WEBVTT 00:00:00.040 --> 00:00:02.460 The following content is provided under a Creative 00:00:02.460 --> 00:00:03.870 Commons license. 00:00:03.870 --> 00:00:06.910 Your support will help MIT OpenCourseWare continue to 00:00:06.910 --> 00:00:10.560 offer high quality educational resources for free. 00:00:10.560 --> 00:00:13.460 To make a donation or view additional materials from 00:00:13.460 --> 00:00:19.280 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:19.280 --> 00:00:20.530 ocw.mit.edu. 00:00:29.270 --> 00:00:30.030 PROFESSOR: Hi. 00:00:30.030 --> 00:00:33.330 Our lesson, today, hopefully will serve two purposes. 00:00:33.330 --> 00:00:36.270 On the one hand, we will give a nice application of the 00:00:36.270 --> 00:00:36.940 chain rule. 00:00:36.940 --> 00:00:40.970 Now that we've had two units devoted to working with the 00:00:40.970 --> 00:00:43.820 chain rule, I thought you might enjoy seeing how it's 00:00:43.820 --> 00:00:48.360 used in places which aren't quite that obvious-- 00:00:48.360 --> 00:00:50.520 places that we wouldn't expect to be used, at least. 00:00:50.520 --> 00:00:54.670 And secondly, I would like to pick as my application one 00:00:54.670 --> 00:00:58.520 which comes up in many, many different contexts. 00:00:58.520 --> 00:01:01.750 And without further ado, the topic I want to cover today is 00:01:01.750 --> 00:01:04.950 called "Integrals Involving Parameters". 00:01:04.950 --> 00:01:06.430 Now, that sounds like a big mouthful. 00:01:06.430 --> 00:01:10.000 Let me motivate that for you, first of all, physically, and 00:01:10.000 --> 00:01:12.440 then in term of a couple of geometric examples. 00:01:12.440 --> 00:01:15.500 You all know from past experience my great ability 00:01:15.500 --> 00:01:18.700 with physical applications, so I won't even try to find 00:01:18.700 --> 00:01:20.050 anything profound here. 00:01:20.050 --> 00:01:24.000 Let me just take a pseudo example, pointing out what 00:01:24.000 --> 00:01:27.570 type of situation we're trying to deal with and leaving it to 00:01:27.570 --> 00:01:30.500 your own backgrounds to see places where the same 00:01:30.500 --> 00:01:33.210 principle could have been applied, but hopefully in a 00:01:33.210 --> 00:01:35.400 more practical, meaningful way for you. 00:01:35.400 --> 00:01:39.390 Imagine for example, that we have a platform that we're 00:01:39.390 --> 00:01:41.830 looking along in the x direction. 00:01:41.830 --> 00:01:46.180 And we've punched holes in this platform, say, and liquid 00:01:46.180 --> 00:01:48.480 is trickling through these various holes. 00:01:48.480 --> 00:01:52.360 The holes are all on a horizontal line this way. 00:01:52.360 --> 00:01:54.870 What we're going to do is we're going to focus our 00:01:54.870 --> 00:01:58.020 attention on a particular particle of the liquid, and 00:01:58.020 --> 00:02:00.510 we're going to watch it as it falls. 00:02:00.510 --> 00:02:04.110 And what we would like to do is find how far that particle 00:02:04.110 --> 00:02:08.520 fell, say, during the first second of its flight. 00:02:08.520 --> 00:02:11.350 Obviously, from a calculus point of view, the first thing 00:02:11.350 --> 00:02:14.610 we have to do is know what the velocity function is, because 00:02:14.610 --> 00:02:17.880 ultimately, we would like to integrate the velocity. 00:02:17.880 --> 00:02:20.590 Now, notice that we would expect the velocity to be 00:02:20.590 --> 00:02:21.770 depending on time. 00:02:21.770 --> 00:02:25.360 If this were a freely falling situation, we'd expect the 00:02:25.360 --> 00:02:27.980 usual gravitational type situation. 00:02:27.980 --> 00:02:30.440 Or whatever the situation happened to be, we would 00:02:30.440 --> 00:02:32.780 expect on the one hand that the velocity does 00:02:32.780 --> 00:02:33.970 depend on of time. 00:02:33.970 --> 00:02:38.030 On the other hand, because of how the streams are flowing-- 00:02:38.030 --> 00:02:40.770 in other words, we don't know what's happening above here 00:02:40.770 --> 00:02:43.760 that's causing the water to shoot out-- we don't know what 00:02:43.760 --> 00:02:46.670 the initial velocity is coming out of each of these holes in 00:02:46.670 --> 00:02:50.420 the sense that a different opening may give rise to a 00:02:50.420 --> 00:02:53.010 different velocity of stream coming out. 00:02:53.010 --> 00:02:55.790 All I'm trying to bring out here is that as we try to 00:02:55.790 --> 00:03:00.470 focus our attention on a particular opening, we find 00:03:00.470 --> 00:03:02.950 that the velocity of the particle that will follow it 00:03:02.950 --> 00:03:08.620 during that first second is a function both of its position 00:03:08.620 --> 00:03:11.420 x-- in other words, x sub 0 in this case, because we're 00:03:11.420 --> 00:03:14.700 focusing at x equals x0 and the time t as t 00:03:14.700 --> 00:03:16.030 goes from 0 to 1. 00:03:16.030 --> 00:03:18.570 And we then simply integrate along the 00:03:18.570 --> 00:03:20.040 vertical direction here. 00:03:20.040 --> 00:03:26.960 We find y as a function of x0 from 0 to 1-- v(x0, t) dt. 00:03:26.960 --> 00:03:29.340 Once we're through integrating, you see, notice 00:03:29.340 --> 00:03:32.610 that the integration is with respect to t. 00:03:32.610 --> 00:03:35.400 So when we're through integrating, this being a 00:03:35.400 --> 00:03:37.450 definite integral, t no longer appears. 00:03:37.450 --> 00:03:41.690 We have a function of x0 alone, saying nothing more 00:03:41.690 --> 00:03:44.220 than the distance that the particle falls during the 00:03:44.220 --> 00:03:48.630 first second is a function of the position of the opening 00:03:48.630 --> 00:03:49.610 along the line here. 00:03:49.610 --> 00:03:52.800 Now, at any rate, the practical application is not 00:03:52.800 --> 00:03:56.160 so much writing down this equation as the 00:03:56.160 --> 00:03:57.690 inverse is the case. 00:03:57.690 --> 00:04:02.650 Namely, in many practical applications, we are given 00:04:02.650 --> 00:04:04.280 this particular integral. 00:04:04.280 --> 00:04:07.850 And for some reason or other, want to determine 00:04:07.850 --> 00:04:09.190 what v itself is. 00:04:09.190 --> 00:04:14.250 In other words, we often want to find the derivative given 00:04:14.250 --> 00:04:15.200 the integral. 00:04:15.200 --> 00:04:16.190 All right. 00:04:16.190 --> 00:04:18.649 Let's just let it go at that for the time being. 00:04:18.649 --> 00:04:22.000 The important point is that I want you to see an example of 00:04:22.000 --> 00:04:22.530 an integral. 00:04:22.530 --> 00:04:25.530 Let me just write this here in more abstract form. 00:04:25.530 --> 00:04:28.680 It appears to be a definite integral, a to b. 00:04:28.680 --> 00:04:32.290 The function inside the integrand is apparently a 00:04:32.290 --> 00:04:35.850 function of two variables, x and y-- 00:04:35.850 --> 00:04:37.650 say, in this case x0 and t. 00:04:37.650 --> 00:04:40.730 One of the variables is a variable of integration-- 00:04:40.730 --> 00:04:41.960 in this case, y-- 00:04:41.960 --> 00:04:44.640 and the other variable is being treated as a constant. 00:04:44.640 --> 00:04:47.660 And that's where the word parameter comes in. x is a 00:04:47.660 --> 00:04:51.600 parameter, meaning a variable constant, in the sense that, 00:04:51.600 --> 00:04:55.930 for this particular problem, x is chosen to be in some domain 00:04:55.930 --> 00:04:57.040 and remains fixed. 00:04:57.040 --> 00:05:01.040 In other words, this is some function of x when we're all 00:05:01.040 --> 00:05:01.970 through here. 00:05:01.970 --> 00:05:02.640 All right? 00:05:02.640 --> 00:05:04.930 As a geometric example, imagine 00:05:04.930 --> 00:05:06.300 the following situation. 00:05:06.300 --> 00:05:09.030 We have a surface w = f(x,y). 00:05:09.030 --> 00:05:14.710 We take the plane x = x0 and intersect this surface with 00:05:14.710 --> 00:05:16.150 that particular plane. 00:05:16.150 --> 00:05:17.770 We get a curve, you see. 00:05:17.770 --> 00:05:22.450 Now, we look at that curve corresponding to two points, p 00:05:22.450 --> 00:05:25.990 and q, where p and q are determined by the y 00:05:25.990 --> 00:05:27.640 values a and b. 00:05:27.640 --> 00:05:30.990 In other words, p corresponds to y = a. 00:05:30.990 --> 00:05:33.100 q corresponds to y = b. 00:05:33.100 --> 00:05:35.800 And now, a very natural question that might come up is 00:05:35.800 --> 00:05:39.380 that we would like to find the area of this particular plane 00:05:39.380 --> 00:05:42.520 region, in other words, the area of this slice 00:05:42.520 --> 00:05:43.985 between p and q. 00:05:43.985 --> 00:05:46.830 Now, you know, the first thing I hope that you'll notice is 00:05:46.830 --> 00:05:49.950 that because this shape of the surface can be in many 00:05:49.950 --> 00:05:53.490 different ways, the particular cross section that we get does 00:05:53.490 --> 00:05:55.690 depend on the choice of x0. 00:05:55.690 --> 00:05:57.320 Different slices-- 00:05:57.320 --> 00:06:00.630 different planes x = x0 will give us different curves of 00:06:00.630 --> 00:06:01.520 intersection. 00:06:01.520 --> 00:06:04.930 The point is that once we have the curve intersection x is 00:06:04.930 --> 00:06:06.730 being treated as the parameter. 00:06:06.730 --> 00:06:10.130 This particular curve is given by what equation? 00:06:10.130 --> 00:06:13.780 f is a function of x0 and y. 00:06:13.780 --> 00:06:18.730 See, x = x0 for every point on this curve. 00:06:18.730 --> 00:06:22.090 And so the area of the region R is the integral from a to b, 00:06:22.090 --> 00:06:24.420 f(x0, y) dy. 00:06:24.420 --> 00:06:26.950 And the question that very often comes up is, how do you 00:06:26.950 --> 00:06:31.180 find the derivative of A sub R with respect to x0, noticing, 00:06:31.180 --> 00:06:34.940 you see, that A sub R is a function of x0 alone, the y 00:06:34.940 --> 00:06:38.070 dropping out between the limits a and b when we perform 00:06:38.070 --> 00:06:40.270 the operation of integration. 00:06:40.270 --> 00:06:43.890 A third place that this type of situation occurs is in 00:06:43.890 --> 00:06:46.100 solving certain differential equations. 00:06:46.100 --> 00:06:49.220 For example, suppose we're given a particular curve and 00:06:49.220 --> 00:06:54.460 that that curve determines a region R between the lines x = 00:06:54.460 --> 00:06:56.820 a and x = b. 00:06:56.820 --> 00:06:58.830 Suppose all we know about the curve is its 00:06:58.830 --> 00:07:00.170 slope at any point. 00:07:00.170 --> 00:07:03.330 We know that its slope at any point is given by dy/dx and 00:07:03.330 --> 00:07:06.500 some function of x and y which we don't necessarily have to 00:07:06.500 --> 00:07:07.550 go into right now. 00:07:07.550 --> 00:07:09.470 And let's suppose, for the sake of argument, that we 00:07:09.470 --> 00:07:12.060 solve this first order differential equation. 00:07:12.060 --> 00:07:14.840 What we'll find, if we're lucky, is that y is some 00:07:14.840 --> 00:07:17.900 function of x and an arbitrary constant. 00:07:17.900 --> 00:07:20.090 Remember, once you have one solution to a differential 00:07:20.090 --> 00:07:23.760 equation, you have an infinite family in terms of a one 00:07:23.760 --> 00:07:27.370 parameter solution to a differential equation. 00:07:27.370 --> 00:07:30.850 In other words, in finding the area of the region R in this 00:07:30.850 --> 00:07:34.690 case, there are many curves that satisfy this particular 00:07:34.690 --> 00:07:35.770 differential equation. 00:07:35.770 --> 00:07:38.820 Until we know what specific points are being referred to 00:07:38.820 --> 00:07:41.510 over here, the best we know for sure is what? 00:07:41.510 --> 00:07:45.420 That these endpoints are a and b, that the integrand is 00:07:45.420 --> 00:07:49.650 f(x,c), and we're integrating that with respect 00:07:49.650 --> 00:07:52.130 to x from a to b. 00:07:52.130 --> 00:07:55.630 I freudianly put a c in here, because I think what I was 00:07:55.630 --> 00:07:58.420 trying to emphasize for you is that when you look at this 00:07:58.420 --> 00:08:01.320 thing, observe that this integral is a 00:08:01.320 --> 00:08:02.870 function of c alone. 00:08:02.870 --> 00:08:05.490 Namely, when you integrate this thing, you integrate it 00:08:05.490 --> 00:08:06.280 with respect to x. 00:08:06.280 --> 00:08:07.120 The x drops out. 00:08:07.120 --> 00:08:08.740 All you have left here is a c. 00:08:08.740 --> 00:08:11.570 A sub R, then, is a function of c. 00:08:11.570 --> 00:08:14.090 And in many cases, what we would like to do is see how 00:08:14.090 --> 00:08:17.320 fast the area changes as a function of c. 00:08:17.320 --> 00:08:20.700 In other words, how do we change the area as a function 00:08:20.700 --> 00:08:23.740 of changing the arbitrary constant c? 00:08:23.740 --> 00:08:27.410 And I have enough exercises in the assignment to give you 00:08:27.410 --> 00:08:28.950 concrete drill on this. 00:08:28.950 --> 00:08:31.910 All I'm trying to give you here is an overview of the 00:08:31.910 --> 00:08:32.960 entire topic. 00:08:32.960 --> 00:08:35.350 And the reason I want to give you this overview is that it's 00:08:35.350 --> 00:08:37.929 hinted at in the textbook, but this topic 00:08:37.929 --> 00:08:38.940 is not covered there. 00:08:38.940 --> 00:08:41.870 In fact, the reason that I had you read that particular 00:08:41.870 --> 00:08:44.920 section of the textbook before the lecture this time-- you 00:08:44.920 --> 00:08:46.700 notice that usually we start with the lecture. 00:08:46.700 --> 00:08:50.640 This time I had you read the textbook first, because the 00:08:50.640 --> 00:08:54.080 way the textbook covered this topic is essentially nothing 00:08:54.080 --> 00:08:56.430 more than the way we tackled a different 00:08:56.430 --> 00:08:58.040 problem last semester. 00:08:58.040 --> 00:09:00.910 And I want you to see that the problem done in the Thomas 00:09:00.910 --> 00:09:03.620 text is not a new problem-- 00:09:03.620 --> 00:09:05.380 it's one that we've done before-- 00:09:05.380 --> 00:09:08.500 but that with the tools that we now have available, we 00:09:08.500 --> 00:09:10.740 could've tackled a more significant problem. 00:09:10.740 --> 00:09:13.770 And that's the one I'm electing to do here and what I 00:09:13.770 --> 00:09:15.890 want to show you the key steps on, because 00:09:15.890 --> 00:09:17.140 they're not in the text. 00:09:17.140 --> 00:09:17.900 But I will leave the 00:09:17.900 --> 00:09:20.230 reinforcement for the exercise. 00:09:20.230 --> 00:09:23.790 At any rate, hopefully now, when you see an integral of 00:09:23.790 --> 00:09:26.300 this type, it will not bother you too much. 00:09:26.300 --> 00:09:28.320 In other words don't worry about how do you integrate a 00:09:28.320 --> 00:09:30.060 function of two independent variables? 00:09:30.060 --> 00:09:33.700 When you see something like this, it means that there is 00:09:33.700 --> 00:09:37.930 some implicitly implied domain for x. 00:09:37.930 --> 00:09:40.800 In other words, we have some function of x. 00:09:40.800 --> 00:09:46.010 Let's say the domain of g might very well be, say, all 00:09:46.010 --> 00:09:49.380 x's between two values, say c and d. 00:09:49.380 --> 00:09:50.840 But who cares about that right now? 00:09:50.840 --> 00:09:53.390 The important point is that g is defined on a certain 00:09:53.390 --> 00:09:54.410 set of values x. 00:09:54.410 --> 00:09:58.870 And what it says is to compute the output of the g machine. 00:09:58.870 --> 00:10:04.470 For the given x, you fix that x and integrate f(x,y) dy 00:10:04.470 --> 00:10:05.880 between a and b. 00:10:05.880 --> 00:10:09.020 Notice, you see, during the integration, x is being 00:10:09.020 --> 00:10:12.430 treated as a constant, so that for all intents and purposes, 00:10:12.430 --> 00:10:14.680 this is an ordinary integral. 00:10:14.680 --> 00:10:17.780 But because x isn't a bona fide constant, meaning what? 00:10:17.780 --> 00:10:21.080 It's a constant only in the sense that once chosen, it 00:10:21.080 --> 00:10:24.180 remains fixed for this particular integration. 00:10:24.180 --> 00:10:26.540 Different values of x will give me 00:10:26.540 --> 00:10:28.030 different integrals here. 00:10:28.030 --> 00:10:30.910 And consequently a very natural question that comes up 00:10:30.910 --> 00:10:34.710 is how does my function g-- which depends on x-- 00:10:34.710 --> 00:10:38.130 how does that vary as x varies? 00:10:38.130 --> 00:10:40.160 In other words, the key question is simply this. 00:10:40.160 --> 00:10:44.060 First of all, given g defined this way, one, does g prime 00:10:44.060 --> 00:10:45.120 even exist? 00:10:45.120 --> 00:10:47.260 Does dg/dx exist? 00:10:47.260 --> 00:10:52.620 And two, if it does exist, what is it? 00:10:52.620 --> 00:10:54.840 In other words, the question that we're raising is if we 00:10:54.840 --> 00:10:58.570 can find g prime of x, how do we do it in terms of looking 00:10:58.570 --> 00:11:00.440 at the right hand side? 00:11:00.440 --> 00:11:02.900 And let me not try to guess the answer here. 00:11:02.900 --> 00:11:05.940 The answer does turn out to be, in this particular case, 00:11:05.940 --> 00:11:07.470 one that you might have guessed. 00:11:07.470 --> 00:11:09.890 But I prefer to show you that we don't have to 00:11:09.890 --> 00:11:11.500 guess, point one. 00:11:11.500 --> 00:11:13.620 Point two, if you do guess, you won't 00:11:13.620 --> 00:11:14.610 always be that lucky. 00:11:14.610 --> 00:11:16.680 That's my finale for today's lecture. 00:11:16.680 --> 00:11:18.760 But let me see if I can survive to get 00:11:18.760 --> 00:11:20.140 to the finale first. 00:11:20.140 --> 00:11:22.510 Let me see how we'll tackle a problem like this. 00:11:22.510 --> 00:11:27.220 First of all, to see if g prime exists at some value x0, 00:11:27.220 --> 00:11:29.140 what we have to do is-- 00:11:29.140 --> 00:11:31.520 way back to the very beginning of Part 1-- 00:11:31.520 --> 00:11:34.930 the same old definition for an ordinary derivative. 00:11:34.930 --> 00:11:39.200 We have to compute the limit of g of x sub 0 plus h minus 00:11:39.200 --> 00:11:43.880 g(x sub 0) over h, taking the limit as h approaches 0. 00:11:43.880 --> 00:11:45.960 Notice what the g machine does... 00:11:45.960 --> 00:11:50.230 What the g machine does is it feeds x into the integrand 00:11:50.230 --> 00:11:54.960 here and integrates this with respect to y from a to b. 00:11:54.960 --> 00:11:59.670 So if the input of my g machine is x0 plus h, that 00:11:59.670 --> 00:12:04.360 means that the x is replaced by x0 plus h here. g of x0 00:12:04.360 --> 00:12:07.210 plus h is simply integral from a to b f of x0 00:12:07.210 --> 00:12:09.140 plus h comma y dy. 00:12:09.140 --> 00:12:13.700 Similarly, g(x0) is integral from a to b f(x0, y) dy. 00:12:13.700 --> 00:12:16.080 I now want to form this difference. 00:12:16.080 --> 00:12:17.190 And noticing-- 00:12:17.190 --> 00:12:19.820 again, this is all calculus of a single variable-- 00:12:19.820 --> 00:12:23.470 that the difference of two definite integrals is the 00:12:23.470 --> 00:12:26.670 definite integral of the difference, I can conclude 00:12:26.670 --> 00:12:31.320 that g(x0 + h) minus g(x0) is simply this single 00:12:31.320 --> 00:12:33.990 integral over here. 00:12:33.990 --> 00:12:37.060 Now, my next step in determining g prime of x0 is I 00:12:37.060 --> 00:12:39.030 must divide this by h. 00:12:39.030 --> 00:12:43.990 Notice, by the way, that h is an arbitrary increment, but 00:12:43.990 --> 00:12:46.240 once chosen, remains fixed. 00:12:46.240 --> 00:12:49.210 Notice that h is a constant as far as this 00:12:49.210 --> 00:12:50.880 integration is concerned. 00:12:50.880 --> 00:12:56.050 Consequently, to divide by h, it is permissible to bring the 00:12:56.050 --> 00:12:57.990 h inside the integrand. 00:12:57.990 --> 00:13:01.250 In other words, technically speaking, the h should be 00:13:01.250 --> 00:13:04.750 here, but since h is a constant with respect to y, 00:13:04.750 --> 00:13:08.000 the integral can have the h brought in. 00:13:08.000 --> 00:13:10.080 Why do I want to bring the h in here? 00:13:10.080 --> 00:13:12.910 Let me again telegraph what I'm leading up to. 00:13:12.910 --> 00:13:16.310 Obviously, when I'm going to compute g prime, my next step 00:13:16.310 --> 00:13:19.000 is to take the limit of this as h approaches 0. 00:13:19.000 --> 00:13:23.230 With h in here, I look at this and what I hope is that by 00:13:23.230 --> 00:13:26.960 this amount of time at least the following minimum amount 00:13:26.960 --> 00:13:29.960 of material has rubbed off on you in a second nature way-- 00:13:29.960 --> 00:13:34.050 that if I look at this expression in brackets as h 00:13:34.050 --> 00:13:38.250 approaches 0, this is precisely the definition of 00:13:38.250 --> 00:13:45.110 what we mean by the partial of f(x, y) with respect to x 00:13:45.110 --> 00:13:46.600 evaluated at (x0, y). 00:13:46.600 --> 00:13:47.460 See, this is what? 00:13:47.460 --> 00:13:50.500 The change in f-- see, y is held constant. 00:13:50.500 --> 00:13:54.110 We're taking this over to change from x0 to x0 plus h 00:13:54.110 --> 00:13:55.150 and dividing by h. 00:13:55.150 --> 00:13:57.760 This is a partial of f with respect to x. 00:13:57.760 --> 00:13:59.900 That's why I want to bring the h inside. 00:13:59.900 --> 00:14:02.780 So now, I say, OK, g prime of x0, by 00:14:02.780 --> 00:14:04.540 definition, is this limit. 00:14:04.540 --> 00:14:07.255 I now want to take the limit of this expression. 00:14:10.770 --> 00:14:14.010 And by the way, notice what I'd love to do now is to jump 00:14:14.010 --> 00:14:17.885 right in here and say, aha, this is just the partial of f 00:14:17.885 --> 00:14:21.310 with respect to x evaluated at (x0, y0). 00:14:21.310 --> 00:14:23.510 But the thing I would like you to notice-- 00:14:23.510 --> 00:14:26.700 and again, going back to Part 1 of course, one of the big 00:14:26.700 --> 00:14:29.420 things that we talked about under the heading of uniform 00:14:29.420 --> 00:14:30.310 convergence. 00:14:30.310 --> 00:14:33.060 There is a very dangerous thing in general to 00:14:33.060 --> 00:14:36.190 interchange the order of limit and integration. 00:14:36.190 --> 00:14:37.110 This says what? 00:14:37.110 --> 00:14:40.580 First perform the integration, and then take the limit. 00:14:40.580 --> 00:14:44.860 What we would like to be able to do is first take the limit 00:14:44.860 --> 00:14:47.030 and then integrate the result. 00:14:47.030 --> 00:14:51.500 Now, we did see that, provided the integrand was continuous, 00:14:51.500 --> 00:14:53.330 these operations were permissible. 00:14:53.330 --> 00:14:55.150 But we'll talk about that a little later. 00:14:55.150 --> 00:14:59.120 For the time being, let's simply summarize by saying if 00:14:59.120 --> 00:15:02.660 the limit operation and the integration operation can be 00:15:02.660 --> 00:15:06.720 interchanged, then the derivative-- 00:15:06.720 --> 00:15:07.630 see, this thing here is what? 00:15:07.630 --> 00:15:09.500 This is my g(x). 00:15:09.500 --> 00:15:12.640 The derivative of g(x) with respect to x has the very 00:15:12.640 --> 00:15:16.080 delightful form that, essentially, all I have to do 00:15:16.080 --> 00:15:19.340 is take the derivative operation, come inside the 00:15:19.340 --> 00:15:23.500 integrand, and replace the derivative with respect to x 00:15:23.500 --> 00:15:26.380 by the partial derivative with respect to x. 00:15:26.380 --> 00:15:29.220 In other words, the derivative of the integral from a to b, 00:15:29.220 --> 00:15:33.890 f(x, y) dy is the integral from a to b, the partial of f 00:15:33.890 --> 00:15:36.250 with respect to x dy-- 00:15:36.250 --> 00:15:39.480 provided, of course, that the limit and the integration are 00:15:39.480 --> 00:15:40.510 interchangeable. 00:15:40.510 --> 00:15:44.820 In particular, this will be true if f and f sub x exist 00:15:44.820 --> 00:15:48.590 and are continuous and see, straighten out the range and 00:15:48.590 --> 00:15:50.830 the domain and what have you once and for all. 00:15:50.830 --> 00:15:53.860 Notice that y is allowed to exist between a and b. 00:15:53.860 --> 00:15:56.690 We've said that x is going to exist on some domain 00:15:56.690 --> 00:15:58.040 between c and d. 00:15:58.040 --> 00:16:01.090 Notice that saying that y is between a and b and x is 00:16:01.090 --> 00:16:04.410 between c and d geometrically says that f is 00:16:04.410 --> 00:16:06.570 defined on a rectangle. 00:16:06.570 --> 00:16:07.320 See? 00:16:07.320 --> 00:16:07.970 OK. 00:16:07.970 --> 00:16:11.270 And then what we're saying is, under these conditions, to 00:16:11.270 --> 00:16:15.620 integrate an integral with a parameter, with respect to 00:16:15.620 --> 00:16:19.360 that parameter, all we have to do is come inside the integral 00:16:19.360 --> 00:16:21.980 sign and differentiate-- 00:16:21.980 --> 00:16:23.220 take the partial derivative-- 00:16:23.220 --> 00:16:26.190 with respect to what is being used as the parameter. 00:16:26.190 --> 00:16:28.540 In this case, it's x, which is the parameter. 00:16:28.540 --> 00:16:30.880 Now, the only danger with this particular thing-- and by the 00:16:30.880 --> 00:16:33.040 way, notice, not only is there a danger here that I'm going 00:16:33.040 --> 00:16:33.380 to mention. 00:16:33.380 --> 00:16:36.260 The danger is this looks so easy, you may be saying, why 00:16:36.260 --> 00:16:37.250 did he do it the hard way? 00:16:37.250 --> 00:16:38.840 Why didn't he just tell us this was the right 00:16:38.840 --> 00:16:40.050 way of doing it? 00:16:40.050 --> 00:16:41.840 And the point is it just happens to be one of those 00:16:41.840 --> 00:16:46.460 coincidences where the rigorous way yields a logical 00:16:46.460 --> 00:16:49.240 answer which is consistent with what is probably our 00:16:49.240 --> 00:16:51.230 intuitive guess. 00:16:51.230 --> 00:16:52.940 But it's not always going to happen that way. 00:16:52.940 --> 00:16:55.545 And the example that I have in mind now goes back to what we 00:16:55.545 --> 00:16:57.180 were talking about the beginning of the lecture. 00:16:57.180 --> 00:16:59.300 Namely, I wanted to give you an application 00:16:59.300 --> 00:17:00.360 of the chain rule. 00:17:00.360 --> 00:17:02.810 And here's where that application comes in. 00:17:02.810 --> 00:17:04.319 I now call this-- 00:17:04.319 --> 00:17:05.130 I don't know what to call it. 00:17:05.130 --> 00:17:07.540 So let's just call it variable limits of integration. 00:17:07.540 --> 00:17:09.480 Same problem as before-- 00:17:09.480 --> 00:17:12.540 it's going to cause the chain rule to come in now. 00:17:12.540 --> 00:17:14.490 The only difference is going to be-- 00:17:14.490 --> 00:17:15.680 and that's just a forewarning. 00:17:15.680 --> 00:17:17.650 You don't have to know that right now. 00:17:17.650 --> 00:17:19.240 I'm going to have the same problem as before. 00:17:19.240 --> 00:17:21.130 What did I have before? 00:17:21.130 --> 00:17:25.280 I had that g(x) was integral f(x, y) dy between the two 00:17:25.280 --> 00:17:27.050 constants a and b. 00:17:27.050 --> 00:17:30.760 Now, I'm going to let my constants of integration also 00:17:30.760 --> 00:17:32.240 depend on the parameter. 00:17:32.240 --> 00:17:35.300 See, all the constants of integration have to be our 00:17:35.300 --> 00:17:39.540 constants as far as y is concerned. 00:17:39.540 --> 00:17:42.080 What I'm saying is what makes this problem differ from the 00:17:42.080 --> 00:17:45.030 previous one is suppose that it happens that instead of 00:17:45.030 --> 00:17:47.950 being given a nice rectangle to play around with, I'm given 00:17:47.950 --> 00:17:50.520 a couple of curves like this in the xy plane. 00:17:50.520 --> 00:17:53.360 See, this would be a of x. 00:17:53.360 --> 00:17:55.860 This would be y equals b(x). 00:17:55.860 --> 00:18:00.610 See, what I'm saying now is that not only does the 00:18:00.610 --> 00:18:04.220 integrand depend on what value of x I pick, but the limits of 00:18:04.220 --> 00:18:07.200 integration as I'm finding a cross-sectional area of a 00:18:07.200 --> 00:18:08.180 surface, you see. 00:18:08.180 --> 00:18:10.990 The limits of the integral themselves depend on the 00:18:10.990 --> 00:18:12.250 choice of x-- 00:18:12.250 --> 00:18:15.820 constant, as far as y is concerned, but depend on x. 00:18:15.820 --> 00:18:20.250 You see, now, what happens is that my parameter appears in 00:18:20.250 --> 00:18:23.670 the limits as well as just in the integrand. 00:18:23.670 --> 00:18:26.430 And now, you see, also what this means is if I try the 00:18:26.430 --> 00:18:30.510 previous approach of computing g(x plus delta x), et cetera, 00:18:30.510 --> 00:18:33.710 I'm in trouble, because the only way I can combine two 00:18:33.710 --> 00:18:36.930 integrals and put them under the same integral sign is if 00:18:36.930 --> 00:18:39.470 they're between the same limits of integration. 00:18:39.470 --> 00:18:43.460 Notice here, for example, that if I replace x by x plus delta 00:18:43.460 --> 00:18:47.080 x, I not only change the integrand, but notice that the 00:18:47.080 --> 00:18:48.590 limits become what? 00:18:48.590 --> 00:18:54.330 a of x plus delta x, b(x plus delta x)-- and those in 00:18:54.330 --> 00:18:56.680 general, unless a and b happen to be constants, 00:18:56.680 --> 00:18:58.420 will vary with x. 00:18:58.420 --> 00:19:00.180 In fact, let's look at it this way. 00:19:00.180 --> 00:19:04.000 This is the problem we should have started with in the sense 00:19:04.000 --> 00:19:06.840 that constant limits of integration are a 00:19:06.840 --> 00:19:08.640 special case of this. 00:19:08.640 --> 00:19:10.880 At any rate, what I wanted to show you was that this 00:19:10.880 --> 00:19:14.100 particular problem can be handled very nicely in terms 00:19:14.100 --> 00:19:15.390 of the chain rule. 00:19:15.390 --> 00:19:18.900 Namely, what we do here is we observe that, first of all, y 00:19:18.900 --> 00:19:20.510 is not really a variable here. 00:19:20.510 --> 00:19:22.310 It's integrated out. 00:19:22.310 --> 00:19:26.240 So what we think of is let's think of x as being some 00:19:26.240 --> 00:19:27.330 variable u. 00:19:27.330 --> 00:19:31.000 Let's think of b(x) as being some variable v. Let's write 00:19:31.000 --> 00:19:33.400 down the function of three independent 00:19:33.400 --> 00:19:35.590 variables u, v, and x. 00:19:35.590 --> 00:19:36.840 OK. 00:19:36.840 --> 00:19:39.240 What will that function be? 00:19:39.240 --> 00:19:43.100 Let u, v, and x be arbitrary, independent variables. 00:19:43.100 --> 00:19:46.460 Look at the integral from u to v f(x, y) dy. 00:19:46.460 --> 00:19:49.210 This is obviously dependent upon u. 00:19:49.210 --> 00:19:52.780 It's dependent upon v. And it's dependent upon x. 00:19:52.780 --> 00:19:55.340 The place that the chain rule comes in is that in our 00:19:55.340 --> 00:19:58.810 particular problem, u and v cannot be arbitrary, but 00:19:58.810 --> 00:20:01.900 rather u must be that particular function a(x), and 00:20:01.900 --> 00:20:05.030 v must be the particular function b(x). 00:20:05.030 --> 00:20:07.480 Consequently g(x) is simply what? 00:20:07.480 --> 00:20:12.720 It's h(u, v, x), where u is a(x) and v is b(x). 00:20:12.720 --> 00:20:16.450 Consequently, to find g prime of x, what we want is h prime 00:20:16.450 --> 00:20:19.240 of x and to find h prime as a function of x. 00:20:19.240 --> 00:20:20.410 See the chain rule here? 00:20:20.410 --> 00:20:22.540 u can be expressed in terms of x. 00:20:22.540 --> 00:20:24.430 v can be expressed in terms of x. 00:20:24.430 --> 00:20:27.180 Obviously, x is already expressed in terms of x. 00:20:27.180 --> 00:20:31.380 So this is really implicitly a function of x alone. 00:20:31.380 --> 00:20:34.300 So by the chain rule, what I'd like to be able to do is to 00:20:34.300 --> 00:20:36.530 combine these three pieces of information to 00:20:36.530 --> 00:20:38.410 find h prime of x. 00:20:38.410 --> 00:20:39.810 And remember how the chain rule works. 00:20:39.810 --> 00:20:41.320 Now, I'm not going to beat that to death. 00:20:41.320 --> 00:20:42.970 We've just had two units on that. 00:20:42.970 --> 00:20:45.290 Let's just say it rather quickly. 00:20:45.290 --> 00:20:49.780 g prime of x is the partial of h with respect to u times u 00:20:49.780 --> 00:20:53.140 prime of x plus the partial of h with respectively to v times 00:20:53.140 --> 00:20:57.030 v prime of x times the partial of h with respect to x times x 00:20:57.030 --> 00:20:58.790 prime of x, which, of course, is just 1. 00:20:58.790 --> 00:21:02.020 In other words, writing this thing out, g prime of x is 00:21:02.020 --> 00:21:04.730 simply this. 00:21:04.730 --> 00:21:08.870 What is the partial of h with respect to u? 00:21:08.870 --> 00:21:11.590 Let's come back here for a second and remember what h is. 00:21:11.590 --> 00:21:12.790 h is this integral. 00:21:12.790 --> 00:21:15.610 I want to take the partial of that with respect to u. 00:21:15.610 --> 00:21:18.950 That means I have to investigate this. 00:21:18.950 --> 00:21:20.450 Now, here's the interesting point. 00:21:20.450 --> 00:21:25.210 Whereas u, v, and x are independent variables, what 00:21:25.210 --> 00:21:27.280 does it mean when you say you're taking the partial with 00:21:27.280 --> 00:21:28.350 respect to u? 00:21:28.350 --> 00:21:33.650 It means that you're treating v and x as constants. 00:21:33.650 --> 00:21:37.370 Now, if v and x are being treated as constants, what I 00:21:37.370 --> 00:21:39.530 have is simply what? 00:21:39.530 --> 00:21:42.920 I'm taking a derivative with respect to a variable where 00:21:42.920 --> 00:21:47.100 the only place the variable appears is as the lower limit 00:21:47.100 --> 00:21:48.410 of the integrand. 00:21:48.410 --> 00:21:50.730 In other words, I claim that that's nothing more 00:21:50.730 --> 00:21:53.460 than minus f(x, u). 00:21:53.460 --> 00:21:54.570 Then I go inside the integrand. 00:21:54.570 --> 00:21:57.620 In other words, I differentiate the integral. 00:21:57.620 --> 00:21:59.530 That leaves me just the integrand-- 00:21:59.530 --> 00:22:05.350 and replace the variable by the variable of 00:22:05.350 --> 00:22:06.580 integration u here. 00:22:06.580 --> 00:22:08.250 And because it's the lower limit, I put 00:22:08.250 --> 00:22:08.960 in the minus sign. 00:22:08.960 --> 00:22:11.280 Now, why did I go through that very fast? 00:22:11.280 --> 00:22:13.840 That's why I had you read this assignment first. 00:22:13.840 --> 00:22:17.300 Notice that the assignment in the textbook does not touch 00:22:17.300 --> 00:22:21.390 what I'm talking about, but rather seems to review that 00:22:21.390 --> 00:22:23.790 topic that we covered under Part 1-- 00:22:23.790 --> 00:22:26.050 that if you wanted to take an ordinary derivative with 00:22:26.050 --> 00:22:31.480 respect to u, integral from u to a, g(y) dy, the answer 00:22:31.480 --> 00:22:33.950 would just be minus g(u). 00:22:33.950 --> 00:22:35.770 And that's exactly what I did in here. 00:22:35.770 --> 00:22:38.200 I treated v and x as constants here. 00:22:38.200 --> 00:22:42.100 In other words, the only variable in here was u. 00:22:42.100 --> 00:22:44.380 That will be emphasized again in the exercises. 00:22:44.380 --> 00:22:47.700 In a similar way, the partial of h with respect to v means 00:22:47.700 --> 00:22:48.920 this thing. 00:22:48.920 --> 00:22:52.020 Notice now that u is being treated as a constant. 00:22:52.020 --> 00:22:54.070 x is being treated as a constant. 00:22:54.070 --> 00:22:58.640 To differentiate this, my variable appears only as an 00:22:58.640 --> 00:23:00.410 upper limit on the integrand. 00:23:00.410 --> 00:23:03.810 That means I come inside the integral sign, replace the 00:23:03.810 --> 00:23:07.800 integral by just the function itself, replacing what? 00:23:13.050 --> 00:23:14.150 Replacing y-- 00:23:14.150 --> 00:23:16.740 that's the only variables of integration-- 00:23:16.740 --> 00:23:21.450 by the upper limit v. In other words, this is f(x, v). 00:23:21.450 --> 00:23:22.030 All right? 00:23:22.030 --> 00:23:25.770 And finally, the partial of h with respect to x is this 00:23:25.770 --> 00:23:26.970 integral here. 00:23:26.970 --> 00:23:30.470 Notice now that u and v are being treated as constants. 00:23:30.470 --> 00:23:33.990 With u and v being treated as constants, that's the special 00:23:33.990 --> 00:23:37.070 case that we started out lecture with, namely, to 00:23:37.070 --> 00:23:40.230 differentiate with respect to a parameter when the parameter 00:23:40.230 --> 00:23:42.360 appears only as part of the integrand. 00:23:42.360 --> 00:23:43.420 So how do we do that? 00:23:43.420 --> 00:23:44.900 We come inside. 00:23:44.900 --> 00:23:47.690 This is the interval from u to v, the partial of f with 00:23:47.690 --> 00:23:50.050 respect to x dy. 00:23:50.050 --> 00:23:52.190 Putting the whole thing together-- 00:23:52.190 --> 00:23:56.760 recalling, among other things, that du/dx, since u is a(x), 00:23:56.760 --> 00:24:01.090 is just a prime of x and that dv/dx is b prime of x-- what 00:24:01.090 --> 00:24:02.440 this says-- 00:24:02.440 --> 00:24:04.940 and again, I want to see the beauty of the chain rule here, 00:24:04.940 --> 00:24:07.610 because at least to my way of thinking, I don't see anything 00:24:07.610 --> 00:24:10.480 at all intuitive about the result I'm going to show you. 00:24:10.480 --> 00:24:14.790 And that is as soon as you make the limits of integration 00:24:14.790 --> 00:24:18.930 variable, to differentiate an integral involving a 00:24:18.930 --> 00:24:19.450 parameter-- 00:24:19.450 --> 00:24:22.970 you see again, what's the parameter here? x. 00:24:22.970 --> 00:24:24.070 We integrate it with respect to y. 00:24:24.070 --> 00:24:27.550 This is a constant as far as y is concerned. 00:24:27.550 --> 00:24:30.500 See, intuitively, you might say, gee, all you've got to do 00:24:30.500 --> 00:24:33.820 is take the derivative sign, bring it inside, and this 00:24:33.820 --> 00:24:35.450 should be the answer. 00:24:35.450 --> 00:24:36.920 See, it's the same as we did before. 00:24:36.920 --> 00:24:39.510 The point is-- and this is where many serious mistakes 00:24:39.510 --> 00:24:43.010 are made in problems involving integrals of this type-- 00:24:43.010 --> 00:24:45.750 is that the reason that our intuitive way happened to be 00:24:45.750 --> 00:24:50.790 right in the simpler case was that these were constants with 00:24:50.790 --> 00:24:51.890 respect to x. 00:24:51.890 --> 00:24:52.930 Now, they're variables. 00:24:52.930 --> 00:24:55.810 Well, it turns out, if you just wrote this thing down, 00:24:55.810 --> 00:24:56.900 you would be wrong. 00:24:56.900 --> 00:24:58.510 What is the correction factor? 00:24:58.510 --> 00:25:00.380 Again, come back to here. 00:25:00.380 --> 00:25:04.760 The correction factor is this here, which we've just started 00:25:04.760 --> 00:25:07.030 to compute. 00:25:07.030 --> 00:25:08.600 Again, just saying it-- 00:25:08.600 --> 00:25:11.640 if you wrote this term down to get the correct answer, you 00:25:11.640 --> 00:25:13.180 would have to tack on what? 00:25:13.180 --> 00:25:16.810 b prime of x times f(x, b(x)). 00:25:16.810 --> 00:25:18.270 That means what? 00:25:18.270 --> 00:25:20.790 You think of a y as being over here. 00:25:20.790 --> 00:25:24.685 For the particular value of x, you replace y by b(x). 00:25:24.685 --> 00:25:27.140 In other words, you look at f(x, y), and every place you 00:25:27.140 --> 00:25:29.130 see a y, replace it by b(x). 00:25:29.130 --> 00:25:31.490 And then subtract that from that a prime of 00:25:31.490 --> 00:25:34.910 x times f(x, a(x)). 00:25:34.910 --> 00:25:38.280 Now again, I suspect that, for many of you, it's the first 00:25:38.280 --> 00:25:40.800 time that you've seen something like this, because I 00:25:40.800 --> 00:25:44.850 say it's a topic which I believe was a natural one to 00:25:44.850 --> 00:25:47.560 occur in the textbook, but for some reason it 00:25:47.560 --> 00:25:48.830 doesn't appear there. 00:25:48.830 --> 00:25:52.470 Because of the importance of the concept, the number of 00:25:52.470 --> 00:25:55.190 times it appears in physical applications, the number of 00:25:55.190 --> 00:25:58.480 times that one has to differentiate with respect to 00:25:58.480 --> 00:25:59.270 an integral-- 00:25:59.270 --> 00:26:02.190 I don't know the physical applications well enough to 00:26:02.190 --> 00:26:05.410 lecture on them, but it does occur in probability theory, 00:26:05.410 --> 00:26:08.660 among other places, it appears in any subject involving 00:26:08.660 --> 00:26:11.040 integral equations and the like-- 00:26:11.040 --> 00:26:14.610 that I wanted to give you the experience of seeing what the 00:26:14.610 --> 00:26:16.960 concept means, to have you hear me say it. 00:26:16.960 --> 00:26:22.320 And then I will spend the exercises trying to drive home 00:26:22.320 --> 00:26:25.180 the computational know how so that you will be able to do 00:26:25.180 --> 00:26:29.280 these things at least in a mechanical way, independently 00:26:29.280 --> 00:26:32.800 of whether the theory made that much sense because of the 00:26:32.800 --> 00:26:35.380 lack of physical example motivation other than what I 00:26:35.380 --> 00:26:36.410 did at the beginning. 00:26:36.410 --> 00:26:38.880 At any rate, keep in mind, though, that in terms of our 00:26:38.880 --> 00:26:41.740 present topic, where we're talking about the chain rule, 00:26:41.740 --> 00:26:45.500 this is a certainly noble application to show an 00:26:45.500 --> 00:26:48.530 important place in the physical world where knowledge 00:26:48.530 --> 00:26:51.340 of the chain rule plays a very important role. 00:26:51.340 --> 00:26:54.220 And with that, I might just as well conclude today's lecture. 00:26:54.220 --> 00:26:55.990 And until next time, good bye. 00:26:58.740 --> 00:27:01.940 Funding for the publication of this video was provided by the 00:27:01.940 --> 00:27:05.990 Gabriella and Paul Rosenbaum Foundation. 00:27:05.990 --> 00:27:10.170 Help OCW continue to provide free and open access to MIT 00:27:10.170 --> 00:27:17.870 courses by making a donation at ocw.mit.edu/donate.