WEBVTT 00:00:00.000 --> 00:00:02.330 The following content is provided under a Creative 00:00:02.330 --> 00:00:03.610 Commons license. 00:00:03.610 --> 00:00:05.990 Your support will help MIT OpenCourseWare 00:00:05.990 --> 00:00:10.010 continue to offer high quality educational resources for free. 00:00:10.010 --> 00:00:12.540 To make a donation, or to view additional materials 00:00:12.540 --> 00:00:15.870 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:15.870 --> 00:00:21.422 at ocw.mit.edu. 00:00:21.422 --> 00:00:23.380 PROFESSOR: What we're going to talk about today 00:00:23.380 --> 00:00:26.750 is a continuation of last time. 00:00:26.750 --> 00:00:29.600 I want to review Newton's method because I want to talk 00:00:29.600 --> 00:00:41.220 to you about its accuracy. 00:00:41.220 --> 00:00:46.090 So if you remember, the way Newton's method works is this. 00:00:46.090 --> 00:00:49.280 If you have a curve and you want to know 00:00:49.280 --> 00:00:52.642 whether it crosses the axis. 00:00:52.642 --> 00:00:54.100 And you don't know where this point 00:00:54.100 --> 00:01:00.100 is, this point which I'll call x here, what you do 00:01:00.100 --> 00:01:01.250 is you take a guess. 00:01:01.250 --> 00:01:03.590 Maybe you take a point x_0 here. 00:01:03.590 --> 00:01:06.640 And then you go down to this point on the graph, 00:01:06.640 --> 00:01:08.602 and you draw the tangent line. 00:01:08.602 --> 00:01:10.560 I'll draw these in a couple of different colors 00:01:10.560 --> 00:01:13.180 so that you can see the difference between them. 00:01:13.180 --> 00:01:14.410 So here's a tangent line. 00:01:14.410 --> 00:01:18.280 It's coming out like that. 00:01:18.280 --> 00:01:20.750 And that one is going to get a little closer 00:01:20.750 --> 00:01:23.090 to our target point. 00:01:23.090 --> 00:01:26.550 But now the trick is, and this is rather hard 00:01:26.550 --> 00:01:30.060 to see because the scale gets small incredibly fast, 00:01:30.060 --> 00:01:32.500 is that if you go right up from that, 00:01:32.500 --> 00:01:34.420 and you do this same trick over again. 00:01:34.420 --> 00:01:38.400 That is, this is your second guess, x_1, and now 00:01:38.400 --> 00:01:41.360 you draw the second tangent line. 00:01:41.360 --> 00:01:44.680 Which is going to come down this way. 00:01:44.680 --> 00:01:46.290 That's really close. 00:01:46.290 --> 00:01:48.570 You can see here on the chalkboard, 00:01:48.570 --> 00:01:51.740 it's practically the same as the dot of x. 00:01:51.740 --> 00:01:54.610 So that's the next guess. 00:01:54.610 --> 00:01:56.860 Which is x_2. 00:01:56.860 --> 00:02:03.620 And I want to analyze, now, how close it gets. 00:02:03.620 --> 00:02:05.990 And just describe to you how it works. 00:02:05.990 --> 00:02:09.060 So let me just remind you of the formulas, too. 00:02:09.060 --> 00:02:11.150 It's worth having them in your head. 00:02:11.150 --> 00:02:19.240 So the formula for the next one is this. 00:02:19.240 --> 00:02:23.570 And then the idea is just to repeat this process. 00:02:23.570 --> 00:02:28.720 Which has a fancy name, in algorithms, which 00:02:28.720 --> 00:02:31.210 is to iterate, if you like. 00:02:31.210 --> 00:02:32.530 So we repeat the process. 00:02:32.530 --> 00:02:35.720 And that means, for example, we generate x_2 from x_1 00:02:35.720 --> 00:02:41.730 by the same formula. 00:02:41.730 --> 00:02:43.170 And we did this last time. 00:02:43.170 --> 00:02:47.930 And, more generally, the (n+1)st is generated from the nth 00:02:47.930 --> 00:02:55.150 guess, by this formula here. 00:02:55.150 --> 00:03:02.820 So what I'd like to do is just draw the picture of one step 00:03:02.820 --> 00:03:03.920 a little bit more closely. 00:03:03.920 --> 00:03:05.580 So I want to blow up the picture, which 00:03:05.580 --> 00:03:10.880 is above me there. 00:03:10.880 --> 00:03:11.950 That's a little too high. 00:03:11.950 --> 00:03:16.750 Where are my erasers? 00:03:16.750 --> 00:03:18.390 Got to get it a little lower than that, 00:03:18.390 --> 00:03:21.890 since I'm going to depict everything above the line here. 00:03:21.890 --> 00:03:24.180 So here's my curve coming down. 00:03:24.180 --> 00:03:30.620 And suppose that x_1 is here, so this is directly 00:03:30.620 --> 00:03:32.170 above it is this point here. 00:03:32.170 --> 00:03:35.040 And then as I drew it, this green 00:03:35.040 --> 00:03:38.530 tangent coming down like that. 00:03:38.530 --> 00:03:42.970 It's a little bit closer, and this was the place, x_2, 00:03:42.970 --> 00:03:46.440 and then here was x, our target, which 00:03:46.440 --> 00:03:49.390 is where the curve crosses as opposed to the straight tangent 00:03:49.390 --> 00:03:52.600 line crossing. 00:03:52.600 --> 00:03:56.310 So that's the picture that I want you to keep in mind. 00:03:56.310 --> 00:04:00.440 And now, we're just going to do a very qualitative kind 00:04:00.440 --> 00:04:03.450 of error analysis. 00:04:03.450 --> 00:04:12.960 So here's our error analysis. 00:04:12.960 --> 00:04:19.355 And we're starting out, the distance between x_1 and x 00:04:19.355 --> 00:04:20.480 is what we want to measure. 00:04:20.480 --> 00:04:23.530 In other words, how close we are to where we're heading. 00:04:23.530 --> 00:04:26.600 And so I've called that, I'm going to call that Error 1. 00:04:26.600 --> 00:04:31.440 That's x - x1, in absolute value. 00:04:31.440 --> 00:04:37.010 And then, E_2 would be x - x_2, in absolute value. 00:04:37.010 --> 00:04:39.100 And so forth. 00:04:39.100 --> 00:04:44.190 And, last time, when I was estimating the size of this-- 00:04:44.190 --> 00:04:45.870 So E_n would be whatever it was. 00:04:45.870 --> 00:04:51.900 Last time, remember, we did it for a specific case. 00:04:51.900 --> 00:04:56.020 So last time, I actually wrote down the numbers. 00:04:56.020 --> 00:04:57.700 And they were these numbers, maybe 00:04:57.700 --> 00:05:00.365 you could call them E_n, which was the absolute value 00:05:00.365 --> 00:05:03.270 of square root of 5 minus x_n. 00:05:03.270 --> 00:05:06.610 These are the sizes that I was writing down last time. 00:05:06.610 --> 00:05:11.620 And I just want to talk about in general what to expect. 00:05:11.620 --> 00:05:13.480 That worked amazingly well, and I 00:05:13.480 --> 00:05:16.990 want to show you that that's true pretty much in general. 00:05:16.990 --> 00:05:23.660 So the first distance, again, is E_1, is this distance here. 00:05:23.660 --> 00:05:25.190 That's the E_1. 00:05:25.190 --> 00:05:27.570 And then this shorter distance, here, 00:05:27.570 --> 00:05:37.410 this little bit, which I'll mark maybe in green, is E_2. 00:05:37.410 --> 00:05:40.970 So how much shorter is E_1 than E_2? 00:05:40.970 --> 00:05:44.080 Well, the idea is pretty simple. 00:05:44.080 --> 00:05:46.895 It's that if this distance and this vertical distance, 00:05:46.895 --> 00:05:49.520 they are probably about the same as the perpendicular distance. 00:05:49.520 --> 00:05:51.550 And this is basically the situation 00:05:51.550 --> 00:05:54.140 of a curve touching a tangent line. 00:05:54.140 --> 00:05:58.010 Then the separation is going to be quadratic. 00:05:58.010 --> 00:06:00.150 And that's basically what's going to happen. 00:06:00.150 --> 00:06:03.780 So, in other words the distance E_2 00:06:03.780 --> 00:06:06.530 is going to be about the square of the distance E_1. 00:06:06.530 --> 00:06:10.579 And that's really the only feature of this 00:06:10.579 --> 00:06:11.620 that I want to point out. 00:06:11.620 --> 00:06:14.175 So, approximately, this is the situation 00:06:14.175 --> 00:06:17.880 that we're going to get. 00:06:17.880 --> 00:06:21.690 And so what that means is, and maybe 00:06:21.690 --> 00:06:24.380 thinking from last time, what we had was 00:06:24.380 --> 00:06:25.760 something roughly like this. 00:06:25.760 --> 00:06:29.060 You have an E_0, you have an E_1, you have an E_2, 00:06:29.060 --> 00:06:31.590 you have an E_3, and so forth. 00:06:31.590 --> 00:06:34.370 Maybe I'll write down E_4 here. 00:06:34.370 --> 00:06:37.330 And last time, this was about 10^(-1). 00:06:37.330 --> 00:06:39.650 So the expectation based on this rule 00:06:39.650 --> 00:06:42.710 is that the next error's the square of the previous one. 00:06:42.710 --> 00:06:44.500 So that's 10^(-2). 00:06:44.500 --> 00:06:47.140 The next one is the square of the previous one. 00:06:47.140 --> 00:06:49.090 So that's 10^(-4). 00:06:49.090 --> 00:06:52.069 And the next one is the square of that, that's 10^(-8). 00:06:52.069 --> 00:06:53.110 And this one is 10^(-16). 00:06:55.980 --> 00:06:59.550 So the thing that's impressive about this list of numbers 00:06:59.550 --> 00:07:02.780 is you can see that the number of digits of accuracy 00:07:02.780 --> 00:07:09.590 doubles at each stage. 00:07:09.590 --> 00:07:20.760 Accuracy doubles at each step. 00:07:20.760 --> 00:07:23.710 The number of digits of accuracy doubles at each step. 00:07:23.710 --> 00:07:28.150 So, very, very quickly you get past the accuracy 00:07:28.150 --> 00:07:31.150 of your calculator, as you saw on your problem set. 00:07:31.150 --> 00:07:34.220 And this thing works beautifully. 00:07:34.220 --> 00:07:42.700 So, let me just summarize by saying that Newton's method 00:07:42.700 --> 00:07:49.680 works very well. 00:07:49.680 --> 00:07:52.250 By which I mean this kind of rate. 00:07:52.250 --> 00:07:55.450 And I want to be just slightly specific. 00:07:55.450 --> 00:08:00.490 If-- there are really two conditions disguised in this, 00:08:00.490 --> 00:08:01.480 that are going on. 00:08:01.480 --> 00:08:09.890 One is that f' has to be, not to big-- to be not small. 00:08:09.890 --> 00:08:19.624 And f'' has to be not too big. 00:08:19.624 --> 00:08:21.290 That's roughly speaking what's going on. 00:08:21.290 --> 00:08:23.320 I'll explain these in just a second. 00:08:23.320 --> 00:08:34.050 And x_0 starts nearby. 00:08:34.050 --> 00:08:37.490 Nearby the target value x. 00:08:37.490 --> 00:08:42.050 So that's really what's going on here. 00:08:42.050 --> 00:08:44.610 So let me just illustrate to you. 00:08:44.610 --> 00:08:48.920 So I'm not going to explain this, except to say the reason 00:08:48.920 --> 00:08:51.120 why this second derivative gets involved 00:08:51.120 --> 00:08:53.360 is that it's how curved the curve is, 00:08:53.360 --> 00:08:55.330 that how far away you get. 00:08:55.330 --> 00:08:57.507 If the second derivative were 0, that 00:08:57.507 --> 00:08:58.840 would be the best possible case. 00:08:58.840 --> 00:09:00.660 Then we would get it on the nose. 00:09:00.660 --> 00:09:02.890 If the second derivative is not too big, 00:09:02.890 --> 00:09:05.360 that means the quadratic part is not too big. 00:09:05.360 --> 00:09:07.820 So we don't get away very far from the green line 00:09:07.820 --> 00:09:14.110 to the curve itself. 00:09:14.110 --> 00:09:17.460 The other thing to say is, as I said, 00:09:17.460 --> 00:09:19.050 that x_0 needs to start nearby. 00:09:19.050 --> 00:09:21.520 So I'll explain that by explaining 00:09:21.520 --> 00:09:23.310 what maybe could go wrong. 00:09:23.310 --> 00:09:40.080 So the ways the method can fail, and one example which actually 00:09:40.080 --> 00:09:45.080 would have happened in our example from last time, 00:09:45.080 --> 00:09:51.750 which was y = x^2 - 5, is suppose we'd started x_0 over 00:09:51.750 --> 00:09:54.840 here. 00:09:54.840 --> 00:09:57.430 Then this thing would have gone off to the left, 00:09:57.430 --> 00:10:00.713 and we would have landed on not the square root 00:10:00.713 --> 00:10:03.030 of 5 but the other root. 00:10:03.030 --> 00:10:11.750 So if it's too far away, then we get the wrong root. 00:10:11.750 --> 00:10:15.590 So that's an example, explaining that the x_0 needs 00:10:15.590 --> 00:10:18.710 to start near the root that we're talking about. 00:10:18.710 --> 00:10:21.460 Otherwise, the method doesn't know 00:10:21.460 --> 00:10:22.940 which root you're asking for. 00:10:22.940 --> 00:10:24.580 It only knows where you started. 00:10:24.580 --> 00:10:27.310 So it may go off to the wrong place. 00:10:27.310 --> 00:10:34.430 OK, it can't read your mind. 00:10:34.430 --> 00:10:35.350 Yes, question. 00:10:35.350 --> 00:10:42.610 STUDENT: [INAUDIBLE] 00:10:42.610 --> 00:10:44.460 PROFESSOR: Oh, good question. 00:10:44.460 --> 00:10:49.550 So the question was, what if the first error is larger than 1. 00:10:49.550 --> 00:10:51.220 Are you in trouble? 00:10:51.220 --> 00:10:56.000 And the answer is, absolutely, yes. 00:10:56.000 --> 00:10:58.110 If you have quadratic behavior, you can see. 00:10:58.110 --> 00:11:00.800 If you have a quadratic nearby, it's 00:11:00.800 --> 00:11:02.820 pretty close to the straight line. 00:11:02.820 --> 00:11:05.900 But far away, a parabola is miles from a straight line. 00:11:05.900 --> 00:11:08.180 It's way, way, way far away. 00:11:08.180 --> 00:11:14.010 So if you're foolish enough to start over here, 00:11:14.010 --> 00:11:17.150 you may have some trouble making progress. 00:11:17.150 --> 00:11:20.430 Actually, it isn't, when I-- that little wiggle 00:11:20.430 --> 00:11:22.010 there just meant proportional to. 00:11:22.010 --> 00:11:24.570 In fact, in the particular case of a parabola, 00:11:24.570 --> 00:11:25.810 it manages to get back. 00:11:25.810 --> 00:11:27.150 It saves itself. 00:11:27.150 --> 00:11:30.010 But there's no guarantee of that sort of thing. 00:11:30.010 --> 00:11:32.790 You really do want to start reasonably close. 00:11:32.790 --> 00:11:33.350 Yep. 00:11:33.350 --> 00:11:40.140 STUDENT: [INAUDIBLE] 00:11:40.140 --> 00:11:42.580 PROFESSOR: What you have to do is you have to watch out. 00:11:42.580 --> 00:11:46.790 That is, it's hard to know what assumptions to make about x_0. 00:11:46.790 --> 00:11:50.250 You plug it into the machine and you see what you get. 00:11:50.250 --> 00:11:53.227 And either it works or it doesn't. 00:11:53.227 --> 00:11:55.560 You can tell that it's marching toward a specific place, 00:11:55.560 --> 00:11:58.400 and you can tell that that place probably is a zero, usually. 00:11:58.400 --> 00:12:00.510 But maybe it's not the one you were looking for. 00:12:00.510 --> 00:12:02.384 So in other words, you have to use your head. 00:12:02.384 --> 00:12:05.230 You run the program and then you see what it does. 00:12:05.230 --> 00:12:08.000 And if you're lucky-- the problem is, if you have no idea 00:12:08.000 --> 00:12:12.270 where the zero is, you may just wander around forever. 00:12:12.270 --> 00:12:14.720 As we'll see in a second. 00:12:14.720 --> 00:12:20.870 So the next example here is the following. 00:12:20.870 --> 00:12:24.590 I said that f' has to be not too small. 00:12:24.590 --> 00:12:26.970 There's a real catastrophe hiding just 00:12:26.970 --> 00:12:28.250 inside this picture. 00:12:28.250 --> 00:12:30.935 Which is the transition between when you find the positive root 00:12:30.935 --> 00:12:32.890 and when you find the negative root here. 00:12:32.890 --> 00:12:35.480 Which is, if you're right down here. 00:12:35.480 --> 00:12:37.860 If you were foolish enough to get 0, 00:12:37.860 --> 00:12:41.060 then what's going to happen is your tangent line 00:12:41.060 --> 00:12:41.940 is horizontal. 00:12:41.940 --> 00:12:44.610 It doesn't even meet the axis. 00:12:44.610 --> 00:12:47.840 So in the formula, you can see that's a catastrophe. 00:12:47.840 --> 00:12:53.030 Because there's an f' in the denominator. 00:12:53.030 --> 00:12:53.530 So that's 0. 00:12:53.530 --> 00:12:54.791 That's undefined. 00:12:54.791 --> 00:12:56.290 It's not surprising, it's consistent 00:12:56.290 --> 00:12:59.480 that the parallel line doesn't meet the axis. 00:12:59.480 --> 00:13:01.940 And you have no x_1. 00:13:01.940 --> 00:13:06.370 So you had-- So if you like, another point here 00:13:06.370 --> 00:13:12.870 is that f' = 0 is a disaster. 00:13:12.870 --> 00:13:24.310 A disaster for the method. 00:13:24.310 --> 00:13:28.620 Because the next-- So say, if f(x_0) = 0, 00:13:28.620 --> 00:13:34.700 then x_1 is undefined. 00:13:34.700 --> 00:13:39.170 And finally, there's one other weird thing that can happen. 00:13:39.170 --> 00:13:43.460 Which is, which I'll just draw a picture of schematically. 00:13:43.460 --> 00:13:47.150 Which you can get from a wiggle. 00:13:47.150 --> 00:13:49.260 So this wiggle has three roots. 00:13:49.260 --> 00:13:52.060 The way I've drawn it. 00:13:52.060 --> 00:13:56.720 And it can be that you can start over here with your x_0. 00:13:56.720 --> 00:14:01.200 And draw your tangent line and go over here to an x_1. 00:14:01.200 --> 00:14:05.510 And then that tangent line will take you right back to the x_0. 00:14:05.510 --> 00:14:09.560 I didn't draw it quite right, but that's about right. 00:14:09.560 --> 00:14:11.330 So it goes over like this. 00:14:11.330 --> 00:14:13.520 So let me draw the two tangent lines, so 00:14:13.520 --> 00:14:14.760 that you can see it properly. 00:14:14.760 --> 00:14:16.614 Sorry, I messed it up. 00:14:16.614 --> 00:14:18.030 So here are the two tangent lines. 00:14:18.030 --> 00:14:20.870 This guy and this guy. 00:14:20.870 --> 00:14:25.950 And it just goes back and forth. x_0 cycles to x_1, 00:14:25.950 --> 00:14:27.520 and x_1 goes back to x_0. 00:14:27.520 --> 00:14:30.200 We have a cycle. 00:14:30.200 --> 00:14:32.090 And it never goes anywhere. 00:14:32.090 --> 00:14:34.915 This is, the grass is always greener. 00:14:34.915 --> 00:14:36.540 It's over here, it thinks, oh, I really 00:14:36.540 --> 00:14:39.000 would prefer to go to this zero and then it 00:14:39.000 --> 00:14:40.840 thinks oh, I want to go back. 00:14:40.840 --> 00:14:43.650 And it goes back and forth, and back and forth. 00:14:43.650 --> 00:14:47.060 Grass is always greener on the other side of the fence. 00:14:47.060 --> 00:14:50.090 Never, never gets anywhere. 00:14:50.090 --> 00:14:52.510 So those are the sorts of things that can go wrong 00:14:52.510 --> 00:14:53.490 with Newton's method. 00:14:53.490 --> 00:14:55.210 Nevertheless, it's fantastic. 00:14:55.210 --> 00:14:59.740 It works very well, in a lot of situations. 00:14:59.740 --> 00:15:03.510 And solves basically any equation that you can imagine, 00:15:03.510 --> 00:15:10.894 numerically. 00:15:10.894 --> 00:15:12.060 Next we're going to move on. 00:15:12.060 --> 00:15:13.518 We're going to move on to something 00:15:13.518 --> 00:15:15.730 which is a little theoretical. 00:15:15.730 --> 00:15:18.170 Which is the mean value theorem. 00:15:18.170 --> 00:15:22.410 And that will allow us in just a day or so 00:15:22.410 --> 00:15:26.150 to launch into the ideas of integration, 00:15:26.150 --> 00:15:31.140 which is the whole second half of the course. 00:15:31.140 --> 00:15:50.200 So let's get started with that. 00:15:50.200 --> 00:15:57.030 The mean value theorem will henceforth be abbreviated MVT. 00:15:57.030 --> 00:15:58.630 So I don't have to write quite as 00:15:58.630 --> 00:16:03.750 much every time I refer to it. 00:16:03.750 --> 00:16:07.290 The mean value theorem, colloquially, 00:16:07.290 --> 00:16:09.530 says the following. 00:16:09.530 --> 00:16:23.040 If you go from Boston to LA, which I think a lot of Red Sox 00:16:23.040 --> 00:16:31.110 fans are going to want to do soon, so that's 3,000 miles. 00:16:31.110 --> 00:16:47.800 In 6 hours, then at some time you 00:16:47.800 --> 00:16:55.640 are going at a certain speed. 00:16:55.640 --> 00:17:01.360 The average of this speed. 00:17:01.360 --> 00:17:08.850 Average, so, speed, which in this case is what? 00:17:08.850 --> 00:17:10.770 So we're going at the average speed. 00:17:10.770 --> 00:17:16.890 That's 3,000 miles times 6 hours, 00:17:16.890 --> 00:17:21.420 so that's 500 miles per hour. 00:17:21.420 --> 00:17:23.340 Exactly. 00:17:23.340 --> 00:17:26.442 So some time on your journey-- of course, some of the time 00:17:26.442 --> 00:17:28.150 you're going more than 500 miles an hour, 00:17:28.150 --> 00:17:29.610 sometimes you are going less. 00:17:29.610 --> 00:17:33.220 And some time you must've been going 500 miles an hour 00:17:33.220 --> 00:17:35.110 exactly. 00:17:35.110 --> 00:17:37.520 That's the mean value theorem. 00:17:37.520 --> 00:17:39.890 The reason why it's called mean value theorem 00:17:39.890 --> 00:17:55.440 is that word mean is the same as the word average. 00:17:55.440 --> 00:18:08.160 So now I'm going to state it in math symbols, the same theorem. 00:18:08.160 --> 00:18:10.470 And it's a formula. 00:18:10.470 --> 00:18:16.275 It says that the difference quotient 00:18:16.275 --> 00:18:20.410 - so this is the distance traveled 00:18:20.410 --> 00:18:25.570 divided by the time elapsed, that's the average speed - 00:18:25.570 --> 00:18:35.320 is equal to the infinitesimal speed for some time in between. 00:18:35.320 --> 00:18:48.590 So some c, which is in between a and b-- I'm not quite done. 00:18:48.590 --> 00:18:53.130 It's a real theorem, it has hypotheses. 00:18:53.130 --> 00:18:56.230 I've told you the conclusion first, 00:18:56.230 --> 00:18:58.030 but there are some hypotheses, they're 00:18:58.030 --> 00:18:59.650 straightforward hypotheses. 00:18:59.650 --> 00:19:03.320 Provided f is differentiable, that is, 00:19:03.320 --> 00:19:12.820 it has a derivative in the interval a < x < b. 00:19:12.820 --> 00:19:21.560 And continuous in a <= x <= b. 00:19:29.110 --> 00:19:32.080 There has to be a sense that you can make out of the speed, 00:19:32.080 --> 00:19:36.140 or the rate of change of f at each intermediate point. 00:19:36.140 --> 00:19:40.200 And in order for the values at the ends to make sense, 00:19:40.200 --> 00:19:41.200 it has to be continuous. 00:19:41.200 --> 00:19:44.470 There has to be a link between the values at the ends 00:19:44.470 --> 00:19:47.409 and what's going on in between. 00:19:47.409 --> 00:19:48.950 If it were discontinuous, there would 00:19:48.950 --> 00:19:52.330 be no relation between the left and right values 00:19:52.330 --> 00:19:55.970 and the rest of the function. 00:19:55.970 --> 00:20:00.160 So here's the theorem, conclusion and its hypothesis. 00:20:00.160 --> 00:20:11.780 And it means what I said more colloquially up above. 00:20:11.780 --> 00:20:14.540 Now, I'm going to prove this theorem immediately. 00:20:14.540 --> 00:20:18.110 At least, give a geometric intuitive argument, 00:20:18.110 --> 00:20:21.220 which is not very different from the one that's given 00:20:21.220 --> 00:20:26.410 in a very systematic treatment. 00:20:26.410 --> 00:20:34.280 So here's the proof of the mean value theorem. 00:20:34.280 --> 00:20:36.860 It's really just a picture. 00:20:36.860 --> 00:20:42.880 So here's a place, and here's another place on the graph. 00:20:42.880 --> 00:20:47.180 And the graph is going along like this, let's say. 00:20:47.180 --> 00:20:50.700 And this line here is the secant line. 00:20:50.700 --> 00:20:55.310 So this is (a, f(a)) down here. 00:20:55.310 --> 00:20:59.320 And this is (b, f(b)) up there. 00:20:59.320 --> 00:21:02.720 And this segment is the secant, its slope 00:21:02.720 --> 00:21:04.480 is the slope that we're aiming for. 00:21:04.480 --> 00:21:09.400 The slope of that line is the left-hand side of this formula 00:21:09.400 --> 00:21:11.190 here. 00:21:11.190 --> 00:21:14.062 So we need to find something with that slope. 00:21:14.062 --> 00:21:16.520 And what we need to find is a tangent line with that slope, 00:21:16.520 --> 00:21:18.340 because what's on the right-hand side 00:21:18.340 --> 00:21:20.220 is the slope of a tangent line. 00:21:20.220 --> 00:21:22.340 So here's how we construct it. 00:21:22.340 --> 00:21:27.240 We take a parallel line, down here. 00:21:27.240 --> 00:21:31.125 And then we just translate it up, leaving it parallel, 00:21:31.125 --> 00:21:32.750 we move it up. 00:21:32.750 --> 00:21:34.080 Towards this one. 00:21:34.080 --> 00:21:38.410 Until it touches. 00:21:38.410 --> 00:21:43.040 And where it touches, at this point of tangency, down there, 00:21:43.040 --> 00:21:47.490 I've just found my value of c. 00:21:47.490 --> 00:21:49.830 And you can see that if the tangent line is parallel 00:21:49.830 --> 00:21:53.180 to this line, that's exactly the equation we want. 00:21:53.180 --> 00:21:59.910 So this thing has slope f'(c). 00:21:59.910 --> 00:22:07.150 And this other one has slope equal to this complicated 00:22:07.150 --> 00:22:15.970 expression, (f(b) - f(a)) / (b - a). 00:22:15.970 --> 00:22:19.840 That is almost the end of the proof. 00:22:19.840 --> 00:22:25.330 There's one problem. 00:22:25.330 --> 00:22:29.880 So, again, we move a parallel line up. 00:22:29.880 --> 00:22:43.310 Move up the parallel line until it touches. 00:22:43.310 --> 00:22:46.300 There's a little subtlety here, which I just want to emphasize. 00:22:46.300 --> 00:22:49.210 Which is that that dotted line keeps on going here. 00:22:49.210 --> 00:22:51.220 But when we bring it up, we're going 00:22:51.220 --> 00:22:54.500 to ignore what's happening outside of a. 00:22:54.500 --> 00:22:56.870 And beyond b, alright? 00:22:56.870 --> 00:23:01.590 So we're just going to ignore the rest of the graph. 00:23:01.590 --> 00:23:06.090 But there is one thing that can go wrong. 00:23:06.090 --> 00:23:16.800 So if it does not touch, then the picture looks likes this. 00:23:16.800 --> 00:23:18.230 Here are the same two points. 00:23:18.230 --> 00:23:20.240 And the graph is all above. 00:23:20.240 --> 00:23:22.060 And we brought up our thing. 00:23:22.060 --> 00:23:23.960 And it went like that. 00:23:23.960 --> 00:23:27.480 So we didn't construct a tangent line. 00:23:27.480 --> 00:23:29.520 If this happens. 00:23:29.520 --> 00:23:31.980 So we're in trouble, in that point. 00:23:31.980 --> 00:23:37.010 In this situation, sorry. 00:23:37.010 --> 00:23:40.860 But there's a trick, which is a straightforward trick. 00:23:40.860 --> 00:23:55.810 Then bring the tangent lines down from the top. 00:23:55.810 --> 00:23:58.040 So parallel lines, sorry, not tangent lines. 00:23:58.040 --> 00:24:06.580 Parallel lines. 00:24:06.580 --> 00:24:11.600 From above. 00:24:11.600 --> 00:24:16.380 So, that's the whole story. 00:24:16.380 --> 00:24:43.460 That's how we cook up this point c, with the right properties. 00:24:43.460 --> 00:24:46.640 I want to point out just one more theoretical thing. 00:24:46.640 --> 00:24:50.087 And then the rest, we're going to be drawing conclusions. 00:24:50.087 --> 00:24:51.670 So there's one more theoretical remark 00:24:51.670 --> 00:24:55.490 about the proof, which is something that is 00:24:55.490 --> 00:24:59.130 fairly important to understand. 00:24:59.130 --> 00:25:01.640 When you understand a proof, you should always 00:25:01.640 --> 00:25:06.170 be thinking about why the hypotheses are necessary. 00:25:06.170 --> 00:25:08.280 Where do I use the hypothesis. 00:25:08.280 --> 00:25:10.890 And I want to give you an example the proof doesn't 00:25:10.890 --> 00:25:15.350 work to show you that the hypothesis is an important one. 00:25:15.350 --> 00:25:17.230 So the example is the following. 00:25:17.230 --> 00:25:18.710 I'll just take a function which is 00:25:18.710 --> 00:25:22.400 two straight lines like this. 00:25:22.400 --> 00:25:27.920 And if you try to perform this trick with these things, 00:25:27.920 --> 00:25:32.380 then it's going to come up and it's going to touch here. 00:25:32.380 --> 00:25:36.070 But the problem is that the tangent line is not 00:25:36.070 --> 00:25:36.740 defined here. 00:25:36.740 --> 00:25:38.660 There are lots of tangents, and there's 00:25:38.660 --> 00:25:40.610 no derivative at this point. 00:25:40.610 --> 00:25:44.860 So the derivative doesn't exist here. 00:25:44.860 --> 00:25:57.230 So this is the claim that one bad point ruins the proof. 00:25:57.230 --> 00:26:08.730 We need f' to exist at all-- so, f'(x) to exist at all x 00:26:08.730 --> 00:26:14.340 in between. 00:26:14.340 --> 00:26:30.980 Can't get away even with one defective point. 00:26:30.980 --> 00:26:40.620 Now it's time to draw some consequences. 00:26:40.620 --> 00:26:45.670 And the main consequence is going 00:26:45.670 --> 00:26:57.880 to have to do with applications to graphing. 00:26:57.880 --> 00:27:01.090 But we'll see tomorrow and for the rest of the course 00:27:01.090 --> 00:27:03.920 that this is even more significant. 00:27:03.920 --> 00:27:09.790 It's significant to all the rest of calculus. 00:27:09.790 --> 00:27:12.440 I'm going to list three consequences which you're 00:27:12.440 --> 00:27:14.890 quite familiar with already. 00:27:14.890 --> 00:27:27.550 So, the first one is if f' is positive, then f is increasing. 00:27:27.550 --> 00:27:40.430 And the second one is if f' is negative, then f is decreasing. 00:27:40.430 --> 00:27:44.130 And the last one seems like the simplest. 00:27:44.130 --> 00:27:48.460 But even this one alone is the key to everything. 00:27:48.460 --> 00:28:03.340 If f' = 0, then f is constant. 00:28:03.340 --> 00:28:13.490 These are three consequences, now, of the mean value theorem. 00:28:13.490 --> 00:28:17.130 And let me show you how they're proved. 00:28:17.130 --> 00:28:22.870 I just told you that they were true, maybe, a while ago. 00:28:22.870 --> 00:28:27.030 And certainly I mentioned the first two. 00:28:27.030 --> 00:28:30.200 The last one was so simple that we maybe just swept it 00:28:30.200 --> 00:28:30.810 under the rug. 00:28:30.810 --> 00:28:36.570 You did use it on a problem set, once or twice. 00:28:36.570 --> 00:28:39.140 But it turns out that this actually requires proof, 00:28:39.140 --> 00:28:48.010 and we're going to give the proof right now. 00:28:48.010 --> 00:28:52.190 The way that the proof goes is simply to write down, 00:28:52.190 --> 00:28:54.500 to rewrite star. 00:28:54.500 --> 00:28:59.440 Rewrite our formula. 00:28:59.440 --> 00:29:05.440 Which says that (f(b) - f(a)) / (b - a) = f'(c). 00:29:10.050 --> 00:29:13.730 And you see I've written it from left to right here 00:29:13.730 --> 00:29:16.110 to say that the right-hand side information 00:29:16.110 --> 00:29:17.590 about the derivative is going to be 00:29:17.590 --> 00:29:19.340 giving the information about the function. 00:29:19.340 --> 00:29:22.230 That's the way I'm going to read it. 00:29:22.230 --> 00:29:26.360 In order to express this, though, I'm 00:29:26.360 --> 00:29:30.760 going to just rewrite it a couple of times here. 00:29:30.760 --> 00:29:37.360 So here's f(a), multiplying through by the denominator. 00:29:37.360 --> 00:29:40.470 And now I'm going to write it in another customary form 00:29:40.470 --> 00:29:42.300 for the mean value theorem. 00:29:42.300 --> 00:29:46.120 Which is f(b) = f(a) + f'(c) (b-a). 00:29:51.300 --> 00:29:53.524 So here's another version. 00:29:53.524 --> 00:29:55.440 I should probably have put this one in the box 00:29:55.440 --> 00:29:59.870 to begin with anyway. 00:29:59.870 --> 00:30:03.800 And, just changing it around algebraically, 00:30:03.800 --> 00:30:07.160 it's this fact here. 00:30:07.160 --> 00:30:13.310 They're the same thing. 00:30:13.310 --> 00:30:17.760 And now with the formula written in this form, 00:30:17.760 --> 00:30:24.060 I claim that I can check these three facts. 00:30:24.060 --> 00:30:26.680 Let's start with the first one. 00:30:26.680 --> 00:30:33.580 I'm going to set things up always so that a < b. 00:30:33.580 --> 00:30:36.840 And that's the setup of the theorem. 00:30:36.840 --> 00:30:42.010 And so that means that b - a is positive. 00:30:42.010 --> 00:30:48.190 Which means that this factor over here is a positive number. 00:30:48.190 --> 00:30:56.000 If f' is positive, which is what happens 00:30:56.000 --> 00:30:58.830 in the first case, that's the assumption that we're making, 00:30:58.830 --> 00:31:01.210 then this is a positive number. 00:31:01.210 --> 00:31:02.750 And so f(b) > f(a). 00:31:08.190 --> 00:31:09.565 Which means that it's increasing. 00:31:09.565 --> 00:31:13.900 It goes up as the value goes up. 00:31:13.900 --> 00:31:20.780 Similarly, if f'(c) is negative, then this is a positive times 00:31:20.780 --> 00:31:22.830 a negative number, this is negative. 00:31:22.830 --> 00:31:25.610 f(b) < f(a). 00:31:25.610 --> 00:31:37.850 So it goes the other way. 00:31:37.850 --> 00:31:39.520 Maybe I'll write this way. 00:31:39.520 --> 00:31:49.910 And finally, if f'(c) = 0, then f(b) = f(a). 00:31:49.910 --> 00:31:52.110 Which if you apply it to all possible ends, 00:31:52.110 --> 00:31:56.110 means if you can do it for every interval, which you can, 00:31:56.110 --> 00:31:57.940 then that means that f is constant. 00:31:57.940 --> 00:32:12.640 It never gets to change values. 00:32:12.640 --> 00:32:17.260 Well you might have believed these facts already. 00:32:17.260 --> 00:32:20.670 But I just want to emphasize to you that this turns out 00:32:20.670 --> 00:32:23.670 to be the one key link between infinitesimals, 00:32:23.670 --> 00:32:27.610 between limits and these actual differences. 00:32:27.610 --> 00:32:30.240 Before, we were saying that the difference quotient 00:32:30.240 --> 00:32:32.530 was approximately equal to the derivative. 00:32:32.530 --> 00:32:35.520 Now we're saying that it's exactly equal to a derivative. 00:32:35.520 --> 00:32:38.930 Although we don't know exactly which point to use. 00:32:38.930 --> 00:32:47.150 It's some point in between. 00:32:47.150 --> 00:32:49.900 I'm going to be deducing some other consequences in a second, 00:32:49.900 --> 00:32:52.187 but let me stop for second to make sure 00:32:52.187 --> 00:32:53.270 that everybody's on board. 00:32:53.270 --> 00:32:56.880 Especially since I've finished the blackboards here. 00:32:56.880 --> 00:32:59.510 Before we-- everybody happy? 00:32:59.510 --> 00:33:00.100 One question. 00:33:00.100 --> 00:33:08.617 STUDENT: [INAUDIBLE] 00:33:08.617 --> 00:33:10.950 PROFESSOR: I'm just going to repeat your question first. 00:33:10.950 --> 00:33:12.950 I'm a little bit confused, you said, 00:33:12.950 --> 00:33:16.230 about what guarantees that there's a point of tangency. 00:33:16.230 --> 00:33:19.030 That's what you said. 00:33:19.030 --> 00:33:20.780 So do you want to elaborate, or do you 00:33:20.780 --> 00:33:23.152 want to want to stop with what you just said? 00:33:23.152 --> 00:33:24.360 What is it that confuses you? 00:33:24.360 --> 00:33:28.764 STUDENT: [INAUDIBLE] 00:33:28.764 --> 00:33:29.430 PROFESSOR: Yeah. 00:33:29.430 --> 00:33:43.500 STUDENT: [INAUDIBLE] 00:33:43.500 --> 00:33:46.305 PROFESSOR: So I'm not claiming that there's only one point. 00:33:46.305 --> 00:33:48.180 This could wiggle a lot of times and it maybe 00:33:48.180 --> 00:33:49.990 touches at ten places. 00:33:49.990 --> 00:33:54.640 In other words, it's OK with me if it touches more than once. 00:33:54.640 --> 00:33:56.830 Then I just have more, the more the merrier. 00:33:56.830 --> 00:33:59.550 In other words, I don't want there necessarily only 00:33:59.550 --> 00:34:00.050 to be one. 00:34:00.050 --> 00:34:02.720 It could come down like this. 00:34:02.720 --> 00:34:05.140 And touch a second time. 00:34:05.140 --> 00:34:09.290 Is that what was concerning you? 00:34:09.290 --> 00:34:11.580 So in mathematics, when we claim that this 00:34:11.580 --> 00:34:14.410 is true for some point, we don't necessarily 00:34:14.410 --> 00:34:16.210 mean that it doesn't work for others. 00:34:16.210 --> 00:34:18.310 In fact, if the function is constant, 00:34:18.310 --> 00:34:25.960 this is 0 and in fact this equation is true for every c. 00:34:25.960 --> 00:34:30.840 That satisfies your question? 00:34:30.840 --> 00:34:33.620 The fact that this point exists actually is a touchy point. 00:34:33.620 --> 00:34:35.320 I just convinced you of it visually. 00:34:35.320 --> 00:34:39.030 It's a geometric issue, whether you're allowed to do this. 00:34:39.030 --> 00:34:41.840 Indeed, it has to do with the existence 00:34:41.840 --> 00:34:44.150 of tangent lines and more analysis 00:34:44.150 --> 00:34:46.120 than we can do in this class. 00:34:46.120 --> 00:34:46.620 Yeah. 00:34:46.620 --> 00:34:47.328 Another question. 00:34:47.328 --> 00:34:48.880 STUDENT: [INAUDIBLE] 00:34:48.880 --> 00:34:50.437 PROFESSOR: Pardon me. 00:34:50.437 --> 00:34:51.270 STUDENT: [INAUDIBLE] 00:34:51.270 --> 00:34:52.686 PROFESSOR: The question is, what's 00:34:52.686 --> 00:34:56.780 the difference between this and the linear approximation. 00:34:56.780 --> 00:35:11.010 And I think, let me see if I can describe that. 00:35:11.010 --> 00:35:12.520 I'll leave the theorem on the board. 00:35:12.520 --> 00:35:14.490 I'm going to get rid of the colloquial version 00:35:14.490 --> 00:35:19.380 of the theorem. 00:35:19.380 --> 00:35:26.200 And I'll try to describe to you the difference between this 00:35:26.200 --> 00:35:32.130 and the linear approximation. 00:35:32.130 --> 00:35:34.220 I was planning to do that in a while, 00:35:34.220 --> 00:35:36.890 but we'll do it right now since that's what you're asking. 00:35:36.890 --> 00:35:37.660 That's fine. 00:35:37.660 --> 00:35:45.570 So here's the situation. 00:35:45.570 --> 00:35:51.660 The linear approximation, so let's say comparison 00:35:51.660 --> 00:35:57.390 with linear approximation. 00:35:57.390 --> 00:35:59.710 They're very closely related. 00:35:59.710 --> 00:36:02.710 The linear approximation says the change in f over the change 00:36:02.710 --> 00:36:06.290 in x, that's the left-hand side of this thing, 00:36:06.290 --> 00:36:09.980 is approximately f'(a). 00:36:09.980 --> 00:36:21.200 For b near a, and b - a = delta x. 00:36:21.200 --> 00:36:23.110 This statement, which is in the box, which 00:36:23.110 --> 00:36:25.980 is sitting right up there, is the statement 00:36:25.980 --> 00:36:31.330 that this change in f is actually equal to something. 00:36:31.330 --> 00:36:33.180 Not approximately equal to it. 00:36:33.180 --> 00:36:37.670 It's equal to f' of some c. 00:36:37.670 --> 00:36:41.330 And the problem here is that we don't know exactly which c. 00:36:41.330 --> 00:36:43.930 This is for some c. 00:36:43.930 --> 00:36:53.190 Between a and b. 00:36:53.190 --> 00:36:54.810 Right, so. 00:36:54.810 --> 00:36:59.190 That's the difference between the two. 00:36:59.190 --> 00:37:19.700 And let me elaborate a little bit. 00:37:19.700 --> 00:37:27.050 If you're trying to understand what (f(b) - f(a)) / (b - 00:37:27.050 --> 00:37:32.840 a) is, the mean value theorem is telling you for sure that 00:37:32.840 --> 00:37:35.530 it's equal to this f'(c). 00:37:35.530 --> 00:37:38.230 So that means it's less than or equal to the largest 00:37:38.230 --> 00:37:44.280 possible value on the-- largest value you can get, for sure. 00:37:44.280 --> 00:37:48.110 And this is on the whole interval. 00:37:48.110 --> 00:37:49.980 And I'm going to include the ends, 00:37:49.980 --> 00:37:54.050 because when you take a max it's sometimes achieved at the ends. 00:37:54.050 --> 00:37:58.770 And similarly, because it's f'(c), it's definitely bigger 00:37:58.770 --> 00:38:07.030 than the min on this same interval here. 00:38:07.030 --> 00:38:13.420 This is all you can say based on the mean value theorem. 00:38:13.420 --> 00:38:14.620 All you know is this. 00:38:14.620 --> 00:38:17.710 And colloquially, what that means 00:38:17.710 --> 00:38:25.690 is that the average speed is between the maximum 00:38:25.690 --> 00:38:29.000 and the minimum. 00:38:29.000 --> 00:38:31.450 Not very surprising. 00:38:31.450 --> 00:38:33.120 The mean value theorem is supposed 00:38:33.120 --> 00:38:36.480 to be very intuitively obvious. 00:38:36.480 --> 00:38:39.370 It's saying the average speed is trapped 00:38:39.370 --> 00:38:42.850 between the maximum speed and the minimum speed. 00:38:42.850 --> 00:38:44.830 For sure, that's something, that's 00:38:44.830 --> 00:38:48.340 why-- incidentally this wasn't really proved when 00:38:48.340 --> 00:38:50.350 Newton and Leibniz were around. 00:38:50.350 --> 00:38:52.860 But, let's write this so that you can read it. 00:38:52.860 --> 00:39:01.350 Average speed is between the max and the min. 00:39:01.350 --> 00:39:04.960 But nobody had any trouble, they didn't disbelieve it 00:39:04.960 --> 00:39:09.750 because it's a very natural thing. 00:39:09.750 --> 00:39:16.980 Now if, for example, I take any kind of linear approximation, 00:39:16.980 --> 00:39:25.670 say, for instance, e^x is approximately 1 + x. 00:39:25.670 --> 00:39:30.620 Then I'm making the guess-- no, I don't want to say this yet. 00:39:30.620 --> 00:39:35.830 That's not going to explain it to you well enough. 00:39:35.830 --> 00:39:38.690 What we're saying, so this is the mean value here. 00:39:38.690 --> 00:39:40.840 This is what the mean value theorem says. 00:39:40.840 --> 00:39:47.270 And here's the linear approximation. 00:39:47.270 --> 00:39:52.140 The linear approximation is saying that the average speed 00:39:52.140 --> 00:39:59.550 is approximately the initial speed, or possibly 00:39:59.550 --> 00:40:01.900 the final speed. 00:40:01.900 --> 00:40:07.140 So if a is the left endpoint, then it's the initial speed. 00:40:07.140 --> 00:40:09.630 If it happens to be the right endpoint, if the value of x 00:40:09.630 --> 00:40:13.460 is to the left then it's the final speed. 00:40:13.460 --> 00:40:16.640 So those are the-- so you can see it's approximately right. 00:40:16.640 --> 00:40:19.180 Because the speed, when you're on a short interval, 00:40:19.180 --> 00:40:20.530 shouldn't be varying very much. 00:40:20.530 --> 00:40:22.696 The max and the min should be pretty close together. 00:40:22.696 --> 00:40:25.630 So that's why the linear approximation is reasonable. 00:40:25.630 --> 00:40:27.800 And this is telling you absolutely, 00:40:27.800 --> 00:40:34.140 it's no less than the min and no more than the max. 00:40:34.140 --> 00:40:34.640 Yeah. 00:40:34.640 --> 00:40:40.965 STUDENT: [INAUDIBLE] 00:40:40.965 --> 00:40:42.090 PROFESSOR: The little kink? 00:40:42.090 --> 00:40:46.074 STUDENT: [INAUDIBLE] 00:40:46.074 --> 00:40:47.740 PROFESSOR: If you approach from the top. 00:40:47.740 --> 00:40:50.507 So if it's still under here I can show you it again. 00:40:50.507 --> 00:40:51.590 Oh yeah, it's still there. 00:40:51.590 --> 00:40:52.090 Good. 00:40:52.090 --> 00:40:54.630 STUDENT: [INAUDIBLE] 00:40:54.630 --> 00:40:56.830 PROFESSOR: Oh, the one with the wiggle on top? 00:40:56.830 --> 00:40:57.872 Yeah, this one you can't. 00:40:57.872 --> 00:40:59.704 Because there's nothing to touch and it also 00:40:59.704 --> 00:41:02.160 fails from the bottom because there's this bad point. 00:41:02.160 --> 00:41:04.217 From the top, it could work. 00:41:04.217 --> 00:41:05.550 It can certainly work both ways. 00:41:05.550 --> 00:41:07.760 So, for example. 00:41:07.760 --> 00:41:09.940 See if you're a machine, you maybe 00:41:09.940 --> 00:41:11.310 don't have a way of doing this. 00:41:11.310 --> 00:41:14.330 But if you're a human being you can spot all the places. 00:41:14.330 --> 00:41:17.990 There are a bunch of spots where the slope is right. 00:41:17.990 --> 00:41:20.770 And it's perfectly OK. 00:41:20.770 --> 00:41:21.780 All of them work. 00:41:21.780 --> 00:41:25.139 STUDENT: [INAUDIBLE] 00:41:25.139 --> 00:41:26.930 PROFESSOR: It's not that the c is the same. 00:41:26.930 --> 00:41:29.470 It's just we've now found one, two, three, four, 00:41:29.470 --> 00:41:31.160 five c's for which it works. 00:41:31.160 --> 00:41:40.390 STUDENT: [INAUDIBLE] PROFESSOR: If you're asked to find a c, 00:41:40.390 --> 00:41:44.897 so first of all that's kind of a phony question. 00:41:44.897 --> 00:41:46.730 There are some questions on your problem set 00:41:46.730 --> 00:41:48.340 which ask you to find a c. 00:41:48.340 --> 00:41:51.850 That actually is struggling to get you to understand what 00:41:51.850 --> 00:41:54.310 the statement of the mean value theorem is, 00:41:54.310 --> 00:41:58.890 but you should not pay a lot of attention to those questions. 00:41:58.890 --> 00:42:01.870 They're not very impressive. 00:42:01.870 --> 00:42:04.860 But, of course, you would have to find all the-- if it asked 00:42:04.860 --> 00:42:06.110 you to find one, you find one. 00:42:06.110 --> 00:42:10.950 If you can find some more, fine. 00:42:10.950 --> 00:42:13.470 You can pick whichever one you want. 00:42:13.470 --> 00:42:16.560 Mean value theorem just doesn't care. 00:42:16.560 --> 00:42:18.190 The mean value theorem doesn't care 00:42:18.190 --> 00:42:21.060 because actually, the mean value theorem is never 00:42:21.060 --> 00:42:28.020 used except to-- in real life, except in this context here. 00:42:28.020 --> 00:42:31.225 You can never nail down which c it 00:42:31.225 --> 00:42:33.140 is, so the only thing you can say 00:42:33.140 --> 00:42:35.980 is that you're going slower than the maximum speed 00:42:35.980 --> 00:42:40.414 and faster than the minimum speed. 00:42:40.414 --> 00:42:41.330 Sorry, say that again? 00:42:41.330 --> 00:42:47.057 STUDENT: [INAUDIBLE] 00:42:47.057 --> 00:42:48.890 PROFESSOR: If you're asked for a specific c, 00:42:48.890 --> 00:42:51.080 you have to find a specific c. 00:42:51.080 --> 00:42:53.070 And it has to be in the range. 00:42:53.070 --> 00:43:04.420 In between, it has to be in here. 00:43:04.420 --> 00:43:07.880 So now I want to tell you about another kind of application, 00:43:07.880 --> 00:43:11.100 which is really just a consequence of what 00:43:11.100 --> 00:43:22.070 I've described here. 00:43:22.070 --> 00:43:26.050 I should emphasize, by the way, this, probably, 00:43:26.050 --> 00:43:27.420 should be doing this. 00:43:27.420 --> 00:43:32.120 I guess we've never used this color here. 00:43:32.120 --> 00:43:32.950 It's popular. 00:43:32.950 --> 00:43:33.640 This is pink. 00:43:33.640 --> 00:43:35.480 So this one is so good. 00:43:35.480 --> 00:43:47.360 So since we're going to do this. 00:43:47.360 --> 00:43:50.960 So the reason why the exclamation points 00:43:50.960 --> 00:43:54.150 are temporary, this is such an obvious fact. 00:43:54.150 --> 00:43:57.857 But this is the way that you're going 00:43:57.857 --> 00:43:59.440 to want to use the mean value theorem, 00:43:59.440 --> 00:44:01.650 and this is the only way you need to understand 00:44:01.650 --> 00:44:02.670 the mean value theorem. 00:44:02.670 --> 00:44:06.520 On your test, or ever in your whole life. 00:44:06.520 --> 00:44:10.630 So this is the way it will be used. 00:44:10.630 --> 00:44:17.140 As I will make very clear when we review for the exam. 00:44:17.140 --> 00:44:18.714 In practice what happens is you even 00:44:18.714 --> 00:44:21.130 forget about the mean value theorem, and what you remember 00:44:21.130 --> 00:44:24.360 is these three properties here. 00:44:24.360 --> 00:44:26.000 Which are themselves consequences 00:44:26.000 --> 00:44:27.720 of the mean value theorem. 00:44:27.720 --> 00:44:31.950 So these are the ones that I want to illustrate now. 00:44:31.950 --> 00:44:35.710 In my next discussion here. 00:44:35.710 --> 00:44:42.420 I'm just going to talk about inequalities. 00:44:42.420 --> 00:44:46.930 Inequalities are relationships between functions. 00:44:46.930 --> 00:44:50.170 And I'm going to prove a couple of them using the properties 00:44:50.170 --> 00:44:52.300 over there, the properties that functions 00:44:52.300 --> 00:44:56.900 with positive derivatives are increasing. 00:44:56.900 --> 00:44:58.900 Here's an example. 00:44:58.900 --> 00:45:08.390 e^x > 1 + x, where x > 0. 00:45:08.390 --> 00:45:10.940 The proof is the following. 00:45:10.940 --> 00:45:16.070 I consider-- So here's a proof. 00:45:16.070 --> 00:45:21.550 I consider the function f(x), which is the difference. 00:45:21.550 --> 00:45:22.490 e^x - (1+x). 00:45:27.360 --> 00:45:37.140 I observe that it starts at f(0) equal to, well, that's e^0 - 00:45:37.140 --> 00:45:42.110 (1+0), which is 0. 00:45:42.110 --> 00:45:45.510 And, it keeps on going. 00:45:45.510 --> 00:45:50.310 f'(x) is e^x, if I differentiate here, the 1 goes away. 00:45:50.310 --> 00:45:54.117 I get minus 1. 00:45:54.117 --> 00:45:55.700 That's the derivative of the function. 00:45:55.700 --> 00:46:03.750 And this function, because e^x > 1 for x positive, is positive. 00:46:03.750 --> 00:46:07.095 As x gets bigger and bigger, this rate of increase 00:46:07.095 --> 00:46:08.290 is positive. 00:46:08.290 --> 00:46:13.940 And therefore, three dots, that's therefore, 00:46:13.940 --> 00:46:23.350 f(x) is bigger than its starting place, for x > 0. 00:46:23.350 --> 00:46:27.100 If it's increasing, then that's-- in particular, 00:46:27.100 --> 00:46:28.730 it's increasing starting from 0. 00:46:28.730 --> 00:46:30.340 So this is true. 00:46:30.340 --> 00:46:36.000 Now, all I have to do is read what this inequality says. 00:46:36.000 --> 00:46:39.860 And what it says is that e^x, just plug in for f(x), 00:46:39.860 --> 00:46:45.130 which is right here, minus (1+x) is greater than the starting 00:46:45.130 --> 00:46:48.590 value, which was 0. 00:46:48.590 --> 00:46:52.050 Now, I put the thing that's negative on the other side. 00:46:52.050 --> 00:47:01.550 So that's the same thing as e^x > 1 + x. 00:47:01.550 --> 00:47:04.330 That's a typical inequality. 00:47:04.330 --> 00:47:11.150 And now, we'll use this principle again. 00:47:11.150 --> 00:47:12.900 Oh gee, I erased the wrong thing. 00:47:12.900 --> 00:47:15.490 I erased the statement and not the proof. 00:47:15.490 --> 00:47:23.340 Well, hide the proof. 00:47:23.340 --> 00:47:27.500 The next thing I want to prove to you is that e^x > 1 + x + 00:47:27.500 --> 00:47:28.000 x^2 / 2. 00:47:33.090 --> 00:47:34.300 So, how do I do that? 00:47:34.300 --> 00:47:42.174 I introduce a function g(x), which is e^x minus this. 00:47:42.174 --> 00:47:44.340 And now, I'm just going to do exactly the same thing 00:47:44.340 --> 00:47:45.780 I did before. 00:47:45.780 --> 00:47:51.870 Which is, I get started with g(0), which is 1 - 1. 00:47:51.870 --> 00:47:53.680 Which is 0. 00:47:53.680 --> 00:48:00.690 And g'(x) is e^x minus - now, look at what happens when I 00:48:00.690 --> 00:48:03.590 differentiate this. 00:48:03.590 --> 00:48:04.870 The 1 goes away. 00:48:04.870 --> 00:48:10.600 The x gives me a 1, and the x^2 / 2 gives me a plus x. 00:48:10.600 --> 00:48:18.190 And this one is positive for x > 0, because of step 1. 00:48:18.190 --> 00:48:21.840 Because of the previous one that I did. 00:48:21.840 --> 00:48:28.340 So this one is increasing. g is increasing. 00:48:28.340 --> 00:48:33.350 Which says that g(x) > g(0). 00:48:33.350 --> 00:48:36.620 And if you just read that off, it's exactly the same 00:48:36.620 --> 00:48:41.690 as our inequality here. e^x > 1 + x + x^2 / 2. 00:48:48.270 --> 00:48:54.010 Now, you can keep on going with this essentially forever. 00:48:54.010 --> 00:48:58.510 And let me just write down what you get. 00:48:58.510 --> 00:49:04.772 You get e^x is greater than 1 plus x plus x^2 / 2, 00:49:04.772 --> 00:49:06.480 the next one turns out to be x^3 / (3*2). 00:49:10.460 --> 00:49:11.730 x^4 / (4*3*2). 00:49:14.850 --> 00:49:16.820 And you can do whatever you want. 00:49:16.820 --> 00:49:19.490 You can do others. 00:49:19.490 --> 00:49:22.860 And this is like the tortoise and the hare. 00:49:22.860 --> 00:49:27.740 This is the tortoise, and this is the hare, it's always ahead. 00:49:27.740 --> 00:49:31.910 But eventually, if you go infinitely far, it catches up. 00:49:31.910 --> 00:49:38.190 So this turns out to be exactly equal to e^x in the limit. 00:49:38.190 --> 00:49:41.280 And we'll talk about that maybe at the end of the course.