1 00:00:00,500 --> 00:00:03,610 The following is provided under a Creative Commons License. 2 00:00:03,610 --> 00:00:05,720 Your support will help MIT OpenCourseWare 3 00:00:05,720 --> 00:00:09,950 continue to offer high quality educational resources for free. 4 00:00:09,950 --> 00:00:12,560 To make a donation or to view additional materials 5 00:00:12,560 --> 00:00:15,910 from hundreds of MIT courses, visit MIT OpenCourseWare 6 00:00:15,910 --> 00:00:19,350 at ocw.mit.edu. 7 00:00:19,350 --> 00:00:25,150 Professor: In today's lecture I want 8 00:00:25,150 --> 00:00:27,715 to develop several more formulas that 9 00:00:27,715 --> 00:00:32,410 will allow us to reach our goal of differentiating everything. 10 00:00:32,410 --> 00:00:41,800 So these are derivative formulas, 11 00:00:41,800 --> 00:00:45,560 and they come in two flavors. 12 00:00:45,560 --> 00:00:52,070 The first kind is specific, so some specific 13 00:00:52,070 --> 00:00:55,460 function we're giving the derivative of. 14 00:00:55,460 --> 00:01:00,740 And that would be, for example, x^n or (1/x) . 15 00:01:00,740 --> 00:01:05,220 Those are the ones that we did a couple of lectures ago. 16 00:01:05,220 --> 00:01:10,510 And then there are general formulas, 17 00:01:10,510 --> 00:01:12,400 and the general ones don't actually 18 00:01:12,400 --> 00:01:14,820 give you a formula for a specific function 19 00:01:14,820 --> 00:01:18,750 but tell you something like, if you take two functions 20 00:01:18,750 --> 00:01:20,840 and add them together, their derivative 21 00:01:20,840 --> 00:01:23,470 is the sum of the derivatives. 22 00:01:23,470 --> 00:01:27,220 Or if you multiply by a constant, for example, 23 00:01:27,220 --> 00:01:31,490 so c times u, the derivative of that 24 00:01:31,490 --> 00:01:38,990 is c times u' where c is constant. 25 00:01:38,990 --> 00:01:41,630 All right, so these kinds of formulas 26 00:01:41,630 --> 00:01:46,070 are very useful, both the specific and the general kind. 27 00:01:46,070 --> 00:02:03,630 For example, we need both kinds for polynomials. 28 00:02:03,630 --> 00:02:05,755 And more generally, pretty much any set of formulas 29 00:02:05,755 --> 00:02:07,713 that we give you, will give you a few functions 30 00:02:07,713 --> 00:02:09,180 to start out with and then you'll 31 00:02:09,180 --> 00:02:16,480 be able to generate lots more by these general formulas. 32 00:02:16,480 --> 00:02:21,000 So today, we wanna concentrate on the trig functions, 33 00:02:21,000 --> 00:02:27,570 and so we'll start out with some specific formulas. 34 00:02:27,570 --> 00:02:29,580 And they're going to be the formulas 35 00:02:29,580 --> 00:02:33,000 for the derivative of the sine function 36 00:02:33,000 --> 00:02:37,910 and the cosine function. 37 00:02:37,910 --> 00:02:41,240 So that's what we'll spend the first part of the lecture on, 38 00:02:41,240 --> 00:02:46,510 and at the same time I hope to get you very used to dealing 39 00:02:46,510 --> 00:02:49,130 with trig functions, although that's something 40 00:02:49,130 --> 00:02:55,630 that you should think of as a gradual process. 41 00:02:55,630 --> 00:02:58,310 Alright, so in order to calculate these, 42 00:02:58,310 --> 00:03:03,720 I'm gonna start over here and just start the calculation. 43 00:03:03,720 --> 00:03:05,270 So here we go. 44 00:03:05,270 --> 00:03:08,110 Let's check what happens with the sine function. 45 00:03:08,110 --> 00:03:15,590 So, I take sin (x + delta x), I subtract sin x 46 00:03:15,590 --> 00:03:22,090 and I divide by delta x. 47 00:03:22,090 --> 00:03:24,270 Right, so this is the difference quotient 48 00:03:24,270 --> 00:03:26,850 and eventually I'm gonna have to take the limit as delta 49 00:03:26,850 --> 00:03:29,320 x goes to 0. 50 00:03:29,320 --> 00:03:31,850 And there's really only one thing 51 00:03:31,850 --> 00:03:36,830 we can do with this to simplify or change it, 52 00:03:36,830 --> 00:03:42,440 and that is to use the sum formula for the sine function. 53 00:03:42,440 --> 00:03:43,560 So, that's this. 54 00:03:43,560 --> 00:03:48,680 That's sin x cos delta x plus-- 55 00:03:54,490 --> 00:03:56,070 Oh, that's not what it is? 56 00:03:56,070 --> 00:04:01,040 OK, so what is it? sin x sin delta x. 57 00:04:01,040 --> 00:04:02,940 OK, good. 58 00:04:02,940 --> 00:04:07,500 Plus cosine. 59 00:04:07,500 --> 00:04:09,290 No? 60 00:04:09,290 --> 00:04:10,560 Oh, OK. 61 00:04:10,560 --> 00:04:14,470 So which is it? 62 00:04:14,470 --> 00:04:16,176 OK. 63 00:04:16,176 --> 00:04:17,300 Alright, let's take a vote. 64 00:04:17,300 --> 00:04:20,020 Is it sine, sine, or is it sine, cosine? 65 00:04:20,020 --> 00:04:22,580 Audience: [INAUDIBLE] 66 00:04:22,580 --> 00:04:31,100 Professor: OK, so is this going to be... cosine. 67 00:04:31,100 --> 00:04:34,450 All right, you better remember these formulas, alright? 68 00:04:34,450 --> 00:04:37,291 OK, turns out that it's sine, cosine. 69 00:04:37,291 --> 00:04:37,790 All right. 70 00:04:37,790 --> 00:04:39,640 Cosine, sine. 71 00:04:39,640 --> 00:04:47,310 So here we go, no gotta do x here, sin (delta x). 72 00:04:47,310 --> 00:04:51,100 Alright, so now there's lots of places to get confused here, 73 00:04:51,100 --> 00:04:55,160 and you're gonna need to make sure you get it right. 74 00:04:55,160 --> 00:04:59,230 Alright, so we're gonna put those in parentheses here. 75 00:04:59,230 --> 00:05:10,310 sin (a + b) is sin a cos b plus cos a sin b. 76 00:05:10,310 --> 00:05:12,740 All right, now that's what I did over here, 77 00:05:12,740 --> 00:05:21,430 except the letter x was a, and the letter b was delta x. 78 00:05:21,430 --> 00:05:23,560 Now that's just the first part. 79 00:05:23,560 --> 00:05:26,980 That's just this part of the expression. 80 00:05:26,980 --> 00:05:29,134 I still have to remember the minus sin x. 81 00:05:29,134 --> 00:05:30,050 That comes at the end. 82 00:05:30,050 --> 00:05:32,120 Minus sin x. 83 00:05:32,120 --> 00:05:37,900 And then, I have to remember the denominator, which is delta x. 84 00:05:37,900 --> 00:05:43,040 OK? 85 00:05:43,040 --> 00:05:47,250 Alright, so now... 86 00:05:47,250 --> 00:05:49,710 The next thing we're gonna do is we're 87 00:05:49,710 --> 00:05:52,280 gonna try to group the terms. 88 00:05:52,280 --> 00:05:58,150 And the difficulty with all such arguments is the following one: 89 00:05:58,150 --> 00:06:02,060 any tricky limit is basically 0 / 0 when 90 00:06:02,060 --> 00:06:03,200 you set delta x equal to 0. 91 00:06:03,200 --> 00:06:06,480 If I set delta x equal to 0, this is sin x - sin x. 92 00:06:06,480 --> 00:06:08,520 So it's a 0 / 0 term. 93 00:06:08,520 --> 00:06:10,040 Here we have various things which 94 00:06:10,040 --> 00:06:12,170 are 0 and various things which are non-zero. 95 00:06:12,170 --> 00:06:17,721 We must group the terms so that a 0 stays over a 0. 96 00:06:17,721 --> 00:06:19,220 Otherwise, we're gonna have no hope. 97 00:06:19,220 --> 00:06:21,990 If we get some 1 / 0 term, we'll get something 98 00:06:21,990 --> 00:06:24,160 meaningless in the limit. 99 00:06:24,160 --> 00:06:27,790 So I claim that the right thing to do here is to notice, 100 00:06:27,790 --> 00:06:31,630 and I'll just point out this one thing. 101 00:06:31,630 --> 00:06:35,580 When delta x goes to 0, this cosine of 0 is 1. 102 00:06:35,580 --> 00:06:39,630 So it doesn't cancel unless we throw in this extra sine term 103 00:06:39,630 --> 00:06:40,130 here. 104 00:06:40,130 --> 00:06:43,620 So I'm going to use this common factor, 105 00:06:43,620 --> 00:06:44,670 and combine those terms. 106 00:06:44,670 --> 00:06:46,480 So this is really the only thing you're 107 00:06:46,480 --> 00:06:48,800 gonna have to check in this particular calculation. 108 00:06:48,800 --> 00:06:50,760 So we have the common factor of sin 109 00:06:50,760 --> 00:06:54,540 x, and that multiplies something that will cancel, 110 00:06:54,540 --> 00:06:59,170 which is (cos delta x - 1) / delta x. 111 00:06:59,170 --> 00:07:03,300 That's the first term, and now what's left, 112 00:07:03,300 --> 00:07:06,470 well there's a cos x that factors out, 113 00:07:06,470 --> 00:07:14,040 and then the other factor is (sin delta x) / (delta x). 114 00:07:14,040 --> 00:07:20,850 OK, now does anyone remember from last time what 115 00:07:20,850 --> 00:07:25,000 this thing goes to? 116 00:07:25,000 --> 00:07:27,580 How many people say 1? 117 00:07:27,580 --> 00:07:29,340 How many people say 0? 118 00:07:29,340 --> 00:07:31,180 All right, it's 0. 119 00:07:31,180 --> 00:07:33,540 That's my favorite number, alright? 120 00:07:33,540 --> 00:07:34,040 0. 121 00:07:34,040 --> 00:07:36,180 It's the easiest number to deal with. 122 00:07:36,180 --> 00:07:39,200 So this goes to 0, and that's what 123 00:07:39,200 --> 00:07:45,980 happens as delta x tends to 0. 124 00:07:45,980 --> 00:07:47,150 How about this one? 125 00:07:47,150 --> 00:07:51,750 This one goes to 1, my second favorite number, almost as 126 00:07:51,750 --> 00:07:54,350 easy to deal with as 0. 127 00:07:54,350 --> 00:07:56,280 And these things are picked for a reason. 128 00:07:56,280 --> 00:07:58,030 They're the simplest numbers to deal with. 129 00:07:58,030 --> 00:08:06,820 So altogether, this thing as delta x goes to 0 goes to what? 130 00:08:06,820 --> 00:08:09,440 I want a single person to answer, a brave volunteer. 131 00:08:09,440 --> 00:08:10,330 Alright, back there. 132 00:08:10,330 --> 00:08:12,070 Student: Cosine 133 00:08:12,070 --> 00:08:14,650 Professor: Cosine, because this factor is 0. 134 00:08:14,650 --> 00:08:17,560 It cancels and this factor has a 1, so it's cosine. 135 00:08:17,560 --> 00:08:20,230 So it's cos x. 136 00:08:20,230 --> 00:08:25,840 So our conclusion over here - and I'll put it in orange - 137 00:08:25,840 --> 00:08:34,920 is that the derivative of the sine is the cosine. 138 00:08:34,920 --> 00:08:38,170 OK, now I still wanna label these very important limit 139 00:08:38,170 --> 00:08:39,220 facts here. 140 00:08:39,220 --> 00:08:41,650 This one we'll call A, and this one we're 141 00:08:41,650 --> 00:08:44,340 going to call B, because we haven't checked them yet. 142 00:08:44,340 --> 00:08:46,340 I promised you I would do that, and I'll 143 00:08:46,340 --> 00:08:48,460 have to do that this time. 144 00:08:48,460 --> 00:08:52,490 So we're relying on those things being true. 145 00:08:52,490 --> 00:08:56,140 Now I'm gonna do the same thing with the cosine function, 146 00:08:56,140 --> 00:08:58,730 except in order to do it I'm gonna have to remember the sum 147 00:08:58,730 --> 00:09:00,930 rule for cosine. 148 00:09:00,930 --> 00:09:03,474 So we're gonna do almost the same calculation here. 149 00:09:03,474 --> 00:09:05,140 We're gonna see that that will work out, 150 00:09:05,140 --> 00:09:12,390 but now you have to remember that cos (a + b) = cos cos, 151 00:09:12,390 --> 00:09:15,350 no it's not cosine^2, because there are two different 152 00:09:15,350 --> 00:09:16,760 quantities here. 153 00:09:16,760 --> 00:09:23,650 It's cos a cos b - sin a sin b. 154 00:09:23,650 --> 00:09:31,280 All right, so you'll have to be willing to call those 155 00:09:31,280 --> 00:09:34,800 forth at will right now. 156 00:09:34,800 --> 00:09:36,460 So let's do the cosine now. 157 00:09:36,460 --> 00:09:45,500 So that's cos (x + delta x) - cos x divided by delta x. 158 00:09:45,500 --> 00:09:47,830 OK, there's the difference quotient 159 00:09:47,830 --> 00:09:49,318 for the cosine function. 160 00:09:49,318 --> 00:09:51,776 And now I'm gonna do the same thing I did before except I'm 161 00:09:51,776 --> 00:09:53,700 going to apply the second rule, that 162 00:09:53,700 --> 00:09:55,770 is the sum rule for cosine. 163 00:09:55,770 --> 00:10:03,401 And that's gonna give me cos x cos delta x - sin x sin delta 164 00:10:03,401 --> 00:10:03,900 x. 165 00:10:03,900 --> 00:10:09,300 And I have to remember again to subtract the cosine divided 166 00:10:09,300 --> 00:10:11,590 by this delta x. 167 00:10:11,590 --> 00:10:16,370 And now I'm going to regroup just the way I did before, 168 00:10:16,370 --> 00:10:22,050 and I get the common factor of cosine multiplying (cos delta x 169 00:10:22,050 --> 00:10:25,180 - 1) / delta x. 170 00:10:25,180 --> 00:10:30,910 And here I get the sin x but actually it's -sin x. 171 00:10:30,910 --> 00:10:36,320 And then I have (sin delta x) / delta x. 172 00:10:36,320 --> 00:10:36,900 All right? 173 00:10:36,900 --> 00:10:38,570 The only difference is this minus sign 174 00:10:38,570 --> 00:10:42,740 which I stuck inside there. 175 00:10:42,740 --> 00:10:44,240 Well that's not the only difference, 176 00:10:44,240 --> 00:10:48,440 but it's a crucial difference. 177 00:10:48,440 --> 00:10:54,700 OK, again by A we get that this is 0 as delta x tends to 0. 178 00:10:54,700 --> 00:10:56,370 And this is 1. 179 00:10:56,370 --> 00:10:59,590 Those are the properties I called A and B. 180 00:10:59,590 --> 00:11:04,840 And so the result here as delta x tends to 0 181 00:11:04,840 --> 00:11:08,470 is that we get negative sin x. 182 00:11:08,470 --> 00:11:11,800 That's the factor. 183 00:11:11,800 --> 00:11:18,880 So this guy is negative sin x. 184 00:11:18,880 --> 00:11:24,560 I'll put a little box around that too. 185 00:11:24,560 --> 00:11:29,040 Alright, now these formulas take a little bit 186 00:11:29,040 --> 00:11:34,220 of getting used to, but before I do 187 00:11:34,220 --> 00:11:38,480 that I'm gonna explain to you the proofs of A and B. 188 00:11:38,480 --> 00:11:44,470 So we'll get ourselves started by mentioning that. 189 00:11:44,470 --> 00:11:47,570 Maybe before I do that though, I want 190 00:11:47,570 --> 00:11:51,710 to show you how A and B fit into the proofs of these theorems. 191 00:11:51,710 --> 00:12:06,030 So, let me just make some remarks here. 192 00:12:06,030 --> 00:12:09,020 So this is just a remark but it's 193 00:12:09,020 --> 00:12:15,600 meant to help you to frame how these proofs worked. 194 00:12:15,600 --> 00:12:17,650 So, first of all, I want to point out 195 00:12:17,650 --> 00:12:19,750 that if you take the rate of change 196 00:12:19,750 --> 00:12:28,450 of sin x, no let's start with cosine 197 00:12:28,450 --> 00:12:30,450 because a little bit less obvious. 198 00:12:30,450 --> 00:12:33,820 If I take the rate of change of cos x, so in other words 199 00:12:33,820 --> 00:12:41,180 this derivative at x = 0, then by definition 200 00:12:41,180 --> 00:12:45,300 this is a certain limit as delta x goes to 0. 201 00:12:45,300 --> 00:12:46,860 So which one is it? 202 00:12:46,860 --> 00:12:51,170 Well I have to evaluate cosine at 0 + delta 203 00:12:51,170 --> 00:12:53,390 x, but that's just delta x. 204 00:12:53,390 --> 00:12:56,240 And I have to subtract cosine at 0. 205 00:12:56,240 --> 00:13:00,040 That's the base point, but that's just 1. 206 00:13:00,040 --> 00:13:03,050 And then I have to divide by delta x. 207 00:13:03,050 --> 00:13:06,850 And lo and behold you can see that this is exactly the limit 208 00:13:06,850 --> 00:13:08,220 that we had over there. 209 00:13:08,220 --> 00:13:15,880 This is the one that we know is 0 by what we call property A. 210 00:13:15,880 --> 00:13:23,150 And similarly, if I take the derivative of sin x at x=0, 211 00:13:23,150 --> 00:13:27,000 then that's going to be the limit as delta x goes to 0 212 00:13:27,000 --> 00:13:30,700 of sin delta x / delta x. 213 00:13:30,700 --> 00:13:35,130 And that's because I should be subtracting sine of 0 214 00:13:35,130 --> 00:13:37,871 but sine of 0 is 0. 215 00:13:37,871 --> 00:13:38,370 Right? 216 00:13:38,370 --> 00:13:48,510 So this is going to be 1 by our property B. And so the remark 217 00:13:48,510 --> 00:13:51,030 that I want to make, in addition to this, 218 00:13:51,030 --> 00:13:55,200 is something about the structure of these two proofs. 219 00:13:55,200 --> 00:14:12,990 Which is the derivatives of sine and cosine at x = 0 220 00:14:12,990 --> 00:14:25,500 give all values of d/dx sin x, d/dx cos x. 221 00:14:25,500 --> 00:14:27,630 So that's really what this argument is showing us, 222 00:14:27,630 --> 00:14:31,420 is that we just need one rate of change at one place 223 00:14:31,420 --> 00:14:38,770 and then we work out all the rest of them. 224 00:14:38,770 --> 00:14:40,990 So that's really the substance of this proof. 225 00:14:40,990 --> 00:14:43,570 That of course really then shows that it boils down 226 00:14:43,570 --> 00:14:48,020 to showing what this rate of change is in these two cases. 227 00:14:48,020 --> 00:14:50,730 So now there's enough suspense that we 228 00:14:50,730 --> 00:15:08,010 want to make sure that we know that those answers are correct. 229 00:15:08,010 --> 00:15:12,180 OK, so let's demonstrate both of them. 230 00:15:12,180 --> 00:15:18,540 I'll start with B. I need to figure out property B. Now, 231 00:15:18,540 --> 00:15:22,470 we only have one alternative as to a type of proof 232 00:15:22,470 --> 00:15:24,710 that we can give of this kind of result, 233 00:15:24,710 --> 00:15:29,070 and that's because we only have one way of describing 234 00:15:29,070 --> 00:15:32,540 sine and cosine functions, that is geometrically. 235 00:15:32,540 --> 00:15:42,620 So we have to give a geometric proof. 236 00:15:42,620 --> 00:15:45,190 And to write down a geometric proof 237 00:15:45,190 --> 00:15:47,150 we are going to have to draw a picture. 238 00:15:47,150 --> 00:15:49,550 And the first step in the proof, really, 239 00:15:49,550 --> 00:15:51,750 is to replace this variable delta 240 00:15:51,750 --> 00:15:55,800 x which is going to 0 with another name which 241 00:15:55,800 --> 00:15:58,370 is suggestive of what we're gonna do which is the letter 242 00:15:58,370 --> 00:16:00,950 theta for an angle. 243 00:16:00,950 --> 00:16:03,560 OK, so let's draw a picture of what 244 00:16:03,560 --> 00:16:05,950 it is that we're going to do. 245 00:16:05,950 --> 00:16:07,980 Here is the circle. 246 00:16:07,980 --> 00:16:10,740 And here is the origin. 247 00:16:10,740 --> 00:16:13,900 And here's some little angle, well I'll 248 00:16:13,900 --> 00:16:16,110 draw it a little larger so it's visible. 249 00:16:16,110 --> 00:16:19,430 Here's theta, alright? 250 00:16:19,430 --> 00:16:21,010 And this is the unit circle. 251 00:16:21,010 --> 00:16:25,910 I won't write that down on here but that's the unit circle. 252 00:16:25,910 --> 00:16:29,780 And now sin theta is this vertical distance here. 253 00:16:29,780 --> 00:16:32,740 Maybe, I'll draw it in a different color 254 00:16:32,740 --> 00:16:34,750 so that we can see it all. 255 00:16:34,750 --> 00:16:38,400 OK so here's this distance. 256 00:16:38,400 --> 00:16:45,820 This distance is sin theta. 257 00:16:45,820 --> 00:16:48,360 OK? 258 00:16:48,360 --> 00:16:51,440 Now almost the only other thing we 259 00:16:51,440 --> 00:16:54,770 have to write down in this picture to have it work out 260 00:16:54,770 --> 00:16:58,500 is that we have to recognize that when theta is the angle, 261 00:16:58,500 --> 00:17:03,279 that's also the arc length of this piece of the circle 262 00:17:03,279 --> 00:17:04,320 when measured in radians. 263 00:17:04,320 --> 00:17:13,696 So this length here is also arc length theta. 264 00:17:13,696 --> 00:17:14,820 That little piece in there. 265 00:17:14,820 --> 00:17:18,580 So maybe I'll use a different color for that to indicate it. 266 00:17:18,580 --> 00:17:25,560 So that's orange and that's this little chunk there. 267 00:17:25,560 --> 00:17:26,870 So those are the two pieces. 268 00:17:26,870 --> 00:17:36,250 Now in order to persuade you now that the limit is 269 00:17:36,250 --> 00:17:37,964 what it's supposed to be, I'm going 270 00:17:37,964 --> 00:17:39,630 to extend the picture just a little bit. 271 00:17:39,630 --> 00:17:42,430 I'm going to double it, just for my own linguistic sake 272 00:17:42,430 --> 00:17:44,240 and so that I can tell you a story. 273 00:17:44,240 --> 00:17:46,690 Alright, so that you'll remember this. 274 00:17:46,690 --> 00:17:49,640 So I'm going to take a theta angle below 275 00:17:49,640 --> 00:17:53,670 and I'll have another copy of sin theta down here. 276 00:17:53,670 --> 00:18:00,790 And now the total picture is really 277 00:18:00,790 --> 00:18:04,721 like a bow and its bow string there. 278 00:18:04,721 --> 00:18:05,220 Alright? 279 00:18:05,220 --> 00:18:11,050 So what we have here is a length of 2 sin theta. 280 00:18:11,050 --> 00:18:13,630 So maybe I'll write it this way, 2 sin theta. 281 00:18:13,630 --> 00:18:15,120 I just doubled it. 282 00:18:15,120 --> 00:18:25,640 And here I have underneath, whoops, I got it backwards. 283 00:18:25,640 --> 00:18:27,040 Sorry about that. 284 00:18:27,040 --> 00:18:29,144 Trying to be fancy with the colored chalk 285 00:18:29,144 --> 00:18:30,310 and I have it reversed here. 286 00:18:30,310 --> 00:18:32,180 So this is not 2 sin theta. 287 00:18:32,180 --> 00:18:33,540 2 sin theta is the vertical. 288 00:18:33,540 --> 00:18:34,910 That's the green. 289 00:18:34,910 --> 00:18:37,170 So let's try that again. 290 00:18:37,170 --> 00:18:41,190 This is 2 sin theta, alright? 291 00:18:41,190 --> 00:18:44,450 And then in the denominator I have the arc length 292 00:18:44,450 --> 00:18:50,680 which is theta is the first half and so double it is 2 theta. 293 00:18:50,680 --> 00:18:51,310 Alright? 294 00:18:51,310 --> 00:18:56,630 So if you like, this is the bow and up here we 295 00:18:56,630 --> 00:19:04,290 have the bow string. 296 00:19:04,290 --> 00:19:07,740 And of course we can cancel the 2's. 297 00:19:07,740 --> 00:19:11,250 That's equal to sin theta / theta. 298 00:19:11,250 --> 00:19:17,900 And so now why does this tend to 1 as theta goes to 0? 299 00:19:17,900 --> 00:19:23,670 Well, it's because as the angle theta gets very small, 300 00:19:23,670 --> 00:19:28,880 this curved piece looks more and more like a straight one. 301 00:19:28,880 --> 00:19:29,640 Alright? 302 00:19:29,640 --> 00:19:32,630 And if you get very, very close here the green segment 303 00:19:32,630 --> 00:19:34,610 and the orange segment would just merge. 304 00:19:34,610 --> 00:19:36,850 They would be practically on top of each other. 305 00:19:36,850 --> 00:19:42,360 And they have closer and closer and closer to the same length. 306 00:19:42,360 --> 00:19:51,790 So that's why this is true. 307 00:19:51,790 --> 00:20:03,200 I guess I'll articulate that by saying that short curves are 308 00:20:03,200 --> 00:20:06,550 nearly straight. 309 00:20:06,550 --> 00:20:10,000 Alright, so that's the principle that we're using. 310 00:20:10,000 --> 00:20:18,970 Or short pieces of curves, if you like, are nearly straight. 311 00:20:18,970 --> 00:20:23,640 So if you like, this is the principle. 312 00:20:23,640 --> 00:20:30,850 So short pieces of curves. 313 00:20:30,850 --> 00:20:31,540 Alright? 314 00:20:31,540 --> 00:20:39,390 So now I also need to give you a proof of A. 315 00:20:39,390 --> 00:20:43,990 And that has to do with this cosine function here. 316 00:20:43,990 --> 00:20:51,905 This is the property A. So I'm going 317 00:20:51,905 --> 00:20:53,487 to do this by flipping it around, 318 00:20:53,487 --> 00:20:56,070 because it turns out that this numerator is a negative number. 319 00:20:56,070 --> 00:20:58,060 If I want to interpret it as a length, 320 00:20:58,060 --> 00:21:00,340 I'm gonna want a positive quantity. 321 00:21:00,340 --> 00:21:04,180 So I'm gonna write down 1 - cos theta here 322 00:21:04,180 --> 00:21:08,320 and then I'm gonna divide by theta there. 323 00:21:08,320 --> 00:21:10,720 Again I'm gonna make some kind of interpretation. 324 00:21:10,720 --> 00:21:15,130 Now this time I'm going to draw the same sort of bow and arrow 325 00:21:15,130 --> 00:21:18,970 arrangement, but maybe I'll exaggerate it a little bit. 326 00:21:18,970 --> 00:21:23,190 So here's the vertex of the sector, 327 00:21:23,190 --> 00:21:31,780 but we'll maybe make it a little longer. 328 00:21:31,780 --> 00:21:35,590 Alright, so here it is, and here was that middle line 329 00:21:35,590 --> 00:21:38,310 which was the unit-- Whoops. 330 00:21:38,310 --> 00:21:40,880 OK, I think I'm going to have to tilt it up. 331 00:21:40,880 --> 00:21:47,000 OK, let's try from here. 332 00:21:47,000 --> 00:21:51,570 Alright, well you know on your pencil and paper 333 00:21:51,570 --> 00:21:53,810 it will look better than it does on my blackboard. 334 00:21:53,810 --> 00:21:55,190 OK, so here we are. 335 00:21:55,190 --> 00:21:56,770 Here's this shape. 336 00:21:56,770 --> 00:22:01,430 Now this angle is supposed to be theta 337 00:22:01,430 --> 00:22:03,350 and this angle is another theta. 338 00:22:03,350 --> 00:22:05,895 So here we have a length which is again theta 339 00:22:05,895 --> 00:22:07,910 and another length which is theta over here. 340 00:22:07,910 --> 00:22:09,670 That's the same as in the other picture, 341 00:22:09,670 --> 00:22:13,290 except we've exaggerated a bit here. 342 00:22:13,290 --> 00:22:15,010 And now we have this vertical line, 343 00:22:15,010 --> 00:22:18,240 which again I'm gonna draw in green, the bow string. 344 00:22:18,240 --> 00:22:24,820 But notice that as the vertex gets farther and farther away, 345 00:22:24,820 --> 00:22:27,019 the curved line gets closer and closer 346 00:22:27,019 --> 00:22:28,060 to being a vertical line. 347 00:22:28,060 --> 00:22:30,670 That's sort of the flip side, by expansion, 348 00:22:30,670 --> 00:22:33,110 of the zoom in principle. 349 00:22:33,110 --> 00:22:35,130 The principle that curves are nearly 350 00:22:35,130 --> 00:22:36,950 straight when you zoom in. 351 00:22:36,950 --> 00:22:39,370 If you zoom out that would mean sending this vertex way, 352 00:22:39,370 --> 00:22:42,200 way out somewhere. 353 00:22:42,200 --> 00:22:44,370 The curved line, the piece of the circle, 354 00:22:44,370 --> 00:22:48,570 gets more and more straight. 355 00:22:48,570 --> 00:22:53,630 And now let me show you where this numerator 1 - cos 356 00:22:53,630 --> 00:22:57,620 theta is on this picture. 357 00:22:57,620 --> 00:23:01,040 So where is it? 358 00:23:01,040 --> 00:23:03,680 Well, this whole distance is 1. 359 00:23:03,680 --> 00:23:07,610 But the distance from the vertex to the green 360 00:23:07,610 --> 00:23:09,870 is cosine of theta. 361 00:23:09,870 --> 00:23:12,430 Right, because this is theta, so dropping down 362 00:23:12,430 --> 00:23:16,830 the perpendicular this distance back to the origin 363 00:23:16,830 --> 00:23:17,630 is cos theta. 364 00:23:17,630 --> 00:23:20,890 So this little tiny, bitty segment 365 00:23:20,890 --> 00:23:26,540 here is basically the gap between the curve 366 00:23:26,540 --> 00:23:28,820 and the vertical segment. 367 00:23:28,820 --> 00:23:35,820 So the gap is equal to 1 - cos theta. 368 00:23:35,820 --> 00:23:40,640 So now you can see that as this point gets farther away, 369 00:23:40,640 --> 00:23:43,670 if this got sent off to the Stata Center, 370 00:23:43,670 --> 00:23:45,840 you would hardly be able to tell the difference. 371 00:23:45,840 --> 00:23:48,120 The bow string would coincide with the bow 372 00:23:48,120 --> 00:23:50,480 and this little gap between the bow string 373 00:23:50,480 --> 00:23:52,430 and the bow would be tending to 0. 374 00:23:52,430 --> 00:23:54,282 And that's the statement that this 375 00:23:54,282 --> 00:23:58,540 tends to 0 as theta tends to 0. 376 00:23:58,540 --> 00:24:00,130 The scaled version of that. 377 00:24:00,130 --> 00:24:01,250 Yeah, question down here. 378 00:24:01,250 --> 00:24:04,800 Student: Doesn't the denominator also tend to 0 though? 379 00:24:04,800 --> 00:24:10,190 Professor: Ah, the question is "doesn't the denominator also 380 00:24:10,190 --> 00:24:11,670 tend to 0?" 381 00:24:11,670 --> 00:24:14,510 And the answer is yes. 382 00:24:14,510 --> 00:24:18,660 In my strange analogy with zooming in, what I did 383 00:24:18,660 --> 00:24:20,390 was I zoomed out the picture. 384 00:24:20,390 --> 00:24:25,450 So in other words, if you imagine you're taking this 385 00:24:25,450 --> 00:24:28,209 and you're putting it under a microscope over here 386 00:24:28,209 --> 00:24:29,750 and you're looking at something where 387 00:24:29,750 --> 00:24:31,910 theta is getting smaller and smaller and smaller 388 00:24:31,910 --> 00:24:33,340 and smaller. 389 00:24:33,340 --> 00:24:34,370 Alright? 390 00:24:34,370 --> 00:24:39,500 But now because I want my picture, I expanded my picture. 391 00:24:39,500 --> 00:24:42,960 So the ratio is the thing that's preserved. 392 00:24:42,960 --> 00:24:50,470 So if I make it so that this gap is tiny... 393 00:24:50,470 --> 00:24:52,410 Let me say this one more time. 394 00:24:52,410 --> 00:24:57,030 I'm afraid I've made life complicated for myself. 395 00:24:57,030 --> 00:25:01,040 If I simply let this theta tend in to 0, 396 00:25:01,040 --> 00:25:04,044 that would be the same effect as making this 397 00:25:04,044 --> 00:25:06,460 closer and closer in and then the vertical would approach. 398 00:25:06,460 --> 00:25:09,040 But I want to keep on blowing up the picture 399 00:25:09,040 --> 00:25:11,940 so that I can see the difference between the vertical 400 00:25:11,940 --> 00:25:13,260 and the curve. 401 00:25:13,260 --> 00:25:16,480 So that's very much like if you are on a video screen 402 00:25:16,480 --> 00:25:18,740 and you zoom in, zoom in, zoom in, and zoom in. 403 00:25:18,740 --> 00:25:20,615 So the question is what would that look like? 404 00:25:20,615 --> 00:25:23,350 That has the same effect as sending this point out 405 00:25:23,350 --> 00:25:27,840 farther and farther in that direction, to the left. 406 00:25:27,840 --> 00:25:29,430 And so I'm just trying to visualize it 407 00:25:29,430 --> 00:25:32,410 for you by leaving the theta at this scale, 408 00:25:32,410 --> 00:25:34,090 but actually the scale of the picture 409 00:25:34,090 --> 00:25:36,290 is then changing when I do that. 410 00:25:36,290 --> 00:25:39,380 So theta is going to 0, but I I'm 411 00:25:39,380 --> 00:25:42,790 rescaling so that it's of a size that we can look at it, 412 00:25:42,790 --> 00:25:46,292 And then imagine what's happening to it. 413 00:25:46,292 --> 00:25:47,750 OK, does that answer your question? 414 00:25:47,750 --> 00:25:50,592 Student: My question then is that seems 415 00:25:50,592 --> 00:25:54,250 to prove that that limit is equal to 0/0. 416 00:25:54,250 --> 00:26:01,510 Professor: It proves more than it is equal to 0/0. 417 00:26:01,510 --> 00:26:04,925 It's the ratio of this little short thing to this longer 418 00:26:04,925 --> 00:26:06,090 thing. 419 00:26:06,090 --> 00:26:08,920 And this is getting much, much shorter than this total length. 420 00:26:08,920 --> 00:26:09,990 You're absolutely right that we're 421 00:26:09,990 --> 00:26:12,573 comparing two quantities which are going to 0, but one of them 422 00:26:12,573 --> 00:26:14,357 is much smaller than the other. 423 00:26:14,357 --> 00:26:16,190 In the other case we compared two quantities 424 00:26:16,190 --> 00:26:18,148 which were both going to 0 and they both end up 425 00:26:18,148 --> 00:26:19,840 being about equal in length. 426 00:26:19,840 --> 00:26:23,060 Here the previous one was this green one. 427 00:26:23,060 --> 00:26:25,000 Here it's this little tiny bit here 428 00:26:25,000 --> 00:26:32,840 and it's way shorter than the 2 theta distance. 429 00:26:32,840 --> 00:26:33,900 Yeah, another question. 430 00:26:33,900 --> 00:26:34,745 Student: cos theta - 1 over cos theta is the same as 1- 431 00:26:34,745 --> 00:26:35,620 cos theta over theta? 432 00:26:35,620 --> 00:26:45,740 Professor: cos theta - 1 over... 433 00:26:45,740 --> 00:26:49,780 Student: [INAUDIBLE] 434 00:26:49,780 --> 00:26:56,690 Professor: So here, what I wrote is (cos delta x - 1) 435 00:26:56,690 --> 00:27:01,620 / delta x, OK, and I claimed that it goes to 0. 436 00:27:01,620 --> 00:27:12,370 Here, I wrote minus that, that is I replaced delta x by theta. 437 00:27:12,370 --> 00:27:22,720 But then I wrote this thing. 438 00:27:22,720 --> 00:27:26,550 So (cos theta - 1) minus 1 is the negative of this. 439 00:27:26,550 --> 00:27:28,070 Alright? 440 00:27:28,070 --> 00:27:29,510 And if I show that this goes to 0, 441 00:27:29,510 --> 00:27:33,162 it's the same as showing the other one goes to 0. 442 00:27:33,162 --> 00:27:33,870 Another question? 443 00:27:33,870 --> 00:27:39,050 Student: [INAUDIBLE] 444 00:27:39,050 --> 00:27:42,640 Professor: So the question is, what about this business 445 00:27:42,640 --> 00:27:44,220 about arc length. 446 00:27:44,220 --> 00:27:48,610 So the word arc length, that orange shape is an arc. 447 00:27:48,610 --> 00:27:51,350 And we're just talking about the length of that arc, 448 00:27:51,350 --> 00:27:52,837 and so we're calling it arc length. 449 00:27:52,837 --> 00:27:54,420 That's what the word arc length means, 450 00:27:54,420 --> 00:27:55,920 it just means the length of the arc. 451 00:27:55,920 --> 00:28:03,620 Student: [INAUDIBLE] 452 00:28:03,620 --> 00:28:06,030 Professor: Why is this length theta? 453 00:28:06,030 --> 00:28:08,380 Ah, OK so this is a very important point, 454 00:28:08,380 --> 00:28:11,480 and in fact it's the very next point that I wanted to make. 455 00:28:11,480 --> 00:28:15,130 Namely, notice that in this calculation 456 00:28:15,130 --> 00:28:19,700 it was very important that we used length. 457 00:28:19,700 --> 00:28:24,140 And that means that the way that we're measuring theta, 458 00:28:24,140 --> 00:28:32,330 is in what are known as radians. 459 00:28:32,330 --> 00:28:36,870 Right, so that applies to both B and A, it's a scale change in A 460 00:28:36,870 --> 00:28:39,470 and doesn't really matter but in B it's very important. 461 00:28:39,470 --> 00:28:42,980 The only way that this orange length 462 00:28:42,980 --> 00:28:46,140 is comparable to this green length, 463 00:28:46,140 --> 00:28:48,600 the vertical is comparable to the arc, 464 00:28:48,600 --> 00:28:53,520 is if we measure them in terms of the same notion of length. 465 00:28:53,520 --> 00:28:56,360 If we measure them in degrees, for example, 466 00:28:56,360 --> 00:28:58,890 it would be completely wrong. 467 00:28:58,890 --> 00:29:02,720 We divide up the angles into 360 degrees, 468 00:29:02,720 --> 00:29:04,320 and that's the wrong unit of measure. 469 00:29:04,320 --> 00:29:07,990 The correct measure is the length along the unit circle, 470 00:29:07,990 --> 00:29:09,590 which is what radians are. 471 00:29:09,590 --> 00:29:21,490 And so this is only true if we use radians. 472 00:29:21,490 --> 00:29:33,650 So again, a little warning here, that this is in radians. 473 00:29:33,650 --> 00:29:41,060 Now here x is in radians. 474 00:29:41,060 --> 00:29:45,690 The formulas are just wrong if you use other units. 475 00:29:45,690 --> 00:29:46,250 Ah yeah? 476 00:29:46,250 --> 00:29:55,480 Student: [INAUDIBLE]. 477 00:29:55,480 --> 00:29:57,290 Professor: OK so the second question is why 478 00:29:57,290 --> 00:30:00,630 is this crazy length here 1. 479 00:30:00,630 --> 00:30:08,000 And the reason is that the relationship 480 00:30:08,000 --> 00:30:11,780 between this picture up here and this picture down here, 481 00:30:11,780 --> 00:30:16,150 is that I'm drawing a different shape. 482 00:30:16,150 --> 00:30:18,800 Namely, what I'm really imagining here 483 00:30:18,800 --> 00:30:21,460 is a much, much smaller theta. 484 00:30:21,460 --> 00:30:22,340 OK? 485 00:30:22,340 --> 00:30:25,430 And then I'm blowing that up in scale. 486 00:30:25,430 --> 00:30:27,880 So this scale of this picture down here 487 00:30:27,880 --> 00:30:31,430 is very different from the scale of the picture up there. 488 00:30:31,430 --> 00:30:34,850 And if the angle is very, very, very small 489 00:30:34,850 --> 00:30:37,835 then one has to be very, very long in order 490 00:30:37,835 --> 00:30:39,430 for me to finish the circle. 491 00:30:39,430 --> 00:30:41,820 So, in other words, this length is 1 492 00:30:41,820 --> 00:30:44,690 because that's what I'm insisting on. 493 00:30:44,690 --> 00:30:47,910 So, I'm claiming that that's how I define this circle, 494 00:30:47,910 --> 00:30:52,490 to be of unit radius. 495 00:30:52,490 --> 00:30:53,200 Another question? 496 00:30:53,200 --> 00:31:05,194 Student: [INAUDIBLE] the ratio between 1 - 497 00:31:05,194 --> 00:31:07,360 cos theta and theta will get closer and closer to 1. 498 00:31:07,360 --> 00:31:08,651 I don't understand [INAUDIBLE]. 499 00:31:08,651 --> 00:31:21,550 Professor: OK, so the question is it's 500 00:31:21,550 --> 00:31:25,810 hard to visualize this fact here. 501 00:31:25,810 --> 00:31:30,829 So let me, let me take you through a couple of steps, 502 00:31:30,829 --> 00:31:33,370 because I think probably other people are also having trouble 503 00:31:33,370 --> 00:31:34,930 with this visualization. 504 00:31:34,930 --> 00:31:36,830 The first part of the visualization I'm 505 00:31:36,830 --> 00:31:39,570 gonna try to demonstrate on this picture up here. 506 00:31:39,570 --> 00:31:41,440 The first part of the visualization 507 00:31:41,440 --> 00:31:45,350 is that I should think of a beak of a bird 508 00:31:45,350 --> 00:31:47,880 closing down, getting narrower and narrower. 509 00:31:47,880 --> 00:31:50,430 So in other words, the angle theta 510 00:31:50,430 --> 00:31:54,050 has to be getting smaller and smaller and smaller. 511 00:31:54,050 --> 00:31:55,780 OK, that's the first step. 512 00:31:55,780 --> 00:31:58,650 So that's the process that we're talking about. 513 00:31:58,650 --> 00:32:03,880 Now, in order to draw that, once theta gets incredibly narrow, 514 00:32:03,880 --> 00:32:06,450 in order to depict that I have to blow the whole picture back 515 00:32:06,450 --> 00:32:07,700 up in order be able to see it. 516 00:32:07,700 --> 00:32:09,430 Otherwise it just disappears on me. 517 00:32:09,430 --> 00:32:12,090 In fact in the limit theta = 0, it's meaningless. 518 00:32:12,090 --> 00:32:13,080 It's just a flat line. 519 00:32:13,080 --> 00:32:15,500 That's the whole problem with these tricky limits. 520 00:32:15,500 --> 00:32:18,100 They're meaningless right at the zero-zero level. 521 00:32:18,100 --> 00:32:22,010 It's only just a little away that they're actually useful, 522 00:32:22,010 --> 00:32:25,890 that you get useful geometric information out of them. 523 00:32:25,890 --> 00:32:27,300 So we're just a little away. 524 00:32:27,300 --> 00:32:31,160 So that's what this picture down below in part A is meant to be. 525 00:32:31,160 --> 00:32:34,135 It's supposed to be that theta is open a tiny crack, just 526 00:32:34,135 --> 00:32:35,161 a little bit. 527 00:32:35,161 --> 00:32:37,285 And the smallest I can draw it on the board for you 528 00:32:37,285 --> 00:32:40,160 to visualize it is using the whole length of the blackboard 529 00:32:40,160 --> 00:32:41,390 here for that. 530 00:32:41,390 --> 00:32:43,390 So I've opened a little tiny bit and by the time 531 00:32:43,390 --> 00:32:45,139 we get to the other end of the blackboard, 532 00:32:45,139 --> 00:32:46,510 of course it's fairly wide. 533 00:32:46,510 --> 00:32:50,291 But this angle theta is a very small angle. 534 00:32:50,291 --> 00:32:50,790 Alright? 535 00:32:50,790 --> 00:32:56,670 So I'm trying to imagine what happens as this collapses. 536 00:32:56,670 --> 00:32:59,670 Now, when I imagine that I have to imagine 537 00:32:59,670 --> 00:33:02,190 a geometric interpretation of both the numerator 538 00:33:02,190 --> 00:33:06,020 and the denominator of this quantity here. 539 00:33:06,020 --> 00:33:08,300 And just see what happens. 540 00:33:08,300 --> 00:33:14,020 Now I claimed the numerator is this little tiny bit over here 541 00:33:14,020 --> 00:33:19,004 and the denominator is actually half of this whole length here. 542 00:33:19,004 --> 00:33:20,420 But the factor of 2 doesn't matter 543 00:33:20,420 --> 00:33:24,250 when you're seeing whether something tends to 0 or not. 544 00:33:24,250 --> 00:33:24,990 Alright? 545 00:33:24,990 --> 00:33:26,656 And I claimed that if you stare at this, 546 00:33:26,656 --> 00:33:28,460 it's clear that this is much shorter 547 00:33:28,460 --> 00:33:32,750 than that vertical curve there. 548 00:33:32,750 --> 00:33:35,140 And I'm claiming, so this is what you have to imagine, 549 00:33:35,140 --> 00:33:38,820 is this as it gets smaller and smaller and smaller still 550 00:33:38,820 --> 00:33:41,590 that has the same effect of this thing going way, way way, 551 00:33:41,590 --> 00:33:45,510 farther away and this vertical curve getting closer and closer 552 00:33:45,510 --> 00:33:47,160 and closer to the green. 553 00:33:47,160 --> 00:33:52,530 And so that the gap between them gets tiny and goes to 0. 554 00:33:52,530 --> 00:33:53,440 Alright? 555 00:33:53,440 --> 00:33:56,580 So not only does it go to 0, that's not enough for us, 556 00:33:56,580 --> 00:34:01,540 but it also goes to 0 faster than this theta goes to 0. 557 00:34:01,540 --> 00:34:05,280 And I hope the evidence is pretty strong here 558 00:34:05,280 --> 00:34:10,220 because it's so tiny already at this stage. 559 00:34:10,220 --> 00:34:12,350 Alright. 560 00:34:12,350 --> 00:34:16,140 We are going to move forward and you'll 561 00:34:16,140 --> 00:34:18,420 have to ponder these things some other time. 562 00:34:18,420 --> 00:34:20,640 So I'm gonna give you an even harder thing 563 00:34:20,640 --> 00:34:26,600 to visualize now so be prepared. 564 00:34:26,600 --> 00:34:36,310 OK, so now, the next thing that I'd like to do 565 00:34:36,310 --> 00:34:37,700 is to give you a second proof. 566 00:34:37,700 --> 00:34:39,430 Because it really is important, I 567 00:34:39,430 --> 00:34:47,650 think, to understand this particular fact more thoroughly 568 00:34:47,650 --> 00:34:49,310 and also to get a lot of practice 569 00:34:49,310 --> 00:34:51,450 with sines and cosines. 570 00:34:51,450 --> 00:34:55,930 So I'm gonna give you a geometric proof 571 00:34:55,930 --> 00:35:11,010 of the formula for sine here, for the derivative of sine. 572 00:35:11,010 --> 00:35:13,420 So here we go. 573 00:35:13,420 --> 00:35:26,280 This is a geometric proof of this fact. 574 00:35:26,280 --> 00:35:29,400 This is for all theta. 575 00:35:29,400 --> 00:35:33,100 So far we only did it for theta = 0 576 00:35:33,100 --> 00:35:36,360 and now we're going to do it for all theta. 577 00:35:36,360 --> 00:35:38,930 So this is a different proof, but it uses 578 00:35:38,930 --> 00:35:42,420 exactly the same principles. 579 00:35:42,420 --> 00:35:45,390 Right? 580 00:35:45,390 --> 00:35:51,350 So, I want do this by drawing another picture, 581 00:35:51,350 --> 00:35:54,790 and the picture is going to describe 582 00:35:54,790 --> 00:35:59,370 y, which is sin theta, which is if you 583 00:35:59,370 --> 00:36:22,160 like the vertical position of some circular motion. 584 00:36:22,160 --> 00:36:27,170 So I'm imagining that something is going around in a circle. 585 00:36:27,170 --> 00:36:30,620 Some particle is going around in a circle. 586 00:36:30,620 --> 00:36:36,377 And so here's the circle, here the origin. 587 00:36:36,377 --> 00:36:37,460 This is the unit distance. 588 00:36:37,460 --> 00:36:43,640 And right now it happens to be at this location P. Maybe 589 00:36:43,640 --> 00:36:46,160 we'll put P a little over here. 590 00:36:46,160 --> 00:36:50,260 And here's the angle theta. 591 00:36:50,260 --> 00:36:51,560 And now we're going to move it. 592 00:36:51,560 --> 00:36:54,490 We're going to vary theta and we're 593 00:36:54,490 --> 00:36:56,920 interested in the rate of change of y. 594 00:36:56,920 --> 00:37:00,290 So y is the height of P but we're 595 00:37:00,290 --> 00:37:01,870 gonna move it to another location. 596 00:37:01,870 --> 00:37:07,400 We'll move it along the circle to Q. Right? 597 00:37:07,400 --> 00:37:09,200 So here it is. 598 00:37:09,200 --> 00:37:12,360 Here's the thing. 599 00:37:12,360 --> 00:37:14,450 So how far did we move it? 600 00:37:14,450 --> 00:37:18,570 Well we moved it by an angle delta theta. 601 00:37:18,570 --> 00:37:20,540 So we started theta, theta is going 602 00:37:20,540 --> 00:37:22,313 to be fixed in this argument, and we're 603 00:37:22,313 --> 00:37:23,990 going to move a little bit delta theta. 604 00:37:23,990 --> 00:37:26,180 And now we're just gonna try to figure out 605 00:37:26,180 --> 00:37:28,510 how far the thing moved. 606 00:37:28,510 --> 00:37:30,440 Well, in order to do that we've got 607 00:37:30,440 --> 00:37:34,590 to keep track of the height, the vertical displacement here. 608 00:37:34,590 --> 00:37:38,334 So we're going to draw this right angle here, this 609 00:37:38,334 --> 00:37:42,440 is the position R. And then this distance here 610 00:37:42,440 --> 00:37:45,500 is the change in y. 611 00:37:45,500 --> 00:37:46,000 Alright? 612 00:37:46,000 --> 00:37:50,190 So the picture is we have something 613 00:37:50,190 --> 00:37:52,110 moving around a unit circle. 614 00:37:52,110 --> 00:37:53,680 A point moving around a unit circle. 615 00:37:53,680 --> 00:37:57,635 It starts at P, it moves to Q. It moves from angle theta 616 00:37:57,635 --> 00:37:59,590 to angle theta plus delta theta. 617 00:37:59,590 --> 00:38:05,770 And the issue is how much does y move? 618 00:38:05,770 --> 00:38:07,340 And the formula for y is sin theta. 619 00:38:07,340 --> 00:38:29,710 So that's telling us the rate of change of sin theta. 620 00:38:29,710 --> 00:38:34,500 Alright, well so let's just try to think a little bit 621 00:38:34,500 --> 00:38:35,980 about what this is. 622 00:38:35,980 --> 00:38:37,736 So, first of all, I've already said this 623 00:38:37,736 --> 00:38:39,300 and I'm going to repeat it here. 624 00:38:39,300 --> 00:38:41,650 Delta y is PR. 625 00:38:41,650 --> 00:38:44,490 It's going from P and going straight up to R. 626 00:38:44,490 --> 00:38:47,080 That's how far y moves. 627 00:38:47,080 --> 00:38:49,640 That's the change in y That's what I said up 628 00:38:49,640 --> 00:38:52,910 in the right hand corner there. 629 00:38:52,910 --> 00:38:53,470 Oops. 630 00:38:53,470 --> 00:38:56,430 I said PR but I wrote PQ. 631 00:38:56,430 --> 00:38:59,130 Alright, that's not a good idea. 632 00:38:59,130 --> 00:38:59,630 Alright. 633 00:38:59,630 --> 00:39:03,090 So delta Y is PR. 634 00:39:03,090 --> 00:39:07,160 And now I want to draw the diagram again one time. 635 00:39:07,160 --> 00:39:16,030 So here's Q, here's R, and here's P, and here's 636 00:39:16,030 --> 00:39:17,300 my triangle. 637 00:39:17,300 --> 00:39:24,300 And now what I'd like to do is draw this curve here 638 00:39:24,300 --> 00:39:26,970 which is a piece of the arc of the circle. 639 00:39:26,970 --> 00:39:29,560 But really what I want to keep in mind 640 00:39:29,560 --> 00:39:33,300 is something that I did also in all these other arguments. 641 00:39:33,300 --> 00:39:35,330 Which is, maybe I should have called 642 00:39:35,330 --> 00:39:38,011 this orange, that I'm gonna think of the straight line 643 00:39:38,011 --> 00:39:38,510 between. 644 00:39:38,510 --> 00:39:41,165 So it's the straight line approximation to the curve 645 00:39:41,165 --> 00:39:45,009 that we're always interested in. 646 00:39:45,009 --> 00:39:46,550 So the straight line is much simpler, 647 00:39:46,550 --> 00:39:48,610 because then we just have a triangle here. 648 00:39:48,610 --> 00:39:52,017 And in fact it's a right triangle. 649 00:39:52,017 --> 00:39:54,350 Right, so we have the geometry of a right triangle which 650 00:39:54,350 --> 00:39:59,210 is going to now let us do all of our calculations. 651 00:39:59,210 --> 00:40:03,640 OK, so now the key step is this same principle 652 00:40:03,640 --> 00:40:07,560 that we already used which is that short pieces of curves 653 00:40:07,560 --> 00:40:09,040 are nearly straight. 654 00:40:09,040 --> 00:40:12,000 So that means that this piece of the circular arc here from P 655 00:40:12,000 --> 00:40:17,323 to Q is practically the same as the straight segment from P 656 00:40:17,323 --> 00:40:24,290 to Q. So, that's this principle that - well, 657 00:40:24,290 --> 00:40:27,580 let's put it over here - Is that PQ 658 00:40:27,580 --> 00:40:33,190 is practically the same as the straight segment from P to Q. 659 00:40:33,190 --> 00:40:35,820 So how are we going to use that? 660 00:40:35,820 --> 00:40:37,750 We want to use that quantitatively 661 00:40:37,750 --> 00:40:39,080 in the following way. 662 00:40:39,080 --> 00:40:42,350 What we want to notice is that the distance from P to Q 663 00:40:42,350 --> 00:40:46,120 is approximately delta theta. 664 00:40:46,120 --> 00:40:46,620 Right? 665 00:40:46,620 --> 00:40:49,056 Because the arc length along that curve, 666 00:40:49,056 --> 00:40:50,680 the length of the curve is delta theta. 667 00:40:50,680 --> 00:40:55,050 So the length of the green which is PQ is almost delta theta. 668 00:40:55,050 --> 00:41:01,690 So this is essentially delta theta, this distance here. 669 00:41:01,690 --> 00:41:05,360 Now the second step, which is a little trickier, 670 00:41:05,360 --> 00:41:08,980 is that we have to work out what this angle is. 671 00:41:08,980 --> 00:41:11,640 So our goal, and I'm gonna put it one step below because I'm 672 00:41:11,640 --> 00:41:14,280 gonna put the geometric reasoning in between, 673 00:41:14,280 --> 00:41:20,980 is I need to figure out what the angle QPR is. 674 00:41:20,980 --> 00:41:23,340 If I can figure out what this angle is, 675 00:41:23,340 --> 00:41:26,630 then I'll be able to figure out what this vertical distance is 676 00:41:26,630 --> 00:41:28,840 because I'll know the hypotenuse and I'll 677 00:41:28,840 --> 00:41:30,900 know the angle so I'll be able to figure out what 678 00:41:30,900 --> 00:41:36,610 the side of the triangle is. 679 00:41:36,610 --> 00:41:40,220 So now let me show you why that's possible to do. 680 00:41:40,220 --> 00:41:43,400 So in order to do that first of all I'm gonna trade the boards 681 00:41:43,400 --> 00:41:50,600 and show you where the line PQ is. 682 00:41:50,600 --> 00:41:54,370 So the line PQ is here. 683 00:41:54,370 --> 00:41:56,470 That's the whole thing. 684 00:41:56,470 --> 00:42:00,190 And the key point about this line that I need you to realize 685 00:42:00,190 --> 00:42:04,230 is that it's practically perpendicular, 686 00:42:04,230 --> 00:42:08,910 it's almost perpendicular, to this ray here. 687 00:42:08,910 --> 00:42:09,550 Alright? 688 00:42:09,550 --> 00:42:12,540 It's not quite because the distance between P to Q 689 00:42:12,540 --> 00:42:13,262 is non-zero. 690 00:42:13,262 --> 00:42:14,720 So it isn't quite, but in the limit 691 00:42:14,720 --> 00:42:17,070 it's going to be perpendicular. 692 00:42:17,070 --> 00:42:18,100 Exactly perpendicular. 693 00:42:18,100 --> 00:42:20,980 The tangent line to the circle. 694 00:42:20,980 --> 00:42:26,820 So the key thing that I'm going to use 695 00:42:26,820 --> 00:42:35,131 is that PQ is almost perpendicular to OP. 696 00:42:35,131 --> 00:42:35,630 Alright? 697 00:42:35,630 --> 00:42:37,710 The ray from the origin is basically 698 00:42:37,710 --> 00:42:39,900 perpendicular to that green line. 699 00:42:39,900 --> 00:42:42,540 And then the second thing I'm going to use 700 00:42:42,540 --> 00:42:53,121 is something that's obvious which is that PR is vertical. 701 00:42:53,121 --> 00:42:53,620 OK? 702 00:42:53,620 --> 00:42:58,080 So those are the two pieces of geometry that I need to see. 703 00:42:58,080 --> 00:43:02,680 And now notice what's happening upstairs on the picture here 704 00:43:02,680 --> 00:43:05,050 in the upper right. 705 00:43:05,050 --> 00:43:09,550 What I have is the angle theta is the angle 706 00:43:09,550 --> 00:43:12,910 between the horizontal and OP. 707 00:43:12,910 --> 00:43:14,350 That's angle theta. 708 00:43:14,350 --> 00:43:17,990 If I rotate it by ninety degree, the horizontal 709 00:43:17,990 --> 00:43:18,880 becomes vertical. 710 00:43:18,880 --> 00:43:21,730 It becomes PR and the other thing 711 00:43:21,730 --> 00:43:24,810 rotated by 90 degrees becomes the green line. 712 00:43:24,810 --> 00:43:30,080 So the angle that I'm talking about I get by taking this guy 713 00:43:30,080 --> 00:43:32,470 and rotating it by 90 degrees. 714 00:43:32,470 --> 00:43:33,800 It's the same angle. 715 00:43:33,800 --> 00:43:38,230 So that means that this angle here is essentially theta. 716 00:43:38,230 --> 00:43:39,880 That's what this angle is. 717 00:43:39,880 --> 00:43:41,840 Let me repeat that one more time. 718 00:43:41,840 --> 00:43:43,290 We started out with an angle that 719 00:43:43,290 --> 00:43:46,540 looks like this, which is the horizontal-- that's 720 00:43:46,540 --> 00:43:48,600 the origin straight out horizontally. 721 00:43:48,600 --> 00:43:50,560 That's the thing labeled 1. 722 00:43:50,560 --> 00:43:54,805 That distance there. 723 00:43:54,805 --> 00:43:56,430 That's my right arm which is down here. 724 00:43:56,430 --> 00:44:01,180 My left arm is pointing up and it's going from the origin 725 00:44:01,180 --> 00:44:06,430 to the point P. So here's the horizontal 726 00:44:06,430 --> 00:44:09,370 and the angle between them is theta. 727 00:44:09,370 --> 00:44:13,930 And now, what I claim is that if I rotate by 90 degrees up, 728 00:44:13,930 --> 00:44:16,840 like this, without changing anything - 729 00:44:16,840 --> 00:44:18,860 so that was what I did - the horizontal 730 00:44:18,860 --> 00:44:21,160 will become a vertical. 731 00:44:21,160 --> 00:44:22,990 That's PR. 732 00:44:22,990 --> 00:44:25,030 That's going up, PR. 733 00:44:25,030 --> 00:44:32,080 And if I rotate OP 90 degrees, that's exactly PQ. 734 00:44:32,080 --> 00:44:33,540 OK? 735 00:44:33,540 --> 00:44:42,560 So let me draw it on there one time. 736 00:44:42,560 --> 00:44:45,220 Let's do it with some arrows here. 737 00:44:45,220 --> 00:44:49,620 So I started out with this and then, we'll 738 00:44:49,620 --> 00:44:56,470 label this as orange, OK so red to orange, 739 00:44:56,470 --> 00:45:01,840 and then I rotate by 90 degrees and the red 740 00:45:01,840 --> 00:45:07,060 becomes this starting from P and the orange rotates around 90 741 00:45:07,060 --> 00:45:11,370 degrees and becomes this thing here. 742 00:45:11,370 --> 00:45:12,190 Alright? 743 00:45:12,190 --> 00:45:16,252 So this angle here is the same as the other one 744 00:45:16,252 --> 00:45:18,460 which I've just drawn. 745 00:45:18,460 --> 00:45:27,030 Different vertices for the angles. 746 00:45:27,030 --> 00:45:28,210 OK? 747 00:45:28,210 --> 00:45:31,100 Well I didn't say that all arguments 748 00:45:31,100 --> 00:45:36,450 were supposed to be easy. 749 00:45:36,450 --> 00:45:38,180 Alright, so I claim that the conclusion 750 00:45:38,180 --> 00:45:43,200 is that this angle is approximately theta. 751 00:45:43,200 --> 00:45:45,670 And now we can finish our calculation, 752 00:45:45,670 --> 00:45:48,300 because we have something with the hypotenuse being delta 753 00:45:48,300 --> 00:45:53,340 theta and the angle being theta and so this segment here PR is 754 00:45:53,340 --> 00:45:56,830 approximately the hypotenuse length 755 00:45:56,830 --> 00:46:02,430 times the cosine of the angle. 756 00:46:02,430 --> 00:46:05,740 And that is exactly what we wanted. 757 00:46:05,740 --> 00:46:10,830 If we divide, we divide by delta theta, we get (delta y) 758 00:46:10,830 --> 00:46:17,030 / (delta theta) is approximately cos theta. 759 00:46:17,030 --> 00:46:20,700 And that's the same thing as... 760 00:46:20,700 --> 00:46:22,370 So what we want in the limit is exactly 761 00:46:22,370 --> 00:46:24,900 the delta theta going to 0 of (delta y) / (delta theta) 762 00:46:24,900 --> 00:46:28,020 is equal to cos theta. 763 00:46:28,020 --> 00:46:32,270 So we get an approximation on a scale that we can visualize 764 00:46:32,270 --> 00:46:39,590 and in the limit the formula is exact. 765 00:46:39,590 --> 00:46:44,060 OK, so that is a geometric argument for the same result. 766 00:46:44,060 --> 00:46:47,940 Namely that the derivative of sine is cosine. 767 00:46:47,940 --> 00:46:48,440 Yeah? 768 00:46:48,440 --> 00:46:51,590 Student: [INAUDIBLE]. 769 00:46:51,590 --> 00:46:54,420 Professor: You will have to do some kind of geometric proofs 770 00:46:54,420 --> 00:46:55,840 sometimes. 771 00:46:55,840 --> 00:46:59,730 When you'll really need this is probably in 18.02. 772 00:46:59,730 --> 00:47:03,020 So you'll need to make reasoning like this. 773 00:47:03,020 --> 00:47:05,730 This is, for example, the way that you actually develop 774 00:47:05,730 --> 00:47:08,200 the theory of arc length. 775 00:47:08,200 --> 00:47:13,250 Dealing with delta x's and delta y's is a common tool. 776 00:47:13,250 --> 00:47:17,810 Alright, I have one more thing that I 777 00:47:17,810 --> 00:47:25,070 want to talk about today, which is some general rules. 778 00:47:25,070 --> 00:47:28,230 We took a little bit more time than I expected with this. 779 00:47:28,230 --> 00:47:31,780 So what I'm gonna do is just tell you the rules 780 00:47:31,780 --> 00:47:36,330 and we'll discuss them in a few days. 781 00:47:36,330 --> 00:47:50,180 So let me tell you the general rules. 782 00:47:50,180 --> 00:48:00,170 So these were the specific ones and here are some general ones. 783 00:48:00,170 --> 00:48:08,490 So the first one is called the product rule. 784 00:48:08,490 --> 00:48:11,010 And what it says is that if you take the product of two 785 00:48:11,010 --> 00:48:13,630 functions and differentiate them, 786 00:48:13,630 --> 00:48:18,180 you get the derivative of one times the other plus 787 00:48:18,180 --> 00:48:22,060 the other times the derivative of the one. 788 00:48:22,060 --> 00:48:24,350 Now the way that you should remember this, 789 00:48:24,350 --> 00:48:27,970 and the way that I'll carry out the proof, 790 00:48:27,970 --> 00:48:40,010 is that you should think of it is you change one at a time. 791 00:48:40,010 --> 00:48:44,080 And this is a very useful way of thinking about differentiation 792 00:48:44,080 --> 00:48:46,530 when you have things which depend 793 00:48:46,530 --> 00:48:49,660 on more than one function. 794 00:48:49,660 --> 00:48:53,750 So this is a general procedure. 795 00:48:53,750 --> 00:48:59,050 The second formula that I wanted to mention 796 00:48:59,050 --> 00:49:07,470 is called the quotient rule and that says the following. 797 00:49:07,470 --> 00:49:13,220 That (u / v) prime has a formula as well. 798 00:49:13,220 --> 00:49:21,280 And the formula is (u'v - uv' ) / v^2. 799 00:49:21,280 --> 00:49:23,240 So this is our second formula. 800 00:49:23,240 --> 00:49:31,360 Let me just mention, both of them are extremely valuable 801 00:49:31,360 --> 00:49:33,170 and you'll use them all the time. 802 00:49:33,170 --> 00:49:43,600 This one of course only works when v is not 0. 803 00:49:43,600 --> 00:49:46,926 Alright, so because we're out of time 804 00:49:46,926 --> 00:49:48,300 we're not gonna prove these today 805 00:49:48,300 --> 00:49:50,508 but we'll prove these next time and you're definitely 806 00:49:50,508 --> 00:49:53,760 going to be responsible for these kinds of proofs.