WEBVTT 00:00:00.090 --> 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 --> 00:00:16.190 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.190 --> 00:00:21.740 at ocw.mit.edu. 00:00:21.740 --> 00:00:22.240 PROF. 00:00:22.240 --> 00:00:24.660 JERISON: So, we're ready to begin Lecture 10, 00:00:24.660 --> 00:00:28.860 and what I'm going to begin with is by finishing up 00:00:28.860 --> 00:00:33.880 some things from last time. 00:00:33.880 --> 00:00:42.460 We'll talk about approximations, and I 00:00:42.460 --> 00:00:47.650 want to fill in a number of comments 00:00:47.650 --> 00:00:52.440 and get you a little bit more oriented in the point of view 00:00:52.440 --> 00:00:55.980 that I'm trying to express about approximations. 00:00:55.980 --> 00:00:58.790 So, first of all, I want to remind you 00:00:58.790 --> 00:01:03.800 of the actual applied example that I wrote down last time. 00:01:03.800 --> 00:01:08.730 So that was this business here. 00:01:08.730 --> 00:01:11.590 There was something from special relativity. 00:01:11.590 --> 00:01:15.990 And the approximation that we used was the linear 00:01:15.990 --> 00:01:22.170 approximation, with a -1/2 power that comes out to be T(1 + 1/2 00:01:22.170 --> 00:01:22.670 v^2/c^2). 00:01:27.340 --> 00:01:30.850 I want to reiterate why this is a useful way of thinking 00:01:30.850 --> 00:01:31.930 of things. 00:01:31.930 --> 00:01:34.830 And why this is that this comes up in real life. 00:01:34.830 --> 00:01:37.090 Why this is maybe more important than everything 00:01:37.090 --> 00:01:39.820 that I've taught you about technically so far. 00:01:39.820 --> 00:01:46.750 So, first of all, what this is telling us is the change in T 00:01:46.750 --> 00:01:50.430 divided by T, if you do the arithmetic here and subtract T, 00:01:50.430 --> 00:01:55.970 that's using the change in T is T' - T here. 00:01:55.970 --> 00:02:01.440 If you work that out, this is approximately the same as 1/2 00:02:01.440 --> 00:02:04.140 v^2/c^2. 00:02:04.140 --> 00:02:06.270 So what is this saying? 00:02:06.270 --> 00:02:08.580 This is saying that if you have this satellite, which 00:02:08.580 --> 00:02:12.610 is going at speed v, and little c is the speed of light, 00:02:12.610 --> 00:02:17.060 then the change in the watch down here on earth, 00:02:17.060 --> 00:02:20.770 relative to the time on the satellite, 00:02:20.770 --> 00:02:24.050 is going to be proportional to this ratio here. 00:02:24.050 --> 00:02:25.710 So, physically, this makes sense. 00:02:25.710 --> 00:02:28.120 This is time divided by time. 00:02:28.120 --> 00:02:30.670 And this is velocity squared divided by velocity squared. 00:02:30.670 --> 00:02:32.620 So, in each case, the units divide out. 00:02:32.620 --> 00:02:34.880 So this is a dimensionless quantity. 00:02:34.880 --> 00:02:36.960 And this is a dimensionless quantity. 00:02:36.960 --> 00:02:40.840 And the only point here that we're trying to make 00:02:40.840 --> 00:02:43.440 is just this notion of proportionality. 00:02:43.440 --> 00:02:45.470 So I want to write this down. 00:02:45.470 --> 00:02:48.010 Just-- in summary. 00:02:48.010 --> 00:02:50.210 So the error fraction, if you like, 00:02:50.210 --> 00:02:53.630 which is sort of the number of significant digits 00:02:53.630 --> 00:03:03.090 that we have in our measurement, is proportional, in this case, 00:03:03.090 --> 00:03:04.860 to this quantity. 00:03:04.860 --> 00:03:09.030 It happens to be proportional to this quantity here. 00:03:09.030 --> 00:03:16.980 And the factor is, happens to be, 1/2. 00:03:16.980 --> 00:03:21.350 So these proportionality factors are what we're looking for. 00:03:21.350 --> 00:03:22.404 Their rates of change. 00:03:22.404 --> 00:03:23.820 Their rates of change of something 00:03:23.820 --> 00:03:25.550 with respect to something else. 00:03:25.550 --> 00:03:28.800 Now, on your homework, you have something 00:03:28.800 --> 00:03:30.740 rather similar to this. 00:03:30.740 --> 00:03:38.460 So in Problem, on Part 2B, Part II, Problem 1, 00:03:38.460 --> 00:03:41.580 there's the speed of a pitch, right? 00:03:41.580 --> 00:03:44.100 And the speed of the pitch is changing depending 00:03:44.100 --> 00:03:45.600 on how high the mound is. 00:03:45.600 --> 00:03:47.460 And the point here is that that's 00:03:47.460 --> 00:03:50.150 approximately proportional to the change 00:03:50.150 --> 00:03:52.320 in the height of the mound. 00:03:52.320 --> 00:03:54.290 In that problem, we had this delta h, 00:03:54.290 --> 00:03:56.830 that was the x variable in that problem. 00:03:56.830 --> 00:03:58.330 And what you're trying to figure out 00:03:58.330 --> 00:04:02.904 is what the constant of proportionality is. 00:04:02.904 --> 00:04:04.820 That's what you're aiming for in this problem. 00:04:04.820 --> 00:04:08.070 So there's a linear relationship, approximately, 00:04:08.070 --> 00:04:11.890 to all intents and purposes this is an equality. 00:04:11.890 --> 00:04:13.290 Because the lower order terms are 00:04:13.290 --> 00:04:14.720 unimportant for the problem. 00:04:14.720 --> 00:04:18.020 Just as over here, this function is a little bit complicated. 00:04:18.020 --> 00:04:19.800 This function is a little more simple. 00:04:19.800 --> 00:04:22.800 For the purposes of this problem, they are the same. 00:04:22.800 --> 00:04:27.370 Because the errors are negligible 00:04:27.370 --> 00:04:29.760 for the particular problem that we're working on. 00:04:29.760 --> 00:04:34.210 So we might as well work with the simpler relationship. 00:04:34.210 --> 00:04:37.621 And similarly, over here, so you could do this with, 00:04:37.621 --> 00:04:39.120 in this case with square roots, it's 00:04:39.120 --> 00:04:41.980 not so hard here with reciprocals of square roots. 00:04:41.980 --> 00:04:45.300 It's also not terribly hard to do it numerically. 00:04:45.300 --> 00:04:47.850 And the reason why we're not doing it numerically 00:04:47.850 --> 00:04:51.260 is that, as I say, this is something that 00:04:51.260 --> 00:04:53.560 happens all across engineering. 00:04:53.560 --> 00:04:55.640 People are looking for these linear relationships 00:04:55.640 --> 00:05:00.240 between the change in some input and the change in the output. 00:05:00.240 --> 00:05:03.210 And if you don't make these simplifications, 00:05:03.210 --> 00:05:06.760 then when you get, say, a dozen of them together, 00:05:06.760 --> 00:05:09.540 you can't figure out what's going on. 00:05:09.540 --> 00:05:12.640 In this case the design of the satellite, it's very important. 00:05:12.640 --> 00:05:15.160 The speed actually isn't just one speed. 00:05:15.160 --> 00:05:19.070 Because it's the relative speed of you to the satellite. 00:05:19.070 --> 00:05:21.349 And you might be-- it depends on your angle of sight 00:05:21.349 --> 00:05:22.890 with the satellite what the speed is. 00:05:22.890 --> 00:05:24.300 So it varies quite a bit. 00:05:24.300 --> 00:05:26.280 So you really need this rule of thumb. 00:05:26.280 --> 00:05:28.372 Then there are all kinds of other considerations 00:05:28.372 --> 00:05:29.080 in this question. 00:05:29.080 --> 00:05:30.852 Like, for example, there's the fact 00:05:30.852 --> 00:05:33.530 that we're sitting on Earth and so we're rotating around 00:05:33.530 --> 00:05:36.250 on what's called a non-inertial frame. 00:05:36.250 --> 00:05:38.721 So there's the question of that acceleration. 00:05:38.721 --> 00:05:41.220 There's the question that the gravity that I experience here 00:05:41.220 --> 00:05:44.830 on Earth is not the same as up at the satellite. 00:05:44.830 --> 00:05:47.870 And that also creates a difference in time, 00:05:47.870 --> 00:05:49.580 as a result of general relativity. 00:05:49.580 --> 00:05:53.400 So all of these considerations come down 00:05:53.400 --> 00:05:56.030 to formulas which are this complicated or maybe 00:05:56.030 --> 00:05:56.910 a tiny bit more. 00:05:56.910 --> 00:05:58.330 Not really that much. 00:05:58.330 --> 00:06:00.700 And then people simplify them enormously to these very 00:06:00.700 --> 00:06:02.090 simple-minded rules. 00:06:02.090 --> 00:06:05.130 And they don't keep track of what's going on. 00:06:05.130 --> 00:06:06.830 So in order to design the system, 00:06:06.830 --> 00:06:08.960 you must make these simplifications, 00:06:08.960 --> 00:06:12.630 otherwise you can't even think about what's going on. 00:06:12.630 --> 00:06:13.860 This comes up in everything. 00:06:13.860 --> 00:06:17.422 In weather forecasting, economic forecasting. 00:06:17.422 --> 00:06:19.880 Figuring out whether there's going to be an asteroid that's 00:06:19.880 --> 00:06:22.140 going to hit the Earth. 00:06:22.140 --> 00:06:23.970 Every single one of these things involves 00:06:23.970 --> 00:06:27.900 dozens of these considerations. 00:06:27.900 --> 00:06:29.951 OK, there was a question that I saw, here. 00:06:29.951 --> 00:06:30.450 Yes. 00:06:30.450 --> 00:06:38.821 STUDENT: [INAUDIBLE] 00:06:38.821 --> 00:06:39.320 PROF. 00:06:39.320 --> 00:06:40.120 JERISON: Yeah. 00:06:40.120 --> 00:06:41.950 Basically, any problem where you have 00:06:41.950 --> 00:06:43.840 a derivative, the rate of change also 00:06:43.840 --> 00:06:46.017 depends upon what the base point is. 00:06:46.017 --> 00:06:46.850 That's the question. 00:06:46.850 --> 00:06:49.070 You're saying, doesn't this delta v also 00:06:49.070 --> 00:06:51.050 depend, I had a base point in that problem. 00:06:51.050 --> 00:06:53.290 I happened to decide that pitchers pitch on average 00:06:53.290 --> 00:06:55.350 about 90 miles an hour. 00:06:55.350 --> 00:06:58.327 Whereas, in fact, some pitchers pitch at 100 miles an hour, 00:06:58.327 --> 00:07:00.410 some pitch at 80 miles an hour, and of course they 00:07:00.410 --> 00:07:02.040 vary the speed of the pitch. 00:07:02.040 --> 00:07:03.510 And so, this varies a little bit. 00:07:03.510 --> 00:07:05.426 In fact, that's sort of a second order effect. 00:07:05.426 --> 00:07:08.220 It does change the constant of proportionality. 00:07:08.220 --> 00:07:11.790 It's a rate of change at a different base point. 00:07:11.790 --> 00:07:14.230 Which we're considering fixed. 00:07:14.230 --> 00:07:17.060 In fact, that's sort of a second order effect. 00:07:17.060 --> 00:07:19.700 When you actually do the computations, what you discover 00:07:19.700 --> 00:07:21.940 is that it doesn't make that much difference. 00:07:21.940 --> 00:07:22.630 To the a. 00:07:22.630 --> 00:07:25.040 And that's something that you get from experience. 00:07:25.040 --> 00:07:28.980 That it turns out, which things matter and which things don't. 00:07:28.980 --> 00:07:31.820 And yet again, that's exactly the same sort of consideration 00:07:31.820 --> 00:07:34.472 but at the next order of what I'm talking about here is. 00:07:34.472 --> 00:07:36.430 You have to have enough experience with numbers 00:07:36.430 --> 00:07:38.992 to know that if you take, if you vary something a little bit 00:07:38.992 --> 00:07:40.950 it's not going to change the answer that you're 00:07:40.950 --> 00:07:42.680 looking for very much. 00:07:42.680 --> 00:07:45.530 And that's exactly the point that I'm making. 00:07:45.530 --> 00:07:51.830 So I can't make them all at once, all such points. 00:07:51.830 --> 00:07:55.290 So that's my pitch for understanding things 00:07:55.290 --> 00:07:56.470 from this point of view. 00:07:56.470 --> 00:08:02.000 Now, we're going to go on, now, to quadratic approximations, 00:08:02.000 --> 00:08:13.290 which are a little more complicated. 00:08:13.290 --> 00:08:15.920 So, we talked a little bit about this last time 00:08:15.920 --> 00:08:16.850 but I didn't finish. 00:08:16.850 --> 00:08:19.930 So I want to finish this up. 00:08:19.930 --> 00:08:22.320 And the first thing that I should say 00:08:22.320 --> 00:08:30.740 is that you use these when the linear approximation is not 00:08:30.740 --> 00:08:35.680 enough. 00:08:35.680 --> 00:08:38.270 OK, so, that's something that you really 00:08:38.270 --> 00:08:40.390 need to get a little experience with. 00:08:40.390 --> 00:08:43.810 In economics, I told you they use logarithms. 00:08:43.810 --> 00:08:46.030 So sometimes they use log-linear functions. 00:08:46.030 --> 00:08:48.010 Sometimes they use log-quadratic functions 00:08:48.010 --> 00:08:49.860 when the log-linear ones don't work. 00:08:49.860 --> 00:08:53.472 So most modeling in economics is with log-quadratic functions. 00:08:53.472 --> 00:08:55.680 And if you've made it any more complicated than that, 00:08:55.680 --> 00:08:56.740 it's useless. 00:08:56.740 --> 00:08:57.850 And it's a mess. 00:08:57.850 --> 00:08:58.810 And people don't do it. 00:08:58.810 --> 00:09:02.790 So they stick with the quadratic ones, typically. 00:09:02.790 --> 00:09:07.010 So the basic formula here, and I'm going to take the base 00:09:07.010 --> 00:09:14.380 point to be 0, is that f(x) is approximately f(0) + f'(0) x. 00:09:14.380 --> 00:09:16.100 That's the linear part. 00:09:16.100 --> 00:09:17.174 Plus this extra term. 00:09:17.174 --> 00:09:18.090 Which is f''(0)/2 x^2. 00:09:21.440 --> 00:09:27.740 And this is supposed to work for x near 0. 00:09:27.740 --> 00:09:34.310 So I've chosen the base point as simply as possible. 00:09:34.310 --> 00:09:38.060 So here's more or less where we left off last time. 00:09:38.060 --> 00:09:41.710 And one thing that I said I was going to explain, 00:09:41.710 --> 00:09:48.390 which I will now, is why it's 1/2 f''(0). 00:09:48.390 --> 00:09:51.610 So we need to know that. 00:09:51.610 --> 00:09:54.770 So let's work that out here first of all. 00:09:54.770 --> 00:09:57.290 So I'm just going to do it by example. 00:09:57.290 --> 00:09:59.860 So if you like, the answer is just, well, 00:09:59.860 --> 00:10:03.600 what happens when you have a parabola? 00:10:03.600 --> 00:10:06.640 A parabola's a quadratic. 00:10:06.640 --> 00:10:09.250 It had better-- its quadratic approximation had better 00:10:09.250 --> 00:10:10.314 be itself. 00:10:10.314 --> 00:10:11.480 It's got to be the best one. 00:10:11.480 --> 00:10:13.010 So it's got to be itself. 00:10:13.010 --> 00:10:15.240 So this formula, if it's going to work, 00:10:15.240 --> 00:10:21.890 has to work on the nose, for quadratic functions. 00:10:21.890 --> 00:10:23.770 So, let's take a look. 00:10:23.770 --> 00:10:28.130 If I differentiate, I get b + 2cx. 00:10:28.130 --> 00:10:32.230 If I differentiate a second time, I get 2c. 00:10:32.230 --> 00:10:34.580 And now let's plug it in. 00:10:34.580 --> 00:10:39.450 Well, we can recover, what is it that we want to recover? 00:10:39.450 --> 00:10:42.150 We want to recover these numbers a, b and c 00:10:42.150 --> 00:10:45.930 using the derivatives evaluated at 0. 00:10:45.930 --> 00:10:49.500 So let's see. 00:10:49.500 --> 00:10:51.210 It's pretty easy, actually. 00:10:51.210 --> 00:10:52.569 f(0) = a. 00:10:52.569 --> 00:10:53.360 That's on the nose. 00:10:53.360 --> 00:10:58.070 If you plug in x = 0 here, these terms drop out and you get a. 00:10:58.070 --> 00:11:02.220 And now, f'(0), whoops that was wrong. 00:11:02.220 --> 00:11:05.100 So I wrote f' but what I meant was f. 00:11:05.100 --> 00:11:07.950 So f(0) is a. 00:11:07.950 --> 00:11:09.200 Let's back up. 00:11:09.200 --> 00:11:13.130 f(0) is a, so if I plug in x = 0 I get a. 00:11:13.130 --> 00:11:18.220 Now, f'(0), that's this next formula here, 00:11:18.220 --> 00:11:21.180 f'(0), I plug in 0 here, and I get b. 00:11:21.180 --> 00:11:22.220 That's also good. 00:11:22.220 --> 00:11:24.220 And that's exactly what the linear approximation 00:11:24.220 --> 00:11:25.540 is supposed to be. 00:11:25.540 --> 00:11:28.480 But now you notice, f'' is 2c. 00:11:28.480 --> 00:11:34.440 So to recover c, I better take half of it. 00:11:34.440 --> 00:11:35.120 And that's it. 00:11:35.120 --> 00:11:38.010 That's the reason. 00:11:38.010 --> 00:11:40.690 There's no chance that any other formula could work. 00:11:40.690 --> 00:11:45.050 And this one does. 00:11:45.050 --> 00:11:50.310 So that's the explanation for the formula. 00:11:50.310 --> 00:11:54.215 And now I remind you that I had a collection of basic formulas 00:11:54.215 --> 00:11:55.090 written on the board. 00:11:55.090 --> 00:12:00.700 And I want to just make sure we know all of them again. 00:12:00.700 --> 00:12:07.580 So, first of all, there was sine x is approximately x. 00:12:07.580 --> 00:12:12.610 Cosine x is approximately 1 - 1/2 x^2. 00:12:12.610 --> 00:12:19.230 And e^x is approximately 1 + x + 1/2 x^2. 00:12:19.230 --> 00:12:22.230 So those were three that I mentioned last time. 00:12:22.230 --> 00:12:28.980 And, again, this is all for x near 0. 00:12:28.980 --> 00:12:31.160 All for x near 0 only. 00:12:31.160 --> 00:12:35.360 These are wildly wrong far away, but near 0 they're 00:12:35.360 --> 00:12:37.730 nice, good, quadratic approximations. 00:12:37.730 --> 00:12:39.300 Now, the other two approximations 00:12:39.300 --> 00:12:44.960 that I want to mention are the logarithm, 00:12:44.960 --> 00:12:46.960 and we use the base point shifted. 00:12:46.960 --> 00:12:50.500 So we can put it at x near 0. 00:12:50.500 --> 00:12:54.545 And this one - sorry, this is an approximately equals sign there 00:12:54.545 --> 00:12:59.410 - turns out to be x - 1/2 x^2. 00:12:59.410 --> 00:13:08.979 And the last one is (1+x)^r, which turns out to be 1 + rx + 00:13:08.979 --> 00:13:09.520 r(r-1)/2 x^2. 00:13:14.480 --> 00:13:19.980 Now, eventually, your mind will converge on all of these 00:13:19.980 --> 00:13:23.380 and you'll find them relatively easy to memorize. 00:13:23.380 --> 00:13:25.820 But it'll take some getting used to. 00:13:25.820 --> 00:13:29.670 And I'm not claiming that you should recognize them 00:13:29.670 --> 00:13:32.370 and understand them all now. 00:13:32.370 --> 00:13:35.630 But I'm going to put a giant box around this. 00:13:35.630 --> 00:13:41.793 STUDENT: [INAUDIBLE] 00:13:41.793 --> 00:13:42.293 PROF. 00:13:42.293 --> 00:13:42.834 JERISON: Yes. 00:13:42.834 --> 00:13:44.543 So the question was, you get all of these 00:13:44.543 --> 00:13:45.834 if you use that equation there. 00:13:45.834 --> 00:13:47.500 That's exactly what I'm going to do. 00:13:47.500 --> 00:13:52.350 So I already did it actually for these three, last time. 00:13:52.350 --> 00:13:55.880 But I didn't do it yet for these two. 00:13:55.880 --> 00:13:58.490 But I will do it in about two minutes. 00:13:58.490 --> 00:14:00.405 Well, maybe five minutes. 00:14:00.405 --> 00:14:10.410 But first I want to explain just a few things about these. 00:14:10.410 --> 00:14:12.850 They all follow from the basic formula. 00:14:12.850 --> 00:14:16.780 In fact, that one deserves a pink box too, doesn't it. 00:14:16.780 --> 00:14:18.020 That one's pretty important. 00:14:18.020 --> 00:14:19.770 Alright. 00:14:19.770 --> 00:14:23.020 Yeah. 00:14:23.020 --> 00:14:25.100 Maybe even some little sparkles. 00:14:25.100 --> 00:14:32.830 Alright. 00:14:32.830 --> 00:14:33.430 OK. 00:14:33.430 --> 00:14:36.390 So that's pretty important. 00:14:36.390 --> 00:14:39.900 Almost as important as the more basic one without this term 00:14:39.900 --> 00:14:41.190 here. 00:14:41.190 --> 00:14:47.480 So now, let me just tell you a little bit more 00:14:47.480 --> 00:14:54.740 about the significance. 00:14:54.740 --> 00:14:56.641 Again, this is just to reinforce something 00:14:56.641 --> 00:14:57.640 that we've already done. 00:14:57.640 --> 00:14:59.264 But it's closely related to what you're 00:14:59.264 --> 00:15:00.880 doing on your problem set. 00:15:00.880 --> 00:15:05.230 So it's worth your while to recall this. 00:15:05.230 --> 00:15:10.870 So, there's this expression that we were dealing with. 00:15:10.870 --> 00:15:13.370 And we talked about it in lecture. 00:15:13.370 --> 00:15:19.610 And we showed that this tends to e as k goes to infinity. 00:15:19.610 --> 00:15:21.110 So that's what we showed in lecture. 00:15:21.110 --> 00:15:25.310 And the way that we did that was, we took the logarithm 00:15:25.310 --> 00:15:27.320 and we wrote it as k times, sorry, 00:15:27.320 --> 00:15:29.640 the logarithm of 1 + 1/k. 00:15:32.530 --> 00:15:35.780 And then we evaluated the limit of this. 00:15:35.780 --> 00:15:38.000 And I want to do this limit again, 00:15:38.000 --> 00:15:40.220 using linear approximation. 00:15:40.220 --> 00:15:42.660 To show you how easy it is if you just remember 00:15:42.660 --> 00:15:44.820 the linear approximation. 00:15:44.820 --> 00:15:47.720 And then we'll explain where the quadratic approximation comes 00:15:47.720 --> 00:15:48.390 in. 00:15:48.390 --> 00:15:51.770 So I claim that this is approximately 00:15:51.770 --> 00:15:56.470 equal to k times 1/k. 00:15:59.300 --> 00:16:00.750 Now, why is that? 00:16:00.750 --> 00:16:04.350 Well, that's just this linear approximation. 00:16:04.350 --> 00:16:05.500 So what did I use here? 00:16:05.500 --> 00:16:14.760 I used log of 1+x is approximately x, for x = 1/k. 00:16:14.760 --> 00:16:18.600 That's what I used in this approximation here. 00:16:18.600 --> 00:16:20.330 And that's the linear approximation 00:16:20.330 --> 00:16:24.010 to the natural logarithm. 00:16:24.010 --> 00:16:27.250 And this number is relatively easy to evaluate. 00:16:27.250 --> 00:16:28.150 I know how to do it. 00:16:28.150 --> 00:16:31.010 It's equal to 1. 00:16:31.010 --> 00:16:34.680 That's the same, well, so where does this work? 00:16:34.680 --> 00:16:37.450 This works where this thing is near 0. 00:16:37.450 --> 00:16:41.610 Which is when k is going to infinity. 00:16:41.610 --> 00:16:44.170 This thing is working only when k is going to infinity. 00:16:44.170 --> 00:16:46.660 So what it's really saying, this approximation formula, 00:16:46.660 --> 00:16:50.690 it's really saying that as we go to infinity, in k, 00:16:50.690 --> 00:16:54.720 this thing is going to 1. 00:16:54.720 --> 00:16:58.890 As k goes to infinity. 00:16:58.890 --> 00:17:00.410 So that's what it's saying. 00:17:00.410 --> 00:17:01.740 That's the substance there. 00:17:01.740 --> 00:17:05.200 And that's how we want to use it, in many instances. 00:17:05.200 --> 00:17:06.580 Just to evaluate limits. 00:17:06.580 --> 00:17:09.180 We also want to realize that it's nearby 00:17:09.180 --> 00:17:12.800 when k is pretty large, like 100 or something like that. 00:17:12.800 --> 00:17:16.660 Now, so that's the idea of the linear approximation. 00:17:16.660 --> 00:17:21.920 Now, if you want to get the rate of convergence here, 00:17:21.920 --> 00:17:27.950 so the rate of what's called convergence. 00:17:27.950 --> 00:17:34.260 So convergence means how fast this is going towards that. 00:17:34.260 --> 00:17:36.220 What I have to do is take the difference. 00:17:36.220 --> 00:17:40.670 I have to take ln a_k, and I have to subtract 1 from it. 00:17:40.670 --> 00:17:42.420 And I know that this is going to 0, 00:17:42.420 --> 00:17:47.870 and the question is how big is this. 00:17:47.870 --> 00:17:51.760 We want it to be very small. 00:17:51.760 --> 00:17:54.490 And the answer we're going to get, 00:17:54.490 --> 00:18:03.570 so the answer just uses the quadratic approximation. 00:18:03.570 --> 00:18:06.520 So if I just have a little bit more detail, then 00:18:06.520 --> 00:18:08.720 this expression here, in other words, 00:18:08.720 --> 00:18:11.210 I have the next higher-order term. 00:18:11.210 --> 00:18:15.420 This is like 1/k, this is like 1 / k^2. 00:18:15.420 --> 00:18:18.890 Then I can understand how big the difference 00:18:18.890 --> 00:18:24.260 is between the expression that I've got and its limit. 00:18:24.260 --> 00:18:26.200 And so that's what's on your homework. 00:18:26.200 --> 00:18:31.610 This is on your problem set. 00:18:31.610 --> 00:18:35.050 OK, so that is more or less an explanation for one 00:18:35.050 --> 00:18:38.860 of the things that quadratic approximations are good for. 00:18:38.860 --> 00:18:42.900 And I'm going to give you one more illustration. 00:18:42.900 --> 00:18:45.360 One more illustration. 00:18:45.360 --> 00:18:47.472 And then we'll actually check these formulas. 00:18:47.472 --> 00:18:48.430 Yeah, another question. 00:18:48.430 --> 00:18:55.520 STUDENT: [INAUDIBLE] 00:18:55.520 --> 00:18:56.020 PROF. 00:18:56.020 --> 00:18:58.470 JERISON: That's a very good question here. 00:18:58.470 --> 00:19:01.260 When they, which in this case means maybe, 00:19:01.260 --> 00:19:07.510 me, when I give you a question, does one 00:19:07.510 --> 00:19:13.030 specify whether you want to use a linear or a quadratic 00:19:13.030 --> 00:19:14.430 approximation. 00:19:14.430 --> 00:19:17.910 The answer is, in real life when you're 00:19:17.910 --> 00:19:22.470 faced with a problem like this, where some satellite is 00:19:22.470 --> 00:19:24.700 orbiting and you want to know the effects of gravity 00:19:24.700 --> 00:19:28.304 or something like that, nobody is going to tell you anything. 00:19:28.304 --> 00:19:29.720 They're not even going to tell you 00:19:29.720 --> 00:19:32.480 whether a linear approximation is relevant, or a quadratic 00:19:32.480 --> 00:19:33.740 or anything. 00:19:33.740 --> 00:19:36.170 So you're on your own. 00:19:36.170 --> 00:19:39.830 When I give you a question, at least for right now, 00:19:39.830 --> 00:19:42.640 I'm always going to tell you. 00:19:42.640 --> 00:19:44.640 But as time goes on I'd like you to get 00:19:44.640 --> 00:19:48.360 used to when it's enough to get away 00:19:48.360 --> 00:19:49.840 with a linear approximation. 00:19:49.840 --> 00:19:53.540 And you should only use a quadratic approximation 00:19:53.540 --> 00:19:55.690 if somebody forces you to. 00:19:55.690 --> 00:19:57.960 You should always start trying with a linear one. 00:19:57.960 --> 00:20:00.610 Because the quadratic ones are much more complicated as you'll 00:20:00.610 --> 00:20:03.390 see in this next example. 00:20:03.390 --> 00:20:05.320 OK, so the example that I want to use 00:20:05.320 --> 00:20:07.740 is, you're going to be stuck with it because I'm 00:20:07.740 --> 00:20:09.110 asking for the quadratic. 00:20:09.110 --> 00:20:16.190 So we're going to find the quadratic approximation 00:20:16.190 --> 00:20:23.740 near-- for x near 0. 00:20:23.740 --> 00:20:24.525 To what? 00:20:24.525 --> 00:20:26.360 Well, this is the same function that we 00:20:26.360 --> 00:20:30.840 used in the last lecture. 00:20:30.840 --> 00:20:32.880 I think this was it. 00:20:32.880 --> 00:20:34.090 e^(-3x) (1+x)^(-1/2). 00:20:37.460 --> 00:20:40.670 OK. 00:20:40.670 --> 00:20:45.060 So, unfortunately, I stuck it in the wrong place 00:20:45.060 --> 00:20:47.620 to be able to fit this very long formula here. 00:20:47.620 --> 00:20:51.100 So I'm going to switch it. 00:20:51.100 --> 00:20:57.070 I'm just going to write it here. 00:20:57.070 --> 00:21:00.140 And we're going to just do the approximation. 00:21:00.140 --> 00:21:03.830 So we're going to say quadratic, in parentheses. 00:21:03.830 --> 00:21:08.450 And we'll say x near 0. 00:21:08.450 --> 00:21:12.160 So that's what I want. 00:21:12.160 --> 00:21:15.424 So now, here's what I have to do. 00:21:15.424 --> 00:21:17.590 Well, I have to write in the quadratic approximation 00:21:17.590 --> 00:21:25.710 for e^(-3x), and I'm going to use this formula right here. 00:21:25.710 --> 00:21:33.970 And so that's 1 + (-3x) + (-3x)^2 / 2. 00:21:33.970 --> 00:21:36.530 And the other factor, I'm going to have 00:21:36.530 --> 00:21:40.910 to use this formula down here. 00:21:40.910 --> 00:21:43.460 Because r is - 1/2. 00:21:43.460 --> 00:21:50.300 And so that's 1 - 1/2 x + 1/2 (-1/2)(-3/2)x^2. 00:21:59.090 --> 00:22:09.970 So this is the r term, and this is the r - 1 term. 00:22:09.970 --> 00:22:11.630 And now I'm going to do something 00:22:11.630 --> 00:22:15.550 which is the only good thing about quadratic approximations. 00:22:15.550 --> 00:22:17.660 They're messy, they're long, there's nothing 00:22:17.660 --> 00:22:19.450 particularly good about them. 00:22:19.450 --> 00:22:21.670 But there is one good thing about them. 00:22:21.670 --> 00:22:25.610 Which is that you always get to ignore the higher order terms. 00:22:25.610 --> 00:22:29.420 So even though this looks like a very ugly multiplication, 00:22:29.420 --> 00:22:31.590 I can do it in my head. 00:22:31.590 --> 00:22:33.940 Just watching it. 00:22:33.940 --> 00:22:38.710 Because I get a 1 * 1, I'm forced with that term here. 00:22:38.710 --> 00:22:41.200 And then I get the cross terms which are linear, 00:22:41.200 --> 00:22:44.150 which is -3x - 1/2 x. 00:22:44.150 --> 00:22:46.460 We already did that when we calculated 00:22:46.460 --> 00:22:48.550 the linear approximation, so that's 00:22:48.550 --> 00:22:51.670 this times the 1 and this times that 1. 00:22:51.670 --> 00:22:55.230 And now I have three cross terms which are quadratic. 00:22:55.230 --> 00:22:58.880 So one of them is these two linear terms are multiplying. 00:22:58.880 --> 00:23:02.160 So that's plus 3/2 x^2. 00:23:02.160 --> 00:23:05.210 That's -3 times -1/2. 00:23:05.210 --> 00:23:08.160 And then there's this term, multiplying the 1, 00:23:08.160 --> 00:23:11.730 that's plus 9/2 x^2. 00:23:11.730 --> 00:23:15.080 And then there's one last term, which is this monster here. 00:23:15.080 --> 00:23:23.260 Multiplying 1, and that is -3/8. 00:23:23.260 --> 00:23:31.430 So the great thing is, we drop x^3, x^4, etc., terms. 00:23:31.430 --> 00:23:37.930 Yeah? 00:23:37.930 --> 00:23:39.656 STUDENT: [INAUDIBLE] 00:23:39.656 --> 00:23:40.156 PROF. 00:23:40.156 --> 00:23:42.500 JERISON: OK, well so copy it down. 00:23:42.500 --> 00:23:45.620 And you work it out as I'm doing it now. 00:23:45.620 --> 00:23:47.930 So what I did is, I multiplied 1 by 1. 00:23:47.930 --> 00:23:50.290 I'm using the distributive law here. 00:23:50.290 --> 00:23:51.560 That was this one. 00:23:51.560 --> 00:23:55.120 I multiplied this 3x by this one, that was that term. 00:23:55.120 --> 00:23:58.550 I multiplied this by this, that's that term. 00:23:58.550 --> 00:24:01.120 And then I multiplied this by this. 00:24:01.120 --> 00:24:03.860 In other words, two x terms that gave me an x^2 00:24:03.860 --> 00:24:06.940 and a (-3)(-1/2). 00:24:06.940 --> 00:24:08.499 And I'm going to stop at that point. 00:24:08.499 --> 00:24:10.290 Because the point is it's just all the rest 00:24:10.290 --> 00:24:12.230 of the terms that come up. 00:24:12.230 --> 00:24:14.320 Now, the reason, the only reason why it's easy, 00:24:14.320 --> 00:24:16.319 is that I only have to go up to x squared terms. 00:24:16.319 --> 00:24:21.462 I don't have to do the higher ones. 00:24:21.462 --> 00:24:22.795 Another question, way back here. 00:24:22.795 --> 00:24:23.545 Yeah, right there. 00:24:23.545 --> 00:24:28.696 STUDENT: [INAUDIBLE] 00:24:28.696 --> 00:24:29.196 PROF. 00:24:29.196 --> 00:24:29.696 JERISON: OK. 00:24:29.696 --> 00:24:38.690 So somebody can check my arithmetic, too. 00:24:38.690 --> 00:24:39.470 Good. 00:24:39.470 --> 00:24:40.361 STUDENT: [INAUDIBLE] 00:24:40.361 --> 00:24:40.861 PROF. 00:24:40.861 --> 00:24:43.466 JERISON: Why do I get to drop all the higher-order terms. 00:24:43.466 --> 00:24:46.210 So, that's because the situation where 00:24:46.210 --> 00:24:50.200 I'm going to apply this is the situation in which x is, 00:24:50.200 --> 00:24:52.610 say, 1/100. 00:24:52.610 --> 00:24:55.030 So here's about 1/100. 00:24:55.030 --> 00:24:57.210 Here's something which is on the order of 1/100. 00:24:57.210 --> 00:25:00.130 This is on the order of 1/100^2. 00:25:00.130 --> 00:25:02.450 1/100^2, all of these terms. 00:25:02.450 --> 00:25:07.400 Now, these cubic and quartic terms are of the order 00:25:07.400 --> 00:25:10.520 of 1/100^3. 00:25:10.520 --> 00:25:11.420 That's 10^(-6). 00:25:11.420 --> 00:25:12.420 6. 00:25:12.420 --> 00:25:14.730 And the point is that I'm not claiming 00:25:14.730 --> 00:25:16.230 that I have an exact answer. 00:25:16.230 --> 00:25:19.950 And I'm going to drop things of that order of magnitude. 00:25:19.950 --> 00:25:22.890 So I'm saving everything up to 4 decimal places. 00:25:22.890 --> 00:25:29.300 I'm throwing away things which are 6 decimal places out. 00:25:29.300 --> 00:25:30.870 Does that answer your question? 00:25:30.870 --> 00:25:35.211 STUDENT: [INAUDIBLE] 00:25:35.211 --> 00:25:35.710 PROF. 00:25:35.710 --> 00:25:36.370 JERISON: So. 00:25:36.370 --> 00:25:39.770 That's the situation, and now you can combine the terms. 00:25:39.770 --> 00:25:43.780 I mean, it's not very impressive here. 00:25:43.780 --> 00:25:54.300 This is equal to 1 - 7/2 x, maybe, plus 51/8 x^2. 00:25:54.300 --> 00:25:57.530 If I've made that-- if those minus signs hadn't canceled, 00:25:57.530 --> 00:25:59.560 I would have gotten the wrong answer here. 00:25:59.560 --> 00:26:00.140 Anyway. 00:26:00.140 --> 00:26:03.430 So, this is a 2 here, sorry. 00:26:03.430 --> 00:26:04.920 7/2. 00:26:04.920 --> 00:26:07.864 This is the linear approximation we got last time 00:26:07.864 --> 00:26:09.530 and here's the extra information that we 00:26:09.530 --> 00:26:11.540 got from this calculation. 00:26:11.540 --> 00:26:17.650 Which is this 51/8 term. 00:26:17.650 --> 00:26:19.620 Right, you have to accept that there's 00:26:19.620 --> 00:26:21.890 a certain degree of complexity to this problem 00:26:21.890 --> 00:26:23.840 and the answer is sufficiently complicated 00:26:23.840 --> 00:26:25.740 so it can't be less arithmetic because we 00:26:25.740 --> 00:26:29.380 get this peculiar 51/8 there, right. 00:26:29.380 --> 00:26:31.950 So one of the things to realize is 00:26:31.950 --> 00:26:35.120 that these kinds of problems, because they involve 00:26:35.120 --> 00:26:37.135 many, many terms are always going 00:26:37.135 --> 00:26:43.960 to involve a little bit of complicated arithmetic. 00:26:43.960 --> 00:26:46.910 Last little bit, I did promise you 00:26:46.910 --> 00:26:51.360 that I was going to derive these two relations, as I said. 00:26:51.360 --> 00:26:53.020 Did the ones in the left column. 00:26:53.020 --> 00:26:56.640 So let's carry that out. 00:26:56.640 --> 00:27:01.400 And as someone just pointed out, it all comes from this formula 00:27:01.400 --> 00:27:01.900 here. 00:27:01.900 --> 00:27:07.710 So let's just check it. 00:27:07.710 --> 00:27:12.160 So we'll start with the log function. 00:27:12.160 --> 00:27:18.810 This is the function, f, and then f' is 1/(1+x). 00:27:18.810 --> 00:27:24.620 And f'', so this is f', this is f'', is - -1 / (1+x)^2. 00:27:28.160 --> 00:27:31.730 And now I have to plug in x = 0. 00:27:31.730 --> 00:27:35.180 So at x = 0 this is ln 1, which is 0. 00:27:35.180 --> 00:27:37.860 So this is at x = 0. 00:27:37.860 --> 00:27:41.850 I'm getting 0 here, I plug in 0 and I get 1. 00:27:41.850 --> 00:27:45.720 And here, I plug in 0 and I get -1. 00:27:45.720 --> 00:27:48.050 So now I go and I look up at that formula, 00:27:48.050 --> 00:27:50.270 which is way in that upper corner there. 00:27:50.270 --> 00:27:53.420 And I see that the coefficient on the constant is 0. 00:27:53.420 --> 00:27:55.260 The coefficient on x is 1. 00:27:55.260 --> 00:27:59.290 And then the other coefficient, the very last one, is -1/2. 00:27:59.290 --> 00:28:01.010 So this is the -1 here. 00:28:01.010 --> 00:28:05.340 And then in the formula, there's a 2 in the denominator. 00:28:05.340 --> 00:28:08.400 So it's half of whatever I get for this second derivative, 00:28:08.400 --> 00:28:10.590 at 0. 00:28:10.590 --> 00:28:13.100 So this is the approximation formula, which 00:28:13.100 --> 00:28:17.260 is way up in that corner there. 00:28:17.260 --> 00:28:21.290 Similarly, if I do it for (1+x)^r, 00:28:21.290 --> 00:28:25.990 I have to differentiate that, I get r(1+x)^(r-1), 00:28:25.990 --> 00:28:27.300 and then r(r-1)(x+1)^(r-2). 00:28:31.270 --> 00:28:33.830 So here are the derivatives. 00:28:33.830 --> 00:28:41.110 And so if I evaluate them at x = 0, I get 1. 00:28:41.110 --> 00:28:43.240 That's 1^r is 1. 00:28:43.240 --> 00:28:46.910 And here I get r. 00:28:46.910 --> 00:28:49.580 1^(r-1) times r. 00:28:49.580 --> 00:28:54.010 And here, I plug in x = 0 and I get r(r-1). 00:28:59.790 --> 00:29:02.880 So again, the pattern is right above it here. 00:29:02.880 --> 00:29:05.300 The 1 is there, the r is there. 00:29:05.300 --> 00:29:08.920 And then instead of r(r-1), I have half that. 00:29:08.920 --> 00:29:21.480 For the coefficient. 00:29:21.480 --> 00:29:22.660 So these are just examples. 00:29:22.660 --> 00:29:24.700 Obviously if we had a more complicated function, 00:29:24.700 --> 00:29:26.050 we might carry this out. 00:29:26.050 --> 00:29:28.210 But as a practical matter, we try 00:29:28.210 --> 00:29:30.810 to stick with the ones in the pink box 00:29:30.810 --> 00:29:42.460 and just use algebra to get other formulas. 00:29:42.460 --> 00:29:46.430 So I want to shift gears now and treat the subject that was 00:29:46.430 --> 00:29:48.910 supposed to be this lecture. 00:29:48.910 --> 00:29:51.230 And we're not quite caught up, but we 00:29:51.230 --> 00:29:54.890 will try to do our best to do as much as we can today. 00:29:54.890 --> 00:29:59.880 So the next topic is curve sketching. 00:29:59.880 --> 00:30:18.120 And so let's get started with that. 00:30:18.120 --> 00:30:24.950 So now, happily in this subject, there are more pictures 00:30:24.950 --> 00:30:26.880 and it's a little bit more geometric. 00:30:26.880 --> 00:30:30.420 And there's relatively little computation. 00:30:30.420 --> 00:30:33.600 So let's hope we can do this. 00:30:33.600 --> 00:30:36.890 So I want to-- so here we go, we'll 00:30:36.890 --> 00:30:44.930 start with curve sketching. 00:30:44.930 --> 00:30:47.520 And the goal here-- 00:30:47.520 --> 00:30:56.410 STUDENT: [INAUDIBLE] 00:30:56.410 --> 00:30:56.910 PROF. 00:30:56.910 --> 00:31:00.880 JERISON: So that's like 'liner', the last time. 00:31:00.880 --> 00:31:09.600 That's kind of sketchy spelling, isn't it? 00:31:09.600 --> 00:31:12.800 Yeah, there are certain kinds of things which I can't spell. 00:31:12.800 --> 00:31:16.640 But, all right. 00:31:16.640 --> 00:31:18.741 Sketching. 00:31:18.741 --> 00:31:19.240 Alright. 00:31:19.240 --> 00:31:21.350 So here's our goal. 00:31:21.350 --> 00:31:38.280 Our goal is to draw the graph of f, using f' and f''. 00:31:42.380 --> 00:31:47.190 Whether they're positive or negative. 00:31:47.190 --> 00:31:48.440 So that's it. 00:31:48.440 --> 00:31:52.400 This is the goal here. 00:31:52.400 --> 00:31:57.020 However, there is a big warning that I want to give you. 00:31:57.020 --> 00:32:04.000 And this is one that unfortunately 00:32:04.000 --> 00:32:07.005 I now have to make you unlearn, especially 00:32:07.005 --> 00:32:08.380 those that you that have actually 00:32:08.380 --> 00:32:10.700 had a little bit of calculus before, I 00:32:10.700 --> 00:32:12.929 want to make you unlearn some of your instincts 00:32:12.929 --> 00:32:13.720 that you developed. 00:32:13.720 --> 00:32:15.511 So this will be harder for those of you who 00:32:15.511 --> 00:32:19.840 have actually done this before. 00:32:19.840 --> 00:32:22.650 But for the rest of you, it will be relatively easy. 00:32:22.650 --> 00:32:35.490 Which is, don't abandon your precalculus skills. 00:32:35.490 --> 00:32:42.380 And common sense. 00:32:42.380 --> 00:32:46.760 So there's a great deal of common sense in this. 00:32:46.760 --> 00:32:50.555 And it actually trumps some of the calculus. 00:32:50.555 --> 00:32:56.190 The calculus just fills in what you didn't quite know yet. 00:32:56.190 --> 00:32:59.660 So I will try to illustrate this. 00:32:59.660 --> 00:33:01.390 And because we're running a bit late, 00:33:01.390 --> 00:33:03.910 I won't get to the some of the main punchlines 00:33:03.910 --> 00:33:05.970 until next lecture. 00:33:05.970 --> 00:33:07.380 But I want you to do it. 00:33:07.380 --> 00:33:09.412 So for now, I'm just going to tell you 00:33:09.412 --> 00:33:10.620 about the general principles. 00:33:10.620 --> 00:33:14.100 And in the process I'm going to introduce the terminology. 00:33:14.100 --> 00:33:17.167 Just, the words that we need to use to describe what 00:33:17.167 --> 00:33:18.000 is that we're doing. 00:33:18.000 --> 00:33:19.375 And there's also a certain amount 00:33:19.375 --> 00:33:22.160 of carelessness with that in many of the treatments 00:33:22.160 --> 00:33:23.070 that you'll see. 00:33:23.070 --> 00:33:24.750 And a lot of hastiness. 00:33:24.750 --> 00:33:29.350 So just be a little patient and we will do this. 00:33:29.350 --> 00:33:33.740 So, the first principle is the following. 00:33:33.740 --> 00:33:40.670 If f' is positive, then f is increasing. 00:33:40.670 --> 00:33:44.530 That's a straightforward idea, and it's closely related 00:33:44.530 --> 00:33:46.730 to this tangent line approximation 00:33:46.730 --> 00:33:49.220 or the linear approximation that I just did. 00:33:49.220 --> 00:33:50.230 You can just imagine. 00:33:50.230 --> 00:33:53.050 Here's the tangent line, here's the function. 00:33:53.050 --> 00:33:55.680 And if the tangent line is pointing up, 00:33:55.680 --> 00:33:58.490 then the function is also going up, too. 00:33:58.490 --> 00:34:00.310 So that's all that's going on here. 00:34:00.310 --> 00:34:10.370 Similarly, if f' is negative, then f is decreasing. 00:34:10.370 --> 00:34:12.220 And that's the basic idea. 00:34:12.220 --> 00:34:17.170 Now, the second step is also fairly straightforward. 00:34:17.170 --> 00:34:21.130 It's just a second-order effect of the same type. 00:34:21.130 --> 00:34:29.190 If you have f'' as positive, then that means that f' is 00:34:29.190 --> 00:34:33.150 increasing. 00:34:33.150 --> 00:34:36.740 That's the same principle applied one step up. 00:34:36.740 --> 00:34:37.320 Right? 00:34:37.320 --> 00:34:41.880 Because if f'' is positive, that means it's the derivative 00:34:41.880 --> 00:34:42.710 of f'. 00:34:42.710 --> 00:34:45.970 So it's the same principle just repeated. 00:34:45.970 --> 00:34:49.710 And now I just want to draw a picture of this. 00:34:49.710 --> 00:34:52.470 Here's a picture of it, I claim. 00:34:52.470 --> 00:34:54.790 And it looks like something's going down. 00:34:54.790 --> 00:34:56.430 And I did that on purpose. 00:34:56.430 --> 00:34:58.650 But there is something that's increasing here. 00:34:58.650 --> 00:35:02.470 Which is, the slope is very steep negative here. 00:35:02.470 --> 00:35:06.330 And it's less steep negative over here. 00:35:06.330 --> 00:35:09.790 So we have the slope which is some negative number, say, -4. 00:35:09.790 --> 00:35:13.500 And here it's -3. 00:35:13.500 --> 00:35:14.800 So it's increasing. 00:35:14.800 --> 00:35:18.250 It's getting less negative, and maybe eventually it'll 00:35:18.250 --> 00:35:19.680 curve up the other way. 00:35:19.680 --> 00:35:23.180 And this is a picture of what I'm talking about here. 00:35:23.180 --> 00:35:25.630 That's what it means to say that f' is increasing. 00:35:25.630 --> 00:35:28.020 The slope is getting larger. 00:35:28.020 --> 00:35:34.930 And the way to describe a curve like this is that it's concave. 00:35:34.930 --> 00:35:41.620 So f is concave up. 00:35:41.620 --> 00:35:49.320 And similarly, f'' negative is going to be the same thing as f 00:35:49.320 --> 00:35:59.770 concave-- or implies f concave down. 00:35:59.770 --> 00:36:04.850 So those are the ways in which derivatives will help us 00:36:04.850 --> 00:36:08.270 qualitatively to draw graphs. 00:36:08.270 --> 00:36:09.880 But as I said before, we still have 00:36:09.880 --> 00:36:13.360 to use a little bit of common sense when we draw the graphs. 00:36:13.360 --> 00:36:15.710 These are just the additional bits of help 00:36:15.710 --> 00:36:17.850 that we have from calculus. 00:36:17.850 --> 00:36:25.410 In drawing pictures. 00:36:25.410 --> 00:36:30.650 So I'm going to go through one example 00:36:30.650 --> 00:36:36.190 to introduce all the notations. 00:36:36.190 --> 00:36:42.210 And then eventually, so probably at the beginning of next time, 00:36:42.210 --> 00:36:45.340 I'll give you a systematic strategy 00:36:45.340 --> 00:36:48.600 that's going to work when what I'm describing now 00:36:48.600 --> 00:36:52.360 goes wrong, or a little bit wrong. 00:36:52.360 --> 00:36:58.730 So let's begin with a straightforward example. 00:36:58.730 --> 00:37:03.280 So, the first example that I'll give you is the function f(x) = 00:37:03.280 --> 00:37:04.690 3x - x^3. 00:37:07.520 --> 00:37:11.130 Just, as I said, to be able to introduce all the notations. 00:37:11.130 --> 00:37:17.640 Now, if you differentiate it, you get 3 - 3x^2. 00:37:17.640 --> 00:37:20.884 And I can factor that. 00:37:20.884 --> 00:37:21.800 This is 3 (1-x) (1+x). 00:37:26.440 --> 00:37:27.750 OK? 00:37:27.750 --> 00:37:33.140 And so, I can decide whether the derivative 00:37:33.140 --> 00:37:37.390 is positive or negative. 00:37:37.390 --> 00:37:38.830 Easily enough. 00:37:38.830 --> 00:37:50.120 Namely, just staring at this, I can see that when -1 < x < 1, 00:37:50.120 --> 00:37:53.550 in that range there, both these numbers, both these factors, 00:37:53.550 --> 00:37:55.550 are positive. 00:37:55.550 --> 00:37:59.290 1-x is a positive number and 1 1+x is a positive number. 00:37:59.290 --> 00:38:04.710 So, in this range, f'(x) is positive. 00:38:04.710 --> 00:38:09.700 So this thing is, so f is increasing. 00:38:09.700 --> 00:38:13.310 And similarly, in the other ranges, 00:38:13.310 --> 00:38:15.770 if x is very, very large, this becomes, 00:38:15.770 --> 00:38:18.190 if it crosses 1, in fact, this becomes, 00:38:18.190 --> 00:38:21.310 this factor becomes negative and this one stays positive. 00:38:21.310 --> 00:38:29.680 So when x > 1, we have that f'(x) is negative. 00:38:29.680 --> 00:38:35.510 And so f is decreasing. 00:38:35.510 --> 00:38:39.540 And the same thing goes for the other side. 00:38:39.540 --> 00:38:42.500 When it's less than -1, that also works this way. 00:38:42.500 --> 00:38:46.420 Because when it's less than -1, this factor is positive. 00:38:46.420 --> 00:38:50.300 But the other one is negative. 00:38:50.300 --> 00:38:56.290 So in both of these cases, we get that it's decreasing. 00:38:56.290 --> 00:39:07.300 So now, here's the schematic picture of this function. 00:39:07.300 --> 00:39:12.950 So here's -1, here's 1. 00:39:12.950 --> 00:39:19.840 It's going to go down, up, down. 00:39:19.840 --> 00:39:21.590 That's what it's doing. 00:39:21.590 --> 00:39:23.710 Maybe I'll just leave it alone like this. 00:39:23.710 --> 00:39:28.120 That's what it looks like. 00:39:28.120 --> 00:39:31.400 So, this is the kind of information 00:39:31.400 --> 00:39:32.980 we can get right off the bat. 00:39:32.980 --> 00:39:38.089 And you notice immediately that it's very important, 00:39:38.089 --> 00:39:39.880 from the features of the function, the sort 00:39:39.880 --> 00:39:42.060 of key features of the function that we see here, 00:39:42.060 --> 00:39:44.690 are these two places. 00:39:44.690 --> 00:39:49.270 Maybe I'll even mark them in a, like this. 00:39:49.270 --> 00:39:59.090 And these things are turning points. 00:39:59.090 --> 00:40:00.430 So what are they? 00:40:00.430 --> 00:40:03.920 Well, they're just the points where 00:40:03.920 --> 00:40:05.749 the derivative changes sign. 00:40:05.749 --> 00:40:07.790 Where it's negative here and it's positive there, 00:40:07.790 --> 00:40:09.850 so there it must be 0. 00:40:09.850 --> 00:40:12.830 So we have a definition, and this is the most important 00:40:12.830 --> 00:40:21.110 definition in this subject, which is that is if f'(x_0) = 00:40:21.110 --> 00:40:33.370 0, we call x_0 a critical point. 00:40:33.370 --> 00:40:36.620 The word 'turning point' is not used just because, in fact, it 00:40:36.620 --> 00:40:38.880 doesn't have to turn around at those points. 00:40:38.880 --> 00:40:42.710 But certainly, if it turns around then this will happen. 00:40:42.710 --> 00:40:44.980 And we also have another notation, 00:40:44.980 --> 00:40:50.580 which is the number y_0 which is f(x_0) is 00:40:50.580 --> 00:40:59.100 called a critical value. 00:40:59.100 --> 00:41:02.230 So these are the key numbers that we're 00:41:02.230 --> 00:41:05.340 going to have to work out in order to understand 00:41:05.340 --> 00:41:18.750 what the function looks like. 00:41:18.750 --> 00:41:27.990 So what I'm going to do is just plot them. 00:41:27.990 --> 00:41:30.440 We're going to plot the critical points and the values. 00:41:30.440 --> 00:41:34.650 Well, we found the critical points relatively easily. 00:41:34.650 --> 00:41:37.240 I didn't write it down here but it's pretty obvious. 00:41:37.240 --> 00:41:42.353 If you set f(x) = 0, that implies that (1 - 00:41:42.353 --> 00:41:50.570 x)(1 + x) = 0, which implies that x is plus or minus 1. 00:41:50.570 --> 00:41:52.710 So those are known as the critical points. 00:41:52.710 --> 00:41:56.420 And now, in order to get the critical values here, 00:41:56.420 --> 00:42:02.280 I have to plug in f(1), for instance, the function is 3x - 00:42:02.280 --> 00:42:07.480 x^2, so there's this 3 * 1 - 1^3, which is 2. 00:42:07.480 --> 00:42:17.180 And f(-1), which is 3(-1) - (-1)^3, which is -2. 00:42:17.180 --> 00:42:21.780 And so I can plot the function here. 00:42:21.780 --> 00:42:26.160 So here's the point -1 and here's, up here, is 2. 00:42:26.160 --> 00:42:28.620 So this is-- whoops, which one is it? 00:42:28.620 --> 00:42:29.160 Yeah. 00:42:29.160 --> 00:42:32.560 This is -1, so it's down here. 00:42:32.560 --> 00:42:35.160 So it's (-1, -2). 00:42:35.160 --> 00:42:41.570 And then over here, I have the point (1, 2). 00:42:41.570 --> 00:42:44.570 Alright, now, what information do 00:42:44.570 --> 00:42:48.930 I get from - so I've now plotted two, I claim, 00:42:48.930 --> 00:42:50.630 very interesting points. 00:42:50.630 --> 00:42:55.000 What information do I get from this? 00:42:55.000 --> 00:42:59.680 The answer is, I know something very nearby. 00:42:59.680 --> 00:43:01.830 Because I've already checked that the thing 00:43:01.830 --> 00:43:04.770 is coming down from the left, and coming back up. 00:43:04.770 --> 00:43:07.370 And so it must be shaped like this. 00:43:07.370 --> 00:43:08.330 Over here. 00:43:08.330 --> 00:43:11.620 The tangent line is 0, it's going to be level there. 00:43:11.620 --> 00:43:14.490 And similarly over here, it's going to do that. 00:43:14.490 --> 00:43:22.211 So this is what we know so far, based on what we've computed. 00:43:22.211 --> 00:43:22.710 Question. 00:43:22.710 --> 00:43:36.916 STUDENT: [INAUDIBLE] 00:43:36.916 --> 00:43:37.416 PROF. 00:43:37.416 --> 00:43:38.150 JERISON: The question is, what happens 00:43:38.150 --> 00:43:39.233 if there's a sharp corner. 00:43:39.233 --> 00:43:43.350 The answer is, calculus is-- it's not 00:43:43.350 --> 00:43:45.050 called a critical point. 00:43:45.050 --> 00:43:47.090 It's a something else. 00:43:47.090 --> 00:43:50.440 And it's a very important point, too. 00:43:50.440 --> 00:43:52.570 And we will be discussing those kinds of points. 00:43:52.570 --> 00:43:54.540 There are much more dramatic instances of that. 00:43:54.540 --> 00:43:56.640 That's part of what we're going to say. 00:43:56.640 --> 00:43:58.670 But I just want to save that, all right. 00:43:58.670 --> 00:44:02.490 We will be discussing. 00:44:02.490 --> 00:44:03.010 Yeah. 00:44:03.010 --> 00:44:03.540 Question. 00:44:03.540 --> 00:44:08.516 STUDENT: [INAUDIBLE] PROF. 00:44:08.516 --> 00:44:10.316 JERISON: The question that was asked 00:44:10.316 --> 00:44:12.070 was, how did I know at the critical point 00:44:12.070 --> 00:44:17.190 that it's concave down over here and concave up over here. 00:44:17.190 --> 00:44:21.590 The answer is that I actually did not 00:44:21.590 --> 00:44:24.300 use the second derivative yet. 00:44:24.300 --> 00:44:26.510 What I used is another piece of information. 00:44:26.510 --> 00:44:28.760 I used the information that I derived over here. 00:44:28.760 --> 00:44:32.500 That f' is positive, where f' is positive 00:44:32.500 --> 00:44:33.670 and where it's negative. 00:44:33.670 --> 00:44:36.980 So what I know is that the graph is going down 00:44:36.980 --> 00:44:39.470 to the left of -1. 00:44:39.470 --> 00:44:41.530 It's going up to the right, here. 00:44:41.530 --> 00:44:44.920 It's going up here and it's going down there. 00:44:44.920 --> 00:44:47.060 I did not use the second derivative. 00:44:47.060 --> 00:44:49.960 I used the first derivative. 00:44:49.960 --> 00:44:51.840 OK, but I didn't just use the fact 00:44:51.840 --> 00:44:55.834 that there was a turning point here. 00:44:55.834 --> 00:44:57.250 So, actually, I was using the fact 00:44:57.250 --> 00:44:58.416 that it was a turning point. 00:44:58.416 --> 00:45:01.080 I wasn't using the fact that it had the second derivative, 00:45:01.080 --> 00:45:01.730 though. 00:45:01.730 --> 00:45:02.230 OK. 00:45:02.230 --> 00:45:03.120 For now. 00:45:03.120 --> 00:45:09.500 You can also see it by the second derivative as well. 00:45:09.500 --> 00:45:13.710 So now, the next thing that I'd like to do, 00:45:13.710 --> 00:45:16.170 I need to finish off this graph. 00:45:16.170 --> 00:45:19.950 And I just want to do it a little bit carefully here. 00:45:19.950 --> 00:45:22.710 In the order that is reasonable. 00:45:22.710 --> 00:45:26.830 Now, you might happen to notice, and there's 00:45:26.830 --> 00:45:31.660 nothing wrong with this. 00:45:31.660 --> 00:45:34.826 So let's even fill in a guess. 00:45:34.826 --> 00:45:36.700 In order to fill in a guess, though, and have 00:45:36.700 --> 00:45:38.190 it be even vaguely right, I do have 00:45:38.190 --> 00:45:39.606 to notice that this thing crosses, 00:45:39.606 --> 00:45:41.800 this function crosses the origin. 00:45:41.800 --> 00:45:48.120 The function f(x) = 3x - x^3 happens to also have 00:45:48.120 --> 00:45:50.780 the property that f(0) = 0. 00:45:50.780 --> 00:45:52.560 Again, common sense. 00:45:52.560 --> 00:45:54.590 You're allowed to use your common sense. 00:45:54.590 --> 00:45:58.320 You're allowed to notice a value of the function and put it in. 00:45:58.320 --> 00:46:00.270 So there's nothing wrong with that. 00:46:00.270 --> 00:46:03.960 If you happen to have such a value. 00:46:03.960 --> 00:46:06.650 So, now we can guess what our function is going to look like. 00:46:06.650 --> 00:46:10.010 It's going to maybe come down like this. 00:46:10.010 --> 00:46:11.670 Come up like this. 00:46:11.670 --> 00:46:13.130 And come down like this. 00:46:13.130 --> 00:46:15.230 That could be what it looks like. 00:46:15.230 --> 00:46:16.800 But, you know, another possibility 00:46:16.800 --> 00:46:19.820 is it sort of comes along here and goes out that way. 00:46:19.820 --> 00:46:22.260 Comes along here and goes out that way, who knows? 00:46:22.260 --> 00:46:25.070 It happens, by the way, that it's an odd function. 00:46:25.070 --> 00:46:25.570 Right? 00:46:25.570 --> 00:46:26.640 Those are all odd powers. 00:46:26.640 --> 00:46:28.710 So, actually, it's symmetric on the right half 00:46:28.710 --> 00:46:29.860 and the left half. 00:46:29.860 --> 00:46:31.289 And crosses at 0. 00:46:31.289 --> 00:46:32.830 So everything that we do on the right 00:46:32.830 --> 00:46:34.996 is going to be the same as what happens on the left. 00:46:34.996 --> 00:46:36.620 That's another piece of common sense. 00:46:36.620 --> 00:46:38.120 You want to make use of that as much 00:46:38.120 --> 00:46:40.060 as possible, whenever you're drawing anything. 00:46:40.060 --> 00:46:42.340 Don't want to throw out information. 00:46:42.340 --> 00:46:46.540 So this function happens to be odd. 00:46:46.540 --> 00:46:48.700 Odd, and f(0) = 0. 00:46:48.700 --> 00:46:52.520 I'm considering those to be kinds of precalculus skills 00:46:52.520 --> 00:47:00.610 that I want you to use as much as you can. 00:47:00.610 --> 00:47:02.880 So now, here's the first feature which 00:47:02.880 --> 00:47:08.880 is unfortunately ignored in most discussions of functions. 00:47:08.880 --> 00:47:11.340 And it's strange, because nowadays we 00:47:11.340 --> 00:47:13.380 have graphing things. 00:47:13.380 --> 00:47:19.080 And it's really the only part of the exercise 00:47:19.080 --> 00:47:22.480 that you couldn't do, at least on this relatively simpleminded 00:47:22.480 --> 00:47:28.180 level, with a graphing calculator. 00:47:28.180 --> 00:47:33.790 And that is what I would call the ends of the problem. 00:47:33.790 --> 00:47:37.000 So what happens off the screen, is the question. 00:47:37.000 --> 00:47:39.880 And that basically is the theoretical part of the problem 00:47:39.880 --> 00:47:41.430 that you have to address. 00:47:41.430 --> 00:47:42.920 You can program this. 00:47:42.920 --> 00:47:45.410 You can draw all the pictures that you want. 00:47:45.410 --> 00:47:48.190 But what you won't see is what's off the screen. 00:47:48.190 --> 00:47:50.260 You need to know something to figure out 00:47:50.260 --> 00:47:51.780 what's off the screen. 00:47:51.780 --> 00:47:54.560 So, in this case, I'm talking about what's off the screen 00:47:54.560 --> 00:48:01.930 going to the right, or going to the left. 00:48:01.930 --> 00:48:06.100 So let's check the ends. 00:48:06.100 --> 00:48:07.680 So here, let's just take a look. 00:48:07.680 --> 00:48:12.280 We have the function f(x), which is, sorry, 3x - x^3. 00:48:12.280 --> 00:48:14.620 Again this is a precalculus sort of thing. 00:48:14.620 --> 00:48:16.230 And we're imagining now, let's just 00:48:16.230 --> 00:48:18.360 do x goes to plus infinity. 00:48:18.360 --> 00:48:19.670 So what happens here. 00:48:19.670 --> 00:48:25.460 When x is gigantic, this term is completely negligible. 00:48:25.460 --> 00:48:30.480 And it just behaves like -x^3, which goes to minus infinity 00:48:30.480 --> 00:48:32.600 as x goes to plus infinity. 00:48:32.600 --> 00:48:40.330 And similarly, f(x) goes to plus infinity 00:48:40.330 --> 00:48:46.150 if x goes to minus infinity. 00:48:46.150 --> 00:48:50.060 Now let me pull down this picture again, and show you 00:48:50.060 --> 00:48:53.140 what piece of the information we've got. 00:48:53.140 --> 00:48:55.610 We now know that it is heading up this way. 00:48:55.610 --> 00:48:58.560 It doesn't go like this, it goes up like that. 00:48:58.560 --> 00:49:00.220 And I'm going to put an arrow for it, 00:49:00.220 --> 00:49:02.580 And it's going down like this. 00:49:02.580 --> 00:49:06.500 Heading down to minus infinity as x goes out farther 00:49:06.500 --> 00:49:07.310 to the right. 00:49:07.310 --> 00:49:15.120 And going out to plus infinity as x goes farther to the left. 00:49:15.120 --> 00:49:19.070 So now there's hardly anything left 00:49:19.070 --> 00:49:21.910 of this function to describe. 00:49:21.910 --> 00:49:27.120 There's really nothing left except maybe decoration. 00:49:27.120 --> 00:49:29.650 And we kind of like that decoration, 00:49:29.650 --> 00:49:32.560 so we will pay attention to it. 00:49:32.560 --> 00:49:35.580 And to do that, we'll have to check the second derivative. 00:49:35.580 --> 00:49:40.150 So if we differentiate a second time, the first derivative was, 00:49:40.150 --> 00:49:42.680 remember, 3 - 3x^2. 00:49:42.680 --> 00:49:53.540 So the second derivative is -6x. 00:49:53.540 --> 00:50:02.470 So now we notice that f''(x) is negative if x is positive. 00:50:02.470 --> 00:50:08.310 And f''(x) is positive if x is negative. 00:50:08.310 --> 00:50:13.170 And so in this part it's concave down. 00:50:13.170 --> 00:50:18.990 And in this part it's concave up. 00:50:18.990 --> 00:50:22.390 And now I'm going to switch the boards so that you'll, and draw 00:50:22.390 --> 00:50:24.860 it. 00:50:24.860 --> 00:50:30.630 And you see that it was begging to be this way. 00:50:30.630 --> 00:50:32.960 So we'll fill in the rest of it here. 00:50:32.960 --> 00:50:35.560 Maybe in a nice color here. 00:50:35.560 --> 00:50:38.100 So this is the whole graph and this is the correct graph. 00:50:38.100 --> 00:50:41.450 It comes down in one swoop down here, and comes up here. 00:50:41.450 --> 00:50:46.620 And then it changes to concave down right at the origin. 00:50:46.620 --> 00:50:49.250 So this point is of interest, not only 00:50:49.250 --> 00:50:51.750 because it's the place where it crosses the axis, 00:50:51.750 --> 00:51:00.690 but it's also what's called an inflection point. 00:51:00.690 --> 00:51:02.970 Inflection point, that's a point where-- 00:51:02.970 --> 00:51:07.620 because f'' at that place is equal to 0. 00:51:07.620 --> 00:51:10.970 So it's a place where the second derivative is 0. 00:51:10.970 --> 00:51:15.160 We also consider those to be interesting points. 00:51:15.160 --> 00:51:21.010 Now, so let me just making one closing remark here. 00:51:21.010 --> 00:51:26.590 Which is that all of this information fits together. 00:51:26.590 --> 00:51:29.610 And we're going to have much, much harder examples of this 00:51:29.610 --> 00:51:32.570 where you'll actually have to think about what's going on. 00:51:32.570 --> 00:51:35.190 But there's a lot of stuff protecting you. 00:51:35.190 --> 00:51:39.050 And functions will behave themselves 00:51:39.050 --> 00:51:40.920 and turn around appropriately. 00:51:40.920 --> 00:51:43.473 Anyway, we'll talk about it next time.