1 00:00:00,000 --> 00:00:08,850 PROFESSOR: Welcome to recitation. 2 00:00:08,850 --> 00:00:10,660 Today in this video what we're going to do 3 00:00:10,660 --> 00:00:13,250 is look at how we can determine the graph 4 00:00:13,250 --> 00:00:15,930 of a derivative of a function from the graph 5 00:00:15,930 --> 00:00:17,310 of the function itself. 6 00:00:17,310 --> 00:00:19,950 So I've given a function here. 7 00:00:19,950 --> 00:00:23,330 We're calling it just y equals f of x-- or this is the curve, 8 00:00:23,330 --> 00:00:24,260 y equals f of x. 9 00:00:24,260 --> 00:00:26,860 So we're thinking about a function f of x. 10 00:00:26,860 --> 00:00:29,280 I'm not giving you the equation for the function. 11 00:00:29,280 --> 00:00:31,090 I'm just giving you the graph. 12 00:00:31,090 --> 00:00:32,780 And what I'd like you to do, what 13 00:00:32,780 --> 00:00:34,490 I'd like us to do in this time, is 14 00:00:34,490 --> 00:00:36,820 to figure out what the curve y equals 15 00:00:36,820 --> 00:00:38,570 f prime of x will look like. 16 00:00:38,570 --> 00:00:41,500 So that's our objective. 17 00:00:41,500 --> 00:00:44,940 So what we'll do first is try and figure out the things 18 00:00:44,940 --> 00:00:47,030 that we know about f prime of x. 19 00:00:47,030 --> 00:00:49,600 So what I want to remind you is that when 20 00:00:49,600 --> 00:00:52,250 you think about a function's derivative, 21 00:00:52,250 --> 00:00:54,350 remember its derivative's output is 22 00:00:54,350 --> 00:00:56,870 measuring the slope of the tangent line at each point. 23 00:00:56,870 --> 00:00:58,810 So that's what we're interested in finding, 24 00:00:58,810 --> 00:01:01,310 is understanding the slope of the tangent line 25 00:01:01,310 --> 00:01:04,970 of this curve at each x-value. 26 00:01:04,970 --> 00:01:06,890 So it's always easiest when you're 27 00:01:06,890 --> 00:01:09,160 thinking about a derivative to find the places where 28 00:01:09,160 --> 00:01:11,196 the slope of the tangent line is 0. 29 00:01:11,196 --> 00:01:12,570 Because those are the only places 30 00:01:12,570 --> 00:01:15,990 where you can hope to change the sign on the derivative. 31 00:01:15,990 --> 00:01:19,130 So what we'd like to do is first identify, on this curve, 32 00:01:19,130 --> 00:01:22,260 where the tangent line has slope equal to 0. 33 00:01:22,260 --> 00:01:25,390 And I think there are two places we can find it fairly easily. 34 00:01:25,390 --> 00:01:29,080 That would be at whatever this x value is, that slope there 35 00:01:29,080 --> 00:01:29,580 is 0. 36 00:01:29,580 --> 00:01:32,640 It's going to be a horizontal tangent line. 37 00:01:32,640 --> 00:01:34,700 And then whatever this x value is. 38 00:01:34,700 --> 00:01:36,410 The slope there is also 0. 39 00:01:36,410 --> 00:01:37,779 Horizontal tangent line. 40 00:01:37,779 --> 00:01:40,320 But there's a third place where the slope of the tangent line 41 00:01:40,320 --> 00:01:43,510 is 0, and that's kind of hidden right in here. 42 00:01:43,510 --> 00:01:45,650 And actually, I've drawn in-- maybe you 43 00:01:45,650 --> 00:01:47,900 think there are a few more-- but we're 44 00:01:47,900 --> 00:01:50,430 going to assume that this function is always continuing 45 00:01:50,430 --> 00:01:52,050 down through this region. 46 00:01:52,050 --> 00:01:56,590 So there are three places where the tangent line is horizontal. 47 00:01:56,590 --> 00:02:01,384 So I can even sort of draw them lightly through here. 48 00:02:01,384 --> 00:02:03,050 You have three horizontal tangent lines. 49 00:02:03,050 --> 00:02:06,230 So at those points, we know that the derivative's value is 50 00:02:06,230 --> 00:02:08,860 equal to 0, the output is equal to 0. 51 00:02:08,860 --> 00:02:12,630 And now what we can determine is, between those regions, 52 00:02:12,630 --> 00:02:15,590 where are the values of the derivative positive and 53 00:02:15,590 --> 00:02:16,560 negative? 54 00:02:16,560 --> 00:02:18,426 So what I'm going to do is below here, 55 00:02:18,426 --> 00:02:20,050 I'm just going to make a line and we're 56 00:02:20,050 --> 00:02:22,210 going to sort of keep track of what 57 00:02:22,210 --> 00:02:24,000 the signs of the derivative are. 58 00:02:24,000 --> 00:02:26,770 So let me just draw. 59 00:02:26,770 --> 00:02:34,691 This would be sort of our sign on f prime. 60 00:02:34,691 --> 00:02:35,190 OK. 61 00:02:35,190 --> 00:02:37,210 So that's going to tell us what our signs are. 62 00:02:37,210 --> 00:02:41,100 So right below, we'll keep track. 63 00:02:41,100 --> 00:02:42,980 So here, this, I'll just come straight down. 64 00:02:42,980 --> 00:02:44,938 Here we know the sign of f prime is equal to 0. 65 00:02:44,938 --> 00:02:46,100 OK? 66 00:02:46,100 --> 00:02:48,480 We know it's equal to 0 there. 67 00:02:48,480 --> 00:02:51,002 We know it's also equal to 0 here, 68 00:02:51,002 --> 00:02:52,585 and we know it's also equal to 0 here. 69 00:02:52,585 --> 00:02:54,510 OK? 70 00:02:54,510 --> 00:02:57,060 And now the question is, what is the sign 71 00:02:57,060 --> 00:02:58,890 of f prime in this region? 72 00:02:58,890 --> 00:03:00,950 So to the left of whatever that x value is. 73 00:03:00,950 --> 00:03:03,990 What is the sign of f prime in this region, in this region, 74 00:03:03,990 --> 00:03:05,190 and then to the right? 75 00:03:05,190 --> 00:03:07,650 So there are really-- we can divide up the x-values 76 00:03:07,650 --> 00:03:11,460 as left of whatever that x-value is, in between these two 77 00:03:11,460 --> 00:03:13,750 values, in between these two values, 78 00:03:13,750 --> 00:03:15,184 and to the right of this x-value. 79 00:03:15,184 --> 00:03:16,600 That's really, really what we need 80 00:03:16,600 --> 00:03:20,080 to do to determine what the signs of f prime are. 81 00:03:20,080 --> 00:03:22,350 So again, what we want to do to understand 82 00:03:22,350 --> 00:03:24,490 f prime is we look at the slope of the tangent line 83 00:03:24,490 --> 00:03:26,670 of the curve y equals f of x. 84 00:03:26,670 --> 00:03:32,030 So let's pick a place in this region left of where it's 0, 85 00:03:32,030 --> 00:03:34,500 say right here, and let's look at the tangent line. 86 00:03:34,500 --> 00:03:39,060 The tangent line has what kind of slope? 87 00:03:39,060 --> 00:03:40,870 Well, it has a positive slope. 88 00:03:40,870 --> 00:03:43,560 And in fact, if you look along here, you see all of the slopes 89 00:03:43,560 --> 00:03:45,100 are positive. 90 00:03:45,100 --> 00:03:49,507 So f prime is bigger than 0 here. 91 00:03:49,507 --> 00:03:51,090 And now I'm just going to record that. 92 00:03:51,090 --> 00:03:52,798 I'm going to keep that in mind as a plus. 93 00:03:52,798 --> 00:03:55,010 The sign is positive there. 94 00:03:55,010 --> 00:03:59,550 Now, if I look right of where f prime equals 0, 95 00:03:59,550 --> 00:04:01,190 if I look for x-values to the right, 96 00:04:01,190 --> 00:04:02,660 I see that as I move to the right, 97 00:04:02,660 --> 00:04:04,369 the tangent line is curving down. 98 00:04:04,369 --> 00:04:05,660 So let me do it with the chalk. 99 00:04:05,660 --> 00:04:09,100 You see the tangent line looks, has a slope negative slope. 100 00:04:09,100 --> 00:04:14,020 If I draw one point in, it looks something like that. 101 00:04:14,020 --> 00:04:15,650 So the slope is negative there. 102 00:04:15,650 --> 00:04:16,940 So here I can record that. 103 00:04:16,940 --> 00:04:21,400 The sign of f prime is a minus sign there. 104 00:04:21,400 --> 00:04:23,540 Now, if I look between these two x-values, which 105 00:04:23,540 --> 00:04:26,810 I'm saying here it's 0 and here it's 0 for the x values, 106 00:04:26,810 --> 00:04:31,080 and I take a take a point, we notice the sign is negative 107 00:04:31,080 --> 00:04:31,900 there, also. 108 00:04:31,900 --> 00:04:35,290 So in fact, the sign of f prime changed 109 00:04:35,290 --> 00:04:39,380 at this zero of f prime, but it stays the same 110 00:04:39,380 --> 00:04:40,970 around this zero of f prime. 111 00:04:40,970 --> 00:04:43,400 So it's negative and then it goes to negative again. 112 00:04:43,400 --> 00:04:45,620 It's negative, then 0, then negative. 113 00:04:45,620 --> 00:04:47,890 And then if I look to the right of this x-value 114 00:04:47,890 --> 00:04:52,790 and I take a point, I see that the slope of the tangent line 115 00:04:52,790 --> 00:04:53,689 is positive. 116 00:04:53,689 --> 00:04:55,105 And so the sign there is positive. 117 00:04:57,870 --> 00:05:02,370 So we have the derivative is positive, and then 0, and then 118 00:05:02,370 --> 00:05:05,230 negative, and then 0, and then negative, and then 0, and then 119 00:05:05,230 --> 00:05:06,070 positive. 120 00:05:06,070 --> 00:05:07,640 So there's a lot going on. 121 00:05:07,640 --> 00:05:11,000 But I, if I want to plot, now, y equals f prime of x, I 122 00:05:11,000 --> 00:05:14,600 have some sort of launching point by which to do that. 123 00:05:14,600 --> 00:05:17,060 So what I can do is, I know that the derivative 0-- 124 00:05:17,060 --> 00:05:19,360 I'm going to draw the derivative in blue, 125 00:05:19,360 --> 00:05:23,380 here-- the derivative is 0, its output is 0 at these places. 126 00:05:23,380 --> 00:05:26,724 So I'm going to put those points on. 127 00:05:26,724 --> 00:05:29,140 And then if I were just trying to get a rough idea of what 128 00:05:29,140 --> 00:05:32,910 happens, the derivative is positive left of this x value. 129 00:05:32,910 --> 00:05:35,376 So it's certainly coming down. 130 00:05:35,376 --> 00:05:36,413 It's coming down. 131 00:05:36,413 --> 00:05:41,130 Oops, let me make these a little darker. 132 00:05:41,130 --> 00:05:43,190 It's coming down because it's positive. 133 00:05:43,190 --> 00:05:46,610 It's coming down to 0-- it has to stay above the x-axis, 134 00:05:46,610 --> 00:05:48,185 but it has to head towards 0. 135 00:05:48,185 --> 00:05:49,677 Right? 136 00:05:49,677 --> 00:05:51,260 What does that actually correspond to? 137 00:05:51,260 --> 00:05:53,000 Well, look at what the slopes are doing. 138 00:05:53,000 --> 00:05:55,070 The slopes of these tangent lines, 139 00:05:55,070 --> 00:05:58,250 as I move in the x-direction, the slope-- let me just 140 00:05:58,250 --> 00:06:01,630 keep my hand, watch what my hand is doing-- the slope is always 141 00:06:01,630 --> 00:06:04,960 positive, but it's becoming less and less vertical, right? 142 00:06:04,960 --> 00:06:06,710 It's headed towards horizontal. 143 00:06:06,710 --> 00:06:09,470 So the slope that was steeper over here 144 00:06:09,470 --> 00:06:11,610 is becoming less steep. 145 00:06:11,610 --> 00:06:15,120 The steepness is really the magnitude of the derivative. 146 00:06:15,120 --> 00:06:18,370 That's really measuring how far it is, the output is, from 0. 147 00:06:18,370 --> 00:06:20,610 So as the derivative becomes less steep, 148 00:06:20,610 --> 00:06:23,780 the derivative's values have to be headed closer to 0. 149 00:06:23,780 --> 00:06:26,649 Now, what happens when the derivative is equal to 0 here? 150 00:06:26,649 --> 00:06:28,940 Well, all of a sudden the slopes are becoming negative. 151 00:06:28,940 --> 00:06:31,735 So the outputs of the derivative are negative. 152 00:06:31,735 --> 00:06:34,150 It's going down. 153 00:06:34,150 --> 00:06:37,140 But then once it hits here, again, notice what happens. 154 00:06:37,140 --> 00:06:39,780 The derivative is 0 again, and notice how I get there. 155 00:06:39,780 --> 00:06:41,494 The derivative's negative, and then it 156 00:06:41,494 --> 00:06:44,757 starts to-- the slopes of these tangent lines start 157 00:06:44,757 --> 00:06:45,465 to get shallower. 158 00:06:45,465 --> 00:06:47,350 Right? 159 00:06:47,350 --> 00:06:49,062 They were steep and then somewhere they 160 00:06:49,062 --> 00:06:50,020 start to get shallower. 161 00:06:50,020 --> 00:06:53,270 So there's someplace sort of in the x-values between here 162 00:06:53,270 --> 00:06:55,100 and here where the derivative is as 163 00:06:55,100 --> 00:06:58,750 steep as it gets in this region, and then gets less steep. 164 00:06:58,750 --> 00:07:00,800 The steepest point is that point where 165 00:07:00,800 --> 00:07:04,020 you have the biggest magnitude in that region for f prime. 166 00:07:04,020 --> 00:07:06,650 So that's where it's going to be furthest from 0. 167 00:07:06,650 --> 00:07:09,020 So if I'm guessing, it looks like right 168 00:07:09,020 --> 00:07:12,720 around here the tangent line is as steep as it ever 169 00:07:12,720 --> 00:07:15,480 gets in that region, between these two zeros, 170 00:07:15,480 --> 00:07:16,980 and then it gets less steep. 171 00:07:16,980 --> 00:07:19,110 So I'd say, right around there we 172 00:07:19,110 --> 00:07:21,094 should say, OK, that's as low as it goes 173 00:07:21,094 --> 00:07:22,552 and now it's going to come back up. 174 00:07:22,552 --> 00:07:24,390 OK? 175 00:07:24,390 --> 00:07:25,850 So hopefully that makes sense. 176 00:07:25,850 --> 00:07:27,640 We'll get to see it again, here. 177 00:07:27,640 --> 00:07:30,430 Between these two zeros the same kind of thing happens. 178 00:07:30,430 --> 00:07:33,350 But notice-- this is, we have to be careful-- we shouldn't 179 00:07:33,350 --> 00:07:36,306 go through 0 here because the derivative's output, 180 00:07:36,306 --> 00:07:37,180 the sign is negative. 181 00:07:37,180 --> 00:07:39,092 Right? 182 00:07:39,092 --> 00:07:40,550 Notice, so the tangent line, it was 183 00:07:40,550 --> 00:07:43,380 negative, negative, negative, 0, oh, it's still negative. 184 00:07:43,380 --> 00:07:46,260 So the outputs are still negative, 185 00:07:46,260 --> 00:07:49,870 and they're going to be negative all the way to this zero. 186 00:07:49,870 --> 00:07:53,530 And what we need to see again is the same kind of thing 187 00:07:53,530 --> 00:07:55,040 happens as happened in this region 188 00:07:55,040 --> 00:07:57,240 will happen in this region. 189 00:07:57,240 --> 00:07:59,450 The point being that, again, we're 0 here. 190 00:07:59,450 --> 00:08:00,920 We're 0 here. 191 00:08:00,920 --> 00:08:03,650 So somewhere in the middle, we start at 0, 192 00:08:03,650 --> 00:08:06,260 the tangent lines start to get steeper, then at some point 193 00:08:06,260 --> 00:08:09,090 they stop getting steeper, they start getting shallower. 194 00:08:09,090 --> 00:08:12,370 That place looks maybe right around here. 195 00:08:12,370 --> 00:08:14,770 That's the sort of steepest tangent line, 196 00:08:14,770 --> 00:08:16,180 then it gets less steep. 197 00:08:16,180 --> 00:08:19,480 So that's the place where the derivative's magnitude 198 00:08:19,480 --> 00:08:22,621 is going to be the biggest in this region. 199 00:08:22,621 --> 00:08:24,120 And actually, I've sort of drawn it, 200 00:08:24,120 --> 00:08:26,620 they look like they're about the same steepness at those two 201 00:08:26,620 --> 00:08:30,370 places, so I should probably put the outputs about the same 202 00:08:30,370 --> 00:08:30,870 down here. 203 00:08:30,870 --> 00:08:33,150 Their magnitudes are about the same. 204 00:08:33,150 --> 00:08:36,686 So this has to bounce off, come up here. 205 00:08:36,686 --> 00:08:38,560 I made that a little sharper than I meant to. 206 00:08:38,560 --> 00:08:40,720 OK? 207 00:08:40,720 --> 00:08:41,710 So that's the place. 208 00:08:41,710 --> 00:08:45,100 That's the output here-- or the tangent line, sorry. 209 00:08:45,100 --> 00:08:47,690 The tangent line at this x value is the steepest 210 00:08:47,690 --> 00:08:52,210 that we get in this region, so the output at that x-value 211 00:08:52,210 --> 00:08:53,830 is the lowest we get. 212 00:08:53,830 --> 00:08:55,370 And then, when we're to the right 213 00:08:55,370 --> 00:08:57,680 of this zero for the derivative, we 214 00:08:57,680 --> 00:08:59,610 start seeing the tangent lines positive-- 215 00:08:59,610 --> 00:09:02,706 we pointed that out already-- and it gets more positive. 216 00:09:02,706 --> 00:09:04,580 So it starts at 0, it starts to get positive, 217 00:09:04,580 --> 00:09:06,220 and then it gets more positive. 218 00:09:06,220 --> 00:09:10,730 It's going to do something like that, roughly. 219 00:09:10,730 --> 00:09:13,580 So let me fill in the dotted lines so we can see it clearly. 220 00:09:23,020 --> 00:09:25,430 Well, this is not exact, but this 221 00:09:25,430 --> 00:09:29,130 is a fairly good drawing, I think we can say, 222 00:09:29,130 --> 00:09:30,260 of f prime of x. 223 00:09:30,260 --> 00:09:32,740 y equals f prime of x. 224 00:09:32,740 --> 00:09:34,670 And now I'm going to ask you a question. 225 00:09:34,670 --> 00:09:36,444 I'm going to write it on the board, 226 00:09:36,444 --> 00:09:38,860 and then I'm going to give you a moment to think about it. 227 00:09:38,860 --> 00:09:40,068 So let me write the question. 228 00:09:43,590 --> 00:09:54,085 It's, find a function y equals-- or sorry-- 229 00:09:54,085 --> 00:10:06,200 find a function g of x so that y equals g prime of x looks 230 00:10:06,200 --> 00:10:11,566 like y equals f prime of x. 231 00:10:11,566 --> 00:10:13,440 OK, let me be clear about that, and then I'll 232 00:10:13,440 --> 00:10:15,010 give you a moment to think about it. 233 00:10:15,010 --> 00:10:16,525 So I want you to find a function g 234 00:10:16,525 --> 00:10:20,910 of x so that its derivative's graph, y equals g prime of x, 235 00:10:20,910 --> 00:10:23,500 looks exactly like the graph we've drawn in blue here, 236 00:10:23,500 --> 00:10:25,230 y equals f prime of x. 237 00:10:25,230 --> 00:10:27,470 Now, I don't want you to find something in terms 238 00:10:27,470 --> 00:10:29,460 of x squareds and x cubes. 239 00:10:29,460 --> 00:10:34,620 I don't want you to find an actual g of x equals something 240 00:10:34,620 --> 00:10:35,390 in terms of x. 241 00:10:35,390 --> 00:10:38,260 I want you to just try and find a relationship 242 00:10:38,260 --> 00:10:40,770 that it must have with f. 243 00:10:40,770 --> 00:10:42,929 So I'm going to give me a moment to think about it 244 00:10:42,929 --> 00:10:45,220 and work out your answer, and I'll be back to tell you. 245 00:10:53,780 --> 00:10:54,280 OK. 246 00:10:54,280 --> 00:10:55,170 Welcome back. 247 00:10:55,170 --> 00:10:57,160 So what we're looking for is a function 248 00:10:57,160 --> 00:10:59,840 g of x so that its derivative, when I graph it, 249 00:10:59,840 --> 00:11:02,330 y equals g prime of x, I get exactly the same curve 250 00:11:02,330 --> 00:11:03,500 as the blue one. 251 00:11:03,500 --> 00:11:04,970 The blue one. 252 00:11:04,970 --> 00:11:06,780 And the point is that if you thought 253 00:11:06,780 --> 00:11:08,840 about it for a little bit, what you really 254 00:11:08,840 --> 00:11:13,670 need is a function that looks exactly like this function, 255 00:11:13,670 --> 00:11:17,510 y equals f of x, at all the x-values in terms 256 00:11:17,510 --> 00:11:21,230 of its slopes, but those slopes can happen 257 00:11:21,230 --> 00:11:23,190 shifted up or down anywhere. 258 00:11:23,190 --> 00:11:25,650 So the point is that if I take the function y equals 259 00:11:25,650 --> 00:11:28,920 f of x and I add a constant to it, which shifts 260 00:11:28,920 --> 00:11:32,500 the whole graph up or down, the tangent lines 261 00:11:32,500 --> 00:11:34,570 are unaffected by that shift. 262 00:11:34,570 --> 00:11:36,400 And so I get exactly the same picture 263 00:11:36,400 --> 00:11:39,070 when I take the derivative of that graph. 264 00:11:39,070 --> 00:11:43,070 When I look at that the tangent line slopes of that graph. 265 00:11:43,070 --> 00:11:45,010 So you could draw another picture 266 00:11:45,010 --> 00:11:48,790 and check it for yourself if you didn't feel convinced, 267 00:11:48,790 --> 00:11:51,390 shift this, shift this curve up, and then look 268 00:11:51,390 --> 00:11:53,400 at what the tangent lines do on that curve. 269 00:11:53,400 --> 00:11:55,590 But then you'll see its derivative's outputs 270 00:11:55,590 --> 00:11:57,350 are exactly the same. 271 00:11:57,350 --> 00:11:58,819 So we'll stop there.