WEBVTT 00:00:00.040 --> 00:00:02.400 The following content is provided under a Creative 00:00:02.400 --> 00:00:03.690 Commons license. 00:00:03.690 --> 00:00:06.630 Your support will help MIT OpenCourseWare continue to 00:00:06.630 --> 00:00:09.990 offer high-quality educational resources for free. 00:00:09.990 --> 00:00:12.830 To make a donation, or to view additional materials from 00:00:12.830 --> 00:00:16.760 hundreds of MIT courses, visit MIT OpenCourseWare at 00:00:16.760 --> 00:00:18.010 ocw.mit.edu. 00:00:32.670 --> 00:00:33.470 PROFESSOR: Hi. 00:00:33.470 --> 00:00:36.680 Our lecture today is called 'Rolle's Theorem and Its 00:00:36.680 --> 00:00:37.930 Consequences'. 00:00:37.930 --> 00:00:40.960 And I suppose we could've made a take off on what goes up 00:00:40.960 --> 00:00:44.030 must come down, and say that what Rolle's theorem says 00:00:44.030 --> 00:00:47.970 intuitively is that what goes up smoothly and comes down 00:00:47.970 --> 00:00:51.480 smoothly must level off somewhere. 00:00:51.480 --> 00:00:51.880 OK? 00:00:51.880 --> 00:00:55.280 Now because that may sound too easy to understand, let's 00:00:55.280 --> 00:00:59.350 cloak that in the language of more formal mathematics. 00:00:59.350 --> 00:01:02.110 Rolle's theorem says this. 00:01:02.110 --> 00:01:05.760 Let 'f' be defined and continuous on the closed 00:01:05.760 --> 00:01:08.050 interval from 'a' to 'b'. 00:01:08.050 --> 00:01:08.620 In other words, what? 00:01:08.620 --> 00:01:09.930 The domain of 'f' is the closed 00:01:09.930 --> 00:01:11.140 interval from 'a' to 'b'. 00:01:11.140 --> 00:01:14.340 The graph of 'f' is unbroken on this interval. 00:01:14.340 --> 00:01:18.770 And differentiable in the open interval from 'a' to 'b'. 00:01:18.770 --> 00:01:21.840 In other words, you want the function to be continuous on 00:01:21.840 --> 00:01:25.270 the entire interval, but for differentiability, you only 00:01:25.270 --> 00:01:29.000 require that it be smooth, differentiable, in the 00:01:29.000 --> 00:01:32.010 interior of the interval, in the open interval. 00:01:32.010 --> 00:01:38.000 Suppose also that 'f of a' and 'f of b' are 0. 00:01:38.000 --> 00:01:41.990 Then, what Rolle's theorem says is that 'f prime of c', 00:01:41.990 --> 00:01:48.180 the derivative of 'f of x', must be 0 for some number 'c', 00:01:48.180 --> 00:01:51.680 at least one number 'c', in the open 00:01:51.680 --> 00:01:54.160 interval from 'a' to 'b'. 00:01:54.160 --> 00:01:58.420 Now what this thing says intuitively is simply this, 00:01:58.420 --> 00:02:04.440 suppose you have a curve that is unbroken for all values of 00:02:04.440 --> 00:02:07.130 'x' between 'a' and 'b' inclusively. 00:02:07.130 --> 00:02:09.580 Suppose the curve is smooth. 00:02:09.580 --> 00:02:14.780 Suppose the curve starts here and ends here, then what we're 00:02:14.780 --> 00:02:18.330 saying is there must be some point in here where the curve 00:02:18.330 --> 00:02:19.030 levels off. 00:02:19.030 --> 00:02:21.800 In other words, someplace where you have a horizontal 00:02:21.800 --> 00:02:24.800 tangent, which is what 'f prime of c' equals 0 means. 00:02:24.800 --> 00:02:29.000 In this particular diagram, this would be the value of 'c' 00:02:29.000 --> 00:02:30.510 that we're talking about. 00:02:30.510 --> 00:02:35.080 By the way, I think the proof is intuitively clear. 00:02:35.080 --> 00:02:39.510 Namely, if the curve never leaves the x-axis, then it's 00:02:39.510 --> 00:02:42.660 leveled off for the entire domain. 00:02:42.660 --> 00:02:45.937 And if the curve does leave the x-axis, for example, if 00:02:45.937 --> 00:02:50.400 the curve starts to rise, OK, since it must eventually get 00:02:50.400 --> 00:02:55.320 back to the x-axis when 'b' is 0, it must ultimately begin to 00:02:55.320 --> 00:02:56.240 fall again. 00:02:56.240 --> 00:02:59.860 Well, if the curve goes from rising to falling, 00:02:59.860 --> 00:03:01.040 it must have what? 00:03:01.040 --> 00:03:04.010 Since it's coming up and then going down, it must attain a 00:03:04.010 --> 00:03:05.520 maximum value. 00:03:05.520 --> 00:03:09.030 Because the curve is unbroken and smooth, as we saw in our 00:03:09.030 --> 00:03:12.690 previous lecture, the maximum value is characterized by the 00:03:12.690 --> 00:03:15.220 derivative at that point being 0. 00:03:15.220 --> 00:03:18.920 In fact, the analytic proof is precisely what we've just 00:03:18.920 --> 00:03:22.650 said, only translated into more mathematical language. 00:03:22.650 --> 00:03:25.280 By the way, I would like to make a slight aside here, 00:03:25.280 --> 00:03:28.980 because I think it sometimes gets confusing to students to 00:03:28.980 --> 00:03:33.600 see, why do you say that the function has to be continuous 00:03:33.600 --> 00:03:38.000 on the closed interval, but differentiable only in the 00:03:38.000 --> 00:03:39.220 open interval? 00:03:39.220 --> 00:03:42.580 I thought you might like to see a contrived example as to 00:03:42.580 --> 00:03:46.360 what goes wrong if you allow the curve to be broken at the 00:03:46.360 --> 00:03:47.220 end points. 00:03:47.220 --> 00:03:49.520 See, all I'm thinking of is something like this. 00:03:49.520 --> 00:03:52.880 Suppose I say, look, let's define a curve as follows. 00:03:52.880 --> 00:03:56.510 At 'x' equals 'a' and 'x' equals 'b', the 00:03:56.510 --> 00:03:59.040 curve will be 0. 00:03:59.040 --> 00:04:01.550 So in other words, it'll cross the x-axis. 00:04:01.550 --> 00:04:04.980 Then immediately, for 'x' greater than 'a', the curve 00:04:04.980 --> 00:04:09.090 jumps up to here, comes down along this line, and then, you 00:04:09.090 --> 00:04:13.480 see, when 'x' equals 'b', it jumps down here again. 00:04:13.480 --> 00:04:15.560 In other words, why not let the curve 00:04:15.560 --> 00:04:16.950 be defined as follows? 00:04:16.950 --> 00:04:20.170 It will be 0 at these two endpoints. 00:04:20.170 --> 00:04:23.650 It'll be this curve on the open interval. 00:04:23.650 --> 00:04:28.010 Notice in this contrived example that 'f of a' and 'f 00:04:28.010 --> 00:04:31.780 of b' are both equal to 0, but there is no place in the open 00:04:31.780 --> 00:04:35.880 interval where the curve has a horizontal tangent line. 00:04:35.880 --> 00:04:38.790 In other words, the significance is you've got to 00:04:38.790 --> 00:04:41.830 be sure that the curve doesn't get broken at the ends, 00:04:41.830 --> 00:04:43.940 because with these gaps, all sorts of 00:04:43.940 --> 00:04:45.780 crazy things can happen. 00:04:45.780 --> 00:04:49.180 Now, just as in our previous lecture, there are some rather 00:04:49.180 --> 00:04:53.960 important cautions that have to be taken in understanding 00:04:53.960 --> 00:04:55.210 Rolle's theorem. 00:04:55.210 --> 00:04:59.020 As simple as it is, we have to be sure that we understand 00:04:59.020 --> 00:05:02.070 exactly what's really happening here. 00:05:02.070 --> 00:05:06.260 The first caution is that Rolle's theorem is what we 00:05:06.260 --> 00:05:09.920 mathematicians call an 'existence theorem'. 00:05:09.920 --> 00:05:13.800 It says, under certain conditions, there exists at 00:05:13.800 --> 00:05:17.080 least one number 'c' that has a certain property. 00:05:17.080 --> 00:05:19.790 It doesn't tell us how many "c's" there are. 00:05:19.790 --> 00:05:21.870 It doesn't tell us where to find them. 00:05:21.870 --> 00:05:23.040 It just says, what? 00:05:23.040 --> 00:05:25.720 There exists at least one such 'c'. 00:05:25.720 --> 00:05:28.770 And the point is, you must be careful to remember-- 00:05:28.770 --> 00:05:29.930 so let's take an example. 00:05:29.930 --> 00:05:31.970 Here's 'a', here's 'b'. 00:05:31.970 --> 00:05:34.826 If the function is continuous and smooth, in other words, if 00:05:34.826 --> 00:05:37.950 the graph is continuous and smooth, all we're saying is 00:05:37.950 --> 00:05:40.710 that at least one number between 'a' and 'b', the 00:05:40.710 --> 00:05:43.520 curve, must possess a horizontal tangent. 00:05:43.520 --> 00:05:44.640 There may be more than one. 00:05:44.640 --> 00:05:46.480 You see, the curve, for example, could do 00:05:46.480 --> 00:05:48.250 something like this. 00:05:48.250 --> 00:05:51.640 See, in other words, here is one value of 'c', which we'll 00:05:51.640 --> 00:05:54.060 call 'c1', horizontal tangent here. 00:05:54.060 --> 00:05:57.730 Here's another value, which we'll call 'c2', horizontal 00:05:57.730 --> 00:05:59.280 tangent here. 00:05:59.280 --> 00:06:03.100 See, again, the meaning of at least one. 00:06:03.100 --> 00:06:11.370 Secondly, we must make sure that we remember that the 00:06:11.370 --> 00:06:12.290 curve is smooth. 00:06:12.290 --> 00:06:12.780 Meaning what? 00:06:12.780 --> 00:06:14.570 That the function is differentiable. 00:06:14.570 --> 00:06:15.980 Now I'm taking the liberty of drawing 00:06:15.980 --> 00:06:17.880 these things in freehand. 00:06:17.880 --> 00:06:21.760 There's some mixed emotions here, if I draw the diagrams 00:06:21.760 --> 00:06:24.260 too smoothly all the time, you lose the significance of 00:06:24.260 --> 00:06:26.200 what's going on because of the picture. 00:06:26.200 --> 00:06:28.020 And if I draw them freehand all the time, you won't 00:06:28.020 --> 00:06:29.230 understand what I'm doing, because I 00:06:29.230 --> 00:06:30.380 don't draw very well. 00:06:30.380 --> 00:06:32.270 But I think here we can get away with this. 00:06:32.270 --> 00:06:33.650 What I'm driving at is this. 00:06:33.650 --> 00:06:36.710 Let's suppose you have 'a' and 'b' here. 00:06:36.710 --> 00:06:38.810 Let's suppose that the function, the curve that we're 00:06:38.810 --> 00:06:41.230 drawing, passes through these two points. 00:06:41.230 --> 00:06:44.260 But suppose there happens to be a sharp corner in here. 00:06:44.260 --> 00:06:46.530 Maybe the curve goes like this, it goes up like this, 00:06:46.530 --> 00:06:50.010 then very abruptly comes down like this. 00:06:50.010 --> 00:06:51.550 Notice, what? 00:06:51.550 --> 00:06:54.270 That the curve is continuous. 00:06:54.270 --> 00:06:57.380 It does reach a local maximum. 00:06:57.380 --> 00:07:01.890 But the point is, for this particular value of 'c', 'f 00:07:01.890 --> 00:07:05.320 prime of c' is not 0 by default. 00:07:05.320 --> 00:07:10.320 Namely, 'f prime of c' doesn't even exist. 00:07:10.320 --> 00:07:15.230 So in other words, Rolle's theorem doesn't apply if you 00:07:15.230 --> 00:07:16.700 don't have differentiabilities. 00:07:16.700 --> 00:07:19.890 I want to make sure you see where each of the parts of the 00:07:19.890 --> 00:07:22.600 hypotheses for the theorem are used. 00:07:22.600 --> 00:07:25.470 By the way, here's another interesting result, which has 00:07:25.470 --> 00:07:27.200 nothing to do with the statement of the theorem, but 00:07:27.200 --> 00:07:31.140 again, another piece of evidence as to why we like to 00:07:31.140 --> 00:07:34.700 shy away from functions which are not single value. 00:07:34.700 --> 00:07:37.700 See, for example, suppose you allowed the function to be 00:07:37.700 --> 00:07:41.520 multivalued, and you say, OK, I want the curve to pass 00:07:41.520 --> 00:07:42.960 through here and here. 00:07:42.960 --> 00:07:44.890 And I want it to be smooth. 00:07:44.890 --> 00:07:48.180 But the curve does not have to be single value. 00:07:48.180 --> 00:07:49.660 Notice what you can do. 00:07:49.660 --> 00:07:52.070 You could have a curve that does something like this. 00:07:55.490 --> 00:07:57.210 I don't know. 00:07:57.210 --> 00:08:00.190 Now you see, there will be a point 'c' where the 00:08:00.190 --> 00:08:02.390 derivative will be 0. 00:08:02.390 --> 00:08:04.790 Even as badly as I've drawn this, I think roughly 00:08:04.790 --> 00:08:09.430 speaking, we can see that 'c' would be something like this. 00:08:09.430 --> 00:08:12.130 Notice, however, that in Rolle's theorem, the 00:08:12.130 --> 00:08:13.150 statement is what? 00:08:13.150 --> 00:08:16.490 That 'c' must be on the open interval from 'a' to 'b'. 00:08:16.490 --> 00:08:20.010 If the function is not single value, as long as the curve is 00:08:20.010 --> 00:08:23.330 smooth, there will be places where the curve levels off. 00:08:23.330 --> 00:08:27.130 But the x-coordinate of the point at which the curve 00:08:27.130 --> 00:08:30.530 levels off may not be in the interval-- 00:08:30.530 --> 00:08:34.179 may not be, it could be, but it might not be, I should 00:08:34.179 --> 00:08:37.220 say-- in the open interval from 'a' to 'b'. 00:08:37.220 --> 00:08:39.070 This is a very, very important result. 00:08:39.070 --> 00:08:43.070 In fact, I'll have reason to make reference to this in a 00:08:43.070 --> 00:08:45.700 little while later in the lecture. 00:08:45.700 --> 00:08:47.660 I couldn't make reference to it later earlier in the 00:08:47.660 --> 00:08:49.570 lecture, I guess. 00:08:49.570 --> 00:08:54.200 The fourth assumption here is also an aside, and it's one 00:08:54.200 --> 00:08:55.360 that's rather crucial. 00:08:55.360 --> 00:08:59.470 In most textbooks in which Rolle's theorem is stated, the 00:08:59.470 --> 00:09:00.350 condition is what? 00:09:00.350 --> 00:09:04.180 That 'f of a' equals 'f of b' equals 0. 00:09:04.180 --> 00:09:09.340 It turns out that this is too restrictive, that essentially, 00:09:09.340 --> 00:09:13.630 all you need is 'f of a' equals 'f of b'. 00:09:13.630 --> 00:09:17.200 What I mean by that is, let's suppose 'f of a' is not 0. 00:09:17.200 --> 00:09:19.810 Let's suppose this height represents 'f of a'. 00:09:19.810 --> 00:09:24.300 What I'm saying is suppose that 'f of a' and 00:09:24.300 --> 00:09:27.210 'f of b' are equal. 00:09:27.210 --> 00:09:29.760 What that means is, if I want to think of a new axis, which 00:09:29.760 --> 00:09:32.330 I call the 'x sub 1' axis-- 00:09:32.330 --> 00:09:35.980 see, with respect to the 'x sub 1' axis, the curve crosses 00:09:35.980 --> 00:09:37.550 the axis at these two points. 00:09:37.550 --> 00:09:40.235 In other words, notice that as long as these two points are 00:09:40.235 --> 00:09:43.160 at the same level, the same argument that we used to prove 00:09:43.160 --> 00:09:46.970 Rolle's theorem goes through unimpeded over here. 00:09:46.970 --> 00:09:48.170 Namely, we say what? 00:09:48.170 --> 00:09:52.490 What goes up smoothly and comes down smoothly-- 00:09:52.490 --> 00:09:54.670 because it has to come down, because it comes back to the 00:09:54.670 --> 00:10:00.540 same level here-- must reach a point someplace in here where 00:10:00.540 --> 00:10:02.280 it levels off. 00:10:02.280 --> 00:10:07.100 So these are the four cautions that I'd like you to look at 00:10:07.100 --> 00:10:09.200 when we view Rolle's theorem. 00:10:09.200 --> 00:10:12.110 Now, if somebody were to say to us, what's so important 00:10:12.110 --> 00:10:14.290 about Rolle's theorem? 00:10:14.290 --> 00:10:17.250 And this happens so often in mathematics that frequently, 00:10:17.250 --> 00:10:21.510 the most important thing about one particular theorem, is 00:10:21.510 --> 00:10:24.420 that it may be the building block by which a more 00:10:24.420 --> 00:10:26.670 important, or more useful, or more 00:10:26.670 --> 00:10:28.910 practical theorem is derived. 00:10:28.910 --> 00:10:32.680 And in this respect, I would say for my own opinion, that 00:10:32.680 --> 00:10:35.830 the most important application of Rolle's theorem is that it 00:10:35.830 --> 00:10:39.560 facilitates a very famous result known as the 'Mean 00:10:39.560 --> 00:10:40.590 Value theorem'. 00:10:40.590 --> 00:10:43.790 As our course proceeds, from time to time we will have 00:10:43.790 --> 00:10:47.330 ample reason to back track and make references to the mean 00:10:47.330 --> 00:10:48.390 value theorem. 00:10:48.390 --> 00:10:51.640 I intend not to make too deep references to the mean value 00:10:51.640 --> 00:10:54.190 theorem now, because what I would like to do is to 00:10:54.190 --> 00:10:57.220 establish the result, give you enough of an intuitive feeling 00:10:57.220 --> 00:11:00.960 so that you can tuck it under your belt without feeling too 00:11:00.960 --> 00:11:04.880 overwhelmed by it, and just enough applications of it so 00:11:04.880 --> 00:11:07.450 that we can get into the next phase of our calculus course. 00:11:07.450 --> 00:11:10.080 But the mean value theorem is another one of these things 00:11:10.080 --> 00:11:12.680 where if you state the thing analytically, and have no 00:11:12.680 --> 00:11:15.490 feeling for what's going on pictorially, the thing can 00:11:15.490 --> 00:11:17.180 become overwhelming. 00:11:17.180 --> 00:11:19.940 Let's, in fact, do it in an overwhelming way, and then 00:11:19.940 --> 00:11:22.040 show what the thing means pictorially. 00:11:22.040 --> 00:11:25.520 Notice again how this thing now starts off the same way as 00:11:25.520 --> 00:11:26.560 Rolle's theorem. 00:11:26.560 --> 00:11:29.560 Let 'f' be continuous on the closed interval from 'a' to 00:11:29.560 --> 00:11:34.150 'b', and differentiable in the open interval from 'a' to 'b'. 00:11:34.150 --> 00:11:38.110 By the way, this is just an idiosyncrasy of mine, I don't 00:11:38.110 --> 00:11:39.050 know if it's standard. 00:11:39.050 --> 00:11:42.070 When I talk about the closed interval, I have the habit of 00:11:42.070 --> 00:11:44.050 saying 'on' the interval. 00:11:44.050 --> 00:11:47.280 When I talk about the open interval, I like to say 'in' 00:11:47.280 --> 00:11:49.840 the interval, to sort of emphasize the interior. 00:11:49.840 --> 00:11:53.250 It's just a vocabulary trait, and don't read 00:11:53.250 --> 00:11:54.080 too much into this. 00:11:54.080 --> 00:11:55.160 Don't be upset by it. 00:11:55.160 --> 00:11:58.060 But it's continuous on the closed interval, 00:11:58.060 --> 00:12:00.050 differentiable in the open interval. 00:12:00.050 --> 00:12:02.360 Now it's again an existence theorem. 00:12:02.360 --> 00:12:04.750 It says then there exists a number 'c'. 00:12:04.750 --> 00:12:06.580 When I say there exists, it means what? 00:12:06.580 --> 00:12:10.190 There is at least one number 'c' in the open interval from 00:12:10.190 --> 00:12:11.080 'a' to 'b'-- 00:12:11.080 --> 00:12:13.240 and this is the part that looks kind of tough-- 00:12:13.240 --> 00:12:19.030 such that ''f of b' minus 'f of a'' divided by 'b - a' is 00:12:19.030 --> 00:12:20.600 'f prime of c'. 00:12:20.600 --> 00:12:24.040 And this somehow or other may seem at first glance to be 00:12:24.040 --> 00:12:27.290 more ominous than the intuitive feeling about 00:12:27.290 --> 00:12:28.520 Rolle's theorem. 00:12:28.520 --> 00:12:31.590 By the way, as the name implies, where by 'mean' we 00:12:31.590 --> 00:12:34.900 don't mean nasty, we mean average, if you'd like to see 00:12:34.900 --> 00:12:37.240 what this thing means, and I'll draw you a picture in a 00:12:37.240 --> 00:12:44.020 second, all it says is that if a particle is moving from 00:12:44.020 --> 00:12:48.790 point 'a' to point 'b', say, at at least one point during 00:12:48.790 --> 00:12:51.750 its trip, the instantaneous speed must 00:12:51.750 --> 00:12:54.980 equal the average speed. 00:12:54.980 --> 00:12:55.210 You see? 00:12:55.210 --> 00:12:57.380 Because after all, if you're always going less than your 00:12:57.380 --> 00:13:00.490 average speed, how could you have had an average speed as 00:13:00.490 --> 00:13:02.010 high as your average speed? 00:13:02.010 --> 00:13:04.250 And if you're always going less than your average speed, 00:13:04.250 --> 00:13:08.270 how could you have had, you see, an average speed equaling 00:13:08.270 --> 00:13:09.660 this, what it did? 00:13:09.660 --> 00:13:13.610 So that somehow or other, all you're saying is that somehow 00:13:13.610 --> 00:13:17.050 or other, the instantaneous speed at a particular instance 00:13:17.050 --> 00:13:21.160 must equal your average speed someplace along the path. 00:13:21.160 --> 00:13:24.550 Now what that means pictorially is this-- 00:13:24.550 --> 00:13:27.750 again, I'll chance a freehand diagram-- 00:13:27.750 --> 00:13:31.320 suppose our curve is 'y' equals 'f of x'. 00:13:31.320 --> 00:13:35.230 See, I've drawn it to be smooth. 00:13:35.230 --> 00:13:39.380 Suppose it's continuous and smooth on this open interval 00:13:39.380 --> 00:13:41.680 from 'a' to 'b'. 00:13:41.680 --> 00:13:45.530 Now what is, if I think of a particle moving from point 'p' 00:13:45.530 --> 00:13:50.270 to point 'q', how do I identify the average speed? 00:13:50.270 --> 00:13:53.200 The average speed is the slope of the straight line that 00:13:53.200 --> 00:13:55.530 joins 'p' to 'q'. 00:13:55.530 --> 00:13:57.880 On the other hand, what is the instantaneous speed? 00:13:57.880 --> 00:14:00.280 If we think of it in terms of the picture, it's the slope of 00:14:00.280 --> 00:14:03.620 the line tangent to the curve at a particular point. 00:14:03.620 --> 00:14:06.420 So in other words, what we're saying is this. 00:14:06.420 --> 00:14:11.270 You see, if we were to take the line 'PQ', and we shift it 00:14:11.270 --> 00:14:15.070 parallel to itself, I think you can sense that the points 00:14:15.070 --> 00:14:19.070 'P' and 'Q', if we labeled 'P' and 'Q' the points at which 00:14:19.070 --> 00:14:22.170 this chord intersects the curve, the points 'P' and 'Q' 00:14:22.170 --> 00:14:24.440 will roll in closer and closer together. 00:14:24.440 --> 00:14:29.760 Ultimately, the line will fail to intersect the curve, and at 00:14:29.760 --> 00:14:32.340 the transition point, if the curve is smooth, all we're 00:14:32.340 --> 00:14:36.780 saying is that the last point at which that line touches the 00:14:36.780 --> 00:14:41.460 curve as we move it out, OK, that the line would be tangent 00:14:41.460 --> 00:14:44.060 to the curve at that particular point. 00:14:44.060 --> 00:14:45.430 You see, all we're saying is what? 00:14:45.430 --> 00:14:48.650 That someplace between here and here there must be a point 00:14:48.650 --> 00:14:51.760 where the tangent line to the curve is parallel 00:14:51.760 --> 00:14:53.310 to the chord 'PQ'. 00:14:53.310 --> 00:14:56.320 And now, we have all the ingredients that we need to 00:14:56.320 --> 00:14:59.500 see what the mean value theorem says geometrically. 00:14:59.500 --> 00:15:01.550 Let's call the tangent line 'l'. 00:15:01.550 --> 00:15:04.820 First of all, what is the slope of the line 'PQ'? 00:15:04.820 --> 00:15:07.120 Well, it's a straight line. 00:15:07.120 --> 00:15:09.610 The slope of a straight line is 'delta y' 00:15:09.610 --> 00:15:11.090 divided by 'delta x'. 00:15:11.090 --> 00:15:13.840 Well, notice that this height here is by 00:15:13.840 --> 00:15:15.680 definition 'f of b'. 00:15:15.680 --> 00:15:17.880 This height here is 'f of a'. 00:15:17.880 --> 00:15:22.820 So this height here is just 'f of b' minus 'f of a'. 00:15:22.820 --> 00:15:26.770 This length here is just 'b - a'. 00:15:26.770 --> 00:15:30.710 So the slope, 'delta y' divided by 'delta x', is just 00:15:30.710 --> 00:15:37.180 ''f of b' minus 'f of a'', over 'b - a'. 00:15:37.180 --> 00:15:40.010 On the other hand, what is the slope of the line 'l'? 00:15:40.010 --> 00:15:44.340 By definition, it's 'f prime of x' evaluated at 'x' equals 00:15:44.340 --> 00:15:47.100 'c', that's 'f prime of c'. 00:15:47.100 --> 00:15:49.240 Now what does it mean in terms of slopes for 00:15:49.240 --> 00:15:50.700 two lines to be parallel? 00:15:50.700 --> 00:15:53.630 It means that their slopes are equal. 00:15:53.630 --> 00:15:55.170 And where is 'c'? 00:15:55.170 --> 00:16:02.240 'c' is someplace in the open interval from 'a' to 'b'. 00:16:02.240 --> 00:16:07.730 Now the reason I call this intuitively an extension of 00:16:07.730 --> 00:16:09.490 Rolle's theorem-- 00:16:09.490 --> 00:16:12.270 and by the way, you'll notice that what I have to say is a 00:16:12.270 --> 00:16:16.490 much simpler demonstration than what's given in the book. 00:16:16.490 --> 00:16:19.460 But before you think I'm being egotistic about this, let me 00:16:19.460 --> 00:16:22.610 point out, as is so often the case that wherever my 00:16:22.610 --> 00:16:25.930 demonstrations are easier than the one in the book, I'm 00:16:25.930 --> 00:16:28.520 losing something in my presentation. 00:16:28.520 --> 00:16:31.410 Either I haven't shown the most analytic representation, 00:16:31.410 --> 00:16:34.760 or I'm overlooking a particular complicated side 00:16:34.760 --> 00:16:36.760 effect that might occur. 00:16:36.760 --> 00:16:39.330 But disregarding that for the moment, you see what I'm 00:16:39.330 --> 00:16:42.790 saying is this, let's suppose, for the sake of argument, we 00:16:42.790 --> 00:16:47.720 visualize the line 'PQ' as being our new x-axis. 00:16:47.720 --> 00:16:49.710 I'll call that the x1-axis. 00:16:49.710 --> 00:16:53.110 And now let's take a line perpendicular to 'PQ' and call 00:16:53.110 --> 00:16:56.740 that our new y-axis, the y1-axis. 00:16:56.740 --> 00:16:58.870 Now look at the curve that we've drawn. 00:16:58.870 --> 00:17:05.079 With respect to the y1- x1-axis, notice that the curve 00:17:05.079 --> 00:17:06.819 is smooth, right? 00:17:06.819 --> 00:17:08.329 It's unbroken. 00:17:08.329 --> 00:17:13.050 And it cuts the x1-axis at two points. 00:17:13.050 --> 00:17:16.270 Now if we apply Rolle's theorem with respect to the 00:17:16.270 --> 00:17:19.970 x1- y1-axis, we say, look, here's a curve 00:17:19.970 --> 00:17:21.770 which cuts the x-axis-- 00:17:21.770 --> 00:17:22.660 the x1-axis-- 00:17:22.660 --> 00:17:23.790 at two points. 00:17:23.790 --> 00:17:26.560 It's smooth. 00:17:26.560 --> 00:17:28.740 Therefore, it must level off someplace. 00:17:28.740 --> 00:17:31.370 In other words, there must be some point on this curve where 00:17:31.370 --> 00:17:36.480 the tangent line to the curve is parallel to the x1-axis. 00:17:36.480 --> 00:17:39.670 That's exactly, you see, what this thing here says. 00:17:39.670 --> 00:17:42.660 That's another geometric interpretation that indicates 00:17:42.660 --> 00:17:44.620 how Rolle's theorem might be used. 00:17:44.620 --> 00:17:49.000 However, there is a very, very subtle flaw in what I've said. 00:17:49.000 --> 00:17:52.240 One that is so subtle that you may not even notice it until I 00:17:52.240 --> 00:17:54.830 point it out to you, and even after I point it out, there's 00:17:54.830 --> 00:17:56.760 a chance you may not realize what I've said. 00:17:56.760 --> 00:17:59.910 Because it's a point that I know took me a long, long time 00:17:59.910 --> 00:18:01.390 to discover for myself. 00:18:01.390 --> 00:18:03.470 And it all hinges on the concept of 00:18:03.470 --> 00:18:05.200 single valuedness again. 00:18:05.200 --> 00:18:07.840 The trouble with this interpretation is the 00:18:07.840 --> 00:18:09.380 following-- 00:18:09.380 --> 00:18:12.770 and by the way, let me point out, I'm not knocking my 00:18:12.770 --> 00:18:15.390 interpretation, I think it's still a tremendous way of 00:18:15.390 --> 00:18:18.110 visualizing the result, but from an analytical point of 00:18:18.110 --> 00:18:20.600 view, why we have to be careful. 00:18:20.600 --> 00:18:23.830 Let's suppose that my curve 'y' equals 'f of x' happens to 00:18:23.830 --> 00:18:27.110 look something like this, OK? 00:18:27.110 --> 00:18:30.200 Happens to look something like this. 00:18:30.200 --> 00:18:33.160 Notice, barring any bad drawing that I've done here, 00:18:33.160 --> 00:18:37.130 that this curve is single value, that no line parallel 00:18:37.130 --> 00:18:40.670 to the y-axis cuts this curve in more than one place. 00:18:40.670 --> 00:18:43.150 Now here's my 'a' and here's my 'b'. 00:18:45.670 --> 00:18:49.820 And so I say, OK, by Rolle's theorem, if I look at this as 00:18:49.820 --> 00:18:52.740 being the x-axis and this as being the y-axis-- 00:18:52.740 --> 00:18:55.500 in other words, the x1- y1-axis again-- 00:18:55.500 --> 00:18:58.930 I say to myself, look, here's a smooth curve, it cuts the 00:18:58.930 --> 00:19:04.620 x-axis in two points, therefore, someplace between 00:19:04.620 --> 00:19:07.210 these two points, there must be a place where the curve 00:19:07.210 --> 00:19:11.120 levels off, et cetera, et cetera, et cetera. 00:19:11.120 --> 00:19:15.370 And the interesting point is to notice that a given curve, 00:19:15.370 --> 00:19:18.850 as to whether it's single valued or not, is dependent 00:19:18.850 --> 00:19:21.830 upon the orientation of the axes. 00:19:21.830 --> 00:19:24.040 In other words, notice that I've drawn this particular 00:19:24.040 --> 00:19:28.100 curve so it is single valued with respect to the xy-plane. 00:19:28.100 --> 00:19:31.580 On the other hand, with respect to the x1- 00:19:31.580 --> 00:19:36.020 y1-coordinate system, this curve is not single valued. 00:19:36.020 --> 00:19:40.920 Namely, observe how a line parallel to the y1-axis can 00:19:40.920 --> 00:19:45.630 intersect this curve at more than one point. 00:19:45.630 --> 00:19:48.590 In other words, whether a curve is single valued or not 00:19:48.590 --> 00:19:51.570 is not an absolute property independent of 00:19:51.570 --> 00:19:52.970 the coordinate system. 00:19:52.970 --> 00:19:57.160 So again, if I could be sure that when I rotated my 00:19:57.160 --> 00:20:01.460 coordinate axes the original single valued curve was still 00:20:01.460 --> 00:20:04.010 single valued, then my above proof 00:20:04.010 --> 00:20:05.250 would have been rigorous. 00:20:05.250 --> 00:20:07.410 But of course, I can't be sure of that. 00:20:07.410 --> 00:20:11.700 By the way, the technique used in the book is quite standard, 00:20:11.700 --> 00:20:14.270 and what it does is the following, it still utilizes 00:20:14.270 --> 00:20:17.630 Rolle's theorem, but the technique behind the proof in 00:20:17.630 --> 00:20:18.780 the book is this. 00:20:18.780 --> 00:20:23.160 The function that we set up is the vertical distance between 00:20:23.160 --> 00:20:27.930 the chord and the curve, as we move along this way. 00:20:27.930 --> 00:20:32.090 And notice that that distance is 0 at these two endpoints. 00:20:32.090 --> 00:20:33.450 OK? 00:20:33.450 --> 00:20:37.510 And therefore, Rolle's theorem applies to that function. 00:20:37.510 --> 00:20:39.710 And the whole idea is something like this. 00:20:39.710 --> 00:20:40.950 All we say is-- 00:20:40.950 --> 00:20:43.780 and the analytic part proves this-- 00:20:43.780 --> 00:20:47.190 all we say is look, the point at which this chord would have 00:20:47.190 --> 00:20:50.960 been tangent to the curve is the place where the vertical 00:20:50.960 --> 00:20:55.360 distance between the chord and the curve is what? 00:20:55.360 --> 00:20:56.730 Maximum. 00:20:56.730 --> 00:20:58.420 And we won't go into that right now, that 00:20:58.420 --> 00:21:00.250 is done in the text. 00:21:00.250 --> 00:21:03.520 All I wanted to do, as I always will do when possible, 00:21:03.520 --> 00:21:06.580 is that whenever the rigorous proof seems far more 00:21:06.580 --> 00:21:10.520 complicated than proofs which are more intuitive, I will not 00:21:10.520 --> 00:21:13.830 take the time in general, in our lectures, to give the more 00:21:13.830 --> 00:21:14.700 rigorous proof. 00:21:14.700 --> 00:21:17.590 What I will take the time to do is to show why the less 00:21:17.590 --> 00:21:19.610 rigorous proof has pitfalls. 00:21:19.610 --> 00:21:21.640 Well, enough said about the statement of 00:21:21.640 --> 00:21:22.950 the mean value theorem. 00:21:22.950 --> 00:21:26.080 Time is getting very short, and we don't need much more 00:21:26.080 --> 00:21:29.260 time to make the home run ball pitch that we 00:21:29.260 --> 00:21:30.520 want to make now. 00:21:30.520 --> 00:21:35.040 And the idea is this, that the most important analytical 00:21:35.040 --> 00:21:39.740 reason for having the mean value theorem is, for those of 00:21:39.740 --> 00:21:43.560 us who like to use our geometric intuition, it turns 00:21:43.560 --> 00:21:47.080 out that almost every geometrically obvious fact 00:21:47.080 --> 00:21:52.200 that has a proper analytic counterpart has the property 00:21:52.200 --> 00:21:54.640 that the analytic counterpart is proven by 00:21:54.640 --> 00:21:56.130 the mean value theorem. 00:21:56.130 --> 00:21:58.320 See, let me give you a simple for instance. 00:21:58.320 --> 00:22:02.700 In fact, in the text book this is called the first corollary 00:22:02.700 --> 00:22:04.530 to the mean value theorem. 00:22:04.530 --> 00:22:07.810 Suppose we have a function capital 'F of x', and we know 00:22:07.810 --> 00:22:11.330 that the derivative is always equal to 0. 00:22:11.330 --> 00:22:14.730 The claim is that 'F of x' itself must be a constant. 00:22:14.730 --> 00:22:17.160 By the way, two cautions here. 00:22:17.160 --> 00:22:20.140 Don't say that we've had this result before. 00:22:20.140 --> 00:22:23.030 The result that we had before was actually 00:22:23.030 --> 00:22:24.660 the converse of this. 00:22:24.660 --> 00:22:27.790 The result that we had before was the one that said what? 00:22:27.790 --> 00:22:32.570 If 'F of x' is a constant, then its derivative is 0. 00:22:32.570 --> 00:22:34.800 Now we're saying the opposite--not the opposite, 00:22:34.800 --> 00:22:35.400 but the converse. 00:22:35.400 --> 00:22:38.000 Now we're saying, look, if the derivative is always 0-- 00:22:38.000 --> 00:22:39.980 notice the use of my identity symbol here-- 00:22:39.980 --> 00:22:44.680 if the derivative is 0 for all values of 'x', then the 00:22:44.680 --> 00:22:46.480 function must've been a constant. 00:22:46.480 --> 00:22:48.660 Now, you know, geometrically this is a very 00:22:48.660 --> 00:22:50.090 simple thing to visualize. 00:22:50.090 --> 00:22:53.160 You say, look, the derivative is the slope. 00:22:53.160 --> 00:22:56.150 And if you're saying that the slope of the curve is always 00:22:56.150 --> 00:23:00.300 horizontal, the curve itself must be a straight line. 00:23:00.300 --> 00:23:02.960 And if the curve is a straight line, that's exactly what you 00:23:02.960 --> 00:23:05.590 mean by saying that the function is a constant. 00:23:05.590 --> 00:23:08.500 How do we prove this using the mean value theorem? 00:23:08.500 --> 00:23:11.250 See, and I just wanted to go through a proof here once, 00:23:11.250 --> 00:23:14.970 just so to get the idea of what a proof means. 00:23:14.970 --> 00:23:16.350 You see, to show that something is a 00:23:16.350 --> 00:23:17.930 constant should mean what? 00:23:17.930 --> 00:23:23.430 That if 'a' is unequal to 'b' for any two values 'a' and 00:23:23.430 --> 00:23:25.850 'b', 'F of a'-- 00:23:25.850 --> 00:23:28.570 well, I'm using capital 'F' here--capital 'F of a' has to 00:23:28.570 --> 00:23:30.210 equal capital 'F of b'. 00:23:30.210 --> 00:23:32.500 That's what you mean for a function to be a constant. 00:23:32.500 --> 00:23:36.530 No matter what the input is, the outputs are always equal. 00:23:36.530 --> 00:23:39.310 By the way, if 'a' equals 'b', it's trivial that 'F of a' 00:23:39.310 --> 00:23:40.350 equals 'F of b'. 00:23:40.350 --> 00:23:43.500 But essentially, to prove that capital F is a constant, this 00:23:43.500 --> 00:23:44.810 is what I have to prove. 00:23:44.810 --> 00:23:47.740 That if 'a' is different from 'b', no matter what 'a' and 00:23:47.740 --> 00:23:50.960 'b' I use, that 'F of a' equals 'F of b'. 00:23:50.960 --> 00:23:53.570 And the idea is by the mean value theorem, we say, look, 00:23:53.570 --> 00:23:56.090 what does the mean value theorem say? 00:23:56.090 --> 00:23:59.440 We're assuming now that 'F' is a continuous and 00:23:59.440 --> 00:24:05.420 differentiable function on an interval, OK, from 'a' to 'b'. 00:24:05.420 --> 00:24:08.210 The mean value theorem says under these conditions, there 00:24:08.210 --> 00:24:10.600 exists a number 'c' between 'a' and 00:24:10.600 --> 00:24:12.730 'b' with what property? 00:24:12.730 --> 00:24:20.260 That ''F of b' minus 'F of a'' over 'b - a' is equal to 'F 00:24:20.260 --> 00:24:22.810 prime of c'. 00:24:22.810 --> 00:24:26.260 That's just a statement of the mean value theorem. 00:24:26.260 --> 00:24:29.060 This is always true if the conditions of the mean value 00:24:29.060 --> 00:24:30.150 theorem apply. 00:24:30.150 --> 00:24:33.480 Now all we're saying is, in this particular problem, what 00:24:33.480 --> 00:24:36.130 property that capital 'F' have? 00:24:36.130 --> 00:24:38.670 It had the property that its derivative for all 00:24:38.670 --> 00:24:40.250 values of 'x' was 0. 00:24:40.250 --> 00:24:43.970 In particular then, when 'c' is the value that we're 00:24:43.970 --> 00:24:47.890 talking about, if 'F prime of x' is 0 for all values of 'x', 00:24:47.890 --> 00:24:51.400 in particular, then, it's 0 when 'x' is equal to 'c'. 00:24:51.400 --> 00:24:55.920 In other words, by our given hypothesis, this is 0. 00:24:55.920 --> 00:25:00.380 But if a fraction is 0, its numerator must be 0. 00:25:03.120 --> 00:25:04.300 That says what? 00:25:04.300 --> 00:25:07.920 'F of b' minus 'F of a' is 0. 00:25:07.920 --> 00:25:11.450 See, the only way a quotient can be 0 is for the 00:25:11.450 --> 00:25:12.370 numerator-- 00:25:12.370 --> 00:25:15.860 or the dividend, the divisor, I don't know these formal 00:25:15.860 --> 00:25:20.210 names, they slipped my mind, but the top one has to be 0. 00:25:20.210 --> 00:25:24.110 And if 'F of b' minus 'F of a' is 0, that says 'F of b' 00:25:24.110 --> 00:25:27.730 equals 'F of a', and that's precisely what we had to show 00:25:27.730 --> 00:25:30.830 to show that 'F' was a constant. 00:25:30.830 --> 00:25:31.490 OK? 00:25:31.490 --> 00:25:33.800 So again, notice, it's not that we're saying that the 00:25:33.800 --> 00:25:37.190 mean value theorem is a harder way of proving what we already 00:25:37.190 --> 00:25:40.610 know to be intuitively true, what we're saying is what? 00:25:40.610 --> 00:25:44.890 That we know that many intuitively obvious results 00:25:44.890 --> 00:25:46.380 frequently turn out to be false. 00:25:46.380 --> 00:25:49.760 We would like some analytical way of knowing which of the 00:25:49.760 --> 00:25:52.240 intuitive results are actually correct. 00:25:52.240 --> 00:25:55.125 All I'm saying is the mean value theorem gives us a big 00:25:55.125 --> 00:25:56.080 hint that way. 00:25:56.080 --> 00:25:59.750 By the way, let me close by giving you one more important 00:25:59.750 --> 00:26:01.400 illustration of what we can prove by 00:26:01.400 --> 00:26:03.030 the mean value theorem. 00:26:03.030 --> 00:26:05.810 And this is called a corollary of a corollary, as I'll show 00:26:05.810 --> 00:26:07.240 you what I mean in a minute. 00:26:07.240 --> 00:26:11.900 The next example that I want to use is what it means to say 00:26:11.900 --> 00:26:15.560 suppose I'm given two functions 'f' and 'g', and all 00:26:15.560 --> 00:26:17.790 I know about these two functions is that the 00:26:17.790 --> 00:26:19.970 derivatives are identical. 00:26:19.970 --> 00:26:23.280 In other ways, that 'f' and 'g' have the property that for 00:26:23.280 --> 00:26:26.950 every value of 'x', 'f prime of x' is equal 00:26:26.950 --> 00:26:28.320 to 'g prime of x'. 00:26:28.320 --> 00:26:31.200 By the way, when I say every value of 'x', again, it's 00:26:31.200 --> 00:26:32.890 local versus global. 00:26:32.890 --> 00:26:35.850 It's not necessary that this happens for all 'x', what is 00:26:35.850 --> 00:26:37.240 important is what? 00:26:37.240 --> 00:26:41.385 That 'x' be defined on some interval. 00:26:41.385 --> 00:26:44.340 In other words, even if I know that this property is true for 00:26:44.340 --> 00:26:47.470 some interval, I don't really care what happens outside of 00:26:47.470 --> 00:26:47.760 that interval. 00:26:47.760 --> 00:26:50.660 In terms of local properties, all I'm saying is, all I know 00:26:50.660 --> 00:26:53.300 is that for some interval, maybe the whole axis, doesn't 00:26:53.300 --> 00:26:58.390 make any difference, 'f prime' is identical to 'g prime'. 00:26:58.390 --> 00:27:01.690 Now, you would like to be able to say, maybe, that if 'f 00:27:01.690 --> 00:27:04.300 prime' is equal to 'g prime', 'f' equals 'g'. 00:27:04.300 --> 00:27:05.920 But that's not the case. 00:27:05.920 --> 00:27:08.510 What is the case is that the difference between the two 00:27:08.510 --> 00:27:10.740 functions must be a constant. 00:27:10.740 --> 00:27:14.180 Again, geometrically, what you're saying is what? 00:27:14.180 --> 00:27:20.400 That if you have two curves, which point by point always 00:27:20.400 --> 00:27:21.810 have the same slope-- 00:27:21.810 --> 00:27:23.410 in other words, for each 'x' value, the 00:27:23.410 --> 00:27:24.540 slopes are the same-- 00:27:24.540 --> 00:27:26.590 is just essentially saying that the 00:27:26.590 --> 00:27:28.530 two curves are parallel. 00:27:28.530 --> 00:27:31.130 And if they're parallel curves, what's a way of 00:27:31.130 --> 00:27:32.900 stating that two curves are parallel? 00:27:32.900 --> 00:27:36.910 That one is a constant displacement of the other. 00:27:36.910 --> 00:27:41.300 In other words, the geometric impact of two curves having 00:27:41.300 --> 00:27:44.110 the same derivative is not that the curves are the same, 00:27:44.110 --> 00:27:45.780 but that they're parallel. 00:27:45.780 --> 00:27:50.200 And by the way, the proof of this result is again a 00:27:50.200 --> 00:27:52.290 corollary to the mean value theorem. 00:27:52.290 --> 00:27:54.600 Namely, let's look at the function 'f of 00:27:54.600 --> 00:27:55.970 x' minus 'g of x'. 00:27:55.970 --> 00:27:58.000 Call that capital 'F of x'. 00:27:58.000 --> 00:28:02.100 Let capital 'F of x' be ''f of x' minus 'g of x''. 00:28:02.100 --> 00:28:04.460 Since the derivative of a difference is the difference 00:28:04.460 --> 00:28:08.430 of the derivatives, that would say the derivative of capital 00:28:08.430 --> 00:28:11.970 'F' is the derivative little 'f' minus the 00:28:11.970 --> 00:28:14.360 derivative of 'g'. 00:28:14.360 --> 00:28:15.090 OK? 00:28:15.090 --> 00:28:18.470 Now what do we know about 'f prime' and 'g prime of x'? 00:28:18.470 --> 00:28:21.250 We know that 'f prime of x' equals 'g prime 00:28:21.250 --> 00:28:22.970 of x' for all 'x'. 00:28:22.970 --> 00:28:28.180 Consequently, the difference between these two must be 0. 00:28:28.180 --> 00:28:30.550 Remember, if two functions are identical, their 00:28:30.550 --> 00:28:32.000 difference is 0. 00:28:32.000 --> 00:28:35.510 That says, therefore, that capital 'F prime of x' is 00:28:35.510 --> 00:28:36.920 identically 0. 00:28:36.920 --> 00:28:38.870 And by our previous theorem-- 00:28:38.870 --> 00:28:40.430 notice the beautiful logic of this-- 00:28:40.430 --> 00:28:44.540 from the mean value theorem, we proved that if the 00:28:44.540 --> 00:28:47.400 derivative of a function is identically 0, the function 00:28:47.400 --> 00:28:49.660 must be a constant. 00:28:49.660 --> 00:28:52.260 So we apply that here. 00:28:52.260 --> 00:28:54.020 But what was capital 'F'? 00:28:54.020 --> 00:28:57.410 It was 'little 'f - g'. 00:28:57.410 --> 00:29:00.130 And that proves our desired result. 00:29:00.130 --> 00:29:04.510 Again, what I want you to see here is that we have not done 00:29:04.510 --> 00:29:07.530 anything different with the mean value theorem. 00:29:07.530 --> 00:29:10.140 We're not trying to say we're going to prove results we 00:29:10.140 --> 00:29:11.410 couldn't prove before. 00:29:11.410 --> 00:29:12.230 Rather, what? 00:29:12.230 --> 00:29:15.970 The mean value theorem is our way of showing that certain 00:29:15.970 --> 00:29:19.540 intuitive results hold true analytically, that we can talk 00:29:19.540 --> 00:29:21.920 about parallel curves, and things like this. 00:29:21.920 --> 00:29:25.250 Most important, in terms of summarizing this lecture from 00:29:25.250 --> 00:29:28.650 a point of view of what's coming next, it's crucial to 00:29:28.650 --> 00:29:33.050 observe that this last example is what is going to allow us 00:29:33.050 --> 00:29:35.080 to enter the study of something called the 00:29:35.080 --> 00:29:36.640 'indefinite integral'. 00:29:36.640 --> 00:29:39.820 Or in another manner of speaking, something called the 00:29:39.820 --> 00:29:42.600 inverse of taking a derivative. 00:29:42.600 --> 00:29:46.290 You see, the idea is, notice that in these two examples we 00:29:46.290 --> 00:29:49.490 start with information about the derivative and deduce 00:29:49.490 --> 00:29:51.820 what's true about the original function. 00:29:51.820 --> 00:29:54.820 That's inverting the emphasis of what we've been doing up 00:29:54.820 --> 00:29:57.740 until now, where we've started with the function and 00:29:57.740 --> 00:29:59.800 investigated its derivative. 00:29:59.800 --> 00:30:02.910 To see this in more detail, join me again next time. 00:30:02.910 --> 00:30:04.520 And until next time, goodbye. 00:30:07.500 --> 00:30:10.700 Funding for the publication of this video was provided by the 00:30:10.700 --> 00:30:14.750 Gabriella and Paul Rosenbaum Foundation. 00:30:14.750 --> 00:30:18.930 Help OCW continue to provide free and open access to MIT 00:30:18.930 --> 00:30:23.130 courses by making a donation at ocw.mit.edu/donate.