1 00:00:07,080 --> 00:00:12,640 PROFESSOR: OK, this lecture is about the slopes, the 2 00:00:12,640 --> 00:00:15,690 derivatives, of two of the great functions of 3 00:00:15,690 --> 00:00:18,830 mathematics: sine x and cosine x. 4 00:00:21,600 --> 00:00:23,250 Why do I say great functions? 5 00:00:23,250 --> 00:00:28,670 What sort of motion do we see sines and cosines? 6 00:00:28,670 --> 00:00:31,960 Well, I guess I'm thinking of oscillations. 7 00:00:31,960 --> 00:00:34,470 Things go back and forth. 8 00:00:34,470 --> 00:00:36,270 They go up and down. 9 00:00:36,270 --> 00:00:37,850 They go round in a circle. 10 00:00:37,850 --> 00:00:40,980 Your heart beats and beats and beats. 11 00:00:40,980 --> 00:00:42,530 Your lungs go in and out. 12 00:00:42,530 --> 00:00:46,470 The earth goes around the sun. 13 00:00:46,470 --> 00:00:52,490 So many motions are repeating motions, and that's when sines 14 00:00:52,490 --> 00:00:56,030 and cosines show up. 15 00:00:56,030 --> 00:00:58,810 The opposite is growing motions. 16 00:00:58,810 --> 00:01:03,990 That's where we have powers of x, x cubed, x to the n-th. 17 00:01:03,990 --> 00:01:06,170 Or if we really want the motion to get 18 00:01:06,170 --> 00:01:09,820 going, e to the x. 19 00:01:09,820 --> 00:01:14,010 Or decaying would be e to the minus x. 20 00:01:14,010 --> 00:01:15,760 So there are two kinds here. 21 00:01:15,760 --> 00:01:19,830 We're talking about the ones that repeat and stay level, 22 00:01:19,830 --> 00:01:22,480 and they all involve sines and cosines. 23 00:01:22,480 --> 00:01:26,010 And to make that point, I'm going to have to-- 24 00:01:26,010 --> 00:01:29,910 you know what sines and cosines are for triangles from 25 00:01:29,910 --> 00:01:31,250 trigonometry. 26 00:01:31,250 --> 00:01:33,920 But I have to make those triangles move. 27 00:01:33,920 --> 00:01:38,830 So I'm going to put the triangle in a circle, with one 28 00:01:38,830 --> 00:01:43,400 corner at the center, and another corner on the circle, 29 00:01:43,400 --> 00:01:45,450 and I'm going to move that point. 30 00:01:45,450 --> 00:01:47,820 So it's going to be circular motion. 31 00:01:47,820 --> 00:01:51,550 It's going to be the motion that-- 32 00:01:51,550 --> 00:01:55,890 the perfect model of repeating motion, around 33 00:01:55,890 --> 00:01:57,940 and around the circle. 34 00:01:57,940 --> 00:02:02,690 And then the answer we're going to get is just great. 35 00:02:02,690 --> 00:02:06,170 The derivative of sine x turns out to be cosine x. 36 00:02:06,170 --> 00:02:09,699 And the derivative of cosine x turns out to be minus sine x. 37 00:02:09,699 --> 00:02:11,740 You couldn't ask for more. 38 00:02:11,740 --> 00:02:18,230 So my interest is always to explain those, but then I want 39 00:02:18,230 --> 00:02:20,320 to really-- 40 00:02:20,320 --> 00:02:25,370 we're seeing this limit stuff in taking a derivative, and 41 00:02:25,370 --> 00:02:31,230 here's a chance for me to find a limit. 42 00:02:31,230 --> 00:02:34,900 This turns out to be the crucial quantity: the sine of 43 00:02:34,900 --> 00:02:42,320 an angle divided by the angle, when the angle goes to 0. 44 00:02:42,320 --> 00:02:48,030 Of course, when it's at 0, the sine of 0 is 0, so 45 00:02:48,030 --> 00:02:49,300 we have 0 over 0. 46 00:02:49,300 --> 00:02:51,560 This is the big problem of calculus. 47 00:02:51,560 --> 00:02:54,110 You can't be at the limit, because it's 0 48 00:02:54,110 --> 00:02:56,020 over 0 at that point. 49 00:02:56,020 --> 00:02:58,060 But you can be close to it. 50 00:02:58,060 --> 00:03:01,480 And then if we drew a graph, had a calculator, whatever we 51 00:03:01,480 --> 00:03:05,950 do, we would see that that ratio is very close to 1, but 52 00:03:05,950 --> 00:03:11,770 today we're going to actually prove it from the meaning of 53 00:03:11,770 --> 00:03:13,170 sine theta. 54 00:03:13,170 --> 00:03:15,390 Now remember what that meaning is. 55 00:03:15,390 --> 00:03:18,160 So back to the start of the world. 56 00:03:18,160 --> 00:03:23,040 Actually back to Pythagoras, way, way back. 57 00:03:23,040 --> 00:03:26,780 The key fact is what you remember about right 58 00:03:26,780 --> 00:03:31,580 triangles, a squared plus b squared equals c squared. 59 00:03:31,580 --> 00:03:34,545 That's where everything starts for a right triangle. 60 00:03:37,190 --> 00:03:39,000 I don't know if Pythagoras knew how to prove it. 61 00:03:39,000 --> 00:03:42,270 I think his friends helped him. 62 00:03:42,270 --> 00:03:44,510 A lot of people have suggested proofs. 63 00:03:44,510 --> 00:03:46,570 Einstein gave a proof. 64 00:03:46,570 --> 00:03:49,290 Some US president even gave a proof. 65 00:03:52,330 --> 00:03:57,440 So it's a fundamental fact, and I'm going to divide by c 66 00:03:57,440 --> 00:04:00,940 squared, because I'd like the right hand side to be 1. 67 00:04:00,940 --> 00:04:04,870 So if I divide by c squared, I just have a squared over c 68 00:04:04,870 --> 00:04:10,440 squared plus b squared over c squared is 1. 69 00:04:10,440 --> 00:04:14,840 And I'm going to make that hypotenuse in my picture 1. 70 00:04:14,840 --> 00:04:23,130 So then this will be the a over c, and that ratio of the 71 00:04:23,130 --> 00:04:27,110 near side to the hypotenuse is the cosine. 72 00:04:27,110 --> 00:04:31,940 So what I have here is cosine theta squared. 73 00:04:31,940 --> 00:04:33,440 Let me put theta in there. 74 00:04:33,440 --> 00:04:36,390 Theta is that angle at the center. 75 00:04:36,390 --> 00:04:37,440 And what's b? 76 00:04:37,440 --> 00:04:41,240 So this is a over c. 77 00:04:41,240 --> 00:04:43,110 That's cosine theta. 78 00:04:43,110 --> 00:04:49,670 B over c is this point, and that's sine theta. 79 00:04:49,670 --> 00:04:52,500 And they add to 1. 80 00:04:52,500 --> 00:04:58,250 So that's Pythagoras using sines and cosines. 81 00:04:58,250 --> 00:05:02,590 So this is the cosine. 82 00:05:02,590 --> 00:05:07,300 And this vertical distance is sine theta. 83 00:05:07,300 --> 00:05:11,200 OK, so that's the triangle I like. 84 00:05:11,200 --> 00:05:13,680 That's the triangle that's going to move. 85 00:05:13,680 --> 00:05:17,860 As this point goes around the circle at a steady speed, this 86 00:05:17,860 --> 00:05:20,970 triangle is going to move. 87 00:05:20,970 --> 00:05:25,850 The base will go left and right, left and right. 88 00:05:25,850 --> 00:05:30,370 The height will go up and down, up and down, following 89 00:05:30,370 --> 00:05:32,400 cosine and sine. 90 00:05:32,400 --> 00:05:35,370 And we want to know things about the speed. 91 00:05:35,370 --> 00:05:39,460 OK, so that's circular motion. 92 00:05:39,460 --> 00:05:42,440 Now I've introduced this word radians. 93 00:05:42,440 --> 00:05:49,190 And let me remind you what they are and why we need them. 94 00:05:49,190 --> 00:05:54,230 Why don't we just use 360 degrees for the full circle? 95 00:05:54,230 --> 00:05:56,420 360 degrees. 96 00:05:56,420 --> 00:06:02,280 Well, that's a nice number, 360. 97 00:06:02,280 --> 00:06:05,200 Somebody must have thought it was really nice, and chose it 98 00:06:05,200 --> 00:06:10,750 for measuring angles around the world. 99 00:06:13,650 --> 00:06:21,030 It's nice, but it's not natural. 100 00:06:21,030 --> 00:06:25,370 Somebody thought of it, so it's not good. 101 00:06:25,370 --> 00:06:30,580 What we need is the natural way to measure the angle. 102 00:06:30,580 --> 00:06:35,540 If we don't use the natural way, then this is the sine-- 103 00:06:35,540 --> 00:06:39,060 if I measure this x in degrees, that 104 00:06:39,060 --> 00:06:40,240 formula won't be right. 105 00:06:40,240 --> 00:06:44,240 There will be a miserable factor that I want to be 1. 106 00:06:44,240 --> 00:06:47,210 So I have to measure the angles the right way, and 107 00:06:47,210 --> 00:06:50,310 here's the idea of radians. 108 00:06:50,310 --> 00:06:55,960 The measure of that angle is this 109 00:06:55,960 --> 00:06:59,720 distance around the circle. 110 00:06:59,720 --> 00:07:03,630 That distance I'm going to call theta, and I'm going to 111 00:07:03,630 --> 00:07:05,980 say this angle is theta radians when 112 00:07:05,980 --> 00:07:09,790 that distance is theta. 113 00:07:09,790 --> 00:07:12,770 So that now, what's a full circle? 114 00:07:12,770 --> 00:07:16,870 A full circle would mean the angle went all the way round. 115 00:07:16,870 --> 00:07:20,430 I get the whole circumference, which is 2 pi. 116 00:07:20,430 --> 00:07:24,575 So 360 degrees is 2 pi radians. 117 00:07:27,740 --> 00:07:31,010 So the natural number here is 2 pi. 118 00:07:31,010 --> 00:07:35,150 This can't be helped, it's the right one to use. 119 00:07:35,150 --> 00:07:38,590 Radians are the right way to measure an angle. 120 00:07:38,590 --> 00:07:44,370 So now I'm ready to do the job of finding this derivative. 121 00:07:44,370 --> 00:07:45,620 OK. 122 00:07:47,330 --> 00:07:52,370 Let me start at the key point 0. 123 00:07:52,370 --> 00:07:57,330 If we get this one, we get all the rest easily. 124 00:07:57,330 --> 00:08:00,080 So I'm looking at the graph of the sine curve. 125 00:08:00,080 --> 00:08:02,460 I'm starting at 0. 126 00:08:02,460 --> 00:08:09,450 We know what sine theta looks like, and I'm interested in 127 00:08:09,450 --> 00:08:11,950 the slope, the derivative. 128 00:08:11,950 --> 00:08:15,920 That's what this subject is about, calculus, 129 00:08:15,920 --> 00:08:17,530 differentiating. 130 00:08:17,530 --> 00:08:20,810 So I want to know the slope at that point. 131 00:08:20,810 --> 00:08:25,310 And it's 1. 132 00:08:25,310 --> 00:08:27,260 And how do we show that it's 1? 133 00:08:27,260 --> 00:08:35,520 So now I'm coming to the point where I'm going to give a 134 00:08:35,520 --> 00:08:37,419 proof that is 1. 135 00:08:37,419 --> 00:08:43,624 And the proof isn't just for the sake of formality or rigor 136 00:08:43,624 --> 00:08:45,390 or something. 137 00:08:45,390 --> 00:08:50,120 You really have to understand the sine function, the cosine 138 00:08:50,120 --> 00:08:53,350 function, and this is the heart of it. 139 00:08:53,350 --> 00:08:54,560 OK. 140 00:08:54,560 --> 00:08:56,470 So we want to show that slope is 1. 141 00:08:56,470 --> 00:08:57,720 How am I going to do that? 142 00:09:02,470 --> 00:09:03,510 That's the slope, right? 143 00:09:03,510 --> 00:09:08,400 If I go a tiny amount theta, then I go up sine theta. 144 00:09:08,400 --> 00:09:14,960 So in this average slope, if I take a finite step-- 145 00:09:14,960 --> 00:09:17,440 I could have called it delta theta, but I don't want to 146 00:09:17,440 --> 00:09:19,000 write deltas all the time. 147 00:09:19,000 --> 00:09:23,300 So I just go out a little distance theta and up to the 148 00:09:23,300 --> 00:09:25,190 sine curve. 149 00:09:25,190 --> 00:09:28,890 I stopped at the sine curve by the way. 150 00:09:28,890 --> 00:09:32,730 The straight line is a little above the sine curve here. 151 00:09:32,730 --> 00:09:39,560 And that ratio, up divided by across, that's the delta sine 152 00:09:39,560 --> 00:09:42,610 divided by delta theta. 153 00:09:42,610 --> 00:09:47,530 And because it started at 0, it's just sine theta is the 154 00:09:47,530 --> 00:09:50,070 distance up, and theta is the distance across. 155 00:09:50,070 --> 00:09:53,220 So this is the average slope. 156 00:09:53,220 --> 00:09:56,760 And of course you remember what calculus is doing. 157 00:09:56,760 --> 00:10:01,720 There's always this limiting process where you push things 158 00:10:01,720 --> 00:10:05,300 closer and closer to the point, and you find the slope 159 00:10:05,300 --> 00:10:10,390 at that point, sometimes called the instantaneous 160 00:10:10,390 --> 00:10:13,840 velocity or slope or derivative. 161 00:10:13,840 --> 00:10:15,830 Now here's the way it's going to work. 162 00:10:15,830 --> 00:10:18,940 I'm going to show that sine theta over theta is 163 00:10:18,940 --> 00:10:20,200 always below 1. 164 00:10:20,200 --> 00:10:22,590 So two facts I want to prove. 165 00:10:22,590 --> 00:10:27,440 I want to show that sine theta over theta is less-- 166 00:10:27,440 --> 00:10:30,670 sorry, sine theta over theta-- 167 00:10:30,670 --> 00:10:34,510 well, let me get this right. 168 00:10:34,510 --> 00:10:36,690 I might as well put it the neat way. 169 00:10:36,690 --> 00:10:41,250 I want to show that sine theta is below theta. 170 00:10:41,250 --> 00:10:45,190 This is for theta greater than 0. 171 00:10:45,190 --> 00:10:46,180 That's what I'm doing. 172 00:10:46,180 --> 00:10:47,430 OK. 173 00:10:49,560 --> 00:10:56,550 So that tells me that this curve stays under that 174 00:10:56,550 --> 00:11:00,500 straight line, that 45 degree line, which I'm claiming is 175 00:11:00,500 --> 00:11:02,800 the tangent line. 176 00:11:02,800 --> 00:11:05,830 And it tells me when I divide it by the theta, it tells me 177 00:11:05,830 --> 00:11:08,920 that sine theta over theta is below 1. 178 00:11:08,920 --> 00:11:13,360 But now how much below 1? 179 00:11:13,360 --> 00:11:15,920 Right now if I only know this, I haven't ruled out the 180 00:11:15,920 --> 00:11:18,860 possibility that the slope could be much smaller. 181 00:11:18,860 --> 00:11:21,310 So I need something below it. 182 00:11:21,310 --> 00:11:24,830 And fantastically, the cosine is below it. 183 00:11:24,830 --> 00:11:29,190 So the other thing that I want to prove is that the cosine-- 184 00:11:29,190 --> 00:11:31,590 and I'll let me do it this way. 185 00:11:31,590 --> 00:11:36,310 I'm going to show the tangent of theta is bigger than theta. 186 00:11:36,310 --> 00:11:39,430 Again, some range of thetas. 187 00:11:39,430 --> 00:11:41,570 Positive thetas up to somewhere. 188 00:11:41,570 --> 00:11:44,333 I don't know, I think maybe pi over 2. 189 00:11:44,333 --> 00:11:49,800 But the main point is near 0, that's the main point. 190 00:11:49,800 --> 00:11:51,160 So can I just rewrite-- 191 00:11:51,160 --> 00:11:53,600 do you remember what the tangent is? 192 00:11:53,600 --> 00:11:57,340 Of course, sine theta over cosine theta. 193 00:11:57,340 --> 00:12:02,000 So this is sine theta over cosine 194 00:12:02,000 --> 00:12:03,480 theta, bigger than theta. 195 00:12:03,480 --> 00:12:06,870 We still have to prove this. 196 00:12:06,870 --> 00:12:10,590 And now I want to bring the theta down and move the cosine 197 00:12:10,590 --> 00:12:16,240 up, and that will tell me that sine theta, when I divide by 198 00:12:16,240 --> 00:12:20,510 the theta and multiply by the cosine theta. 199 00:12:23,300 --> 00:12:27,980 So that was the same as that, was the same as that. 200 00:12:27,980 --> 00:12:29,710 And that's what I want. 201 00:12:29,710 --> 00:12:36,210 That tells that this ratio is above the cosine curve. 202 00:12:36,210 --> 00:12:41,800 Do you see that if I can convince you, and convince 203 00:12:41,800 --> 00:12:45,450 myself that these are both true, that this picture is 204 00:12:45,450 --> 00:12:47,215 right, then-- 205 00:12:50,080 --> 00:12:54,520 I haven't gone into gory detail about limits. 206 00:12:54,520 --> 00:12:58,400 If you really insist, I'll do it later. 207 00:12:58,400 --> 00:12:59,650 But whatever. 208 00:13:01,320 --> 00:13:06,400 You can see this has just got to be true, that if this curve 209 00:13:06,400 --> 00:13:14,180 is squeezed between the cosine curve and the 1, then as theta 210 00:13:14,180 --> 00:13:18,230 gets smaller and smaller, it's squeezed to 211 00:13:18,230 --> 00:13:21,780 equal 1 in the limit. 212 00:13:21,780 --> 00:13:25,155 Allow me to say that that's pretty darn clear. 213 00:13:25,155 --> 00:13:26,405 OK. 214 00:13:29,210 --> 00:13:32,410 Whatever limit meets. 215 00:13:32,410 --> 00:13:35,620 So these are the main facts that I need to show. 216 00:13:35,620 --> 00:13:39,760 And I need to show those using trig, right? 217 00:13:39,760 --> 00:13:44,250 I have to draw some graph that convinces you. 218 00:13:44,250 --> 00:13:46,950 And this isn't quite good enough, because I just 219 00:13:46,950 --> 00:13:48,910 sketched a sine graph. 220 00:13:48,910 --> 00:13:51,565 I have to say where does sine theta come from? 221 00:13:51,565 --> 00:13:52,590 OK. 222 00:13:52,590 --> 00:13:57,150 So this will be number one, and this will be number two, 223 00:13:57,150 --> 00:14:03,450 and when we get those two things convincing, then we 224 00:14:03,450 --> 00:14:07,570 know that sine theta over theta is squeezed between and 225 00:14:07,570 --> 00:14:09,380 approaches 1. 226 00:14:09,380 --> 00:14:14,780 And then we'll know the story at the start, and you'll see 227 00:14:14,780 --> 00:14:21,060 that it becomes easy to find these formulas 228 00:14:21,060 --> 00:14:21,980 all along the curve. 229 00:14:21,980 --> 00:14:23,130 OK. 230 00:14:23,130 --> 00:14:24,500 Ready for these two? 231 00:14:24,500 --> 00:14:26,330 Number one and number two. 232 00:14:26,330 --> 00:14:27,580 OK, number one. 233 00:14:30,200 --> 00:14:31,590 Why is sine theta-- 234 00:14:31,590 --> 00:14:35,240 oh, I can probably see it on this picture. 235 00:14:35,240 --> 00:14:37,880 Yeah, I can prove number one on this picture. 236 00:14:37,880 --> 00:14:40,950 Look, that piece was sine theta, right? 237 00:14:45,060 --> 00:14:48,890 And I want to prove that this length-- 238 00:14:48,890 --> 00:14:49,910 what am I trying to prove? 239 00:14:49,910 --> 00:14:51,800 That sine theta is below theta. 240 00:14:51,800 --> 00:14:55,701 Let me write it again what it is to show. 241 00:14:55,701 --> 00:14:59,000 In math it's always a good idea to keep reminding 242 00:14:59,000 --> 00:15:01,170 yourself of what it is you're doing. 243 00:15:01,170 --> 00:15:03,899 Sine theta is below theta. 244 00:15:03,899 --> 00:15:03,958 OK. 245 00:15:03,958 --> 00:15:05,208 So why is it? 246 00:15:09,000 --> 00:15:09,850 And you see it here. 247 00:15:09,850 --> 00:15:12,400 That was sine theta, right? 248 00:15:12,400 --> 00:15:14,260 And where was theta? 249 00:15:14,260 --> 00:15:18,690 Well, because we measured theta in radians, theta is 250 00:15:18,690 --> 00:15:23,080 this curvy distance that's clearly longer. 251 00:15:23,080 --> 00:15:28,250 The shortest way from this to the axis there is straight 252 00:15:28,250 --> 00:15:30,590 down, and that's sine theta. 253 00:15:30,590 --> 00:15:35,560 A slower way is go round and end up at not the nearest 254 00:15:35,560 --> 00:15:38,310 point, and that was theta. 255 00:15:38,310 --> 00:15:39,570 Is that good? 256 00:15:39,570 --> 00:15:44,570 I could sometimes just to be even more convincing, you add 257 00:15:44,570 --> 00:15:48,890 a second angle, and you say OK, there's 2 sine thetas and 258 00:15:48,890 --> 00:15:52,750 here is 2 thetas, and clearly we all know that the shortest 259 00:15:52,750 --> 00:15:57,210 way from there to there is the straight way. 260 00:15:57,210 --> 00:16:03,580 So I regard this as done by that picture. 261 00:16:03,580 --> 00:16:05,960 You see we didn't just make it up. 262 00:16:05,960 --> 00:16:08,800 It went back to the fundamental idea of where sine 263 00:16:08,800 --> 00:16:10,390 theta is in a picture. 264 00:16:10,390 --> 00:16:13,920 Now I need another picture. 265 00:16:13,920 --> 00:16:20,100 Yeah, I need another picture for number two to show that 266 00:16:20,100 --> 00:16:23,340 tan theta is bigger than theta. 267 00:16:23,340 --> 00:16:24,930 That was our other job. 268 00:16:24,930 --> 00:16:30,370 So essentially, I need that same picture again. 269 00:16:30,370 --> 00:16:33,900 Whoops, let me draw that triangle. 270 00:16:37,080 --> 00:16:41,240 Yeah, and it's got a circle. 271 00:16:41,240 --> 00:16:44,320 OK, that's not a bad circle. 272 00:16:44,320 --> 00:16:47,010 It's got an angle theta. 273 00:16:47,010 --> 00:16:48,260 And now I'm going to-- 274 00:16:51,650 --> 00:16:54,140 math has always got some little trick. 275 00:16:54,140 --> 00:16:55,590 So this is it. 276 00:16:55,590 --> 00:17:01,840 Go all the way out, so now the base is 1, and this is still 277 00:17:01,840 --> 00:17:03,090 the angle theta. 278 00:17:06,069 --> 00:17:08,520 And what else do I know on that picture? 279 00:17:08,520 --> 00:17:12,060 Now I've scaled the triangle up from this little one, so 280 00:17:12,060 --> 00:17:13,190 the base is 1. 281 00:17:13,190 --> 00:17:15,589 So what's that height? 282 00:17:15,589 --> 00:17:19,960 Well, the ratio of the opposite side to the near 283 00:17:19,960 --> 00:17:23,819 side, that's what tangent is. 284 00:17:23,819 --> 00:17:25,619 Tangent is the ratio-- 285 00:17:25,619 --> 00:17:27,470 whatever size the triangle-- 286 00:17:27,470 --> 00:17:30,310 is the ratio of the opposite side to the near side. 287 00:17:30,310 --> 00:17:37,180 Sine to cosine, here it's tan theta is that 288 00:17:37,180 --> 00:17:39,690 distance, and to 1. 289 00:17:39,690 --> 00:17:40,630 Good. 290 00:17:40,630 --> 00:17:43,650 OK, but now how am I going to see this? 291 00:17:48,640 --> 00:17:51,870 I have to ask you-- and it's OK-- 292 00:17:51,870 --> 00:17:55,770 to think about area instead of distance for a moment. 293 00:17:55,770 --> 00:17:57,230 What about area? 294 00:17:57,230 --> 00:17:59,130 So what do I see of area? 295 00:17:59,130 --> 00:18:06,480 I see right away that the area of this triangle is smaller 296 00:18:06,480 --> 00:18:08,360 than the area-- 297 00:18:08,360 --> 00:18:10,520 sorry, I shouldn't have called that a triangle. 298 00:18:10,520 --> 00:18:15,020 That's a little piece of pie, a little sector of a circle. 299 00:18:15,020 --> 00:18:17,080 So the area of this shaded-- 300 00:18:17,080 --> 00:18:18,590 did I shade it OK-- 301 00:18:21,510 --> 00:18:22,980 is less than-- 302 00:18:22,980 --> 00:18:26,330 so this is the area of the sector. 303 00:18:26,330 --> 00:18:30,470 Can I just call it the pie, piece of a pie, is less than 304 00:18:30,470 --> 00:18:32,070 the area of the triangle. 305 00:18:34,860 --> 00:18:39,300 But we know what the area of the triangle is. 306 00:18:39,300 --> 00:18:41,290 What's the area of a triangle? 307 00:18:41,290 --> 00:18:42,750 We can do that. 308 00:18:42,750 --> 00:18:45,200 It's the base half, right? 309 00:18:45,200 --> 00:18:48,010 1/2 times the base times the height. 310 00:18:48,010 --> 00:18:51,230 So the area of the triangle is 1/2 times the 311 00:18:51,230 --> 00:18:52,555 base 1 times the height. 312 00:18:52,555 --> 00:18:53,805 OK. 313 00:18:56,280 --> 00:18:58,950 Notice we've got the sign going the right way. 314 00:18:58,950 --> 00:19:01,680 We want tan theta to be bigger than something, 315 00:19:01,680 --> 00:19:03,420 so what do I hope? 316 00:19:03,420 --> 00:19:08,620 I hope that the area of this shaded part, the area of the 317 00:19:08,620 --> 00:19:15,520 circular sector, is 1/2 of theta. 318 00:19:15,520 --> 00:19:18,660 Wouldn't that be wonderful? 319 00:19:18,660 --> 00:19:22,310 If I look at those areas, nobody's in any doubt that 320 00:19:22,310 --> 00:19:26,390 this piece, this sector that's inside the triangle, has an 321 00:19:26,390 --> 00:19:28,530 area less than the area of the triangle. 322 00:19:28,530 --> 00:19:33,560 So now I just have to remember why is the area of this 323 00:19:33,560 --> 00:19:37,120 sector, half of theta. 324 00:19:37,120 --> 00:19:40,680 You know, there's another reason why areas come up right 325 00:19:40,680 --> 00:19:44,600 when we use a radians, when we measure theta with radians. 326 00:19:44,600 --> 00:19:52,050 So remember, just think about this piece of pie compared to 327 00:19:52,050 --> 00:19:53,300 the whole pie. 328 00:19:56,180 --> 00:19:59,660 What's the area of the whole piece of pie? 329 00:19:59,660 --> 00:20:03,740 So I'm explaining 1/2 theta. 330 00:20:03,740 --> 00:20:07,320 The area of the whole pie-- 331 00:20:07,320 --> 00:20:12,350 I'm going to get some terrible pun here on the word pie. 332 00:20:12,350 --> 00:20:15,460 Unintended, forgive it. 333 00:20:15,460 --> 00:20:19,170 The area of that whole circle, the radius is 1, we all know 334 00:20:19,170 --> 00:20:22,030 what the area of a circle is pi r squared. 335 00:20:22,030 --> 00:20:26,020 r is 1, so the area is pi. 336 00:20:26,020 --> 00:20:29,240 My God, I didn't expect that. 337 00:20:29,240 --> 00:20:30,860 Now what about this? 338 00:20:30,860 --> 00:20:34,280 What fraction is this sector? 339 00:20:34,280 --> 00:20:40,020 Well, the whole angle would be 2 pi, and this part of it is 340 00:20:40,020 --> 00:20:49,320 theta, so I have the sector is theta over 2 pi, that's the 341 00:20:49,320 --> 00:20:56,420 angle fraction, times the pi, the whole area. 342 00:20:56,420 --> 00:20:58,190 Do you see it? 343 00:20:58,190 --> 00:21:02,830 This piece of pie, or pizza, whatever-- 344 00:21:02,830 --> 00:21:04,940 yeah, if I'd said pizza, I wouldn't have had that 345 00:21:04,940 --> 00:21:06,840 terrible pun. 346 00:21:06,840 --> 00:21:08,940 Forget it. 347 00:21:08,940 --> 00:21:14,330 So the area of this piece of pizza compared to the whole 348 00:21:14,330 --> 00:21:18,050 one is theta over the whole 2 pi. 349 00:21:21,270 --> 00:21:25,690 Suppose it was a pizza cut in the usual 6 pieces. 350 00:21:25,690 --> 00:21:28,180 Then this would be a 60 degree angle, but 351 00:21:28,180 --> 00:21:30,290 I don't want degrees. 352 00:21:30,290 --> 00:21:34,010 What would be the angle of that piece of pizza that's cut 353 00:21:34,010 --> 00:21:36,510 when the whole pizza's cut in 6? 354 00:21:36,510 --> 00:21:41,750 It would be 1/6 of 360. 355 00:21:41,750 --> 00:21:42,900 That's 60 degrees. 356 00:21:42,900 --> 00:21:44,300 But I don't want degrees. 357 00:21:44,300 --> 00:21:47,580 It's 1/6 of 2 pi. 358 00:21:47,580 --> 00:21:50,130 And this one is theta of 2 pi. 359 00:21:50,130 --> 00:21:52,540 Anyway, the pis cancel. 360 00:21:52,540 --> 00:21:57,090 Theta over 2 is the right answer, and now we can cancel 361 00:21:57,090 --> 00:21:59,640 the 1/2, and we've got what we want. 362 00:22:03,560 --> 00:22:08,660 That's pretty nice when you realize that we were facing 363 00:22:08,660 --> 00:22:12,210 for the first time, more or less, the sort of tough 364 00:22:12,210 --> 00:22:16,970 problem of calculus when I can't really divide theta into 365 00:22:16,970 --> 00:22:18,270 sine theta. 366 00:22:18,270 --> 00:22:22,880 Sine theta, I can't just divide it in. 367 00:22:22,880 --> 00:22:27,770 I have to keep them both approaching 0 over 0, and see 368 00:22:27,770 --> 00:22:29,910 what that ratio is doing. 369 00:22:29,910 --> 00:22:36,800 And now I said to conclude, I'll go back and prove the 370 00:22:36,800 --> 00:22:40,830 slopes, find the slopes at all points. 371 00:22:40,830 --> 00:22:43,140 OK, so at all points-- 372 00:22:43,140 --> 00:22:44,920 now let's start with sine x. 373 00:22:49,150 --> 00:22:51,430 So what am I doing now? 374 00:22:51,430 --> 00:22:55,570 I'm looking at the sine curve. 375 00:22:55,570 --> 00:22:58,560 You remember it went up like this and down like this. 376 00:22:58,560 --> 00:22:59,965 I'm taking any point x. 377 00:23:03,220 --> 00:23:07,870 Suddenly I changed the angle from theta to x, just because 378 00:23:07,870 --> 00:23:09,830 I'm used to functions of x. 379 00:23:09,830 --> 00:23:12,420 We're just talking letters there. 380 00:23:12,420 --> 00:23:16,940 X is good, and this is a graph of sine x. 381 00:23:16,940 --> 00:23:18,327 X is measured in radians still. 382 00:23:18,327 --> 00:23:19,600 OK. 383 00:23:19,600 --> 00:23:23,090 So now what am I doing to find the derivative at some 384 00:23:23,090 --> 00:23:26,400 particular point? 385 00:23:26,400 --> 00:23:28,740 I look at the sine there. 386 00:23:28,740 --> 00:23:32,550 I go a little distance delta x. 387 00:23:32,550 --> 00:23:37,120 I go up to here, and I look to see-- 388 00:23:37,120 --> 00:23:42,950 I want to know the change in sine x divided by 389 00:23:42,950 --> 00:23:46,600 the change in x. 390 00:23:46,600 --> 00:23:48,710 And of course, I'm going to let the piece 391 00:23:48,710 --> 00:23:49,750 get smaller and smaller. 392 00:23:49,750 --> 00:23:52,320 That's what calculus does. 393 00:23:52,320 --> 00:23:55,700 The main point is my x is now here instead of 394 00:23:55,700 --> 00:23:57,120 being at the start. 395 00:23:57,120 --> 00:23:59,510 I've done it for the start, but now I have to do it for 396 00:23:59,510 --> 00:24:00,670 all the other x's. 397 00:24:00,670 --> 00:24:03,170 So there's the x. 398 00:24:03,170 --> 00:24:05,820 There's the x plus delta x, a little bit long. 399 00:24:09,380 --> 00:24:13,230 In other words, can I write this in the familiar way, sine 400 00:24:13,230 --> 00:24:20,275 of x plus delta minus sine there divided by delta x? 401 00:24:20,275 --> 00:24:21,525 OK. 402 00:24:23,230 --> 00:24:28,150 So again, we can't simplify totally by just dividing the 403 00:24:28,150 --> 00:24:31,000 delta x in. 404 00:24:31,000 --> 00:24:33,920 We've got to go back to trigonometry. 405 00:24:33,920 --> 00:24:40,500 Trig had a formula for the sine of a plus b. 406 00:24:40,500 --> 00:24:45,440 Two angles added, then there's a neat formula for it. 407 00:24:45,440 --> 00:24:46,540 So the sine-- 408 00:24:46,540 --> 00:24:50,260 can I remind you of that formula? 409 00:24:50,260 --> 00:24:55,390 It is the sine of the first angle times the cosine of the 410 00:24:55,390 --> 00:25:04,960 second minus the cosine of the first angle times the sine of 411 00:25:04,960 --> 00:25:05,290 the second. 412 00:25:05,290 --> 00:25:06,540 OK? 413 00:25:09,760 --> 00:25:12,095 You remember this, right, from trig? 414 00:25:12,095 --> 00:25:15,310 The sine of a plus b is this neat thing. 415 00:25:15,310 --> 00:25:17,620 Now I have to subtract sine x. 416 00:25:17,620 --> 00:25:20,550 So now can I subtract off sine x? 417 00:25:20,550 --> 00:25:27,650 When I subtract off sine x, then I need a minus 1. 418 00:25:27,650 --> 00:25:31,920 And now I have to divide by delta x. 419 00:25:31,920 --> 00:25:37,270 So I divide this by delta x, and I divide this by delta x. 420 00:25:37,270 --> 00:25:38,520 OK. 421 00:25:42,090 --> 00:25:43,990 This is an expression I can work with. 422 00:25:43,990 --> 00:25:47,850 That's why I had to remember this trig formula to get this 423 00:25:47,850 --> 00:25:49,700 expression that I can work with. 424 00:25:49,700 --> 00:25:51,490 Why do I say I can work with it? 425 00:25:51,490 --> 00:25:56,430 Because this is exactly what I've already pinned down. 426 00:25:56,430 --> 00:25:59,080 Delta x is now headed for 0. 427 00:25:59,080 --> 00:26:03,040 This point is going to come close to this one. 428 00:26:03,040 --> 00:26:07,330 So actually, I've got two terms. This sine delta x over 429 00:26:07,330 --> 00:26:12,590 delta x, what does that do as delta x goes to 0? 430 00:26:12,590 --> 00:26:13,960 It goes to 1. 431 00:26:13,960 --> 00:26:16,900 That was the point of that whole right hand board. 432 00:26:16,900 --> 00:26:21,750 So this thing goes to 1. 433 00:26:21,750 --> 00:26:23,890 Wait a minute. 434 00:26:23,890 --> 00:26:26,380 That's a plus sign. 435 00:26:26,380 --> 00:26:33,530 Everybody watching is going to think, OK, forgot trig. 436 00:26:33,530 --> 00:26:36,600 The sine of the sum of an angle is the sine times the 437 00:26:36,600 --> 00:26:41,400 cosine plus the cosine times the sine. 438 00:26:41,400 --> 00:26:43,190 Sorry about that one too. 439 00:26:43,190 --> 00:26:47,680 OK, so sine of delta x over delta x goes to 0. 440 00:26:47,680 --> 00:26:54,300 And now finally, this goes to 1, and actually I need another 441 00:26:54,300 --> 00:26:55,590 little piece. 442 00:26:55,590 --> 00:26:59,030 I need to know this piece, and I need to know that that 443 00:26:59,030 --> 00:27:02,010 ratio goes to 0. 444 00:27:04,520 --> 00:27:08,750 So I need to go back to that board and look again at the 445 00:27:08,750 --> 00:27:11,770 cosine curve at 0. 446 00:27:11,770 --> 00:27:17,240 Because this will be a slope of the cosine curve at 0. 447 00:27:17,240 --> 00:27:20,840 And the slope comes out 0 for the cosine curve. 448 00:27:20,840 --> 00:27:23,030 The slope for the sine curve came out 1. 449 00:27:23,030 --> 00:27:25,470 Do you see how it's working? 450 00:27:25,470 --> 00:27:29,330 So this is gone because of the 0. 451 00:27:29,330 --> 00:27:31,570 This is the cosine x times 1. 452 00:27:31,570 --> 00:27:36,590 All together I get cosine of x. 453 00:27:36,590 --> 00:27:37,230 Hooray. 454 00:27:37,230 --> 00:27:38,430 That's the goal. 455 00:27:38,430 --> 00:27:44,410 That's the predicted plan, desired formula cos x for the 456 00:27:44,410 --> 00:27:51,070 ratio of delta of sine x over delta x as delta x goes to 0. 457 00:27:51,070 --> 00:27:53,270 Do you see that? 458 00:27:53,270 --> 00:27:56,760 So we used a trig formula, and we got the sine 459 00:27:56,760 --> 00:27:59,340 right a little late. 460 00:27:59,340 --> 00:28:01,910 Well, of course the reason I-- 461 00:28:01,910 --> 00:28:06,920 one reason I goofed was that the other example, the other 462 00:28:06,920 --> 00:28:11,640 case we need for the second formula does 463 00:28:11,640 --> 00:28:13,050 have a minus sign. 464 00:28:13,050 --> 00:28:16,160 And it survives in the end. 465 00:28:16,160 --> 00:28:21,740 So I would do exactly the same thing for the cosines that I 466 00:28:21,740 --> 00:28:22,930 did for the sines. 467 00:28:22,930 --> 00:28:24,710 If there's another board underneath here, 468 00:28:24,710 --> 00:28:25,740 I'm going to do it. 469 00:28:25,740 --> 00:28:28,200 Yeah, there is. 470 00:28:28,200 --> 00:28:38,480 Now I want to know the delta of cosine x over delta x. 471 00:28:38,480 --> 00:28:43,190 That's what we do, we have to simplify that, then we have to 472 00:28:43,190 --> 00:28:44,930 let delta x go to 0. 473 00:28:44,930 --> 00:28:46,780 So what does this mean? 474 00:28:46,780 --> 00:28:52,570 This means the cosine a little bit along minus the cosine at 475 00:28:52,570 --> 00:28:55,950 the point divided by delta x. 476 00:28:55,950 --> 00:29:00,810 Again, we can't do that division just right away, but 477 00:29:00,810 --> 00:29:03,790 we can simplify this by remembering the 478 00:29:03,790 --> 00:29:06,750 formula that cosign-- 479 00:29:06,750 --> 00:29:08,300 now let me try to remember it. 480 00:29:08,300 --> 00:29:17,160 It's a cosine times a cosine for this guy plus a sine-- 481 00:29:17,160 --> 00:29:23,510 no, minus a sine times the sine. 482 00:29:23,510 --> 00:29:28,610 That's the formula that we all remember and go to sleep with. 483 00:29:28,610 --> 00:29:30,160 Now divide by delta x. 484 00:29:30,160 --> 00:29:32,460 Oh, first subtract cosine x. 485 00:29:32,460 --> 00:29:35,909 So there was a cosine x, so I want to subtract one of them. 486 00:29:35,909 --> 00:29:37,370 OK? 487 00:29:37,370 --> 00:29:39,860 And now I have to divide by the delta x. 488 00:29:39,860 --> 00:29:42,260 So I do that there. 489 00:29:42,260 --> 00:29:44,290 I do it here. 490 00:29:44,290 --> 00:29:50,630 And you see that we're in the same happy position. 491 00:29:50,630 --> 00:29:54,140 We're in the happy position that as delta x goes to 0, we 492 00:29:54,140 --> 00:29:55,880 know what this does. 493 00:29:55,880 --> 00:29:57,770 That goes to 1. 494 00:29:57,770 --> 00:30:00,630 We know what this does, or we soon will. 495 00:30:00,630 --> 00:30:05,000 That goes to 0, just the way they did on the 496 00:30:05,000 --> 00:30:07,430 board that got raised. 497 00:30:07,430 --> 00:30:10,160 So that term disappeared just like before. 498 00:30:10,160 --> 00:30:12,340 This term survives. 499 00:30:12,340 --> 00:30:15,620 It's got a 1, it's got now a sine x, and it's got now a 500 00:30:15,620 --> 00:30:16,950 minus sign. 501 00:30:16,950 --> 00:30:20,940 So that's the final result, that the limit 502 00:30:20,940 --> 00:30:24,820 is minus sine x. 503 00:30:24,820 --> 00:30:30,740 That's the slope of the cosine curve. 504 00:30:30,740 --> 00:30:33,050 And you wouldn't want it any other way. 505 00:30:33,050 --> 00:30:35,030 You want that minus sign. 506 00:30:35,030 --> 00:30:38,660 You'll see it with second derivatives. 507 00:30:38,660 --> 00:30:42,190 So it's just terrific that those functions, the 508 00:30:42,190 --> 00:30:45,340 derivative of the sine is the cosine with a plus, the 509 00:30:45,340 --> 00:30:47,310 derivative of the cosine is the sine with a minus. 510 00:30:47,310 --> 00:30:48,570 OK. 511 00:30:48,570 --> 00:30:53,310 And we've almost proved it, we just didn't quite pick up this 512 00:30:53,310 --> 00:30:56,210 point yet, and let me do that. 513 00:30:56,210 --> 00:30:58,190 That will finish this lecture. 514 00:30:58,190 --> 00:31:03,100 Why does that ratio approach zero? 515 00:31:03,100 --> 00:31:05,750 What is that ratio? 516 00:31:05,750 --> 00:31:11,670 That ratio is coming from the cosine curve. 517 00:31:11,670 --> 00:31:16,130 The cosine curve at 0, the way this ratio came from the sine 518 00:31:16,130 --> 00:31:17,440 curve at 0. 519 00:31:17,440 --> 00:31:20,090 Here I'm taking-- 520 00:31:20,090 --> 00:31:21,420 this is delta cosine. 521 00:31:25,080 --> 00:31:29,370 There's lots of ways I can do this, but maybe I'll just do 522 00:31:29,370 --> 00:31:34,580 it the way you see it. 523 00:31:34,580 --> 00:31:37,230 What's the slope of the cosine at 0? 524 00:31:40,340 --> 00:31:45,060 Yeah, I think I can ask that without doing limits, without 525 00:31:45,060 --> 00:31:49,220 doing hard work. 526 00:31:49,220 --> 00:31:53,270 I'll just add the rest of the cosine curve, because we know 527 00:31:53,270 --> 00:31:55,700 it's symmetric. 528 00:31:55,700 --> 00:31:57,410 What's the slope at that point? 529 00:32:01,470 --> 00:32:04,750 This is actually the most important application of 530 00:32:04,750 --> 00:32:09,140 calculus, is to locate the place where a 531 00:32:09,140 --> 00:32:11,470 function has a maximum. 532 00:32:11,470 --> 00:32:14,630 The cosine, its maximum is right there. 533 00:32:14,630 --> 00:32:19,030 Its maximum value is 1, and it happens at theta equals 0. 534 00:32:19,030 --> 00:32:21,440 So the slope at a maximum-- 535 00:32:21,440 --> 00:32:24,280 all right, I'm going to put this-- 536 00:32:24,280 --> 00:32:30,840 I could get this result by these pictures, but let me do 537 00:32:30,840 --> 00:32:33,750 it short circuit. 538 00:32:33,750 --> 00:32:39,305 The slope at the maximum is 0. 539 00:32:39,305 --> 00:32:40,555 OK. 540 00:32:46,160 --> 00:32:47,870 Your intuition tells you that. 541 00:32:47,870 --> 00:32:52,960 If the slope was positive, the function 542 00:32:52,960 --> 00:32:54,370 would still be rising. 543 00:32:54,370 --> 00:32:56,890 It wouldn't be a maximum, it would be going higher. 544 00:32:56,890 --> 00:33:01,390 If the slope was negative, the function would be coming down, 545 00:33:01,390 --> 00:33:04,470 and the maximum would have been earlier. 546 00:33:04,470 --> 00:33:08,870 But here the maximum is right there. 547 00:33:08,870 --> 00:33:11,840 The slope has to be 0 at that point. 548 00:33:11,840 --> 00:33:16,710 And that's the quantity that we were after, because this is 549 00:33:16,710 --> 00:33:20,450 the cosine of delta x. 550 00:33:20,450 --> 00:33:23,060 There is the cosine of delta x. 551 00:33:23,060 --> 00:33:28,970 Here is the 1, here is the delta x, and this ratio is 552 00:33:28,970 --> 00:33:31,220 height over slope. 553 00:33:31,220 --> 00:33:35,310 It gets to height over slope as we get closer and closer. 554 00:33:35,310 --> 00:33:39,510 It's the derivative, and it's 0 at a maximum. 555 00:33:39,510 --> 00:33:47,640 And my notes give another way to convince yourself that 556 00:33:47,640 --> 00:33:52,855 that's 0 by using these facts that we've already got. 557 00:33:52,855 --> 00:33:55,040 OK. 558 00:33:55,040 --> 00:34:00,190 End of the-- so let me just recap one moment, which this 559 00:34:00,190 --> 00:34:02,000 board will do. 560 00:34:02,000 --> 00:34:05,430 We now know the derivatives of two of the great 561 00:34:05,430 --> 00:34:07,380 functions of calculus. 562 00:34:07,380 --> 00:34:11,600 We already know the derivative of x to the n-th, and in the 563 00:34:11,600 --> 00:34:16,739 future is coming e to the x and the logarithm. 564 00:34:16,739 --> 00:34:18,920 Then you've got the big ones. 565 00:34:18,920 --> 00:34:20,469 Thank you. 566 00:34:20,469 --> 00:34:22,280 ANNOUNCER: This has been a production of MIT 567 00:34:22,280 --> 00:34:24,670 OpenCourseWare and Gilbert Strang. 568 00:34:24,670 --> 00:34:26,940 Funding for this video was provided by the Lord 569 00:34:26,940 --> 00:34:28,159 Foundation. 570 00:34:28,159 --> 00:34:31,290 To help OCW continue to provide free and open access 571 00:34:31,290 --> 00:34:34,370 to MIT courses, please make a donation at 572 00:34:34,370 --> 00:34:35,929 ocw.mit.edu/donate.