1 00:00:24,108 --> 00:00:26,110 PROFESSOR: Hi. 2 00:00:26,110 --> 00:00:28,725 Well, I hope you're ready for second derivatives. 3 00:00:31,230 --> 00:00:34,560 We don't go higher than that in many problems, but the 4 00:00:34,560 --> 00:00:36,400 second derivative is an important-- 5 00:00:36,400 --> 00:00:40,400 the derivative of the derivative is an important 6 00:00:40,400 --> 00:00:45,180 thing to know, especially in problems with maximum and 7 00:00:45,180 --> 00:00:49,200 minimum, which is the big application of derivatives, to 8 00:00:49,200 --> 00:00:54,240 locate a maximum or a minimum, and to decide which one it is. 9 00:00:54,240 --> 00:00:59,700 And I can tell you right away, locating a maximum, minimum, 10 00:00:59,700 --> 00:01:02,690 is the first derivative's job. 11 00:01:02,690 --> 00:01:04,950 The first derivative is 0. 12 00:01:04,950 --> 00:01:08,710 If I have a maximum or a minimum, and we'll have 13 00:01:08,710 --> 00:01:17,000 pictures, somewhere in the middle of my function I'll 14 00:01:17,000 --> 00:01:20,220 recognize by derivative equals 0. 15 00:01:20,220 --> 00:01:23,470 Slope equals 0, that the function is leveling off, 16 00:01:23,470 --> 00:01:30,370 either bending down or bending up, maximum or minimum. 17 00:01:30,370 --> 00:01:33,060 OK, and it's the second derivative that 18 00:01:33,060 --> 00:01:34,840 tells me which it is. 19 00:01:34,840 --> 00:01:40,670 The second derivative tells me the bending of the graph. 20 00:01:40,670 --> 00:01:47,870 OK, so we now will have three generations. 21 00:01:47,870 --> 00:01:53,220 The big picture of calculus started with two functions: 22 00:01:53,220 --> 00:01:54,930 the distance and the speed. 23 00:01:54,930 --> 00:02:01,720 And we discussed in detail the connection between them. 24 00:02:01,720 --> 00:02:06,410 How to recover the speed if we know the distance, take the 25 00:02:06,410 --> 00:02:09,030 derivative. 26 00:02:09,030 --> 00:02:13,590 Now comes the derivative of the speed, which in that 27 00:02:13,590 --> 00:02:18,140 language, in the distance-speed-time language, 28 00:02:18,140 --> 00:02:23,800 the second derivative is the acceleration, the rate at 29 00:02:23,800 --> 00:02:26,760 which your speed is changing, the rate at which you're 30 00:02:26,760 --> 00:02:30,510 speeding up or slowing down. 31 00:02:30,510 --> 00:02:33,320 And this is the way I would write that. 32 00:02:33,320 --> 00:02:36,030 If the speed is the first derivative-- 33 00:02:36,030 --> 00:02:37,630 df dt-- 34 00:02:37,630 --> 00:02:42,190 this is the way you write the second derivative, and you say 35 00:02:42,190 --> 00:02:46,750 d second f dt squared. 36 00:02:46,750 --> 00:02:49,040 d second f dt squared. 37 00:02:49,040 --> 00:02:55,300 OK, so that's you could say the physics example: distance, 38 00:02:55,300 --> 00:02:56,080 speed, acceleration. 39 00:02:56,080 --> 00:03:00,480 And I say physics because, of course, acceleration is the a 40 00:03:00,480 --> 00:03:04,610 in Newton's Law f equals ma. 41 00:03:04,610 --> 00:03:10,400 For a graph, like these graphs here, I won't especially use 42 00:03:10,400 --> 00:03:11,710 those physics words. 43 00:03:11,710 --> 00:03:13,860 I'll use graph words. 44 00:03:13,860 --> 00:03:16,850 So I would say function one would be the 45 00:03:16,850 --> 00:03:20,280 height of the graph. 46 00:03:20,280 --> 00:03:25,690 And in this case, that height is y equals x squared, so it's 47 00:03:25,690 --> 00:03:28,880 a simple parabola. 48 00:03:28,880 --> 00:03:31,020 Here would be the slope. 49 00:03:31,020 --> 00:03:35,660 I would use the word "slope" for the second function. 50 00:03:35,660 --> 00:03:42,040 And the slope of y equals x squared we know is 2x, so we 51 00:03:42,040 --> 00:03:44,300 see the slope increasing. 52 00:03:44,300 --> 00:03:49,100 And you see on this picture the slope is increasing. 53 00:03:49,100 --> 00:03:53,340 As x increases, I'm going up more steeply. 54 00:03:53,340 --> 00:03:55,670 Now, it's the second derivative. 55 00:03:55,670 --> 00:03:57,540 And what shall I call that? 56 00:03:57,540 --> 00:03:58,390 Bending. 57 00:03:58,390 --> 00:04:01,930 Bending is the natural word for the second 58 00:04:01,930 --> 00:04:04,320 derivative on a graph. 59 00:04:04,320 --> 00:04:07,310 And what do I-- 60 00:04:07,310 --> 00:04:14,350 the derivative of 2x is 2, a constant, a positive constant, 61 00:04:14,350 --> 00:04:18,730 and that positive constant tells me that the slope is 62 00:04:18,730 --> 00:04:24,850 going upwards and that the curve is bending upwards. 63 00:04:24,850 --> 00:04:30,240 So in this simple case, we connect these three 64 00:04:30,240 --> 00:04:33,040 descriptions of our function. 65 00:04:33,040 --> 00:04:36,100 It's positive. 66 00:04:36,100 --> 00:04:38,490 It's slope is positive. 67 00:04:38,490 --> 00:04:40,860 And its second derivative-- 68 00:04:40,860 --> 00:04:41,420 bending-- 69 00:04:41,420 --> 00:04:42,980 is positive. 70 00:04:42,980 --> 00:04:47,020 And that gives us a function that goes like that. 71 00:04:47,020 --> 00:04:53,790 Now, let me go to a different function. 72 00:04:53,790 --> 00:05:02,720 Let me take a second example now, an example where not 73 00:05:02,720 --> 00:05:03,970 everything is positive. 74 00:05:07,320 --> 00:05:09,290 But let's make it familiar. 75 00:05:09,290 --> 00:05:11,610 Take sine x. 76 00:05:11,610 --> 00:05:15,390 So sine x starts out like that. 77 00:05:15,390 --> 00:05:22,660 So this is a graph of sine x up to 90 degrees, pi over 2, 78 00:05:22,660 --> 00:05:25,680 so that's y equals sine x. 79 00:05:25,680 --> 00:05:31,580 OK, what do you think about its slope? 80 00:05:31,580 --> 00:05:35,290 We know the derivative of sine x, but before we write it 81 00:05:35,290 --> 00:05:38,240 down, look at the graph. 82 00:05:38,240 --> 00:05:40,550 The slope is positive, right? 83 00:05:40,550 --> 00:05:45,550 But the slope actually starts out at 1. 84 00:05:45,550 --> 00:05:48,230 Better make it look a little more realistic. 85 00:05:48,230 --> 00:05:50,600 That's a slope of 1 there. 86 00:05:50,600 --> 00:05:55,780 So the slope starts at 1 and the slope drops to a slope of 87 00:05:55,780 --> 00:05:56,760 0 up there. 88 00:05:56,760 --> 00:05:58,660 So a slope of 1. 89 00:05:58,660 --> 00:06:00,860 I see here is a 1. 90 00:06:00,860 --> 00:06:04,170 Here I'm graphing y prime. 91 00:06:04,170 --> 00:06:09,370 dy dx I sometimes write as y prime, just because it's 92 00:06:09,370 --> 00:06:12,780 shorter, and particularly, it'll be shorter for a second 93 00:06:12,780 --> 00:06:13,620 derivative. 94 00:06:13,620 --> 00:06:23,380 So y prime, we know the derivative of sine x is cos x, 95 00:06:23,380 --> 00:06:27,250 which is pretty neat actually, that we start with a familiar 96 00:06:27,250 --> 00:06:33,400 function, and then we get its twin, its other half. 97 00:06:33,400 --> 00:06:39,040 And the cosine is the slope of the sine curve, and it starts 98 00:06:39,040 --> 00:06:43,860 at 1, a slope of 1, and it comes down to 0, as we know 99 00:06:43,860 --> 00:06:45,110 the cosine does. 100 00:06:48,680 --> 00:06:52,100 So that's a graph of the cosine. 101 00:06:52,100 --> 00:06:56,270 And now, of course, we have three generations. 102 00:06:56,270 --> 00:06:59,620 I'm going to graph y double prime. 103 00:06:59,620 --> 00:07:02,750 Let me put it up here. y double prime, the second 104 00:07:02,750 --> 00:07:06,230 derivative, the derivative of the cosine of x 105 00:07:06,230 --> 00:07:09,960 is minus sine x. 106 00:07:09,960 --> 00:07:11,850 OK, let's just-- 107 00:07:11,850 --> 00:07:15,590 from the picture, what am I seeing here? 108 00:07:15,590 --> 00:07:19,600 I'm seeing a slope of 0. 109 00:07:19,600 --> 00:07:22,960 I'm taking now the slope of the slope. 110 00:07:22,960 --> 00:07:25,680 So here it starts at 0. 111 00:07:25,680 --> 00:07:28,990 The slope is downwards, so the second derivative is going to 112 00:07:28,990 --> 00:07:30,220 be negative. 113 00:07:30,220 --> 00:07:33,570 Oh, and it is negative, minus sign x. 114 00:07:33,570 --> 00:07:46,670 So the slope starts at 0 and ends at minus 1 because that 115 00:07:46,670 --> 00:07:50,440 now comes down at a negative slope. 116 00:07:50,440 --> 00:07:51,480 The slope is negative. 117 00:07:51,480 --> 00:07:59,140 I'm going downhill, and that's a graph of the second 118 00:07:59,140 --> 00:08:01,330 derivative. 119 00:08:01,330 --> 00:08:04,950 And which way is our function bending? 120 00:08:04,950 --> 00:08:07,360 It's bending down. 121 00:08:07,360 --> 00:08:10,890 As I go along, the slope is dropping. 122 00:08:10,890 --> 00:08:14,440 And I see that in the slope curve. 123 00:08:14,440 --> 00:08:15,590 It's falling. 124 00:08:15,590 --> 00:08:20,540 And I see it in the bending curve 125 00:08:20,540 --> 00:08:25,530 because I'm below 0 here. 126 00:08:25,530 --> 00:08:36,640 This is bending down, where that one was bending up. 127 00:08:39,690 --> 00:08:45,230 I could introduce the word convex for something that 128 00:08:45,230 --> 00:08:48,960 bends upwards, and bending down, I could introduce the 129 00:08:48,960 --> 00:08:56,010 word concave. But those are just words. 130 00:08:56,010 --> 00:09:00,030 The graphs are telling us much more than the words do. 131 00:09:00,030 --> 00:09:06,300 OK, so do you see that picture bending down, but going up? 132 00:09:06,300 --> 00:09:12,750 So the slope is positive here, but the second derivative, the 133 00:09:12,750 --> 00:09:14,290 slope is dropping. 134 00:09:14,290 --> 00:09:17,920 So the second derivative-- and you have to pay attention to 135 00:09:17,920 --> 00:09:20,370 keep them straight. 136 00:09:20,370 --> 00:09:24,330 The second derivative is telling us that the original 137 00:09:24,330 --> 00:09:25,640 one is bending down. 138 00:09:25,640 --> 00:09:33,030 OK, let me continue these graphs just a little beyond 90 139 00:09:33,030 --> 00:09:38,090 degrees, pi over 2, because you'll see something 140 00:09:38,090 --> 00:09:39,360 interesting. 141 00:09:39,360 --> 00:09:41,920 So what happens in the next part of the graph? 142 00:09:41,920 --> 00:09:44,710 So this is going-- 143 00:09:44,710 --> 00:09:46,140 the sine curve, of course, 144 00:09:46,140 --> 00:09:50,430 continues on its way downwards. 145 00:09:50,430 --> 00:09:55,250 So the slope is going negative, as I know the cosine 146 00:09:55,250 --> 00:09:58,900 curve will do, as the cosine curve will come like that. 147 00:09:58,900 --> 00:10:03,010 The slope down to minus 1, the slope-- 148 00:10:03,010 --> 00:10:04,060 do you see here? 149 00:10:04,060 --> 00:10:11,320 The slope is negative, so on this slope graph, I'm below 0. 150 00:10:11,320 --> 00:10:15,530 And the slope is 0. 151 00:10:15,530 --> 00:10:21,390 Let me put a little mark at these points here, at these 152 00:10:21,390 --> 00:10:22,640 three points. 153 00:10:26,340 --> 00:10:28,940 Those are important points. 154 00:10:28,940 --> 00:10:35,250 In fact, that is a maximum, of course. 155 00:10:35,250 --> 00:10:38,110 The sine curve hits its maximum at 1. 156 00:10:41,600 --> 00:10:46,170 At that point when it hits its maximum, what's its slope? 157 00:10:46,170 --> 00:10:52,780 When you hit a maximum, you're not going up anymore. 158 00:10:52,780 --> 00:10:55,480 You haven't started down. 159 00:10:55,480 --> 00:10:58,320 The slope is 0 right there. 160 00:10:58,320 --> 00:10:59,900 What's the second derivative? 161 00:10:59,900 --> 00:11:02,260 What's the bending at a maximum? 162 00:11:02,260 --> 00:11:05,550 The bending tells you that the slope is going down, so the 163 00:11:05,550 --> 00:11:07,540 bending is negative. 164 00:11:07,540 --> 00:11:10,110 The bending is negative at a maximum. 165 00:11:12,660 --> 00:11:14,620 Good. 166 00:11:14,620 --> 00:11:20,640 OK, now I'm going to continue this sine curve for another 90 167 00:11:20,640 --> 00:11:25,910 degrees, the cosine curve, and I'll continue the bending 168 00:11:25,910 --> 00:11:31,360 curve, so I have minus sine x, which will go back up. 169 00:11:31,360 --> 00:11:34,130 OK, now what? 170 00:11:34,130 --> 00:11:35,380 Now what? 171 00:11:38,080 --> 00:11:41,420 And then, of course, it would continue along. 172 00:11:41,420 --> 00:11:45,500 OK, there's something interesting happening at 180 173 00:11:45,500 --> 00:11:47,990 degrees, at pi. 174 00:11:47,990 --> 00:11:50,030 Can I identify that point? 175 00:11:50,030 --> 00:11:52,640 So there's 180 degrees. 176 00:11:52,640 --> 00:11:54,150 Something's happening there. 177 00:11:54,150 --> 00:11:56,510 I don't see-- 178 00:11:56,510 --> 00:11:58,890 I don't quite know how to say what yet, but something's 179 00:11:58,890 --> 00:12:00,300 happening there. 180 00:12:00,300 --> 00:12:07,400 It's got to show up here, and it has to show up here. 181 00:12:07,400 --> 00:12:13,450 So whatever is happening is showing up by a point where y 182 00:12:13,450 --> 00:12:17,850 double prime, the second derivative, is 0. 183 00:12:17,850 --> 00:12:23,060 That's my new little observation, not as big a deal 184 00:12:23,060 --> 00:12:25,550 as maximum or minimum. 185 00:12:25,550 --> 00:12:26,990 This was a max here. 186 00:12:29,870 --> 00:12:34,320 And we identified it as a max because the second derivative 187 00:12:34,320 --> 00:12:35,340 was negative. 188 00:12:35,340 --> 00:12:37,790 Now I'm interested in this point. 189 00:12:37,790 --> 00:12:43,140 Can you see what's happening at this point as far as 190 00:12:43,140 --> 00:12:45,500 bending goes? 191 00:12:45,500 --> 00:12:48,890 This curve is bending down. 192 00:12:48,890 --> 00:12:53,560 But when I continue, the bending changes to up. 193 00:12:53,560 --> 00:12:58,640 This is a point where the bending changes. 194 00:12:58,640 --> 00:13:03,260 The second derivative changes sign, and we see it here. 195 00:13:03,260 --> 00:13:10,510 Up to this square point, the bending is below 0. 196 00:13:10,510 --> 00:13:14,260 The bending is downwards as I come to here. 197 00:13:14,260 --> 00:13:17,140 But then there's something rather special that-- 198 00:13:17,140 --> 00:13:20,160 you see, can I try to blow that point up? 199 00:13:20,160 --> 00:13:23,540 Here the bending is down, and there it turns to up, and 200 00:13:23,540 --> 00:13:26,360 right in there with the-- 201 00:13:26,360 --> 00:13:29,140 this is called-- 202 00:13:29,140 --> 00:13:32,110 so this is my final word to introduce-- 203 00:13:32,110 --> 00:13:35,310 inflection point. 204 00:13:35,310 --> 00:13:36,560 Don't ask me why. 205 00:13:40,920 --> 00:13:44,110 An inflection point is a point where the second 206 00:13:44,110 --> 00:13:46,990 derivative is 0. 207 00:13:46,990 --> 00:13:49,050 And what does that mean? 208 00:13:49,050 --> 00:13:54,400 That means at that moment, it stopped bending down, and it's 209 00:13:54,400 --> 00:13:57,000 going to start bending up. 210 00:13:57,000 --> 00:14:00,400 The second derivative is passing through 0. 211 00:14:00,400 --> 00:14:02,660 The sign of bending is changing. 212 00:14:02,660 --> 00:14:07,440 It's changing from concave here to convex there. 213 00:14:07,440 --> 00:14:13,160 That's a significant point on the graph. 214 00:14:13,160 --> 00:14:19,130 Not as big a thing as the max or the min 215 00:14:19,130 --> 00:14:20,250 that we had over there. 216 00:14:20,250 --> 00:14:26,750 So let me draw one more example and identify all these 217 00:14:26,750 --> 00:14:27,760 different points. 218 00:14:27,760 --> 00:14:30,790 OK, so here we go. 219 00:14:30,790 --> 00:14:35,190 I drew it ahead of time because it's got a few loops, 220 00:14:35,190 --> 00:14:38,540 and I wanted to get it in good form. 221 00:14:38,540 --> 00:14:40,350 OK, here it is. 222 00:14:40,350 --> 00:14:44,120 This is my function: x cubed minus x squared. 223 00:14:44,120 --> 00:14:48,980 Well, before I look at the picture, what would be the 224 00:14:48,980 --> 00:14:50,870 first calculus thing I do? 225 00:14:50,870 --> 00:14:52,660 I take the derivative. 226 00:14:52,660 --> 00:14:58,900 y prime is the derivative of x cubed, is three x squared 227 00:14:58,900 --> 00:15:02,860 minus the derivative of x squared, which is 2x. 228 00:15:02,860 --> 00:15:07,690 And now today, I take the derivative of that. 229 00:15:07,690 --> 00:15:11,210 I take the second derivative, y double prime. 230 00:15:11,210 --> 00:15:15,280 So the second derivative is the derivative of this. 231 00:15:15,280 --> 00:15:18,880 x squared is going to give me 2x, and I have a 3, so it's 232 00:15:18,880 --> 00:15:21,070 all together 6x. 233 00:15:21,070 --> 00:15:27,730 And minus 2x, the slope of that is minus 2, right? 234 00:15:27,730 --> 00:15:33,480 Cubic, quadratic, linear, and if I cared about y triple 235 00:15:33,480 --> 00:15:36,750 prime, which I don't, constant. 236 00:15:36,750 --> 00:15:39,640 And then the fourth derivatives and all the rest 237 00:15:39,640 --> 00:15:41,810 would be 0 for this case. 238 00:15:41,810 --> 00:15:47,870 OK, now somehow, those derivatives, those formulas 239 00:15:47,870 --> 00:15:52,820 for y, y prime, y double prime should tell me details about 240 00:15:52,820 --> 00:15:54,950 this graph. 241 00:15:54,950 --> 00:15:58,040 And the first thing I'm interested in and the most 242 00:15:58,040 --> 00:16:00,970 important thing is max and min. 243 00:16:00,970 --> 00:16:04,940 So let me set y prime to be-- 244 00:16:04,940 --> 00:16:09,070 which is 3x squared minus 2x. 245 00:16:09,070 --> 00:16:18,980 I'll set it to be 0 because I want to look for max, or min. 246 00:16:18,980 --> 00:16:23,840 And I look for both at the same time by setting y prime 247 00:16:23,840 --> 00:16:27,950 equals 0, and then I find out which I've got by looking at y 248 00:16:27,950 --> 00:16:28,760 double prime. 249 00:16:28,760 --> 00:16:31,740 So let me set y prime to be 0. 250 00:16:31,740 --> 00:16:33,150 What are the solutions? 251 00:16:33,150 --> 00:16:39,530 Where are the points on the curve where it's stationary? 252 00:16:39,530 --> 00:16:44,600 It's not climbing and it's not dropping? 253 00:16:44,600 --> 00:16:48,020 Well, I see them on the curve here. 254 00:16:48,020 --> 00:16:53,240 That is a point where the slope is 0. 255 00:16:53,240 --> 00:16:55,560 And I see one down here. 256 00:16:55,560 --> 00:17:01,220 There is a point where the slope is 0, but I can find 257 00:17:01,220 --> 00:17:02,580 them with algebra. 258 00:17:02,580 --> 00:17:08,670 I solve 3x squared equals to 2x, and I see 259 00:17:08,670 --> 00:17:10,359 it's a quadratic equation. 260 00:17:10,359 --> 00:17:12,130 I expect to find two roots. 261 00:17:12,130 --> 00:17:17,480 One of them is x equals 0, and the other one is what? 262 00:17:17,480 --> 00:17:24,990 If I cancel those x's to find a non-zero, canceling those 263 00:17:24,990 --> 00:17:31,340 x's leaves me with 3x equals 2 or x equals 2/3. 264 00:17:31,340 --> 00:17:35,570 Yeah, and that's what our graph shows. 265 00:17:35,570 --> 00:17:43,480 OK, now we can see on the graph which is a max 266 00:17:43,480 --> 00:17:45,750 and which is a min. 267 00:17:45,750 --> 00:17:48,030 And by the way, let me just notice, of 268 00:17:48,030 --> 00:17:49,475 course, this is the max. 269 00:17:52,530 --> 00:17:55,790 But let me just notice that it's what I would 270 00:17:55,790 --> 00:17:59,150 call a local maximum. 271 00:17:59,150 --> 00:18:01,870 It's not the absolute top of the function because the 272 00:18:01,870 --> 00:18:05,330 function later on is climbing off to infinity. 273 00:18:05,330 --> 00:18:12,610 This would be way a maximum in its neighborhood, so a 274 00:18:12,610 --> 00:18:16,910 maximum, and it's only a local max. 275 00:18:16,910 --> 00:18:22,890 And what do I expect to see at a maximum at x equals 0? 276 00:18:22,890 --> 00:18:30,220 I expect to see the slope 0 at x equals 0, which it is. 277 00:18:30,220 --> 00:18:31,510 Check. 278 00:18:31,510 --> 00:18:37,420 And at a maximum, I need to know the second derivative. 279 00:18:37,420 --> 00:18:39,460 OK, here's my formula. 280 00:18:39,460 --> 00:18:46,720 At x equals 0, I see y double prime if x is 0 is minus 2. 281 00:18:46,720 --> 00:18:47,970 Good. 282 00:18:49,540 --> 00:18:55,990 Negative second derivative tells me I'm bending down, as 283 00:18:55,990 --> 00:19:01,850 the graph confirms, and the place where the slope is 0 is 284 00:19:01,850 --> 00:19:04,630 a maximum and not a minimum. 285 00:19:04,630 --> 00:19:06,990 What about the other one? 286 00:19:06,990 --> 00:19:08,460 What about at x equals 2/3? 287 00:19:12,320 --> 00:19:16,570 At that point, y double prime, looking at my formula here for 288 00:19:16,570 --> 00:19:20,730 y double prime, is what? 289 00:19:20,730 --> 00:19:24,980 6 times 2/3 is 4 minus 2 is plus 2. 290 00:19:24,980 --> 00:19:26,680 4 minus 2 is 2. 291 00:19:26,680 --> 00:19:28,590 So this will be-- 292 00:19:28,590 --> 00:19:33,120 this is positive, so I'm expecting a min. 293 00:19:33,120 --> 00:19:36,510 At x equals 2/3, I'm expecting a min. 294 00:19:36,510 --> 00:19:39,070 And, of course, it is. 295 00:19:39,070 --> 00:19:42,530 And again, it's only a local minimum. 296 00:19:42,530 --> 00:19:45,490 The derivative can only tell you what's happening very, 297 00:19:45,490 --> 00:19:48,070 very close to that point. 298 00:19:48,070 --> 00:19:51,640 The derivative doesn't know that over here the function is 299 00:19:51,640 --> 00:19:53,670 going further down. 300 00:19:53,670 --> 00:19:58,360 So this is a min, and again, a local min. 301 00:19:58,360 --> 00:20:02,370 OK, those are maximum and minimum 302 00:20:02,370 --> 00:20:04,490 when we know the function. 303 00:20:04,490 --> 00:20:08,255 Oh yeah, I better do the inflection point. 304 00:20:08,255 --> 00:20:11,140 Do you remember what the inflection point is? 305 00:20:11,140 --> 00:20:16,310 The inflection point is when the bending changes from-- 306 00:20:16,310 --> 00:20:18,965 up to here I see that bending down. 307 00:20:21,540 --> 00:20:24,860 From here, I see it bending up. 308 00:20:24,860 --> 00:20:30,260 So I will not be surprised if that's the point where the 309 00:20:30,260 --> 00:20:35,550 bending is changing, and 1/3 is the inflection point. 310 00:20:35,550 --> 00:20:37,860 And now how do we find an inflection point? 311 00:20:37,860 --> 00:20:39,910 How do we identify this point? 312 00:20:39,910 --> 00:20:43,320 Well, y double prime was negative. 313 00:20:43,320 --> 00:20:45,430 y double prime was positive. 314 00:20:45,430 --> 00:20:51,300 At that point, y double prime is 0. 315 00:20:51,300 --> 00:20:53,790 This is an inflection point. 316 00:20:53,790 --> 00:20:55,580 And it is. 317 00:20:55,580 --> 00:21:03,800 At x equals to 1/3, I do have 6 times 1/3. 318 00:21:03,800 --> 00:21:06,640 2 subtract 2, I have 0. 319 00:21:06,640 --> 00:21:10,870 So that is truly an inflection point. 320 00:21:10,870 --> 00:21:13,900 And now I know all the essential 321 00:21:13,900 --> 00:21:16,120 points about the curve. 322 00:21:16,120 --> 00:21:19,240 And these are the quantities-- 323 00:21:19,240 --> 00:21:20,460 oh! 324 00:21:20,460 --> 00:21:24,860 Say you're an economist. You're looking now at the 325 00:21:24,860 --> 00:21:30,070 statistics for the US economy or the world economy. 326 00:21:30,070 --> 00:21:34,300 OK, I suppose we're in a-- 327 00:21:34,300 --> 00:21:41,330 we had a local maximum there, a happy time a little while 328 00:21:41,330 --> 00:21:46,870 ago, but it went downhill, right? 329 00:21:46,870 --> 00:21:53,400 If y is, say, the gross product for the world or gross 330 00:21:53,400 --> 00:21:55,500 national product, it started down. 331 00:21:58,120 --> 00:22:03,110 The slope of that curve was negative. 332 00:22:03,110 --> 00:22:04,740 The bending was even negative. 333 00:22:04,740 --> 00:22:08,350 It was going down faster all the time. 334 00:22:08,350 --> 00:22:14,540 Now, at a certain moment, the economy kept going down, but 335 00:22:14,540 --> 00:22:16,630 you could see some sign of hope. 336 00:22:16,630 --> 00:22:18,320 And what was the sign of hope? 337 00:22:18,320 --> 00:22:23,290 It was the fact that it started bending up. 338 00:22:23,290 --> 00:22:28,470 And probably that's where we are as I'm making this video. 339 00:22:28,470 --> 00:22:33,250 I suspect we're still going down, but we're bending up. 340 00:22:33,250 --> 00:22:40,090 And at some point, hopefully tomorrow, we'll hit minimum 341 00:22:40,090 --> 00:22:42,330 and start really up. 342 00:22:42,330 --> 00:22:43,330 So I don't know. 343 00:22:43,330 --> 00:22:45,130 I would guess we're somewhere in there, and 344 00:22:45,130 --> 00:22:46,880 I don't know where. 345 00:22:46,880 --> 00:22:50,830 If I knew where, mathematics would be even more useful than 346 00:22:50,830 --> 00:22:53,620 it is, which would be hard to do. 347 00:22:53,620 --> 00:22:58,690 OK, so that's an example of how the second 348 00:22:58,690 --> 00:23:00,390 derivative comes in. 349 00:23:00,390 --> 00:23:09,600 Now, I started by giving this lecture the title Max and Min 350 00:23:09,600 --> 00:23:14,900 and saying those are the biggest applications of the 351 00:23:14,900 --> 00:23:16,070 derivative. 352 00:23:16,070 --> 00:23:19,880 Set the derivative to 0 and solve. 353 00:23:19,880 --> 00:23:22,390 Locate maximum points, minimum points. 354 00:23:22,390 --> 00:23:27,920 That's what calculus is most-- 355 00:23:27,920 --> 00:23:33,620 many of the word problems, most of the ones I see in use, 356 00:23:33,620 --> 00:23:37,790 involve derivative equals 0. 357 00:23:37,790 --> 00:23:41,895 OK, so let me take a particular example. 358 00:23:44,730 --> 00:23:49,660 So these were graphs, simple functions which I chose: sine 359 00:23:49,660 --> 00:23:52,920 x, x squared, x cubed minus x squared. 360 00:23:52,920 --> 00:23:58,520 Now let me tell you the problem because this is how 361 00:23:58,520 --> 00:24:00,210 math really comes. 362 00:24:00,210 --> 00:24:03,290 Let me tell you the problem, and let's create the function. 363 00:24:03,290 --> 00:24:05,830 OK, so much it's the problem I faced this 364 00:24:05,830 --> 00:24:09,080 morning and every morning. 365 00:24:09,080 --> 00:24:09,870 I live here. 366 00:24:09,870 --> 00:24:13,230 So OK, so here's home. 367 00:24:13,230 --> 00:24:20,500 And there is a-- the Mass Pike is the fast road to MIT. 368 00:24:20,500 --> 00:24:26,360 So let me put in the Mass Pike here, and let's say that's 369 00:24:26,360 --> 00:24:32,220 MIT, and I'm trying to get there as fast as possible. 370 00:24:32,220 --> 00:24:40,010 OK, so for part of the time, I'm going to have to drive on 371 00:24:40,010 --> 00:24:40,720 city streets. 372 00:24:40,720 --> 00:24:43,690 I do have to drive on city streets, and then I get to go 373 00:24:43,690 --> 00:24:49,440 on the Mass Pike, which is, let's say, twice as fast. The 374 00:24:49,440 --> 00:24:56,680 question is should I go directly over to the fast road 375 00:24:56,680 --> 00:24:59,960 and then take off? 376 00:24:59,960 --> 00:25:01,470 Let's take off on a good morning. 377 00:25:01,470 --> 00:25:04,540 The Mass Pike could be twice as slow, but let's assume 378 00:25:04,540 --> 00:25:07,070 twice as fast. Should I go straight over? 379 00:25:07,070 --> 00:25:09,440 Probably not. 380 00:25:09,440 --> 00:25:11,560 That's not the best way. 381 00:25:11,560 --> 00:25:17,050 I should probably pick up the Mass Pike on some road. 382 00:25:17,050 --> 00:25:23,010 I could go directly to MIT on the city streets at the slow 383 00:25:23,010 --> 00:25:29,150 rate, say 30 miles an hour or 30 kilometers an hour and 60, 384 00:25:29,150 --> 00:25:37,890 so speeds 30 and 60 as my speeds. 385 00:25:37,890 --> 00:25:44,550 OK, so now I should have put in some measure. 386 00:25:44,550 --> 00:25:50,350 Let's call that distance a, whatever it is. 387 00:25:50,350 --> 00:25:52,950 Maybe it's about three miles. 388 00:25:52,950 --> 00:25:54,420 And let me call-- 389 00:25:54,420 --> 00:25:56,170 so that's the direct distance. 390 00:25:56,170 --> 00:25:59,650 If I just went direct to the turnpike, I would go a 391 00:25:59,650 --> 00:26:02,380 distance a at 30 miles an hour, and then 392 00:26:02,380 --> 00:26:03,700 I would go a distance-- 393 00:26:03,700 --> 00:26:06,120 shall I call that b?-- 394 00:26:06,120 --> 00:26:07,660 at 60. 395 00:26:07,660 --> 00:26:10,290 So that's one possibility. 396 00:26:10,290 --> 00:26:13,110 But I think it's not the best. 397 00:26:13,110 --> 00:26:15,410 I think better to-- 398 00:26:15,410 --> 00:26:17,210 and you know better than me. 399 00:26:17,210 --> 00:26:19,380 I think I should probably angle over 400 00:26:19,380 --> 00:26:23,170 here and pick up this-- 401 00:26:23,170 --> 00:26:26,000 my question is where should I join the Mass Pike. 402 00:26:26,000 --> 00:26:28,270 And let's-- 403 00:26:28,270 --> 00:26:31,480 so we get a calculus problem, let's model it. 404 00:26:31,480 --> 00:26:35,730 Suppose that I can join it anywhere I like, not just at a 405 00:26:35,730 --> 00:26:39,930 couple of entrances. 406 00:26:39,930 --> 00:26:41,010 Anywhere. 407 00:26:41,010 --> 00:26:43,790 And the question is where? 408 00:26:43,790 --> 00:26:48,770 So calculus deals with the continuous choice of x. 409 00:26:48,770 --> 00:26:52,180 So that is the unknown. 410 00:26:52,180 --> 00:26:54,830 I could take that as the unknown x. 411 00:26:54,830 --> 00:26:57,510 That was a key step, of course, deciding what should 412 00:26:57,510 --> 00:26:58,510 be the unknown. 413 00:26:58,510 --> 00:27:02,630 I could also have taken this angle as an unknown, and that 414 00:27:02,630 --> 00:27:05,070 would be quite neat, too. 415 00:27:05,070 --> 00:27:07,530 But let me take that x. 416 00:27:07,530 --> 00:27:12,500 So this distance is then b minus x. 417 00:27:12,500 --> 00:27:16,590 So that's what I travel on the Mass Pike, 418 00:27:16,590 --> 00:27:20,340 so my time to minimize. 419 00:27:23,930 --> 00:27:26,480 I'm trying to minimize my time. 420 00:27:26,480 --> 00:27:35,240 OK, so on this Mass Pike when I travel at 60, I have 421 00:27:35,240 --> 00:27:40,770 distance divided by 60 is the time, right? 422 00:27:40,770 --> 00:27:42,860 Am I remembering correctly? 423 00:27:42,860 --> 00:27:44,310 Let's just remember. 424 00:27:44,310 --> 00:27:48,390 Distance is speed times time. 425 00:27:48,390 --> 00:27:50,370 That's the one we know. 426 00:27:50,370 --> 00:27:57,180 And then if I divide by the speed, the time is the 427 00:27:57,180 --> 00:28:00,230 distance divided by the speed, the distance divided by the 428 00:28:00,230 --> 00:28:02,470 speed on the pike. 429 00:28:02,470 --> 00:28:06,750 And now I have the distance on the city streets. 430 00:28:06,750 --> 00:28:13,930 OK, so that speed is going to be 30. 431 00:28:13,930 --> 00:28:17,700 So the time is going to be a bit longer for the distance, 432 00:28:17,700 --> 00:28:19,310 and what is that distance? 433 00:28:19,310 --> 00:28:21,540 OK, that was a. 434 00:28:21,540 --> 00:28:22,790 This was x. 435 00:28:25,330 --> 00:28:30,390 Pythagoras is the great leveler of mathematics. 436 00:28:33,580 --> 00:28:37,400 That's the distance on the city streets. 437 00:28:37,400 --> 00:28:43,430 And now what do I do? 438 00:28:43,430 --> 00:28:45,440 I've got an expression for the time. 439 00:28:45,440 --> 00:28:49,770 This is the quantity I'm trying to minimize. 440 00:28:49,770 --> 00:28:53,820 I minimize it by taking its derivative and set the 441 00:28:53,820 --> 00:28:55,670 derivative to 0. 442 00:28:55,670 --> 00:28:58,150 Take the derivative and set the derivative to 0. 443 00:28:58,150 --> 00:29:01,980 So now this is where I use the formulas of calculus. 444 00:29:01,980 --> 00:29:05,220 So the derivative, now I'm ready to write the derivative, 445 00:29:05,220 --> 00:29:06,730 and I'll set it to 0. 446 00:29:06,730 --> 00:29:10,140 So the derivative of that, b is a constant, so I have minus 447 00:29:10,140 --> 00:29:13,870 1/60; is that OK? 448 00:29:13,870 --> 00:29:17,220 Plus whatever the derivative of this is. 449 00:29:17,220 --> 00:29:20,390 Well, I have 1/30. 450 00:29:20,390 --> 00:29:22,080 I always take the constant first. 451 00:29:22,080 --> 00:29:24,980 Now I have to deal with that expression. 452 00:29:24,980 --> 00:29:28,720 That is some quantity square root. 453 00:29:28,720 --> 00:29:34,420 The square root is the 1/2 power, so I have 1/2 times 454 00:29:34,420 --> 00:29:37,600 this quantity to one lower power. 455 00:29:37,600 --> 00:29:39,790 That's the minus 1/2 power. 456 00:29:39,790 --> 00:29:44,770 That means that I still have a square root, but now it's a 457 00:29:44,770 --> 00:29:46,700 minus 1/2 power. 458 00:29:46,700 --> 00:29:48,560 It's down here. 459 00:29:48,560 --> 00:29:58,490 And then the chain rule says don't forget the derivative of 460 00:29:58,490 --> 00:30:00,180 what's inside, which is 2x. 461 00:30:03,430 --> 00:30:08,050 OK, depending on what order you've seen these videos and 462 00:30:08,050 --> 00:30:13,360 read text, you know the chain rule, or you see it now. 463 00:30:13,360 --> 00:30:17,940 It's a very, very valuable rule to find derivatives as 464 00:30:17,940 --> 00:30:19,910 the function gets complicated. 465 00:30:19,910 --> 00:30:23,460 And the thing to remember, there will be a proper 466 00:30:23,460 --> 00:30:24,840 discussion of the chain rule. 467 00:30:24,840 --> 00:30:26,120 It's so important. 468 00:30:26,120 --> 00:30:29,800 But you're seeing it here that the thing to remember is take 469 00:30:29,800 --> 00:30:33,780 also the derivative of what's inside the a squared plus x 470 00:30:33,780 --> 00:30:36,850 squared, and the derivative of the x squared is the 2x. 471 00:30:36,850 --> 00:30:39,120 OK, and that I have to set to 0. 472 00:30:39,120 --> 00:30:43,180 And, of course, I'm going to cancel the 2's, and 473 00:30:43,180 --> 00:30:44,250 I'll set it to 0. 474 00:30:44,250 --> 00:30:45,990 What does that mean "set to zero"? 475 00:30:45,990 --> 00:30:47,300 Here's something minus. 476 00:30:47,300 --> 00:30:48,920 Here's something plus. 477 00:30:48,920 --> 00:30:51,450 I guess what I really want is to make them equal. 478 00:30:55,120 --> 00:31:03,570 When the 1/60 equals this messier expression, at that 479 00:31:03,570 --> 00:31:07,850 point the minus term cancels the plus term. 480 00:31:07,850 --> 00:31:11,640 I get 0 for the derivative, so I'm looking for 481 00:31:11,640 --> 00:31:14,450 derivative equals 0. 482 00:31:14,450 --> 00:31:17,000 That's my equation now. 483 00:31:17,000 --> 00:31:19,250 OK, now I just have to solve it. 484 00:31:19,250 --> 00:31:22,260 All right, let's see. 485 00:31:22,260 --> 00:31:24,810 If I wanted to solve that, I would probably multiply 486 00:31:24,810 --> 00:31:27,820 through by 60. 487 00:31:27,820 --> 00:31:29,750 Can I do this? 488 00:31:29,750 --> 00:31:32,600 I'll multiply both sides by 60. 489 00:31:32,600 --> 00:31:35,670 That will cancel the 30 and leave an extra 2, so 490 00:31:35,670 --> 00:31:38,910 I'll have a 2x here. 491 00:31:38,910 --> 00:31:42,600 And let me multiply also by this miserable square root 492 00:31:42,600 --> 00:31:45,565 that's in the denominator to get it up there. 493 00:31:51,180 --> 00:31:54,160 I think that's what I've got. 494 00:31:54,160 --> 00:31:57,020 That's the same equation as this one, just simplified. 495 00:31:57,020 --> 00:31:59,010 Multiply through by 60. 496 00:31:59,010 --> 00:32:01,370 Multiply through by square root of a squared plus x 497 00:32:01,370 --> 00:32:04,200 squared, and it's looking good. 498 00:32:04,200 --> 00:32:07,070 All right, how am I going to solve that? 499 00:32:07,070 --> 00:32:11,240 Well, the only mess up is the square root. 500 00:32:11,240 --> 00:32:14,480 Get rid of that by squaring both sides. 501 00:32:14,480 --> 00:32:18,690 So now I square both sides, and I get a squared plus x 502 00:32:18,690 --> 00:32:23,440 squared, and the square of 2x is 4x squared. 503 00:32:23,440 --> 00:32:28,855 All right, now I have an equation that's way better. 504 00:32:32,550 --> 00:32:36,000 In fact, even better if I subtract x 505 00:32:36,000 --> 00:32:37,080 squared from both sides. 506 00:32:37,080 --> 00:32:39,850 My equation is telling me that a squared 507 00:32:39,850 --> 00:32:42,710 should be 3x squared. 508 00:32:42,710 --> 00:32:46,060 In other words, this good x is-- 509 00:32:46,060 --> 00:32:51,540 now I'm ready to take the square root and find x itself. 510 00:32:51,540 --> 00:32:53,310 So put the 3 here. 511 00:32:53,310 --> 00:32:54,830 Take the square root. 512 00:32:54,830 --> 00:32:58,735 I'm getting a over the square root of 3. 513 00:33:02,470 --> 00:33:08,920 So there is a word problem, a minimum problem, where we had 514 00:33:08,920 --> 00:33:14,170 to create the function to minimize, which was the time, 515 00:33:14,170 --> 00:33:18,810 trying to get to work as quickly as possible. 516 00:33:18,810 --> 00:33:24,190 After naming the key quantity x, then taking the derivative, 517 00:33:24,190 --> 00:33:29,150 then simplifying, that's where the little work of calculus 518 00:33:29,150 --> 00:33:32,660 comes in, in the end getting something nice, solving it, 519 00:33:32,660 --> 00:33:34,830 and getting the answer a over square root of 3. 520 00:33:34,830 --> 00:33:40,120 So we now know what to do driving in if there's an 521 00:33:40,120 --> 00:33:42,000 entrance where we want to get it. 522 00:33:42,000 --> 00:33:47,100 And actually, it is a beautiful answer. 523 00:33:47,100 --> 00:33:50,390 If this is a over the square root of 3, this will turn out 524 00:33:50,390 --> 00:33:55,980 to be 30 degrees, pi over 6-- 525 00:33:55,980 --> 00:33:57,640 I think. 526 00:33:57,640 --> 00:34:00,140 Yeah, I think that's right. 527 00:34:00,140 --> 00:34:03,260 So that's the conclusion from calculus. 528 00:34:03,260 --> 00:34:04,990 Drive at a 30-degree angle. 529 00:34:04,990 --> 00:34:07,290 Hope that there's a road going that way-- 530 00:34:07,290 --> 00:34:09,000 sorry about that point-- 531 00:34:09,000 --> 00:34:12,159 and join the turnpike. 532 00:34:12,159 --> 00:34:15,830 And probably the reason for that nice 533 00:34:15,830 --> 00:34:19,239 answer, 30 degrees, came-- 534 00:34:19,239 --> 00:34:23,860 I can't help but imagine that because I chose 30 and 60 535 00:34:23,860 --> 00:34:30,060 here, a ratio of 1:2, and then somehow the fact that the sine 536 00:34:30,060 --> 00:34:34,110 of 30 degrees is 1/2, those two facts 537 00:34:34,110 --> 00:34:35,560 have got to be connected. 538 00:34:35,560 --> 00:34:39,860 So I change these 30 and 60 numbers, I'll change my 539 00:34:39,860 --> 00:34:43,260 answer, but basically, the picture won't change much. 540 00:34:43,260 --> 00:34:51,949 And there's another little point to make to really 541 00:34:51,949 --> 00:34:54,960 complete this problem. 542 00:34:54,960 --> 00:34:58,830 It could have happened that the distance on the turnpike 543 00:34:58,830 --> 00:35:04,140 was very small and that this was a dumb move. 544 00:35:04,140 --> 00:35:06,550 That 30-degree angle could be overshooting 545 00:35:06,550 --> 00:35:11,070 MIT if MIT was there. 546 00:35:11,070 --> 00:35:16,310 So that's a case in which the minimum time didn't happen 547 00:35:16,310 --> 00:35:21,210 where the derivative bottomed out. 548 00:35:21,210 --> 00:35:27,701 If MIT was here, the good idea would be go straight for it. 549 00:35:31,580 --> 00:35:36,180 Yeah, the extra part on the turn-- you wouldn't drive on 550 00:35:36,180 --> 00:35:38,090 the turnpike at all. 551 00:35:38,090 --> 00:35:41,990 And that's a signal that somehow in the graph, which I 552 00:35:41,990 --> 00:35:48,910 didn't graph this function, but if I did, then this stuff 553 00:35:48,910 --> 00:35:52,580 would be locating the minimum of the graph. 554 00:35:52,580 --> 00:35:57,000 But this extra example where you go straight for MIT would 555 00:35:57,000 --> 00:36:00,110 be a case in which the minimum is at the end. 556 00:36:00,110 --> 00:36:02,650 And, of course, that could happen. 557 00:36:02,650 --> 00:36:06,520 You could have a graph that just goes down, and then it 558 00:36:06,520 --> 00:36:09,380 ends, so the minimum is there. 559 00:36:09,380 --> 00:36:11,660 Even though the graph looks like it's still going down, 560 00:36:11,660 --> 00:36:13,290 the graph ended. 561 00:36:13,290 --> 00:36:14,300 What can you do? 562 00:36:14,300 --> 00:36:16,500 That's the best point there is. 563 00:36:16,500 --> 00:36:21,840 OK, so that is a-- 564 00:36:21,840 --> 00:36:26,910 can I recap this lecture coming first over here? 565 00:36:26,910 --> 00:36:30,870 So the lecture is about maximum and minimum, and we 566 00:36:30,870 --> 00:36:35,310 learned which it is by the second derivative. 567 00:36:35,310 --> 00:36:37,490 So then we had examples. 568 00:36:37,490 --> 00:36:40,980 There was an example of a minimum when the second 569 00:36:40,980 --> 00:36:43,030 derivative was positive. 570 00:36:43,030 --> 00:36:47,110 Here was an example of a local maximum when the second 571 00:36:47,110 --> 00:36:50,230 derivative was negative. 572 00:36:50,230 --> 00:36:55,240 Here with the sine and cosine, those are nice examples. 573 00:36:55,240 --> 00:36:58,810 And it takes some patience to go through them. 574 00:36:58,810 --> 00:37:03,750 I suggest you take another simple function, like start 575 00:37:03,750 --> 00:37:06,100 with cosine x. 576 00:37:06,100 --> 00:37:07,290 Find its maximum. 577 00:37:07,290 --> 00:37:08,330 Find its minimum. 578 00:37:08,330 --> 00:37:11,700 Find its inflection points so the inflection points are 579 00:37:11,700 --> 00:37:17,750 where the bending is 0 because it's changing from bending one 580 00:37:17,750 --> 00:37:19,380 way to bending the other way. 581 00:37:19,380 --> 00:37:23,080 We didn't need an inflection test-- 582 00:37:23,080 --> 00:37:27,070 so actually, I didn't complete the lecture, because I didn't 583 00:37:27,070 --> 00:37:30,140 compute the second derivative and show that 584 00:37:30,140 --> 00:37:33,190 this was truly a minimum. 585 00:37:33,190 --> 00:37:34,270 I could have done that. 586 00:37:34,270 --> 00:37:37,040 I would have had to take the derivative of this, which 587 00:37:37,040 --> 00:37:41,900 would be one level messier, and look at its sign. 588 00:37:41,900 --> 00:37:43,610 I wouldn't have to set it to 0. 589 00:37:43,610 --> 00:37:47,310 I would be looking at the sign of the second derivative. 590 00:37:47,310 --> 00:37:52,910 And in this problem, it would be safely come out positive 591 00:37:52,910 --> 00:37:57,210 sign, meaning bending upwards, meaning that this point I've 592 00:37:57,210 --> 00:38:02,540 identified by all these steps was truly the minimum time, 593 00:38:02,540 --> 00:38:04,230 not a maximum. 594 00:38:04,230 --> 00:38:10,120 OK, that's a big part of important calculus 595 00:38:10,120 --> 00:38:11,670 applications. 596 00:38:11,670 --> 00:38:12,870 Thanks. 597 00:38:12,870 --> 00:38:14,650 NARRATOR: This has been a production of MIT 598 00:38:14,650 --> 00:38:17,040 OpenCourseWare and Gilbert Strang. 599 00:38:17,040 --> 00:38:19,310 Funding for this video was provided by the Lord 600 00:38:19,310 --> 00:38:20,530 Foundation. 601 00:38:20,530 --> 00:38:23,660 To help OCW continue to provide free and open access 602 00:38:23,660 --> 00:38:26,740 to MIT courses, please make a donation at 603 00:38:26,740 --> 00:38:28,300 ocw.mit.edu/donate.