1 00:00:25,690 --> 00:00:27,660 PROFESSOR: OK, hi. 2 00:00:27,660 --> 00:00:33,780 This is the second in my videos about the main ideas, 3 00:00:33,780 --> 00:00:36,090 the big picture of calculus. 4 00:00:36,090 --> 00:00:40,180 And this is an important one, because I want to introduce 5 00:00:40,180 --> 00:00:44,480 and compute some derivatives. 6 00:00:44,480 --> 00:00:50,500 And you'll remember the overall situation is we have 7 00:00:50,500 --> 00:00:55,270 pairs of functions, distance and speed, function 1 and 8 00:00:55,270 --> 00:01:00,570 function 2, height of a graph, slope of the graph, height of 9 00:01:00,570 --> 00:01:03,000 a mountain, slope of a mountain. 10 00:01:03,000 --> 00:01:06,650 And it's the connection between those two that 11 00:01:06,650 --> 00:01:08,220 calculus is about. 12 00:01:08,220 --> 00:01:16,770 And so our problem today is, you could imagine we have an 13 00:01:16,770 --> 00:01:19,880 airplane climbing. 14 00:01:19,880 --> 00:01:27,160 Its height is y as it covers a distance x. 15 00:01:27,160 --> 00:01:30,640 And its flight recorder will-- 16 00:01:30,640 --> 00:01:33,000 Well, probably it has two flight recorders. 17 00:01:33,000 --> 00:01:34,890 Let's suppose it has. 18 00:01:34,890 --> 00:01:40,120 Or your car has two recorders. 19 00:01:40,120 --> 00:01:44,330 One records the distance, the height, the total amount 20 00:01:44,330 --> 00:01:48,430 achieved up to that moment, up to that time, t, 21 00:01:48,430 --> 00:01:50,990 or that point, x. 22 00:01:50,990 --> 00:01:56,400 The second recorder would tell you at every instant 23 00:01:56,400 --> 00:01:57,640 what the speed is. 24 00:01:57,640 --> 00:02:00,050 So it would tell you the speed at all times. 25 00:02:00,050 --> 00:02:01,040 Do you see the difference? 26 00:02:01,040 --> 00:02:05,732 The speed is like what's happening at an instant. 27 00:02:05,732 --> 00:02:11,400 The distance or the height, y, is the total accumulation of 28 00:02:11,400 --> 00:02:14,540 how far you've gone, how high you've gone. 29 00:02:14,540 --> 00:02:17,750 And now I'm going to suppose that this speed, this second 30 00:02:17,750 --> 00:02:22,930 function, the recorder is lost. But the information is 31 00:02:22,930 --> 00:02:24,480 there, and how to recover it. 32 00:02:24,480 --> 00:02:25,970 So that's the question. 33 00:02:25,970 --> 00:02:29,130 How, if I have a total record, say of height-- 34 00:02:29,130 --> 00:02:32,940 I'll say mostly with y of x. 35 00:02:32,940 --> 00:02:36,350 I write these two so that you realize that letters are not 36 00:02:36,350 --> 00:02:37,660 what calculus is about. 37 00:02:37,660 --> 00:02:39,020 It's ideas. 38 00:02:39,020 --> 00:02:42,150 And here is a central idea. 39 00:02:42,150 --> 00:02:44,660 if I know the height-- 40 00:02:44,660 --> 00:02:47,780 as I go along, I know the height, it could go down-- 41 00:02:50,780 --> 00:02:55,390 how can I recover from that height what the slope is at 42 00:02:55,390 --> 00:02:57,970 each point? 43 00:02:57,970 --> 00:03:04,330 So here's something rather important, that's the notation 44 00:03:04,330 --> 00:03:08,740 that Leibniz created, and it was a good, good idea for the 45 00:03:08,740 --> 00:03:09,330 derivative. 46 00:03:09,330 --> 00:03:11,790 And you'll see where it comes from. 47 00:03:11,790 --> 00:03:19,190 But somehow I'm dividing distance up by distance 48 00:03:19,190 --> 00:03:26,760 across, and that ratio of up to across is a slope. 49 00:03:26,760 --> 00:03:32,770 So let me develop what we're doing. 50 00:03:32,770 --> 00:03:41,930 So the one thing we can do and now will do is, for the great 51 00:03:41,930 --> 00:03:46,250 functions of calculus, a few very special, 52 00:03:46,250 --> 00:03:48,060 very important functions. 53 00:03:48,060 --> 00:03:53,360 We will actually figure out what the slope is. 54 00:03:53,360 --> 00:03:57,260 These are given by formulas, and I'll find a formula for 55 00:03:57,260 --> 00:04:01,120 the slope, dy/dx equals. 56 00:04:01,120 --> 00:04:04,750 And I won't write it in yet. 57 00:04:04,750 --> 00:04:06,290 Let me keep a little suspense. 58 00:04:06,290 --> 00:04:11,640 But this short list of the great functions is 59 00:04:11,640 --> 00:04:14,800 tremendously valuable. 60 00:04:14,800 --> 00:04:17,620 The process that we go through takes a little time, but once 61 00:04:17,620 --> 00:04:20,240 we do it it's done. 62 00:04:20,240 --> 00:04:23,640 Once we write in the answer here, we know it. 63 00:04:23,640 --> 00:04:31,420 And the point is that other functions of science, of 64 00:04:31,420 --> 00:04:37,380 engineering, of economics, of life, come from these 65 00:04:37,380 --> 00:04:42,410 functions by multiplying-- 66 00:04:42,410 --> 00:04:44,750 I could multiply that times that-- 67 00:04:44,750 --> 00:04:47,330 and then I need a product rule, the rule for the 68 00:04:47,330 --> 00:04:49,760 derivative, the slope of a product. 69 00:04:49,760 --> 00:04:51,490 I could divide that by that. 70 00:04:54,070 --> 00:04:56,090 So I need a quotient rule. 71 00:04:56,090 --> 00:04:58,050 I could do a chain. 72 00:04:58,050 --> 00:05:02,490 And you'll see that's maybe the best and most valuable. 73 00:05:02,490 --> 00:05:07,360 e to the sine x, so I'm putting e to the x together 74 00:05:07,360 --> 00:05:13,700 with sine x in a chain of functions, e to the sine of x. 75 00:05:13,700 --> 00:05:15,230 Then we need a chain rule. 76 00:05:15,230 --> 00:05:17,620 That's all coming. 77 00:05:17,620 --> 00:05:18,460 Let me start-- 78 00:05:18,460 --> 00:05:23,680 Well, let me even start by giving away the main facts for 79 00:05:23,680 --> 00:05:26,930 these three examples, because they're three you want to 80 00:05:26,930 --> 00:05:31,350 remember, and might as well start now. 81 00:05:31,350 --> 00:05:35,120 The x to the nth, so that's something if n is positive. 82 00:05:35,120 --> 00:05:36,680 x to the nth climbs up. 83 00:05:36,680 --> 00:05:42,920 Let me draw a graph here of y equals x squared, because 84 00:05:42,920 --> 00:05:47,940 that's one we'll work out in detail, y equals x squared. 85 00:05:47,940 --> 00:05:50,370 So this direction is x. 86 00:05:50,370 --> 00:05:52,520 This direction is y. 87 00:05:52,520 --> 00:05:54,610 And I want to know the slope. 88 00:05:54,610 --> 00:05:58,680 And the point is that that slope is changing as I go up. 89 00:05:58,680 --> 00:06:01,090 So the slope depends on x. 90 00:06:01,090 --> 00:06:03,220 The slope will be different here. 91 00:06:03,220 --> 00:06:06,330 So it's getting steeper and steeper. 92 00:06:06,330 --> 00:06:08,400 I'll figure out that slope. 93 00:06:08,400 --> 00:06:10,520 For this example, x squared went in 94 00:06:10,520 --> 00:06:12,690 as 2, and it's climbing. 95 00:06:12,690 --> 00:06:20,520 If n was minus 2, just because that's also on our list, n 96 00:06:20,520 --> 00:06:26,120 could be negative, the function would be dropping. 97 00:06:26,120 --> 00:06:31,400 You remember x to the minus 2, that negative exponent means 98 00:06:31,400 --> 00:06:34,550 divide by x squared. 99 00:06:34,550 --> 00:06:37,630 x to the minus 2 is 1 divided by x squared, 100 00:06:37,630 --> 00:06:39,730 and it'll be dropping. 101 00:06:39,730 --> 00:06:43,030 So n could be positive or negative here. 102 00:06:43,030 --> 00:06:45,220 So I tell you the derivative. 103 00:06:45,220 --> 00:06:49,830 The derivative is easy to remember, the set number n. 104 00:06:49,830 --> 00:06:53,020 It's another power of x, and that power is 105 00:06:53,020 --> 00:06:55,610 one less, one down. 106 00:06:55,610 --> 00:06:58,030 You lose one power. 107 00:06:58,030 --> 00:07:01,470 I'm going to go through the steps here for n is equal to 108 00:07:01,470 --> 00:07:07,850 2, so I hope to get the answer 2 times x to the 2 minus 1 109 00:07:07,850 --> 00:07:11,280 will just be 1, 2x. 110 00:07:11,280 --> 00:07:13,860 But what does the slope mean? 111 00:07:13,860 --> 00:07:17,450 That's what this lecture is really telling you. 112 00:07:17,450 --> 00:07:22,830 I'll tell you the answer for if it's sine x going in, 113 00:07:22,830 --> 00:07:29,990 beautifully, the derivative of sine x is cos x, the cosine. 114 00:07:29,990 --> 00:07:33,260 The derivative of the sine curve is the cosine curve. 115 00:07:33,260 --> 00:07:37,860 You couldn't hope for more than that. 116 00:07:37,860 --> 00:07:42,470 And then we'll also, at the same time, find the derivative 117 00:07:42,470 --> 00:07:45,940 of the cosine curve, which is minus the sine curve. 118 00:07:45,940 --> 00:07:49,380 It turns out a minus sine comes in because the cosine 119 00:07:49,380 --> 00:07:52,280 curve drops at the start. 120 00:07:52,280 --> 00:07:55,640 And would you like to know this one? 121 00:07:55,640 --> 00:08:01,480 e to the x, which I will introduce in a lecture coming 122 00:08:01,480 --> 00:08:05,220 very soon, because it's the function of calculus. 123 00:08:05,220 --> 00:08:09,230 And the reason it's so terrific is, the connection 124 00:08:09,230 --> 00:08:15,220 between the function, whatever it is, whatever this number e 125 00:08:15,220 --> 00:08:19,900 is to whatever the xth power means, the slope is the same 126 00:08:19,900 --> 00:08:22,200 as the function, e to the x. 127 00:08:25,810 --> 00:08:28,750 That's amazing. 128 00:08:28,750 --> 00:08:31,300 As the function changes, the slope changes 129 00:08:31,300 --> 00:08:32,550 and they stay equal. 130 00:08:35,210 --> 00:08:39,059 Really, my help is just to say, if you know those three, 131 00:08:39,059 --> 00:08:41,950 you're really off to a good start, plus the 132 00:08:41,950 --> 00:08:43,500 rules that you need. 133 00:08:43,500 --> 00:08:47,000 All right, now I'll tackle this particular one and say, 134 00:08:47,000 --> 00:08:50,690 what does slope mean? 135 00:08:50,690 --> 00:08:56,500 So I'm given the recorder that I have. This 136 00:08:56,500 --> 00:08:57,850 is function 1 here. 137 00:08:57,850 --> 00:09:01,910 This is function 1, the one I know. 138 00:09:01,910 --> 00:09:05,700 And I know it at every point. 139 00:09:05,700 --> 00:09:09,710 If I only had the trip meter after an hour or two hours or 140 00:09:09,710 --> 00:09:15,810 three hours, well, calculus can't do the impossible. 141 00:09:15,810 --> 00:09:20,440 It can't know, if I only knew the distance reached after 142 00:09:20,440 --> 00:09:26,560 each hour, I couldn't tell what it did over that hour, 143 00:09:26,560 --> 00:09:29,500 how often you had to break, how often you accelerated. 144 00:09:29,500 --> 00:09:33,360 I could only find an average speed over that hour. 145 00:09:33,360 --> 00:09:34,820 That would be easy. 146 00:09:34,820 --> 00:09:38,890 So averages don't need calculus. 147 00:09:38,890 --> 00:09:43,140 It's instant stuff, what happens at a moment, what is 148 00:09:43,140 --> 00:09:48,630 that speedometer reading at the moment x, say, x equal 1. 149 00:09:48,630 --> 00:09:50,280 What is the slope? 150 00:09:50,280 --> 00:09:58,600 Yeah, let me put in x equals 1 on this graph and x equals 2. 151 00:09:58,600 --> 00:10:02,720 And now x squared is going to be at height 1. 152 00:10:02,720 --> 00:10:08,320 If x is 1, then x squared is also 1. 153 00:10:08,320 --> 00:10:13,160 If x is 2, x squared will be 4. 154 00:10:13,160 --> 00:10:14,150 What's the average? 155 00:10:14,150 --> 00:10:15,970 Let me just come back one second to 156 00:10:15,970 --> 00:10:17,810 that average business. 157 00:10:17,810 --> 00:10:24,410 The average slope there would be, in a distance across of 1, 158 00:10:24,410 --> 00:10:27,840 I went up by how much? 159 00:10:27,840 --> 00:10:29,590 3. 160 00:10:29,590 --> 00:10:31,680 I went up from 1 to 4. 161 00:10:31,680 --> 00:10:33,650 I have to do a subtraction. 162 00:10:33,650 --> 00:10:39,480 Differences, derivatives, that's the connection here. 163 00:10:39,480 --> 00:10:41,510 So it's 4 minus 1. 164 00:10:41,510 --> 00:10:42,830 That is 3. 165 00:10:42,830 --> 00:10:50,125 So I would say the average is 3/1. 166 00:10:54,120 --> 00:10:56,620 But that's not calculus. 167 00:10:56,620 --> 00:11:00,160 Calculus is looking at the instant thing. 168 00:11:00,160 --> 00:11:03,633 And let me begin at this instant, at that point. 169 00:11:07,090 --> 00:11:11,750 What does the slope look like to you at that point? 170 00:11:11,750 --> 00:11:14,840 At at x equals 0, here's x equals 0, and 171 00:11:14,840 --> 00:11:16,120 here's y equals 0. 172 00:11:16,120 --> 00:11:19,380 We're very much 0. 173 00:11:19,380 --> 00:11:26,600 You see it's climbing, but at that moment, it's like it just 174 00:11:26,600 --> 00:11:30,300 started from a red light, whatever. 175 00:11:30,300 --> 00:11:32,940 The speed is 0 at that point. 176 00:11:32,940 --> 00:11:37,130 And I want to say the slope is 0. 177 00:11:37,130 --> 00:11:38,860 That's flat right there. 178 00:11:38,860 --> 00:11:39,800 That's flat. 179 00:11:39,800 --> 00:11:44,350 If I continued the curve, if I continued the x squared curve 180 00:11:44,350 --> 00:11:47,030 for x negative, it would be the same as for x positive. 181 00:11:47,030 --> 00:11:49,620 Well, it doesn't look very the same. 182 00:11:49,620 --> 00:11:51,680 Let me improve that. 183 00:11:51,680 --> 00:11:55,440 It would start up the same way and be completely symmetric. 184 00:11:55,440 --> 00:12:00,650 Everybody sees that, at that 0 position, the 185 00:12:00,650 --> 00:12:04,050 curve has hit bottom. 186 00:12:04,050 --> 00:12:06,710 Actually, this will be a major, major 187 00:12:06,710 --> 00:12:08,420 application of calculus. 188 00:12:08,420 --> 00:12:12,960 You identify the bottom of a curve by the fact that the 189 00:12:12,960 --> 00:12:14,600 slope is 0. 190 00:12:14,600 --> 00:12:15,540 It's not going up. 191 00:12:15,540 --> 00:12:17,470 It's not going down. 192 00:12:17,470 --> 00:12:19,250 It's 0 at that point. 193 00:12:19,250 --> 00:12:23,830 But now, what do I mean by slope at a point? 194 00:12:23,830 --> 00:12:27,000 Here comes the new idea. 195 00:12:27,000 --> 00:12:30,570 If I go way over to 1, that's too much. 196 00:12:30,570 --> 00:12:32,540 I just want to stay near this point. 197 00:12:32,540 --> 00:12:36,140 I'll go over a little bit, and I call that 198 00:12:36,140 --> 00:12:39,030 little bit delta x. 199 00:12:39,030 --> 00:12:44,300 So that letter, delta, signals to our 200 00:12:44,300 --> 00:12:48,660 minds small, some small. 201 00:12:48,660 --> 00:12:52,230 And actually, probably smaller than I drew it there. 202 00:12:52,230 --> 00:12:54,620 And then, so what's the average? 203 00:12:54,620 --> 00:12:56,780 I'd like to just find the average 204 00:12:56,780 --> 00:13:00,310 speed, or average slope. 205 00:13:00,310 --> 00:13:06,040 If I go over by delta x, and then how high do I go up? 206 00:13:06,040 --> 00:13:08,240 Well, the curve is y equals x squared. 207 00:13:08,240 --> 00:13:10,420 So how high is this? 208 00:13:10,420 --> 00:13:20,120 So the average is up first, divided by across. 209 00:13:20,120 --> 00:13:22,750 Across is our usual delta x. 210 00:13:22,750 --> 00:13:25,830 How far did it go up? 211 00:13:25,830 --> 00:13:30,530 Well, if our curve is x squared and I'm at the point, 212 00:13:30,530 --> 00:13:32,830 delta x, then it's delta x squared. 213 00:13:35,360 --> 00:13:42,011 That's the average over the first piece, over short, over 214 00:13:42,011 --> 00:13:46,760 the first piece of the curve. 215 00:13:46,760 --> 00:13:49,342 Out is-- from here out to delta x. 216 00:13:49,342 --> 00:13:50,592 OK. 217 00:13:52,160 --> 00:13:57,920 Now, again, that's still only an average, because delta x 218 00:13:57,920 --> 00:13:59,480 might have been short. 219 00:13:59,480 --> 00:14:01,370 I want to push it to 0. 220 00:14:01,370 --> 00:14:06,680 That's where calculus comes in, taking the limit of 221 00:14:06,680 --> 00:14:09,930 shorter and shorter and shorter pieces in order to 222 00:14:09,930 --> 00:14:17,120 zoom in on that instant, that moment, that spot where we're 223 00:14:17,120 --> 00:14:19,630 looking at the slope, and where we're expecting the 224 00:14:19,630 --> 00:14:22,500 answer is 0, in this case. 225 00:14:22,500 --> 00:14:26,270 And you see that the average, it happens to 226 00:14:26,270 --> 00:14:28,300 be especially simple. 227 00:14:28,300 --> 00:14:31,180 Delta x squared over delta x is just delta x. 228 00:14:31,180 --> 00:14:38,710 So the average slope is extremely small. 229 00:14:38,710 --> 00:14:43,140 And I'll just complete that thought. 230 00:14:43,140 --> 00:14:55,040 So the instant slope-- instant slope at 0, at x equals 0, I 231 00:14:55,040 --> 00:14:56,860 let this delta x get smaller and smaller. 232 00:14:56,860 --> 00:15:01,440 I get the answer is 0, which is just what I expected it. 233 00:15:01,440 --> 00:15:04,520 And you could say, well, not too exciting. 234 00:15:04,520 --> 00:15:07,110 But it was an easy one to do. 235 00:15:07,110 --> 00:15:10,400 It was the first time that we actually went through the 236 00:15:10,400 --> 00:15:12,260 steps of computing. 237 00:15:12,260 --> 00:15:15,180 This is a, like, a delta y. 238 00:15:15,180 --> 00:15:18,440 This is the delta x. 239 00:15:18,440 --> 00:15:23,140 Instead of 3/1, starting here I had delta x 240 00:15:23,140 --> 00:15:24,290 squared over delta x. 241 00:15:24,290 --> 00:15:25,520 That was easy to see. 242 00:15:25,520 --> 00:15:26,730 It was delta x. 243 00:15:26,730 --> 00:15:31,180 And if I move in closer, that average slope is smaller and 244 00:15:31,180 --> 00:15:35,670 smaller, and the slope at that instant is 0. 245 00:15:35,670 --> 00:15:37,970 No problem. 246 00:15:37,970 --> 00:15:43,340 The travel, the climbing began from rest, but 247 00:15:43,340 --> 00:15:45,220 it picked up speed. 248 00:15:45,220 --> 00:15:48,300 The slope here is certainly not 0. 249 00:15:48,300 --> 00:15:50,200 We'll find that slope. 250 00:15:50,200 --> 00:15:52,740 We need now to find the slope at every point. 251 00:15:52,740 --> 00:15:55,210 OK. 252 00:15:55,210 --> 00:15:57,810 That's a good start. 253 00:15:57,810 --> 00:16:04,170 Now I'm ready to find the slope at any point. 254 00:16:04,170 --> 00:16:09,890 Instead of just x equals 0, which we've now done, I better 255 00:16:09,890 --> 00:16:18,920 draw a new graph of the same picture, climbing up. 256 00:16:18,920 --> 00:16:23,900 Now I'm putting in a little climb at some point x here. 257 00:16:23,900 --> 00:16:26,085 I'm up at a height, x squared. 258 00:16:30,590 --> 00:16:33,310 I'm at that point on the climb. 259 00:16:33,310 --> 00:16:37,410 I'd like to know the slope there, at that point. 260 00:16:37,410 --> 00:16:40,110 How am I going to do it? 261 00:16:40,110 --> 00:16:45,880 I will follow this as the central dogma of calculus, of 262 00:16:45,880 --> 00:16:50,810 differential calculus, function 1 to function 2. 263 00:16:50,810 --> 00:16:56,990 Take a little delta x, go as far as x plus delta x. 264 00:16:56,990 --> 00:16:59,860 That will take you to a higher point on the curve. 265 00:16:59,860 --> 00:17:05,230 That's now the point x plus delta x squared, because our 266 00:17:05,230 --> 00:17:09,730 curve is still y equals x squared in this 267 00:17:09,730 --> 00:17:11,919 nice, simple parabola. 268 00:17:11,919 --> 00:17:13,810 OK. 269 00:17:13,810 --> 00:17:19,829 So now I you look at distance across and distance up. 270 00:17:19,829 --> 00:17:25,210 So delta y is the change up. 271 00:17:25,210 --> 00:17:30,360 Delta x is the across. 272 00:17:30,360 --> 00:17:35,500 And I have to put what is delta y. 273 00:17:35,500 --> 00:17:38,310 I have to write in, what is delta y? 274 00:17:38,310 --> 00:17:40,280 It's this distance up. 275 00:17:40,280 --> 00:17:44,700 It's x plus delta x squared. 276 00:17:44,700 --> 00:17:49,120 That's this height minus this height. 277 00:17:49,120 --> 00:17:51,610 I'm not counting this bit, of course. 278 00:17:51,610 --> 00:17:53,190 It's that that I want. 279 00:17:53,190 --> 00:17:58,040 That's the delta y, is this piece. 280 00:17:58,040 --> 00:18:03,600 So it's up to this, subtract x squared. 281 00:18:03,600 --> 00:18:05,780 That's delta y. 282 00:18:05,780 --> 00:18:07,470 That's important. 283 00:18:07,470 --> 00:18:11,520 Now I divide by delta x. 284 00:18:11,520 --> 00:18:13,820 This is all algebra now. 285 00:18:13,820 --> 00:18:18,320 Calculus is going to come in a moment, but not yet. 286 00:18:18,320 --> 00:18:20,160 For algebra, what do I do? 287 00:18:20,160 --> 00:18:21,870 I multiply this out. 288 00:18:21,870 --> 00:18:23,520 I see that thing squared. 289 00:18:23,520 --> 00:18:26,160 I remember that x is squared. 290 00:18:26,160 --> 00:18:29,190 And then I have this times this twice. 291 00:18:29,190 --> 00:18:34,133 2x delta x's, and then I have delta x squared. 292 00:18:36,750 --> 00:18:38,820 And then I'm subtracting x squared. 293 00:18:38,820 --> 00:18:47,080 So that's delta y written out in full glory. 294 00:18:47,080 --> 00:18:53,080 I wrote it out because now I can simplify by 295 00:18:53,080 --> 00:18:55,400 canceling the x squared. 296 00:18:55,400 --> 00:18:58,200 I'm not surprised. 297 00:18:58,200 --> 00:19:01,840 Now, in this case, I can actually do the division. 298 00:19:01,840 --> 00:19:04,420 Delta x is just there. 299 00:19:04,420 --> 00:19:06,520 So it leaves me with a 2x. 300 00:19:06,520 --> 00:19:08,810 Delta x over delta x is 1. 301 00:19:08,810 --> 00:19:11,810 And then here's a delta x squared over a delta x, so 302 00:19:11,810 --> 00:19:13,940 that leaves me with one delta x. 303 00:19:17,930 --> 00:19:21,400 As you get the hang of calculus, you see the 304 00:19:21,400 --> 00:19:27,590 important things is like this first order, delta x to the 305 00:19:27,590 --> 00:19:28,680 first power. 306 00:19:28,680 --> 00:19:34,370 Delta x squared, that, when divided by delta x, gives us 307 00:19:34,370 --> 00:19:38,130 this, which is going to disappear. 308 00:19:38,130 --> 00:19:39,490 That's the point. 309 00:19:39,490 --> 00:19:48,940 This was the average over a short but still not instant 310 00:19:48,940 --> 00:19:52,080 range, distance. 311 00:19:52,080 --> 00:19:54,870 Now, what happens? 312 00:19:54,870 --> 00:19:56,130 Now dy/dx. 313 00:20:00,180 --> 00:20:09,470 So if this is short, short over short, this is darn short 314 00:20:09,470 --> 00:20:11,160 over darn short. 315 00:20:11,160 --> 00:20:18,610 That d is, well, it's too short to see. 316 00:20:18,610 --> 00:20:23,880 So I don't actually now try to separate a distance dy. 317 00:20:23,880 --> 00:20:31,350 This isn't a true division, because it's effectively 0/0. 318 00:20:31,350 --> 00:20:35,360 And you might say, well, 0/0, what's the meaning? 319 00:20:35,360 --> 00:20:40,620 Well, the meaning of 0/0, in this situation, is, I take the 320 00:20:40,620 --> 00:20:54,140 limit of this one, which does have a meaning, because those 321 00:20:54,140 --> 00:20:55,070 are true numbers. 322 00:20:55,070 --> 00:20:58,320 They're little numbers but they're numbers. 323 00:20:58,320 --> 00:21:07,440 And this was this, so now here's the big step, leaving 324 00:21:07,440 --> 00:21:11,380 algebra behind, going to calculus in order to get 325 00:21:11,380 --> 00:21:13,780 what's happening at a point. 326 00:21:13,780 --> 00:21:16,230 I let delta x go to 0. 327 00:21:16,230 --> 00:21:17,510 And what is that? 328 00:21:17,510 --> 00:21:19,900 So delta y over delta x is this. 329 00:21:19,900 --> 00:21:22,380 What is the dy/dx? 330 00:21:22,380 --> 00:21:28,960 So in the limit, ah, it's not hard. 331 00:21:28,960 --> 00:21:31,270 Here's the 2x. 332 00:21:31,270 --> 00:21:34,030 It's there. 333 00:21:34,030 --> 00:21:35,570 Here's the delta x. 334 00:21:35,570 --> 00:21:39,280 In the limit, it disappears. 335 00:21:39,280 --> 00:21:44,460 So the conclusion is that the derivative is 2x. 336 00:21:48,330 --> 00:21:50,000 So that's function two. 337 00:21:50,000 --> 00:21:51,660 That's function two here. 338 00:21:51,660 --> 00:21:53,450 That's the slope function. 339 00:21:53,450 --> 00:21:55,060 That's the speed function. 340 00:21:55,060 --> 00:21:56,390 Maybe I should draw it. 341 00:21:56,390 --> 00:21:59,370 Can I draw it above and then I'll put the board back up? 342 00:21:59,370 --> 00:22:06,740 So here's a picture of function 2, the derivative, or 343 00:22:06,740 --> 00:22:08,910 the slope, which I was calling s. 344 00:22:08,910 --> 00:22:12,070 So that's the s function, against x. 345 00:22:12,070 --> 00:22:16,320 x is still the thing that's varying, or it could be t, or 346 00:22:16,320 --> 00:22:20,270 it could be whatever letter we've got. 347 00:22:20,270 --> 00:22:23,710 And the answer was 2x for this function. 348 00:22:23,710 --> 00:22:28,970 So if I graph it, it starts at 0, and it climbs 349 00:22:28,970 --> 00:22:32,540 steadily with slope 2. 350 00:22:32,540 --> 00:22:37,184 So that's a graph of s of x. 351 00:22:37,184 --> 00:22:40,460 And for example-- 352 00:22:40,460 --> 00:22:45,490 yeah, so take a couple of points on that graph-- 353 00:22:45,490 --> 00:22:50,520 at x equals 0, the slope is 0. 354 00:22:50,520 --> 00:22:55,520 And we did that first. And we actually got it right. 355 00:22:55,520 --> 00:22:59,680 The slope is 0 at the start, at the bottom of the curve. 356 00:22:59,680 --> 00:23:04,430 At some other point on the curve, what's the slope here? 357 00:23:04,430 --> 00:23:08,240 Ha, yeah, tell me the slope there. 358 00:23:08,240 --> 00:23:12,100 At that point on the curve, an average slope was 3/1, but 359 00:23:12,100 --> 00:23:15,180 that was the slope of this, like, you know-- 360 00:23:19,490 --> 00:23:21,660 sometimes called a chord. 361 00:23:21,660 --> 00:23:26,040 That's over a big jump of 1. 362 00:23:26,040 --> 00:23:29,190 Then I did it over a small jump of delta x, and then I 363 00:23:29,190 --> 00:23:34,790 let delta x go to 0, so it was an instant infinitesimal jump. 364 00:23:34,790 --> 00:23:39,630 So the actual slope, the way to visualize it is that it's 365 00:23:39,630 --> 00:23:41,560 more like that. 366 00:23:41,560 --> 00:23:45,170 That's the line that's really giving the 367 00:23:45,170 --> 00:23:47,860 slope at that point. 368 00:23:47,860 --> 00:23:49,240 That's my best picture. 369 00:23:49,240 --> 00:23:54,920 It's not Rembrandt, but it's got it. 370 00:23:54,920 --> 00:23:57,040 And what is the slope at that point? 371 00:23:57,040 --> 00:24:00,540 Well, that's what our calculation was. 372 00:24:00,540 --> 00:24:02,410 It found the slope at that point. 373 00:24:02,410 --> 00:24:11,250 And at the particular point, x equals 1, the height was 2. 374 00:24:11,250 --> 00:24:12,570 The slope is 2. 375 00:24:12,570 --> 00:24:17,720 The actual tangent line is only-- is there. 376 00:24:17,720 --> 00:24:18,400 You see? 377 00:24:18,400 --> 00:24:20,730 It's up. 378 00:24:20,730 --> 00:24:24,350 Oh, wait a minute. 379 00:24:24,350 --> 00:24:27,950 Yeah, well, the slope is 2. 380 00:24:27,950 --> 00:24:31,230 I don't know. 381 00:24:31,230 --> 00:24:34,510 This goes up to 3. 382 00:24:34,510 --> 00:24:41,300 It's not Rembrandt, but the math is OK. 383 00:24:41,300 --> 00:24:43,070 So what have we done? 384 00:24:43,070 --> 00:24:49,130 We've taken the first small step and literally I could say 385 00:24:49,130 --> 00:24:52,290 small step, almost a play on words because that's the 386 00:24:52,290 --> 00:24:55,180 point, the step is so small-- 387 00:24:55,180 --> 00:24:59,170 to getting these great functions. 388 00:24:59,170 --> 00:25:06,110 Before I close this lecture, can I draw this pair, function 389 00:25:06,110 --> 00:25:14,630 1 and function 2, and just see that the movement of the 390 00:25:14,630 --> 00:25:16,650 curves is what we would expect. 391 00:25:16,650 --> 00:25:23,710 So let me, just for one more good example, great example, 392 00:25:23,710 --> 00:25:26,510 actually, is let me draw. 393 00:25:26,510 --> 00:25:28,990 Here goes x. 394 00:25:28,990 --> 00:25:32,830 In fact, maybe I already drew in the first letter, lecture 395 00:25:32,830 --> 00:25:37,890 that bit out to 90 degrees. 396 00:25:37,890 --> 00:25:41,140 Only if we want a nice formula, we better call that 397 00:25:41,140 --> 00:25:45,000 pi over 2 radians. 398 00:25:45,000 --> 00:25:46,870 And here's a graph of sine x. 399 00:25:46,870 --> 00:25:48,450 This is y. 400 00:25:48,450 --> 00:25:52,380 This is the function 1, sine x. 401 00:25:52,380 --> 00:25:54,530 And what's function 2? 402 00:25:54,530 --> 00:25:57,210 What can we see about function 2? 403 00:26:00,400 --> 00:26:01,650 Again, x. 404 00:26:04,560 --> 00:26:06,410 We see a slope. 405 00:26:06,410 --> 00:26:09,160 This is not the same as x squared. 406 00:26:09,160 --> 00:26:12,230 This starts with a definite slope. 407 00:26:12,230 --> 00:26:17,690 And it turns out this will be one of the most important 408 00:26:17,690 --> 00:26:19,480 limits we'll find. 409 00:26:19,480 --> 00:26:24,760 We'll discover that the first little delta x, which goes up 410 00:26:24,760 --> 00:26:30,610 by sine of delta x, has a slope that gets closer and 411 00:26:30,610 --> 00:26:34,120 closer to 1. 412 00:26:34,120 --> 00:26:36,440 Good. 413 00:26:36,440 --> 00:26:42,210 Luckily, cosine does start at 1, so we're OK so far. 414 00:26:42,210 --> 00:26:45,120 Now the slope is dropping. 415 00:26:45,120 --> 00:26:49,330 And what's the slope at the top of the sine curve? 416 00:26:49,330 --> 00:26:51,810 It's a maximum. 417 00:26:51,810 --> 00:26:56,710 But we identify that by the fact that the slope is 0, 418 00:26:56,710 --> 00:27:01,690 because we know the thing is going to go down here and go 419 00:27:01,690 --> 00:27:02,780 somewhere else. 420 00:27:02,780 --> 00:27:04,410 The slope there is 0. 421 00:27:04,410 --> 00:27:07,400 The tangent line is horizontal. 422 00:27:07,400 --> 00:27:12,980 And that is that point. 423 00:27:12,980 --> 00:27:14,260 It passes through 0. 424 00:27:14,260 --> 00:27:15,980 The slope is dropping. 425 00:27:15,980 --> 00:27:18,030 So this is the slope curve. 426 00:27:18,030 --> 00:27:22,200 And the great thing is that it's the cosine of x. 427 00:27:22,200 --> 00:27:26,090 And what I'm doing now is not proving this fact. 428 00:27:26,090 --> 00:27:28,610 I'm not doing my delta x's. 429 00:27:28,610 --> 00:27:33,160 That's the job I have to do once, and it won't be today, 430 00:27:33,160 --> 00:27:34,445 but I only have to do it once. 431 00:27:37,820 --> 00:27:40,330 But today, I'm just saying it makes sense 432 00:27:40,330 --> 00:27:42,630 the slope is dropping. 433 00:27:42,630 --> 00:27:46,700 In that first part, I'm going up, so the slope is positive 434 00:27:46,700 --> 00:27:49,910 but the slope is dropping. 435 00:27:49,910 --> 00:27:52,650 And then, at this point, it hits 0. 436 00:27:52,650 --> 00:27:54,500 And that's this point. 437 00:27:54,500 --> 00:27:58,730 And then the slope turns negative. 438 00:27:58,730 --> 00:28:00,110 I'm falling. 439 00:28:00,110 --> 00:28:02,830 So the slope goes negative, and actually 440 00:28:02,830 --> 00:28:04,420 it follows the cosine. 441 00:28:04,420 --> 00:28:09,240 So I go along here to that point, and then I can continue 442 00:28:09,240 --> 00:28:13,840 on to this point where it bottoms out again 443 00:28:13,840 --> 00:28:16,760 and then starts up. 444 00:28:16,760 --> 00:28:19,140 So where is that on this curve? 445 00:28:19,140 --> 00:28:23,300 Well, I'd better draw a little further out. 446 00:28:23,300 --> 00:28:27,730 This bottom here would be the-- 447 00:28:27,730 --> 00:28:28,980 This is our pi/2. 448 00:28:31,220 --> 00:28:36,270 This is our pi, 180 degrees, everybody would say. 449 00:28:39,280 --> 00:28:41,510 So what's happening on that curve? 450 00:28:41,510 --> 00:28:47,760 The function is dropping, and actually it's dropping its 451 00:28:47,760 --> 00:28:51,510 fastest. It's dropping its fastest at that point, which 452 00:28:51,510 --> 00:28:53,640 is the slope is minus 1. 453 00:28:53,640 --> 00:28:57,180 And then the slope is still negative, but it's not so 454 00:28:57,180 --> 00:29:02,670 negative, and it comes back up to 0 at 3 pi/2 So this is the 455 00:29:02,670 --> 00:29:05,500 point, 3 pi/2. 456 00:29:05,500 --> 00:29:10,530 And this has come back to 0 at that point. 457 00:29:10,530 --> 00:29:14,500 And then it finishes the whole thing at 2 pi. 458 00:29:14,500 --> 00:29:17,920 This finishes up here back at 1 again. 459 00:29:17,920 --> 00:29:19,170 It's climbing. 460 00:29:22,050 --> 00:29:26,850 All right, climbing, dropping, faster, slower, maximum, 461 00:29:26,850 --> 00:29:33,070 minimum, those are the words that make derivatives 462 00:29:33,070 --> 00:29:36,150 important and useful to learn. 463 00:29:36,150 --> 00:29:40,930 And we've done, in detail, the first of our 464 00:29:40,930 --> 00:29:45,580 great list of functions. 465 00:29:45,580 --> 00:29:46,720 Thanks. 466 00:29:46,720 --> 00:29:48,480 FEMALE SPEAKER: This has been a production of MIT 467 00:29:48,480 --> 00:29:50,870 OpenCourseWare and Gilbert Strang. 468 00:29:50,870 --> 00:29:53,140 Funding for this video was provided by the Lord 469 00:29:53,140 --> 00:29:54,360 Foundation. 470 00:29:54,360 --> 00:29:57,490 To help OCW continue to provide free and open access 471 00:29:57,490 --> 00:30:00,570 to MIT courses, please make a donation at 472 00:30:00,570 --> 00:30:02,130 ocw.mit.edu/donate.