1 00:00:00,000 --> 00:00:03,950 The following content is provided under a Creative 2 00:00:03,950 --> 00:00:05,000 Commons license. 3 00:00:05,000 --> 00:00:06,708 Your support will help MIT OpenCourseWare 4 00:00:06,708 --> 00:00:10,010 continue to offer high quality educational resources for free. 5 00:00:10,010 --> 00:00:13,600 To make a donation, or to view additional materials 6 00:00:13,600 --> 00:00:15,920 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:15,920 --> 00:00:21,576 at ocw.mit.edu. 8 00:00:21,576 --> 00:00:22,076 PROF. 9 00:00:22,076 --> 00:00:26,950 JERISON: We're starting a new unit today. 10 00:00:26,950 --> 00:00:39,050 And, so this is Unit 2, and it's called Applications 11 00:00:39,050 --> 00:00:48,810 of Differentiation. 12 00:00:48,810 --> 00:00:51,200 OK. 13 00:00:51,200 --> 00:00:57,400 So, the first application, and we're going to do two today, 14 00:00:57,400 --> 00:01:04,030 is what are known as linear approximations. 15 00:01:04,030 --> 00:01:06,310 Whoops, that should have two p's in it. 16 00:01:06,310 --> 00:01:12,460 Approximations. 17 00:01:12,460 --> 00:01:16,360 So, that can be summarized with one formula, 18 00:01:16,360 --> 00:01:19,040 but it's going to take us at least half an hour 19 00:01:19,040 --> 00:01:21,960 to explain how this formula is used. 20 00:01:21,960 --> 00:01:24,100 So here's the formula. 21 00:01:24,100 --> 00:01:34,390 It's f(x) is approximately equal to its value at a base point 22 00:01:34,390 --> 00:01:38,260 plus the derivative times x - x_0. 23 00:01:38,260 --> 00:01:38,760 Right? 24 00:01:38,760 --> 00:01:42,720 So this is the main formula. 25 00:01:42,720 --> 00:01:44,310 For right now. 26 00:01:44,310 --> 00:01:52,140 Put it in a box. 27 00:01:52,140 --> 00:01:57,430 And let me just describe what it means, first. 28 00:01:57,430 --> 00:01:59,830 And then I'll describe what it means again, 29 00:01:59,830 --> 00:02:01,780 and several other times. 30 00:02:01,780 --> 00:02:04,690 So, first of all, what it means is 31 00:02:04,690 --> 00:02:11,140 that if you have a curve, which is y = f(x), 32 00:02:11,140 --> 00:02:18,860 it's approximately the same as its tangent line. 33 00:02:18,860 --> 00:02:37,020 So this other side is the equation of the tangent line. 34 00:02:37,020 --> 00:02:43,090 So let's give an example. 35 00:02:43,090 --> 00:02:50,190 I'm going to take the function f(x), which is ln x, 36 00:02:50,190 --> 00:02:53,980 and then its derivative is 1/x. 37 00:02:58,990 --> 00:03:03,800 And, so let's take the base point x_0 = 1. 38 00:03:03,800 --> 00:03:05,470 That's pretty much the only place where 39 00:03:05,470 --> 00:03:08,900 we know the logarithm for sure. 40 00:03:08,900 --> 00:03:13,360 And so, what we plug in here now, are the values. 41 00:03:13,360 --> 00:03:17,890 So f(1) is the log of 0. 42 00:03:17,890 --> 00:03:20,750 Or, sorry, the log of 1, which is 0. 43 00:03:20,750 --> 00:03:28,130 And f'(1), well, that's 1/1, which is 1. 44 00:03:28,130 --> 00:03:31,100 So now we have an approximation formula which, 45 00:03:31,100 --> 00:03:34,020 if I copy down what's right up here, 46 00:03:34,020 --> 00:03:40,560 it's going to be ln x is approximately, so f(0) 47 00:03:40,560 --> 00:03:44,480 is 0, right? 48 00:03:44,480 --> 00:03:49,700 Plus 1 times (x - 1). 49 00:03:49,700 --> 00:03:52,910 So I plugged in here, for x_0, three places. 50 00:03:52,910 --> 00:04:00,670 I evaluated the coefficients and this is the dependent variable. 51 00:04:00,670 --> 00:04:03,720 So, all told, if you like, what I have here 52 00:04:03,720 --> 00:04:11,100 is that the logarithm of x is approximately x - 1. 53 00:04:11,100 --> 00:04:16,660 And let me draw a picture of this. 54 00:04:16,660 --> 00:04:22,310 So here's the graph of ln x. 55 00:04:22,310 --> 00:04:26,920 And then, I'll draw in the tangent line at the place 56 00:04:26,920 --> 00:04:30,350 that we're considering, which is x = 1. 57 00:04:30,350 --> 00:04:33,030 So here's the tangent line. 58 00:04:33,030 --> 00:04:35,195 And I've separated a little bit, but really 59 00:04:35,195 --> 00:04:37,551 I probably should have drawn it a little closer there, 60 00:04:37,551 --> 00:04:38,050 to show you. 61 00:04:38,050 --> 00:04:42,870 The whole point is that these two are nearby. 62 00:04:42,870 --> 00:04:44,400 But they're not nearby everywhere. 63 00:04:44,400 --> 00:04:50,130 So this is the line y = x - 1. 64 00:04:50,130 --> 00:04:51,960 Right, that's the tangent line. 65 00:04:51,960 --> 00:04:55,240 They're nearby only when x is near 1. 66 00:04:55,240 --> 00:04:58,010 So say in this little realm here. 67 00:04:58,010 --> 00:05:05,050 So when x is approximately 1, this is true. 68 00:05:05,050 --> 00:05:06,580 Once you get a little farther away, 69 00:05:06,580 --> 00:05:08,413 this straight line, this straight green line 70 00:05:08,413 --> 00:05:10,540 will separate from the graph. 71 00:05:10,540 --> 00:05:14,610 But near this place they're close together. 72 00:05:14,610 --> 00:05:17,920 So the idea, again, is that the curve, the curved line, 73 00:05:17,920 --> 00:05:19,770 is approximately the tangent line. 74 00:05:19,770 --> 00:05:25,350 And this is one example of it. 75 00:05:25,350 --> 00:05:29,850 All right, so I want to explain this in one more way. 76 00:05:29,850 --> 00:05:32,690 And then we want to discuss it systematically. 77 00:05:32,690 --> 00:05:37,090 So the second way that I want to describe this 78 00:05:37,090 --> 00:05:39,310 requires me to remind you what the definition 79 00:05:39,310 --> 00:05:41,290 of the derivative is. 80 00:05:41,290 --> 00:05:46,370 So, the definition of a derivative 81 00:05:46,370 --> 00:05:53,360 is that it's the limit, as delta x goes to 0, of delta f / delta 82 00:05:53,360 --> 00:05:56,410 x, that's one way of writing it, all right? 83 00:05:56,410 --> 00:06:01,260 And this is the way we defined it. 84 00:06:01,260 --> 00:06:03,860 And one of the things that we did in the first unit 85 00:06:03,860 --> 00:06:09,070 was we looked at this backwards. 86 00:06:09,070 --> 00:06:12,890 We used the derivative knowing the derivatives of functions 87 00:06:12,890 --> 00:06:14,250 to evaluate some limits. 88 00:06:14,250 --> 00:06:17,860 So you were supposed to do that on your. 89 00:06:17,860 --> 00:06:21,200 In our test, there were some examples there, 90 00:06:21,200 --> 00:06:23,200 at least one example, where that was the easiest 91 00:06:23,200 --> 00:06:26,140 way to do the problem. 92 00:06:26,140 --> 00:06:28,850 So in other words, you can read this equation both ways. 93 00:06:28,850 --> 00:06:31,770 This is really, of course, the same equation written twice. 94 00:06:31,770 --> 00:06:34,970 Now, what's new about what we're going to do now 95 00:06:34,970 --> 00:06:40,150 is that we're going to take this expression here, delta f 96 00:06:40,150 --> 00:06:42,910 / delta x, and we're going to say 97 00:06:42,910 --> 00:06:45,810 well, when delta x is fairly near 0, 98 00:06:45,810 --> 00:06:47,360 this expression is going to be fairly 99 00:06:47,360 --> 00:06:49,450 close to the limiting value. 100 00:06:49,450 --> 00:06:53,760 So this is approximately f'(x_0). 101 00:06:53,760 --> 00:07:00,730 So that, I claim, is the same as what's in the box in pink 102 00:07:00,730 --> 00:07:02,810 that I have over here. 103 00:07:02,810 --> 00:07:10,840 So this approximation formula here is the same as this one. 104 00:07:10,840 --> 00:07:13,760 This is an average rate of change, 105 00:07:13,760 --> 00:07:16,420 and this is an infinitesimal rate of change. 106 00:07:16,420 --> 00:07:17,980 And they're nearly the same. 107 00:07:17,980 --> 00:07:19,230 That's the claim. 108 00:07:19,230 --> 00:07:22,950 So you'll have various exercises in which this approximation is 109 00:07:22,950 --> 00:07:25,220 the useful one to use. 110 00:07:25,220 --> 00:07:28,860 And I will, as I said, I'll be illustrating this a little bit 111 00:07:28,860 --> 00:07:29,610 today. 112 00:07:29,610 --> 00:07:34,060 Now, let me just explain why those two formulas in the boxes 113 00:07:34,060 --> 00:07:36,560 are the same. 114 00:07:36,560 --> 00:07:41,110 So let's just start over here and explain that. 115 00:07:41,110 --> 00:07:47,150 So the smaller box is the same thing if I multiply through 116 00:07:47,150 --> 00:07:55,490 by delta x, as delta f is approximately f'(x_0) delta x. 117 00:07:55,490 --> 00:07:57,290 And now if I just write out what this 118 00:07:57,290 --> 00:08:09,820 is, it's f(x), right, minus f(x_0), 119 00:08:09,820 --> 00:08:11,630 I'm going to write it this way. 120 00:08:11,630 --> 00:08:16,450 Which is approximately f'(x_0), and this is x - x_0. 121 00:08:16,450 --> 00:08:25,670 So here I'm using the notations delta x is x - x0. 122 00:08:25,670 --> 00:08:28,160 And so this is the change in f, this 123 00:08:28,160 --> 00:08:32,270 is just rewriting what delta x is. 124 00:08:32,270 --> 00:08:36,480 And now the last step is just to put the constant 125 00:08:36,480 --> 00:08:37,430 on the other side. 126 00:08:37,430 --> 00:08:47,010 So f(x) is approximately f(x_0) + f'(x_0)(x - x_0). 127 00:08:47,010 --> 00:08:51,920 So this is exactly what I had just to begin with, right? 128 00:08:51,920 --> 00:08:53,570 So these two are just algebraically 129 00:08:53,570 --> 00:08:56,690 the same statement. 130 00:08:56,690 --> 00:09:00,660 That's one another way of looking at it. 131 00:09:00,660 --> 00:09:05,100 All right, so now, I want to go through 132 00:09:05,100 --> 00:09:08,590 some systematic discussion here of 133 00:09:08,590 --> 00:09:12,250 several linear approximations, which you're going 134 00:09:12,250 --> 00:09:14,850 to be wanting to memorize. 135 00:09:14,850 --> 00:09:18,020 And rather than it's being hard to memorize these, 136 00:09:18,020 --> 00:09:19,760 it's supposed to remind you. 137 00:09:19,760 --> 00:09:22,440 So that you'll have a lot of extra reinforcement 138 00:09:22,440 --> 00:09:25,240 in remembering derivatives of all kinds. 139 00:09:25,240 --> 00:09:31,240 So, when we carry out these systematic discussions, 140 00:09:31,240 --> 00:09:33,060 we want to make things absolutely as 141 00:09:33,060 --> 00:09:34,550 simple as possible. 142 00:09:34,550 --> 00:09:36,640 And so one of the things that we do 143 00:09:36,640 --> 00:09:40,380 is we always use the base point to be x_0. 144 00:09:40,380 --> 00:09:44,180 So I'm always going to have x_0 = 0 145 00:09:44,180 --> 00:09:48,920 in this standard list of formulas that I'm going to use. 146 00:09:48,920 --> 00:09:52,030 And if I put x_0 = 0, then this formula 147 00:09:52,030 --> 00:09:56,130 becomes f(x), a little bit simpler to read. 148 00:09:56,130 --> 00:09:59,980 It becomes f(x) is f(0) + f'(0) x. 149 00:10:03,520 --> 00:10:05,950 So this is probably the form that you'll 150 00:10:05,950 --> 00:10:10,680 want to remember most. 151 00:10:10,680 --> 00:10:12,580 That's again, just the linear approximation. 152 00:10:12,580 --> 00:10:16,010 But one always has to remember, and this 153 00:10:16,010 --> 00:10:22,650 is a very important thing, this one only worked near x is 1. 154 00:10:22,650 --> 00:10:29,170 This approximation here really only works when x is near x_0. 155 00:10:29,170 --> 00:10:31,600 So that's a little addition that you need to throw in. 156 00:10:31,600 --> 00:10:38,810 So this one works when x is near 0. 157 00:10:38,810 --> 00:10:40,770 You can't expect it to be true far away. 158 00:10:40,770 --> 00:10:42,790 The curve can go anywhere it wants, 159 00:10:42,790 --> 00:10:46,500 when it's far away from the point of tangency. 160 00:10:46,500 --> 00:10:49,060 So, OK, so let's work this out. 161 00:10:49,060 --> 00:10:51,560 Let's do it for the sine function, 162 00:10:51,560 --> 00:10:56,401 for the cosine function, and for e^x, to begin with. 163 00:10:56,401 --> 00:10:56,900 Yeah. 164 00:10:56,900 --> 00:10:57,400 Question. 165 00:10:57,400 --> 00:11:02,522 STUDENT: [INAUDIBLE] 166 00:11:02,522 --> 00:11:03,022 PROF. 167 00:11:03,022 --> 00:11:03,126 JERISON: Yeah. 168 00:11:03,126 --> 00:11:04,125 When does this one work. 169 00:11:04,125 --> 00:11:07,410 Well, so the question was, when does this one work. 170 00:11:07,410 --> 00:11:12,100 Again, this is when x is approximately x_0. 171 00:11:12,100 --> 00:11:18,130 Because it's actually the same as this one over here. 172 00:11:18,130 --> 00:11:20,050 OK. 173 00:11:20,050 --> 00:11:23,010 And indeed, that's what's going on when 174 00:11:23,010 --> 00:11:24,870 we take this limiting value. 175 00:11:24,870 --> 00:11:26,760 Delta x going to 0 is the same. 176 00:11:26,760 --> 00:11:27,980 Delta x small. 177 00:11:27,980 --> 00:11:37,050 So another way of saying it is, the delta x is small. 178 00:11:37,050 --> 00:11:41,030 Now, exactly what we mean by small will also be explained. 179 00:11:41,030 --> 00:11:45,240 But it is a matter to some extent of intuition 180 00:11:45,240 --> 00:11:47,620 as to how much, how good it is. 181 00:11:47,620 --> 00:11:49,650 In practical cases, people will really 182 00:11:49,650 --> 00:11:52,950 care about how small it is before the approximation is 183 00:11:52,950 --> 00:11:53,970 useful. 184 00:11:53,970 --> 00:11:56,710 And that's a serious issue. 185 00:11:56,710 --> 00:12:00,200 All right, so let me carry out these approximations for x. 186 00:12:00,200 --> 00:12:06,710 Again, this is always for x near 0. 187 00:12:06,710 --> 00:12:08,930 So all of these are going to be for x near 0. 188 00:12:08,930 --> 00:12:10,790 So in order to make this computation, 189 00:12:10,790 --> 00:12:15,170 I have to evaluate the function. 190 00:12:15,170 --> 00:12:17,817 I need to plug in two numbers here. 191 00:12:17,817 --> 00:12:19,150 In order to get this expression. 192 00:12:19,150 --> 00:12:23,070 I need to know what f(0) is and I need to know what f'(0) is. 193 00:12:23,070 --> 00:12:26,110 If this is the function f(x), then I'm going to make a little 194 00:12:26,110 --> 00:12:30,615 table over to the right here with f' and then I'm going 195 00:12:30,615 --> 00:12:33,160 to evaluate f(0), and then I'm going to evaluate 196 00:12:33,160 --> 00:12:38,150 f'(0), and then read off what the answers are. 197 00:12:38,150 --> 00:12:41,450 Right, so first of all if the function is sine x, 198 00:12:41,450 --> 00:12:44,170 the derivative is cosine x. 199 00:12:44,170 --> 00:12:49,690 The value of f(0), that's sine of 0, is 0. 200 00:12:49,690 --> 00:12:51,550 The derivative is cosine. 201 00:12:51,550 --> 00:12:54,060 Cosine of 0 is 1. 202 00:12:54,060 --> 00:12:55,350 So there we go. 203 00:12:55,350 --> 00:12:58,850 So now we have the coefficients 0 and 1. 204 00:12:58,850 --> 00:13:01,070 So this number is 0. 205 00:13:01,070 --> 00:13:04,480 And this number is 1. 206 00:13:04,480 --> 00:13:11,310 So what we get here is 0 + 1x, so this is approximately x. 207 00:13:11,310 --> 00:13:18,080 There's the linear approximation to sin x. 208 00:13:18,080 --> 00:13:20,570 Similarly, so now this is a routine matter 209 00:13:20,570 --> 00:13:22,440 to just read this off for this table. 210 00:13:22,440 --> 00:13:23,940 We'll do it for the cosine function. 211 00:13:23,940 --> 00:13:30,230 If you differentiate the cosine, what you get is -sin x. 212 00:13:30,230 --> 00:13:34,940 The value at 0 is 1, so that's cosine of 0 at 1. 213 00:13:34,940 --> 00:13:39,540 The value of this minus sine at 0 is 0. 214 00:13:39,540 --> 00:13:43,890 So this is going back over here, 1 + 0x, 215 00:13:43,890 --> 00:13:48,170 so this is approximately 1. 216 00:13:48,170 --> 00:13:52,300 This linear function happens to be constant. 217 00:13:52,300 --> 00:13:58,730 And finally, if I do need e^x, its derivative is again e^x, 218 00:13:58,730 --> 00:14:02,840 and its value at 0 is 1, the value of the derivative at 0 is 219 00:14:02,840 --> 00:14:04,180 also 1. 220 00:14:04,180 --> 00:14:09,500 So both of the terms here, f(0) and f'(0), they're both 1 221 00:14:09,500 --> 00:14:15,400 and we get 1 + x. 222 00:14:15,400 --> 00:14:18,360 So these are the linear approximations. 223 00:14:18,360 --> 00:14:19,610 You can memorize these. 224 00:14:19,610 --> 00:14:23,940 You'll probably remember them either this way or that way. 225 00:14:23,940 --> 00:14:26,130 This collection of information here 226 00:14:26,130 --> 00:14:28,556 encodes the same collection of information 227 00:14:28,556 --> 00:14:29,430 as we have over here. 228 00:14:29,430 --> 00:14:31,460 For the values of the function and the values 229 00:14:31,460 --> 00:14:36,310 of their derivatives at 0. 230 00:14:36,310 --> 00:14:39,370 So let me just emphasize again the geometric point of view 231 00:14:39,370 --> 00:14:48,840 by drawing pictures of these results. 232 00:14:48,840 --> 00:14:56,605 So first of all, for the sine function, here's the sine 233 00:14:56,605 --> 00:15:03,500 - well, close enough. 234 00:15:03,500 --> 00:15:07,170 So that's - boy, now that is quite some sine, isn't it? 235 00:15:07,170 --> 00:15:10,570 I should try to make the two bumps be the same height, 236 00:15:10,570 --> 00:15:11,870 roughly speaking. 237 00:15:11,870 --> 00:15:15,450 Anyway the tangent line we're talking about is here. 238 00:15:15,450 --> 00:15:17,730 And this is y = x. 239 00:15:17,730 --> 00:15:22,870 And this is the function sine x. 240 00:15:22,870 --> 00:15:28,870 And near 0, those things coincide pretty closely. 241 00:15:28,870 --> 00:15:34,040 The cosine function, I'll put that underneath, I guess. 242 00:15:34,040 --> 00:15:35,110 I think I can fit it. 243 00:15:35,110 --> 00:15:39,390 Make it a little smaller here. 244 00:15:39,390 --> 00:15:44,850 So for the cosine function, we're up here. 245 00:15:44,850 --> 00:15:48,500 It's y = 1. 246 00:15:48,500 --> 00:15:51,990 Well, no wonder the tangent line is constant. 247 00:15:51,990 --> 00:15:54,630 It's horizontal. 248 00:15:54,630 --> 00:15:57,920 The tangent line is horizontal, so the function corresponding 249 00:15:57,920 --> 00:15:59,560 is constant. 250 00:15:59,560 --> 00:16:04,890 So this is y = cos x. 251 00:16:04,890 --> 00:16:14,790 And finally, if I draw y = e^x, that's coming down like this. 252 00:16:14,790 --> 00:16:17,870 And the tangent line is here. 253 00:16:17,870 --> 00:16:19,410 And it's y = 1 + x. 254 00:16:19,410 --> 00:16:24,700 The value is 1 and the slope is 1. 255 00:16:24,700 --> 00:16:28,030 So this is how to remember it graphically if you like. 256 00:16:28,030 --> 00:16:34,810 This analytic picture is extremely important 257 00:16:34,810 --> 00:16:37,950 and will help you to deal with sines, cosines 258 00:16:37,950 --> 00:16:41,090 and exponentials. 259 00:16:41,090 --> 00:16:41,730 Yes, question. 260 00:16:41,730 --> 00:16:45,806 STUDENT: [INAUDIBLE] 261 00:16:45,806 --> 00:16:46,306 PROF. 262 00:16:46,306 --> 00:16:48,181 JERISON: The question is what do you normally 263 00:16:48,181 --> 00:16:50,350 use linear approximations for. 264 00:16:50,350 --> 00:16:51,140 Good question. 265 00:16:51,140 --> 00:16:52,260 We're getting there. 266 00:16:52,260 --> 00:16:54,210 First, we're getting a little library of them 267 00:16:54,210 --> 00:16:56,220 and I'll give you a few examples. 268 00:16:56,220 --> 00:17:02,540 OK, so now, I need to finish the catalog 269 00:17:02,540 --> 00:17:05,700 with two more examples which are just a little bit, slightly 270 00:17:05,700 --> 00:17:07,620 more challenging. 271 00:17:07,620 --> 00:17:09,860 And a little bit less obvious. 272 00:17:09,860 --> 00:17:22,356 So, the next couple that we're going to do are ln(1+x) and (1 273 00:17:22,356 --> 00:17:25,530 + x)^r. 274 00:17:25,530 --> 00:17:28,130 OK, these are the last two that we're going to write down. 275 00:17:28,130 --> 00:17:30,850 And that you need to think about. 276 00:17:30,850 --> 00:17:34,930 Now, the procedure is the same as over here. 277 00:17:34,930 --> 00:17:39,230 Namely, I have to write down f' and I have to write down 278 00:17:39,230 --> 00:17:41,992 f'(0) and I have to write down f'(0). 279 00:17:41,992 --> 00:17:43,450 And then I'll have the coefficients 280 00:17:43,450 --> 00:17:46,970 to be able to fill in what the approximation is. 281 00:17:46,970 --> 00:17:51,840 So f' = 1 / (1+x), in the case of the logarithm. 282 00:17:51,840 --> 00:17:57,010 And f(0), if I plug in, that's log of 1, which is 0. 283 00:17:57,010 --> 00:18:01,190 And f' if I plug in 0 here, I get 1. 284 00:18:01,190 --> 00:18:04,850 And similarly if I do it for this one, I get r(1+x)^(r-1). 285 00:18:07,850 --> 00:18:12,320 And when I plug in f(0), I get 1^r, which is 1. 286 00:18:12,320 --> 00:18:18,850 And here I get r (1)^(r-1), which is r. 287 00:18:18,850 --> 00:18:22,830 So the corresponding statement here is that ln(1+x) is 288 00:18:22,830 --> 00:18:24,790 approximately x. 289 00:18:24,790 --> 00:18:31,140 And (1+x)^r is approximately 1 + rx. 290 00:18:31,140 --> 00:18:35,660 That's 0 + 1x and here we have 1 + rx. 291 00:18:41,370 --> 00:18:44,320 And now, I do want to make a connection, 292 00:18:44,320 --> 00:18:47,120 explain to you what's going on here and the connection 293 00:18:47,120 --> 00:18:48,750 with the first example. 294 00:18:48,750 --> 00:18:50,800 We already did the logarithm once. 295 00:18:50,800 --> 00:18:53,520 And let's just point out that these two computations 296 00:18:53,520 --> 00:18:57,470 are the same, or practically the same. 297 00:18:57,470 --> 00:19:02,580 Here I use the base point 1, but because of my, 298 00:19:02,580 --> 00:19:05,420 sort of, convenient form, which will end up, 299 00:19:05,420 --> 00:19:07,180 I claim, being much more convenient 300 00:19:07,180 --> 00:19:09,310 for pretty much every purpose, we 301 00:19:09,310 --> 00:19:14,510 want to do these things near x is approximately 0. 302 00:19:14,510 --> 00:19:19,040 You cannot expand the logarithm and understand a tangent line 303 00:19:19,040 --> 00:19:22,730 for it at x equals 0, because it goes down to minus infinity. 304 00:19:22,730 --> 00:19:27,260 Similarly, if you try to graph (1+x)^r, 305 00:19:27,260 --> 00:19:30,690 x^r without the 1 here, you'll discover that sometimes 306 00:19:30,690 --> 00:19:33,260 the slope is infinite, and so forth. 307 00:19:33,260 --> 00:19:35,500 So this is a bad choice of point. 308 00:19:35,500 --> 00:19:39,030 1 is a much better choice of a place to expand around. 309 00:19:39,030 --> 00:19:42,150 And then we shift things so that it looks like it's x = 0, 310 00:19:42,150 --> 00:19:43,600 by shifting by the 1. 311 00:19:43,600 --> 00:19:50,950 So the connection with the previous example is that 312 00:19:50,950 --> 00:19:57,020 the-- what we wrote before I could write as ln u = u - 1. 313 00:19:57,020 --> 00:20:00,890 Right, that's just recopying what I have over here. 314 00:20:00,890 --> 00:20:04,930 Except with the letter u rather than the letter x. 315 00:20:04,930 --> 00:20:12,790 And then I plug in, u = 1 + x. 316 00:20:12,790 --> 00:20:14,920 And then that, if I copy it down, 317 00:20:14,920 --> 00:20:16,880 you see that I have a u in place of 1+x, 318 00:20:16,880 --> 00:20:19,020 that's the same as this. 319 00:20:19,020 --> 00:20:22,880 And if I write out u-1, if I subtract 1 from u, 320 00:20:22,880 --> 00:20:23,924 that means that it's x. 321 00:20:23,924 --> 00:20:25,840 So that's what's on the right-hand side there. 322 00:20:25,840 --> 00:20:27,720 So these are the same computation, 323 00:20:27,720 --> 00:20:38,860 I've just changed the variable. 324 00:20:38,860 --> 00:20:44,220 So now I want to try to address the question that was 325 00:20:44,220 --> 00:20:47,380 asked about how this is used. 326 00:20:47,380 --> 00:20:49,370 And what the importance is. 327 00:20:49,370 --> 00:20:58,010 And what I'm going to do is just give you one example here. 328 00:20:58,010 --> 00:21:02,690 And then try to emphasize. 329 00:21:02,690 --> 00:21:05,930 The first way in which this is a useful idea. 330 00:21:05,930 --> 00:21:10,460 So, or maybe this is the second example. 331 00:21:10,460 --> 00:21:13,330 If you like. 332 00:21:13,330 --> 00:21:16,460 So we'll call this Example 2, maybe. 333 00:21:16,460 --> 00:21:19,070 So let's just take the logarithm of 1.1. 334 00:21:19,070 --> 00:21:22,220 Just a second. 335 00:21:22,220 --> 00:21:25,710 Let's take the logarithm of 1.1. 336 00:21:25,710 --> 00:21:30,150 So I claim that, according to our rules, I can glance at this 337 00:21:30,150 --> 00:21:33,680 and I can immediately see that it's approximately 1/10. 338 00:21:33,680 --> 00:21:35,710 So what did I use here? 339 00:21:35,710 --> 00:21:42,630 I used that ln(1+x) is approximately x, 340 00:21:42,630 --> 00:21:46,281 and the value of x that I used was 1/10. 341 00:21:46,281 --> 00:21:46,780 Right? 342 00:21:46,780 --> 00:21:48,850 So that is the formula, so I should 343 00:21:48,850 --> 00:21:54,560 put a box around these two formulas too. 344 00:21:54,560 --> 00:21:57,730 That's this formula here, applied with x = 1/10. 345 00:21:57,730 --> 00:22:01,680 And I'm claiming that 1/10 is a sufficiently small number, 346 00:22:01,680 --> 00:22:08,845 sufficiently close to 0, that this is an OK statement. 347 00:22:08,845 --> 00:22:10,220 So the first question that I want 348 00:22:10,220 --> 00:22:12,000 to ask you is, which do you think 349 00:22:12,000 --> 00:22:14,470 is a more complicated thing. 350 00:22:14,470 --> 00:22:19,244 The left-hand side or the right-hand side. 351 00:22:19,244 --> 00:22:21,160 I claim that this is a more complicated thing, 352 00:22:21,160 --> 00:22:24,070 you'd have to go to a calculator to punch out and figure out 353 00:22:24,070 --> 00:22:25,170 what this thing is. 354 00:22:25,170 --> 00:22:26,230 This is easy. 355 00:22:26,230 --> 00:22:28,730 You know what a tenth is. 356 00:22:28,730 --> 00:22:31,530 So the distinction that I want to make 357 00:22:31,530 --> 00:22:37,190 is that this half, this part, this is hard. 358 00:22:37,190 --> 00:22:40,700 And this is easy. 359 00:22:40,700 --> 00:22:43,090 Now, that may look contradictory, 360 00:22:43,090 --> 00:22:45,610 but I want to just do it right above as well. 361 00:22:45,610 --> 00:22:48,940 This is hard. 362 00:22:48,940 --> 00:22:52,160 And this is easy. 363 00:22:52,160 --> 00:22:52,870 OK. 364 00:22:52,870 --> 00:22:56,850 This looks uglier, but actually this is the hard one. 365 00:22:56,850 --> 00:22:58,650 And this is giving us information about it. 366 00:22:58,650 --> 00:23:00,930 Now, let me show you why that's true. 367 00:23:00,930 --> 00:23:02,530 Look down this column here. 368 00:23:02,530 --> 00:23:05,230 These are the hard ones, hard functions. 369 00:23:05,230 --> 00:23:07,360 These are the easy functions. 370 00:23:07,360 --> 00:23:09,970 What's easier than this? 371 00:23:09,970 --> 00:23:11,260 Nothing. 372 00:23:11,260 --> 00:23:11,760 OK. 373 00:23:11,760 --> 00:23:12,590 Well, yeah, 0. 374 00:23:12,590 --> 00:23:14,480 That's easier. 375 00:23:14,480 --> 00:23:16,060 Over here it gets even worse. 376 00:23:16,060 --> 00:23:21,090 These are the hard functions and these are the easy ones. 377 00:23:21,090 --> 00:23:24,910 So that's the main advantage of linear approximation 378 00:23:24,910 --> 00:23:27,730 is you get something much simpler to deal with. 379 00:23:27,730 --> 00:23:30,380 And if you've made a valid approximation 380 00:23:30,380 --> 00:23:33,510 you can make much progress on problems. 381 00:23:33,510 --> 00:23:35,780 OK, we'll be doing some more examples, 382 00:23:35,780 --> 00:23:38,990 but I saw some more questions before I made that point. 383 00:23:38,990 --> 00:23:39,490 Yeah. 384 00:23:39,490 --> 00:23:42,373 STUDENT: [INAUDIBLE] 385 00:23:42,373 --> 00:23:42,873 PROF. 386 00:23:42,873 --> 00:23:46,080 JERISON: Is this ln of 1.1 or what? 387 00:23:46,080 --> 00:23:48,410 STUDENT: [INAUDIBLE] 388 00:23:48,410 --> 00:23:49,115 PROF. 389 00:23:49,115 --> 00:23:52,580 JERISON: This is a parens there. 390 00:23:52,580 --> 00:23:56,430 It's ln of 1.1, it's the digital number, right. 391 00:23:56,430 --> 00:23:59,820 I guess I've never used that before a decimal point, have I? 392 00:23:59,820 --> 00:24:04,834 I don't know. 393 00:24:04,834 --> 00:24:05,500 Other questions. 394 00:24:05,500 --> 00:24:11,910 STUDENT: [INAUDIBLE] 395 00:24:11,910 --> 00:24:12,410 PROF. 396 00:24:12,410 --> 00:24:12,990 JERISON: OK. 397 00:24:12,990 --> 00:24:14,900 So let's continue here. 398 00:24:14,900 --> 00:24:18,570 Let me give you some more examples, where it becomes 399 00:24:18,570 --> 00:24:21,460 even more vivid if you like. 400 00:24:21,460 --> 00:24:24,340 That this approximation is giving us something 401 00:24:24,340 --> 00:24:30,190 a little simpler to deal with. 402 00:24:30,190 --> 00:24:34,960 So here's Example 3. 403 00:24:34,960 --> 00:24:48,400 I want to, I'll find the linear approximation near x = 0. 404 00:24:48,400 --> 00:24:52,340 I also - when I write this expression near x = 0, 405 00:24:52,340 --> 00:24:55,020 that's the same thing as this. 406 00:24:55,020 --> 00:24:58,940 That's the same thing as saying x is approximately 0 - 407 00:24:58,940 --> 00:25:06,360 of the function e^(-3x) divided by square root 1+x. 408 00:25:09,170 --> 00:25:17,390 So here's a function. 409 00:25:17,390 --> 00:25:17,890 OK. 410 00:25:17,890 --> 00:25:21,990 Now, what I claim I want to use for the purposes 411 00:25:21,990 --> 00:25:26,830 of this approximation, are just the sum 412 00:25:26,830 --> 00:25:32,110 of the approximation formulas that we've already derived. 413 00:25:32,110 --> 00:25:33,850 And just to combine them algebraically. 414 00:25:33,850 --> 00:25:35,350 So I'm not going to do any calculus, 415 00:25:35,350 --> 00:25:37,080 I'm just going to remember. 416 00:25:37,080 --> 00:25:41,580 So with e^(-3x), it's pretty clear that I should be using 417 00:25:41,580 --> 00:25:44,570 this formula for e^x. 418 00:25:44,570 --> 00:25:47,820 For the other one, it may be slightly less obvious but we 419 00:25:47,820 --> 00:25:53,470 have powers of 1+x over here. 420 00:25:53,470 --> 00:25:55,510 So let's plug those in. 421 00:25:55,510 --> 00:26:04,640 I'll put this up so that you can remember it. 422 00:26:04,640 --> 00:26:10,810 And we're going to carry out this approximation. 423 00:26:10,810 --> 00:26:16,380 So, first of all, I'm going to write this so that it's 424 00:26:16,380 --> 00:26:17,910 slightly more suggestive. 425 00:26:17,910 --> 00:26:23,630 Namely, I'm going to write it as a product. 426 00:26:23,630 --> 00:26:27,220 And there you can now see the exponent. 427 00:26:27,220 --> 00:26:31,870 In this case, r = 1/2, eh -1/2, that we're going to use. 428 00:26:31,870 --> 00:26:32,900 OK. 429 00:26:32,900 --> 00:26:39,880 So now I have e^(-3x) (1+x)^(-1/2), 430 00:26:39,880 --> 00:26:41,940 and that's going to be approximately-- 431 00:26:41,940 --> 00:26:44,220 well I'm going to use this formula. 432 00:26:44,220 --> 00:26:48,760 I have to use it correctly. x is replaced by -3x, so this is 1 - 433 00:26:48,760 --> 00:26:50,160 3x. 434 00:26:50,160 --> 00:26:52,270 And then over here, I can just copy 435 00:26:52,270 --> 00:26:57,900 verbatim the other approximation formula with r = -1/2. 436 00:26:57,900 --> 00:27:05,670 So this is times 1 - 1/2 x. 437 00:27:05,670 --> 00:27:11,080 And now I'm going to carry out the multiplication. 438 00:27:11,080 --> 00:27:17,100 So this is 1 - 3x - 1/2 x + 3/2 x^2. 439 00:27:27,310 --> 00:27:32,090 So now, here's our formula. 440 00:27:32,090 --> 00:27:34,340 So now this isn't where things stop. 441 00:27:34,340 --> 00:27:36,960 And indeed, in this kind of arithmetic 442 00:27:36,960 --> 00:27:39,420 that I'm describing now, things are 443 00:27:39,420 --> 00:27:43,780 easier than they are in ordinary algebra, in arithmetic. 444 00:27:43,780 --> 00:27:47,770 The reason is that there's another step, which 445 00:27:47,770 --> 00:27:49,200 I'm now going to perform. 446 00:27:49,200 --> 00:27:54,602 Which is that I'm going to throw away this term here. 447 00:27:54,602 --> 00:27:55,560 I'm going to ignore it. 448 00:27:55,560 --> 00:27:57,480 In fact, I didn't even have to work it out. 449 00:27:57,480 --> 00:27:59,070 Because I'm going to throw it away. 450 00:27:59,070 --> 00:28:01,520 So the reason is that already, when 451 00:28:01,520 --> 00:28:03,424 I passed from this expression to this one, 452 00:28:03,424 --> 00:28:05,340 that is from this type of thing to this thing, 453 00:28:05,340 --> 00:28:07,940 I was already throwing away quadratic and higher-ordered 454 00:28:07,940 --> 00:28:09,460 terms. 455 00:28:09,460 --> 00:28:12,650 So this isn't the only quadratic term. 456 00:28:12,650 --> 00:28:13,640 There are tons of them. 457 00:28:13,640 --> 00:28:14,920 I have to ignore all of them if I'm 458 00:28:14,920 --> 00:28:16,128 going to ignore some of them. 459 00:28:16,128 --> 00:28:20,240 And in fact, I only want to be left with the linear stuff. 460 00:28:20,240 --> 00:28:23,220 Because that's all I'm really getting a valid computation 461 00:28:23,220 --> 00:28:24,080 for. 462 00:28:24,080 --> 00:28:28,060 So, this is approximately 1 minus, so let's see. 463 00:28:28,060 --> 00:28:32,450 It's a total of 7/2 x. 464 00:28:32,450 --> 00:28:36,410 And this is the answer. 465 00:28:36,410 --> 00:28:38,290 This is the linear part. 466 00:28:38,290 --> 00:28:42,800 So the x^2 term is negligible. 467 00:28:42,800 --> 00:28:46,680 So we drop x^2 term. 468 00:28:46,680 --> 00:28:55,712 Terms, and higher. 469 00:28:55,712 --> 00:28:57,420 All of those terms should be lower-order. 470 00:28:57,420 --> 00:29:00,180 If you imagine x is 1/10, or maybe 1/100, 471 00:29:00,180 --> 00:29:04,470 then these terms will end up being much smaller. 472 00:29:04,470 --> 00:29:08,970 So we have a rather crude approach. 473 00:29:08,970 --> 00:29:10,530 And that's really the simplicity, 474 00:29:10,530 --> 00:29:15,360 and that's the savings. 475 00:29:15,360 --> 00:29:21,020 So now, since this unit is called Applications, 476 00:29:21,020 --> 00:29:24,380 and these are indeed applications to math, 477 00:29:24,380 --> 00:29:30,360 I also wanted to give you a real-life application. 478 00:29:30,360 --> 00:29:34,290 Or a place where linear approximations come up 479 00:29:34,290 --> 00:29:46,590 in real life. 480 00:29:46,590 --> 00:29:50,560 So maybe we'll call this Example 4. 481 00:29:50,560 --> 00:29:57,270 This is supposedly a real-life example. 482 00:29:57,270 --> 00:30:06,580 I'll try to persuade you that it is. 483 00:30:06,580 --> 00:30:09,840 So I like this example because it's got a lot of math, 484 00:30:09,840 --> 00:30:11,170 as well as physics in it. 485 00:30:11,170 --> 00:30:17,000 So here I am, on the surface of the earth. 486 00:30:17,000 --> 00:30:24,610 And here is a satellite going this way. 487 00:30:24,610 --> 00:30:30,790 At some velocity, v. And this satellite 488 00:30:30,790 --> 00:30:33,630 has a clock on it because this is a GPS satellite. 489 00:30:33,630 --> 00:30:37,720 And it has a time, T, OK? 490 00:30:37,720 --> 00:30:41,160 But I have a watch, in fact it's right here. 491 00:30:41,160 --> 00:30:44,030 And I have a time which I keep. 492 00:30:44,030 --> 00:30:48,170 Which is T', And there's an interesting relationship 493 00:30:48,170 --> 00:30:56,650 between T and T', which is called time dilation. 494 00:30:56,650 --> 00:31:04,860 And this is from special relativity. 495 00:31:04,860 --> 00:31:06,470 And it's the following formula. 496 00:31:06,470 --> 00:31:13,720 T' = T divided by the square root of 1 - v^2/c^2, 497 00:31:13,720 --> 00:31:17,230 where v is the velocity of the satellite, 498 00:31:17,230 --> 00:31:22,960 and c is the speed of light. 499 00:31:22,960 --> 00:31:28,080 So now I'd like to get a rough idea of how different 500 00:31:28,080 --> 00:31:34,980 my watch is from the clock on the satellite. 501 00:31:34,980 --> 00:31:38,540 So I'm going to use this same approximation, 502 00:31:38,540 --> 00:31:40,990 we've already used it once. 503 00:31:40,990 --> 00:31:42,010 I'm going to write t. 504 00:31:42,010 --> 00:31:43,990 But now let me just remind you. 505 00:31:43,990 --> 00:31:46,809 The situation here is, we have something of the form 506 00:31:46,809 --> 00:31:47,350 (1-u)^(-1/2). 507 00:31:52,410 --> 00:31:55,760 That's what's happening when I multiply through here. 508 00:31:55,760 --> 00:31:59,500 So with u = v^2 / c^2. 509 00:32:02,080 --> 00:32:05,240 So in real life, of course, the expression 510 00:32:05,240 --> 00:32:07,780 that you're going to use the linear approximation on 511 00:32:07,780 --> 00:32:10,280 isn't necessarily itself linear. 512 00:32:10,280 --> 00:32:11,990 It can be any physical quantity. 513 00:32:11,990 --> 00:32:15,940 So in this case it's v squared over c squared. 514 00:32:15,940 --> 00:32:18,189 And now the approximation formula 515 00:32:18,189 --> 00:32:20,230 says that if this is approximately equal to, well 516 00:32:20,230 --> 00:32:21,540 again it's the same rule. 517 00:32:21,540 --> 00:32:25,870 There's an r and then x is -u, so this is - - 1/2, 518 00:32:25,870 --> 00:32:34,610 so it's 1 + 1/2 u. 519 00:32:34,610 --> 00:32:40,350 So this is approximately, by the same rule, this is T, 520 00:32:40,350 --> 00:32:55,800 T' is approximately t T(1 + 1/2 v^2/c^2) Now, 521 00:32:55,800 --> 00:32:58,150 I promised you that this would be a real-life problem. 522 00:32:58,150 --> 00:33:02,520 So the question is when people were designing these GPS 523 00:33:02,520 --> 00:33:06,666 systems, they run clocks in the satellites. 524 00:33:06,666 --> 00:33:08,790 You're down there, you're making your measurements, 525 00:33:08,790 --> 00:33:12,270 you're talking to the satellite by-- 526 00:33:12,270 --> 00:33:15,310 or you're receiving its signals from its radio. 527 00:33:15,310 --> 00:33:19,010 The question is, is this going to cause problems 528 00:33:19,010 --> 00:33:23,670 in the transmission. 529 00:33:23,670 --> 00:33:25,580 And there are dozens of such problems 530 00:33:25,580 --> 00:33:27,180 that you have to check for. 531 00:33:27,180 --> 00:33:29,950 So in this case, what actually happened 532 00:33:29,950 --> 00:33:35,010 is that v is about 4 kilometers per second. 533 00:33:35,010 --> 00:33:38,740 That's how fast the GPS satellites actually go. 534 00:33:38,740 --> 00:33:41,430 In fact, they had to decide to put them at a certain altitude 535 00:33:41,430 --> 00:33:43,950 and they could've tweaked this if they had put them 536 00:33:43,950 --> 00:33:46,040 at different places. 537 00:33:46,040 --> 00:33:55,330 Anyway, the speed of light is 3 * 10^5 kilometers per second. 538 00:33:55,330 --> 00:34:01,100 So this number, v^2 / c^2 is approximately 10^(-10). 539 00:34:05,710 --> 00:34:11,160 Now, if you actually keep track of how much of an error 540 00:34:11,160 --> 00:34:15,530 that would make in a GPS location, what you would find 541 00:34:15,530 --> 00:34:17,820 is maybe it's a millimeter or something like that. 542 00:34:17,820 --> 00:34:20,080 So in fact it doesn't matter. 543 00:34:20,080 --> 00:34:21,380 So that's nice. 544 00:34:21,380 --> 00:34:23,180 But in fact the engineers who were 545 00:34:23,180 --> 00:34:26,870 designing these systems actually did use this very computation. 546 00:34:26,870 --> 00:34:29,270 Exactly this. 547 00:34:29,270 --> 00:34:31,640 And the way that they used it was, 548 00:34:31,640 --> 00:34:35,190 they decided that because the clocks were different, 549 00:34:35,190 --> 00:34:38,740 when the satellite broadcasts its radio frequency, 550 00:34:38,740 --> 00:34:40,350 that frequency would be shifted. 551 00:34:40,350 --> 00:34:41,500 Would be offset. 552 00:34:41,500 --> 00:34:44,426 And they decided that the fidelity was so important 553 00:34:44,426 --> 00:34:46,050 that they would send the satellites off 554 00:34:46,050 --> 00:34:49,120 with this kind of, exactly this, offset. 555 00:34:49,120 --> 00:34:51,460 To compensate for the way the signal is. 556 00:34:51,460 --> 00:34:53,360 So from the point of view of good reception 557 00:34:53,360 --> 00:34:56,950 on your little GPS device, they changed the frequency at which 558 00:34:56,950 --> 00:35:00,160 the transmitter in the satellites, 559 00:35:00,160 --> 00:35:04,990 according to exactly this rule. 560 00:35:04,990 --> 00:35:08,120 And incidentally, the reason why they didn't-- they ignored 561 00:35:08,120 --> 00:35:11,010 higher-order terms, the sort of quadratic terms, 562 00:35:11,010 --> 00:35:17,460 is that if you take u^2 that's a size 10^(-20). 563 00:35:17,460 --> 00:35:20,104 And that really is totally negligible. 564 00:35:20,104 --> 00:35:22,020 That doesn't matter to any measurement at all. 565 00:35:22,020 --> 00:35:25,210 That's on the order of nanometers, 566 00:35:25,210 --> 00:35:30,200 and it's not important for any of the uses to which GPS 567 00:35:30,200 --> 00:35:32,510 is put. 568 00:35:32,510 --> 00:35:40,470 OK, so that's a real example of a use of linear approximations. 569 00:35:40,470 --> 00:35:42,720 So. let's take a little pause here. 570 00:35:42,720 --> 00:35:44,850 I'm going to switch gears and talk 571 00:35:44,850 --> 00:35:46,610 about quadratic approximations. 572 00:35:46,610 --> 00:35:48,900 But before I do that, let's have some more questions. 573 00:35:48,900 --> 00:35:49,400 Yeah. 574 00:35:49,400 --> 00:36:03,780 STUDENT: [INAUDIBLE] 575 00:36:03,780 --> 00:36:04,566 PROF. 576 00:36:04,566 --> 00:36:08,040 JERISON: OK, so the question was asked, 577 00:36:08,040 --> 00:36:11,580 suppose I did this by different method. 578 00:36:11,580 --> 00:36:15,840 Suppose I applied the original formula here. 579 00:36:15,840 --> 00:36:18,050 Namely, I define the function f(x), 580 00:36:18,050 --> 00:36:22,140 which was this function here. 581 00:36:22,140 --> 00:36:25,050 And then I plugged in its value at x = 0 582 00:36:25,050 --> 00:36:28,000 and the value of its derivative at x = 0. 583 00:36:28,000 --> 00:36:32,510 So the answer is, yes, it's also true that if I call this 584 00:36:32,510 --> 00:36:37,940 function f f(x), then it must be true that the linear 585 00:36:37,940 --> 00:36:45,910 approximation is f(x_0) plus f' of - I'm sorry, it's at 0, 586 00:36:45,910 --> 00:36:49,340 so it's f(0), f'(0) times x. 587 00:36:49,340 --> 00:36:50,550 So that should be true. 588 00:36:50,550 --> 00:36:52,810 That's the formula that we're using. 589 00:36:52,810 --> 00:36:57,170 It's up there in the pink also. 590 00:36:57,170 --> 00:36:58,590 So this is the formula. 591 00:36:58,590 --> 00:37:00,650 So now, what about f(0)? 592 00:37:00,650 --> 00:37:04,350 Well, if I plug in 0 here, I get 1 * 1. 593 00:37:04,350 --> 00:37:05,940 So this thing is 1. 594 00:37:05,940 --> 00:37:07,550 So that's no surprise. 595 00:37:07,550 --> 00:37:11,260 And that's what I got. 596 00:37:11,260 --> 00:37:15,600 If I computed f', by the product rule 597 00:37:15,600 --> 00:37:19,150 it would be an annoying, somewhat long, computation. 598 00:37:19,150 --> 00:37:21,510 And because of what we just done, 599 00:37:21,510 --> 00:37:23,130 we know what it has to be. 600 00:37:23,130 --> 00:37:25,990 It has to be negative 7/2. 601 00:37:25,990 --> 00:37:28,280 Because this is a shortcut for doing it. 602 00:37:28,280 --> 00:37:29,900 This is faster than doing that. 603 00:37:29,900 --> 00:37:32,190 But of course, that's a legal way of doing it. 604 00:37:32,190 --> 00:37:33,780 When you get to second derivatives, 605 00:37:33,780 --> 00:37:36,210 you'll quickly discover that this method that I've just 606 00:37:36,210 --> 00:37:38,950 described is complicated, but far 607 00:37:38,950 --> 00:37:41,330 superior to differentiating this expression twice. 608 00:37:41,330 --> 00:37:46,087 STUDENT: [INAUDIBLE] PROF. 609 00:37:46,087 --> 00:37:48,420 JERISON: Would you have to throw away an x^2 term if you 610 00:37:48,420 --> 00:37:49,560 differentiated? 611 00:37:49,560 --> 00:37:50,470 No. 612 00:37:50,470 --> 00:37:53,220 And in fact, we didn't really have to do that here. 613 00:37:53,220 --> 00:37:55,385 If you differentiate and then plug in x = 0. 614 00:37:55,385 --> 00:37:57,510 So if you differentiate this and you plug in x = 0, 615 00:37:57,510 --> 00:37:58,970 you get -7/2. 616 00:37:58,970 --> 00:38:01,349 You differentiate this and you plug in x = 0, 617 00:38:01,349 --> 00:38:03,140 this term still drops out because it's just 618 00:38:03,140 --> 00:38:05,370 a 3x when you differentiate. 619 00:38:05,370 --> 00:38:08,270 And then you plug in x = 0, it's gone too. 620 00:38:08,270 --> 00:38:10,650 And similarly, if you're up here, it goes away 621 00:38:10,650 --> 00:38:12,410 and similarly over here it goes away. 622 00:38:12,410 --> 00:38:18,555 So the higher-order terms never influence this computation 623 00:38:18,555 --> 00:38:19,055 here. 624 00:38:19,055 --> 00:38:27,430 This just captures the linear features of the function. 625 00:38:27,430 --> 00:38:30,980 So now I want to go on to quadratic approximation. 626 00:38:30,980 --> 00:38:44,500 And now we're going to elaborate on this formula. 627 00:38:44,500 --> 00:38:46,040 So, linear approximation. 628 00:38:46,040 --> 00:38:49,840 Well, that should have been linear approximation. 629 00:38:49,840 --> 00:38:50,530 Liner. 630 00:38:50,530 --> 00:38:51,680 That's interesting. 631 00:38:51,680 --> 00:38:54,070 OK, so that was wrong. 632 00:38:54,070 --> 00:38:59,700 But now we're going to change it to quadratic. 633 00:38:59,700 --> 00:39:04,280 So, suppose we talk about a quadratic approximation here. 634 00:39:04,280 --> 00:39:07,450 Now, the quadratic approximation is 635 00:39:07,450 --> 00:39:15,430 going to be just an elaboration, one more step of detail. 636 00:39:15,430 --> 00:39:16,270 From the linear. 637 00:39:16,270 --> 00:39:18,060 In other words, it's an extension 638 00:39:18,060 --> 00:39:20,230 of the linear approximation. 639 00:39:20,230 --> 00:39:24,320 And so we're adding one more term here. 640 00:39:24,320 --> 00:39:26,650 And the extra term turns out to be related 641 00:39:26,650 --> 00:39:28,990 to the second derivative. 642 00:39:28,990 --> 00:39:34,340 But there's a factor of 2. 643 00:39:34,340 --> 00:39:39,090 So this is the formula for the quadratic approximation. 644 00:39:39,090 --> 00:39:46,450 And this chunk of it, of course, is the linear part. 645 00:39:46,450 --> 00:39:54,190 This time I'll spell 'linear' correctly. 646 00:39:54,190 --> 00:39:56,030 So the linear part is the first piece. 647 00:39:56,030 --> 00:40:05,050 And the quadratic part is the second piece. 648 00:40:05,050 --> 00:40:09,630 I want to develop this same catalog of functions 649 00:40:09,630 --> 00:40:11,140 as I had before. 650 00:40:11,140 --> 00:40:14,640 In other words, I want to extend our formulas 651 00:40:14,640 --> 00:40:19,660 to the higher-order terms. 652 00:40:19,660 --> 00:40:26,070 And if you do that for this example here, 653 00:40:26,070 --> 00:40:28,180 maybe I'll even illustrate with this example 654 00:40:28,180 --> 00:40:31,050 before I go on, if you do it with this example 655 00:40:31,050 --> 00:40:39,320 here, just to give you a flavor for what goes on, 656 00:40:39,320 --> 00:40:41,140 what turns out to be the case. 657 00:40:41,140 --> 00:40:45,390 So this is the linear version. 658 00:40:45,390 --> 00:40:48,220 And now I'm going to compare it to the quadratic version. 659 00:40:48,220 --> 00:40:55,540 So the quadratic version turns out to be this. 660 00:40:55,540 --> 00:40:58,760 That's what turns out to be the quadratic approximation. 661 00:40:58,760 --> 00:41:03,100 And when I use this example here, 662 00:41:03,100 --> 00:41:09,400 so this is 1.1, which is the same as ln of 1 + 1/10, right? 663 00:41:09,400 --> 00:41:17,430 So that's approximately 1/10 - 1/2 (1/10)^2. 664 00:41:17,430 --> 00:41:19,170 So 1/200. 665 00:41:19,170 --> 00:41:21,960 So that turns out, instead of being 666 00:41:21,960 --> 00:41:29,160 1/10, that's point, what is it, .095 or something like that. 667 00:41:29,160 --> 00:41:31,370 It's a little bit less. 668 00:41:31,370 --> 00:41:36,240 It's not .1, but it's pretty close. 669 00:41:36,240 --> 00:41:39,350 So if you like, the correction is 670 00:41:39,350 --> 00:41:48,900 lower in the decimal expansion. 671 00:41:48,900 --> 00:41:53,650 Now let me actually check a few of these. 672 00:41:53,650 --> 00:41:54,940 I'll carry them out. 673 00:41:54,940 --> 00:41:58,670 And what I'm going to probably save for next time 674 00:41:58,670 --> 00:42:08,020 is explaining to you, so this is why this factor of 1/2, 675 00:42:08,020 --> 00:42:10,610 and we're going to do this later. 676 00:42:10,610 --> 00:42:11,530 Do this next time. 677 00:42:11,530 --> 00:42:17,230 You can certainly do well to stick with this presentation 678 00:42:17,230 --> 00:42:18,470 for one more lecture. 679 00:42:18,470 --> 00:42:22,210 So we can see this reinforced. 680 00:42:22,210 --> 00:42:32,580 So now I'm going to work out these derivatives 681 00:42:32,580 --> 00:42:34,630 of the higher-order terms. 682 00:42:34,630 --> 00:42:39,450 And let me do it for the x approximately 0 case. 683 00:42:39,450 --> 00:42:47,990 So first of all, I want to add in the extra term here. 684 00:42:47,990 --> 00:42:50,830 Here's the extra term. 685 00:42:50,830 --> 00:42:53,780 For the quadratic part. 686 00:42:53,780 --> 00:42:57,050 And now in order to figure out what's going on, 687 00:42:57,050 --> 00:43:03,350 I'm going to need to compute, also, second derivatives. 688 00:43:03,350 --> 00:43:05,150 So here I need a second derivative. 689 00:43:05,150 --> 00:43:11,465 And I need to throw in the value of that second derivative at 0. 690 00:43:11,465 --> 00:43:13,340 So this is what I'm going to need to compute. 691 00:43:13,340 --> 00:43:17,449 So if I do it, for example, for the sine function, 692 00:43:17,449 --> 00:43:18,740 I already have the linear part. 693 00:43:18,740 --> 00:43:20,290 I need this last bit. 694 00:43:20,290 --> 00:43:22,570 So I differentiate the sine function twice 695 00:43:22,570 --> 00:43:25,180 and I get, I claim minus the sine function. 696 00:43:25,180 --> 00:43:26,900 The first derivative is the cosine 697 00:43:26,900 --> 00:43:29,250 and the cosine derivative is minus the sine. 698 00:43:29,250 --> 00:43:34,180 And when I evaluate it at 0, I get, lo and behold, 0. 699 00:43:34,180 --> 00:43:35,540 Sine of 0 is 0. 700 00:43:35,540 --> 00:43:40,361 So actually the quadratic approximation is the same. 701 00:43:40,361 --> 00:43:40,860 0x^2. 702 00:43:40,860 --> 00:43:43,070 There's no x^2 term here. 703 00:43:43,070 --> 00:43:46,510 So that's why this is such a terrific approximation. 704 00:43:46,510 --> 00:43:48,890 It's also the quadratic approximation. 705 00:43:48,890 --> 00:43:53,460 For the cosine function, if you differentiate twice, 706 00:43:53,460 --> 00:43:56,300 you get the derivative is minus the sign and derivative 707 00:43:56,300 --> 00:44:00,170 of that is minus the cosine. 708 00:44:00,170 --> 00:44:03,060 So that's f''. 709 00:44:03,060 --> 00:44:09,600 And now, if I evaluate that at 0, I get -1. 710 00:44:09,600 --> 00:44:11,530 And so the term that I have to plug in here, 711 00:44:11,530 --> 00:44:15,240 this -1 is the coefficient that appears right here. 712 00:44:15,240 --> 00:44:23,350 So I need a -1/2 x^2 extra. 713 00:44:23,350 --> 00:44:26,100 And if you do it for the e^x, you get an e^x, 714 00:44:26,100 --> 00:44:39,450 and you got a 1 and so you get 1/2 x^2 here. 715 00:44:39,450 --> 00:44:42,329 I'm going to finish these two in just a second, 716 00:44:42,329 --> 00:44:43,745 but I first want to tell you about 717 00:44:43,745 --> 00:44:56,480 the geometric significance of this quadratic term. 718 00:44:56,480 --> 00:44:58,790 So here we go. 719 00:44:58,790 --> 00:45:18,430 Geometric significance (of the quadratic term). 720 00:45:18,430 --> 00:45:21,100 So the geometric significance is best 721 00:45:21,100 --> 00:45:25,670 to describe just by drawing a picture here. 722 00:45:25,670 --> 00:45:29,300 And I'm going to draw the picture of the cosine function. 723 00:45:29,300 --> 00:45:34,270 And remember we already had the tangent line. 724 00:45:34,270 --> 00:45:38,620 So the tangent line was this horizontal here. 725 00:45:38,620 --> 00:45:40,350 And that was y = 1. 726 00:45:40,350 --> 00:45:42,880 But you can see intuitively, that doesn't even 727 00:45:42,880 --> 00:45:46,130 tell you whether this function is above or below 1 there. 728 00:45:46,130 --> 00:45:47,437 Doesn't tell you much. 729 00:45:47,437 --> 00:45:50,020 It's sort of begging for there to be a little more information 730 00:45:50,020 --> 00:45:52,470 to tell us what the function is doing nearby. 731 00:45:52,470 --> 00:45:57,470 And indeed, that's what this second expression does for us. 732 00:45:57,470 --> 00:46:00,850 It's some kind of parabola underneath here. 733 00:46:00,850 --> 00:46:05,420 So this is y = 1 - 1/2 x^2. 734 00:46:05,420 --> 00:46:08,890 Which is a much better fit to the curve 735 00:46:08,890 --> 00:46:12,740 than the horizontal line. 736 00:46:12,740 --> 00:46:23,750 And this is, if you like, this is the best fit parabola. 737 00:46:23,750 --> 00:46:28,510 So it's going to be the closest parabola to the curve. 738 00:46:28,510 --> 00:46:31,370 And that's more or less the significance. 739 00:46:31,370 --> 00:46:34,600 It's much, much closer. 740 00:46:34,600 --> 00:46:40,220 All right, I want to give you, well, 741 00:46:40,220 --> 00:46:43,040 I think we'll save these other derivations for next time 742 00:46:43,040 --> 00:46:44,880 because I think we're out of time now. 743 00:46:44,880 --> 00:46:47,110 So we'll do these next time.