represents the integral of f of x dx,"},{"Start":"01:24.470 ","End":"01:28.775","Text":"and clearly because g of x is always above f of x,"},{"Start":"01:28.775 ","End":"01:30.770","Text":"at least on the interval a b,"},{"Start":"01:30.770 ","End":"01:36.680","Text":"then we can see that the green is less than or equal to the purple diagonal shaded."},{"Start":"01:36.680 ","End":"01:39.780","Text":"This means that this is less than or equal to this."},{"Start":"01:39.780 ","End":"01:44.090","Text":"Which says that the integral from a to b of f of"},{"Start":"01:44.090 ","End":"01:51.590","Text":"x dx is less than or equal to the integral from a to b of g of x dx."},{"Start":"01:51.590 ","End":"01:57.920","Text":"By the way, if we look over here that actually g at some point is less than or"},{"Start":"01:57.920 ","End":"02:00.890","Text":"equal to f. But that doesn\u0027t matter because we only"},{"Start":"02:00.890 ","End":"02:04.670","Text":"care about what happens in the interval from a to b."},{"Start":"02:04.670 ","End":"02:07.910","Text":"This is an illustration of what I wrote here,"},{"Start":"02:07.910 ","End":"02:09.995","Text":"and I think it\u0027s fairly straightforward."},{"Start":"02:09.995 ","End":"02:13.990","Text":"Of course, it\u0027ll be clear in the examples that follow,"},{"Start":"02:13.990 ","End":"02:16.570","Text":"there at least a couple of examples that use this."},{"Start":"02:16.570 ","End":"02:19.460","Text":"This can be used for an inequality on an integral."},{"Start":"02:19.460 ","End":"02:23.270","Text":"For example, if we have difficulty in computing f of x,"},{"Start":"02:23.270 ","End":"02:24.920","Text":"we can compute g of x,"},{"Start":"02:24.920 ","End":"02:28.175","Text":"then we can get an upper bound for f, and the other hand,"},{"Start":"02:28.175 ","End":"02:32.870","Text":"if we can\u0027t do g of x and we know how to compute f of x so we can get a lower bound for"},{"Start":"02:32.870 ","End":"02:35.210","Text":"g. It\u0027s good for either getting an upper bound"},{"Start":"02:35.210 ","End":"02:37.805","Text":"or a lower bound depending on which 1 we know."},{"Start":"02:37.805 ","End":"02:41.390","Text":"I just want to say another thing that it works also with"},{"Start":"02:41.390 ","End":"02:45.275","Text":"3 integrals and I\u0027ll just write it down."},{"Start":"02:45.275 ","End":"02:48.589","Text":"Here is the generalization to 3 functions."},{"Start":"02:48.589 ","End":"02:51.160","Text":"If instead of f and g have f and g and h,"},{"Start":"02:51.160 ","End":"02:52.940","Text":"and this is less than or equal to this,"},{"Start":"02:52.940 ","End":"02:57.230","Text":"less than or equal to this on the interval from a to b inclusive."},{"Start":"02:57.230 ","End":"03:02.030","Text":"Then we can conclude that the integral of f is less than or equal to integral of g,"},{"Start":"03:02.030 ","End":"03:05.630","Text":"which is less than or equal to the integral of h. This is often used"},{"Start":"03:05.630 ","End":"03:09.530","Text":"when the middle 1 g is unknown or difficult to compute."},{"Start":"03:09.530 ","End":"03:15.065","Text":"But we can compute f and h. Then we get an estimate on g of x."},{"Start":"03:15.065 ","End":"03:17.585","Text":"It\u0027s integral between something and something,"},{"Start":"03:17.585 ","End":"03:20.945","Text":"and you\u0027ll see there\u0027s these 1 exercise they\u0027re using this."},{"Start":"03:20.945 ","End":"03:23.990","Text":"Other than telling you to go ahead and do the exercises,"},{"Start":"03:23.990 ","End":"03:27.000","Text":"the solved ones. I\u0027m done here."}],"ID":8608},{"Watched":false,"Name":"Exercise 1","Duration":"3m 26s","ChapterTopicVideoID":8332,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8332.jpeg","UploadDate":"2017-01-28T12:21:21.8930000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.520","Text":"In this exercise, we have to prove certain inequalities that some definite"},{"Start":"00:05.520 ","End":"00:11.355","Text":"integral is less than or equal to a number and greater or equal to something else."},{"Start":"00:11.355 ","End":"00:16.015","Text":"This exercise is usually solved by means of the following theorem"},{"Start":"00:16.015 ","End":"00:20.255","Text":"that whenever we have a definite integral on an interval,"},{"Start":"00:20.255 ","End":"00:25.400","Text":"then we can bound it above by b minus a times capital M"},{"Start":"00:25.400 ","End":"00:28.080","Text":"and below by b minus a times little m,"},{"Start":"00:28.080 ","End":"00:30.165","Text":"where the little m and big M"},{"Start":"00:30.165 ","End":"00:33.740","Text":"are just the minimum and the maximum of the function on the interval."},{"Start":"00:33.740 ","End":"00:37.025","Text":"So we already have some of the things here."},{"Start":"00:37.025 ","End":"00:38.965","Text":"We know what f of x is."},{"Start":"00:38.965 ","End":"00:44.450","Text":"Our function is 1 over 1 plus x^4. We also have"},{"Start":"00:44.450 ","End":"00:51.360","Text":"a, which is minus 1, and we have that b is equal to 3."},{"Start":"00:51.360 ","End":"00:55.660","Text":"What we need are little m and big M."}],"ID":8609},{"Watched":false,"Name":"Exercise 2","Duration":"3m 19s","ChapterTopicVideoID":8333,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8333.jpeg","UploadDate":"2017-01-28T12:21:29.5400000","DurationForVideoObject":"PT3M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"In this exercise, we\u0027re given a definite integral and we have"},{"Start":"00:03.930 ","End":"00:07.424","Text":"to show that it\u0027s between something and something."},{"Start":"00:07.424 ","End":"00:10.005","Text":"In other words, we possibly can\u0027t compute it,"},{"Start":"00:10.005 ","End":"00:12.990","Text":"but we can still give an estimate of what it can be at most,"},{"Start":"00:12.990 ","End":"00:14.610","Text":"and what good can be at least."},{"Start":"00:14.610 ","End":"00:17.940","Text":"This is based on the theorem that,"},{"Start":"00:17.940 ","End":"00:20.835","Text":"if we have a function on a, b,"},{"Start":"00:20.835 ","End":"00:24.750","Text":"then the integral is going to be less than or equal to,"},{"Start":"00:24.750 ","End":"00:30.390","Text":"b minus a times big m and bigger or equal to b minus a times little m,"},{"Start":"00:30.390 ","End":"00:33.060","Text":"little m and big m are just shortcuts for"},{"Start":"00:33.060 ","End":"00:37.630","Text":"the minimum and the maximum of f of x on the interval a."}],"ID":8610},{"Watched":false,"Name":"Exercise 3","Duration":"2m 31s","ChapterTopicVideoID":8334,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8334.jpeg","UploadDate":"2017-01-28T12:21:52.2470000","DurationForVideoObject":"PT2M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"Here we have to prove that this definite integral satisfies a couple of inequalities,"},{"Start":"00:05.340 ","End":"00:08.250","Text":"that it\u0027s less than or equal to this and bigger or equal to this."},{"Start":"00:08.250 ","End":"00:13.395","Text":"The theory we need here is this 1 that I\u0027ve already put on the page,"},{"Start":"00:13.395 ","End":"00:17.025","Text":"is that the integral is less than or equal to,"},{"Start":"00:17.025 ","End":"00:22.875","Text":"basically it\u0027s the difference between the upper and lower limits times both M and m,"},{"Start":"00:22.875 ","End":"00:29.100","Text":"where M is the maximum and m is the minimum of the function f on the interval a,"},{"Start":"00:29.100 ","End":"00:31.050","Text":"b in our case 0 and 2."},{"Start":"00:31.050 ","End":"00:35.610","Text":"In our case, the f of x is e^ x squared."},{"Start":"00:35.610 ","End":"00:36.894","Text":"That\u0027s the function."},{"Start":"00:36.894 ","End":"00:41.030","Text":"The end points a is 0, b is 2,"},{"Start":"00:41.030 ","End":"00:47.030","Text":"and I guess it\u0027s useful to write down what is b minus a, which is 2."},{"Start":"00:47.030 ","End":"00:49.445","Text":"Now, the way I look for extrema,"},{"Start":"00:49.445 ","End":"00:50.989","Text":"which is minimum and maximum,"},{"Start":"00:50.989 ","End":"00:52.130","Text":"is in 2 places."},{"Start":"00:52.130 ","End":"00:56.930","Text":"I first differentiate f prime of x and set it to"},{"Start":"00:56.930 ","End":"00:59.870","Text":"0 and see what x that gives us and we\u0027re looking"},{"Start":"00:59.870 ","End":"01:03.050","Text":"for x is in the interior between 0 and 2 and secondly,"},{"Start":"01:03.050 ","End":"01:04.850","Text":"we\u0027ll take the end points 0 and 2."},{"Start":"01:04.850 ","End":"01:08.160","Text":"First, this suspect for critical point."},{"Start":"01:08.160 ","End":"01:11.940","Text":"We get the derivative of e^ x squared,"},{"Start":"01:11.940 ","End":"01:19.685","Text":"is e^ x squared times inner derivative 2x is equal to 0."},{"Start":"01:19.685 ","End":"01:22.459","Text":"Now e to the something is never 0, always positive."},{"Start":"01:22.459 ","End":"01:25.475","Text":"This can only happen when x equals 0,"},{"Start":"01:25.475 ","End":"01:27.889","Text":"but it\u0027s not in the interior,"},{"Start":"01:27.889 ","End":"01:30.020","Text":"it\u0027s on the endpoints, so I can ignore it,"},{"Start":"01:30.020 ","End":"01:33.590","Text":"but I still get it because I get it by taking the endpoints."},{"Start":"01:33.590 ","End":"01:36.519","Text":"What I need to check is f of 0"},{"Start":"01:36.519 ","End":"01:42.050","Text":"and f of 2 and let\u0027s see which of these is the largest and which is the smallest?"},{"Start":"01:42.050 ","End":"01:46.790","Text":"If x is 0, f of x is e^0, which is 1."},{"Start":"01:46.790 ","End":"01:51.620","Text":"If f is 2, we get 2 squared is 4, e^4."},{"Start":"01:51.620 ","End":"01:56.640","Text":"Now, it\u0027s clear that the e^4 is much bigger than 1,"},{"Start":"01:56.640 ","End":"02:01.350","Text":"so this 1 is going to be my M and 1 will be"},{"Start":"02:01.350 ","End":"02:07.195","Text":"m. All I have left to do is to substitute in this formula here."},{"Start":"02:07.195 ","End":"02:16.880","Text":"What I get is b minus a is 2 times m. I get 2 times 1 is less than or equal to"},{"Start":"02:16.880 ","End":"02:23.240","Text":"the integral e^ x squared dx and it\u0027s less than or equal to"},{"Start":"02:23.240 ","End":"02:29.990","Text":"the same 2 times M and M is e^4 and now I\u0027ve got exactly what\u0027s written here,"},{"Start":"02:29.990 ","End":"02:32.340","Text":"so we are done."}],"ID":8611},{"Watched":false,"Name":"Exercise 4","Duration":"4m 16s","ChapterTopicVideoID":8335,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8335.jpeg","UploadDate":"2017-01-28T12:22:41.6400000","DurationForVideoObject":"PT4M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"In this exercise, we have to prove that"},{"Start":"00:02.010 ","End":"00:07.035","Text":"this definite integral satisfies certain inequalities."},{"Start":"00:07.035 ","End":"00:09.330","Text":"We have to show that it\u0027s less than 1"},{"Start":"00:09.330 ","End":"00:14.010","Text":"and bigger than 1/2 e to the minus 10."},{"Start":"00:14.010 ","End":"00:17.025","Text":"Then it fits the pattern of the theorem,"},{"Start":"00:17.025 ","End":"00:19.335","Text":"which says that when we have a definite integral,"},{"Start":"00:19.335 ","End":"00:21.090","Text":"we can estimate it to be less than"},{"Start":"00:21.090 ","End":"00:22.470","Text":"or equal to something and bigger"},{"Start":"00:22.470 ","End":"00:23.745","Text":"or equal to something else."},{"Start":"00:23.745 ","End":"00:27.090","Text":"Here, the a and the b are 0 and 10."},{"Start":"00:27.090 ","End":"00:30.240","Text":"The m and the big M are going to be"},{"Start":"00:30.240 ","End":"00:32.730","Text":"the minimum and maximum of the function"},{"Start":"00:32.730 ","End":"00:34.440","Text":"on the interval from 0 to 10."},{"Start":"00:34.440 ","End":"00:36.000","Text":"We\u0027re going to find those out."},{"Start":"00:36.000 ","End":"00:37.860","Text":"Well, let\u0027s write down what we have."},{"Start":"00:37.860 ","End":"00:40.790","Text":"The function of x that we have is"},{"Start":"00:40.790 ","End":"00:45.455","Text":"e to the minus x over x plus 10."},{"Start":"00:45.455 ","End":"00:48.740","Text":"We have a lower limit a which is 0,"},{"Start":"00:48.740 ","End":"00:52.309","Text":"and upper limit b which is 10."},{"Start":"00:52.309 ","End":"00:54.440","Text":"We also might as well"},{"Start":"00:54.440 ","End":"00:56.510","Text":"compute b minus a already,"},{"Start":"00:56.510 ","End":"01:00.410","Text":"we\u0027ll need that 10 minus 0 is 10."},{"Start":"01:00.410 ","End":"01:01.910","Text":"What remains is to find"},{"Start":"01:01.910 ","End":"01:03.020","Text":"the minimum and the maximum."},{"Start":"01:03.020 ","End":"01:04.235","Text":"In other words, the extrema."},{"Start":"01:04.235 ","End":"01:05.510","Text":"The way we find the extrema"},{"Start":"01:05.510 ","End":"01:08.690","Text":"is first we look for derivative equal to 0."},{"Start":"01:08.690 ","End":"01:11.750","Text":"It has to be in the interior between 0 and 10."},{"Start":"01:11.750 ","End":"01:15.184","Text":"Let\u0027s see, we have to differentiate a quotient."},{"Start":"01:15.184 ","End":"01:17.120","Text":"Better write the quotient rule quickly,"},{"Start":"01:17.120 ","End":"01:19.310","Text":"u is e to the minus x,"},{"Start":"01:19.310 ","End":"01:20.630","Text":"v is x plus 10."},{"Start":"01:20.630 ","End":"01:28.670","Text":"I get u prime is minus e to the minus x times v,"},{"Start":"01:28.670 ","End":"01:31.940","Text":"which is x plus 10 minus u,"},{"Start":"01:31.940 ","End":"01:33.770","Text":"which is e to the minus x"},{"Start":"01:33.770 ","End":"01:37.009","Text":"times v prime which is 1,"},{"Start":"01:37.009 ","End":"01:41.265","Text":"all this over x plus 10 squared."},{"Start":"01:41.265 ","End":"01:43.385","Text":"This has to equal 0."},{"Start":"01:43.385 ","End":"01:45.410","Text":"Well, the denominator"},{"Start":"01:45.410 ","End":"01:47.330","Text":"is positive and defined,"},{"Start":"01:47.330 ","End":"01:49.250","Text":"so it must be that the numerator\u0027s 0."},{"Start":"01:49.250 ","End":"01:53.450","Text":"The numerator equals 0 minus e to the minus x."},{"Start":"01:53.450 ","End":"01:53.930","Text":"You know what?"},{"Start":"01:53.930 ","End":"01:56.075","Text":"I can take this outside the brackets."},{"Start":"01:56.075 ","End":"02:01.295","Text":"I get x plus 10 plus 1 equals 0."},{"Start":"02:01.295 ","End":"02:03.200","Text":"Now, this is never 0,"},{"Start":"02:03.200 ","End":"02:05.180","Text":"e to the something is always positive."},{"Start":"02:05.180 ","End":"02:06.350","Text":"So x plus 10 plus 1,"},{"Start":"02:06.350 ","End":"02:08.659","Text":"which is x plus 11, must be 0;"},{"Start":"02:08.659 ","End":"02:12.050","Text":"x equals minus 11."},{"Start":"02:12.050 ","End":"02:15.710","Text":"Minus 11 is completely out of range."},{"Start":"02:15.710 ","End":"02:17.720","Text":"It\u0027s not between 0 and 10."},{"Start":"02:17.720 ","End":"02:20.150","Text":"So this is no good for me."},{"Start":"02:20.150 ","End":"02:23.780","Text":"My only suspects or possible extrema"},{"Start":"02:23.780 ","End":"02:26.195","Text":"are the endpoints, 0 and 10."},{"Start":"02:26.195 ","End":"02:29.195","Text":"What I need to do is check what is f of 0."},{"Start":"02:29.195 ","End":"02:31.580","Text":"I need to check what is f of 10"},{"Start":"02:31.580 ","End":"02:33.650","Text":"and see which is the big 1,"},{"Start":"02:33.650 ","End":"02:34.730","Text":"which is a small 1,"},{"Start":"02:34.730 ","End":"02:37.250","Text":"which is little m and which is big M."},{"Start":"02:37.250 ","End":"02:40.490","Text":"Now, f of 0, I plug in 0 here."},{"Start":"02:40.490 ","End":"02:45.565","Text":"So f of 0 is 1/0 plus 10, which is 1/10;"},{"Start":"02:45.565 ","End":"02:49.460","Text":"f of 10 is e to the minus 10"},{"Start":"02:49.460 ","End":"02:53.015","Text":"over 10 plus 10, which is 20."},{"Start":"02:53.015 ","End":"02:54.620","Text":"Now, which is smaller?"},{"Start":"02:54.620 ","End":"02:57.980","Text":"Well, e to the minus 10 is less than 1."},{"Start":"02:57.980 ","End":"03:00.665","Text":"This is less than a 20th."},{"Start":"03:00.665 ","End":"03:03.740","Text":"In fact, this is an incredibly small number."},{"Start":"03:03.740 ","End":"03:08.000","Text":"This is my minimum called little m,"},{"Start":"03:08.000 ","End":"03:10.850","Text":"and this is the maximum, big M."},{"Start":"03:10.850 ","End":"03:14.465","Text":"Now, I can just plug in the equation here."},{"Start":"03:14.465 ","End":"03:19.350","Text":"What I get is that b minus a is 10."},{"Start":"03:19.350 ","End":"03:23.030","Text":"I get 10 times little m,"},{"Start":"03:23.030 ","End":"03:26.540","Text":"which is e to the minus 10/20"},{"Start":"03:26.540 ","End":"03:31.695","Text":"is less than or equal to the integral."},{"Start":"03:31.695 ","End":"03:33.750","Text":"I\u0027ll write that in a second."},{"Start":"03:33.750 ","End":"03:37.100","Text":"This will be less than or equal to big M,"},{"Start":"03:37.100 ","End":"03:39.890","Text":"also 10, which is the b minus a,"},{"Start":"03:39.890 ","End":"03:43.250","Text":"and big M which is 1/10."},{"Start":"03:43.250 ","End":"03:49.165","Text":"The integral is e to the minus x over x plus 10."},{"Start":"03:49.165 ","End":"03:52.190","Text":"I don\u0027t want to write this whole integral again."},{"Start":"03:52.190 ","End":"03:53.795","Text":"Let\u0027s just simplify this."},{"Start":"03:53.795 ","End":"03:58.970","Text":"The 10 with the 20 goes twice,"},{"Start":"03:58.970 ","End":"04:02.270","Text":"and 10 with the 10 disappears."},{"Start":"04:02.270 ","End":"04:05.360","Text":"Well, you can see that this is 1/2 e"},{"Start":"04:05.360 ","End":"04:09.465","Text":"to the minus 10 and this is 1."},{"Start":"04:09.465 ","End":"04:13.310","Text":"I think we can give ourselves a checkmark"},{"Start":"04:13.310 ","End":"04:15.230","Text":"that we\u0027ve succeeded in proving it,"},{"Start":"04:15.230 ","End":"04:17.670","Text":"and we are done."}],"ID":8612},{"Watched":false,"Name":"Exercise 5","Duration":"4m 52s","ChapterTopicVideoID":8345,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8345.jpeg","UploadDate":"2017-01-28T12:24:48.8530000","DurationForVideoObject":"PT4M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Here we have another 1 of those exercises where we\u0027re given an integral,"},{"Start":"00:03.060 ","End":"00:07.920","Text":"and we have to show that it\u0027s bounded above and below by certain constants."},{"Start":"00:07.920 ","End":"00:12.210","Text":"We use the theorem that the definite integral is"},{"Start":"00:12.210 ","End":"00:17.700","Text":"bounded below by b minus a times the minimum of the function,"},{"Start":"00:17.700 ","End":"00:21.090","Text":"and above by b minus a times the maximum of the function,"},{"Start":"00:21.090 ","End":"00:22.730","Text":"where b and a are the limits."},{"Start":"00:22.730 ","End":"00:24.625","Text":"I\u0027m going to use this."},{"Start":"00:24.625 ","End":"00:28.590","Text":"In our case, f of x is 1,"},{"Start":"00:28.590 ","End":"00:33.405","Text":"over 3 plus 4 sine squared of x."},{"Start":"00:33.405 ","End":"00:38.610","Text":"Our a is equal to 0, the lower limit."},{"Start":"00:38.610 ","End":"00:41.534","Text":"B is Pi over 2."},{"Start":"00:41.534 ","End":"00:44.900","Text":"I guess it\u0027s useful to write down b minus a,"},{"Start":"00:44.900 ","End":"00:49.925","Text":"also, so b minus a just squeeze it in."},{"Start":"00:49.925 ","End":"00:53.750","Text":"Pi over 2, minus 0 is Pi over 2."},{"Start":"00:53.750 ","End":"00:56.360","Text":"Now how do we look for minimum and maximum,"},{"Start":"00:56.360 ","End":"00:57.755","Text":"also known as extrema?"},{"Start":"00:57.755 ","End":"01:00.980","Text":"There are 2 places we look on a closed interval."},{"Start":"01:00.980 ","End":"01:05.900","Text":"First of all, we try differentiating and setting to 0 and see if these points,"},{"Start":"01:05.900 ","End":"01:06.995","Text":"which are called critical points,"},{"Start":"01:06.995 ","End":"01:09.920","Text":"are in the interval in the interior,"},{"Start":"01:09.920 ","End":"01:11.665","Text":"and we also take the endpoints."},{"Start":"01:11.665 ","End":"01:17.810","Text":"Let\u0027s first start with finding critical points f prime of x equals 0."},{"Start":"01:17.810 ","End":"01:19.815","Text":"Let\u0027s see. I need to differentiate."},{"Start":"01:19.815 ","End":"01:26.580","Text":"The derivative of 1 over something is minus 1 over that something squared."},{"Start":"01:26.580 ","End":"01:31.785","Text":"I\u0027ll start with 3 plus 4 sine squared x,"},{"Start":"01:31.785 ","End":"01:38.985","Text":"all squared, minus, above I need the derivative of the denominator."},{"Start":"01:38.985 ","End":"01:41.060","Text":"The derivative would be,"},{"Start":"01:41.060 ","End":"01:43.475","Text":"let\u0027s see, 3 gives me nothing."},{"Start":"01:43.475 ","End":"01:45.350","Text":"This gives me 4."},{"Start":"01:45.350 ","End":"01:50.180","Text":"Then sine squared gives me 2 sine x,"},{"Start":"01:50.180 ","End":"01:52.940","Text":"but times the derivative of sine x,"},{"Start":"01:52.940 ","End":"01:56.315","Text":"which is cosine x."},{"Start":"01:56.315 ","End":"02:00.370","Text":"This has to be equal to 0 now."},{"Start":"02:00.370 ","End":"02:02.765","Text":"The numerator has to be 0."},{"Start":"02:02.765 ","End":"02:04.970","Text":"I can ignore the constant."},{"Start":"02:04.970 ","End":"02:10.444","Text":"Basically, what it tells me is that sine squared x cosine x is 0."},{"Start":"02:10.444 ","End":"02:13.100","Text":"Now, 1 of these has to be 0,"},{"Start":"02:13.100 ","End":"02:15.470","Text":"so either sine squared x is 0,"},{"Start":"02:15.470 ","End":"02:17.420","Text":"in which case sine x is 0."},{"Start":"02:17.420 ","End":"02:24.560","Text":"I say that sine x equals 0 or cosine x equals 0."},{"Start":"02:24.560 ","End":"02:29.210","Text":"But I\u0027m looking for x in the interval from 0 to Pi over 2."},{"Start":"02:29.210 ","End":"02:34.430","Text":"In other words, I\u0027ll just remind you that x has to be between 0 and Pi over 2."},{"Start":"02:34.430 ","End":"02:37.145","Text":"Or if you like this is 90 degrees."},{"Start":"02:37.145 ","End":"02:39.800","Text":"When is the sine equal to 0,"},{"Start":"02:39.800 ","End":"02:44.385","Text":"and when is the cosine equal to 0 from 0-90?"},{"Start":"02:44.385 ","End":"02:46.175","Text":"Well, this we know already."},{"Start":"02:46.175 ","End":"02:47.910","Text":"Sine of 0 is 0,"},{"Start":"02:47.910 ","End":"02:50.130","Text":"and cosine of 90 is 0."},{"Start":"02:50.130 ","End":"02:53.730","Text":"Basically, x has to be either 0 or Pi over 2."},{"Start":"02:53.730 ","End":"03:00.515","Text":"This gives us that x equals 0 or x equals Pi over 2."},{"Start":"03:00.515 ","End":"03:02.210","Text":"Now, these are not in the interior."},{"Start":"03:02.210 ","End":"03:04.100","Text":"They\u0027re on the edge, so we ignore them."},{"Start":"03:04.100 ","End":"03:08.410","Text":"But ironically, these are exactly the points that we take when we take the endpoints."},{"Start":"03:08.410 ","End":"03:12.590","Text":"Whether you choose to take them as critical points or as endpoints, these are the points,"},{"Start":"03:12.590 ","End":"03:14.660","Text":"0 and Pi over 2,"},{"Start":"03:14.660 ","End":"03:19.135","Text":"and we just substitute them into the function to see which is larger, which is smaller."},{"Start":"03:19.135 ","End":"03:21.360","Text":"Let\u0027s see. F of 0 equals,"},{"Start":"03:21.360 ","End":"03:25.680","Text":"and I want f of Pi over 2 equals."},{"Start":"03:25.680 ","End":"03:27.075","Text":"Start with the 0."},{"Start":"03:27.075 ","End":"03:31.770","Text":"If x is 0, sine of 0 is 0 times 4,"},{"Start":"03:31.770 ","End":"03:35.380","Text":"is still 0, so we get 1/3."},{"Start":"03:35.630 ","End":"03:39.555","Text":"Let\u0027s see. F of Pi over 2,"},{"Start":"03:39.555 ","End":"03:42.780","Text":"sine of Pi over 2 is 1."},{"Start":"03:42.780 ","End":"03:47.775","Text":"Here we get 3 plus 4 is 7."},{"Start":"03:47.775 ","End":"03:51.850","Text":"Here we get 1 over 7."},{"Start":"03:52.010 ","End":"03:55.060","Text":"This 1 is obviously smaller than this,"},{"Start":"03:55.060 ","End":"03:58.735","Text":"so this 1 is going to be my little m,"},{"Start":"03:58.735 ","End":"04:03.755","Text":"and this 1 is going to be big M, minimum, maximum."},{"Start":"04:03.755 ","End":"04:06.245","Text":"Now we\u0027ll have to do is plug it in here,"},{"Start":"04:06.245 ","End":"04:08.825","Text":"where b minus a is Pi over 2."},{"Start":"04:08.825 ","End":"04:11.015","Text":"What I get is Pi over 2,"},{"Start":"04:11.015 ","End":"04:19.295","Text":"times little m is 1/17 is less than or equal to the integral of f of x d_x,"},{"Start":"04:19.295 ","End":"04:26.350","Text":"which is d_x over 3 plus 4 sine squared x."},{"Start":"04:26.350 ","End":"04:30.415","Text":"That\u0027s less than or equal to the same Pi over 2."},{"Start":"04:30.415 ","End":"04:33.740","Text":"But this time times big M, which is 1/3."},{"Start":"04:33.740 ","End":"04:40.070","Text":"Now, I\u0027m not going to continue with this simplification because obviously Pi over 2,"},{"Start":"04:40.070 ","End":"04:43.910","Text":"times 1/17 is the same as Pi over 14,"},{"Start":"04:43.910 ","End":"04:45.290","Text":"and this Pi over 2, times 1,"},{"Start":"04:45.290 ","End":"04:48.040","Text":"over 3 is Pi over 6."},{"Start":"04:48.040 ","End":"04:53.280","Text":"Looks like we\u0027ve exactly proved what we had to prove. We\u0027re done."}],"ID":8613},{"Watched":false,"Name":"Exercise 6","Duration":"3m 26s","ChapterTopicVideoID":8346,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8346.jpeg","UploadDate":"2017-01-28T12:25:05.9700000","DurationForVideoObject":"PT3M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this exercise, we have one of those inequalities of integrals to show"},{"Start":"00:04.560 ","End":"00:06.240","Text":"that the integral is less than or equal to"},{"Start":"00:06.240 ","End":"00:08.940","Text":"something, and bigger or equal to something else."},{"Start":"00:08.940 ","End":"00:11.280","Text":"Previously we used a certain theorem,"},{"Start":"00:11.280 ","End":"00:13.875","Text":"but here we\u0027re going to use a different tool to solve."},{"Start":"00:13.875 ","End":"00:18.180","Text":"We\u0027re going to use the theorem that if you have 2 functions, f and g,"},{"Start":"00:18.180 ","End":"00:21.270","Text":"and one of them is less than or equal to the other on an interval,"},{"Start":"00:21.270 ","End":"00:23.325","Text":"say x goes from a to b,"},{"Start":"00:23.325 ","End":"00:27.675","Text":"then the same inequality less than or equal to will hold for the integrals."},{"Start":"00:27.675 ","End":"00:29.280","Text":"In other words, the set of f and g,"},{"Start":"00:29.280 ","End":"00:34.120","Text":"I can put the integral from a to b of f of x and the integral from a to b of g of x."},{"Start":"00:34.120 ","End":"00:38.975","Text":"Now, we can extend this theorem to say that if we had a third one, h of x,"},{"Start":"00:38.975 ","End":"00:43.830","Text":"then that would also hold true, because g would be like my f and h would be like"},{"Start":"00:43.830 ","End":"00:49.730","Text":"my g. I could put that this is equal to a to b of h of x dx."},{"Start":"00:49.730 ","End":"00:52.075","Text":"It works for 3 or any number in the chain."},{"Start":"00:52.075 ","End":"00:55.430","Text":"Now, what I need to decide is, what is going to be my f,"},{"Start":"00:55.430 ","End":"00:57.095","Text":"my g, and my h."},{"Start":"00:57.095 ","End":"01:00.860","Text":"The first thing is that the sign, and I don\u0027t care what\u0027s"},{"Start":"01:00.860 ","End":"01:05.210","Text":"inside of it, is always between 1 and minus 1."},{"Start":"01:05.210 ","End":"01:07.805","Text":"I\u0027m going to write that also here is a useful thing."},{"Start":"01:07.805 ","End":"01:14.145","Text":"The sine of an angle Alpha is always between minus 1 and 1, whatever Alpha is."},{"Start":"01:14.145 ","End":"01:16.350","Text":"If I use that fact here,"},{"Start":"01:16.350 ","End":"01:18.965","Text":"then what I\u0027m going to take as my g,"},{"Start":"01:18.965 ","End":"01:25.160","Text":"the middle function will be sine of natural log of x over x plus 1,"},{"Start":"01:25.160 ","End":"01:31.880","Text":"is going to be always less than or equal to 1 and bigger or equal to minus 1."},{"Start":"01:31.880 ","End":"01:36.140","Text":"My apologies. I think I meant to put x plus 1 here,"},{"Start":"01:36.140 ","End":"01:37.810","Text":"so let me just change that."},{"Start":"01:37.810 ","End":"01:40.315","Text":"Now, we\u0027re okay with the definition."},{"Start":"01:40.315 ","End":"01:42.890","Text":"Let\u0027s see what we can do with this."},{"Start":"01:42.890 ","End":"01:45.935","Text":"I want to get from here to this integral."},{"Start":"01:45.935 ","End":"01:49.760","Text":"I\u0027m almost there, I just need an extra X in front."},{"Start":"01:49.760 ","End":"01:55.760","Text":"We can multiply an inequality by a positive quantity without changing the inequality."},{"Start":"01:55.760 ","End":"01:58.445","Text":"I\u0027m going to multiply everything by x,"},{"Start":"01:58.445 ","End":"02:01.250","Text":"so I get minus x is"},{"Start":"02:01.250 ","End":"02:10.355","Text":"less than or equal to x sine of natural log of x plus 1 over x plus 1,"},{"Start":"02:10.355 ","End":"02:12.530","Text":"which is less than or equal to x."},{"Start":"02:12.530 ","End":"02:15.910","Text":"It\u0027s actually true from the 0 or positive."},{"Start":"02:15.910 ","End":"02:17.900","Text":"Now, this is going to be my f, g,"},{"Start":"02:17.900 ","End":"02:21.690","Text":"and h. Let\u0027s take the integral of all of these from 0-1,"},{"Start":"02:21.690 ","End":"02:28.220","Text":"so we get the integral of minus x dx from 0-1 is"},{"Start":"02:28.220 ","End":"02:35.645","Text":"less than or equal to the integral from 0-1 of x sine of natural log of x plus 1,"},{"Start":"02:35.645 ","End":"02:38.900","Text":"over x plus 1 dx,"},{"Start":"02:38.900 ","End":"02:44.965","Text":"which is less than or equal to the integral of x dx from 0-1."},{"Start":"02:44.965 ","End":"02:47.930","Text":"Let\u0027s see what each of these things at the side is."},{"Start":"02:47.930 ","End":"02:55.845","Text":"The integral of minus x is equal to minus x squared over 2,"},{"Start":"02:55.845 ","End":"02:59.515","Text":"and this goes from 0-1."},{"Start":"02:59.515 ","End":"03:02.010","Text":"When we put 0, that\u0027s 0 and put 1,"},{"Start":"03:02.010 ","End":"03:05.610","Text":"it\u0027s minus 1/2, so this is minus 1/2."},{"Start":"03:05.610 ","End":"03:10.620","Text":"Here, I get not minus x squared over 2 plus x squared over 2."},{"Start":"03:10.620 ","End":"03:12.150","Text":"From 0 to 1,"},{"Start":"03:12.150 ","End":"03:14.850","Text":"this gives me 1/2 minus 0,"},{"Start":"03:14.850 ","End":"03:17.265","Text":"so this is 1/2."},{"Start":"03:17.265 ","End":"03:19.415","Text":"Now, this integral here,"},{"Start":"03:19.415 ","End":"03:21.170","Text":"if I just put it over here,"},{"Start":"03:21.170 ","End":"03:24.395","Text":"then I get exactly what is written here,"},{"Start":"03:24.395 ","End":"03:27.720","Text":"and so we are done."}],"ID":8614},{"Watched":false,"Name":"Exercise 7","Duration":"3m 8s","ChapterTopicVideoID":8347,"CourseChapterTopicPlaylistID":65396,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8347.jpeg","UploadDate":"2017-01-28T12:25:21.1000000","DurationForVideoObject":"PT3M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.900","Text":"Here, we have to show that this definite integral is less than or equal to this constant."},{"Start":"00:06.900 ","End":"00:08.970","Text":"We\u0027re going to use a theorem,"},{"Start":"00:08.970 ","End":"00:10.920","Text":"we\u0027ve used it before once,"},{"Start":"00:10.920 ","End":"00:13.635","Text":"that if we have 2 functions, f and g,"},{"Start":"00:13.635 ","End":"00:18.855","Text":"and f is less than or equal to g on a closed interval from a to b,"},{"Start":"00:18.855 ","End":"00:21.600","Text":"then the same inequality holds for the integrals."},{"Start":"00:21.600 ","End":"00:25.230","Text":"In other words, the integral from a to b of f of x dx is also going"},{"Start":"00:25.230 ","End":"00:29.055","Text":"to be less than or equal to the integral from a to b of g of x dx."},{"Start":"00:29.055 ","End":"00:30.750","Text":"We\u0027re going to use that here,"},{"Start":"00:30.750 ","End":"00:35.280","Text":"and we\u0027re going to use the fact that for any x,"},{"Start":"00:35.280 ","End":"00:42.720","Text":"that the arctangent of x is always less than or equal to pi over 2."},{"Start":"00:42.720 ","End":"00:46.745","Text":"It\u0027s actually also bigger or equal to minus pi over 2,"},{"Start":"00:46.745 ","End":"00:48.110","Text":"but I don\u0027t care about that,"},{"Start":"00:48.110 ","End":"00:50.375","Text":"I just care about this inequality."},{"Start":"00:50.375 ","End":"00:54.410","Text":"Now, x could be anything from minus infinity to infinity."},{"Start":"00:54.410 ","End":"00:57.905","Text":"In particular, it could be this mass here."},{"Start":"00:57.905 ","End":"01:02.210","Text":"What I\u0027m saying is that the arctangent of"},{"Start":"01:02.210 ","End":"01:06.560","Text":"anything is going to be less than or equal to pi over 2."},{"Start":"01:06.560 ","End":"01:10.290","Text":"In particular, the arctangent of sine x over"},{"Start":"01:10.290 ","End":"01:14.525","Text":"x plus 4 is going to be less than or equal to pi over 2."},{"Start":"01:14.525 ","End":"01:19.210","Text":"Now, I can multiply both sides by a positive number."},{"Start":"01:19.210 ","End":"01:21.095","Text":"I can get from here,"},{"Start":"01:21.095 ","End":"01:27.155","Text":"that x squared times arctangent of sine x over x plus 4"},{"Start":"01:27.155 ","End":"01:33.680","Text":"is going to be less than or equal to pi over 2 times x squared."},{"Start":"01:33.680 ","End":"01:35.340","Text":"There\u0027s a slight cheating here,"},{"Start":"01:35.340 ","End":"01:37.759","Text":"x squared is not always positive."},{"Start":"01:37.759 ","End":"01:42.440","Text":"It could be 0, but the inequality also holds if x is 0 because then we just"},{"Start":"01:42.440 ","End":"01:47.690","Text":"get that 0 is less than or equal to 0, which is obvious."},{"Start":"01:47.690 ","End":"01:52.280","Text":"Yeah, it works. x anywhere from 0 to pi, this inequality works."},{"Start":"01:52.280 ","End":"01:56.800","Text":"Then we\u0027ll take 0 and pi as our a and b here."},{"Start":"01:56.800 ","End":"02:07.265","Text":"We get that the integral from 0 to pi of x squared arctangent of sine x"},{"Start":"02:07.265 ","End":"02:12.890","Text":"over x plus 4 dx is less than or equal to the integral"},{"Start":"02:12.890 ","End":"02:19.825","Text":"from 0 to pi of pi over 2x squared dx."},{"Start":"02:19.825 ","End":"02:22.830","Text":"Let\u0027s compute this integral."},{"Start":"02:22.830 ","End":"02:24.170","Text":"This is equal to,"},{"Start":"02:24.170 ","End":"02:28.145","Text":"and I can take pi over 2 outside the brackets,"},{"Start":"02:28.145 ","End":"02:33.710","Text":"and then the integral of x squared is x cubed over 3."},{"Start":"02:33.710 ","End":"02:39.605","Text":"But I have to take this between the limits of 0 and pi and see what we get."},{"Start":"02:39.605 ","End":"02:43.130","Text":"Well, when I substitute 0, I get 0."},{"Start":"02:43.130 ","End":"02:46.104","Text":"Really all I care is to substitute pi,"},{"Start":"02:46.104 ","End":"02:51.525","Text":"so I get pi over 2 times pi cubed over 3."},{"Start":"02:51.525 ","End":"02:55.200","Text":"I\u0027ll put the minus 0 just to show you I did do a subtraction."},{"Start":"02:55.200 ","End":"02:57.915","Text":"pi times pi cubed is pi^4,"},{"Start":"02:57.915 ","End":"02:59.445","Text":"2 times 3 is 6."},{"Start":"02:59.445 ","End":"03:03.225","Text":"This gives us pi^4 over 6."},{"Start":"03:03.225 ","End":"03:05.360","Text":"It really looks a lot like this."},{"Start":"03:05.360 ","End":"03:06.530","Text":"We\u0027ve got it."},{"Start":"03:06.530 ","End":"03:09.270","Text":"I\u0027ll put a checkmark, and we\u0027re done."}],"ID":8615}],"Thumbnail":null,"ID":65396},{"Name":"Riemann Sum and Integrability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Riemann sum","Duration":"12m 6s","ChapterTopicVideoID":8354,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8354.jpeg","UploadDate":"2019-12-10T12:46:08.4530000","DurationForVideoObject":"PT12M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"In the middle of the chapter on the definite integral,"},{"Start":"00:03.075 ","End":"00:06.450","Text":"you suddenly find exercises which asks you to"},{"Start":"00:06.450 ","End":"00:10.860","Text":"compute a limit of a series and you wonder how are the 2 related?"},{"Start":"00:10.860 ","End":"00:14.024","Text":"Well, turns out that they are intimately related,"},{"Start":"00:14.024 ","End":"00:17.760","Text":"and I\u0027ll show you how this is so."},{"Start":"00:17.760 ","End":"00:22.185","Text":"Let\u0027s start with an example where we have a function,"},{"Start":"00:22.185 ","End":"00:26.219","Text":"let\u0027s say y equals f of x,"},{"Start":"00:26.219 ","End":"00:30.255","Text":"and it\u0027s defined from 0-1."},{"Start":"00:30.255 ","End":"00:32.705","Text":"Now, the definite integral,"},{"Start":"00:32.705 ","End":"00:40.250","Text":"which is the integral from 0-1 of f of xdx,"},{"Start":"00:40.250 ","End":"00:48.215","Text":"is simply the area under this curve I\u0027m showing you the boundaries of."},{"Start":"00:48.215 ","End":"00:51.430","Text":"Without doing an integration,"},{"Start":"00:51.430 ","End":"00:56.090","Text":"we can actually try and estimate it by"},{"Start":"00:56.090 ","End":"01:01.460","Text":"dividing the interval from 0-1 into n slices let say,"},{"Start":"01:01.460 ","End":"01:07.040","Text":"where these points are 1,"},{"Start":"01:07.040 ","End":"01:15.445","Text":"2, 3, and finally n, which is 1."},{"Start":"01:15.445 ","End":"01:23.790","Text":"The idea is to try and approximate the area under the curve by means of rectangles."},{"Start":"01:23.790 ","End":"01:27.070","Text":"We have 1 rectangle here,"},{"Start":"01:27.440 ","End":"01:31.210","Text":"we have another rectangle,"},{"Start":"01:32.540 ","End":"01:38.925","Text":"let me just draw some lines here,"},{"Start":"01:38.925 ","End":"01:42.370","Text":"not the greatest, but it\u0027ll do."},{"Start":"01:42.930 ","End":"01:48.720","Text":"We have series of rectangles and this 1,"},{"Start":"01:48.720 ","End":"01:57.645","Text":"and this 1, and perhaps I\u0027ll do 1 more here."},{"Start":"01:57.645 ","End":"02:02.460","Text":"This will be 1 before n is n minus 1,"},{"Start":"02:02.460 ","End":"02:08.700","Text":"and so this is our last rectangle."},{"Start":"02:08.700 ","End":"02:13.070","Text":"Now, I want to approximate this by saying that this is"},{"Start":"02:13.070 ","End":"02:17.705","Text":"going to be the sum of the areas of the rectangles."},{"Start":"02:17.705 ","End":"02:21.660","Text":"Let me draw some more horizontal lines."},{"Start":"02:21.660 ","End":"02:25.670","Text":"Here, it\u0027s all the way to the top of the rectangle."},{"Start":"02:25.670 ","End":"02:31.460","Text":"Now, what is the area of the first rectangle?"},{"Start":"02:31.460 ","End":"02:34.880","Text":"Well, the base is 1, what\u0027s the height?"},{"Start":"02:34.880 ","End":"02:39.470","Text":"The height here will be f of 1 because this is the function y equals f of x,"},{"Start":"02:39.470 ","End":"02:40.850","Text":"and let me just mark all these,"},{"Start":"02:40.850 ","End":"02:43.025","Text":"this is f of 2,"},{"Start":"02:43.025 ","End":"02:45.880","Text":"this would be f of 3,"},{"Start":"02:45.880 ","End":"02:51.180","Text":"f of n or f of 1."},{"Start":"02:51.760 ","End":"02:55.065","Text":"Now, we\u0027ve got all the heights of the rectangles,"},{"Start":"02:55.065 ","End":"03:00.675","Text":"the first rectangle would be 1,"},{"Start":"03:00.675 ","End":"03:06.900","Text":"the area times the height times f of 1."},{"Start":"03:06.900 ","End":"03:13.775","Text":"Second 1, the base is also 1 because these are equal steps,"},{"Start":"03:13.775 ","End":"03:16.700","Text":"each of these steps is 1."},{"Start":"03:16.700 ","End":"03:20.510","Text":"It\u0027s 1 times the height,"},{"Start":"03:20.510 ","End":"03:26.390","Text":"which is f of 2 plus 1 times"},{"Start":"03:26.390 ","End":"03:33.800","Text":"f of 3, and so on."},{"Start":"03:33.800 ","End":"03:39.050","Text":"The final 1 is equal to the base,"},{"Start":"03:39.050 ","End":"03:43.670","Text":"which is still 1, times the height,"},{"Start":"03:43.670 ","End":"03:48.710","Text":"which is f of 1 or n."},{"Start":"03:48.710 ","End":"03:53.090","Text":"This is an approximation to the area under the curve,"},{"Start":"03:53.090 ","End":"04:03.115","Text":"and this is approximately equal to integral from 0-1 of f of xdx."},{"Start":"04:03.115 ","End":"04:07.205","Text":"What we\u0027re going to do is take the limit as n goes to infinity,"},{"Start":"04:07.205 ","End":"04:12.110","Text":"and then these will approximate much more closely definer,"},{"Start":"04:12.110 ","End":"04:18.830","Text":"you\u0027d make the rectangles the thinner then better the approximation is until at infinity,"},{"Start":"04:18.830 ","End":"04:20.110","Text":"we actually get the integral,"},{"Start":"04:20.110 ","End":"04:27.625","Text":"and this is how the integral was developed by Mathematician Reimann."},{"Start":"04:27.625 ","End":"04:33.830","Text":"Let me just indicate that this is the sum of"},{"Start":"04:33.830 ","End":"04:41.695","Text":"the areas of the rectangles,"},{"Start":"04:41.695 ","End":"04:44.339","Text":"and this is what this equals."},{"Start":"04:44.339 ","End":"04:52.005","Text":"Now, I\u0027m going to rewrite this in a more convenient form,"},{"Start":"04:52.005 ","End":"04:56.405","Text":"1 thing, I can take 1 outside the brackets."},{"Start":"04:56.405 ","End":"05:06.545","Text":"I have 1 times f of 1 plus f of 2,"},{"Start":"05:06.545 ","End":"05:11.090","Text":"plus f of 3 plus,"},{"Start":"05:11.090 ","End":"05:16.160","Text":"and so on The last term is f of n, which is 1,"},{"Start":"05:16.160 ","End":"05:19.430","Text":"but I prefer to leave it this way because then we can see 1, 2, 3,"},{"Start":"05:19.430 ","End":"05:23.555","Text":"and so on up to n. Now,"},{"Start":"05:23.555 ","End":"05:27.620","Text":"the business of area under a curve is just really an illustration."},{"Start":"05:27.620 ","End":"05:32.180","Text":"In actual fact, the definition of the integral according to Reimann"},{"Start":"05:32.180 ","End":"05:37.460","Text":"is the limit of this as n goes to infinity."},{"Start":"05:37.460 ","End":"05:39.650","Text":"I\u0027m going to take the limit and we\u0027re actually"},{"Start":"05:39.650 ","End":"05:43.050","Text":"going to use Reimann\"s definition of the integral."},{"Start":"05:46.430 ","End":"05:55.070","Text":"This is a formula we\u0027re going to use is that the limit as n goes to infinity of"},{"Start":"05:55.070 ","End":"06:00.605","Text":"1 times f of 1 plus"},{"Start":"06:00.605 ","End":"06:06.470","Text":"f of 2 both with the 3 etc.,"},{"Start":"06:06.470 ","End":"06:12.920","Text":"plus f of n is equal to,"},{"Start":"06:12.920 ","End":"06:15.260","Text":"by definition, by Reimann,"},{"Start":"06:15.260 ","End":"06:21.890","Text":"to the integral from 0-1 of f of xdx."},{"Start":"06:21.890 ","End":"06:25.010","Text":"Well, Reimann didn\u0027t do it exactly in this form,"},{"Start":"06:25.010 ","End":"06:30.005","Text":"but this thing, mathematician."},{"Start":"06:30.005 ","End":"06:34.985","Text":"I\u0027m going to frame this because it\u0027s going to be our definition."},{"Start":"06:34.985 ","End":"06:37.055","Text":"Coming up next."},{"Start":"06:37.055 ","End":"06:39.950","Text":"I\u0027m going to give a lot of examples of this,"},{"Start":"06:39.950 ","End":"06:43.730","Text":"and after that, I\u0027ll show how to generalize this because,"},{"Start":"06:43.730 ","End":"06:47.490","Text":"of course, many integrals are not from 0-1,"},{"Start":"06:47.490 ","End":"06:49.340","Text":"but in general from A-B,"},{"Start":"06:49.340 ","End":"06:51.725","Text":"so that\u0027s coming up."},{"Start":"06:51.725 ","End":"06:56.420","Text":"The first example will be to take f of x equals x squared,"},{"Start":"06:56.420 ","End":"06:58.985","Text":"and this is what we get."},{"Start":"06:58.985 ","End":"07:02.760","Text":"We get same thing here,"},{"Start":"07:02.760 ","End":"07:06.810","Text":"but instead of f, I\u0027m using the x squared function."},{"Start":"07:06.810 ","End":"07:12.620","Text":"It\u0027s 1 squared, 2 squared, 3 squared,"},{"Start":"07:12.620 ","End":"07:20.290","Text":"perhaps I\u0027ll put back in here the 3 up to n squared."},{"Start":"07:20.290 ","End":"07:22.220","Text":"There are many other examples,"},{"Start":"07:22.220 ","End":"07:24.935","Text":"we could take the x cubed function."},{"Start":"07:24.935 ","End":"07:28.580","Text":"Let me show you all that I\u0027ve got here,"},{"Start":"07:28.580 ","End":"07:32.450","Text":"I\u0027ve got the x cubed function, the cosine function,"},{"Start":"07:32.450 ","End":"07:35.840","Text":"the exp1ntial function, and the square root function."},{"Start":"07:35.840 ","End":"07:38.630","Text":"In each case, I get the,"},{"Start":"07:38.630 ","End":"07:44.000","Text":"apply the formula, and I have 1 up to n."},{"Start":"07:44.000 ","End":"07:47.110","Text":"In fact, the general pattern,"},{"Start":"07:47.110 ","End":"07:49.250","Text":"when I see x squared here,"},{"Start":"07:49.250 ","End":"07:55.880","Text":"I look at the x and I replace x by 1,"},{"Start":"07:55.880 ","End":"07:58.030","Text":"2, 3 up to n,"},{"Start":"07:58.030 ","End":"08:00.470","Text":"and the same is squared, so it\u0027s squared."},{"Start":"08:00.470 ","End":"08:02.315","Text":"If it\u0027s the x cubed function,"},{"Start":"08:02.315 ","End":"08:04.235","Text":"I\u0027ll replace x by 1,"},{"Start":"08:04.235 ","End":"08:08.090","Text":"2, 3, n, and they\u0027re all cubed."},{"Start":"08:08.090 ","End":"08:10.730","Text":"If I have the cosine function,"},{"Start":"08:10.730 ","End":"08:16.010","Text":"then x is always replaced by a sum,"},{"Start":"08:16.010 ","End":"08:18.800","Text":"1, 2, and when we use the same function,"},{"Start":"08:18.800 ","End":"08:22.325","Text":"cosine, so cosine for each 1, and so on."},{"Start":"08:22.325 ","End":"08:25.985","Text":"If it\u0027s e to the x, then it\u0027s e to the 1,"},{"Start":"08:25.985 ","End":"08:27.830","Text":"e to the 2,"},{"Start":"08:27.830 ","End":"08:29.930","Text":"e to the 3,"},{"Start":"08:29.930 ","End":"08:32.300","Text":"e to the n."},{"Start":"08:32.300 ","End":"08:35.030","Text":"The last 1 is the square root function so again,"},{"Start":"08:35.030 ","End":"08:38.060","Text":"x is replaced by the square root of various things,"},{"Start":"08:38.060 ","End":"08:44.045","Text":"1, same 2, 3, and n."},{"Start":"08:44.045 ","End":"08:52.310","Text":"That\u0027s basically it for the integral from 0-1."},{"Start":"08:52.310 ","End":"09:01.985","Text":"Next, I\u0027m going to generalize this so that instead of the integral from 0-1,"},{"Start":"09:01.985 ","End":"09:08.970","Text":"we go to the integral from A-B,"},{"Start":"09:08.970 ","End":"09:12.510","Text":"and middle step from 0-b,"},{"Start":"09:12.510 ","End":"09:14.730","Text":"that\u0027s sometimes useful also,"},{"Start":"09:14.730 ","End":"09:16.680","Text":"that\u0027s coming up next."},{"Start":"09:16.680 ","End":"09:20.930","Text":"Just to remind you, I\u0027ve copied the formula again,"},{"Start":"09:20.930 ","End":"09:24.820","Text":"and I\u0027m going to generalize it first of all,"},{"Start":"09:24.820 ","End":"09:27.490","Text":"so that it doesn\u0027t go from 0-1,"},{"Start":"09:27.490 ","End":"09:30.290","Text":"but from 0 to any number b."},{"Start":"09:30.290 ","End":"09:33.120","Text":"That\u0027s a very simple alteration,"},{"Start":"09:33.120 ","End":"09:34.780","Text":"in fact, I\u0027m going to alter it right here."},{"Start":"09:34.780 ","End":"09:37.645","Text":"If I want the integral from 0-b,"},{"Start":"09:37.645 ","End":"09:43.390","Text":"the only changes I need to make is to change this to a b also, and here,"},{"Start":"09:43.390 ","End":"09:46.645","Text":"I put b or 1b won\u0027t hurt,"},{"Start":"09:46.645 ","End":"09:51.405","Text":"2b up to n_b,"},{"Start":"09:51.405 ","End":"09:55.130","Text":"and if it\u0027s 3 here, it\u0027s 3_b and so on."},{"Start":"09:55.130 ","End":"09:57.290","Text":"That\u0027s just a generalization."},{"Start":"09:57.290 ","End":"10:00.910","Text":"In 1 of the exercises I think I saw something from 0-Pi,"},{"Start":"10:00.910 ","End":"10:03.050","Text":"and you can use this formula,"},{"Start":"10:03.050 ","End":"10:05.150","Text":"and it\u0027s based on the same principles,"},{"Start":"10:05.150 ","End":"10:07.625","Text":"but this time the interval from 0-b,"},{"Start":"10:07.625 ","End":"10:11.370","Text":"the width of each rectangle is b, and so on."},{"Start":"10:11.370 ","End":"10:13.625","Text":"Just accept it as a formula."},{"Start":"10:13.625 ","End":"10:19.240","Text":"The next thing I want to do is generalize not from 0-b,"},{"Start":"10:19.240 ","End":"10:28.310","Text":"but I want it to be from a-b of f of xdx."},{"Start":"10:28.310 ","End":"10:30.780","Text":"This is going to be a little bit different,"},{"Start":"10:30.780 ","End":"10:33.559","Text":"let me just write it and then I\u0027ll explain."},{"Start":"10:33.559 ","End":"10:37.370","Text":"Very well, here\u0027s the formula,"},{"Start":"10:37.370 ","End":"10:42.950","Text":"and notice that it contains a term Delta,"},{"Start":"10:42.950 ","End":"10:49.085","Text":"which is just used as an abbreviation for b minus a/ n,"},{"Start":"10:49.085 ","End":"10:50.870","Text":"b is here, a is here,"},{"Start":"10:50.870 ","End":"10:55.505","Text":"the difference is b minus a and divided by the n from here."},{"Start":"10:55.505 ","End":"10:57.650","Text":"Of course, I didn\u0027t have to use Delta,"},{"Start":"10:57.650 ","End":"10:59.720","Text":"I could write b minus a here,"},{"Start":"10:59.720 ","End":"11:01.475","Text":"b minus a here,"},{"Start":"11:01.475 ","End":"11:03.065","Text":"and so on and so on."},{"Start":"11:03.065 ","End":"11:06.230","Text":"It looks a bit messier because then this term, for example,"},{"Start":"11:06.230 ","End":"11:15.295","Text":"looks like f of a plus 2b minus a,"},{"Start":"11:15.295 ","End":"11:18.270","Text":"and it\u0027s just too many b minus a\u0027s everywhere,"},{"Start":"11:18.270 ","End":"11:20.120","Text":"and that\u0027s why I write it this way."},{"Start":"11:20.120 ","End":"11:23.660","Text":"Although strictly speaking, since n is a variable,"},{"Start":"11:23.660 ","End":"11:25.160","Text":"Delta is different for each,"},{"Start":"11:25.160 ","End":"11:27.575","Text":"and I should put a subscript Delta n,"},{"Start":"11:27.575 ","End":"11:31.040","Text":"Delta n, Delta n, and Delta n,"},{"Start":"11:31.040 ","End":"11:36.605","Text":"and that will be better where Delta n is just b minus a."},{"Start":"11:36.605 ","End":"11:41.025","Text":"You will see this in at least 1 of the examples,"},{"Start":"11:41.025 ","End":"11:47.100","Text":"the use of the integral from a-b so I won\u0027t go anymore into it here."},{"Start":"11:48.040 ","End":"11:51.395","Text":"I just framed this because it"},{"Start":"11:51.395 ","End":"11:55.100","Text":"deserves to be framed as part of this formula that goes with this."},{"Start":"11:55.100 ","End":"12:01.255","Text":"I should say where Delta n equals."},{"Start":"12:01.255 ","End":"12:04.580","Text":"That\u0027s it, I\u0027m d1."},{"Start":"12:04.580 ","End":"12:07.050","Text":"Do the exercises."}],"ID":8591},{"Watched":false,"Name":"Riemann Sum and Integrability","Duration":"11m 20s","ChapterTopicVideoID":23299,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23299.jpeg","UploadDate":"2021-01-10T13:18:45.2330000","DurationForVideoObject":"PT11M20S","Description":null,"VideoComments":[],"Subtitles":[],"ID":24156},{"Watched":false,"Name":"Exercise 1","Duration":"5m 37s","ChapterTopicVideoID":23300,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23300.jpeg","UploadDate":"2021-01-10T13:25:17.5470000","DurationForVideoObject":"PT5M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.425","Text":"In this exercise, we have to compute the integral from 1-2 of f of x dx."},{"Start":"00:07.425 ","End":"00:09.360","Text":"This is a 2 in 1 exercise."},{"Start":"00:09.360 ","End":"00:12.120","Text":"One time we\u0027ll do it for this f of x 1 over"},{"Start":"00:12.120 ","End":"00:16.650","Text":"x and in the next one we\u0027ll do it as 1 over x squared."},{"Start":"00:16.650 ","End":"00:19.665","Text":"We\u0027re going to use partitions p_n,"},{"Start":"00:19.665 ","End":"00:25.710","Text":"which is x naught to x_n where x_i is 2 to the i over n. In other words,"},{"Start":"00:25.710 ","End":"00:27.390","Text":"we have 2 to the 0 over n,"},{"Start":"00:27.390 ","End":"00:28.665","Text":"2 to the 1 over n,"},{"Start":"00:28.665 ","End":"00:30.465","Text":"up to 2 to the n over n,"},{"Start":"00:30.465 ","End":"00:32.010","Text":"which comes out to be,"},{"Start":"00:32.010 ","End":"00:34.530","Text":"2 to the 0 over n is 1."},{"Start":"00:34.530 ","End":"00:35.964","Text":"I give an example."},{"Start":"00:35.964 ","End":"00:41.400","Text":"If n is 4, then we have 2 to the power of 0 quarters,"},{"Start":"00:41.400 ","End":"00:45.890","Text":"1/4, 2/4, 3/4, and 4/4, which is just 2."},{"Start":"00:45.890 ","End":"00:48.799","Text":"In both cases, we\u0027ll be taking a Riemann sum,"},{"Start":"00:48.799 ","End":"00:50.540","Text":"reminding you what that is."},{"Start":"00:50.540 ","End":"00:55.520","Text":"The sum of f of c_i x_i minus x_i minus 1."},{"Start":"00:55.520 ","End":"00:58.170","Text":"This bit we call Delta x_i,"},{"Start":"00:58.170 ","End":"01:05.715","Text":"and c_i will be sample point in the sub-interval x_i minus 1 x_i."},{"Start":"01:05.715 ","End":"01:13.120","Text":"In part a, we\u0027re going to choose c_i to be the left interval point."},{"Start":"01:13.970 ","End":"01:20.245","Text":"F of c_i will be 1 over c_i times x_i minus x_i minus 1."},{"Start":"01:20.245 ","End":"01:22.400","Text":"Now what is this equal to?"},{"Start":"01:22.400 ","End":"01:25.370","Text":"C_i, we said is x_i minus 1,"},{"Start":"01:25.370 ","End":"01:29.065","Text":"so it\u0027s 1 over x_i minus 1 x_i minus x_i minus 1."},{"Start":"01:29.065 ","End":"01:34.250","Text":"Here we have x_i over x_i minus 1, minus 1."},{"Start":"01:34.250 ","End":"01:37.295","Text":"We\u0027ve divided this into this."},{"Start":"01:37.295 ","End":"01:40.645","Text":"Now what is x_i over x_i minus 1?"},{"Start":"01:40.645 ","End":"01:45.150","Text":"This is 2 to the i over n. This is 2 to the i minus 1 over n,"},{"Start":"01:45.150 ","End":"01:47.900","Text":"together if you subtract the exponents,"},{"Start":"01:47.900 ","End":"01:52.975","Text":"we get 2 to the 1 over n. Note that this doesn\u0027t depend on i."},{"Start":"01:52.975 ","End":"01:58.195","Text":"What we have is the sum i from 1 to n of 2 to the 1 over n minus 1."},{"Start":"01:58.195 ","End":"02:00.530","Text":"This is not dependent on i,"},{"Start":"02:00.530 ","End":"02:06.320","Text":"so this is just equal to n times the constant term, which is this."},{"Start":"02:06.320 ","End":"02:13.430","Text":"Now what we want is the limit as n goes to infinity of this."},{"Start":"02:13.430 ","End":"02:18.665","Text":"What we can do is instead of a discrete limit where n goes to infinity,"},{"Start":"02:18.665 ","End":"02:26.840","Text":"we could substitute 1 over n as x and take a continuous limit as x goes to positive 0,"},{"Start":"02:26.840 ","End":"02:28.890","Text":"x goes to 0 from above."},{"Start":"02:28.890 ","End":"02:33.095","Text":"1 over n is x, so n is 1 over x."},{"Start":"02:33.095 ","End":"02:35.000","Text":"If this limit exists,"},{"Start":"02:35.000 ","End":"02:36.785","Text":"then this will be equal to this."},{"Start":"02:36.785 ","End":"02:40.175","Text":"This is a case of 0 over 0."},{"Start":"02:40.175 ","End":"02:42.470","Text":"We could use L\u0027Hopital\u0027s Rule,"},{"Start":"02:42.470 ","End":"02:46.790","Text":"this we could try and differentiate numerator and denominator."},{"Start":"02:46.790 ","End":"02:51.110","Text":"Derivative of 2 to the x is 2 to the x, natural log of 2."},{"Start":"02:51.110 ","End":"02:52.670","Text":"In general, for a to the x,"},{"Start":"02:52.670 ","End":"02:55.940","Text":"it\u0027s a to the x natural log of a minus"},{"Start":"02:55.940 ","End":"02:59.705","Text":"1 doesn\u0027t give us anything and derivative of x is 1."},{"Start":"02:59.705 ","End":"03:02.630","Text":"Here we can substitute x equals 0,"},{"Start":"03:02.630 ","End":"03:05.015","Text":"2 to the 0 is 1 over 1."},{"Start":"03:05.015 ","End":"03:12.035","Text":"We\u0027re just left with the integral being natural log of 2, that\u0027s part a."},{"Start":"03:12.035 ","End":"03:15.870","Text":"Now let\u0027s get back and do part b."},{"Start":"03:15.920 ","End":"03:21.130","Text":"Again, we have similar expression except instead of 1 over x,"},{"Start":"03:21.130 ","End":"03:22.360","Text":"we have 1 over x squared,"},{"Start":"03:22.360 ","End":"03:23.860","Text":"so it\u0027s 1 over c_i squared,"},{"Start":"03:23.860 ","End":"03:27.340","Text":"and again, we\u0027ll choose the left end point."},{"Start":"03:27.340 ","End":"03:32.020","Text":"What we get similar to here, it\u0027s a little trickier."},{"Start":"03:32.020 ","End":"03:38.950","Text":"What we\u0027ll do is we\u0027ll split up the x_i minus 1 squared into x_i minus 1 and x_i minus 1."},{"Start":"03:38.950 ","End":"03:40.870","Text":"Now this bit here,"},{"Start":"03:40.870 ","End":"03:45.790","Text":"just like here, is 2 to the 1 over n minus 1,"},{"Start":"03:45.790 ","End":"03:51.385","Text":"x_i minus 1 is 2 to the i minus 1 over n. We can pull the constant part,"},{"Start":"03:51.385 ","End":"03:54.940","Text":"constant with respect to i in front of the sum."},{"Start":"03:54.940 ","End":"03:56.650","Text":"We have this sum,"},{"Start":"03:56.650 ","End":"03:59.915","Text":"put this in the numerator minus here."},{"Start":"03:59.915 ","End":"04:02.405","Text":"Now this is a geometric series."},{"Start":"04:02.405 ","End":"04:10.805","Text":"The first term when i equals 1 is just equal to 2 to the power of minus 0 over n,"},{"Start":"04:10.805 ","End":"04:12.695","Text":"which is just 1."},{"Start":"04:12.695 ","End":"04:15.470","Text":"The common quotient, if you think about it,"},{"Start":"04:15.470 ","End":"04:17.705","Text":"when you increase i by 1,"},{"Start":"04:17.705 ","End":"04:25.160","Text":"then this power increases by minus 1 over n. So altogether the common quotient or"},{"Start":"04:25.160 ","End":"04:33.170","Text":"factor is 2 to the minus 1 over n. We also know that there are n terms in the series."},{"Start":"04:33.170 ","End":"04:36.080","Text":"We use the formula for this,"},{"Start":"04:36.080 ","End":"04:41.190","Text":"which is a_1, 1 minus q to the n over 1 minus q."},{"Start":"04:41.200 ","End":"04:48.169","Text":"This is equal to q to the n is just 2 to the minus 1,"},{"Start":"04:48.169 ","End":"04:55.700","Text":"q as is this over this is 2 to the 1 over n,"},{"Start":"04:55.700 ","End":"04:59.120","Text":"1 into 2 to the 1 over n. You can see that."},{"Start":"04:59.120 ","End":"05:02.060","Text":"This, if you multiply by 2 to the 1 over n,"},{"Start":"05:02.060 ","End":"05:03.770","Text":"just gives us 1."},{"Start":"05:03.770 ","End":"05:05.285","Text":"I\u0027ll highlight it."},{"Start":"05:05.285 ","End":"05:10.490","Text":"This over this gives us this."},{"Start":"05:10.490 ","End":"05:16.730","Text":"For the rest of it, 1 minus 2 to the minus 1 is 1 minus 1/2 which is 1/2."},{"Start":"05:16.730 ","End":"05:20.840","Text":"What we want is the limit as n goes to infinity of"},{"Start":"05:20.840 ","End":"05:25.340","Text":"this 2 to the 1 over n. When n goes to infinity,"},{"Start":"05:25.340 ","End":"05:27.920","Text":"goes to 2 to the 0, which is 1."},{"Start":"05:27.920 ","End":"05:33.610","Text":"This part disappears and all we\u0027re left with is the 1/2."},{"Start":"05:33.610 ","End":"05:37.990","Text":"That\u0027s the answer for part b, and we\u0027re done."}],"ID":24157},{"Watched":false,"Name":"Exercise 2","Duration":"3m 41s","ChapterTopicVideoID":23301,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23301.jpeg","UploadDate":"2021-01-10T13:28:24.1000000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.075","Text":"This is a 2-part exercise about the properties of integrable functions,"},{"Start":"00:06.075 ","End":"00:10.410","Text":"f and g are integrable functions on the interval a, b."},{"Start":"00:10.410 ","End":"00:13.035","Text":"We have to prove both of the following."},{"Start":"00:13.035 ","End":"00:18.225","Text":"A, if f is less than or equal to g on the whole interval,"},{"Start":"00:18.225 ","End":"00:22.860","Text":"then the integral of f is less than or equal to the integral of g."},{"Start":"00:22.860 ","End":"00:28.815","Text":"In part B, if f is sandwiched between 2 constants,"},{"Start":"00:28.815 ","End":"00:31.740","Text":"m and M on the interval,"},{"Start":"00:31.740 ","End":"00:33.870","Text":"then the following inequality holds."},{"Start":"00:33.870 ","End":"00:36.990","Text":"Little m times b minus a is less than or equal"},{"Start":"00:36.990 ","End":"00:40.950","Text":"to the integral less than or equal to M times b minus a."},{"Start":"00:40.950 ","End":"00:50.900","Text":"With part A, let h be the function g minus f so that h is bigger or equal to 0 on a, b."},{"Start":"00:50.900 ","End":"00:53.030","Text":"I mean, if f is less than or equal to g,"},{"Start":"00:53.030 ","End":"00:55.115","Text":"then g minus f bigger or equal to 0."},{"Start":"00:55.115 ","End":"00:57.310","Text":"So h is bigger or equal to 0."},{"Start":"00:57.310 ","End":"00:59.630","Text":"By the linearity of the integral,"},{"Start":"00:59.630 ","End":"01:04.040","Text":"h is integrable because if you subtract 2 integral functions"},{"Start":"01:04.040 ","End":"01:09.730","Text":"in general, multiplying by a constant or adding and subtracting preserves integrability."},{"Start":"01:09.730 ","End":"01:14.790","Text":"The integral is equal to the integral of g minus the integral of f."},{"Start":"01:14.790 ","End":"01:17.390","Text":"Now let\u0027s define a trivial partition of a,"},{"Start":"01:17.390 ","End":"01:20.465","Text":"b, we\u0027ll just take the 2 points, a and b."},{"Start":"01:20.465 ","End":"01:21.800","Text":"Call this 1, x-naught,"},{"Start":"01:21.800 ","End":"01:23.854","Text":"call this 1, x_1,"},{"Start":"01:23.854 ","End":"01:25.490","Text":"and that\u0027s a partition."},{"Start":"01:25.490 ","End":"01:27.275","Text":"It only has 1 integral in it."},{"Start":"01:27.275 ","End":"01:28.910","Text":"But fair enough."},{"Start":"01:28.910 ","End":"01:34.400","Text":"Now the integral on the interval of the function"},{"Start":"01:34.400 ","End":"01:40.855","Text":"is the supremum of the lowest sums over all the partitions."},{"Start":"01:40.855 ","End":"01:47.360","Text":"In particular, the integral is bigger or equal to any given lower sum,"},{"Start":"01:47.360 ","End":"01:50.095","Text":"like the lower sum for p-naught."},{"Start":"01:50.095 ","End":"01:58.910","Text":"The lowest sum is just Sigma of all the infimum times the width of the sub-intervals."},{"Start":"01:58.910 ","End":"02:00.650","Text":"This partition is trivial."},{"Start":"02:00.650 ","End":"02:02.135","Text":"It only has 1 interval,"},{"Start":"02:02.135 ","End":"02:04.805","Text":"its width is b minus a."},{"Start":"02:04.805 ","End":"02:08.900","Text":"The infimum is the infimum over all of h is going to be"},{"Start":"02:08.900 ","End":"02:13.790","Text":"bigger or equal to 0 because the function h is bigger or equal to 0."},{"Start":"02:13.790 ","End":"02:16.505","Text":"Hence, so is its infimum."},{"Start":"02:16.505 ","End":"02:20.060","Text":"So we have bigger or equal to 0 times bigger than 0,"},{"Start":"02:20.060 ","End":"02:22.795","Text":"which is bigger or equal to 0."},{"Start":"02:22.795 ","End":"02:25.760","Text":"The integral of a which is bigger or equal to this,"},{"Start":"02:25.760 ","End":"02:27.050","Text":"which is bigger or equal to 0."},{"Start":"02:27.050 ","End":"02:30.004","Text":"The integral of h is bigger or equal to 0,"},{"Start":"02:30.004 ","End":"02:34.550","Text":"which means that the integral of g minus the integral of f is bigger or"},{"Start":"02:34.550 ","End":"02:38.900","Text":"equal to 0 because we have this and just bring this to the other side,"},{"Start":"02:38.900 ","End":"02:41.570","Text":"the flip side, we got to integral of f less than or"},{"Start":"02:41.570 ","End":"02:45.055","Text":"equal to the integral of g. That concludes part A."},{"Start":"02:45.055 ","End":"02:47.560","Text":"Now for part B,"},{"Start":"02:47.560 ","End":"02:49.495","Text":"we have to prove this."},{"Start":"02:49.495 ","End":"02:56.245","Text":"We can consider the constants as constant functions and that would still hold."},{"Start":"02:56.245 ","End":"03:00.210","Text":"By the above, the integral on a,"},{"Start":"03:00.210 ","End":"03:03.970","Text":"b of m has got to be less than or equal to the integral of f,"},{"Start":"03:03.970 ","End":"03:07.345","Text":"which in turn is less than or equal to the integral of M."},{"Start":"03:07.345 ","End":"03:12.370","Text":"Note that if we have the integral of a constant on the interval a,"},{"Start":"03:12.370 ","End":"03:16.360","Text":"b, we can bring the constant out in front unless the integral"},{"Start":"03:16.360 ","End":"03:21.385","Text":"of 1 and the integral of 1 is just b minus a."},{"Start":"03:21.385 ","End":"03:26.450","Text":"This is equal to c times b minus a, whatever c is."},{"Start":"03:26.450 ","End":"03:30.000","Text":"We get that this integral is m times b minus a."},{"Start":"03:30.000 ","End":"03:32.625","Text":"This integral is M times b minus a."},{"Start":"03:32.625 ","End":"03:35.660","Text":"This integral is sandwiched between these 2."},{"Start":"03:35.660 ","End":"03:38.015","Text":"That\u0027s what we have to prove."},{"Start":"03:38.015 ","End":"03:41.430","Text":"That completes this exercise."}],"ID":24158},{"Watched":false,"Name":"Exercise 3","Duration":"2m 9s","ChapterTopicVideoID":23302,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23302.jpeg","UploadDate":"2021-01-10T13:30:57.0000000","DurationForVideoObject":"PT2M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In this exercise, we recall"},{"Start":"00:02.340 ","End":"00:05.010","Text":"the First Mean Value Theorem for Integrals."},{"Start":"00:05.010 ","End":"00:07.950","Text":"We have a function f on a, b,"},{"Start":"00:07.950 ","End":"00:09.480","Text":"which is continuous."},{"Start":"00:09.480 ","End":"00:11.520","Text":"Then we\u0027re guaranteed that there exist"},{"Start":"00:11.520 ","End":"00:14.295","Text":"some point c in the open interval a, b,"},{"Start":"00:14.295 ","End":"00:16.950","Text":"such that the integral of f on ab"},{"Start":"00:16.950 ","End":"00:20.460","Text":"is f of c times b minus a."},{"Start":"00:20.460 ","End":"00:24.015","Text":"Now, our task is to show that the theorem fails,"},{"Start":"00:24.015 ","End":"00:27.470","Text":"if we replace the word continuous"},{"Start":"00:27.470 ","End":"00:29.705","Text":"with the word integrable."},{"Start":"00:29.705 ","End":"00:31.790","Text":"What does it mean the theorem failed?"},{"Start":"00:31.790 ","End":"00:34.405","Text":"It means at least 1 counterexample."},{"Start":"00:34.405 ","End":"00:35.700","Text":"That\u0027s what we\u0027re going do,"},{"Start":"00:35.700 ","End":"00:37.740","Text":"produce a counterexample."},{"Start":"00:37.740 ","End":"00:40.820","Text":"Here\u0027s 1 possibility out of many."},{"Start":"00:40.820 ","End":"00:43.500","Text":"Let\u0027s define function f."},{"Start":"00:43.500 ","End":"00:44.280","Text":"You know what?"},{"Start":"00:44.280 ","End":"00:46.205","Text":"I\u0027ll give you the picture."},{"Start":"00:46.205 ","End":"00:51.350","Text":"It\u0027s 0 when x is between 0 and a 1/2 inclusive,"},{"Start":"00:51.350 ","End":"00:54.415","Text":"otherwise, it\u0027s 1."},{"Start":"00:54.415 ","End":"00:58.650","Text":"Now, certainly this function is integrable."},{"Start":"00:58.650 ","End":"01:02.655","Text":"1 way to see that is to break it up into 2 sub-intervals."},{"Start":"01:02.655 ","End":"01:05.235","Text":"It\u0027s integrable from 0 to a 1/2,"},{"Start":"01:05.235 ","End":"01:08.180","Text":"and it\u0027s integrable from a 1/2 to 1,"},{"Start":"01:08.180 ","End":"01:10.805","Text":"so it\u0027s integrable from 0 to 1."},{"Start":"01:10.805 ","End":"01:13.990","Text":"It\u0027s also integrable because it\u0027s monotone."},{"Start":"01:13.990 ","End":"01:17.550","Text":"This integral on 0 to 1 is equal"},{"Start":"01:17.550 ","End":"01:19.245","Text":"to the sum of the 2 integrals."},{"Start":"01:19.245 ","End":"01:21.360","Text":"But the integral from 0 to a 1/2"},{"Start":"01:21.360 ","End":"01:24.135","Text":"is just 0 because the function is 0."},{"Start":"01:24.135 ","End":"01:27.050","Text":"The integral from 0 to 1 therefore"},{"Start":"01:27.050 ","End":"01:30.260","Text":"is integral from a 1/2 to 1, which is this,"},{"Start":"01:30.260 ","End":"01:32.720","Text":"and it\u0027s equal to a 1/2 times 1,"},{"Start":"01:32.720 ","End":"01:33.905","Text":"which is a 1/2."},{"Start":"01:33.905 ","End":"01:36.470","Text":"Now, if this theorem were true,"},{"Start":"01:36.470 ","End":"01:40.160","Text":"then we would expect to find some c"},{"Start":"01:40.160 ","End":"01:43.129","Text":"between 0 and 1 such that the integral"},{"Start":"01:43.129 ","End":"01:48.695","Text":"is f of c times 1 minus 0, b minus a."},{"Start":"01:48.695 ","End":"01:52.850","Text":"In other words, 1/2 is equal to f of c."},{"Start":"01:52.850 ","End":"01:54.710","Text":"But if you look at this graph,"},{"Start":"01:54.710 ","End":"01:56.930","Text":"nowhere does f of c equals a 1/2,"},{"Start":"01:56.930 ","End":"01:59.780","Text":"it doesn\u0027t cross this horizontal line."},{"Start":"01:59.780 ","End":"02:03.245","Text":"It\u0027s either 0 or it\u0027s 1, it\u0027s never a 1/2,"},{"Start":"02:03.245 ","End":"02:07.040","Text":"and so the theorem in the modified form fails."},{"Start":"02:07.040 ","End":"02:10.020","Text":"We are done."}],"ID":24159},{"Watched":false,"Name":"Exercise 4","Duration":"3m 33s","ChapterTopicVideoID":23303,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23303.jpeg","UploadDate":"2021-01-10T13:34:21.4430000","DurationForVideoObject":"PT3M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.410","Text":"In this exercise, we have a bounded function f on the interval a,"},{"Start":"00:04.410 ","End":"00:10.770","Text":"b, and we have a partition P of the interval."},{"Start":"00:10.770 ","End":"00:14.190","Text":"There\u0027s 2 questions. Question 1,"},{"Start":"00:14.190 ","End":"00:17.130","Text":"can we always choose tags,"},{"Start":"00:17.130 ","End":"00:24.255","Text":"remember, the tags mean that c_i is in the interval x_i minus 1 x_i."},{"Start":"00:24.255 ","End":"00:28.170","Text":"Can we always choose tags for P such that"},{"Start":"00:28.170 ","End":"00:33.210","Text":"the Riemann sum relative to the partition and the tags"},{"Start":"00:33.210 ","End":"00:42.315","Text":"for f is the same as the lower Riemann sum for P and the function f, that\u0027s A."},{"Start":"00:42.315 ","End":"00:49.670","Text":"Then B, does the answer change if we know that f is also continuous,"},{"Start":"00:49.670 ","End":"00:53.600","Text":"meaning not just bounded, but also continuous."},{"Start":"00:53.600 ","End":"00:58.115","Text":"For part A, turns out the answer is no."},{"Start":"00:58.115 ","End":"01:03.560","Text":"It is not always and just need 1 counterexample,"},{"Start":"01:03.560 ","End":"01:07.595","Text":"which will be a variation on the Dirichlet function."},{"Start":"01:07.595 ","End":"01:13.160","Text":"We define something 1 way for the rationals and another way for the irrationals."},{"Start":"01:13.160 ","End":"01:21.935","Text":"In this case, it\u0027s always 1 when x is rational and f of x equals x when x is irrational."},{"Start":"01:21.935 ","End":"01:27.200","Text":"Just note that this f is never 0 because f"},{"Start":"01:27.200 ","End":"01:32.765","Text":"of 0 is also 1 and not 0 because 0 is rational."},{"Start":"01:32.765 ","End":"01:39.040","Text":"As a partition will take the trivial partition consisting of just the 2 endpoints."},{"Start":"01:39.040 ","End":"01:43.400","Text":"Note that m_1 which is the infimum,"},{"Start":"01:43.400 ","End":"01:49.265","Text":"is 0, even though there is no point at which it\u0027s 0."},{"Start":"01:49.265 ","End":"01:52.710","Text":"The infimum is 0 because, for example,"},{"Start":"01:52.710 ","End":"01:57.320","Text":"f of 1 is 1."},{"Start":"01:57.320 ","End":"02:00.200","Text":"The infimum of 1 is 0,"},{"Start":"02:00.200 ","End":"02:03.725","Text":"and the infimum overall has to be less than or equal to 0."},{"Start":"02:03.725 ","End":"02:07.695","Text":"It is 0 because everything\u0027s bigger equal to 0."},{"Start":"02:07.695 ","End":"02:10.890","Text":"M_1 is 0, f of c_1 is not 0."},{"Start":"02:10.890 ","End":"02:19.505","Text":"The lower Riemann sum is little m_1 times Delta x_1, which is 0."},{"Start":"02:19.505 ","End":"02:24.140","Text":"But the Riemann sum for the partition and"},{"Start":"02:24.140 ","End":"02:29.855","Text":"the tag is f of c_1 times 1 minus 0."},{"Start":"02:29.855 ","End":"02:32.135","Text":"Whatever c_1 you choose,"},{"Start":"02:32.135 ","End":"02:36.760","Text":"f of c_1 is not 0 because f is never 0."},{"Start":"02:36.760 ","End":"02:39.600","Text":"This is 1 times f of c_1 is f of c_1,"},{"Start":"02:39.600 ","End":"02:44.580","Text":"which is not 0, whereas this is 0."},{"Start":"02:44.580 ","End":"02:46.310","Text":"They\u0027re not the same."},{"Start":"02:46.310 ","End":"02:48.080","Text":"This is not equal to this."},{"Start":"02:48.080 ","End":"02:51.365","Text":"This is 0, this is not 0."},{"Start":"02:51.365 ","End":"02:54.650","Text":"For part B, the answer is yes."},{"Start":"02:54.650 ","End":"02:59.870","Text":"You see the snag here was that the infimum was 0,"},{"Start":"02:59.870 ","End":"03:02.525","Text":"but the infimum was never attained."},{"Start":"03:02.525 ","End":"03:06.365","Text":"There\u0027s no point at which f of c was equal to the infimum,"},{"Start":"03:06.365 ","End":"03:08.870","Text":"but if you have continuity,"},{"Start":"03:08.870 ","End":"03:11.540","Text":"then by the extreme value theorem,"},{"Start":"03:11.540 ","End":"03:13.100","Text":"the infimum is always attained,"},{"Start":"03:13.100 ","End":"03:18.865","Text":"meaning you can choose the c_i such that f of c_i equals m_i."},{"Start":"03:18.865 ","End":"03:25.320","Text":"In that case, the sum of m_i Delta x_i is the sum of f of c_i Delta x_i,"},{"Start":"03:25.320 ","End":"03:30.740","Text":"so that the lower Riemann sum equals the Riemann sum,"},{"Start":"03:30.740 ","End":"03:33.720","Text":"and we are done."}],"ID":24160},{"Watched":false,"Name":"Exercise 5","Duration":"9m 37s","ChapterTopicVideoID":23304,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23304.jpeg","UploadDate":"2021-01-10T13:36:42.1100000","DurationForVideoObject":"PT9M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"Before we get started properly with this exercise,"},{"Start":"00:03.600 ","End":"00:06.630","Text":"I want to remind you what a convex function is."},{"Start":"00:06.630 ","End":"00:10.905","Text":"If we have a function f defined on some interval I,"},{"Start":"00:10.905 ","End":"00:12.855","Text":"doesn\u0027t matter what the endpoints are."},{"Start":"00:12.855 ","End":"00:20.795","Text":"If it has the property that for any 2 numbers a and b in the interval,"},{"Start":"00:20.795 ","End":"00:23.525","Text":"usually we assume a is to the left of b,"},{"Start":"00:23.525 ","End":"00:27.350","Text":"and for any t in the interval 0,"},{"Start":"00:27.350 ","End":"00:29.300","Text":"1, the following inequality,"},{"Start":"00:29.300 ","End":"00:31.445","Text":"holds f of ta,"},{"Start":"00:31.445 ","End":"00:33.725","Text":"plus 1 minus tb,"},{"Start":"00:33.725 ","End":"00:37.265","Text":"is less than or equal to t times f of a,"},{"Start":"00:37.265 ","End":"00:42.720","Text":"plus 1 minus t f of b in the diagram here, instead of a and b,"},{"Start":"00:42.720 ","End":"00:47.604","Text":"we have x_1 and x_2 where otherwise illustrates what I\u0027m saying,"},{"Start":"00:47.604 ","End":"00:52.010","Text":"really means that this point is lower than this point."},{"Start":"00:52.010 ","End":"00:54.935","Text":"That\u0027s what the inequality is saying."},{"Start":"00:54.935 ","End":"00:57.860","Text":"This is less than or equal to this."},{"Start":"00:57.860 ","End":"01:02.275","Text":"Convex means that it\u0027s holds water if you like."},{"Start":"01:02.275 ","End":"01:04.395","Text":"Now, that we\u0027ve defined convex,"},{"Start":"01:04.395 ","End":"01:09.395","Text":"let\u0027s get on to the exercise 2 parts, a and B."},{"Start":"01:09.395 ","End":"01:14.300","Text":"In a, we have a convex function on all the reals."},{"Start":"01:14.300 ","End":"01:18.870","Text":"We have to prove that if we have any finite set t_1,"},{"Start":"01:18.870 ","End":"01:20.310","Text":"and so on up to t_n,"},{"Start":"01:20.310 ","End":"01:22.775","Text":"assume that n is 2 or more,"},{"Start":"01:22.775 ","End":"01:26.855","Text":"which satisfies that the sum of these t_i is 1."},{"Start":"01:26.855 ","End":"01:29.270","Text":"Then we have the inequality,"},{"Start":"01:29.270 ","End":"01:31.505","Text":"what\u0027s written here I don\u0027t read it out,"},{"Start":"01:31.505 ","End":"01:35.015","Text":"and we\u0027re given a hint to use induction on n,"},{"Start":"01:35.015 ","End":"01:40.140","Text":"and that\u0027s solve a and then we\u0027ll get back to b and read it."},{"Start":"01:40.460 ","End":"01:42.830","Text":"We do it by induction."},{"Start":"01:42.830 ","End":"01:45.020","Text":"Start off with n equals 2,"},{"Start":"01:45.020 ","End":"01:47.870","Text":"and instead of being given t_1,"},{"Start":"01:47.870 ","End":"01:52.630","Text":"t_2, you could replace that by t and 1 minus t,"},{"Start":"01:52.630 ","End":"01:55.170","Text":"because the sum of the t_i is 1,"},{"Start":"01:55.170 ","End":"01:57.750","Text":"t_1 plus t_2 is 1,"},{"Start":"01:57.750 ","End":"02:01.860","Text":"then t_2 is 1 minus t_1,"},{"Start":"02:01.860 ","End":"02:03.210","Text":"so this could be t,"},{"Start":"02:03.210 ","End":"02:08.060","Text":"and 1 minus t. By the definition of convex,"},{"Start":"02:08.060 ","End":"02:12.600","Text":"we have that f of ta_1 plus 1 minus ta_2,"},{"Start":"02:12.600 ","End":"02:15.225","Text":"less than or equal to, we won\u0027t read it out,"},{"Start":"02:15.225 ","End":"02:20.380","Text":"and replace t by t_1 and 1 minus t by t_2."},{"Start":"02:20.380 ","End":"02:25.640","Text":"Then we have exactly what it is that we want to show in here."},{"Start":"02:25.640 ","End":"02:28.275","Text":"It just translates exactly to this."},{"Start":"02:28.275 ","End":"02:30.175","Text":"Now, the induction step."},{"Start":"02:30.175 ","End":"02:33.190","Text":"Assuming it\u0027s true for some n,"},{"Start":"02:33.190 ","End":"02:34.480","Text":"which is 2 or more,"},{"Start":"02:34.480 ","End":"02:37.460","Text":"let\u0027s prove that it\u0027s true for n plus 1."},{"Start":"02:37.460 ","End":"02:41.715","Text":"This time we have numbers a_1, a_2, a_n,"},{"Start":"02:41.715 ","End":"02:46.640","Text":"a_n plus 1, and t_1 up to t_n plus 1."},{"Start":"02:46.640 ","End":"02:49.225","Text":"These are in the interval 0,1,"},{"Start":"02:49.225 ","End":"02:51.610","Text":"and their sum is 1."},{"Start":"02:51.610 ","End":"02:55.575","Text":"Now, if t_1 is equal to 1,"},{"Start":"02:55.575 ","End":"02:58.380","Text":"and since they all add up to 1,"},{"Start":"02:58.380 ","End":"03:00.490","Text":"and they\u0027re all non-negative,"},{"Start":"03:00.490 ","End":"03:03.670","Text":"then all the other t\u0027s would have to be 0,"},{"Start":"03:03.670 ","End":"03:07.885","Text":"and then the inequality would become true trivially,"},{"Start":"03:07.885 ","End":"03:11.790","Text":"because each side would just be f of 1,"},{"Start":"03:11.790 ","End":"03:15.840","Text":"times a_1 equals 1 times f of a_1."},{"Start":"03:15.840 ","End":"03:18.330","Text":"Anyway it\u0027s trivial."},{"Start":"03:18.330 ","End":"03:23.460","Text":"Let\u0027s assume that t_1 is not equal to 1,"},{"Start":"03:23.460 ","End":"03:25.935","Text":"and that means that it\u0027s less than 1."},{"Start":"03:25.935 ","End":"03:28.119","Text":"Then a bit of algebra,"},{"Start":"03:28.119 ","End":"03:30.430","Text":"we can rewrite the sum from 1 to n,"},{"Start":"03:30.430 ","End":"03:35.375","Text":"plus 1, as just the first term, t_i a_i,"},{"Start":"03:35.375 ","End":"03:39.435","Text":"and remember that t_1 is not equal to 1,"},{"Start":"03:39.435 ","End":"03:43.860","Text":"plus the sum from 2 to n,"},{"Start":"03:43.860 ","End":"03:45.295","Text":"plus 1 of t_i,"},{"Start":"03:45.295 ","End":"03:51.025","Text":"except that I\u0027m dividing and multiplying by 1 minus t_1."},{"Start":"03:51.025 ","End":"03:52.905","Text":"Like I just said,"},{"Start":"03:52.905 ","End":"03:54.450","Text":"t_1 is not equal to 1,"},{"Start":"03:54.450 ","End":"03:57.490","Text":"so we\u0027re not dividing by 0."},{"Start":"03:57.560 ","End":"04:01.350","Text":"Now, apply f to both sides."},{"Start":"04:01.350 ","End":"04:05.290","Text":"This expression is the case where n equals 2,"},{"Start":"04:05.290 ","End":"04:08.780","Text":"so we can say that this is less than or equal to t_1,"},{"Start":"04:08.780 ","End":"04:10.355","Text":"times f of a_1,"},{"Start":"04:10.355 ","End":"04:12.515","Text":"plus 1 minus t_1,"},{"Start":"04:12.515 ","End":"04:14.660","Text":"f of the rest of it."},{"Start":"04:14.660 ","End":"04:19.665","Text":"Now, these coefficients, t_i over 1 minus t_1,"},{"Start":"04:19.665 ","End":"04:24.420","Text":"they all add up to 1, because t_2,"},{"Start":"04:24.420 ","End":"04:28.155","Text":"plus and so on up to t n plus 1,"},{"Start":"04:28.155 ","End":"04:32.600","Text":"is 1 minus t_1 because it\u0027s everything in the sum except for t_1,"},{"Start":"04:32.600 ","End":"04:34.490","Text":"so it adds up to 1 minus t_1, 2."},{"Start":"04:34.490 ","End":"04:36.215","Text":"When we divide by 1 minus t_1,"},{"Start":"04:36.215 ","End":"04:39.380","Text":"this whole thing adds up to 1."},{"Start":"04:39.380 ","End":"04:42.485","Text":"Also, the number of terms here is"},{"Start":"04:42.485 ","End":"04:48.140","Text":"n. If it was from 1 to n plus 1 that\u0027d be n plus 1 terms 1 missing,"},{"Start":"04:48.140 ","End":"04:49.715","Text":"so is only n terms."},{"Start":"04:49.715 ","End":"04:53.630","Text":"We can use the induction hypothesis for"},{"Start":"04:53.630 ","End":"04:58.775","Text":"n. This is basically the inequality we have for n,"},{"Start":"04:58.775 ","End":"05:02.070","Text":"except that the indices are just shifted."},{"Start":"05:02.070 ","End":"05:03.510","Text":"It goes from 2 to n plus 1."},{"Start":"05:03.510 ","End":"05:05.160","Text":"Also instead of the t_i,"},{"Start":"05:05.160 ","End":"05:07.170","Text":"we have t_i prime,"},{"Start":"05:07.170 ","End":"05:09.900","Text":"which is t_i over 1 minus t_1."},{"Start":"05:09.900 ","End":"05:12.120","Text":"But this is essentially the case,"},{"Start":"05:12.120 ","End":"05:16.625","Text":"n. Now, let\u0027s continue the main flow from above."},{"Start":"05:16.625 ","End":"05:20.495","Text":"We got up to this is less than or equal to this."},{"Start":"05:20.495 ","End":"05:23.360","Text":"Now, we have an estimation for this."},{"Start":"05:23.360 ","End":"05:25.355","Text":"That\u0027s less than or equal to this."},{"Start":"05:25.355 ","End":"05:31.890","Text":"We get that this is less than or equal to this part here as is,"},{"Start":"05:31.890 ","End":"05:33.990","Text":"1 minus t_1, as is,"},{"Start":"05:33.990 ","End":"05:37.515","Text":"f this is replaced by this."},{"Start":"05:37.515 ","End":"05:43.055","Text":"Once we have this, we can then cancel the 1 minus t_1s here."},{"Start":"05:43.055 ","End":"05:45.620","Text":"What we get is,"},{"Start":"05:45.620 ","End":"05:48.005","Text":"if you look at it, just the sum,"},{"Start":"05:48.005 ","End":"05:51.350","Text":"this time from not from 2 to n plus 1, but from 1 to n,"},{"Start":"05:51.350 ","End":"05:55.400","Text":"plus 1 of t_i, f of a_i."},{"Start":"05:55.400 ","End":"05:58.310","Text":"This is exactly the case n plus 1,"},{"Start":"05:58.310 ","End":"06:02.910","Text":"so this concludes part a of the exercise."},{"Start":"06:02.910 ","End":"06:05.200","Text":"Now part B here,"},{"Start":"06:05.200 ","End":"06:06.845","Text":"we don\u0027t have to scroll back."},{"Start":"06:06.845 ","End":"06:10.655","Text":"I copied the essence of what we have to prove here."},{"Start":"06:10.655 ","End":"06:13.295","Text":"It\u0027s called Jensen\u0027s inequality."},{"Start":"06:13.295 ","End":"06:14.930","Text":"f and g are continuous,"},{"Start":"06:14.930 ","End":"06:19.530","Text":"f is convex, and we have to prove this inequality."},{"Start":"06:19.530 ","End":"06:25.605","Text":"Let P be a partition of 0,1, I should have said,"},{"Start":"06:25.605 ","End":"06:29.680","Text":"and C is a set of tags or sample points,"},{"Start":"06:29.680 ","End":"06:32.565","Text":"each c_i and its interval."},{"Start":"06:32.565 ","End":"06:35.529","Text":"We have the Riemann sum for g,"},{"Start":"06:35.529 ","End":"06:41.860","Text":"which is equal to the sum of g of c_i times delta x_i."},{"Start":"06:41.860 ","End":"06:45.520","Text":"These delta x_i are going to be the t_i,"},{"Start":"06:45.520 ","End":"06:47.615","Text":"as in part A."},{"Start":"06:47.615 ","End":"06:52.560","Text":"Now, the sum of the delta x_i is b minus a in general,"},{"Start":"06:52.560 ","End":"06:55.125","Text":"and in our case 1 minus 0,"},{"Start":"06:55.125 ","End":"06:58.005","Text":"so the sum of t_i is equal to 1."},{"Start":"06:58.005 ","End":"07:00.015","Text":"If we use part A,"},{"Start":"07:00.015 ","End":"07:03.810","Text":"with the a_i being g of c_i,"},{"Start":"07:03.810 ","End":"07:09.555","Text":"then we get that f of the sum of g of c_i t_i,"},{"Start":"07:09.555 ","End":"07:12.615","Text":"which is the sum of t_i a_i,"},{"Start":"07:12.615 ","End":"07:17.510","Text":"is less than or equal to the sum of f of a_i times t_i."},{"Start":"07:17.510 ","End":"07:21.710","Text":"I just copied the formula from part A."},{"Start":"07:21.710 ","End":"07:24.214","Text":"You can see it really is the same."},{"Start":"07:24.214 ","End":"07:27.665","Text":"If we let a_i equals g of c_i,"},{"Start":"07:27.665 ","End":"07:31.945","Text":"and just to reverse the order by putting the t_i\u0027s in front."},{"Start":"07:31.945 ","End":"07:36.945","Text":"This, if you remember that t_i is delta x_i."},{"Start":"07:36.945 ","End":"07:42.990","Text":"This is exactly the expression for the Riemann sum for the function,"},{"Start":"07:42.990 ","End":"07:45.180","Text":"g, with the partition P,"},{"Start":"07:45.180 ","End":"07:53.590","Text":"and tag set C. Because f of g of c_i is f compose g of c_i,"},{"Start":"07:53.590 ","End":"07:59.270","Text":"the right-hand side part is the Riemann sum for f"},{"Start":"07:59.270 ","End":"08:04.820","Text":"compose g with the partition P and tags set C. Next,"},{"Start":"08:04.820 ","End":"08:12.390","Text":"we take the limit as the mesh norm goes to 0 of both sides."},{"Start":"08:12.390 ","End":"08:16.145","Text":"Now, I\u0027ll justify this in a moment why we\u0027re allowed to do this."},{"Start":"08:16.145 ","End":"08:20.060","Text":"Because of the continuity of f,"},{"Start":"08:20.060 ","End":"08:25.430","Text":"we can exchange the limit with the function."},{"Start":"08:25.430 ","End":"08:28.310","Text":"Just like it works with regular limits,"},{"Start":"08:28.310 ","End":"08:31.220","Text":"it works with the limit as mesh goes to 0,"},{"Start":"08:31.220 ","End":"08:33.950","Text":"so you switch the f with the limit,"},{"Start":"08:33.950 ","End":"08:37.325","Text":"and that means that we can get f out here,"},{"Start":"08:37.325 ","End":"08:44.805","Text":"and then the limit as mesh P goes to 0 of the Riemann sum for g here."},{"Start":"08:44.805 ","End":"08:54.260","Text":"Here, the limit as the mesh of P goes to 0 of the Riemann sum for f compose g. Now,"},{"Start":"08:54.260 ","End":"08:57.335","Text":"about the justification for all this, the functions,"},{"Start":"08:57.335 ","End":"08:59.719","Text":"g and f compose g are both continuous,"},{"Start":"08:59.719 ","End":"09:02.180","Text":"and so they\u0027re integrable."},{"Start":"09:02.180 ","End":"09:06.355","Text":"Not only does this limit exist,"},{"Start":"09:06.355 ","End":"09:10.110","Text":"but the limit is equal to the integral."},{"Start":"09:10.110 ","End":"09:12.340","Text":"That\u0027s what we\u0027ll write now."},{"Start":"09:12.340 ","End":"09:17.345","Text":"The limit is equal to the integral here of g,"},{"Start":"09:17.345 ","End":"09:22.895","Text":"and here the limit is equal to the integral of f composed with g,"},{"Start":"09:22.895 ","End":"09:28.250","Text":"which again, I\u0027ll rewrite f compose g of x is f of g of x."},{"Start":"09:28.250 ","End":"09:31.850","Text":"Now that this is less than or equal to this,"},{"Start":"09:31.850 ","End":"09:33.725","Text":"is what we had to show."},{"Start":"09:33.725 ","End":"09:35.390","Text":"That is part B,"},{"Start":"09:35.390 ","End":"09:37.980","Text":"and so we are done."}],"ID":24161},{"Watched":false,"Name":"Exercise 6","Duration":"2m 37s","ChapterTopicVideoID":8336,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8336.jpeg","UploadDate":"2017-01-28T12:21:03.0270000","DurationForVideoObject":"PT2M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"In this exercise, we have to evaluate the following limit as n goes to infinity."},{"Start":"00:05.310 ","End":"00:06.825","Text":"Have copied it over here."},{"Start":"00:06.825 ","End":"00:12.420","Text":"This is 1 of those limits that is solved by means of an integral using this formula."},{"Start":"00:12.420 ","End":"00:17.910","Text":"We have to somehow transform this algebraically until it starts looking like this."},{"Start":"00:17.910 ","End":"00:20.190","Text":"We can already see there\u0027s a 1, 2,"},{"Start":"00:20.190 ","End":"00:24.000","Text":"3 up to n, but we just have to get it into separate functions."},{"Start":"00:24.000 ","End":"00:25.350","Text":"What I suggest is this."},{"Start":"00:25.350 ","End":"00:29.280","Text":"Let\u0027s first of all take the 1 over n outside the brackets here."},{"Start":"00:29.280 ","End":"00:34.920","Text":"We\u0027ll get the limit as n goes to infinity of 1 over"},{"Start":"00:34.920 ","End":"00:41.410","Text":"n and I\u0027ll take a square bracket and get 1 to the 4th plus 2 to the 4th,"},{"Start":"00:41.410 ","End":"00:46.220","Text":"plus n to the 4th over n to the 4th."},{"Start":"00:46.220 ","End":"00:50.300","Text":"It\u0027s good for me that they are either all saying power of 4,"},{"Start":"00:50.300 ","End":"00:53.360","Text":"because now I can divide each 1 of them by n to the 4th,"},{"Start":"00:53.360 ","End":"00:55.145","Text":"basically is what I\u0027m saying."},{"Start":"00:55.145 ","End":"00:58.850","Text":"I\u0027ve got 1 to the 4th over n to the 4th plus 2 to"},{"Start":"00:58.850 ","End":"01:02.720","Text":"the 4th over n to the 4th plus etc.,"},{"Start":"01:02.720 ","End":"01:05.825","Text":"n to the 4th over n to the 4th."},{"Start":"01:05.825 ","End":"01:09.445","Text":"Why did I put a 2 instead of an n? That\u0027s better."},{"Start":"01:09.445 ","End":"01:12.395","Text":"Now using laws of exponents,"},{"Start":"01:12.395 ","End":"01:15.440","Text":"I can take each 1 of these to the power of 4."},{"Start":"01:15.440 ","End":"01:23.000","Text":"This is 1 over n to the power of 4 plus 2 over n to the power of 4 plus etc.,"},{"Start":"01:23.000 ","End":"01:27.175","Text":"plus n over n to the power of 4."},{"Start":"01:27.175 ","End":"01:29.600","Text":"You see how this is starting to look like this."},{"Start":"01:29.600 ","End":"01:34.055","Text":"All I have to do now is to define f of x is equal to"},{"Start":"01:34.055 ","End":"01:38.870","Text":"x to the 4th and then this is what I get, exactly this."},{"Start":"01:38.870 ","End":"01:47.360","Text":"Then I have the limit as n goes to infinity of 1 over n times f of 1 over n,"},{"Start":"01:47.360 ","End":"01:49.190","Text":"because that\u0027s 1 over n to the 4th,"},{"Start":"01:49.190 ","End":"01:53.640","Text":"plus f of 2 over n plus, and so on,"},{"Start":"01:53.640 ","End":"02:01.760","Text":"f of n over n. If I take this f and put it in this formula,"},{"Start":"02:01.760 ","End":"02:06.200","Text":"what I get is that this thing is going to be equal to"},{"Start":"02:06.200 ","End":"02:11.735","Text":"the integral from 0-1 of f of x dx,"},{"Start":"02:11.735 ","End":"02:15.795","Text":"which is x to the 4th dx."},{"Start":"02:15.795 ","End":"02:18.079","Text":"This is easy enough to compute."},{"Start":"02:18.079 ","End":"02:24.030","Text":"This will be x to the 4th will give me x to the 5th over 5."},{"Start":"02:24.030 ","End":"02:26.635","Text":"Evaluate it between 0 and 1,"},{"Start":"02:26.635 ","End":"02:28.150","Text":"0 gives me nothing,"},{"Start":"02:28.150 ","End":"02:30.415","Text":"1 gives 1 to the 5th over 5,"},{"Start":"02:30.415 ","End":"02:32.995","Text":"so this is just equal to1/5,"},{"Start":"02:32.995 ","End":"02:35.920","Text":"and that will be the answer."},{"Start":"02:35.920 ","End":"02:38.330","Text":"We are done."}],"ID":24162},{"Watched":false,"Name":"Exercise 7","Duration":"2m 26s","ChapterTopicVideoID":8337,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8337.jpeg","UploadDate":"2017-01-28T12:21:15.2970000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Here we have one of those limit of a series,"},{"Start":"00:03.570 ","End":"00:07.020","Text":"which is based on the following formula,"},{"Start":"00:07.020 ","End":"00:10.815","Text":"which converts one of these series into an integral."},{"Start":"00:10.815 ","End":"00:12.660","Text":"I copied it over here."},{"Start":"00:12.660 ","End":"00:15.450","Text":"I want this to look more like this,"},{"Start":"00:15.450 ","End":"00:18.030","Text":"and do algebraic manipulations till it looks very"},{"Start":"00:18.030 ","End":"00:21.075","Text":"much like the left-hand side and then I\u0027ll use the right-hand side."},{"Start":"00:21.075 ","End":"00:23.895","Text":"The first thing I\u0027m going to do is take n outside"},{"Start":"00:23.895 ","End":"00:28.500","Text":"the brackets on each denominator and that will get me closer to this n here."},{"Start":"00:28.500 ","End":"00:32.765","Text":"I get the limit as n goes to infinity."},{"Start":"00:32.765 ","End":"00:36.290","Text":"Now, if I take n outside the brackets here,"},{"Start":"00:36.290 ","End":"00:45.815","Text":"I\u0027m left with 1 plus 1 over n. Here I\u0027m left with n times 1 plus 2 over n. The last one,"},{"Start":"00:45.815 ","End":"00:51.800","Text":"1 over n times 1 plus n over n. After this,"},{"Start":"00:51.800 ","End":"00:53.795","Text":"I can take the 1 over n,"},{"Start":"00:53.795 ","End":"00:56.150","Text":"which appears in each one outside the brackets."},{"Start":"00:56.150 ","End":"01:02.540","Text":"I can say that this is the limit as n goes to infinity of 1 over n. Let\u0027s see."},{"Start":"01:02.540 ","End":"01:04.100","Text":"After I\u0027ve taken all these n\u0027s out,"},{"Start":"01:04.100 ","End":"01:11.210","Text":"I have 1 over 1 plus 1 over n plus 1 over 1 plus 2 over"},{"Start":"01:11.210 ","End":"01:18.870","Text":"n plus 1 over 1 plus n of n. You see I have 1 over n,"},{"Start":"01:18.870 ","End":"01:23.240","Text":"2 over n up to n over n. Here what I have to do is let"},{"Start":"01:23.240 ","End":"01:28.605","Text":"the function f of x equal 1 over 1 plus x."},{"Start":"01:28.605 ","End":"01:32.120","Text":"Then if I let x equal 1 over n 2 over n and so on,"},{"Start":"01:32.120 ","End":"01:33.575","Text":"this is what I\u0027ll get."},{"Start":"01:33.575 ","End":"01:36.830","Text":"According to this formula, if this is my f,"},{"Start":"01:36.830 ","End":"01:43.160","Text":"then this is equal to the integral from 0 to 1 of f of x,"},{"Start":"01:43.160 ","End":"01:45.740","Text":"which is this dx."},{"Start":"01:45.740 ","End":"01:49.385","Text":"Now, this integral is straightforward,"},{"Start":"01:49.385 ","End":"01:53.825","Text":"it\u0027s natural log of 1 plus x."},{"Start":"01:53.825 ","End":"01:56.240","Text":"Normally we put an absolute value,"},{"Start":"01:56.240 ","End":"02:00.920","Text":"but 1 plus x is going to be positive when x is between 0 and 1."},{"Start":"02:00.920 ","End":"02:04.870","Text":"Then we\u0027re going to evaluate this between 0 and 1."},{"Start":"02:04.870 ","End":"02:07.460","Text":"This equals, if x equals 1,"},{"Start":"02:07.460 ","End":"02:12.005","Text":"I get natural log of 1 plus 1 is 2."},{"Start":"02:12.005 ","End":"02:14.210","Text":"If x is 0,"},{"Start":"02:14.210 ","End":"02:18.610","Text":"I get 1 plus 0 is 1 minus natural log of 1."},{"Start":"02:18.610 ","End":"02:21.105","Text":"But natural log of 1 is 0."},{"Start":"02:21.105 ","End":"02:27.780","Text":"Our final answer is natural log of 2. We\u0027re done."}],"ID":24163},{"Watched":false,"Name":"Exercise 8","Duration":"3m 28s","ChapterTopicVideoID":8338,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8338.jpeg","UploadDate":"2017-01-28T12:21:51.3800000","DurationForVideoObject":"PT3M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"Here again, we have 1 of those limit of"},{"Start":"00:02.760 ","End":"00:06.735","Text":"a series that is computed by means of an integral."},{"Start":"00:06.735 ","End":"00:10.590","Text":"This is what I\u0027m given and I\u0027m going to try and convert"},{"Start":"00:10.590 ","End":"00:13.320","Text":"the left-hand side to look something like this so"},{"Start":"00:13.320 ","End":"00:16.515","Text":"I can use the integral. Well, let\u0027s see."},{"Start":"00:16.515 ","End":"00:20.900","Text":"I would like to have 1 here."},{"Start":"00:20.900 ","End":"00:23.730","Text":"I have n on the top already so my idea is to"},{"Start":"00:23.730 ","End":"00:27.060","Text":"take n squared out of each denominator, I\u0027ll show you."},{"Start":"00:27.060 ","End":"00:30.375","Text":"Limit as n goes to infinity."},{"Start":"00:30.375 ","End":"00:32.580","Text":"If I take n squared out of each 1,"},{"Start":"00:32.580 ","End":"00:40.800","Text":"I\u0027ll get n over n squared times 1 plus 1 squared plus"},{"Start":"00:40.800 ","End":"00:49.160","Text":"n over n squared 1 plus 2 squared over n squared plus,"},{"Start":"00:49.160 ","End":"00:53.705","Text":"and so on and so on plus n over"},{"Start":"00:53.705 ","End":"01:00.370","Text":"n squared 1 plus n squared over n squared."},{"Start":"01:00.370 ","End":"01:03.845","Text":"Now note that in each case,"},{"Start":"01:03.845 ","End":"01:06.650","Text":"if I take n over n squared,"},{"Start":"01:06.650 ","End":"01:09.485","Text":"it\u0027s just like leaving n in the bottom,"},{"Start":"01:09.485 ","End":"01:13.715","Text":"because n over n squared is just 1."},{"Start":"01:13.715 ","End":"01:17.060","Text":"I\u0027m going to get 1 in each of these so I could"},{"Start":"01:17.060 ","End":"01:20.690","Text":"take that 1 completely outside the brackets"},{"Start":"01:20.690 ","End":"01:28.275","Text":"and get limit as n goes to infinity of 1 times,"},{"Start":"01:28.275 ","End":"01:34.655","Text":"now I\u0027m going to get 1 over 1 plus 1 squared over n squared"},{"Start":"01:34.655 ","End":"01:42.170","Text":"plus 1 over 1 plus 2 squared over n squared plus,"},{"Start":"01:42.170 ","End":"01:43.999","Text":"and so on and so on,"},{"Start":"01:43.999 ","End":"01:49.925","Text":"1 over 1 plus n squared over n squared."},{"Start":"01:49.925 ","End":"01:54.175","Text":"The final thing I\u0027ll do that will make it very close to this,"},{"Start":"01:54.175 ","End":"01:57.050","Text":"I\u0027ll just instead of 1 squared over n squared,"},{"Start":"01:57.050 ","End":"02:00.020","Text":"I can write it as 1 over n all squared and similarly,"},{"Start":"02:00.020 ","End":"02:03.560","Text":"2 squared over n squared is 2 over n squared so what I get is"},{"Start":"02:03.560 ","End":"02:08.960","Text":"1 over 1 plus 1 squared plus"},{"Start":"02:08.960 ","End":"02:14.975","Text":"1 over 1 plus 2 squared plus and so on,"},{"Start":"02:14.975 ","End":"02:19.435","Text":"1 over 1 plus n squared."},{"Start":"02:19.435 ","End":"02:21.420","Text":"Now look, I have 1 over n,"},{"Start":"02:21.420 ","End":"02:24.885","Text":"2 over n up to n over n. Clearly,"},{"Start":"02:24.885 ","End":"02:31.650","Text":"the function we need is 1 over 1 plus x squared and then when x is 1,"},{"Start":"02:31.650 ","End":"02:33.035","Text":"2, and so on,"},{"Start":"02:33.035 ","End":"02:36.859","Text":"we get this series here and then we can use this formula."},{"Start":"02:36.859 ","End":"02:45.665","Text":"This thing is going to equal the integral from 0-1 of f of xdx."},{"Start":"02:45.665 ","End":"02:50.660","Text":"So it\u0027s dx over 1 plus x squared."},{"Start":"02:50.660 ","End":"02:54.095","Text":"Now this is an immediate integral."},{"Start":"02:54.095 ","End":"02:58.130","Text":"This is the arc tangent so this is equal to"},{"Start":"02:58.130 ","End":"03:05.375","Text":"arc tangent of x evaluated between 0 and 1,"},{"Start":"03:05.375 ","End":"03:11.935","Text":"which makes it arc tangent of 1 minus arc tangent of 0."},{"Start":"03:11.935 ","End":"03:16.610","Text":"Arc tangent of 1 is Pi/4 because tangent of 45 degrees is"},{"Start":"03:16.610 ","End":"03:22.089","Text":"1 and 45 degrees is Pi/4 and the arc tangent of 0 is 0."},{"Start":"03:22.089 ","End":"03:25.365","Text":"Finally, we get just Pi/4,"},{"Start":"03:25.365 ","End":"03:29.410","Text":"final answer, and we are done."}],"ID":24164},{"Watched":false,"Name":"Exercise 9","Duration":"3m 40s","ChapterTopicVideoID":8339,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8339.jpeg","UploadDate":"2017-01-28T12:22:23.6570000","DurationForVideoObject":"PT3M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"Here we have another 1 of those exercises which is the limit of a series,"},{"Start":"00:04.515 ","End":"00:06.075","Text":"just copied it over here,"},{"Start":"00:06.075 ","End":"00:09.210","Text":"and once again, we\u0027re going to use this formula which"},{"Start":"00:09.210 ","End":"00:12.825","Text":"expresses the limit of a series in terms of an integral."},{"Start":"00:12.825 ","End":"00:16.665","Text":"What we have to do is to get this thing to look a bit like this,"},{"Start":"00:16.665 ","End":"00:18.720","Text":"and then we can use an integral."},{"Start":"00:18.720 ","End":"00:21.270","Text":"Let\u0027s see it, let\u0027s start doing some algebra."},{"Start":"00:21.270 ","End":"00:24.525","Text":"This thing will equal the limit."},{"Start":"00:24.525 ","End":"00:30.840","Text":"What I\u0027m going to do now is take n squared outside the expression under the root sign."},{"Start":"00:30.840 ","End":"00:34.710","Text":"For example, here I\u0027m going to write it as the square root of n"},{"Start":"00:34.710 ","End":"00:39.595","Text":"squared and out of the brackets this becomes 1 plus 1 over n squared."},{"Start":"00:39.595 ","End":"00:48.425","Text":"Second 1 becomes the square root of n squared times 1 plus 2 squared over n squared,"},{"Start":"00:48.425 ","End":"00:52.400","Text":"and so on and so on until we get to the last 1,"},{"Start":"00:52.400 ","End":"01:00.270","Text":"which is 1 over the square root of n squared times 1 plus n squared over n squared."},{"Start":"01:00.890 ","End":"01:04.695","Text":"In general, the square root of n squared is n,"},{"Start":"01:04.695 ","End":"01:06.470","Text":"you would say absolute value of n,"},{"Start":"01:06.470 ","End":"01:07.730","Text":"but n is positive."},{"Start":"01:07.730 ","End":"01:11.690","Text":"So what I can do now is take that outside limit,"},{"Start":"01:11.690 ","End":"01:13.850","Text":"n goes to infinity."},{"Start":"01:13.850 ","End":"01:18.530","Text":"I can break this up into square root of n squared times square root of 1 over n squared."},{"Start":"01:18.530 ","End":"01:26.195","Text":"The square root of n squared is n. It\u0027s 1 over n square root of 1 plus 1 over n squared"},{"Start":"01:26.195 ","End":"01:34.760","Text":"plus 1 over n square root of 1 plus 2 squared over n squared,"},{"Start":"01:34.760 ","End":"01:42.620","Text":"and so on, until we get n square root of 1 plus n squared over n squared."},{"Start":"01:42.620 ","End":"01:46.475","Text":"Each time we\u0027re getting closer and closer to an expression involving 1 over n,"},{"Start":"01:46.475 ","End":"01:48.110","Text":"and the 1 over n out front."},{"Start":"01:48.110 ","End":"01:50.750","Text":"Now already I can take the 1 over n outside,"},{"Start":"01:50.750 ","End":"01:57.515","Text":"so I get limit as n goes to infinity of 1 over n and I get 1 over,"},{"Start":"01:57.515 ","End":"01:59.030","Text":"basically get these things."},{"Start":"01:59.030 ","End":"02:00.740","Text":"But I also want to rewrite them."},{"Start":"02:00.740 ","End":"02:02.930","Text":"Instead of 1 squared over n squared,"},{"Start":"02:02.930 ","End":"02:05.425","Text":"I can write 1 over n squared."},{"Start":"02:05.425 ","End":"02:07.725","Text":"Here, the n is gone in front."},{"Start":"02:07.725 ","End":"02:10.070","Text":"I forgot the square root, sorry,"},{"Start":"02:10.070 ","End":"02:15.920","Text":"the square root of 1 plus 2 over n squared,"},{"Start":"02:15.920 ","End":"02:26.710","Text":"and so on, up to 1 over the square root of 1 plus n over n squared."},{"Start":"02:26.710 ","End":"02:32.270","Text":"Now, this is looking very much like this."},{"Start":"02:32.270 ","End":"02:40.295","Text":"If I take the function of x as 1 over the square root of 1 plus x squared,"},{"Start":"02:40.295 ","End":"02:42.710","Text":"then what I get here is exactly this."},{"Start":"02:42.710 ","End":"02:45.875","Text":"Now I have the f for using this formula,"},{"Start":"02:45.875 ","End":"02:52.070","Text":"and so what I get is I can convert this now to an integral from 0-1,"},{"Start":"02:52.070 ","End":"03:01.205","Text":"I\u0027m reading from here of f of x dx of 1 over square root of 1 plus x squared dx."},{"Start":"03:01.205 ","End":"03:04.925","Text":"This is the point at which you take out your integral tables."},{"Start":"03:04.925 ","End":"03:06.590","Text":"The integral of this,"},{"Start":"03:06.590 ","End":"03:14.005","Text":"it\u0027s the natural log of x plus square root of 1 plus x squared."},{"Start":"03:14.005 ","End":"03:18.285","Text":"Of course, I want to take this between 0 and 1."},{"Start":"03:18.285 ","End":"03:21.395","Text":"Now, if I put in 1 and I get,"},{"Start":"03:21.395 ","End":"03:22.730","Text":"this gives square root of 2,"},{"Start":"03:22.730 ","End":"03:27.020","Text":"so I have natural log of 1 plus square root of 2."},{"Start":"03:27.020 ","End":"03:28.525","Text":"If I put 0 in,"},{"Start":"03:28.525 ","End":"03:30.780","Text":"1 plus 0 squared is 1,"},{"Start":"03:30.780 ","End":"03:33.110","Text":"square root of 1 is 1 and this is 0."},{"Start":"03:33.110 ","End":"03:36.005","Text":"I get natural log of 1, which is 0,"},{"Start":"03:36.005 ","End":"03:39.110","Text":"to say that this is the answer,"},{"Start":"03:39.110 ","End":"03:41.670","Text":"and we are done."}],"ID":24165},{"Watched":false,"Name":"Exercise 10","Duration":"3m 54s","ChapterTopicVideoID":8340,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8340.jpeg","UploadDate":"2017-01-28T12:22:50.5870000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.050","Text":"Here we have another 1 of those series limits which we solve by means of integrals."},{"Start":"00:07.050 ","End":"00:10.695","Text":"Specifically, we have this formula which should be familiar to you."},{"Start":"00:10.695 ","End":"00:11.970","Text":"I\u0027ve copied this here,"},{"Start":"00:11.970 ","End":"00:16.125","Text":"but with a slight alteration instead of 2n and I put n plus n,"},{"Start":"00:16.125 ","End":"00:17.895","Text":"so it goes in a pattern."},{"Start":"00:17.895 ","End":"00:22.350","Text":"The denominator I have written with square roots instead of exponents."},{"Start":"00:22.350 ","End":"00:28.290","Text":"To the 3 over 2 is n square root of n. What I want to do is make this look like this,"},{"Start":"00:28.290 ","End":"00:29.790","Text":"and use an integral."},{"Start":"00:29.790 ","End":"00:32.265","Text":"Do some algebraic manipulations."},{"Start":"00:32.265 ","End":"00:36.920","Text":"First thing I\u0027ll do is take 1 over n outside the brackets."},{"Start":"00:36.920 ","End":"00:38.335","Text":"It\u0027ll look like this."},{"Start":"00:38.335 ","End":"00:44.930","Text":"I get the limit as n goes to infinity of 1 over n,"},{"Start":"00:44.930 ","End":"00:52.055","Text":"the square root of n plus 1 plus the square root of n plus 2 plus, and so on,"},{"Start":"00:52.055 ","End":"00:59.285","Text":"square root of n plus n. All this over the square root of n,"},{"Start":"00:59.285 ","End":"01:01.015","Text":"because the n is now here."},{"Start":"01:01.015 ","End":"01:06.140","Text":"Now, what I can do is I can divide each 1 of these square roots by this."},{"Start":"01:06.140 ","End":"01:09.890","Text":"Of course, I\u0027m going to use the algebraic rule that the square root of"},{"Start":"01:09.890 ","End":"01:14.840","Text":"a over b equals the square root of a over the square root of b,"},{"Start":"01:14.840 ","End":"01:16.700","Text":"or the other way around also."},{"Start":"01:16.700 ","End":"01:20.180","Text":"I\u0027ve got the limit as n goes to infinity,"},{"Start":"01:20.180 ","End":"01:25.700","Text":"1 over n times the square root of n plus 1 over"},{"Start":"01:25.700 ","End":"01:31.265","Text":"n plus the square root of n plus 2 over n plus,"},{"Start":"01:31.265 ","End":"01:39.395","Text":"and so on, plus the square root of n plus n over n. Next thing I\u0027m going to do is,"},{"Start":"01:39.395 ","End":"01:40.640","Text":"under each square root,"},{"Start":"01:40.640 ","End":"01:42.445","Text":"I\u0027m going to do the division."},{"Start":"01:42.445 ","End":"01:48.260","Text":"Now, what I have is the limit of 1 over n times the square root of"},{"Start":"01:48.260 ","End":"01:54.965","Text":"1 plus 1 over n plus the square root of 1 plus 2 over n plus,"},{"Start":"01:54.965 ","End":"01:59.840","Text":"and so on, the square root of 1 plus n over n. Now,"},{"Start":"01:59.840 ","End":"02:03.530","Text":"this is really looking like this because I can see the 1 over n, 2 over n,"},{"Start":"02:03.530 ","End":"02:05.960","Text":"n over n. In this case,"},{"Start":"02:05.960 ","End":"02:09.635","Text":"all I have to do is take f of x to equal"},{"Start":"02:09.635 ","End":"02:15.365","Text":"the square root of 1 plus x to make this come out like this,"},{"Start":"02:15.365 ","End":"02:20.450","Text":"exactly where f of 1 over n is 1 plus 1 over n and so on."},{"Start":"02:20.450 ","End":"02:24.655","Text":"That means I can express this now as an integral."},{"Start":"02:24.655 ","End":"02:34.475","Text":"Now, I\u0027ve got the integral from 0 to 1 of the square root of 1 plus x dx."},{"Start":"02:34.475 ","End":"02:38.405","Text":"I\u0027ll do this integral at the side, the indefinite integral."},{"Start":"02:38.405 ","End":"02:44.015","Text":"The integral of 1 plus x square root dx,"},{"Start":"02:44.015 ","End":"02:49.310","Text":"is the integral of 1 plus x to the power of a half dx,"},{"Start":"02:49.310 ","End":"02:52.560","Text":"which is, raise the power by 1 and divide."},{"Start":"02:52.560 ","End":"02:54.875","Text":"If I raise it by 1 at 3 over 2."},{"Start":"02:54.875 ","End":"03:02.070","Text":"I get 1 plus x to the power of 3 over 2 over 3 over 2 plus C,"},{"Start":"03:02.070 ","End":"03:03.360","Text":"which I don\u0027t really need."},{"Start":"03:03.360 ","End":"03:05.840","Text":"I can leave it like this."},{"Start":"03:05.840 ","End":"03:11.000","Text":"I prefer to take the 2/3 at the top rather than leave 3 over 2 at the bottom."},{"Start":"03:11.000 ","End":"03:14.855","Text":"It\u0027s 2/3,1 plus x to the 3 over 2."},{"Start":"03:14.855 ","End":"03:17.840","Text":"Now, go back to here."},{"Start":"03:17.840 ","End":"03:22.490","Text":"What I get is that this is 2/3 of"},{"Start":"03:22.490 ","End":"03:28.610","Text":"1 plus x to the 3 over 2 taken between 0 and 1."},{"Start":"03:28.610 ","End":"03:31.625","Text":"When x is 1,"},{"Start":"03:31.625 ","End":"03:38.955","Text":"then I get 2/3 of 2 to the power of 3 over 2."},{"Start":"03:38.955 ","End":"03:45.960","Text":"When x is 0, I get just 2/3 because 1 to the anything is 1,"},{"Start":"03:45.960 ","End":"03:48.480","Text":"so it\u0027s minus 2/3."},{"Start":"03:48.480 ","End":"03:50.960","Text":"That\u0027s the answer. I could simplify it a bit,"},{"Start":"03:50.960 ","End":"03:55.440","Text":"but basically, this is my answer and we are done."}],"ID":24166},{"Watched":false,"Name":"Exercise 11","Duration":"1m 12s","ChapterTopicVideoID":8348,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8348.jpeg","UploadDate":"2017-01-28T12:27:02.7570000","DurationForVideoObject":"PT1M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.815","Text":"Here we have to evaluate another 1 of those limit of a series which can be done"},{"Start":"00:04.815 ","End":"00:10.365","Text":"in terms of an integral using this formula that we\u0027ve seen several times."},{"Start":"00:10.365 ","End":"00:12.930","Text":"I copied this here, but not exactly."},{"Start":"00:12.930 ","End":"00:16.200","Text":"I took the 1 over n outside,"},{"Start":"00:16.200 ","End":"00:19.290","Text":"and I also put brackets around the 1 over n,"},{"Start":"00:19.290 ","End":"00:20.760","Text":"2 over n, and so on."},{"Start":"00:20.760 ","End":"00:23.130","Text":"Would look a bit more like this."},{"Start":"00:23.130 ","End":"00:25.125","Text":"In fact, it looks a whole lot like this."},{"Start":"00:25.125 ","End":"00:29.700","Text":"If we just take f of x is equal to sine x,"},{"Start":"00:29.700 ","End":"00:31.890","Text":"then it\u0027s exactly like this."},{"Start":"00:31.890 ","End":"00:33.660","Text":"According to this formula,"},{"Start":"00:33.660 ","End":"00:36.300","Text":"I can now replace this by the-."}],"ID":24167},{"Watched":false,"Name":"Exercise 12","Duration":"8m 52s","ChapterTopicVideoID":8349,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8349.jpeg","UploadDate":"2017-01-28T12:27:43.9100000","DurationForVideoObject":"PT8M52S","Description":null,"VideoComments":[],"Subtitles":[],"ID":24168},{"Watched":false,"Name":"Exercise 13","Duration":"4m 20s","ChapterTopicVideoID":8341,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8341.jpeg","UploadDate":"2017-01-28T12:24:06.6230000","DurationForVideoObject":"PT4M20S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this exercise, we have again the limit of a series,"},{"Start":"00:04.350 ","End":"00:06.780","Text":"which we\u0027re going to compute with the help of an integral."},{"Start":"00:06.780 ","End":"00:08.010","Text":"But this time it\u0027s written with"},{"Start":"00:08.010 ","End":"00:12.060","Text":"the Sigma notation and our formula doesn\u0027t have Sigma in it."},{"Start":"00:12.060 ","End":"00:13.650","Text":"It has dot, dot, dot."},{"Start":"00:13.650 ","End":"00:16.440","Text":"Why don\u0027t we write this out in this form,"},{"Start":"00:16.440 ","End":"00:20.970","Text":"the limit as n goes to infinity,"},{"Start":"00:20.970 ","End":"00:27.050","Text":"natural log of the nth root of 1 plus 1 over"},{"Start":"00:27.050 ","End":"00:36.810","Text":"n plus natural log of the nth root of 1 plus 2 over n plus dot,"},{"Start":"00:36.810 ","End":"00:38.690","Text":"dot, dot, so on, and so on,"},{"Start":"00:38.690 ","End":"00:42.140","Text":"and so on up to natural logarithm of"},{"Start":"00:42.140 ","End":"00:48.330","Text":"the nth root of 1 plus n over n. Instead of k, I\u0027m going 1,"},{"Start":"00:48.330 ","End":"00:50.495","Text":"2 and so on up to n,"},{"Start":"00:50.495 ","End":"00:53.110","Text":"because k goes from 1 to n. Now,"},{"Start":"00:53.110 ","End":"00:56.810","Text":"our job is to make this look a lot like this and"},{"Start":"00:56.810 ","End":"01:00.665","Text":"then we can find the function f. The first thing is that"},{"Start":"01:00.665 ","End":"01:05.570","Text":"the natural log of the nth root of some number a"},{"Start":"01:05.570 ","End":"01:11.090","Text":"is equal to the natural log of a to the power of 1 over n,"},{"Start":"01:11.090 ","End":"01:14.210","Text":"because that\u0027s what the nth root means and the natural log of a to"},{"Start":"01:14.210 ","End":"01:17.420","Text":"the power of b is b natural log of a."},{"Start":"01:17.420 ","End":"01:22.220","Text":"In other words, the exponent comes before so it\u0027s 1 over n natural log of a."},{"Start":"01:22.220 ","End":"01:25.114","Text":"Now, if I do this everywhere here,"},{"Start":"01:25.114 ","End":"01:32.825","Text":"what I will get is the limit as n goes to infinity of 1 over n,"},{"Start":"01:32.825 ","End":"01:39.365","Text":"natural log of 1 plus 1 over n plus 1 over n,"},{"Start":"01:39.365 ","End":"01:45.155","Text":"natural log of 1 plus 2 over n plus etc.,"},{"Start":"01:45.155 ","End":"01:55.130","Text":"plus natural log 1 over n of 1 plus n over n. Next thing to do clearly is to take"},{"Start":"01:55.130 ","End":"02:00.695","Text":"the 1 over n in front so we get the lim as n goes to infinity of"},{"Start":"02:00.695 ","End":"02:07.370","Text":"1 over n. Here I have natural log of 1 plus 1 over n,"},{"Start":"02:07.370 ","End":"02:11.095","Text":"plus natural log of 1 plus 2 over n,"},{"Start":"02:11.095 ","End":"02:18.770","Text":"plus and so on up to natural log of 1 plus n over n. Now,"},{"Start":"02:18.770 ","End":"02:20.825","Text":"you see here, 1 over n, 2 over n,"},{"Start":"02:20.825 ","End":"02:23.855","Text":"n over n, they\u0027re here, here, and here."},{"Start":"02:23.855 ","End":"02:31.325","Text":"If I take f of x to be equal to natural log of 1 plus x,"},{"Start":"02:31.325 ","End":"02:37.940","Text":"then what I get is exactly this and so using the equality I can convert this to"},{"Start":"02:37.940 ","End":"02:45.500","Text":"an integration problem and have it as integral from 0 to 1 of f of x dx."},{"Start":"02:45.500 ","End":"02:50.840","Text":"In other words, of natural log of 1 plus x dx."},{"Start":"02:50.840 ","End":"02:55.145","Text":"Let me write a result at the side which we\u0027ve already done once."},{"Start":"02:55.145 ","End":"03:05.270","Text":"The integral of natural log of x dx is equal to x natural log of x minus x."},{"Start":"03:05.270 ","End":"03:07.880","Text":"This was some previous exercise."},{"Start":"03:07.880 ","End":"03:10.850","Text":"Now, what we have here is not natural log of x,"},{"Start":"03:10.850 ","End":"03:13.400","Text":"it\u0027s natural log of 1 plus x,"},{"Start":"03:13.400 ","End":"03:16.190","Text":"but the internal derivative of this is 1,"},{"Start":"03:16.190 ","End":"03:23.540","Text":"so it will work just the same if I replace x by 1 plus x. I\u0027m replacing x by 1 plus x,"},{"Start":"03:23.540 ","End":"03:34.234","Text":"so I get 1 plus x natural log of 1 plus x minus 1 plus x."},{"Start":"03:34.234 ","End":"03:38.345","Text":"This whole thing is taken between 0 and 1."},{"Start":"03:38.345 ","End":"03:41.120","Text":"Let\u0027s see what this comes out to be."},{"Start":"03:41.120 ","End":"03:44.239","Text":"When we put x equals 1,"},{"Start":"03:44.239 ","End":"03:52.050","Text":"we get 2 natural log of 2 minus 2 minus,"},{"Start":"03:52.050 ","End":"03:59.100","Text":"lets put in 0, 1 natural log of 1 minus 1."},{"Start":"03:59.100 ","End":"04:05.115","Text":"This equals natural log of 1 is 0,"},{"Start":"04:05.115 ","End":"04:13.580","Text":"so I get 2 natural log of 2 minus 2 minus minus 1 is minus 2 plus 1,"},{"Start":"04:13.580 ","End":"04:15.530","Text":"which is minus 1."},{"Start":"04:15.530 ","End":"04:20.940","Text":"This is the answer and we are done."}],"ID":24169},{"Watched":false,"Name":"Exercise 14 part 1","Duration":"2m 7s","ChapterTopicVideoID":8350,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8350.jpeg","UploadDate":"2017-01-28T12:28:40.8700000","DurationForVideoObject":"PT2M7S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.405","Text":"In this exercise, we\u0027re given an integral,"},{"Start":"00:03.405 ","End":"00:06.420","Text":"and we have to evaluate it as the limit of"},{"Start":"00:06.420 ","End":"00:09.915","Text":"a series by means of the Riemann sum definition,"},{"Start":"00:09.915 ","End":"00:12.690","Text":"which is what is written over here."},{"Start":"00:12.690 ","End":"00:14.955","Text":"Let\u0027s just get to it."},{"Start":"00:14.955 ","End":"00:17.670","Text":"The integral from 0 to 1 of xdx,"},{"Start":"00:17.670 ","End":"00:22.350","Text":"we take f of x is equal to just x here."},{"Start":"00:22.350 ","End":"00:29.820","Text":"What we get is the limit as n goes to infinity of 1 over n times,"},{"Start":"00:29.820 ","End":"00:31.980","Text":"now f of x is equal to x,"},{"Start":"00:31.980 ","End":"00:35.310","Text":"so f of 1 over n is 1 over n, and so on,"},{"Start":"00:35.310 ","End":"00:42.630","Text":"plus 2 over n plus, etc., up to n over n. Continuing."},{"Start":"00:42.630 ","End":"00:45.169","Text":"Limit, n goes to infinity,"},{"Start":"00:45.169 ","End":"00:47.150","Text":"1 over n. This stage,"},{"Start":"00:47.150 ","End":"00:48.769","Text":"I\u0027m going to put a common denominator."},{"Start":"00:48.769 ","End":"00:50.960","Text":"They\u0027re all of the same denominator n,"},{"Start":"00:50.960 ","End":"00:54.945","Text":"so I can put 1 big denominator and n here,"},{"Start":"00:54.945 ","End":"00:58.225","Text":"and 1 plus 2 plus, etc.,"},{"Start":"00:58.225 ","End":"01:03.290","Text":"plus n. Now, we were nice enough to be given this hint,"},{"Start":"01:03.290 ","End":"01:10.970","Text":"and what we get is the limit as n goes to infinity of 1 over n. Now,"},{"Start":"01:10.970 ","End":"01:13.880","Text":"all of this is replaced by n,"},{"Start":"01:13.880 ","End":"01:16.775","Text":"n plus 1 over 2."},{"Start":"01:16.775 ","End":"01:21.649","Text":"Let\u0027s see now, this n will cancel with this n,"},{"Start":"01:21.649 ","End":"01:27.560","Text":"so we just get the limit as n goes to infinity."},{"Start":"01:27.560 ","End":"01:31.550","Text":"Oh, I\u0027m sorry. There\u0027s an extra n here because that\u0027s"},{"Start":"01:31.550 ","End":"01:36.270","Text":"this n. We have the limit of n goes to infinity."},{"Start":"01:36.270 ","End":"01:39.030","Text":"Let\u0027s take the half outside,"},{"Start":"01:39.030 ","End":"01:44.595","Text":"and we get n plus 1 over n. This is the limit."},{"Start":"01:44.595 ","End":"01:47.025","Text":"I can take the half out front,"},{"Start":"01:47.025 ","End":"01:52.935","Text":"and here I have 1 plus 1 over n as n goes to infinity."},{"Start":"01:52.935 ","End":"01:58.305","Text":"It just comes out to be 1/2 of 1 plus 1 over infinity,"},{"Start":"01:58.305 ","End":"02:03.735","Text":"and then this is equal to just 1/2 because this, obviously, is 0."},{"Start":"02:03.735 ","End":"02:06.150","Text":"This is our answer,"},{"Start":"02:06.150 ","End":"02:08.230","Text":"and we are done."}],"ID":24170},{"Watched":false,"Name":"Exercise 14 part 2","Duration":"32s","ChapterTopicVideoID":8342,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8342.jpeg","UploadDate":"2017-01-28T12:24:09.6770000","DurationForVideoObject":"PT32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"But wait, I just thought we should check"},{"Start":"00:02.340 ","End":"00:05.190","Text":"our solution because we know how to do definite integrals."},{"Start":"00:05.190 ","End":"00:08.970","Text":"Let\u0027s take this as the integral from 0 to 1 of"},{"Start":"00:08.970 ","End":"00:14.790","Text":"x dx is equal the integral of x is x squared over 2,"},{"Start":"00:14.790 ","End":"00:17.505","Text":"which we have to take between 0 and 1."},{"Start":"00:17.505 ","End":"00:19.949","Text":"So it\u0027s 1 squared over 2,"},{"Start":"00:19.949 ","End":"00:22.290","Text":"minus 0 squared over 2."},{"Start":"00:22.290 ","End":"00:24.240","Text":"1 squared over 2 is a 1/2."},{"Start":"00:24.240 ","End":"00:25.950","Text":"0 squared is nothing."},{"Start":"00:25.950 ","End":"00:29.295","Text":"So this is a 1/2 and this agrees with what we have here."},{"Start":"00:29.295 ","End":"00:33.700","Text":"Yes, we are done and verified."}],"ID":24171},{"Watched":false,"Name":"Exercise 15 part 1","Duration":"2m 52s","ChapterTopicVideoID":8343,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8343.jpeg","UploadDate":"2017-01-28T12:24:37.2400000","DurationForVideoObject":"PT2M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.090","Text":"Here we have to evaluate this integral from 0 to 1 using the Riemann sum definition,"},{"Start":"00:06.090 ","End":"00:09.330","Text":"which is exactly what is written in the box here."},{"Start":"00:09.330 ","End":"00:13.590","Text":"In our case, f of x is x squared."},{"Start":"00:13.590 ","End":"00:19.770","Text":"I\u0027m going to use this definition to say that this is equal to the limit as n goes to"},{"Start":"00:19.770 ","End":"00:27.525","Text":"infinity of 1 over n times f of 1 over n is 1 over n squared."},{"Start":"00:27.525 ","End":"00:31.070","Text":"Then we get 2 over n squared, and then,"},{"Start":"00:31.070 ","End":"00:35.590","Text":"and so on up to n over n squared."},{"Start":"00:35.590 ","End":"00:38.505","Text":"This equals the limit."},{"Start":"00:38.505 ","End":"00:41.150","Text":"I\u0027m going to do 2 steps in 1 here."},{"Start":"00:41.150 ","End":"00:44.360","Text":"I\u0027m going to square all these things so I get 1 squared over n squared,"},{"Start":"00:44.360 ","End":"00:45.635","Text":"2 squared over n squared,"},{"Start":"00:45.635 ","End":"00:47.360","Text":"n squared over n squared."},{"Start":"00:47.360 ","End":"00:52.110","Text":"The n squared will just come out on the bottom and I\u0027ll get 1 squared"},{"Start":"00:52.110 ","End":"00:57.289","Text":"plus 2 squared plus and so on plus n squared."},{"Start":"00:57.289 ","End":"01:02.020","Text":"Now, I\u0027m going to use this formula above here. This was nice."},{"Start":"01:02.020 ","End":"01:06.860","Text":"They gave us a hint that this sum is 1/6 of n,"},{"Start":"01:06.860 ","End":"01:08.945","Text":"n plus 1 to n plus 1."},{"Start":"01:08.945 ","End":"01:13.580","Text":"I\u0027m going to write that this equals the limit as n goes to infinity."},{"Start":"01:13.580 ","End":"01:17.330","Text":"But besides this, I also have 1 over n and an n squared here."},{"Start":"01:17.330 ","End":"01:21.840","Text":"I have 1 over n cubed from here and here."},{"Start":"01:21.840 ","End":"01:26.120","Text":"This from the formula here is 1/6 n,"},{"Start":"01:26.120 ","End":"01:29.735","Text":"n plus 1 2n plus 1."},{"Start":"01:29.735 ","End":"01:35.615","Text":"This is equal to the limit as n goes to infinity."},{"Start":"01:35.615 ","End":"01:40.320","Text":"Now the 1/6 I can take out front, so it\u0027s 1/6."},{"Start":"01:40.430 ","End":"01:45.410","Text":"I can split the n cubed into n times"},{"Start":"01:45.410 ","End":"01:50.060","Text":"n times n. What I\u0027m saying is I can take what\u0027s written here,"},{"Start":"01:50.060 ","End":"01:54.410","Text":"n n plus 1 2n plus 1 and"},{"Start":"01:54.410 ","End":"01:59.900","Text":"the n cubed can go here as n times n times n. That\u0027s what the n cubed means,"},{"Start":"01:59.900 ","End":"02:08.075","Text":"3 factors n. Now I can write this as 1/6 of the limit as n goes to infinity,"},{"Start":"02:08.075 ","End":"02:18.015","Text":"of n over n times n plus 1 over n times 2n plus 1 over n. 1/6 the limit,"},{"Start":"02:18.015 ","End":"02:19.770","Text":"n over n is just 1,"},{"Start":"02:19.770 ","End":"02:21.720","Text":"n plus 1 over n,"},{"Start":"02:21.720 ","End":"02:24.345","Text":"this is 1 plus 1 over n,"},{"Start":"02:24.345 ","End":"02:29.010","Text":"and this is 2 plus 1 over n. Finally"},{"Start":"02:29.010 ","End":"02:34.955","Text":"equals 1/6 times 1 plus 1 over infinity,"},{"Start":"02:34.955 ","End":"02:36.800","Text":"2 plus 1 over infinity."},{"Start":"02:36.800 ","End":"02:38.075","Text":"This is just symbolic."},{"Start":"02:38.075 ","End":"02:40.955","Text":"I mean that 1 over n basically goes to 0."},{"Start":"02:40.955 ","End":"02:44.330","Text":"I\u0027ve got 1/6 times 1 times 2,"},{"Start":"02:44.330 ","End":"02:46.985","Text":"which is 1/6 times 2,"},{"Start":"02:46.985 ","End":"02:52.860","Text":"which is 1/3 and this is our answer."}],"ID":24172},{"Watched":false,"Name":"Exercise 15 part 2","Duration":"31s","ChapterTopicVideoID":8344,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8344.jpeg","UploadDate":"2017-01-28T12:24:41.5870000","DurationForVideoObject":"PT31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"I was just thinking, why don\u0027t we check if this answer is right,"},{"Start":"00:02.970 ","End":"00:04.725","Text":"we know how to do integration."},{"Start":"00:04.725 ","End":"00:11.130","Text":"Let\u0027s see, the integral from 0-1 of x squared dx"},{"Start":"00:11.130 ","End":"00:18.780","Text":"is equal to x cubed over 3 between 0 and 1."},{"Start":"00:18.780 ","End":"00:21.060","Text":"This is equal to 1 cubed over 3,"},{"Start":"00:21.060 ","End":"00:26.565","Text":"which is 1/3 minus 0, which is 1/3."},{"Start":"00:26.565 ","End":"00:28.905","Text":"This equals this, so yes,"},{"Start":"00:28.905 ","End":"00:32.019","Text":"we even have checked our result."}],"ID":24173},{"Watched":false,"Name":"Exercise 16","Duration":"3m 16s","ChapterTopicVideoID":8351,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8351.jpeg","UploadDate":"2017-01-28T12:28:57.4700000","DurationForVideoObject":"PT3M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.589","Text":"Here we have to evaluate this integral using the Riemann sum definition,"},{"Start":"00:04.589 ","End":"00:07.050","Text":"which is exactly what is written over here."},{"Start":"00:07.050 ","End":"00:13.290","Text":"The integral from 0 to 1 of x cubed dx will equal."},{"Start":"00:13.290 ","End":"00:17.415","Text":"If we just take f of x is equal to x cubed here,"},{"Start":"00:17.415 ","End":"00:21.210","Text":"what we get is the limit as n goes to infinity,"},{"Start":"00:21.210 ","End":"00:24.150","Text":"I\u0027m just filling in here that if f x cubed."},{"Start":"00:24.150 ","End":"00:29.895","Text":"We have 1 over n, f of 1 over n is 1 over n cubed."},{"Start":"00:29.895 ","End":"00:31.020","Text":"The whole thing is cubed,"},{"Start":"00:31.020 ","End":"00:33.210","Text":"but I can put this cube on the bottom."},{"Start":"00:33.210 ","End":"00:35.535","Text":"Actually, I should write this as 1 cubed."},{"Start":"00:35.535 ","End":"00:41.070","Text":"Here we have 2 cubed over n cubed plus etc.,"},{"Start":"00:41.070 ","End":"00:46.070","Text":"plus n cubed over n cubed."},{"Start":"00:46.070 ","End":"00:48.800","Text":"Now we can put a common denominator here,"},{"Start":"00:48.800 ","End":"00:52.885","Text":"so we get the limit and still goes to infinity,"},{"Start":"00:52.885 ","End":"00:57.245","Text":"1 over n. Now I want to write this with"},{"Start":"00:57.245 ","End":"01:02.300","Text":"n cubed on the bottom and 1 cubed plus 2 cubed plus etc.,"},{"Start":"01:02.300 ","End":"01:04.370","Text":"plus n cubed on the top."},{"Start":"01:04.370 ","End":"01:07.780","Text":"They were nice enough to give us this hint."},{"Start":"01:07.780 ","End":"01:10.490","Text":"What\u0027s on the top is equal to what\u0027s here."},{"Start":"01:10.490 ","End":"01:14.015","Text":"I have the 1 over n with the n cubed,"},{"Start":"01:14.015 ","End":"01:17.180","Text":"and that is equal to 1 over n^4."},{"Start":"01:17.180 ","End":"01:21.649","Text":"That takes care of this and this is 1 over n^4,"},{"Start":"01:21.649 ","End":"01:24.920","Text":"and this is taken care of here,"},{"Start":"01:24.920 ","End":"01:30.325","Text":"1/4, n squared, n plus 1 squared."},{"Start":"01:30.325 ","End":"01:35.375","Text":"Let\u0027s just continue. The rest of it is really just mostly algebra simplification."},{"Start":"01:35.375 ","End":"01:41.270","Text":"I can take the quarter out in front and the limit as n goes to infinity."},{"Start":"01:41.270 ","End":"01:44.380","Text":"Now, basically, I can just write it all out in factors,"},{"Start":"01:44.380 ","End":"01:51.345","Text":"n squared n plus 1 squared is n times n times n plus 1 times n plus 1,"},{"Start":"01:51.345 ","End":"01:56.935","Text":"and n^4 is just n times n times n times n,"},{"Start":"01:56.935 ","End":"02:01.265","Text":"and now I can just write it as separate product of different things."},{"Start":"02:01.265 ","End":"02:10.850","Text":"N over n times n over n times n plus 1 over n times n plus 1 over n. Now this is 1,"},{"Start":"02:10.850 ","End":"02:15.425","Text":"and this is 1, so we get 1/4 of the limit."},{"Start":"02:15.425 ","End":"02:17.300","Text":"As I said, this is 1 and this is 1,"},{"Start":"02:17.300 ","End":"02:19.070","Text":"so we don\u0027t need to take those."},{"Start":"02:19.070 ","End":"02:26.275","Text":"This is n plus 1 over n is 1 plus 1 over n. This is also 1 plus 1 over n,"},{"Start":"02:26.275 ","End":"02:28.200","Text":"n goes to infinity."},{"Start":"02:28.200 ","End":"02:30.520","Text":"Finally, this equals 1/4."},{"Start":"02:30.520 ","End":"02:34.190","Text":"Now we\u0027ve seen this thing enough times already to know that when n goes to infinity,"},{"Start":"02:34.190 ","End":"02:36.020","Text":"1 over n goes to 0."},{"Start":"02:36.020 ","End":"02:39.045","Text":"This is just 1, this is just 1,"},{"Start":"02:39.045 ","End":"02:42.665","Text":"and we end up with just 1/4, and we\u0027re done."},{"Start":"02:42.665 ","End":"02:47.030","Text":"But wouldn\u0027t it be nice to check if our answer is correct?"},{"Start":"02:47.030 ","End":"02:50.000","Text":"Let\u0027s do the integral, the regular way."},{"Start":"02:50.000 ","End":"02:55.490","Text":"We have the integral from 0 to 1 of x cubed dx."},{"Start":"02:55.490 ","End":"02:58.490","Text":"The indefinite is x^4 over 4,"},{"Start":"02:58.490 ","End":"03:01.100","Text":"but we take it between 0 and 1."},{"Start":"03:01.100 ","End":"03:06.965","Text":"What we get is 1^4 over 4 minus 0^4 over 4."},{"Start":"03:06.965 ","End":"03:09.695","Text":"This, of course, is 0, 1^4 is 1."},{"Start":"03:09.695 ","End":"03:12.215","Text":"This is just equal to 1/4,"},{"Start":"03:12.215 ","End":"03:14.565","Text":"and this corroborates this result."},{"Start":"03:14.565 ","End":"03:17.620","Text":"Now we\u0027re done and confirmed."}],"ID":24174},{"Watched":false,"Name":"Exercise 17","Duration":"6m 51s","ChapterTopicVideoID":8352,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8352.jpeg","UploadDate":"2017-01-28T12:29:33.3900000","DurationForVideoObject":"PT6M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.985","Text":"Here we have to use the Riemann sum definition"},{"Start":"00:02.985 ","End":"00:06.240","Text":"of the integral to evaluate this definite integral."},{"Start":"00:06.240 ","End":"00:09.570","Text":"But notice that this time the integral is from"},{"Start":"00:09.570 ","End":"00:13.935","Text":"0 to Pi and we\u0027re so used to doing it from 0 to 1."},{"Start":"00:13.935 ","End":"00:16.110","Text":"I have the formula for 0 to 1,"},{"Start":"00:16.110 ","End":"00:19.350","Text":"but this is not going to be any good to me unless I generalize it."},{"Start":"00:19.350 ","End":"00:22.305","Text":"I\u0027m going to show you what is the generalization."},{"Start":"00:22.305 ","End":"00:24.435","Text":"Here\u0027s how we generalize it,"},{"Start":"00:24.435 ","End":"00:27.735","Text":"for in general from 0 to b,"},{"Start":"00:27.735 ","End":"00:29.505","Text":"instead of 0 to 1."},{"Start":"00:29.505 ","End":"00:31.605","Text":"I make this also a b."},{"Start":"00:31.605 ","End":"00:33.990","Text":"Instead of 1, 2, and so on up to n,"},{"Start":"00:33.990 ","End":"00:35.640","Text":"I have 1 times b,"},{"Start":"00:35.640 ","End":"00:37.740","Text":"I have 2b, 3b,"},{"Start":"00:37.740 ","End":"00:39.675","Text":"and so on up to nb."},{"Start":"00:39.675 ","End":"00:41.500","Text":"This is very similar to this,"},{"Start":"00:41.500 ","End":"00:44.825","Text":"but it stretches everything by a factor of b."},{"Start":"00:44.825 ","End":"00:51.075","Text":"In our case, we will take it with b is equal to Pi,"},{"Start":"00:51.075 ","End":"00:52.515","Text":"because that\u0027s what we have here,"},{"Start":"00:52.515 ","End":"00:54.330","Text":"and we\u0027ll use the second formula,"},{"Start":"00:54.330 ","End":"00:56.520","Text":"and I don\u0027t need the first one."},{"Start":"00:56.520 ","End":"00:58.970","Text":"This is the one that I\u0027m going to use."},{"Start":"00:58.970 ","End":"01:01.160","Text":"Of course, as f of x,"},{"Start":"01:01.160 ","End":"01:03.930","Text":"I\u0027m going to take sine of x."},{"Start":"01:03.930 ","End":"01:07.140","Text":"The next step is to take the limit."},{"Start":"01:07.140 ","End":"01:10.410","Text":"Now n goes to infinity, b is Pi,"},{"Start":"01:10.410 ","End":"01:12.240","Text":"so it\u0027s Pi over n,"},{"Start":"01:12.240 ","End":"01:16.170","Text":"and now I have f which is sine and b is Pi,"},{"Start":"01:16.170 ","End":"01:20.870","Text":"so sine of Pi over n plus sine 2 pi over n,"},{"Start":"01:20.870 ","End":"01:28.110","Text":"and so on up to sine of n Pi over n. Now,"},{"Start":"01:28.110 ","End":"01:29.890","Text":"what is this going to equal to?"},{"Start":"01:29.890 ","End":"01:34.475","Text":"If I rewrite it with Alpha equals Pi over n,"},{"Start":"01:34.475 ","End":"01:37.140","Text":"I want to get it to look like this because we\u0027ve got a hint."},{"Start":"01:37.140 ","End":"01:39.380","Text":"If Alpha is Pi over n,"},{"Start":"01:39.380 ","End":"01:44.090","Text":"then what I have is the limit as n goes to infinity."},{"Start":"01:44.090 ","End":"01:48.290","Text":"This Pi over n I\u0027ll keep as Pi over n. But here I\u0027ll write"},{"Start":"01:48.290 ","End":"01:55.135","Text":"sine Alpha plus sine 2 Alpha plus etc,"},{"Start":"01:55.135 ","End":"01:59.215","Text":"until I get sine of n Alpha."},{"Start":"01:59.215 ","End":"02:02.960","Text":"Now I can use this formula here,"},{"Start":"02:02.960 ","End":"02:12.540","Text":"and we will get limit as n goes to infinity Pi over n sine n over"},{"Start":"02:12.540 ","End":"02:16.815","Text":"2 Alpha sine n plus"},{"Start":"02:16.815 ","End":"02:23.185","Text":"1/2 Alpha over sine of Alpha over 2."},{"Start":"02:23.185 ","End":"02:25.100","Text":"You know what, I\u0027ve changed my mind."},{"Start":"02:25.100 ","End":"02:29.150","Text":"I want to replace this Pi over n also with Alpha,"},{"Start":"02:29.150 ","End":"02:31.385","Text":"so that will be an Alpha here."},{"Start":"02:31.385 ","End":"02:33.965","Text":"Let me just expand this a bit."},{"Start":"02:33.965 ","End":"02:39.290","Text":"I\u0027ve got the limit as n goes to infinity of"},{"Start":"02:39.290 ","End":"02:46.930","Text":"Alpha times sine n over 2 Alpha sine."},{"Start":"02:46.930 ","End":"02:52.655","Text":"Let me write this as n over 2 Alpha plus Alpha over 2."},{"Start":"02:52.655 ","End":"02:55.970","Text":"Just split, this is n over 2 plus 1/2."},{"Start":"02:55.970 ","End":"03:01.795","Text":"On the denominator still sine of Alpha over 2."},{"Start":"03:01.795 ","End":"03:06.225","Text":"Take a look at this n over 2 times Alpha,"},{"Start":"03:06.225 ","End":"03:16.405","Text":"n over 2 times Alpha is n over 2 times Pi over n. The n\u0027s cancel and this is Pi over 2."},{"Start":"03:16.405 ","End":"03:20.540","Text":"The sine of n over 2 Alpha,"},{"Start":"03:20.540 ","End":"03:21.950","Text":"which is what I have here,"},{"Start":"03:21.950 ","End":"03:24.560","Text":"is the sine of Pi over 2,"},{"Start":"03:24.560 ","End":"03:25.820","Text":"which is 90 degrees,"},{"Start":"03:25.820 ","End":"03:28.470","Text":"which is equal to 1."},{"Start":"03:32.590 ","End":"03:35.735","Text":"Now this same computation,"},{"Start":"03:35.735 ","End":"03:38.650","Text":"the n over 2 Alpha is Pi over 2,"},{"Start":"03:38.650 ","End":"03:44.420","Text":"means that I can replace this thing also by Pi over 2."},{"Start":"03:44.420 ","End":"03:46.915","Text":"Let\u0027s see where does that get us."},{"Start":"03:46.915 ","End":"03:49.430","Text":"There\u0027s 1 other change I\u0027d like to make."},{"Start":"03:49.430 ","End":"03:52.430","Text":"I\u0027d like to bypass n altogether."},{"Start":"03:52.430 ","End":"03:59.585","Text":"Notice that Alpha is Pi over n. When n goes to infinity,"},{"Start":"03:59.585 ","End":"04:02.050","Text":"Alpha is going to 0."},{"Start":"04:02.050 ","End":"04:06.500","Text":"I\u0027m also going to replace this with Alpha goes to 0."},{"Start":"04:06.500 ","End":"04:10.355","Text":"Now it\u0027s time to just rewrite what we\u0027ve got so far."},{"Start":"04:10.355 ","End":"04:18.420","Text":"We have the limit Alpha goes to 0 of Alpha times sine of"},{"Start":"04:18.420 ","End":"04:28.540","Text":"Pi over 2 plus Alpha over 2 over sine Alpha over 2."},{"Start":"04:30.980 ","End":"04:34.710","Text":"Since Alpha goes to 0,"},{"Start":"04:34.710 ","End":"04:38.235","Text":"this part, Alpha over 2 also goes to 0."},{"Start":"04:38.235 ","End":"04:43.315","Text":"This part also goes to 1 because sine of Pi over 2 is 1."},{"Start":"04:43.315 ","End":"04:47.660","Text":"Basically what I\u0027m left with is the limit as Alpha goes to"},{"Start":"04:47.660 ","End":"04:53.550","Text":"0 of Alpha over sine of Alpha over 2."},{"Start":"04:53.550 ","End":"04:56.540","Text":"Now I\u0027m going to use the famous trick"},{"Start":"04:56.540 ","End":"05:00.905","Text":"for the limit of sine something over the same something."},{"Start":"05:00.905 ","End":"05:03.590","Text":"Only I don\u0027t quite have the same something"},{"Start":"05:03.590 ","End":"05:07.790","Text":"because here I have Alpha over 2 and here I only have Alpha."},{"Start":"05:07.790 ","End":"05:09.500","Text":"That\u0027s not really a problem."},{"Start":"05:09.500 ","End":"05:10.985","Text":"We know these tricks."},{"Start":"05:10.985 ","End":"05:15.090","Text":"What I do is I take the limit as the Alpha goes to 0,"},{"Start":"05:15.090 ","End":"05:18.480","Text":"and I first of all put Alpha over 2 here,"},{"Start":"05:18.480 ","End":"05:21.490","Text":"over sine of Alpha over 2."},{"Start":"05:21.490 ","End":"05:24.440","Text":"But I\u0027ve divided the numerator by 2,"},{"Start":"05:24.440 ","End":"05:26.150","Text":"which means I\u0027ve divided everything by 2."},{"Start":"05:26.150 ","End":"05:28.550","Text":"I\u0027ve got to compensate so I can write a 2 here,"},{"Start":"05:28.550 ","End":"05:32.970","Text":"but I can take the 2 out in front of the limit. Alpha goes to 0,"},{"Start":"05:32.970 ","End":"05:35.070","Text":"the same thing as Alpha over 2 goes to 0."},{"Start":"05:35.070 ","End":"05:43.085","Text":"This is the famous limit of t goes to 0 of t over sine t is 1."},{"Start":"05:43.085 ","End":"05:45.960","Text":"This whole limit is 1."},{"Start":"05:45.960 ","End":"05:51.990","Text":"It\u0027s equal to 2 times 1 and the answer is 2."},{"Start":"05:51.990 ","End":"05:54.980","Text":"This is our final answer."},{"Start":"05:54.980 ","End":"05:59.345","Text":"But we\u0027re not quite done because I\u0027d like to do a verification."},{"Start":"05:59.345 ","End":"06:07.725","Text":"If you recall the original integral was from 0 to Pi of sine x dx."},{"Start":"06:07.725 ","End":"06:11.285","Text":"Let\u0027s see if we can get this by directly doing the integration."},{"Start":"06:11.285 ","End":"06:14.750","Text":"The integral of sine is minus cosine."},{"Start":"06:14.750 ","End":"06:19.255","Text":"We\u0027ve got minus cosine x from 0 to Pi."},{"Start":"06:19.255 ","End":"06:22.460","Text":"I actually like using my trick of when I have a minus I"},{"Start":"06:22.460 ","End":"06:27.275","Text":"just reverse the order from Pi to 0."},{"Start":"06:27.275 ","End":"06:30.425","Text":"If you subtract the other way it\u0027s a minus."},{"Start":"06:30.425 ","End":"06:37.680","Text":"This equals cosine 0 minus cosine Pi."},{"Start":"06:37.680 ","End":"06:43.320","Text":"Cosine 0 is 1 and cosine Pi is minus 1."},{"Start":"06:43.320 ","End":"06:48.915","Text":"1 minus minus 1 is 2 and thus we corroborate this."},{"Start":"06:48.915 ","End":"06:52.710","Text":"Everything is fine and we really are done."}],"ID":24175},{"Watched":false,"Name":"Exercise 18","Duration":"21m 56s","ChapterTopicVideoID":8353,"CourseChapterTopicPlaylistID":65397,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8353.jpeg","UploadDate":"2017-01-28T12:31:24.4630000","DurationForVideoObject":"PT21M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"Here we have to find this definite integral,"},{"Start":"00:03.690 ","End":"00:05.235","Text":"but not the regular way,"},{"Start":"00:05.235 ","End":"00:08.410","Text":"but using the Riemann sum definition."},{"Start":"00:08.410 ","End":"00:10.935","Text":"There are many versions of this."},{"Start":"00:10.935 ","End":"00:14.070","Text":"In previous exercises, we mostly dealt with"},{"Start":"00:14.070 ","End":"00:17.385","Text":"the case where we had an integral from 0 to 1."},{"Start":"00:17.385 ","End":"00:24.210","Text":"Then we generalized it to go from 0 to b. I remember we had a case 0 to Pi."},{"Start":"00:24.210 ","End":"00:26.250","Text":"Now we have from 2-5."},{"Start":"00:26.250 ","End":"00:28.860","Text":"We have to generalize it still further and give"},{"Start":"00:28.860 ","End":"00:32.520","Text":"a formula for the integral from a to b more generally."},{"Start":"00:32.520 ","End":"00:36.120","Text":"I\u0027m going to give you one version of this definition."},{"Start":"00:36.120 ","End":"00:38.445","Text":"There are several similar versions."},{"Start":"00:38.445 ","End":"00:41.870","Text":"Here it goes. Here it is,"},{"Start":"00:41.870 ","End":"00:44.840","Text":"the formula for the Riemann sum definition,"},{"Start":"00:44.840 ","End":"00:46.490","Text":"one version of it."},{"Start":"00:46.490 ","End":"00:48.725","Text":"Let\u0027s look at what it is."},{"Start":"00:48.725 ","End":"00:51.900","Text":"We have a symbol, Delta n,"},{"Start":"00:51.900 ","End":"00:55.625","Text":"which is just an abbreviation so it doesn\u0027t get too messy,"},{"Start":"00:55.625 ","End":"00:59.930","Text":"which is b minus a over n. If I put b minus a over n everywhere,"},{"Start":"00:59.930 ","End":"01:00.950","Text":"it looks a bit messy."},{"Start":"01:00.950 ","End":"01:05.315","Text":"Let\u0027s call that Delta n. Where b is this,"},{"Start":"01:05.315 ","End":"01:11.405","Text":"a is this, and n is what we\u0027re going to take the limit of as we go to infinity."},{"Start":"01:11.405 ","End":"01:16.580","Text":"Here. Also, we have a similar thing."},{"Start":"01:16.580 ","End":"01:18.920","Text":"We have f everywhere and we have a plus."},{"Start":"01:18.920 ","End":"01:23.270","Text":"Let\u0027s write this as 1 Delta n. You can see a pattern."},{"Start":"01:23.270 ","End":"01:25.565","Text":"It\u0027s 1 Delta n plus 2 Delta n,"},{"Start":"01:25.565 ","End":"01:28.055","Text":"may be plus 3 Delta n and so on,"},{"Start":"01:28.055 ","End":"01:33.270","Text":"up to n Delta n. We take the f of these at the map,"},{"Start":"01:33.270 ","End":"01:37.264","Text":"multiply by Delta n and take the limit as n goes to infinity."},{"Start":"01:37.264 ","End":"01:41.915","Text":"This is a bit messy to deal with,"},{"Start":"01:41.915 ","End":"01:46.070","Text":"with the dot, dot, dot, etc, notation."},{"Start":"01:46.070 ","End":"01:49.745","Text":"I\u0027d like to replace this with a Sigma notation."},{"Start":"01:49.745 ","End":"01:51.125","Text":"It\u0027ll be easier to handle."},{"Start":"01:51.125 ","End":"01:54.460","Text":"Let me bring you now the Sigma notation."},{"Start":"01:54.460 ","End":"01:58.020","Text":"Here it is with the Sigma notation."},{"Start":"01:58.020 ","End":"01:59.780","Text":"The limit is the same as the limit."},{"Start":"01:59.780 ","End":"02:01.460","Text":"Delta n is Delta n,"},{"Start":"02:01.460 ","End":"02:03.050","Text":"but instead of plus, plus,"},{"Start":"02:03.050 ","End":"02:06.665","Text":"plus, etc, I\u0027ve got the sum."},{"Start":"02:06.665 ","End":"02:09.540","Text":"Notice that what I\u0027ve done is,"},{"Start":"02:09.540 ","End":"02:12.510","Text":"I\u0027ve taken whatever is k,"},{"Start":"02:12.510 ","End":"02:16.574","Text":"which goes from 1 to n, is here\u0027s 1,"},{"Start":"02:16.574 ","End":"02:22.490","Text":"2, maybe 3 somewhere up to n. This running index from 1 to n,"},{"Start":"02:22.490 ","End":"02:27.905","Text":"I have replaced it by the Sigma from k to n. Otherwise it\u0027s the same as this."},{"Start":"02:27.905 ","End":"02:29.960","Text":"To give myself some extra space,"},{"Start":"02:29.960 ","End":"02:32.015","Text":"I\u0027m going to get rid of this one now,"},{"Start":"02:32.015 ","End":"02:35.070","Text":"and put this one in its place."},{"Start":"02:35.510 ","End":"02:39.540","Text":"This is the formula we\u0027re going to use, the Sigma version."},{"Start":"02:39.540 ","End":"02:42.155","Text":"Let\u0027s see what we have in our case."},{"Start":"02:42.155 ","End":"02:46.790","Text":"In our case we have that a is 2,"},{"Start":"02:46.790 ","End":"02:52.205","Text":"b is 5, b minus a is 3,"},{"Start":"02:52.205 ","End":"03:01.130","Text":"5 minus 2, so Delta n is 3 over n. Finally f,"},{"Start":"03:01.130 ","End":"03:04.590","Text":"our function f of x is what\u0027s written here,"},{"Start":"03:04.590 ","End":"03:08.490","Text":"2x squared plus 3x."},{"Start":"03:08.490 ","End":"03:12.275","Text":"Now we have all we need to substitute into here."},{"Start":"03:12.275 ","End":"03:17.235","Text":"This limit becomes the limit."},{"Start":"03:17.235 ","End":"03:21.600","Text":"As n goes to infinity, let\u0027s see,"},{"Start":"03:21.600 ","End":"03:28.950","Text":"Delta n is 3 over n. Then we have a Sigma."},{"Start":"03:28.950 ","End":"03:34.860","Text":"K goes from 1 up to n of,"},{"Start":"03:34.860 ","End":"03:37.575","Text":"you may want to write it first in the f form,"},{"Start":"03:37.575 ","End":"03:47.070","Text":"f of a is 2 plus k times Delta n,"},{"Start":"03:47.070 ","End":"03:49.270","Text":"which is b minus"},{"Start":"03:55.730 ","End":"04:03.410","Text":"3 over n. Permit me to write this k times 3 over n as 3k over"},{"Start":"04:03.410 ","End":"04:11.170","Text":"n. I\u0027m going to erase the k times 3 over n and instead write 3k over n. Same thing."},{"Start":"04:11.170 ","End":"04:16.470","Text":"Now, let\u0027s expand according to what f is, which is here."},{"Start":"04:16.470 ","End":"04:21.800","Text":"We get the limit as n goes to infinity,"},{"Start":"04:21.800 ","End":"04:31.910","Text":"3 over n times the sum from k equals 1 to k equals n of 2x squared."},{"Start":"04:31.910 ","End":"04:34.710","Text":"This here is my x."},{"Start":"04:35.510 ","End":"04:38.040","Text":"What I need is 2x squared."},{"Start":"04:38.040 ","End":"04:47.000","Text":"It\u0027s twice 2 plus 3k over n squared plus 3 times"},{"Start":"04:47.000 ","End":"04:55.280","Text":"2 plus"},{"Start":"04:56.390 ","End":"04:59.910","Text":"3k over n. But now,"},{"Start":"04:59.910 ","End":"05:05.755","Text":"I\u0027d better put a square brackets around this because it\u0027s all part of the Sigma."},{"Start":"05:05.755 ","End":"05:07.820","Text":"Now I\u0027m going to scroll down,"},{"Start":"05:07.820 ","End":"05:13.550","Text":"but I just want to point out that there is a hint here or 2 formulas."},{"Start":"05:13.550 ","End":"05:16.440","Text":"We\u0027re going to come back later to use the hint."},{"Start":"05:16.440 ","End":"05:19.455","Text":"Let\u0027s scroll a bit."},{"Start":"05:19.455 ","End":"05:25.490","Text":"Next thing is to expand what\u0027s inside the brackets."},{"Start":"05:25.490 ","End":"05:29.895","Text":"Some stuff I just have to copy, like this."},{"Start":"05:29.895 ","End":"05:35.790","Text":"Sum, k goes from 1 to n. Now,"},{"Start":"05:35.790 ","End":"05:40.730","Text":"this thing squared, using the a plus b squared formula,"},{"Start":"05:40.730 ","End":"05:43.955","Text":"which you should know, is a squared."},{"Start":"05:43.955 ","End":"05:49.220","Text":"The first one squared is 4 plus twice this times this,"},{"Start":"05:49.220 ","End":"05:51.874","Text":"twice 2 times 3 is 12."},{"Start":"05:51.874 ","End":"05:57.000","Text":"I get 12 k over n plus this one squared,"},{"Start":"05:57.000 ","End":"06:00.390","Text":"which will give me 9 to the 3 squared,"},{"Start":"06:00.390 ","End":"06:04.600","Text":"k squared over n squared."},{"Start":"06:05.510 ","End":"06:09.615","Text":"Here it\u0027s just the same, 3,"},{"Start":"06:09.615 ","End":"06:16.545","Text":"2 plus 3k over n. Now what I want to do is collect terms."},{"Start":"06:16.545 ","End":"06:20.894","Text":"First I have to copy the common part,"},{"Start":"06:20.894 ","End":"06:27.695","Text":"3 over n, sum k from 1 to n of."},{"Start":"06:27.695 ","End":"06:31.820","Text":"Now what I\u0027d like to do is collect it as"},{"Start":"06:31.820 ","End":"06:37.920","Text":"a quadratic expression in k. Let\u0027s see where we have k squared from."},{"Start":"06:39.800 ","End":"06:45.565","Text":"Let me just mark them. This is where I\u0027ll get the k squared from."},{"Start":"06:45.565 ","End":"06:48.975","Text":"From here and here I\u0027ll just get the k,"},{"Start":"06:48.975 ","End":"06:55.409","Text":"and the rest of it will just be the free numbers without k. For k squared,"},{"Start":"06:55.409 ","End":"07:00.915","Text":"I have 9 over n squared,"},{"Start":"07:00.915 ","End":"07:08.560","Text":"so 9 over n squared times k squared."},{"Start":"07:09.080 ","End":"07:11.130","Text":"From here and here,"},{"Start":"07:11.130 ","End":"07:12.480","Text":"let\u0027s see what I get."},{"Start":"07:12.480 ","End":"07:14.660","Text":"Sorry, this is not 9, this is 18,"},{"Start":"07:14.660 ","End":"07:16.495","Text":"of course, because there\u0027s a 2."},{"Start":"07:16.495 ","End":"07:19.130","Text":"I see we have to be very careful here."},{"Start":"07:19.130 ","End":"07:21.425","Text":"Now the term with the k,"},{"Start":"07:21.425 ","End":"07:25.135","Text":"here I have 2 times 12 is 24."},{"Start":"07:25.135 ","End":"07:29.220","Text":"Here I have 3 times 3 is 9,"},{"Start":"07:29.220 ","End":"07:35.250","Text":"24 plus 9 is 33."},{"Start":"07:35.250 ","End":"07:40.635","Text":"I get 33 times"},{"Start":"07:40.635 ","End":"07:46.360","Text":"k over n. I\u0027ll write it as 33 over n times k. Finally,"},{"Start":"07:46.360 ","End":"07:50.045","Text":"the terms without k are the ones that I didn\u0027t underline."},{"Start":"07:50.045 ","End":"07:52.025","Text":"2 times 4 is 8,"},{"Start":"07:52.025 ","End":"07:54.185","Text":"3 times 2 is 6."},{"Start":"07:54.185 ","End":"07:58.025","Text":"8 plus 6 is 14."},{"Start":"07:58.025 ","End":"08:00.930","Text":"I get 14."},{"Start":"08:05.120 ","End":"08:12.140","Text":"That\u0027s it. But I\u0027d like to write it as 14 times 1,"},{"Start":"08:12.140 ","End":"08:17.430","Text":"just to emphasize that I have k squared k and 1."},{"Start":"08:18.740 ","End":"08:21.630","Text":"That\u0027s the free number."},{"Start":"08:21.630 ","End":"08:23.100","Text":"Coefficients of k squared,"},{"Start":"08:23.100 ","End":"08:25.050","Text":"coefficients of k, coefficient of 1."},{"Start":"08:25.050 ","End":"08:27.165","Text":"If you like, k to the 0."},{"Start":"08:27.165 ","End":"08:34.025","Text":"Now, what I\u0027m going to do is split this up into 3 separate sums,"},{"Start":"08:34.025 ","End":"08:37.550","Text":"because I have a plus here and the Sigma can"},{"Start":"08:37.550 ","End":"08:41.690","Text":"take the sum separately when you have pluses or minuses."},{"Start":"08:43.070 ","End":"08:46.815","Text":"The first sum I get is,"},{"Start":"08:46.815 ","End":"08:49.170","Text":"I\u0027ll scroll a bit more,"},{"Start":"08:49.170 ","End":"08:53.724","Text":"there we go, so we get 3 limits."},{"Start":"08:53.724 ","End":"08:59.065","Text":"The first one will be limit as n goes to infinity for the first part."},{"Start":"08:59.065 ","End":"09:03.730","Text":"But I can take 18 out of n squared in front,"},{"Start":"09:03.730 ","End":"09:05.170","Text":"because it doesn\u0027t have k in it."},{"Start":"09:05.170 ","End":"09:11.660","Text":"I have 3 over n times 18 over n squared."},{"Start":"09:11.660 ","End":"09:18.985","Text":"The sum from k equals 1 to n of k squared."},{"Start":"09:18.985 ","End":"09:21.010","Text":"That\u0027s the first limit."},{"Start":"09:21.010 ","End":"09:23.785","Text":"The second one is from the second term,"},{"Start":"09:23.785 ","End":"09:30.155","Text":"plus another limit as n goes to infinity of 3 over"},{"Start":"09:30.155 ","End":"09:37.040","Text":"n. I can take the 33 over n outside the brackets."},{"Start":"09:37.040 ","End":"09:45.110","Text":"I get the sum from k equals"},{"Start":"09:45.110 ","End":"09:55.810","Text":"1 to n of just k. The last term is the limit as n goes to infinity,"},{"Start":"09:55.810 ","End":"10:02.295","Text":"3 over n. Then 14 comes outside the bracket,"},{"Start":"10:02.295 ","End":"10:08.460","Text":"times 14 times the sum k equals"},{"Start":"10:08.460 ","End":"10:15.690","Text":"1 to n of just 1 of 1."},{"Start":"10:15.690 ","End":"10:18.710","Text":"A constant. It doesn\u0027t contain k. If you like,"},{"Start":"10:18.710 ","End":"10:21.610","Text":"I can write it as k to the 0 and then you will have a k."},{"Start":"10:21.610 ","End":"10:25.280","Text":"By that we can take a sum of a constant, it\u0027s defined."},{"Start":"10:25.280 ","End":"10:27.700","Text":"Let\u0027s see what that gives us."},{"Start":"10:27.700 ","End":"10:29.915","Text":"I think at this point,"},{"Start":"10:29.915 ","End":"10:33.045","Text":"we should go and look at the hints."},{"Start":"10:33.045 ","End":"10:34.300","Text":"I\u0027m going back up."},{"Start":"10:34.300 ","End":"10:39.065","Text":"Oops."},{"Start":"10:39.065 ","End":"10:44.020","Text":"It went down back to the previous exercise."},{"Start":"10:44.020 ","End":"10:50.890","Text":"The hint says that this is equal to 1/2 and n plus 1 but what is this?"},{"Start":"10:50.890 ","End":"10:58.750","Text":"This is just the sum from k equals 1 to n of k,"},{"Start":"10:58.750 ","End":"11:01.135","Text":"which is what we had below."},{"Start":"11:01.135 ","End":"11:03.610","Text":"See, k goes from 1 to n,"},{"Start":"11:03.610 ","End":"11:05.575","Text":"k itself 1 and so on."},{"Start":"11:05.575 ","End":"11:14.270","Text":"This 1 is the sum from k equals 1 to n of k squared."},{"Start":"11:14.700 ","End":"11:18.879","Text":"I can take these 2 expressions,"},{"Start":"11:18.879 ","End":"11:21.730","Text":"I\u0027ll memorize them, 1/2 of n, n plus 1,"},{"Start":"11:21.730 ","End":"11:25.180","Text":"and 1/6 n, n plus 1, 2n plus 1."},{"Start":"11:25.180 ","End":"11:32.245","Text":"I\u0027m going to go back down there. Let\u0027s see."},{"Start":"11:32.245 ","End":"11:35.065","Text":"The sum of k squared,"},{"Start":"11:35.065 ","End":"11:37.675","Text":"well, I remember, it\u0027ll come when I come to it."},{"Start":"11:37.675 ","End":"11:42.535","Text":"Its limit as n goes to infinity."},{"Start":"11:42.535 ","End":"11:44.110","Text":"I can combine this,"},{"Start":"11:44.110 ","End":"11:47.090","Text":"54 over n cubed."},{"Start":"11:48.540 ","End":"11:52.345","Text":"The formula for this, as I recall,"},{"Start":"11:52.345 ","End":"11:58.269","Text":"was n times n pus 1"},{"Start":"11:58.269 ","End":"12:05.560","Text":"times 2n plus"},{"Start":"12:05.560 ","End":"12:10.580","Text":"1/6."},{"Start":"12:12.870 ","End":"12:16.130","Text":"I hope I remembered right."},{"Start":"12:16.710 ","End":"12:25.060","Text":"This limit becomes the limit as n goes to infinity."},{"Start":"12:25.060 ","End":"12:30.385","Text":"Combining these, I get 99 over n squared."},{"Start":"12:30.385 ","End":"12:33.715","Text":"If I remember correctly, this was n,"},{"Start":"12:33.715 ","End":"12:39.415","Text":"n plus 1 over 2."},{"Start":"12:39.415 ","End":"12:46.165","Text":"Finally, the Sigma of a constants we have lim, limit,"},{"Start":"12:46.165 ","End":"12:51.730","Text":"n goes to infinity of 42 over n. Now,"},{"Start":"12:51.730 ","End":"12:55.690","Text":"what is the sum from k goes from 1 to n of 1?"},{"Start":"12:55.690 ","End":"12:59.935","Text":"It\u0027s just 1 plus 1 pus 1 plus 1 plus 1, n times."},{"Start":"12:59.935 ","End":"13:06.985","Text":"This is equal to n. Don\u0027t be confused by the fact that k doesn\u0027t appear here."},{"Start":"13:06.985 ","End":"13:09.190","Text":"When k is 1, it\u0027s 1."},{"Start":"13:09.190 ","End":"13:10.720","Text":"When k is 2, it\u0027s 1."},{"Start":"13:10.720 ","End":"13:12.820","Text":"When k is 3, it\u0027s 1, and so on."},{"Start":"13:12.820 ","End":"13:14.845","Text":"I have n terms, all of them 1."},{"Start":"13:14.845 ","End":"13:21.940","Text":"It\u0027s n times 1, which is n. It\u0027s long,"},{"Start":"13:21.940 ","End":"13:26.210","Text":"it\u0027s involved, but I think we\u0027re coming close to the end."},{"Start":"13:26.520 ","End":"13:29.800","Text":"There is some canceling I can do."},{"Start":"13:29.800 ","End":"13:32.080","Text":"This n will cancel with"},{"Start":"13:32.080 ","End":"13:40.780","Text":"this n. This n will cancel with 1 of these n\u0027s and the n squared,"},{"Start":"13:40.780 ","End":"13:42.985","Text":"so I can just erase the 2,"},{"Start":"13:42.985 ","End":"13:47.005","Text":"make it just plain n. Here,"},{"Start":"13:47.005 ","End":"13:52.450","Text":"I can cancel n with 1 of these,"},{"Start":"13:52.450 ","End":"13:53.920","Text":"so if I cancel n here,"},{"Start":"13:53.920 ","End":"13:58.165","Text":"I have to change the 3 to a 2 because I\u0027ve taken 1 of the n\u0027s,"},{"Start":"13:58.165 ","End":"14:03.380","Text":"and 6 into 54 goes 9 times."},{"Start":"14:03.450 ","End":"14:06.895","Text":"Let\u0027s continue and see what we get."},{"Start":"14:06.895 ","End":"14:12.610","Text":"We get the limit as n goes to infinity."},{"Start":"14:12.610 ","End":"14:15.265","Text":"Now, here I have n squared."},{"Start":"14:15.265 ","End":"14:19.310","Text":"We have n plus 1,"},{"Start":"14:20.370 ","End":"14:28.914","Text":"2n plus 1 over n squared,"},{"Start":"14:28.914 ","End":"14:32.060","Text":"and we have a 9 here as well,"},{"Start":"14:32.460 ","End":"14:39.535","Text":"plus the limit, n goes to infinity."},{"Start":"14:39.535 ","End":"14:49.340","Text":"Let\u0027s see. Here, we have 99/2 times n plus 1,"},{"Start":"14:52.080 ","End":"15:02.220","Text":"and here we have just 42."},{"Start":"15:02.220 ","End":"15:05.650","Text":"Continuing, we\u0027re really going to get there soon."},{"Start":"15:07.310 ","End":"15:10.635","Text":"I can take the constant out front,"},{"Start":"15:10.635 ","End":"15:16.045","Text":"so I get 9 times the limit as n goes to infinity,"},{"Start":"15:16.045 ","End":"15:23.920","Text":"and I can also split this n squared into n times n. I can"},{"Start":"15:23.920 ","End":"15:33.100","Text":"get n plus 1."},{"Start":"15:33.100 ","End":"15:36.910","Text":"You know what, I\u0027m actually going to do the dividing. Hang on."},{"Start":"15:36.910 ","End":"15:48.300","Text":"This n squared, I can write it as n times n. Now,"},{"Start":"15:48.300 ","End":"15:50.810","Text":"I can take n plus 1,"},{"Start":"15:50.810 ","End":"15:57.075","Text":"this bit here, and that\u0027s 1 plus 1."},{"Start":"15:57.075 ","End":"16:05.590","Text":"The 2n plus 1 is 2 plus 1. That\u0027s the first term."},{"Start":"16:05.590 ","End":"16:10.945","Text":"The second term limit as n goes to infinity."},{"Start":"16:10.945 ","End":"16:19.930","Text":"Once again, I can take 99/2 outside the limit because it\u0027s a constant,"},{"Start":"16:19.930 ","End":"16:25.130","Text":"and n plus 1 is 1 plus 1."},{"Start":"16:27.030 ","End":"16:33.320","Text":"This limit of a constant is just the constant, is 42."},{"Start":"16:35.370 ","End":"16:38.410","Text":"For some people, this is meaningful."},{"Start":"16:38.410 ","End":"16:40.930","Text":"This number, never mind."},{"Start":"16:40.930 ","End":"16:46.760","Text":"The meaning of life in some comedy."},{"Start":"16:55.770 ","End":"17:00.500","Text":"Continuing, we\u0027re almost there."},{"Start":"17:02.220 ","End":"17:09.175","Text":"1, this thing goes to 0 when n goes to infinity."},{"Start":"17:09.175 ","End":"17:13.405","Text":"This 1 also goes to 0."},{"Start":"17:13.405 ","End":"17:18.760","Text":"This 1 goes to 0."},{"Start":"17:18.760 ","End":"17:25.659","Text":"Basically, all we are left with is numbers."},{"Start":"17:25.659 ","End":"17:30.430","Text":"Because this limit then becomes the limit of 2, which is just 2,"},{"Start":"17:30.430 ","End":"17:35.500","Text":"so what we get is 9 times"},{"Start":"17:35.500 ","End":"17:41.920","Text":"2 plus 99/2 times,"},{"Start":"17:41.920 ","End":"17:46.550","Text":"and this limit becomes 1, plus 42."},{"Start":"17:47.430 ","End":"17:50.995","Text":"This equals 18"},{"Start":"17:50.995 ","End":"18:00.100","Text":"plus 49.5 plus 42."},{"Start":"18:00.100 ","End":"18:02.620","Text":"42 plus 18 is 60."},{"Start":"18:02.620 ","End":"18:05.095","Text":"60 plus 40 is 100."},{"Start":"18:05.095 ","End":"18:09.950","Text":"We get 109.5."},{"Start":"18:12.000 ","End":"18:14.935","Text":"We finally got the answer."},{"Start":"18:14.935 ","End":"18:21.680","Text":"But as usual, we\u0027re going to check by doing the actual integral."},{"Start":"18:22.740 ","End":"18:26.650","Text":"Feel free to turn this off at any time,"},{"Start":"18:26.650 ","End":"18:30.910","Text":"but I\u0027m just going to have to do a check doing integration the regular way."},{"Start":"18:30.910 ","End":"18:34.690","Text":"Now, you\u0027ll really appreciate the rules"},{"Start":"18:34.690 ","End":"18:39.535","Text":"for integration rather than having to do it this way."},{"Start":"18:39.535 ","End":"18:44.860","Text":"Scroll down a bit and take a different color."},{"Start":"18:44.860 ","End":"18:53.410","Text":"Let\u0027s take blue and do the integral from 2-5 of 2x"},{"Start":"18:53.410 ","End":"19:08.800","Text":"squared plus 3x"},{"Start":"19:08.800 ","End":"19:15.110","Text":"dx."},{"Start":"19:24.960 ","End":"19:30.130","Text":"This equals, raise the power by 1 and divide by it."},{"Start":"19:30.130 ","End":"19:33.895","Text":"We get 2/3x cubed."},{"Start":"19:33.895 ","End":"19:35.365","Text":"This 3 is this 3."},{"Start":"19:35.365 ","End":"19:37.450","Text":"Then here 1 becomes 2,"},{"Start":"19:37.450 ","End":"19:40.450","Text":"so it\u0027s 3, but divided by that 2,"},{"Start":"19:40.450 ","End":"19:46.360","Text":"x squared, all this between 2 and 5."},{"Start":"19:46.360 ","End":"19:53.020","Text":"What do we get? If we put in 5,"},{"Start":"19:53.020 ","End":"20:00.690","Text":"we get 2/3 of 125,"},{"Start":"20:00.690 ","End":"20:02.325","Text":"which is 5 cubed,"},{"Start":"20:02.325 ","End":"20:08.000","Text":"plus 3 over 2 times 5 squared, which is 25."},{"Start":"20:08.000 ","End":"20:12.340","Text":"Then subtract, what happens when we put in 2."},{"Start":"20:12.340 ","End":"20:17.240","Text":"If we put in 2, we get 2/3 times 8."},{"Start":"20:19.100 ","End":"20:21.540","Text":"2 squared is 4,"},{"Start":"20:21.540 ","End":"20:27.400","Text":"so it\u0027s 3/2 times 4."},{"Start":"20:33.380 ","End":"20:39.940","Text":"I can take 2/3 of 125 minus 8."},{"Start":"20:41.120 ","End":"20:45.485","Text":"I\u0027ll take the 2/3 separately and the 3/2 separately,"},{"Start":"20:45.485 ","End":"20:52.499","Text":"and the 3/2 of 25 minus 4."},{"Start":"20:55.710 ","End":"21:04.760","Text":"Now, this bit is equal"},{"Start":"21:04.760 ","End":"21:15.770","Text":"to 2/3 times 117 plus 3 over 2 times 21,"},{"Start":"21:15.770 ","End":"21:21.990","Text":"which is 2/3 of 117,"},{"Start":"21:21.990 ","End":"21:27.340","Text":"117/3 is 39 times 2 is 78,"},{"Start":"21:27.800 ","End":"21:32.835","Text":"and 3/2 times 21 is 63/2,"},{"Start":"21:32.835 ","End":"21:40.680","Text":"which is 31.5, and 78 plus 31.5,"},{"Start":"21:40.680 ","End":"21:43.450","Text":"I make it 109.5."},{"Start":"21:47.100 ","End":"21:50.080","Text":"It looks the same to me."},{"Start":"21:50.080 ","End":"21:55.740","Text":"I would say that we have this checked and now, we\u0027re really done."}],"ID":24176}],"Thumbnail":null,"ID":65397},{"Name":"Fundamental Theorm of Calculus","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Fundamental Theorem of Calculus","Duration":"3m 39s","ChapterTopicVideoID":8359,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/8359.jpeg","UploadDate":"2019-12-10T12:43:28.5670000","DurationForVideoObject":"PT3M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"There is a theorem called the fundamental theorem of the calculus,"},{"Start":"00:03.630 ","End":"00:05.520","Text":"and it has 2 parts."},{"Start":"00:05.520 ","End":"00:09.060","Text":"On this clip, we\u0027ll be talking about the first part."},{"Start":"00:09.060 ","End":"00:11.024","Text":"It goes like this."},{"Start":"00:11.024 ","End":"00:13.275","Text":"We have a function f,"},{"Start":"00:13.275 ","End":"00:17.505","Text":"which is continuous on an interval,"},{"Start":"00:17.505 ","End":"00:21.345","Text":"and we define another function,"},{"Start":"00:21.345 ","End":"00:23.790","Text":"call it I for integral,"},{"Start":"00:23.790 ","End":"00:30.540","Text":"and I of x is the integral from a to x."},{"Start":"00:30.540 ","End":"00:36.115","Text":"The interval contains some point a."},{"Start":"00:36.115 ","End":"00:44.900","Text":"We take the integral from that a to a variable x and get a function of x of f. Now,"},{"Start":"00:44.900 ","End":"00:46.910","Text":"I can\u0027t reuse the letter x,"},{"Start":"00:46.910 ","End":"00:49.535","Text":"so I\u0027ll take letter t dt."},{"Start":"00:49.535 ","End":"00:53.254","Text":"This gives us a function I of x."},{"Start":"00:53.254 ","End":"00:56.915","Text":"Then it turns out that I is differentiable,"},{"Start":"00:56.915 ","End":"01:01.280","Text":"and, in fact, the derivative I prime of"},{"Start":"01:01.280 ","End":"01:08.320","Text":"x is just equal to the original function f of x."},{"Start":"01:08.320 ","End":"01:13.520","Text":"An alternate way of phrasing this without using"},{"Start":"01:13.520 ","End":"01:17.210","Text":"the intermediate function I is just to directly say that"},{"Start":"01:17.210 ","End":"01:21.815","Text":"this integral derivative is f of x."},{"Start":"01:21.815 ","End":"01:25.160","Text":"Yet another way is to use the other notation,"},{"Start":"01:25.160 ","End":"01:28.670","Text":"the Leibniz\u0027s notation for derivative"},{"Start":"01:28.670 ","End":"01:35.120","Text":"and to say d by dx of this is also equal to f of x."},{"Start":"01:35.120 ","End":"01:38.285","Text":"Informally, what it says is, if I take an"},{"Start":"01:38.285 ","End":"01:42.050","Text":"integral of a function and then differentiate it,"},{"Start":"01:42.050 ","End":"01:44.645","Text":"I get back to the original function."},{"Start":"01:44.645 ","End":"01:54.970","Text":"As an example, let\u0027s take the integral from 1 to x of t^4 dt,"},{"Start":"01:54.970 ","End":"01:59.165","Text":"and then the derivative of this."},{"Start":"01:59.165 ","End":"02:01.670","Text":"Now, according to the theorem,"},{"Start":"02:01.670 ","End":"02:05.870","Text":"if I think of f of t as being t^4,"},{"Start":"02:05.870 ","End":"02:09.305","Text":"then what this says is that I take the same function,"},{"Start":"02:09.305 ","End":"02:15.690","Text":"but just with the letter x instead of the letter t, so this should equal x^4."},{"Start":"02:15.690 ","End":"02:18.790","Text":"Of course, in this case, we could check and say"},{"Start":"02:18.790 ","End":"02:24.300","Text":"the integral would be t^5 over 5,"},{"Start":"02:24.300 ","End":"02:30.870","Text":"evaluated from 1 to x, and then derivative,"},{"Start":"02:30.870 ","End":"02:40.830","Text":"this would be x^5 over 5 minus 1^5 over 5 derivative."},{"Start":"02:40.830 ","End":"02:47.870","Text":"This is, indeed, equal to x^4 because I get 5x^4 over 5,"},{"Start":"02:47.870 ","End":"02:49.234","Text":"and the 5 cancels,"},{"Start":"02:49.234 ","End":"02:52.010","Text":"and this is a constant, so it just drops out."},{"Start":"02:52.010 ","End":"02:57.120","Text":"Example number 2, the integral from 4 to"},{"Start":"02:57.120 ","End":"03:03.180","Text":"x of sine t over t dt."},{"Start":"03:03.180 ","End":"03:06.160","Text":"I want the derivative of that."},{"Start":"03:06.160 ","End":"03:07.790","Text":"Again, by the theorem,"},{"Start":"03:07.790 ","End":"03:10.130","Text":"I have here a function of t,"},{"Start":"03:10.130 ","End":"03:14.554","Text":"and so all I do is replace the t with x,"},{"Start":"03:14.554 ","End":"03:23.390","Text":"and so the answer will be sine x over x. I don\u0027t know if this can be checked easily."},{"Start":"03:23.390 ","End":"03:28.940","Text":"But in any event, the theorem assures us that, even if we can\u0027t check like here,"},{"Start":"03:28.940 ","End":"03:31.550","Text":"that this will be the answer."},{"Start":"03:31.550 ","End":"03:34.700","Text":"That\u0027s all I want to say for now about the fundamental theorem"},{"Start":"03:34.700 ","End":"03:38.940","Text":"of calculus, first part. So we\u0027re done."}],"ID":24136},{"Watched":false,"Name":"First Fundamental Theorem I","Duration":"5m 53s","ChapterTopicVideoID":23290,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23290.jpeg","UploadDate":"2021-01-10T12:15:24.1870000","DurationForVideoObject":"PT5M53S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"Now we come to what are called the fundamental"},{"Start":"00:03.060 ","End":"00:07.455","Text":"theorem of calculus and there are 2 of them,"},{"Start":"00:07.455 ","End":"00:10.590","Text":"first and the second fundamental theorem."},{"Start":"00:10.590 ","End":"00:13.890","Text":"Each of them, in a way says that differentiation and"},{"Start":"00:13.890 ","End":"00:17.410","Text":"integration are opposite of each other."},{"Start":"00:17.410 ","End":"00:22.260","Text":"We\u0027ll also see some connection between the definite and indefinite integrals."},{"Start":"00:22.260 ","End":"00:26.695","Text":"Let\u0027s start with the first fundamental theorem of the calculus and state it."},{"Start":"00:26.695 ","End":"00:29.510","Text":"F is integrable on the interval a,"},{"Start":"00:29.510 ","End":"00:32.525","Text":"b and for each x in the interval,"},{"Start":"00:32.525 ","End":"00:39.440","Text":"we define big F of x to be the integral from a to x of f of t dt,"},{"Start":"00:39.440 ","End":"00:42.260","Text":"we have to use a different letter than x."},{"Start":"00:42.260 ","End":"00:43.505","Text":"We use t here,"},{"Start":"00:43.505 ","End":"00:46.010","Text":"in that case there\u0027s 2 things we\u0027re going to say."},{"Start":"00:46.010 ","End":"00:50.465","Text":"First is that big F is continuous on this interval."},{"Start":"00:50.465 ","End":"00:55.565","Text":"Furthermore, if little f happens to be continuous at a point,"},{"Start":"00:55.565 ","End":"00:59.510","Text":"then big F is differentiable at that point,"},{"Start":"00:59.510 ","End":"01:05.030","Text":"and the derivative of F is little f. Now the proof."},{"Start":"01:05.030 ","End":"01:07.550","Text":"First thing to note is that big F is"},{"Start":"01:07.550 ","End":"01:12.580","Text":"well-defined because little f is integrable on the whole of a,"},{"Start":"01:12.580 ","End":"01:18.005","Text":"b it\u0027s also integrable on the subinterval just from a to x,"},{"Start":"01:18.005 ","End":"01:20.575","Text":"so this is defined."},{"Start":"01:20.575 ","End":"01:24.695","Text":"Now because it\u0027s integrable, it\u0027s bounded."},{"Start":"01:24.695 ","End":"01:28.040","Text":"That\u0027s part of the definition of integrable and so we can"},{"Start":"01:28.040 ","End":"01:32.900","Text":"define big M to be the supremum of the absolute value of f of x,"},{"Start":"01:32.900 ","End":"01:36.110","Text":"where x is in the interval a, b."},{"Start":"01:36.110 ","End":"01:40.460","Text":"Now we\u0027re going to show not only that big F is continuous on a b,"},{"Start":"01:40.460 ","End":"01:42.890","Text":"it\u0027s actually uniformly continuous."},{"Start":"01:42.890 ","End":"01:44.910","Text":"We might need that in future."},{"Start":"01:44.910 ","End":"01:46.470","Text":"Let\u0027s take 2 points,"},{"Start":"01:46.470 ","End":"01:49.550","Text":"x and y in the interval a,"},{"Start":"01:49.550 ","End":"01:53.210","Text":"b and say x is smaller than y so let\u0027s"},{"Start":"01:53.210 ","End":"01:58.910","Text":"estimate the absolute value of f of y minus f of x is"},{"Start":"01:58.910 ","End":"02:03.109","Text":"the absolute value of the integral from x to y"},{"Start":"02:03.109 ","End":"02:09.905","Text":"because we take the integral from a to y and subtract the integral from a to x."},{"Start":"02:09.905 ","End":"02:12.710","Text":"Then we just get the integral from x to y."},{"Start":"02:12.710 ","End":"02:16.910","Text":"It\u0027s the additivity property of integrals."},{"Start":"02:16.910 ","End":"02:19.460","Text":"Then there\u0027s this other property that"},{"Start":"02:19.460 ","End":"02:22.250","Text":"the absolute value of the integral is"},{"Start":"02:22.250 ","End":"02:25.430","Text":"less than or equal to the integral of the absolute value."},{"Start":"02:25.430 ","End":"02:35.909","Text":"This is bounded by M. M is the supremum so it\u0027s less than or equal to M times y minus x."},{"Start":"02:35.909 ","End":"02:42.155","Text":"This is less than or equal to this which means that if we want to use the epsilon Delta,"},{"Start":"02:42.155 ","End":"02:48.980","Text":"if we choose absolute value of x minus y less than epsilon over M,"},{"Start":"02:48.980 ","End":"02:54.320","Text":"then absolute value of f of x minus f of y is less than epsilon."},{"Start":"02:54.320 ","End":"02:57.940","Text":"This is like the Delta for the epsilon."},{"Start":"02:57.940 ","End":"03:03.770","Text":"The way it\u0027s set up is the definition for a uniformly continuous and of course,"},{"Start":"03:03.770 ","End":"03:06.530","Text":"uniform continuity implies continuity,"},{"Start":"03:06.530 ","End":"03:09.215","Text":"so yeah, f is continuous."},{"Start":"03:09.215 ","End":"03:11.870","Text":"Now on to part 2,"},{"Start":"03:11.870 ","End":"03:16.025","Text":"we have to show that if f is continuous at the point x_0,"},{"Start":"03:16.025 ","End":"03:20.360","Text":"then big F is differentiable at that point and there is"},{"Start":"03:20.360 ","End":"03:25.460","Text":"a big F is little f. Given that f is continuous here,"},{"Start":"03:25.460 ","End":"03:29.960","Text":"we\u0027re going to show that the limit as x goes to x_0."},{"Start":"03:29.960 ","End":"03:34.645","Text":"Well, this is petty definition of differentiable at x_0,"},{"Start":"03:34.645 ","End":"03:37.910","Text":"and that the derivative of big F is little f. All we have to do"},{"Start":"03:37.910 ","End":"03:42.020","Text":"is show that this limit holds and we\u0027ll do it"},{"Start":"03:42.020 ","End":"03:49.400","Text":"using the epsilon Delta definition of a limit so that epsilon greater than 0 be given."},{"Start":"03:49.400 ","End":"03:52.720","Text":"Since little f is continuous at x_0,"},{"Start":"03:52.720 ","End":"03:58.020","Text":"we can choose Delta such that if x minus x_0 is less than Delta,"},{"Start":"03:58.020 ","End":"04:01.065","Text":"then f of x minus f of x_0 is less than epsilon."},{"Start":"04:01.065 ","End":"04:06.300","Text":"Now, this is the Delta we need I claim for the above limit."},{"Start":"04:06.300 ","End":"04:08.960","Text":"There are 2 limits here and each 1 has an epsilon Delta,"},{"Start":"04:08.960 ","End":"04:15.250","Text":"but I claim it\u0027s the same Delta that will serve us here also. Let\u0027s see."},{"Start":"04:15.250 ","End":"04:21.140","Text":"The absolute value of this minus this is equal to 1 over x"},{"Start":"04:21.140 ","End":"04:26.990","Text":"minus x_0 times the integral of f of t dt, this part."},{"Start":"04:26.990 ","End":"04:32.150","Text":"Recall that big F of x is the integral from a to x,"},{"Start":"04:32.150 ","End":"04:34.520","Text":"and then take away the integral from a to x_0,"},{"Start":"04:34.520 ","End":"04:37.340","Text":"so it gives us the integral from x_0 to x by"},{"Start":"04:37.340 ","End":"04:40.550","Text":"the additivity and there\u0027s been here just copied."},{"Start":"04:40.550 ","End":"04:43.655","Text":"Now this equals the first part,"},{"Start":"04:43.655 ","End":"04:51.260","Text":"like here and the second part we can rewrite using the fact that the integral of"},{"Start":"04:51.260 ","End":"04:53.990","Text":"a constant is just that"},{"Start":"04:53.990 ","End":"04:58.670","Text":"constant times the difference between the upper limit and the lower limit."},{"Start":"04:58.670 ","End":"05:00.020","Text":"It was 1 here."},{"Start":"05:00.020 ","End":"05:03.805","Text":"The integral from x_0 to x of 1 is x minus x_0,"},{"Start":"05:03.805 ","End":"05:12.285","Text":"and the x minus x_0 cancels with x minus x_0 so f of x_0 is equal to this and then we can"},{"Start":"05:12.285 ","End":"05:15.470","Text":"just take the 1 over x minus x_0 out the brackets"},{"Start":"05:15.470 ","End":"05:20.635","Text":"and the integral out of the brackets and we get this."},{"Start":"05:20.635 ","End":"05:26.179","Text":"Now we can use that absolute value property of the integrals that we used"},{"Start":"05:26.179 ","End":"05:32.675","Text":"earlier and put the absolute value inside the integral and make it less than or equal to."},{"Start":"05:32.675 ","End":"05:37.065","Text":"Now, this is less than epsilon."},{"Start":"05:37.065 ","End":"05:44.665","Text":"This is less than the integral of epsilon dt times 1 over x minus x_0."},{"Start":"05:44.665 ","End":"05:51.095","Text":"This cancels, this becomes epsilon times x minus x_0 so in the end we just get epsilon,"},{"Start":"05:51.095 ","End":"05:54.540","Text":"and that\u0027s all we need to conclude this proof."}],"ID":24137},{"Watched":false,"Name":"First Fundamental Theorem II","Duration":"10m 26s","ChapterTopicVideoID":23291,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23291.jpeg","UploadDate":"2021-01-10T12:21:01.9330000","DurationForVideoObject":"PT10M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"We just proved the first fundamental theorem of the calculus,"},{"Start":"00:04.350 ","End":"00:06.230","Text":"abbreviated like so,"},{"Start":"00:06.230 ","End":"00:09.150","Text":"and now let\u0027s do some sample problems with it."},{"Start":"00:09.150 ","End":"00:10.845","Text":"Here is the first 1,"},{"Start":"00:10.845 ","End":"00:13.830","Text":"p is a fixed number and f is"},{"Start":"00:13.830 ","End":"00:19.015","Text":"a continuous function which satisfies the following equation."},{"Start":"00:19.015 ","End":"00:23.120","Text":"For all x, f of x plus p equals f of x."},{"Start":"00:23.120 ","End":"00:28.695","Text":"This is what we call a periodic function with period p. If you jump along p,"},{"Start":"00:28.695 ","End":"00:30.865","Text":"you end up with the same value."},{"Start":"00:30.865 ","End":"00:37.535","Text":"Anyway, we have to show that the integral from x to x plus p of f of t,"},{"Start":"00:37.535 ","End":"00:39.995","Text":"dt has the same value."},{"Start":"00:39.995 ","End":"00:43.370","Text":"Whatever x is, it comes out the same."},{"Start":"00:43.370 ","End":"00:48.500","Text":"A picture will probably explain it the best, at least intuitively."},{"Start":"00:48.500 ","End":"00:52.460","Text":"Here we have a periodic function and p would"},{"Start":"00:52.460 ","End":"00:56.660","Text":"be the distance from here to here or from here to here."},{"Start":"00:56.660 ","End":"00:58.625","Text":"If we take any x,"},{"Start":"00:58.625 ","End":"01:01.790","Text":"let\u0027s say this is our x and this is x plus p,"},{"Start":"01:01.790 ","End":"01:04.250","Text":"then we have 1 complete period and we take"},{"Start":"01:04.250 ","End":"01:08.060","Text":"the area under the graph here, that\u0027s 1 thing."},{"Start":"01:08.060 ","End":"01:13.345","Text":"But if we take a different x and move p along and take the area under the graph,"},{"Start":"01:13.345 ","End":"01:16.220","Text":"really, we should get the same thing."},{"Start":"01:16.220 ","End":"01:21.020","Text":"It\u0027s fairly clear that if we move along with any length p,"},{"Start":"01:21.020 ","End":"01:22.480","Text":"we\u0027re going to get the same area."},{"Start":"01:22.480 ","End":"01:23.570","Text":"That\u0027s what this says."},{"Start":"01:23.570 ","End":"01:25.280","Text":"Now we have to do it formally,"},{"Start":"01:25.280 ","End":"01:26.695","Text":"not just with a picture."},{"Start":"01:26.695 ","End":"01:29.780","Text":"In the first fundamental theorem of the calculus,"},{"Start":"01:29.780 ","End":"01:32.270","Text":"if we take a equals 0,"},{"Start":"01:32.270 ","End":"01:35.030","Text":"a is that lower limit of integration."},{"Start":"01:35.030 ","End":"01:37.895","Text":"Remember, we take the integral from a to x."},{"Start":"01:37.895 ","End":"01:40.220","Text":"It doesn\u0027t matter, just pick anything."},{"Start":"01:40.220 ","End":"01:44.075","Text":"If I have f of x is the integral from 0 to x of f of t, dt,"},{"Start":"01:44.075 ","End":"01:47.975","Text":"then because little f is continuous everywhere,"},{"Start":"01:47.975 ","End":"01:49.760","Text":"by that theorem, part 2,"},{"Start":"01:49.760 ","End":"01:56.060","Text":"f prime, big F prime is differentiable everywhere and the derivative of big F is"},{"Start":"01:56.060 ","End":"01:59.940","Text":"little f. Now let\u0027s define G of"},{"Start":"01:59.940 ","End":"02:06.440","Text":"x to be this integral from x to x plus p of f of t,"},{"Start":"02:06.440 ","End":"02:10.590","Text":"dt for any given x. G of x,"},{"Start":"02:10.590 ","End":"02:14.840","Text":"you could write it by the additivity property for integrals as the"},{"Start":"02:14.840 ","End":"02:19.445","Text":"integral from 0 to x plus p minus the integral from 0 to x."},{"Start":"02:19.445 ","End":"02:24.320","Text":"What remains is just from x to x plus p. This is equal to,"},{"Start":"02:24.320 ","End":"02:27.305","Text":"by definition of f,"},{"Start":"02:27.305 ","End":"02:32.180","Text":"this is f of x plus p minus f of"},{"Start":"02:32.180 ","End":"02:39.260","Text":"x. G prime of x is going to be f prime of x plus p,"},{"Start":"02:39.260 ","End":"02:41.840","Text":"talking about the derivative of this part."},{"Start":"02:41.840 ","End":"02:44.780","Text":"But we have to multiply by the inner derivative,"},{"Start":"02:44.780 ","End":"02:46.500","Text":"which happens to be 1."},{"Start":"02:46.500 ","End":"02:48.230","Text":"But if it had been like 2x here,"},{"Start":"02:48.230 ","End":"02:50.135","Text":"then we would have got something not 1."},{"Start":"02:50.135 ","End":"02:51.740","Text":"Minus the derivative of this."},{"Start":"02:51.740 ","End":"02:53.735","Text":"Well, that\u0027s just f prime of x."},{"Start":"02:53.735 ","End":"02:58.130","Text":"Now look, we have F prime and here and F prime here."},{"Start":"02:58.130 ","End":"03:00.635","Text":"F prime of this and f prime of that."},{"Start":"03:00.635 ","End":"03:06.320","Text":"But in general, F prime of x_0 is little f of x_0."},{"Start":"03:06.320 ","End":"03:12.555","Text":"1 time, we could take x_0 as x plus p,"},{"Start":"03:12.555 ","End":"03:17.010","Text":"and 1 time we could take x_0 as x."},{"Start":"03:17.010 ","End":"03:24.200","Text":"This 1 would give us f of x plus p and this 1 would give us f of x."},{"Start":"03:24.200 ","End":"03:27.860","Text":"We have f of x plus p minus f of x."},{"Start":"03:27.860 ","End":"03:30.230","Text":"But our function is periodic."},{"Start":"03:30.230 ","End":"03:33.740","Text":"F of x plus p equals f of x for any x."},{"Start":"03:33.740 ","End":"03:35.945","Text":"This minus this is 0,"},{"Start":"03:35.945 ","End":"03:40.880","Text":"and this is true for any x. G prime of x is 0 for any x,"},{"Start":"03:40.880 ","End":"03:46.274","Text":"meaning identically 0, which means that G of x is a constant function."},{"Start":"03:46.274 ","End":"03:50.059","Text":"Now, what does it mean that G of x is a constant function?"},{"Start":"03:50.059 ","End":"03:56.510","Text":"It means that this integral is the same value for all x."},{"Start":"03:56.510 ","End":"04:01.300","Text":"That is exactly what we had to show and so we\u0027re d1."},{"Start":"04:01.300 ","End":"04:04.140","Text":"Now we come to problem number 2."},{"Start":"04:04.140 ","End":"04:10.285","Text":"Here, f is a continuous function on the interval from 0 to Pi over 2."},{"Start":"04:10.285 ","End":"04:17.135","Text":"We\u0027re told that the integral on this interval of f of t, dt is 0."},{"Start":"04:17.135 ","End":"04:24.215","Text":"We have to show that there is some point c between 0 and Pi over 2,"},{"Start":"04:24.215 ","End":"04:31.080","Text":"such that f of c is equal to 2 cosine 2c."},{"Start":"04:31.190 ","End":"04:36.080","Text":"For the solution, what we do is define a function F on"},{"Start":"04:36.080 ","End":"04:42.620","Text":"the same interval by F of x is the integral from 0 to x of f of t,"},{"Start":"04:42.620 ","End":"04:46.055","Text":"dt minus sine 2x."},{"Start":"04:46.055 ","End":"04:49.235","Text":"This is not something you would think to do."},{"Start":"04:49.235 ","End":"04:51.725","Text":"It was reverse engineered."},{"Start":"04:51.725 ","End":"04:56.885","Text":"You could see that this is the antiderivative of"},{"Start":"04:56.885 ","End":"05:02.550","Text":"2 cosine 2x and this is the antiderivative of f of x."},{"Start":"05:02.550 ","End":"05:07.890","Text":"Anyway, we\u0027ll just take it as pulled out of thin air."},{"Start":"05:07.890 ","End":"05:11.855","Text":"We take f of x to be this and we\u0027ll see that this does the trick for us."},{"Start":"05:11.855 ","End":"05:15.320","Text":"By the first fundamental theorem of the calculus,"},{"Start":"05:15.320 ","End":"05:20.760","Text":"F is differentiable on the interval."},{"Start":"05:20.810 ","End":"05:23.910","Text":"Well, it\u0027s not immediately clear."},{"Start":"05:23.910 ","End":"05:28.925","Text":"It\u0027s because f is continuous so this integral"},{"Start":"05:28.925 ","End":"05:34.380","Text":"is differentiable and sine 2x is also differentiable."},{"Start":"05:34.380 ","End":"05:39.210","Text":"First, you subtract differentiable minus differentiable, it\u0027s still differentiable."},{"Start":"05:39.210 ","End":"05:46.785","Text":"Now, I claim that F of 0 is 0 and F of Pi over 2 is 0."},{"Start":"05:46.785 ","End":"05:49.245","Text":"F of 0,"},{"Start":"05:49.245 ","End":"05:54.945","Text":"we just put 0 here instead of x. Integral from 0 to 0 is of course 0."},{"Start":"05:54.945 ","End":"05:57.935","Text":"If we put Pi over 2 here,"},{"Start":"05:57.935 ","End":"06:00.530","Text":"the integral from 0 to Pi over 2 of f of t,"},{"Start":"06:00.530 ","End":"06:03.395","Text":"dt is given to be 0."},{"Start":"06:03.395 ","End":"06:12.760","Text":"All we have to do is the integral of sine of 2x from 0 to Pi over 2."},{"Start":"06:12.950 ","End":"06:15.090","Text":"It\u0027s not hard to show."},{"Start":"06:15.090 ","End":"06:17.730","Text":"I\u0027m presuming you know basic integration."},{"Start":"06:17.730 ","End":"06:22.990","Text":"The integral of minus sine 2x is a half cosine 2x."},{"Start":"06:22.990 ","End":"06:31.290","Text":"Basically, when you plug in 0 and Pi over 2 to 2x,"},{"Start":"06:31.290 ","End":"06:35.710","Text":"then we get 0 and Pi."},{"Start":"06:36.980 ","End":"06:43.550","Text":"Take 2. F of 0 is the integral from 0 to 0 of f of t,"},{"Start":"06:43.550 ","End":"06:47.600","Text":"dt, so that\u0027s of course 0 and sine of twice 0 is 0,"},{"Start":"06:47.600 ","End":"06:49.025","Text":"so that\u0027s this 1."},{"Start":"06:49.025 ","End":"06:50.765","Text":"As for the other 1,"},{"Start":"06:50.765 ","End":"06:53.270","Text":"the integral from 0 to Pi over 2 of f of t,"},{"Start":"06:53.270 ","End":"06:56.000","Text":"dt is given to be 0."},{"Start":"06:56.000 ","End":"06:58.070","Text":"If you plug Pi over 2 in here,"},{"Start":"06:58.070 ","End":"07:04.820","Text":"we get minus sine of Pi and sine of Pi is 0 so this 1 equals 0."},{"Start":"07:04.820 ","End":"07:07.895","Text":"Now we\u0027re going to use Rolle\u0027s theorem."},{"Start":"07:07.895 ","End":"07:14.860","Text":"The function is differentiable and its 0 both at 0 and at Pi over 2,"},{"Start":"07:14.860 ","End":"07:20.645","Text":"so it\u0027s going to be some point c in the open interval between them by"},{"Start":"07:20.645 ","End":"07:27.939","Text":"Rolle such that the derivative F prime of c is 0."},{"Start":"07:27.939 ","End":"07:33.270","Text":"But if f is this,"},{"Start":"07:33.270 ","End":"07:40.820","Text":"then F prime is just f of t minus 2 cosine 2x."},{"Start":"07:40.820 ","End":"07:42.110","Text":"Plug in the c,"},{"Start":"07:42.110 ","End":"07:46.775","Text":"we get f of c minus 2 cosine 2c, and that\u0027s got to be 0."},{"Start":"07:46.775 ","End":"07:52.010","Text":"Replace x by c. If this minus this is 0,"},{"Start":"07:52.010 ","End":"07:54.455","Text":"then this is equal to this."},{"Start":"07:54.455 ","End":"07:58.340","Text":"This is what we had to show and so we\u0027re d1."},{"Start":"07:58.340 ","End":"08:02.165","Text":"Now, the third and last problem in this clip,"},{"Start":"08:02.165 ","End":"08:06.710","Text":"we have to show that the limit as x goes to 0 of 1"},{"Start":"08:06.710 ","End":"08:11.570","Text":"over x cubed integral from 0 to x t squared over 1 plus t^4,"},{"Start":"08:11.570 ","End":"08:14.935","Text":"dt is equal to 1/3."},{"Start":"08:14.935 ","End":"08:19.300","Text":"We\u0027re going to use the fundamental theorem of the calculus for this."},{"Start":"08:19.300 ","End":"08:21.840","Text":"We see that there\u0027s a function here,"},{"Start":"08:21.840 ","End":"08:26.335","Text":"call that f. f of x is x squared over 1 plus x^4."},{"Start":"08:26.335 ","End":"08:31.810","Text":"We\u0027ll let big F of x be the integral from 0 to x of this,"},{"Start":"08:31.810 ","End":"08:36.610","Text":"which is little f. Now we can write that in full, like so."},{"Start":"08:36.610 ","End":"08:38.990","Text":"Little f is continuous."},{"Start":"08:38.990 ","End":"08:41.875","Text":"By the first fundamental theorem,"},{"Start":"08:41.875 ","End":"08:45.820","Text":"the second part, F is differentiable and not only that,"},{"Start":"08:45.820 ","End":"08:51.490","Text":"but the derivative of F is f. Let L be this limit,"},{"Start":"08:51.490 ","End":"08:53.170","Text":"the 1 that we have to find."},{"Start":"08:53.170 ","End":"08:55.525","Text":"Now we have to show that L equals 1/3."},{"Start":"08:55.525 ","End":"09:01.430","Text":"As the limit as x goes to 0 of f of x over x cubed,"},{"Start":"09:01.430 ","End":"09:04.865","Text":"I\u0027m just putting the x cubed here to the denominator."},{"Start":"09:04.865 ","End":"09:08.554","Text":"Now this is a limit of type 0 over 0."},{"Start":"09:08.554 ","End":"09:10.175","Text":"Let\u0027s look at the denominator."},{"Start":"09:10.175 ","End":"09:14.630","Text":"When x goes to 0, then x cubed goes to 0. That\u0027s the denominator."},{"Start":"09:14.630 ","End":"09:16.375","Text":"As to the numerator,"},{"Start":"09:16.375 ","End":"09:21.835","Text":"when x goes to 0, we can just plug it in and you get F of 0."},{"Start":"09:21.835 ","End":"09:27.049","Text":"F of 0 is the integral from 0 to 0, so it\u0027s 0."},{"Start":"09:27.049 ","End":"09:29.720","Text":"Then what do we do if we have a 0 over 0 limit?"},{"Start":"09:29.720 ","End":"09:32.075","Text":"Usually, we try L\u0027Hopital\u0027s rule."},{"Start":"09:32.075 ","End":"09:33.200","Text":"it doesn\u0027t always work,"},{"Start":"09:33.200 ","End":"09:35.545","Text":"but at least we can try it."},{"Start":"09:35.545 ","End":"09:38.630","Text":"L is equal to the limit of what we get when we"},{"Start":"09:38.630 ","End":"09:43.760","Text":"differentiate the numerator and differentiate the denominator,"},{"Start":"09:43.760 ","End":"09:46.980","Text":"but all this is on the condition that this limit exists."},{"Start":"09:46.980 ","End":"09:48.950","Text":"Let\u0027s evaluate it."},{"Start":"09:48.950 ","End":"09:55.370","Text":"Big F prime is little f and the derivative of x cubed is 3x squared."},{"Start":"09:55.370 ","End":"09:57.455","Text":"Now f of x,"},{"Start":"09:57.455 ","End":"09:58.720","Text":"we have what it is,"},{"Start":"09:58.720 ","End":"10:00.010","Text":"it\u0027s equal to this."},{"Start":"10:00.010 ","End":"10:01.660","Text":"Plug that here."},{"Start":"10:01.660 ","End":"10:06.115","Text":"Then x squared cancels top and bottom."},{"Start":"10:06.115 ","End":"10:13.545","Text":"We got 1 over 3 and then the 1 plus x^4 push into the denominator."},{"Start":"10:13.545 ","End":"10:19.900","Text":"At this point, we can substitute x equals 0 and we get 1 over 3 times 1,"},{"Start":"10:19.900 ","End":"10:21.505","Text":"and it\u0027s just 1/3."},{"Start":"10:21.505 ","End":"10:26.900","Text":"That concludes this exercise and this clip."}],"ID":24138},{"Watched":false,"Name":"Second Fundamental Theorem","Duration":"8m 11s","ChapterTopicVideoID":23292,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23292.jpeg","UploadDate":"2021-01-10T12:22:10.6170000","DurationForVideoObject":"PT8M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"Continuing with the fundamental theorems of calculus,"},{"Start":"00:03.750 ","End":"00:05.040","Text":"well, there\u0027s 2 of them."},{"Start":"00:05.040 ","End":"00:06.885","Text":"We just did the first 1."},{"Start":"00:06.885 ","End":"00:09.030","Text":"Now the second 1."},{"Start":"00:09.030 ","End":"00:14.925","Text":"What it says is as follows that f be integrable on the closed interval a, b."},{"Start":"00:14.925 ","End":"00:19.110","Text":"Suppose that there is a differentiable function big F,"},{"Start":"00:19.110 ","End":"00:26.640","Text":"such that the derivative of big F is little f. Then the claim is that the integral"},{"Start":"00:26.640 ","End":"00:35.400","Text":"from a to b of f is just big F at b minus big F at a."},{"Start":"00:35.400 ","End":"00:38.960","Text":"By the way, big F is an indefinite integral"},{"Start":"00:38.960 ","End":"00:42.680","Text":"of little f and antiderivative is an indefinite integral."},{"Start":"00:42.680 ","End":"00:46.205","Text":"This shows us the relationship between definite and indefinite."},{"Start":"00:46.205 ","End":"00:48.650","Text":"To compute the definite integral,"},{"Start":"00:48.650 ","End":"00:54.635","Text":"we take the indefinite integral evaluated at the endpoints and subtract."},{"Start":"00:54.635 ","End":"00:57.205","Text":"Now let\u0027s get to the proof."},{"Start":"00:57.205 ","End":"00:59.270","Text":"To show that these 2 are equal,"},{"Start":"00:59.270 ","End":"01:02.960","Text":"it\u0027s sufficient to show that the difference between the left side and"},{"Start":"01:02.960 ","End":"01:08.590","Text":"the right side is less than Epsilon and absolute value for any positive Epsilon,"},{"Start":"01:08.590 ","End":"01:10.850","Text":"and then these 2 must be equal."},{"Start":"01:10.850 ","End":"01:13.485","Text":"We\u0027ll use Riemann\u0027s criterion here."},{"Start":"01:13.485 ","End":"01:16.750","Text":"There\u0027s the partition P of the interval a,"},{"Start":"01:16.750 ","End":"01:23.885","Text":"b such that the upper sum for P and f minus the lower sum is less than Epsilon."},{"Start":"01:23.885 ","End":"01:27.570","Text":"I should say, let Epsilon bigger than 0 be given."},{"Start":"01:27.570 ","End":"01:29.130","Text":"That\u0027s our Epsilon."},{"Start":"01:29.130 ","End":"01:33.105","Text":"We find P for this Epsilon by Riemann\u0027s criterion."},{"Start":"01:33.105 ","End":"01:36.405","Text":"Let\u0027s say that our P is x naught to x_n."},{"Start":"01:36.405 ","End":"01:40.480","Text":"The first 1 is a, and the last 1 is b. I will apply"},{"Start":"01:40.480 ","End":"01:47.200","Text":"the mean value theorem for big F on each of these x_i minus 1_xi."},{"Start":"01:47.200 ","End":"01:52.460","Text":"That means there exists a point c_i in this interval,"},{"Start":"01:52.460 ","End":"01:56.185","Text":"actually, in the open interval anyway,"},{"Start":"01:56.185 ","End":"02:02.260","Text":"such that the value of big F at the right endpoint minus the value of f at"},{"Start":"02:02.260 ","End":"02:05.110","Text":"the left endpoint is the derivative at"},{"Start":"02:05.110 ","End":"02:09.160","Text":"this inner point c_i times the width of the interval."},{"Start":"02:09.160 ","End":"02:14.845","Text":"The derivative of big F is little f. This just comes out to be f of c_i times Delta x_i."},{"Start":"02:14.845 ","End":"02:17.620","Text":"Definition of Delta x_i is this."},{"Start":"02:17.620 ","End":"02:20.320","Text":"Now f of c_i,"},{"Start":"02:20.320 ","End":"02:24.145","Text":"it\u0027s got to be between the infimum and the supremum."},{"Start":"02:24.145 ","End":"02:26.500","Text":"This is the supremum of the values of f,"},{"Start":"02:26.500 ","End":"02:29.890","Text":"is the infimum, so any particular 1 is between the 2."},{"Start":"02:29.890 ","End":"02:34.265","Text":"If we sum from i equals 1 to n,"},{"Start":"02:34.265 ","End":"02:37.635","Text":"then we get Delta x_i is here,"},{"Start":"02:37.635 ","End":"02:38.865","Text":"here, and here."},{"Start":"02:38.865 ","End":"02:43.385","Text":"We have f of c_i is between m_i and big M_i."},{"Start":"02:43.385 ","End":"02:46.030","Text":"We\u0027re just summing and multiplying by Delta x_i."},{"Start":"02:46.030 ","End":"02:50.560","Text":"Now, this bit here is L of P and f,"},{"Start":"02:50.560 ","End":"02:53.240","Text":"the lower sum for the partition P,"},{"Start":"02:53.240 ","End":"02:59.090","Text":"and the other 1 that\u0027s in this color is the upper sum for P and f."},{"Start":"02:59.090 ","End":"03:06.060","Text":"Each of these terms"},{"Start":"03:06.060 ","End":"03:10.425","Text":"is equal to these cause using this formula here."},{"Start":"03:10.425 ","End":"03:12.290","Text":"This, if you think about it,"},{"Start":"03:12.290 ","End":"03:14.990","Text":"each term is minus, then plus."},{"Start":"03:14.990 ","End":"03:17.075","Text":"It\u0027s F of x_1,"},{"Start":"03:17.075 ","End":"03:18.635","Text":"minus F of x_0,"},{"Start":"03:18.635 ","End":"03:21.685","Text":"and F of x_2 minus F of x_1."},{"Start":"03:21.685 ","End":"03:28.150","Text":"Each term appears once plus and once minus except for the last term,"},{"Start":"03:28.150 ","End":"03:31.420","Text":"which appears only in plus and the first term only in minus."},{"Start":"03:31.420 ","End":"03:36.480","Text":"What we have, we\u0027ve evaluated each of these 3 colors."},{"Start":"03:36.480 ","End":"03:38.340","Text":"We have that this L of P,"},{"Start":"03:38.340 ","End":"03:40.365","Text":"f is the least,"},{"Start":"03:40.365 ","End":"03:42.450","Text":"this is the most, and in-between,"},{"Start":"03:42.450 ","End":"03:44.815","Text":"we have F of b minus F of a."},{"Start":"03:44.815 ","End":"03:48.230","Text":"We also have another triple inequality,"},{"Start":"03:48.230 ","End":"03:52.300","Text":"is that the integral of f between a and b"},{"Start":"03:52.300 ","End":"03:56.395","Text":"is sandwiched between the lower sum and the upper sum."},{"Start":"03:56.395 ","End":"03:59.410","Text":"Both of these middle things are both"},{"Start":"03:59.410 ","End":"04:04.090","Text":"sandwiched in the same 2 bits of bread in this metaphor."},{"Start":"04:04.090 ","End":"04:11.930","Text":"This means that this minus this and absolute value is less than or equal to,"},{"Start":"04:11.930 ","End":"04:14.824","Text":"maybe it\u0027s not immediately obvious, I should explain."},{"Start":"04:14.824 ","End":"04:18.050","Text":"The reason that we get this is if you have"},{"Start":"04:18.050 ","End":"04:23.215","Text":"2 numbers that are both sandwiched between the same pair a and b,"},{"Start":"04:23.215 ","End":"04:29.330","Text":"then the distance between these 2 and absolute value is at most b minus a."},{"Start":"04:29.330 ","End":"04:31.850","Text":"If we apply that to this and this,"},{"Start":"04:31.850 ","End":"04:36.290","Text":"we get that this minus this is less than or equal to this,"},{"Start":"04:36.290 ","End":"04:38.915","Text":"and this already we said is less than Epsilon."},{"Start":"04:38.915 ","End":"04:40.430","Text":"That completes the proof,"},{"Start":"04:40.430 ","End":"04:42.845","Text":"and next we\u0027ll do an example problem."},{"Start":"04:42.845 ","End":"04:47.300","Text":"The problem is, provide a simpler proof of"},{"Start":"04:47.300 ","End":"04:54.820","Text":"the second fundamental theorem of the calculus if we\u0027re given that f is continuous."},{"Start":"04:54.820 ","End":"04:59.645","Text":"We\u0027re given a hint to use the first fundamental theorem of the calculus."},{"Start":"04:59.645 ","End":"05:02.000","Text":"Perhaps this is not clear what it means,"},{"Start":"05:02.000 ","End":"05:06.650","Text":"so I\u0027ll remind you what the second fundamental theorem of the calculus says."},{"Start":"05:06.650 ","End":"05:08.800","Text":"We just heard it a moment ago."},{"Start":"05:08.800 ","End":"05:12.575","Text":"It talks about a function f that is integrable."},{"Start":"05:12.575 ","End":"05:15.125","Text":"The problem says that if, hypothetically,"},{"Start":"05:15.125 ","End":"05:17.840","Text":"instead of the word integrable here,"},{"Start":"05:17.840 ","End":"05:20.585","Text":"we put the word continuous,"},{"Start":"05:20.585 ","End":"05:23.800","Text":"then the proof would go much more simply."},{"Start":"05:23.800 ","End":"05:26.750","Text":"The hint is to use the first fundamental theorem,"},{"Start":"05:26.750 ","End":"05:28.865","Text":"so I also brought that here."},{"Start":"05:28.865 ","End":"05:37.735","Text":"Now the solution, define a function G of x to be the integral from a to x of f of t, dt."},{"Start":"05:37.735 ","End":"05:41.929","Text":"I should have said that x is in the interval a, b,"},{"Start":"05:41.929 ","End":"05:44.615","Text":"just like in the case of big F. Now,"},{"Start":"05:44.615 ","End":"05:49.205","Text":"f is continuous given that\u0027s what the setup is."},{"Start":"05:49.205 ","End":"05:54.590","Text":"By the second part of the first fundamental theorem,"},{"Start":"05:54.590 ","End":"05:57.710","Text":"the function G is differentiable on a,b."},{"Start":"05:57.710 ","End":"06:03.965","Text":"G is our particular case of the big F that\u0027s in the theorem,"},{"Start":"06:03.965 ","End":"06:06.210","Text":"and like it says here,"},{"Start":"06:06.210 ","End":"06:09.660","Text":"that big F prime is little f, in our case,"},{"Start":"06:09.660 ","End":"06:17.810","Text":"it\u0027s G prime is f. Let big F be a differentiable function such that f prime equals f,"},{"Start":"06:17.810 ","End":"06:19.370","Text":"as in the theorem."},{"Start":"06:19.370 ","End":"06:23.750","Text":"We have to show this is true because the theorem says that, if this,"},{"Start":"06:23.750 ","End":"06:27.020","Text":"then this, so we\u0027ll take this is the if part,"},{"Start":"06:27.020 ","End":"06:29.750","Text":"and we need to show this part."},{"Start":"06:29.750 ","End":"06:35.190","Text":"Note that F prime equals f. That\u0027s given here."},{"Start":"06:35.190 ","End":"06:38.940","Text":"Also, we know that f is G prime."},{"Start":"06:38.940 ","End":"06:40.395","Text":"That\u0027s from here."},{"Start":"06:40.395 ","End":"06:43.980","Text":"Big F prime and big G prime are the same,"},{"Start":"06:43.980 ","End":"06:46.470","Text":"and so F prime minus G prime,"},{"Start":"06:46.470 ","End":"06:50.700","Text":"which is the same as F minus G all prime by linearity of the derivative."},{"Start":"06:50.700 ","End":"06:55.670","Text":"This equals 0. If the derivative of a function is 0,"},{"Start":"06:55.670 ","End":"06:57.320","Text":"then it\u0027s a constant function,"},{"Start":"06:57.320 ","End":"07:04.670","Text":"meaning that there is a constant real number such that f minus G is a constant,"},{"Start":"07:04.670 ","End":"07:08.470","Text":"or in other words, F is G plus C. Now,"},{"Start":"07:08.470 ","End":"07:10.970","Text":"remember, we\u0027re trying to prove this line."},{"Start":"07:10.970 ","End":"07:13.025","Text":"I\u0027ll start from the right-hand side."},{"Start":"07:13.025 ","End":"07:16.800","Text":"F of b minus F of a is equal to,"},{"Start":"07:16.800 ","End":"07:18.750","Text":"from here F is G plus C,"},{"Start":"07:18.750 ","End":"07:27.845","Text":"so it\u0027s G of b plus C minus G of a plus C. The C cancels and this is G of b minus G of a."},{"Start":"07:27.845 ","End":"07:32.360","Text":"Now, by the definition of G, which is here,"},{"Start":"07:32.360 ","End":"07:38.180","Text":"this is equal to the integral from a to b minus the integral from a to a,"},{"Start":"07:38.180 ","End":"07:40.145","Text":"of f in each case."},{"Start":"07:40.145 ","End":"07:43.540","Text":"The integral from a to a is 0,"},{"Start":"07:43.540 ","End":"07:49.195","Text":"and therefore, we\u0027re just left with the integral from a to b of f of t, dt."},{"Start":"07:49.195 ","End":"07:52.320","Text":"Now, the letter t is not important."},{"Start":"07:52.320 ","End":"07:54.065","Text":"It\u0027s a dummy variable,"},{"Start":"07:54.065 ","End":"07:58.160","Text":"so we could just replace t by x."},{"Start":"07:58.160 ","End":"08:01.670","Text":"That would give us what we want to prove,"},{"Start":"08:01.670 ","End":"08:05.600","Text":"which is this equals this, and we\u0027ve done it."},{"Start":"08:05.600 ","End":"08:11.100","Text":"That concludes this example problem and the clip."}],"ID":24139},{"Watched":false,"Name":"Exercise 1","Duration":"4m 59s","ChapterTopicVideoID":23293,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23293.jpeg","UploadDate":"2021-01-10T12:22:51.5000000","DurationForVideoObject":"PT4M59S","Description":null,"VideoComments":[],"Subtitles":[],"ID":24140},{"Watched":false,"Name":"Exercise 2","Duration":"2m 33s","ChapterTopicVideoID":23294,"CourseChapterTopicPlaylistID":65398,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.com/Images/Videos_Thumbnails/23294.jpeg","UploadDate":"2021-01-10T12:23:11.2230000","DurationForVideoObject":"PT2M33S","Description":null,"VideoComments":[],"Subtitles":[],"ID":24141},{"Watched":false,"Name":"Exercise 3","Duration":"4m 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