WEBVTT 00:00:00.000 --> 00:00:08.850 PROFESSOR: Welcome to recitation. 00:00:08.850 --> 00:00:10.660 Today in this video what we're going to do 00:00:10.660 --> 00:00:13.250 is look at how we can determine the graph 00:00:13.250 --> 00:00:15.930 of a derivative of a function from the graph 00:00:15.930 --> 00:00:17.310 of the function itself. 00:00:17.310 --> 00:00:19.950 So I've given a function here. 00:00:19.950 --> 00:00:23.330 We're calling it just y equals f of x-- or this is the curve, 00:00:23.330 --> 00:00:24.260 y equals f of x. 00:00:24.260 --> 00:00:26.860 So we're thinking about a function f of x. 00:00:26.860 --> 00:00:29.280 I'm not giving you the equation for the function. 00:00:29.280 --> 00:00:31.090 I'm just giving you the graph. 00:00:31.090 --> 00:00:32.780 And what I'd like you to do, what 00:00:32.780 --> 00:00:34.490 I'd like us to do in this time, is 00:00:34.490 --> 00:00:36.820 to figure out what the curve y equals 00:00:36.820 --> 00:00:38.570 f prime of x will look like. 00:00:38.570 --> 00:00:41.500 So that's our objective. 00:00:41.500 --> 00:00:44.940 So what we'll do first is try and figure out the things 00:00:44.940 --> 00:00:47.030 that we know about f prime of x. 00:00:47.030 --> 00:00:49.600 So what I want to remind you is that when 00:00:49.600 --> 00:00:52.250 you think about a function's derivative, 00:00:52.250 --> 00:00:54.350 remember its derivative's output is 00:00:54.350 --> 00:00:56.870 measuring the slope of the tangent line at each point. 00:00:56.870 --> 00:00:58.810 So that's what we're interested in finding, 00:00:58.810 --> 00:01:01.310 is understanding the slope of the tangent line 00:01:01.310 --> 00:01:04.970 of this curve at each x-value. 00:01:04.970 --> 00:01:06.890 So it's always easiest when you're 00:01:06.890 --> 00:01:09.160 thinking about a derivative to find the places where 00:01:09.160 --> 00:01:11.196 the slope of the tangent line is 0. 00:01:11.196 --> 00:01:12.570 Because those are the only places 00:01:12.570 --> 00:01:15.990 where you can hope to change the sign on the derivative. 00:01:15.990 --> 00:01:19.130 So what we'd like to do is first identify, on this curve, 00:01:19.130 --> 00:01:22.260 where the tangent line has slope equal to 0. 00:01:22.260 --> 00:01:25.390 And I think there are two places we can find it fairly easily. 00:01:25.390 --> 00:01:29.080 That would be at whatever this x value is, that slope there 00:01:29.080 --> 00:01:29.580 is 0. 00:01:29.580 --> 00:01:32.640 It's going to be a horizontal tangent line. 00:01:32.640 --> 00:01:34.700 And then whatever this x value is. 00:01:34.700 --> 00:01:36.410 The slope there is also 0. 00:01:36.410 --> 00:01:37.779 Horizontal tangent line. 00:01:37.779 --> 00:01:40.320 But there's a third place where the slope of the tangent line 00:01:40.320 --> 00:01:43.510 is 0, and that's kind of hidden right in here. 00:01:43.510 --> 00:01:45.650 And actually, I've drawn in-- maybe you 00:01:45.650 --> 00:01:47.900 think there are a few more-- but we're 00:01:47.900 --> 00:01:50.430 going to assume that this function is always continuing 00:01:50.430 --> 00:01:52.050 down through this region. 00:01:52.050 --> 00:01:56.590 So there are three places where the tangent line is horizontal. 00:01:56.590 --> 00:02:01.384 So I can even sort of draw them lightly through here. 00:02:01.384 --> 00:02:03.050 You have three horizontal tangent lines. 00:02:03.050 --> 00:02:06.230 So at those points, we know that the derivative's value is 00:02:06.230 --> 00:02:08.860 equal to 0, the output is equal to 0. 00:02:08.860 --> 00:02:12.630 And now what we can determine is, between those regions, 00:02:12.630 --> 00:02:15.590 where are the values of the derivative positive and 00:02:15.590 --> 00:02:16.560 negative? 00:02:16.560 --> 00:02:18.426 So what I'm going to do is below here, 00:02:18.426 --> 00:02:20.050 I'm just going to make a line and we're 00:02:20.050 --> 00:02:22.210 going to sort of keep track of what 00:02:22.210 --> 00:02:24.000 the signs of the derivative are. 00:02:24.000 --> 00:02:26.770 So let me just draw. 00:02:26.770 --> 00:02:34.691 This would be sort of our sign on f prime. 00:02:34.691 --> 00:02:35.190 OK. 00:02:35.190 --> 00:02:37.210 So that's going to tell us what our signs are. 00:02:37.210 --> 00:02:41.100 So right below, we'll keep track. 00:02:41.100 --> 00:02:42.980 So here, this, I'll just come straight down. 00:02:42.980 --> 00:02:44.938 Here we know the sign of f prime is equal to 0. 00:02:44.938 --> 00:02:46.100 OK? 00:02:46.100 --> 00:02:48.480 We know it's equal to 0 there. 00:02:48.480 --> 00:02:51.002 We know it's also equal to 0 here, 00:02:51.002 --> 00:02:52.585 and we know it's also equal to 0 here. 00:02:52.585 --> 00:02:54.510 OK? 00:02:54.510 --> 00:02:57.060 And now the question is, what is the sign 00:02:57.060 --> 00:02:58.890 of f prime in this region? 00:02:58.890 --> 00:03:00.950 So to the left of whatever that x value is. 00:03:00.950 --> 00:03:03.990 What is the sign of f prime in this region, in this region, 00:03:03.990 --> 00:03:05.190 and then to the right? 00:03:05.190 --> 00:03:07.650 So there are really-- we can divide up the x-values 00:03:07.650 --> 00:03:11.460 as left of whatever that x-value is, in between these two 00:03:11.460 --> 00:03:13.750 values, in between these two values, 00:03:13.750 --> 00:03:15.184 and to the right of this x-value. 00:03:15.184 --> 00:03:16.600 That's really, really what we need 00:03:16.600 --> 00:03:20.080 to do to determine what the signs of f prime are. 00:03:20.080 --> 00:03:22.350 So again, what we want to do to understand 00:03:22.350 --> 00:03:24.490 f prime is we look at the slope of the tangent line 00:03:24.490 --> 00:03:26.670 of the curve y equals f of x. 00:03:26.670 --> 00:03:32.030 So let's pick a place in this region left of where it's 0, 00:03:32.030 --> 00:03:34.500 say right here, and let's look at the tangent line. 00:03:34.500 --> 00:03:39.060 The tangent line has what kind of slope? 00:03:39.060 --> 00:03:40.870 Well, it has a positive slope. 00:03:40.870 --> 00:03:43.560 And in fact, if you look along here, you see all of the slopes 00:03:43.560 --> 00:03:45.100 are positive. 00:03:45.100 --> 00:03:49.507 So f prime is bigger than 0 here. 00:03:49.507 --> 00:03:51.090 And now I'm just going to record that. 00:03:51.090 --> 00:03:52.798 I'm going to keep that in mind as a plus. 00:03:52.798 --> 00:03:55.010 The sign is positive there. 00:03:55.010 --> 00:03:59.550 Now, if I look right of where f prime equals 0, 00:03:59.550 --> 00:04:01.190 if I look for x-values to the right, 00:04:01.190 --> 00:04:02.660 I see that as I move to the right, 00:04:02.660 --> 00:04:04.369 the tangent line is curving down. 00:04:04.369 --> 00:04:05.660 So let me do it with the chalk. 00:04:05.660 --> 00:04:09.100 You see the tangent line looks, has a slope negative slope. 00:04:09.100 --> 00:04:14.020 If I draw one point in, it looks something like that. 00:04:14.020 --> 00:04:15.650 So the slope is negative there. 00:04:15.650 --> 00:04:16.940 So here I can record that. 00:04:16.940 --> 00:04:21.400 The sign of f prime is a minus sign there. 00:04:21.400 --> 00:04:23.540 Now, if I look between these two x-values, which 00:04:23.540 --> 00:04:26.810 I'm saying here it's 0 and here it's 0 for the x values, 00:04:26.810 --> 00:04:31.080 and I take a take a point, we notice the sign is negative 00:04:31.080 --> 00:04:31.900 there, also. 00:04:31.900 --> 00:04:35.290 So in fact, the sign of f prime changed 00:04:35.290 --> 00:04:39.380 at this zero of f prime, but it stays the same 00:04:39.380 --> 00:04:40.970 around this zero of f prime. 00:04:40.970 --> 00:04:43.400 So it's negative and then it goes to negative again. 00:04:43.400 --> 00:04:45.620 It's negative, then 0, then negative. 00:04:45.620 --> 00:04:47.890 And then if I look to the right of this x-value 00:04:47.890 --> 00:04:52.790 and I take a point, I see that the slope of the tangent line 00:04:52.790 --> 00:04:53.689 is positive. 00:04:53.689 --> 00:04:55.105 And so the sign there is positive. 00:04:57.870 --> 00:05:02.370 So we have the derivative is positive, and then 0, and then 00:05:02.370 --> 00:05:05.230 negative, and then 0, and then negative, and then 0, and then 00:05:05.230 --> 00:05:06.070 positive. 00:05:06.070 --> 00:05:07.640 So there's a lot going on. 00:05:07.640 --> 00:05:11.000 But I, if I want to plot, now, y equals f prime of x, I 00:05:11.000 --> 00:05:14.600 have some sort of launching point by which to do that. 00:05:14.600 --> 00:05:17.060 So what I can do is, I know that the derivative 0-- 00:05:17.060 --> 00:05:19.360 I'm going to draw the derivative in blue, 00:05:19.360 --> 00:05:23.380 here-- the derivative is 0, its output is 0 at these places. 00:05:23.380 --> 00:05:26.724 So I'm going to put those points on. 00:05:26.724 --> 00:05:29.140 And then if I were just trying to get a rough idea of what 00:05:29.140 --> 00:05:32.910 happens, the derivative is positive left of this x value. 00:05:32.910 --> 00:05:35.376 So it's certainly coming down. 00:05:35.376 --> 00:05:36.413 It's coming down. 00:05:36.413 --> 00:05:41.130 Oops, let me make these a little darker. 00:05:41.130 --> 00:05:43.190 It's coming down because it's positive. 00:05:43.190 --> 00:05:46.610 It's coming down to 0-- it has to stay above the x-axis, 00:05:46.610 --> 00:05:48.185 but it has to head towards 0. 00:05:48.185 --> 00:05:49.677 Right? 00:05:49.677 --> 00:05:51.260 What does that actually correspond to? 00:05:51.260 --> 00:05:53.000 Well, look at what the slopes are doing. 00:05:53.000 --> 00:05:55.070 The slopes of these tangent lines, 00:05:55.070 --> 00:05:58.250 as I move in the x-direction, the slope-- let me just 00:05:58.250 --> 00:06:01.630 keep my hand, watch what my hand is doing-- the slope is always 00:06:01.630 --> 00:06:04.960 positive, but it's becoming less and less vertical, right? 00:06:04.960 --> 00:06:06.710 It's headed towards horizontal. 00:06:06.710 --> 00:06:09.470 So the slope that was steeper over here 00:06:09.470 --> 00:06:11.610 is becoming less steep. 00:06:11.610 --> 00:06:15.120 The steepness is really the magnitude of the derivative. 00:06:15.120 --> 00:06:18.370 That's really measuring how far it is, the output is, from 0. 00:06:18.370 --> 00:06:20.610 So as the derivative becomes less steep, 00:06:20.610 --> 00:06:23.780 the derivative's values have to be headed closer to 0. 00:06:23.780 --> 00:06:26.649 Now, what happens when the derivative is equal to 0 here? 00:06:26.649 --> 00:06:28.940 Well, all of a sudden the slopes are becoming negative. 00:06:28.940 --> 00:06:31.735 So the outputs of the derivative are negative. 00:06:31.735 --> 00:06:34.150 It's going down. 00:06:34.150 --> 00:06:37.140 But then once it hits here, again, notice what happens. 00:06:37.140 --> 00:06:39.780 The derivative is 0 again, and notice how I get there. 00:06:39.780 --> 00:06:41.494 The derivative's negative, and then it 00:06:41.494 --> 00:06:44.757 starts to-- the slopes of these tangent lines start 00:06:44.757 --> 00:06:45.465 to get shallower. 00:06:45.465 --> 00:06:47.350 Right? 00:06:47.350 --> 00:06:49.062 They were steep and then somewhere they 00:06:49.062 --> 00:06:50.020 start to get shallower. 00:06:50.020 --> 00:06:53.270 So there's someplace sort of in the x-values between here 00:06:53.270 --> 00:06:55.100 and here where the derivative is as 00:06:55.100 --> 00:06:58.750 steep as it gets in this region, and then gets less steep. 00:06:58.750 --> 00:07:00.800 The steepest point is that point where 00:07:00.800 --> 00:07:04.020 you have the biggest magnitude in that region for f prime. 00:07:04.020 --> 00:07:06.650 So that's where it's going to be furthest from 0. 00:07:06.650 --> 00:07:09.020 So if I'm guessing, it looks like right 00:07:09.020 --> 00:07:12.720 around here the tangent line is as steep as it ever 00:07:12.720 --> 00:07:15.480 gets in that region, between these two zeros, 00:07:15.480 --> 00:07:16.980 and then it gets less steep. 00:07:16.980 --> 00:07:19.110 So I'd say, right around there we 00:07:19.110 --> 00:07:21.094 should say, OK, that's as low as it goes 00:07:21.094 --> 00:07:22.552 and now it's going to come back up. 00:07:22.552 --> 00:07:24.390 OK? 00:07:24.390 --> 00:07:25.850 So hopefully that makes sense. 00:07:25.850 --> 00:07:27.640 We'll get to see it again, here. 00:07:27.640 --> 00:07:30.430 Between these two zeros the same kind of thing happens. 00:07:30.430 --> 00:07:33.350 But notice-- this is, we have to be careful-- we shouldn't 00:07:33.350 --> 00:07:36.306 go through 0 here because the derivative's output, 00:07:36.306 --> 00:07:37.180 the sign is negative. 00:07:37.180 --> 00:07:39.092 Right? 00:07:39.092 --> 00:07:40.550 Notice, so the tangent line, it was 00:07:40.550 --> 00:07:43.380 negative, negative, negative, 0, oh, it's still negative. 00:07:43.380 --> 00:07:46.260 So the outputs are still negative, 00:07:46.260 --> 00:07:49.870 and they're going to be negative all the way to this zero. 00:07:49.870 --> 00:07:53.530 And what we need to see again is the same kind of thing 00:07:53.530 --> 00:07:55.040 happens as happened in this region 00:07:55.040 --> 00:07:57.240 will happen in this region. 00:07:57.240 --> 00:07:59.450 The point being that, again, we're 0 here. 00:07:59.450 --> 00:08:00.920 We're 0 here. 00:08:00.920 --> 00:08:03.650 So somewhere in the middle, we start at 0, 00:08:03.650 --> 00:08:06.260 the tangent lines start to get steeper, then at some point 00:08:06.260 --> 00:08:09.090 they stop getting steeper, they start getting shallower. 00:08:09.090 --> 00:08:12.370 That place looks maybe right around here. 00:08:12.370 --> 00:08:14.770 That's the sort of steepest tangent line, 00:08:14.770 --> 00:08:16.180 then it gets less steep. 00:08:16.180 --> 00:08:19.480 So that's the place where the derivative's magnitude 00:08:19.480 --> 00:08:22.621 is going to be the biggest in this region. 00:08:22.621 --> 00:08:24.120 And actually, I've sort of drawn it, 00:08:24.120 --> 00:08:26.620 they look like they're about the same steepness at those two 00:08:26.620 --> 00:08:30.370 places, so I should probably put the outputs about the same 00:08:30.370 --> 00:08:30.870 down here. 00:08:30.870 --> 00:08:33.150 Their magnitudes are about the same. 00:08:33.150 --> 00:08:36.686 So this has to bounce off, come up here. 00:08:36.686 --> 00:08:38.560 I made that a little sharper than I meant to. 00:08:38.560 --> 00:08:40.720 OK? 00:08:40.720 --> 00:08:41.710 So that's the place. 00:08:41.710 --> 00:08:45.100 That's the output here-- or the tangent line, sorry. 00:08:45.100 --> 00:08:47.690 The tangent line at this x value is the steepest 00:08:47.690 --> 00:08:52.210 that we get in this region, so the output at that x-value 00:08:52.210 --> 00:08:53.830 is the lowest we get. 00:08:53.830 --> 00:08:55.370 And then, when we're to the right 00:08:55.370 --> 00:08:57.680 of this zero for the derivative, we 00:08:57.680 --> 00:08:59.610 start seeing the tangent lines positive-- 00:08:59.610 --> 00:09:02.706 we pointed that out already-- and it gets more positive. 00:09:02.706 --> 00:09:04.580 So it starts at 0, it starts to get positive, 00:09:04.580 --> 00:09:06.220 and then it gets more positive. 00:09:06.220 --> 00:09:10.730 It's going to do something like that, roughly. 00:09:10.730 --> 00:09:13.580 So let me fill in the dotted lines so we can see it clearly. 00:09:23.020 --> 00:09:25.430 Well, this is not exact, but this 00:09:25.430 --> 00:09:29.130 is a fairly good drawing, I think we can say, 00:09:29.130 --> 00:09:30.260 of f prime of x. 00:09:30.260 --> 00:09:32.740 y equals f prime of x. 00:09:32.740 --> 00:09:34.670 And now I'm going to ask you a question. 00:09:34.670 --> 00:09:36.444 I'm going to write it on the board, 00:09:36.444 --> 00:09:38.860 and then I'm going to give you a moment to think about it. 00:09:38.860 --> 00:09:40.068 So let me write the question. 00:09:43.590 --> 00:09:54.085 It's, find a function y equals-- or sorry-- 00:09:54.085 --> 00:10:06.200 find a function g of x so that y equals g prime of x looks 00:10:06.200 --> 00:10:11.566 like y equals f prime of x. 00:10:11.566 --> 00:10:13.440 OK, let me be clear about that, and then I'll 00:10:13.440 --> 00:10:15.010 give you a moment to think about it. 00:10:15.010 --> 00:10:16.525 So I want you to find a function g 00:10:16.525 --> 00:10:20.910 of x so that its derivative's graph, y equals g prime of x, 00:10:20.910 --> 00:10:23.500 looks exactly like the graph we've drawn in blue here, 00:10:23.500 --> 00:10:25.230 y equals f prime of x. 00:10:25.230 --> 00:10:27.470 Now, I don't want you to find something in terms 00:10:27.470 --> 00:10:29.460 of x squareds and x cubes. 00:10:29.460 --> 00:10:34.620 I don't want you to find an actual g of x equals something 00:10:34.620 --> 00:10:35.390 in terms of x. 00:10:35.390 --> 00:10:38.260 I want you to just try and find a relationship 00:10:38.260 --> 00:10:40.770 that it must have with f. 00:10:40.770 --> 00:10:42.929 So I'm going to give me a moment to think about it 00:10:42.929 --> 00:10:45.220 and work out your answer, and I'll be back to tell you. 00:10:53.780 --> 00:10:54.280 OK. 00:10:54.280 --> 00:10:55.170 Welcome back. 00:10:55.170 --> 00:10:57.160 So what we're looking for is a function 00:10:57.160 --> 00:10:59.840 g of x so that its derivative, when I graph it, 00:10:59.840 --> 00:11:02.330 y equals g prime of x, I get exactly the same curve 00:11:02.330 --> 00:11:03.500 as the blue one. 00:11:03.500 --> 00:11:04.970 The blue one. 00:11:04.970 --> 00:11:06.780 And the point is that if you thought 00:11:06.780 --> 00:11:08.840 about it for a little bit, what you really 00:11:08.840 --> 00:11:13.670 need is a function that looks exactly like this function, 00:11:13.670 --> 00:11:17.510 y equals f of x, at all the x-values in terms 00:11:17.510 --> 00:11:21.230 of its slopes, but those slopes can happen 00:11:21.230 --> 00:11:23.190 shifted up or down anywhere. 00:11:23.190 --> 00:11:25.650 So the point is that if I take the function y equals 00:11:25.650 --> 00:11:28.920 f of x and I add a constant to it, which shifts 00:11:28.920 --> 00:11:32.500 the whole graph up or down, the tangent lines 00:11:32.500 --> 00:11:34.570 are unaffected by that shift. 00:11:34.570 --> 00:11:36.400 And so I get exactly the same picture 00:11:36.400 --> 00:11:39.070 when I take the derivative of that graph. 00:11:39.070 --> 00:11:43.070 When I look at that the tangent line slopes of that graph. 00:11:43.070 --> 00:11:45.010 So you could draw another picture 00:11:45.010 --> 00:11:48.790 and check it for yourself if you didn't feel convinced, 00:11:48.790 --> 00:11:51.390 shift this, shift this curve up, and then look 00:11:51.390 --> 00:11:53.400 at what the tangent lines do on that curve. 00:11:53.400 --> 00:11:55.590 But then you'll see its derivative's outputs 00:11:55.590 --> 00:11:57.350 are exactly the same. 00:11:57.350 --> 00:11:58.819 So we'll stop there.