1 00:00:00 --> 00:00:00 2 00:00:00 --> 00:00:02 The following content is provided under a Creative 3 00:00:02 --> 00:00:03 Commons license. 4 00:00:03 --> 00:00:06 Your support will help MIT OpenCourseWare continue to 5 00:00:06 --> 00:00:10 offer high quality educational resources for free. 6 00:00:10 --> 00:00:12 To make a donation or to view additional materials from 7 00:00:12 --> 00:00:16 hundreds of MIT courses, visit MIT OpenCourseWare 8 00:00:16 --> 00:00:22 at ocw.mit.edu. 9 00:00:22 --> 00:00:22 PROF. 10 00:00:22 --> 00:00:25 JERISON: So, we're ready to begin Lecture 10, and what 11 00:00:25 --> 00:00:29 I'm going to begin with is by finishing up some 12 00:00:29 --> 00:00:33 things from last time. 13 00:00:33 --> 00:00:42 We'll talk about approximations, and I want to 14 00:00:42 --> 00:00:50 fill in a number of comments and get you a little bit more 15 00:00:50 --> 00:00:54 oriented in the point of view that I'm trying to express 16 00:00:54 --> 00:00:55 about approximations. 17 00:00:55 --> 00:00:59 So, first of all, I want to remind you of the actual 18 00:00:59 --> 00:01:03 applied example that I wrote down last time. 19 00:01:03 --> 00:01:08 So that was this business here. 20 00:01:08 --> 00:01:11 There was something from special relativity. 21 00:01:11 --> 00:01:15 And the approximation that we used was the linear 22 00:01:15 --> 00:01:19 approximation, with a - 1/2 power that comes 23 00:01:19 --> 00:01:21 out to be t( 1 24 00:01:21 --> 00:01:23 1/2 v^2 / C^2). 25 00:01:23 --> 00:01:27 26 00:01:27 --> 00:01:30 I want to reiterate why this is a useful way 27 00:01:30 --> 00:01:31 of thinking of things. 28 00:01:31 --> 00:01:34 And why this is that this comes up in real life. 29 00:01:34 --> 00:01:37 Why this is maybe more important than everything 30 00:01:37 --> 00:01:39 that I've taught you about technically so far. 31 00:01:39 --> 00:01:47 So, first of all, what this is telling us is the change in t / 32 00:01:47 --> 00:01:51 t, if you do the arithmetic here and subtract t that's 33 00:01:51 --> 00:01:55 using the change in t is t' - t here. 34 00:01:55 --> 00:01:59 If you work that out, this is approximately the 35 00:01:59 --> 00:02:02 same as 1/2 (v^2 / C^2). 36 00:02:04 --> 00:02:06 So what is this saying? 37 00:02:06 --> 00:02:08 This is saying that if you have this satellite, which is going 38 00:02:08 --> 00:02:14 at speed v, and little c is the speed of light, then the change 39 00:02:14 --> 00:02:19 in the watch down here on earth, relative to the time on 40 00:02:19 --> 00:02:22 the satellite, is going to be proportional to 41 00:02:22 --> 00:02:24 this ratio here. 42 00:02:24 --> 00:02:25 So, physically, this makes sense. 43 00:02:25 --> 00:02:28 This is time divided by time. 44 00:02:28 --> 00:02:30 And this is velocity squared divided by velocity squared. 45 00:02:30 --> 00:02:32 So, in each case, the units divide out. 46 00:02:32 --> 00:02:34 So this is a dimensionless quantity. 47 00:02:34 --> 00:02:36 And this is a dimensionless quantity. 48 00:02:36 --> 00:02:41 And the only point here that we're trying to make is just 49 00:02:41 --> 00:02:43 this notion of proportionality. 50 00:02:43 --> 00:02:45 So I want to write this down. 51 00:02:45 --> 00:02:48 Just, in summary. 52 00:02:48 --> 00:02:51 So the error fraction, if you like, which is sort of the 53 00:02:51 --> 00:02:55 number of significant digits that we have in our 54 00:02:55 --> 00:03:04 measurement, is proportional, in this case, to this quantity. 55 00:03:04 --> 00:03:09 It happens to be proportional to this quantity here. 56 00:03:09 --> 00:03:16 And the factor is, happens to be, 1/2. 57 00:03:16 --> 00:03:20 So these proportionality factors are what 58 00:03:20 --> 00:03:21 we're looking for. 59 00:03:21 --> 00:03:22 Their rates of change. 60 00:03:22 --> 00:03:24 Their rates of change of something with respect 61 00:03:24 --> 00:03:25 to something else. 62 00:03:25 --> 00:03:29 Now, on your homework, you have something rather 63 00:03:29 --> 00:03:30 similar to this. 64 00:03:30 --> 00:03:39 So in Problem, on Part 2b, Part II, Problem 1, there's the 65 00:03:39 --> 00:03:41 speed of a pitch, right? 66 00:03:41 --> 00:03:44 And the speed of the pitch is changing depending on 67 00:03:44 --> 00:03:45 how high the mound is. 68 00:03:45 --> 00:03:48 And the point here is that that's approximately 69 00:03:48 --> 00:03:52 proportional to the change in the height of the mound. 70 00:03:52 --> 00:03:55 In that problem, we had this delta h, that was the x 71 00:03:55 --> 00:03:56 variable in that problem. 72 00:03:56 --> 00:03:59 And what you're trying to figure out is what the constant 73 00:03:59 --> 00:04:03 of proportionality is. 74 00:04:03 --> 00:04:04 That's what you're aiming for in this problem. 75 00:04:04 --> 00:04:08 So there's a linear relationship, approximately, 76 00:04:08 --> 00:04:11 to all intents and purposes this is an equality. 77 00:04:11 --> 00:04:14 Because the lower order terms are unimportant 78 00:04:14 --> 00:04:14 for the problem. 79 00:04:14 --> 00:04:16 Just as over here, this function is a little 80 00:04:16 --> 00:04:18 bit complicated. 81 00:04:18 --> 00:04:19 This function is a little more simple. 82 00:04:19 --> 00:04:22 For the purposes of this problem, they are the same. 83 00:04:22 --> 00:04:28 Because the errors are negligible for the particular 84 00:04:28 --> 00:04:29 problem that we're working on. 85 00:04:29 --> 00:04:34 So we might as well work with the simpler relationship. 86 00:04:34 --> 00:04:37 And similarly, over here, so you could do this with, in 87 00:04:37 --> 00:04:40 this case with square roots, it's not so hard here with 88 00:04:40 --> 00:04:41 reciprocals of square roots. 89 00:04:41 --> 00:04:45 It's also not terribly hard to do it numerically. 90 00:04:45 --> 00:04:48 And the reason why we're not doing it numerically is 91 00:04:48 --> 00:04:51 that, as I say, this is something that happens 92 00:04:51 --> 00:04:53 all across engineering. 93 00:04:53 --> 00:04:56 People are looking for these linear relationships between 94 00:04:56 --> 00:05:00 the change in some input and the change in the output. 95 00:05:00 --> 00:05:03 And if you don't make these simplifications, then when you 96 00:05:03 --> 00:05:07 get, say, a dozen of them together, you can't figure 97 00:05:07 --> 00:05:09 out what's going on. 98 00:05:09 --> 00:05:12 In this case the design of the satellite, it's very important. 99 00:05:12 --> 00:05:15 The speed actually isn't just one speed. 100 00:05:15 --> 00:05:19 Because it's the relative speed of u to the satellite. 101 00:05:19 --> 00:05:21 And you might be, it depends on your angle of sight with the 102 00:05:21 --> 00:05:22 satellite what the speed is. 103 00:05:22 --> 00:05:24 So it varies quite a bit. 104 00:05:24 --> 00:05:26 So you really need this rule of thumb. 105 00:05:26 --> 00:05:28 Then there are all kinds of other considerations 106 00:05:28 --> 00:05:29 in this question. 107 00:05:29 --> 00:05:31 Like, for example, there's the fact that we're sitting on 108 00:05:31 --> 00:05:34 Earth and so we're rotating around on what's called 109 00:05:34 --> 00:05:36 a non-inertial frame. 110 00:05:36 --> 00:05:38 So there's the question of that acceleration. 111 00:05:38 --> 00:05:41 There's the question that the gravity that I experience here 112 00:05:41 --> 00:05:44 on Earth is not the same as up at the satellite. 113 00:05:44 --> 00:05:48 And that also creates a difference in time, as a 114 00:05:48 --> 00:05:49 result of general relativity. 115 00:05:49 --> 00:05:54 So all of these considerations come down to formulas which 116 00:05:54 --> 00:05:56 are this complicated or maybe a tiny bit more. 117 00:05:56 --> 00:05:58 Not really that much. 118 00:05:58 --> 00:06:00 And then people simplify them enormously to these very 119 00:06:00 --> 00:06:02 simple-minded rules. 120 00:06:02 --> 00:06:05 And they don't keep track of what's going on. 121 00:06:05 --> 00:06:08 So in order to design the system, you must make these 122 00:06:08 --> 00:06:10 simplifications, otherwise you can't even think 123 00:06:10 --> 00:06:12 about what's going on. 124 00:06:12 --> 00:06:13 This comes up in everything. 125 00:06:13 --> 00:06:17 In weather forecasting, economic forecasting. 126 00:06:17 --> 00:06:19 Figuring out whether there's going to be an asteroid that's 127 00:06:19 --> 00:06:22 going to hit the Earth. 128 00:06:22 --> 00:06:24 Every single one of these things involves dozens of 129 00:06:24 --> 00:06:27 these considerations. 130 00:06:27 --> 00:06:30 OK, there was a question that I saw, here. 131 00:06:30 --> 00:06:30 Yes. 132 00:06:30 --> 00:06:38 STUDENT: [INAUDIBLE] 133 00:06:38 --> 00:06:39 PROF. 134 00:06:39 --> 00:06:40 JERISON: Yeah. 135 00:06:40 --> 00:06:42 Basically, any problem where you have a derivative, the rate 136 00:06:42 --> 00:06:46 of change also depends upon what the base point is. 137 00:06:46 --> 00:06:46 That's the question. 138 00:06:46 --> 00:06:50 You're saying, doesn't this delta v also depend, I had a 139 00:06:50 --> 00:06:51 base point in that problem. 140 00:06:51 --> 00:06:53 I happened to decide that pitchers pitch on average 141 00:06:53 --> 00:06:55 about 90 miles an hour. 142 00:06:55 --> 00:06:58 Whereas, in fact, some pitchers pitch at 100 miles an hour, 143 00:06:58 --> 00:07:00 some pitch at 80 miles an hour, and of course they vary 144 00:07:00 --> 00:07:02 the speed of the pitch. 145 00:07:02 --> 00:07:03 And so, this varies a little bit. 146 00:07:03 --> 00:07:05 In fact, that's sort of a second order effect. 147 00:07:05 --> 00:07:08 It does change the constant of proportionality. 148 00:07:08 --> 00:07:11 It's a rate of change at a different base point. 149 00:07:11 --> 00:07:14 Which we're considering fixed. 150 00:07:14 --> 00:07:17 In fact, that's sort of a second order effect. 151 00:07:17 --> 00:07:19 When you actually do the computations, what you discover 152 00:07:19 --> 00:07:21 is that it doesn't make that much difference. 153 00:07:21 --> 00:07:22 To the a. 154 00:07:22 --> 00:07:25 And that's something that you get from experience. 155 00:07:25 --> 00:07:28 That it turns out, which things matter and which things don't. 156 00:07:28 --> 00:07:31 And yet again, that's exactly the same sort of consideration 157 00:07:31 --> 00:07:34 but at the next order of what I'm talking about here is. 158 00:07:34 --> 00:07:36 You have to have enough experience with numbers to know 159 00:07:36 --> 00:07:39 that if you take, if you vary something a little bit it's not 160 00:07:39 --> 00:07:42 going to change the answer that you're looking for very much. 161 00:07:42 --> 00:07:45 And that's exactly the point that I'm making. 162 00:07:45 --> 00:07:51 So I can't make them all at once, all such points. 163 00:07:51 --> 00:07:55 So that's my pitch for understanding things from 164 00:07:55 --> 00:07:56 this point of view. 165 00:07:56 --> 00:08:02 Now, we're going to go on, now, to quadratic approximations, 166 00:08:02 --> 00:08:13 which are a little more complicated. 167 00:08:13 --> 00:08:15 So, we talked a little bit about this last time 168 00:08:15 --> 00:08:16 but I didn't finish. 169 00:08:16 --> 00:08:19 So I want to finish this up. 170 00:08:19 --> 00:08:23 And the first thing that I should say is that you 171 00:08:23 --> 00:08:35 use the when the linear approximation is not enough. 172 00:08:35 --> 00:08:38 OK, so, that's something that you really need to get a 173 00:08:38 --> 00:08:40 little experience with. 174 00:08:40 --> 00:08:43 In economics, I told you they use logarithms. 175 00:08:43 --> 00:08:46 So sometimes they use log linear functions. 176 00:08:46 --> 00:08:48 Sometimes they use log quadratic functions when the 177 00:08:48 --> 00:08:49 log linear ones don't work. 178 00:08:49 --> 00:08:52 So most modeling in economics is with log 179 00:08:52 --> 00:08:53 quadratic functions. 180 00:08:53 --> 00:08:55 And if you've made it any more complicated than 181 00:08:55 --> 00:08:56 that, it's useless. 182 00:08:56 --> 00:08:57 And it's a mess. 183 00:08:57 --> 00:08:58 And people don't do it. 184 00:08:58 --> 00:09:02 So they stick with the quadratic ones, typically. 185 00:09:02 --> 00:09:07 So the basic formula here, and I'm going to take the base 186 00:09:07 --> 00:09:11 point to be 0, is that f ( x ) is approximately f ( 0 ) 187 00:09:12 --> 00:09:14 f' ( 0 )x. 188 00:09:14 --> 00:09:16 That's the linear part. 189 00:09:16 --> 00:09:17 Plus this extra term. 190 00:09:17 --> 00:09:20 Which is f'' ( 0 ) / 2x^2. 191 00:09:21 --> 00:09:27 And this is supposed to work for x near 0. 192 00:09:27 --> 00:09:34 So it shows in the base point as simply as possible. 193 00:09:34 --> 00:09:38 So here's more or less where we left off last time. 194 00:09:38 --> 00:09:42 And one thing that I said I was going to explain, which I will 195 00:09:42 --> 00:09:48 now, is why it's (1/2) f'' ( 0 ). 196 00:09:48 --> 00:09:51 So we need to know that. 197 00:09:51 --> 00:09:54 So let's work that out here first of all. 198 00:09:54 --> 00:09:57 So I'm just going to do it by example. 199 00:09:57 --> 00:10:00 So if you like, the answer is just, well, what happens 200 00:10:00 --> 00:10:03 when you have a parabola? 201 00:10:03 --> 00:10:06 A parabola's a quadratic. 202 00:10:06 --> 00:10:09 It had better, its quadratic approximation had 203 00:10:09 --> 00:10:10 better be itself. 204 00:10:10 --> 00:10:11 It's got to be the best one. 205 00:10:11 --> 00:10:13 So it's got to be itself. 206 00:10:13 --> 00:10:15 So this formula, if it's going to work, has 207 00:10:15 --> 00:10:19 to work on the nose. 208 00:10:19 --> 00:10:21 For quadratic functions. 209 00:10:21 --> 00:10:23 So, let's take a look. 210 00:10:23 --> 00:10:26 If I differentiate, I get b 211 00:10:26 --> 00:10:28 2cx. 212 00:10:28 --> 00:10:32 If I differentiate a second time, I get 2c. 213 00:10:32 --> 00:10:34 And now let's plug it in. 214 00:10:34 --> 00:10:39 Well, we can recover, what is it that we want to recover? 215 00:10:39 --> 00:10:42 We want to recover these numbers a, b and c using the 216 00:10:42 --> 00:10:45 derivatives evaluated at 0. 217 00:10:45 --> 00:10:49 So let's see. 218 00:10:49 --> 00:10:52 It's pretty easy, actually. f ( 0 ) = a. 219 00:10:52 --> 00:10:53 That's on the nose. 220 00:10:53 --> 00:10:56 If you plug in x = 0 here, these terms drop 221 00:10:56 --> 00:10:58 out and you get a. 222 00:10:58 --> 00:11:02 And now, f' ( 0 ), whoops that was wrong. 223 00:11:02 --> 00:11:05 So I wrote f' but what I meant was f. 224 00:11:05 --> 00:11:07 So f ( 0 ) is a. 225 00:11:07 --> 00:11:13 Let's back up. f ( 0 ) is a, so if I plug in x = 0 I get a. 226 00:11:13 --> 00:11:18 Now, f' ( 0 ), that's this next formula here, f' ( 0 ), I 227 00:11:18 --> 00:11:21 plug in 0 here, and I get b. 228 00:11:21 --> 00:11:22 That's also good. 229 00:11:22 --> 00:11:24 And that's exactly what the linear approximation 230 00:11:24 --> 00:11:25 is supposed to be. 231 00:11:25 --> 00:11:28 But now you notice, f'' is 2c. 232 00:11:28 --> 00:11:34 So to recover c, I better take half of it. 233 00:11:34 --> 00:11:35 And that's it. 234 00:11:35 --> 00:11:38 That's the reason. 235 00:11:38 --> 00:11:40 There's no chance that any other formula could work. 236 00:11:40 --> 00:11:45 And this one does. 237 00:11:45 --> 00:11:50 So that's the explanation for the formula. 238 00:11:50 --> 00:11:54 And now I remind you that I had a collection of basic formulas 239 00:11:54 --> 00:11:55 written on the board. 240 00:11:55 --> 00:12:00 And I want to just make sure we know all of them again. 241 00:12:00 --> 00:12:07 So, first of all, there was sine x is approximately x. 242 00:12:07 --> 00:12:10 Cosine x is approximately 1 - 1/2 x^2. 243 00:12:12 --> 00:12:15 And e ^ x is approximately 1 244 00:12:15 --> 00:12:17 x 245 00:12:17 --> 00:12:17 1/2 x^2. 246 00:12:19 --> 00:12:22 So those were three that I mentioned last time. 247 00:12:22 --> 00:12:28 And, again, this is all for x near 0. 248 00:12:28 --> 00:12:31 All for x near 0 only. 249 00:12:31 --> 00:12:35 These are wildly wrong, Far away, but near 0 they're nice, 250 00:12:35 --> 00:12:37 good, quadratic approximations. 251 00:12:37 --> 00:12:39 Now, the other two approximations that I want to 252 00:12:39 --> 00:12:46 mention are the logarithm and we use the base point shifted. 253 00:12:46 --> 00:12:50 So we can put it at x near 0. 254 00:12:50 --> 00:12:52 And this one - sorry, this is an approximately 255 00:12:52 --> 00:12:55 equals sign there. 256 00:12:55 --> 00:12:57 Turns out to be x - 1/2 x^2. 257 00:12:59 --> 00:13:02 And the last one is one (1 258 00:13:02 --> 00:13:07 x) ^ r, which turns out to be 1 259 00:13:07 --> 00:13:08 rx 260 00:13:09 --> 00:13:12 r ( r - 1) / 2x^2. 261 00:13:14 --> 00:13:20 Now, eventually, your mind will converge on all of these and 262 00:13:20 --> 00:13:23 you'll find them relatively easy to memorize. 263 00:13:23 --> 00:13:25 But it'll take some getting used to. 264 00:13:25 --> 00:13:30 And I'm not claiming that you should recognize them and 265 00:13:30 --> 00:13:32 understand them all now. 266 00:13:32 --> 00:13:35 But I'm going to put a giant box around this. 267 00:13:35 --> 00:13:42 STUDENT: [INAUDIBLE] 268 00:13:42 --> 00:13:42 PROF. 269 00:13:42 --> 00:13:42 JERISON: Yes. 270 00:13:42 --> 00:13:45 So the question was, you get all of these if you use 271 00:13:45 --> 00:13:45 that equation there. 272 00:13:45 --> 00:13:48 That's exactly what are you going to do. so I already 273 00:13:48 --> 00:13:52 did it actually for these three, last time. 274 00:13:52 --> 00:13:55 But I didn't do it yet for these two. 275 00:13:55 --> 00:13:58 But I will do it in about two minutes. 276 00:13:58 --> 00:14:00 Well, maybe five minutes. 277 00:14:00 --> 00:14:10 But first I want to explain just a few things about these. 278 00:14:10 --> 00:14:12 They all follow from the basic formula. 279 00:14:12 --> 00:14:16 In fact, that one deserves a pink box too, doesn't it. 280 00:14:16 --> 00:14:18 That one's pretty important. 281 00:14:18 --> 00:14:19 Alright. 282 00:14:19 --> 00:14:23 Yeah. 283 00:14:23 --> 00:14:25 Maybe even some little sparkles. 284 00:14:25 --> 00:14:32 Alright. 285 00:14:32 --> 00:14:33 OK. 286 00:14:33 --> 00:14:36 So that's pretty important. 287 00:14:36 --> 00:14:39 Almost as important as the more basic one without 288 00:14:39 --> 00:14:41 this term here. 289 00:14:41 --> 00:14:47 So now, let me just tell you a little bit more about 290 00:14:47 --> 00:14:54 the significance. 291 00:14:54 --> 00:14:56 Again, this is just to reinforce something that 292 00:14:56 --> 00:14:57 we've already done. 293 00:14:57 --> 00:14:59 But it's closely related to what you're doing 294 00:14:59 --> 00:15:00 on your problem set. 295 00:15:00 --> 00:15:05 So it's worth your while to recall this. 296 00:15:05 --> 00:15:10 So, there's this expression that we were dealing with. 297 00:15:10 --> 00:15:13 And we talked about it in lecture. 298 00:15:13 --> 00:15:19 And we showed that this tends to e as k goes to infinity. 299 00:15:19 --> 00:15:21 So that's what we showed in lecture. 300 00:15:21 --> 00:15:25 And the way that we did that was, we took the logarithm 301 00:15:25 --> 00:15:29 and we wrote it as k times, sorry, the ln of 1 302 00:15:29 --> 00:15:32 (1 / k). 303 00:15:32 --> 00:15:35 And then we evaluated the limit of this. 304 00:15:35 --> 00:15:38 And I want to do this limit again, using 305 00:15:38 --> 00:15:40 linear approximation. 306 00:15:40 --> 00:15:41 To show you how easy it is. 307 00:15:41 --> 00:15:44 If you just remember the linear approximation. 308 00:15:44 --> 00:15:46 And then we'll explain where the quadratic 309 00:15:46 --> 00:15:48 approximation comes in. 310 00:15:48 --> 00:15:52 So I claim that this is approximately equal 311 00:15:52 --> 00:15:59 to k ( 1 / k). 312 00:15:59 --> 00:16:00 Now, why is that? 313 00:16:00 --> 00:16:04 Well, that's just this linear approximation. 314 00:16:04 --> 00:16:05 So what did I use here? 315 00:16:05 --> 00:16:07 I used ln of 1 316 00:16:07 --> 00:16:10 x is approximately x. 317 00:16:10 --> 00:16:14 For x = 1 / k. 318 00:16:14 --> 00:16:18 That's what I used in this approximation here. 319 00:16:18 --> 00:16:20 And that's the linear approximation to the 320 00:16:20 --> 00:16:24 natural logarithm. 321 00:16:24 --> 00:16:27 And this number is relatively easy to evaluate. 322 00:16:27 --> 00:16:28 I know how to do it. 323 00:16:28 --> 00:16:31 It's equal to 1. 324 00:16:31 --> 00:16:34 That's the same, well, so where does this work? 325 00:16:34 --> 00:16:37 This works where this thing is near 0. 326 00:16:37 --> 00:16:41 Which is when k is going to infinity. 327 00:16:41 --> 00:16:44 This thing is working only when k is going to infinity. 328 00:16:44 --> 00:16:46 So what it's really saying, this approximation formula, 329 00:16:46 --> 00:16:50 it's really saying that as we go to infinity, in k, 330 00:16:50 --> 00:16:54 this thing is going to 1. 331 00:16:54 --> 00:16:58 As k goes to infinity. 332 00:16:58 --> 00:17:00 So that's what it's saying. 333 00:17:00 --> 00:17:01 That's the substance there. 334 00:17:01 --> 00:17:05 And that's how we want to use it, in many instances. 335 00:17:05 --> 00:17:06 Just to evaluate limits. 336 00:17:06 --> 00:17:09 We also want to realize that it's nearby when k 337 00:17:09 --> 00:17:12 is pretty large, like 100 or something like that. 338 00:17:12 --> 00:17:16 Now, so that's the idea of the linear approximation. 339 00:17:16 --> 00:17:21 Now, if you want to get the rate of convergence here, 340 00:17:21 --> 00:17:27 so the rate of what's called convergence. 341 00:17:27 --> 00:17:34 So convergence means how fast this is going towards that. 342 00:17:34 --> 00:17:36 What I have to do is take the difference. 343 00:17:36 --> 00:17:40 I have to take ln ak, and I have to subtract 1 from it. 344 00:17:40 --> 00:17:43 And I know that this is going to 0, and the question 345 00:17:43 --> 00:17:47 is how big is this. 346 00:17:47 --> 00:17:51 We want it to be very small. 347 00:17:51 --> 00:17:56 And the answer we're going to get, so the answer just uses 348 00:17:56 --> 00:18:03 the quadratic approximation. 349 00:18:03 --> 00:18:06 So if I just have a little bit more detail, then this 350 00:18:06 --> 00:18:10 expression here, in other words, I have the next 351 00:18:10 --> 00:18:11 higher order term. 352 00:18:11 --> 00:18:13 This is like 1 / k, this is like 1 / k^2. 353 00:18:15 --> 00:18:21 Then I can understand how big the difference is between the 354 00:18:21 --> 00:18:24 expression that I've got and its limit. 355 00:18:24 --> 00:18:26 And so that's what's on your homework. 356 00:18:26 --> 00:18:31 This is on your problem set. 357 00:18:31 --> 00:18:35 OK, so that is more or less an explanation for one of the 358 00:18:35 --> 00:18:38 things that quadratic approximations are good for. 359 00:18:38 --> 00:18:42 And I'm going to give you one more illustration. 360 00:18:42 --> 00:18:45 One more illustration. 361 00:18:45 --> 00:18:47 And then we'll actually check these formulas. 362 00:18:47 --> 00:18:48 Yeah, another question. 363 00:18:48 --> 00:18:55 STUDENT: [INAUDIBLE] 364 00:18:55 --> 00:18:56 PROF. 365 00:18:56 --> 00:18:58 JERISON: That's a very good question here. 366 00:18:58 --> 00:19:02 When they, which in this case means maybe, me, when I give 367 00:19:02 --> 00:19:10 you a question, does one specify whether you want 368 00:19:10 --> 00:19:14 to use a linear or a quadratic approximation. 369 00:19:14 --> 00:19:18 The answer is, in real life when you're faced with a 370 00:19:18 --> 00:19:23 problem like this, where some satellite is orbiting and you 371 00:19:23 --> 00:19:25 want to know the effects of gravity or something like 372 00:19:25 --> 00:19:28 that, nobody is going to tell you anything. 373 00:19:28 --> 00:19:30 They're not even going to tell you whether a linear 374 00:19:30 --> 00:19:33 approximation is relevant, or a quadratic or anything. 375 00:19:33 --> 00:19:36 So you're on your own. 376 00:19:36 --> 00:19:40 When I give you a question, at least for right now, I'm 377 00:19:40 --> 00:19:42 always going to tell you. 378 00:19:42 --> 00:19:47 But as time goes on I'd like you to get used to when it's 379 00:19:47 --> 00:19:49 enough to get away with a linear approximation. 380 00:19:49 --> 00:19:53 And you should only use a quadratic approximation if 381 00:19:53 --> 00:19:55 somebody forces you to. 382 00:19:55 --> 00:19:57 You should always start trying with a linear one. 383 00:19:57 --> 00:20:00 Because the quadratic ones are much more complicated as you'll 384 00:20:00 --> 00:20:03 see in this next example. 385 00:20:03 --> 00:20:06 OK, so the example that I want to use is, you're going to be 386 00:20:06 --> 00:20:09 stuck with it because I'm asking for the quadratic. 387 00:20:09 --> 00:20:16 So we're going to find the quadratic approximation 388 00:20:16 --> 00:20:23 near, for x near 0. 389 00:20:23 --> 00:20:24 To what? 390 00:20:24 --> 00:20:29 Well, this is the same function that we used 391 00:20:29 --> 00:20:30 in the last lecture. 392 00:20:30 --> 00:20:34 I think this was it. e ^ - 3x (1 393 00:20:34 --> 00:20:37 x) ^ - 1/2. 394 00:20:37 --> 00:20:40 OK. 395 00:20:40 --> 00:20:45 So, unfortunately, I stuck it in the wrong place 396 00:20:45 --> 00:20:47 to be able to fit this very long formula here. 397 00:20:47 --> 00:20:51 So I'm going to switch it. 398 00:20:51 --> 00:20:57 I'm just going to write it here. 399 00:20:57 --> 00:21:00 And we're going to just do the approximation. 400 00:21:00 --> 00:21:03 So we're going to say quadratic, in parentheses. 401 00:21:03 --> 00:21:08 And we'll say x near 0. 402 00:21:08 --> 00:21:12 So that's what I want. 403 00:21:12 --> 00:21:15 So now, here's what I have to do. 404 00:21:15 --> 00:21:18 Well, I have to write in the quadratic approximation for e 405 00:21:18 --> 00:21:25 ^ - 3x, and I'm going to use this formula right here. 406 00:21:25 --> 00:21:27 And so that's (1 407 00:21:27 --> 00:21:29 (- 3x) 408 00:21:30 --> 00:21:33 (- 3x)^2 / 2). 409 00:21:33 --> 00:21:39 And the other factor, I'm going to have to use this formula 410 00:21:39 --> 00:21:43 down here. because r is - 1/2. 411 00:21:43 --> 00:21:48 And so that's (1 - 1/2 x 412 00:21:48 --> 00:21:53 1/2 ( - 1/2)( - 3/2)x^2). 413 00:21:59 --> 00:22:09 So this is the r term, and this is the r - 1 term. 414 00:22:09 --> 00:22:12 And now I'm going to do something which is the 415 00:22:12 --> 00:22:15 only good thing about quadratic approximations. 416 00:22:15 --> 00:22:18 They're messy, they're long, there's nothing particularly 417 00:22:18 --> 00:22:19 good about them. 418 00:22:19 --> 00:22:21 But there is one good thing about them. 419 00:22:21 --> 00:22:25 Which is that you always get to ignore the higher order terms. 420 00:22:25 --> 00:22:29 So even though this looks like a very ugly multiplication, 421 00:22:29 --> 00:22:31 I can do it in my head. 422 00:22:31 --> 00:22:33 Just watching it. 423 00:22:33 --> 00:22:38 Because I get a 1 * 1, I'm forced with that term here. 424 00:22:38 --> 00:22:41 And then I get the cross terms which are linear, 425 00:22:41 --> 00:22:44 which is - 3x - 1/2 x. 426 00:22:44 --> 00:22:46 We already did that when we calculated the linear 427 00:22:46 --> 00:22:50 approximation, so that's this times the 1 and 428 00:22:50 --> 00:22:51 this times that 1. 429 00:22:51 --> 00:22:54 And now I have three cross-terms which 430 00:22:54 --> 00:22:55 are quadratic. 431 00:22:55 --> 00:22:58 So one of them is these two linear terms are multiplying. 432 00:22:58 --> 00:23:00 So that's plus 3/2 x^2. 433 00:23:02 --> 00:23:05 That's (- 3)( - 1/2). 434 00:23:05 --> 00:23:07 And then there's this term, multiplying the 435 00:23:07 --> 00:23:10 1, that's plus 9/2 x^2. 436 00:23:11 --> 00:23:15 And then there's one last term, which is this monster here. 437 00:23:15 --> 00:23:23 Multiplying 1, and that is - 3/8. 438 00:23:23 --> 00:23:31 So the great thing is, we drop x^3, x ^ 4, etc., terms. 439 00:23:31 --> 00:23:37 Yeah? 440 00:23:37 --> 00:23:39 STUDENT: [INAUDIBLE] 441 00:23:39 --> 00:23:40 PROF. 442 00:23:40 --> 00:23:42 JERISON: OK, well so copy it down. 443 00:23:42 --> 00:23:45 And you work it out as I'm doing it now. 444 00:23:45 --> 00:23:47 So what I did is, I multiplied 1 by 1. 445 00:23:47 --> 00:23:50 I'm using the distributive law here. 446 00:23:50 --> 00:23:51 That was this one. 447 00:23:51 --> 00:23:55 I multiplied this 3x by this one, that was that term. 448 00:23:55 --> 00:23:58 I multiplied this by this, that's that term. 449 00:23:58 --> 00:24:01 And then I multiplied this by this. 450 00:24:01 --> 00:24:03 In other words, 2 x terms that gave me an x^2 451 00:24:03 --> 00:24:06 and a (- 3)( - 1/2). 452 00:24:06 --> 00:24:08 And I'm going to stop at that point. 453 00:24:08 --> 00:24:10 Because the point is it's just all the rest of 454 00:24:10 --> 00:24:12 the terms that come up. 455 00:24:12 --> 00:24:14 Now, the reason, the only reason why it's easy is 456 00:24:14 --> 00:24:16 that I only have to go up to x squared term. 457 00:24:16 --> 00:24:21 I don't have to do the higher ones. 458 00:24:21 --> 00:24:22 Another question, way back here. 459 00:24:22 --> 00:24:23 Yeah, right there. 460 00:24:23 --> 00:24:29 STUDENT: [INAUDIBLE] 461 00:24:29 --> 00:24:29 PROF. 462 00:24:29 --> 00:24:29 JERISON: OK. 463 00:24:29 --> 00:24:38 So somebody can check my arithmetic, too. 464 00:24:38 --> 00:24:39 Good. 465 00:24:39 --> 00:24:40 STUDENT: [INAUDIBLE] 466 00:24:40 --> 00:24:40 PROF. 467 00:24:40 --> 00:24:43 JERISON: Why do I get to drop all the higher-order terms. 468 00:24:43 --> 00:24:46 So, that's because the situation where I'm going to 469 00:24:46 --> 00:24:52 apply this is the situation in which x is, say, 1/100. 470 00:24:52 --> 00:24:55 So if here's about 1/100. 471 00:24:55 --> 00:24:57 Here's something which is on the order of 100. 472 00:24:57 --> 00:24:58 This is on the order of 1/100^2. 473 00:25:00 --> 00:25:02 1/100^2, all of these terms. 474 00:25:02 --> 00:25:06 Now, these cubic and quartic terms are of 475 00:25:06 --> 00:25:09 the order of 1/100^3. 476 00:25:10 --> 00:25:12 That's 10 ^ - 6. 477 00:25:12 --> 00:25:14 And the point is that I'm not claiming that I 478 00:25:14 --> 00:25:16 have an exact answer. 479 00:25:16 --> 00:25:19 And I'm going to drop things of that order of magnitude. 480 00:25:19 --> 00:25:22 So I'm saving everything up to 4 decimal places. 481 00:25:22 --> 00:25:29 I'm throwing away things which are 6 decimal places out. 482 00:25:29 --> 00:25:30 Does that answer your question? 483 00:25:30 --> 00:25:35 STUDENT: [INAUDIBLE] 484 00:25:35 --> 00:25:35 PROF. 485 00:25:35 --> 00:25:36 JERISON: So. 486 00:25:36 --> 00:25:39 That's the situation, and now you can combine the terms. 487 00:25:39 --> 00:25:43 I mean, it's not very impressive here. 488 00:25:43 --> 00:25:48 This is equal to 1 - 7/2 x, maybe 489 00:25:50 --> 00:25:53 51/8 x^2. 490 00:25:54 --> 00:25:57 If I've made that, if those minus signs hadn't canceled, 491 00:25:57 --> 00:25:59 I would have gotten the wrong answer here. 492 00:25:59 --> 00:26:00 Anyway. 493 00:26:00 --> 00:26:03 So, this is a 2 here, sorry. 494 00:26:03 --> 00:26:04 7/2. 495 00:26:04 --> 00:26:07 This is the linear approximation we got last 496 00:26:07 --> 00:26:09 time and here's the extra information that we got 497 00:26:09 --> 00:26:11 from this calculation. 498 00:26:11 --> 00:26:17 Which is this 51/8 term. 499 00:26:17 --> 00:26:20 Right, you have to accept that there's a certain degree of 500 00:26:20 --> 00:26:23 complexity to this problem and the answer is sufficiently 501 00:26:23 --> 00:26:26 complicated so it can't be less arithmetic because we get this 502 00:26:26 --> 00:26:29 peculiar 51/8 there, right. 503 00:26:29 --> 00:26:33 So one of the things to realize is that these kinds of 504 00:26:33 --> 00:26:37 problems, because they involve many, many terms are always 505 00:26:37 --> 00:26:43 going to involve a little bit of complicated arithmetic. 506 00:26:43 --> 00:26:48 Last little bit, I did promise you that I was going to derive 507 00:26:48 --> 00:26:51 these two relations, as I said. 508 00:26:51 --> 00:26:53 Did the ones in the left column. 509 00:26:53 --> 00:26:56 So let's carry that out. 510 00:26:56 --> 00:27:00 And as someone just pointed out, it all comes from 511 00:27:00 --> 00:27:01 this formula here. 512 00:27:01 --> 00:27:07 So let's just check it. 513 00:27:07 --> 00:27:12 So we'll start with the ln function. 514 00:27:12 --> 00:27:16 This is the function, f, and then f' is 1/1 515 00:27:17 --> 00:27:18 x. 516 00:27:18 --> 00:27:24 And f'', so this is f', this is f'', is - 1/1 517 00:27:24 --> 00:27:25 x^2. 518 00:27:25 --> 00:27:28 519 00:27:28 --> 00:27:31 And now I have to plug in x = 0. 520 00:27:31 --> 00:27:35 So at x = 0 this is ln 1, which is 0. 521 00:27:35 --> 00:27:37 So this is at x = 0. 522 00:27:37 --> 00:27:41 I'm getting 0 here, I plug in 0 and I get 1. 523 00:27:41 --> 00:27:45 And here, I plug in 0 and I get - 1. 524 00:27:45 --> 00:27:49 So now I go and I look up at that formula, which is way 525 00:27:49 --> 00:27:50 in that upper corner there. 526 00:27:50 --> 00:27:53 And I see that the coefficient on the constant is 0. 527 00:27:53 --> 00:27:55 The coefficient on x is 1. 528 00:27:55 --> 00:27:59 And then the other coefficient, the very last one, is - 1/2. 529 00:27:59 --> 00:28:01 So this is the - 1 here. 530 00:28:01 --> 00:28:05 And then in the formula, there's a 2 in the denominator. 531 00:28:05 --> 00:28:07 So it's half of whatever I get for this second 532 00:28:07 --> 00:28:10 derivative, at 0. 533 00:28:10 --> 00:28:13 So this is the approximation formula, which is way up 534 00:28:13 --> 00:28:17 in that corner there. 535 00:28:17 --> 00:28:19 Similarly, if I do it for (1 536 00:28:19 --> 00:28:23 x) ^ r, I have to differentiate that I get r( 1 537 00:28:23 --> 00:28:28 x) ^ r - 1, and then r ( r - 1( x 538 00:28:29 --> 00:28:31 1) ^ r - 2. 539 00:28:31 --> 00:28:33 So here are the derivatives. 540 00:28:33 --> 00:28:41 And so if I evaluate them at x = 0, I get 1. 541 00:28:41 --> 00:28:43 That's 1 ^ r = 1. 542 00:28:43 --> 00:28:47 And here I get r. (1 ^ r - 1)r. 543 00:28:49 --> 00:28:59 And here, I plug in x = 0 and I get r ( r - 1). 544 00:28:59 --> 00:29:02 So again, the pattern is right above it here. 545 00:29:02 --> 00:29:05 The 1 is there, the r is there. 546 00:29:05 --> 00:29:08 And then instead of r ( r - 1), I have half that. 547 00:29:08 --> 00:29:21 For the coefficient. 548 00:29:21 --> 00:29:22 So these are just examples. 549 00:29:22 --> 00:29:24 Obviously if we had a more complicated functional, 550 00:29:24 --> 00:29:26 we might carry this out. 551 00:29:26 --> 00:29:29 But as a practical matter, we try to stick with the ones in 552 00:29:29 --> 00:29:42 the pink box and just use algebra to get other formulas. 553 00:29:42 --> 00:29:46 So I want to shift gears now and treat the subject that was 554 00:29:46 --> 00:29:48 supposed to be this lecture. 555 00:29:48 --> 00:29:53 And we're not quite caught up, but we will try to do our best 556 00:29:53 --> 00:29:54 to do as much as we can today. 557 00:29:54 --> 00:29:59 So the next topic is curve sketching. 558 00:29:59 --> 00:30:18 And so let's get started with that. 559 00:30:18 --> 00:30:23 So now, happily in this subject, there are more 560 00:30:23 --> 00:30:26 pictures and it's a little bit more geometric. 561 00:30:26 --> 00:30:30 And there's relatively little computation. 562 00:30:30 --> 00:30:33 So let's hope we can do this. 563 00:30:33 --> 00:30:37 So I want to, so here we go, we'll start with 564 00:30:37 --> 00:30:44 curve sketching. 565 00:30:44 --> 00:30:47 And the goal here 566 00:30:47 --> 00:30:56 STUDENT: [INAUDIBLE] 567 00:30:56 --> 00:30:56 PROF. 568 00:30:56 --> 00:31:00 JERISON: So that's like, liner, the last time. 569 00:31:00 --> 00:31:09 That's kind of sketchy spelling, isn't it? 570 00:31:09 --> 00:31:12 Yeah, there are certain kinds of things which I can't spell. 571 00:31:12 --> 00:31:16 But, alright. 572 00:31:16 --> 00:31:18 Sketching. 573 00:31:18 --> 00:31:19 Alright. 574 00:31:19 --> 00:31:21 So here's our goal. 575 00:31:21 --> 00:31:38 Our goal is to draw the graph of f, using f' and f''. 576 00:31:38 --> 00:31:42 577 00:31:42 --> 00:31:47 Whether they're positive or negative. 578 00:31:47 --> 00:31:48 So that's it. 579 00:31:48 --> 00:31:52 This is the goal here. 580 00:31:52 --> 00:31:57 However, there is a big warning that I want to give you. 581 00:31:57 --> 00:32:06 And this is one that unfortunately I now have to 582 00:32:06 --> 00:32:08 make you unlearn, especially those that you that have 583 00:32:08 --> 00:32:11 actually had a little bit of calculus before, I want to make 584 00:32:11 --> 00:32:13 you unlearn some of your instincts that you developed. 585 00:32:13 --> 00:32:15 So this will be harder for those of you who have 586 00:32:15 --> 00:32:19 actually done this before. 587 00:32:19 --> 00:32:22 But for the rest of you, it will be relatively easy. 588 00:32:22 --> 00:32:35 Which is, don't abandon your precalculus skills. 589 00:32:35 --> 00:32:42 And common sense. 590 00:32:42 --> 00:32:46 So there's a great deal of common sense in this. 591 00:32:46 --> 00:32:50 And it actually trumps some of the calculus. 592 00:32:50 --> 00:32:56 The calculus just fills in what you didn't quite know yet. 593 00:32:56 --> 00:32:59 So I will try to illustrate this. 594 00:32:59 --> 00:33:02 And because we're running a bit late, I won't get to the some 595 00:33:02 --> 00:33:05 of the main punchlines until next lecture. 596 00:33:05 --> 00:33:07 But I want you to do it. 597 00:33:07 --> 00:33:09 So for now, I'm just going to tell you about the 598 00:33:09 --> 00:33:10 general principles. 599 00:33:10 --> 00:33:14 And in the process I'm going to introduce the terminology. 600 00:33:14 --> 00:33:17 Just, the words that we need to use to describe what 601 00:33:17 --> 00:33:18 is that we're doing. 602 00:33:18 --> 00:33:19 And there's also a certain amount of carelessness 603 00:33:19 --> 00:33:23 with that in many of the treatments that you'll see. 604 00:33:23 --> 00:33:24 And a lot of hastiness. 605 00:33:24 --> 00:33:29 So just be a little patient and we will do this. 606 00:33:29 --> 00:33:33 So, the first principle is the following. 607 00:33:33 --> 00:33:40 If f' is positive, then f is increasing. 608 00:33:40 --> 00:33:44 That's a straightforward idea, and it's closely related to 609 00:33:44 --> 00:33:48 this tangent line approximation or the linear approximation 610 00:33:48 --> 00:33:49 that I just did. 611 00:33:49 --> 00:33:50 You can just imagine. 612 00:33:50 --> 00:33:53 Here's the tangent line, here's the function. 613 00:33:53 --> 00:33:56 And if the tangent line is pointing up, then the function 614 00:33:56 --> 00:33:58 is also going up, too. 615 00:33:58 --> 00:34:00 So that's all that's going on here. 616 00:34:00 --> 00:34:10 Similarly, if f' is negative, then f is decreasing. 617 00:34:10 --> 00:34:12 And that's the basic idea. 618 00:34:12 --> 00:34:17 Now, the second step is also fairly straightforward. 619 00:34:17 --> 00:34:21 It's just a second order effect of the same type. 620 00:34:21 --> 00:34:27 If you have f'' as positive, then that means that 621 00:34:27 --> 00:34:33 f' is increasing. 622 00:34:33 --> 00:34:36 That's the same principle applied one step up. 623 00:34:36 --> 00:34:37 Right? 624 00:34:37 --> 00:34:41 Because if f'' is positive, that means it's the 625 00:34:41 --> 00:34:42 derivative of f'. 626 00:34:42 --> 00:34:45 So it's the same principle just repeated. 627 00:34:45 --> 00:34:49 And now I just want to draw a picture of this. 628 00:34:49 --> 00:34:52 Here's a picture of it, I claim. 629 00:34:52 --> 00:34:54 And it looks like something's going down. 630 00:34:54 --> 00:34:56 And I did that on purpose. 631 00:34:56 --> 00:34:58 But there is something that's increasing here. 632 00:34:58 --> 00:35:02 Which is, the slope is very steep negative here. 633 00:35:02 --> 00:35:06 And it's less steep negative over here. 634 00:35:06 --> 00:35:10 So we have the slope which is some negative number, say, - 4. 635 00:35:10 --> 00:35:13 And here it's - 3. 636 00:35:13 --> 00:35:14 So it's increasing. 637 00:35:14 --> 00:35:18 It's getting less negative, and maybe eventually it'll 638 00:35:18 --> 00:35:19 curve up the other way. 639 00:35:19 --> 00:35:23 And this is a picture of what I'm talking about here. 640 00:35:23 --> 00:35:25 That's what it means to say that f' is increasing. 641 00:35:25 --> 00:35:28 The slope is getting larger. 642 00:35:28 --> 00:35:34 And the way to describe a curve like this is that it's concave. 643 00:35:34 --> 00:35:41 So f is concave up. 644 00:35:41 --> 00:35:49 And similarly, f'' negative is going to be the same thing as f 645 00:35:49 --> 00:35:59 concave, or implies f concave down. 646 00:35:59 --> 00:36:04 So those are the ways in which derivatives will help us 647 00:36:04 --> 00:36:08 qualitatively to draw graphs. 648 00:36:08 --> 00:36:10 But as I said before, we still have to use a little bit of 649 00:36:10 --> 00:36:13 common sense when we draw the graphs. 650 00:36:13 --> 00:36:15 These are just the additional bits of help that we 651 00:36:15 --> 00:36:17 have from calculus. 652 00:36:17 --> 00:36:25 In drawing pictures. 653 00:36:25 --> 00:36:34 So I'm going to go through one example to introduce 654 00:36:34 --> 00:36:36 all the notations. 655 00:36:36 --> 00:36:40 And then eventually, so probably at the beginning of 656 00:36:40 --> 00:36:45 next time, I'll give you a systematic strategy that's 657 00:36:45 --> 00:36:49 going to work when what I'm describing now goes wrong, 658 00:36:49 --> 00:36:52 or a little bit wrong. 659 00:36:52 --> 00:36:58 So let's begin with a straightforward example. 660 00:36:58 --> 00:37:00 So, the first example that I'll give you is the 661 00:37:00 --> 00:37:04 function f (x) = 3x - x^3. 662 00:37:04 --> 00:37:07 663 00:37:07 --> 00:37:11 Just, as I said, to be able to introduce all the notations. 664 00:37:11 --> 00:37:16 Now, if you differentiate it, you get 3 - 3x^2. 665 00:37:17 --> 00:37:20 And I can factor that. 666 00:37:20 --> 00:37:23 This is 3 ( 1 - x)( 1 667 00:37:24 --> 00:37:26 x). 668 00:37:26 --> 00:37:27 OK? 669 00:37:27 --> 00:37:33 And so, I can decide whether the derivative 670 00:37:33 --> 00:37:37 is positive or negative. 671 00:37:37 --> 00:37:38 Easily enough. 672 00:37:38 --> 00:37:50 Namely, just staring at this, I can see that when -1 < x < 1, 673 00:37:50 --> 00:37:53 in that range there, both these numbers, both these 674 00:37:53 --> 00:37:55 factors, are positive. 675 00:37:55 --> 00:37:57 1 - x is a positive number and 1 676 00:37:58 --> 00:37:59 x is a positive number. 677 00:37:59 --> 00:38:04 So, in this range, f' ( x ) is positive. 678 00:38:04 --> 00:38:09 So this thing is, so f is increasing. 679 00:38:09 --> 00:38:14 And similarly, in the other ranges, if x is very, very 680 00:38:14 --> 00:38:17 large, this becomes, if it crosses 1, in fact, this 681 00:38:17 --> 00:38:19 becomes, this factor becomes negative and this 682 00:38:19 --> 00:38:21 one stays positive. 683 00:38:21 --> 00:38:29 So when x > 1, we have that f ' (x) is negative. 684 00:38:29 --> 00:38:35 And so f is decreasing. 685 00:38:35 --> 00:38:39 And the same thing goes for the other side. 686 00:38:39 --> 00:38:42 When it's less than - 1, that also works this way. 687 00:38:42 --> 00:38:45 Because when it's less than - 1, this number 688 00:38:45 --> 00:38:46 factors positive. 689 00:38:46 --> 00:38:50 But the other one is negative. 690 00:38:50 --> 00:38:56 So in both of these cases, we get that it's decreasing. 691 00:38:56 --> 00:39:07 So now, here's the schematic picture of this function. 692 00:39:07 --> 00:39:12 So here's - 1, here's 1. 693 00:39:12 --> 00:39:19 It's going to go down, up, down. 694 00:39:19 --> 00:39:21 That's what it's doing. 695 00:39:21 --> 00:39:23 Maybe I'll just leave it alone like this. 696 00:39:23 --> 00:39:28 That's what it looks like. 697 00:39:28 --> 00:39:31 So, this is the kind of information we can get 698 00:39:31 --> 00:39:32 right off the bat. 699 00:39:32 --> 00:39:38 And you notice immediately that it's very important, from the 700 00:39:38 --> 00:39:40 features of the function, the sort of key features of the 701 00:39:40 --> 00:39:44 function that we see here, are these two places. 702 00:39:44 --> 00:39:49 Maybe I'll even mark them in a, like this. 703 00:39:49 --> 00:39:59 And these things are turning points. 704 00:39:59 --> 00:40:00 So what are they? 705 00:40:00 --> 00:40:03 Well, they're just the points where the 706 00:40:03 --> 00:40:05 derivative changes sign. 707 00:40:05 --> 00:40:07 Where it's negative here and it's positive there, 708 00:40:07 --> 00:40:09 so there it must be 0. 709 00:40:09 --> 00:40:12 So we have a definition, and this is the most important 710 00:40:12 --> 00:40:21 definition in this subject, which is that is if f' (x0) = 711 00:40:21 --> 00:40:33 0, we call x0 a critical point. 712 00:40:33 --> 00:40:36 The word 'turning point' is not used just because, in fact, 713 00:40:36 --> 00:40:38 it doesn't have to turn around at those points. 714 00:40:38 --> 00:40:42 But certainly, if it turns around then this will happen. 715 00:40:42 --> 00:40:46 And we also have another notation, which is the number 716 00:40:46 --> 00:40:59 y0 which is f ( x0 ) is called a critical value. 717 00:40:59 --> 00:41:02 So these are the key numbers that we're going to have to 718 00:41:02 --> 00:41:18 work out in order to understand what the function looks like. 719 00:41:18 --> 00:41:27 So what I'm going to do is just plot them. 720 00:41:27 --> 00:41:30 We're going to plot the critical points and the values. 721 00:41:30 --> 00:41:34 Well, we found the critical points relatively easily. 722 00:41:34 --> 00:41:37 I didn't write it down here but it's pretty obvious. 723 00:41:37 --> 00:41:43 If you set f(x) = 0, that implies that (1 - x)( 1 724 00:41:44 --> 00:41:47 x) = 0, which implies that x is 725 00:41:47 --> 00:41:50 or - 1. 726 00:41:50 --> 00:41:52 So those are known as the critical points. 727 00:41:52 --> 00:41:56 And now, in order to get the critical values here, I have to 728 00:41:56 --> 00:42:02 plug in f (1), for instance, the function is 3x - x^2, so 729 00:42:02 --> 00:42:07 there's this 3 * 1 - 1^3, which is 2. 730 00:42:07 --> 00:42:17 And f ( - 1), which is 3 ( - 1) - (- 1)^3, which is - 2. 731 00:42:17 --> 00:42:21 And so I can plot the function here. 732 00:42:21 --> 00:42:26 So here's the point - 1 and here's, up here, is 2. 733 00:42:26 --> 00:42:28 So this is - whoops, which one is it? 734 00:42:28 --> 00:42:29 Yeah. 735 00:42:29 --> 00:42:32 This is - 1, so it's down here. 736 00:42:32 --> 00:42:35 So it's (- 1, - 2). 737 00:42:35 --> 00:42:41 And then over here, I have the point (1, 2). 738 00:42:41 --> 00:42:47 Alright, now, what information do I get from - so I've 739 00:42:47 --> 00:42:50 now plotted two, I claim, very interesting points. 740 00:42:50 --> 00:42:55 What information do I get from this? 741 00:42:55 --> 00:42:59 The answer is, I know something very nearby. 742 00:42:59 --> 00:43:02 Because I've already checked that the thing is coming down 743 00:43:02 --> 00:43:04 from the left, and coming back up. 744 00:43:04 --> 00:43:07 And so it must be shaped like this. 745 00:43:07 --> 00:43:08 Over here. 746 00:43:08 --> 00:43:11 The tangent line is 0, it's going to be level there. 747 00:43:11 --> 00:43:14 And similarly over here, it's going to do that. 748 00:43:14 --> 00:43:22 So this is what we know so far, based on what we've computed. 749 00:43:22 --> 00:43:22 Question. 750 00:43:22 --> 00:43:37 STUDENT: [INAUDIBLE] 751 00:43:37 --> 00:43:37 PROF. 752 00:43:37 --> 00:43:38 JERISON: The question is, what happens if there's 753 00:43:38 --> 00:43:38 a sharp corner. 754 00:43:38 --> 00:43:45 The answer is, calculus, it's not called a critical point. 755 00:43:45 --> 00:43:47 It's a something else. 756 00:43:47 --> 00:43:50 And it's a very important point, too. 757 00:43:50 --> 00:43:52 And we will be discussing those kinds of points. 758 00:43:52 --> 00:43:54 There are much more dramatic instances of that. 759 00:43:54 --> 00:43:56 That's part of what we're going to say. 760 00:43:56 --> 00:43:58 But I just want to save that, alright. 761 00:43:58 --> 00:44:02 We will be discussing. 762 00:44:02 --> 00:44:03 Yeah. 763 00:44:03 --> 00:44:03 Question. 764 00:44:03 --> 00:44:08 STUDENT: [INAUDIBLE] 765 00:44:08 --> 00:44:08 PROF. 766 00:44:08 --> 00:44:11 JERISON: The question that was asked was, how did I know at 767 00:44:11 --> 00:44:14 the critical point that it's concave down over here and 768 00:44:14 --> 00:44:17 concave up over here. 769 00:44:17 --> 00:44:22 The answer is that I actually did not use the second 770 00:44:22 --> 00:44:24 derivative yet. 771 00:44:24 --> 00:44:26 What I used is another piece of information. 772 00:44:26 --> 00:44:28 I used the information that I derived over here. 773 00:44:28 --> 00:44:32 That f' is positive, where f' is positive and 774 00:44:32 --> 00:44:33 where it's negative. 775 00:44:33 --> 00:44:36 So what I know is that the graph is going down 776 00:44:36 --> 00:44:39 to the left of - 1. 777 00:44:39 --> 00:44:41 It's going up to the right, here. 778 00:44:41 --> 00:44:44 It's going up here and it's going down there. 779 00:44:44 --> 00:44:47 I did not use the second derivative. 780 00:44:47 --> 00:44:49 I used the first derivative. 781 00:44:49 --> 00:44:52 OK, but I didn't just use the fact that there was 782 00:44:52 --> 00:44:56 a turning point here. 783 00:44:56 --> 00:44:57 So, actually, I was using the fact that it was 784 00:44:57 --> 00:44:58 a turning point. 785 00:44:58 --> 00:45:00 I wasn't using the fact that it had the second 786 00:45:00 --> 00:45:01 derivative, though. 787 00:45:01 --> 00:45:01 OK. 788 00:45:01 --> 00:45:03 For now. 789 00:45:03 --> 00:45:09 You can also see it by the second derivative as well. 790 00:45:09 --> 00:45:14 So now, the next thing that I'd like to do, I need to 791 00:45:14 --> 00:45:16 finish off this graph. 792 00:45:16 --> 00:45:19 And I just want to do it a little bit carefully here. 793 00:45:19 --> 00:45:22 In the order that is reasonable. 794 00:45:22 --> 00:45:27 Now, you might happen to notice, and there's nothing 795 00:45:27 --> 00:45:35 wrong with this, so let's even fill in a guess. 796 00:45:35 --> 00:45:37 In order to fill in a guess, though, and have it be even 797 00:45:37 --> 00:45:39 vaguely right, I do have to notice that this thing 798 00:45:39 --> 00:45:41 crosses, this function crosses the origin. 799 00:45:41 --> 00:45:48 The function f(x) = 3x - x^3 happens also have the 800 00:45:48 --> 00:45:50 property that f ( 0 ) = 0. 801 00:45:50 --> 00:45:52 Again, common sense. 802 00:45:52 --> 00:45:54 You're allowed to use your common sense. 803 00:45:54 --> 00:45:56 You're allowed to notice a value of the function 804 00:45:56 --> 00:45:58 and put it in. 805 00:45:58 --> 00:46:00 So there's nothing wrong with that. 806 00:46:00 --> 00:46:03 If you happen to have such a value. 807 00:46:03 --> 00:46:06 So, now we can guess what our function is going to look like. 808 00:46:06 --> 00:46:10 It's going to maybe come down like this. 809 00:46:10 --> 00:46:11 Come up like this. 810 00:46:11 --> 00:46:13 And come down like this. 811 00:46:13 --> 00:46:15 That could be what it looks like. 812 00:46:15 --> 00:46:17 But, you know, another possibility is it sort 813 00:46:17 --> 00:46:19 of comes along here and goes out that way. 814 00:46:19 --> 00:46:22 Comes along here and goes out that way, who knows? 815 00:46:22 --> 00:46:25 It happens, by the way, that it's an odd function. 816 00:46:25 --> 00:46:25 Right? 817 00:46:25 --> 00:46:26 Those are all odd powers. 818 00:46:26 --> 00:46:28 So, actually, it's symmetric on the right half 819 00:46:28 --> 00:46:29 and the left half. 820 00:46:29 --> 00:46:31 And crosses at 0. 821 00:46:31 --> 00:46:33 So everything that we do on the right is going to be the same 822 00:46:33 --> 00:46:34 as what happens on the left. 823 00:46:34 --> 00:46:36 That's another piece of common sense. 824 00:46:36 --> 00:46:39 You want to make use of that as much as possible, whenever 825 00:46:39 --> 00:46:40 you're drawing anything. 826 00:46:40 --> 00:46:42 Don't want to throw out information. 827 00:46:42 --> 00:46:46 So this function happens to be odd. 828 00:46:46 --> 00:46:48 Odd, and f ( 0 ) = 0. 829 00:46:48 --> 00:46:52 I'm considering those to be kinds of precalculus skills 830 00:46:52 --> 00:47:00 that I want you to use as much as you can. 831 00:47:00 --> 00:47:03 So now, here's the first feature which is unfortunately 832 00:47:03 --> 00:47:08 ignored in most discussions of functions. 833 00:47:08 --> 00:47:11 And it's strange, because nowadays we have 834 00:47:11 --> 00:47:13 graphing things. 835 00:47:13 --> 00:47:19 And it's really the only part of the exercise that you 836 00:47:19 --> 00:47:26 couldn't do, at least on this relatively simpleminded level, 837 00:47:26 --> 00:47:28 with a graphing calculator. 838 00:47:28 --> 00:47:33 And that is what I would call the ends of the problem. 839 00:47:33 --> 00:47:37 So what happens off the screen, is the question. 840 00:47:37 --> 00:47:39 And that basically is the theoretical part of the problem 841 00:47:39 --> 00:47:41 that you have to address. 842 00:47:41 --> 00:47:42 You can program this. 843 00:47:42 --> 00:47:45 You can draw all the pictures that you want. 844 00:47:45 --> 00:47:48 But what you won't see is what's off the screen. 845 00:47:48 --> 00:47:50 You need to know something to figure out what's 846 00:47:50 --> 00:47:51 off the screen. 847 00:47:51 --> 00:47:54 So, in this case, I'm talking about what's off the screen 848 00:47:54 --> 00:48:01 going to the right, or going to the left. 849 00:48:01 --> 00:48:06 So let's check the ends. 850 00:48:06 --> 00:48:07 So here, let's just take a look. 851 00:48:07 --> 00:48:12 We have the function f(x), which is, sorry, 3x - x^3. 852 00:48:12 --> 00:48:14 Again this is a precalculus sort of thing. 853 00:48:14 --> 00:48:16 And we're imagining now, let's just do x goes 854 00:48:16 --> 00:48:18 to plus infinity. 855 00:48:18 --> 00:48:19 So what happens here. 856 00:48:19 --> 00:48:25 When x is gigantic, this term is completely negligible. 857 00:48:25 --> 00:48:30 And it just behaves like - x^3, which goes to minus infinity 858 00:48:30 --> 00:48:32 as x goes to plus infinity. 859 00:48:32 --> 00:48:41 And similarly, f (x) goes to plus infinity if x 860 00:48:41 --> 00:48:46 goes to minus infinity. 861 00:48:46 --> 00:48:49 Now let me pull down this picture again, and show 862 00:48:49 --> 00:48:53 you what piece of the information we've got. 863 00:48:53 --> 00:48:55 We now know that it is heading up this way. 864 00:48:55 --> 00:48:58 It doesn't go like this, it goes up like that. 865 00:48:58 --> 00:49:00 And I'm going to put an arrow for it, And it's 866 00:49:00 --> 00:49:02 going down like this. 867 00:49:02 --> 00:49:06 Heading down to minus infinity as x goes out farther 868 00:49:06 --> 00:49:07 to the right. 869 00:49:07 --> 00:49:15 And going out to plus infinity as x goes farther to the left. 870 00:49:15 --> 00:49:21 So now there's hardly anything left of this function 871 00:49:21 --> 00:49:21 to describe. 872 00:49:21 --> 00:49:27 There's really nothing left except maybe decoration. 873 00:49:27 --> 00:49:31 And we kind of like that decoration, so we will 874 00:49:31 --> 00:49:32 pay attention to it. 875 00:49:32 --> 00:49:35 And to do that, we'll have to check the second derivative. 876 00:49:35 --> 00:49:39 So if we differentiate a second time, the first derivative 877 00:49:39 --> 00:49:41 was, remember, 3 - 3x^2. 878 00:49:42 --> 00:49:53 So the second derivative is - 6x. 879 00:49:53 --> 00:50:02 So now we notice that f'' (x) is negative if x is positive. 880 00:50:02 --> 00:50:04 And f'' ( x) 881 00:50:04 --> 00:50:08 is positive if x is negative. 882 00:50:08 --> 00:50:13 And so in this part it's concave down. 883 00:50:13 --> 00:50:18 And in this part it's concave up. 884 00:50:18 --> 00:50:21 And now I'm going to switch the boards so that 885 00:50:21 --> 00:50:24 you'll, and draw it. 886 00:50:24 --> 00:50:30 And you see that it was begging to be this way. 887 00:50:30 --> 00:50:32 So we'll fill in the rest of it here. 888 00:50:32 --> 00:50:35 Maybe in a nice color here. 889 00:50:35 --> 00:50:38 So this is the whole graph and this is the correct graph. 890 00:50:38 --> 00:50:41 It comes down in one swoop down here, and comes up here. 891 00:50:41 --> 00:50:46 And then it changes to concave down right at the origin. 892 00:50:46 --> 00:50:49 So this point is of interest, not only because it's the place 893 00:50:49 --> 00:50:52 where it crosses the axis, but it's also what's called 894 00:50:52 --> 00:51:00 an inflection point. 895 00:51:00 --> 00:51:04 Inflection point, that's a point where because f'' at 896 00:51:04 --> 00:51:07 that place is equal to 0. 897 00:51:07 --> 00:51:10 So it's a place where the second derivative is 0. 898 00:51:10 --> 00:51:15 We also consider those to be interesting points. 899 00:51:15 --> 00:51:21 Now, so let me just making one closing remark here. 900 00:51:21 --> 00:51:26 Which is that all of this information fits together. 901 00:51:26 --> 00:51:29 And we're going to have much, much harder examples of this 902 00:51:29 --> 00:51:32 where you'll actually have to think about what's going on. 903 00:51:32 --> 00:51:35 But there's a lot of stuff protecting you. 904 00:51:35 --> 00:51:39 And functions will behave themselves and turn 905 00:51:39 --> 00:51:40 around appropriately. 906 00:51:40 --> 00:51:43 Anyway, we'll talk about it next time. 907 00:51:43 --> 00:51:43