1 00:00:00,090 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,610 Commons license. 3 00:00:03,610 --> 00:00:05,990 Your support will help MIT OpenCourseWare 4 00:00:05,990 --> 00:00:10,010 continue to offer high quality educational resources for free. 5 00:00:10,010 --> 00:00:12,540 To make a donation or to view additional materials 6 00:00:12,540 --> 00:00:16,190 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,190 --> 00:00:21,740 at ocw.mit.edu. 8 00:00:21,740 --> 00:00:22,240 PROF. 9 00:00:22,240 --> 00:00:24,660 JERISON: So, we're ready to begin Lecture 10, 10 00:00:24,660 --> 00:00:28,860 and what I'm going to begin with is by finishing up 11 00:00:28,860 --> 00:00:33,880 some things from last time. 12 00:00:33,880 --> 00:00:42,460 We'll talk about approximations, and I 13 00:00:42,460 --> 00:00:47,650 want to fill in a number of comments 14 00:00:47,650 --> 00:00:52,440 and get you a little bit more oriented in the point of view 15 00:00:52,440 --> 00:00:55,980 that I'm trying to express about approximations. 16 00:00:55,980 --> 00:00:58,790 So, first of all, I want to remind you 17 00:00:58,790 --> 00:01:03,800 of the actual applied example that I wrote down last time. 18 00:01:03,800 --> 00:01:08,730 So that was this business here. 19 00:01:08,730 --> 00:01:11,590 There was something from special relativity. 20 00:01:11,590 --> 00:01:15,990 And the approximation that we used was the linear 21 00:01:15,990 --> 00:01:22,170 approximation, with a -1/2 power that comes out to be T(1 + 1/2 22 00:01:22,170 --> 00:01:22,670 v^2/c^2). 23 00:01:27,340 --> 00:01:30,850 I want to reiterate why this is a useful way of thinking 24 00:01:30,850 --> 00:01:31,930 of things. 25 00:01:31,930 --> 00:01:34,830 And why this is that this comes up in real life. 26 00:01:34,830 --> 00:01:37,090 Why this is maybe more important than everything 27 00:01:37,090 --> 00:01:39,820 that I've taught you about technically so far. 28 00:01:39,820 --> 00:01:46,750 So, first of all, what this is telling us is the change in T 29 00:01:46,750 --> 00:01:50,430 divided by T, if you do the arithmetic here and subtract T, 30 00:01:50,430 --> 00:01:55,970 that's using the change in T is T' - T here. 31 00:01:55,970 --> 00:02:01,440 If you work that out, this is approximately the same as 1/2 32 00:02:01,440 --> 00:02:04,140 v^2/c^2. 33 00:02:04,140 --> 00:02:06,270 So what is this saying? 34 00:02:06,270 --> 00:02:08,580 This is saying that if you have this satellite, which 35 00:02:08,580 --> 00:02:12,610 is going at speed v, and little c is the speed of light, 36 00:02:12,610 --> 00:02:17,060 then the change in the watch down here on earth, 37 00:02:17,060 --> 00:02:20,770 relative to the time on the satellite, 38 00:02:20,770 --> 00:02:24,050 is going to be proportional to this ratio here. 39 00:02:24,050 --> 00:02:25,710 So, physically, this makes sense. 40 00:02:25,710 --> 00:02:28,120 This is time divided by time. 41 00:02:28,120 --> 00:02:30,670 And this is velocity squared divided by velocity squared. 42 00:02:30,670 --> 00:02:32,620 So, in each case, the units divide out. 43 00:02:32,620 --> 00:02:34,880 So this is a dimensionless quantity. 44 00:02:34,880 --> 00:02:36,960 And this is a dimensionless quantity. 45 00:02:36,960 --> 00:02:40,840 And the only point here that we're trying to make 46 00:02:40,840 --> 00:02:43,440 is just this notion of proportionality. 47 00:02:43,440 --> 00:02:45,470 So I want to write this down. 48 00:02:45,470 --> 00:02:48,010 Just-- in summary. 49 00:02:48,010 --> 00:02:50,210 So the error fraction, if you like, 50 00:02:50,210 --> 00:02:53,630 which is sort of the number of significant digits 51 00:02:53,630 --> 00:03:03,090 that we have in our measurement, is proportional, in this case, 52 00:03:03,090 --> 00:03:04,860 to this quantity. 53 00:03:04,860 --> 00:03:09,030 It happens to be proportional to this quantity here. 54 00:03:09,030 --> 00:03:16,980 And the factor is, happens to be, 1/2. 55 00:03:16,980 --> 00:03:21,350 So these proportionality factors are what we're looking for. 56 00:03:21,350 --> 00:03:22,404 Their rates of change. 57 00:03:22,404 --> 00:03:23,820 Their rates of change of something 58 00:03:23,820 --> 00:03:25,550 with respect to something else. 59 00:03:25,550 --> 00:03:28,800 Now, on your homework, you have something 60 00:03:28,800 --> 00:03:30,740 rather similar to this. 61 00:03:30,740 --> 00:03:38,460 So in Problem, on Part 2B, Part II, Problem 1, 62 00:03:38,460 --> 00:03:41,580 there's the speed of a pitch, right? 63 00:03:41,580 --> 00:03:44,100 And the speed of the pitch is changing depending 64 00:03:44,100 --> 00:03:45,600 on how high the mound is. 65 00:03:45,600 --> 00:03:47,460 And the point here is that that's 66 00:03:47,460 --> 00:03:50,150 approximately proportional to the change 67 00:03:50,150 --> 00:03:52,320 in the height of the mound. 68 00:03:52,320 --> 00:03:54,290 In that problem, we had this delta h, 69 00:03:54,290 --> 00:03:56,830 that was the x variable in that problem. 70 00:03:56,830 --> 00:03:58,330 And what you're trying to figure out 71 00:03:58,330 --> 00:04:02,904 is what the constant of proportionality is. 72 00:04:02,904 --> 00:04:04,820 That's what you're aiming for in this problem. 73 00:04:04,820 --> 00:04:08,070 So there's a linear relationship, approximately, 74 00:04:08,070 --> 00:04:11,890 to all intents and purposes this is an equality. 75 00:04:11,890 --> 00:04:13,290 Because the lower order terms are 76 00:04:13,290 --> 00:04:14,720 unimportant for the problem. 77 00:04:14,720 --> 00:04:18,020 Just as over here, this function is a little bit complicated. 78 00:04:18,020 --> 00:04:19,800 This function is a little more simple. 79 00:04:19,800 --> 00:04:22,800 For the purposes of this problem, they are the same. 80 00:04:22,800 --> 00:04:27,370 Because the errors are negligible 81 00:04:27,370 --> 00:04:29,760 for the particular problem that we're working on. 82 00:04:29,760 --> 00:04:34,210 So we might as well work with the simpler relationship. 83 00:04:34,210 --> 00:04:37,621 And similarly, over here, so you could do this with, 84 00:04:37,621 --> 00:04:39,120 in this case with square roots, it's 85 00:04:39,120 --> 00:04:41,980 not so hard here with reciprocals of square roots. 86 00:04:41,980 --> 00:04:45,300 It's also not terribly hard to do it numerically. 87 00:04:45,300 --> 00:04:47,850 And the reason why we're not doing it numerically 88 00:04:47,850 --> 00:04:51,260 is that, as I say, this is something that 89 00:04:51,260 --> 00:04:53,560 happens all across engineering. 90 00:04:53,560 --> 00:04:55,640 People are looking for these linear relationships 91 00:04:55,640 --> 00:05:00,240 between the change in some input and the change in the output. 92 00:05:00,240 --> 00:05:03,210 And if you don't make these simplifications, 93 00:05:03,210 --> 00:05:06,760 then when you get, say, a dozen of them together, 94 00:05:06,760 --> 00:05:09,540 you can't figure out what's going on. 95 00:05:09,540 --> 00:05:12,640 In this case the design of the satellite, it's very important. 96 00:05:12,640 --> 00:05:15,160 The speed actually isn't just one speed. 97 00:05:15,160 --> 00:05:19,070 Because it's the relative speed of you to the satellite. 98 00:05:19,070 --> 00:05:21,349 And you might be-- it depends on your angle of sight 99 00:05:21,349 --> 00:05:22,890 with the satellite what the speed is. 100 00:05:22,890 --> 00:05:24,300 So it varies quite a bit. 101 00:05:24,300 --> 00:05:26,280 So you really need this rule of thumb. 102 00:05:26,280 --> 00:05:28,372 Then there are all kinds of other considerations 103 00:05:28,372 --> 00:05:29,080 in this question. 104 00:05:29,080 --> 00:05:30,852 Like, for example, there's the fact 105 00:05:30,852 --> 00:05:33,530 that we're sitting on Earth and so we're rotating around 106 00:05:33,530 --> 00:05:36,250 on what's called a non-inertial frame. 107 00:05:36,250 --> 00:05:38,721 So there's the question of that acceleration. 108 00:05:38,721 --> 00:05:41,220 There's the question that the gravity that I experience here 109 00:05:41,220 --> 00:05:44,830 on Earth is not the same as up at the satellite. 110 00:05:44,830 --> 00:05:47,870 And that also creates a difference in time, 111 00:05:47,870 --> 00:05:49,580 as a result of general relativity. 112 00:05:49,580 --> 00:05:53,400 So all of these considerations come down 113 00:05:53,400 --> 00:05:56,030 to formulas which are this complicated or maybe 114 00:05:56,030 --> 00:05:56,910 a tiny bit more. 115 00:05:56,910 --> 00:05:58,330 Not really that much. 116 00:05:58,330 --> 00:06:00,700 And then people simplify them enormously to these very 117 00:06:00,700 --> 00:06:02,090 simple-minded rules. 118 00:06:02,090 --> 00:06:05,130 And they don't keep track of what's going on. 119 00:06:05,130 --> 00:06:06,830 So in order to design the system, 120 00:06:06,830 --> 00:06:08,960 you must make these simplifications, 121 00:06:08,960 --> 00:06:12,630 otherwise you can't even think about what's going on. 122 00:06:12,630 --> 00:06:13,860 This comes up in everything. 123 00:06:13,860 --> 00:06:17,422 In weather forecasting, economic forecasting. 124 00:06:17,422 --> 00:06:19,880 Figuring out whether there's going to be an asteroid that's 125 00:06:19,880 --> 00:06:22,140 going to hit the Earth. 126 00:06:22,140 --> 00:06:23,970 Every single one of these things involves 127 00:06:23,970 --> 00:06:27,900 dozens of these considerations. 128 00:06:27,900 --> 00:06:29,951 OK, there was a question that I saw, here. 129 00:06:29,951 --> 00:06:30,450 Yes. 130 00:06:30,450 --> 00:06:38,821 STUDENT: [INAUDIBLE] 131 00:06:38,821 --> 00:06:39,320 PROF. 132 00:06:39,320 --> 00:06:40,120 JERISON: Yeah. 133 00:06:40,120 --> 00:06:41,950 Basically, any problem where you have 134 00:06:41,950 --> 00:06:43,840 a derivative, the rate of change also 135 00:06:43,840 --> 00:06:46,017 depends upon what the base point is. 136 00:06:46,017 --> 00:06:46,850 That's the question. 137 00:06:46,850 --> 00:06:49,070 You're saying, doesn't this delta v also 138 00:06:49,070 --> 00:06:51,050 depend, I had a base point in that problem. 139 00:06:51,050 --> 00:06:53,290 I happened to decide that pitchers pitch on average 140 00:06:53,290 --> 00:06:55,350 about 90 miles an hour. 141 00:06:55,350 --> 00:06:58,327 Whereas, in fact, some pitchers pitch at 100 miles an hour, 142 00:06:58,327 --> 00:07:00,410 some pitch at 80 miles an hour, and of course they 143 00:07:00,410 --> 00:07:02,040 vary the speed of the pitch. 144 00:07:02,040 --> 00:07:03,510 And so, this varies a little bit. 145 00:07:03,510 --> 00:07:05,426 In fact, that's sort of a second order effect. 146 00:07:05,426 --> 00:07:08,220 It does change the constant of proportionality. 147 00:07:08,220 --> 00:07:11,790 It's a rate of change at a different base point. 148 00:07:11,790 --> 00:07:14,230 Which we're considering fixed. 149 00:07:14,230 --> 00:07:17,060 In fact, that's sort of a second order effect. 150 00:07:17,060 --> 00:07:19,700 When you actually do the computations, what you discover 151 00:07:19,700 --> 00:07:21,940 is that it doesn't make that much difference. 152 00:07:21,940 --> 00:07:22,630 To the a. 153 00:07:22,630 --> 00:07:25,040 And that's something that you get from experience. 154 00:07:25,040 --> 00:07:28,980 That it turns out, which things matter and which things don't. 155 00:07:28,980 --> 00:07:31,820 And yet again, that's exactly the same sort of consideration 156 00:07:31,820 --> 00:07:34,472 but at the next order of what I'm talking about here is. 157 00:07:34,472 --> 00:07:36,430 You have to have enough experience with numbers 158 00:07:36,430 --> 00:07:38,992 to know that if you take, if you vary something a little bit 159 00:07:38,992 --> 00:07:40,950 it's not going to change the answer that you're 160 00:07:40,950 --> 00:07:42,680 looking for very much. 161 00:07:42,680 --> 00:07:45,530 And that's exactly the point that I'm making. 162 00:07:45,530 --> 00:07:51,830 So I can't make them all at once, all such points. 163 00:07:51,830 --> 00:07:55,290 So that's my pitch for understanding things 164 00:07:55,290 --> 00:07:56,470 from this point of view. 165 00:07:56,470 --> 00:08:02,000 Now, we're going to go on, now, to quadratic approximations, 166 00:08:02,000 --> 00:08:13,290 which are a little more complicated. 167 00:08:13,290 --> 00:08:15,920 So, we talked a little bit about this last time 168 00:08:15,920 --> 00:08:16,850 but I didn't finish. 169 00:08:16,850 --> 00:08:19,930 So I want to finish this up. 170 00:08:19,930 --> 00:08:22,320 And the first thing that I should say 171 00:08:22,320 --> 00:08:30,740 is that you use these when the linear approximation is not 172 00:08:30,740 --> 00:08:35,680 enough. 173 00:08:35,680 --> 00:08:38,270 OK, so, that's something that you really 174 00:08:38,270 --> 00:08:40,390 need to get a little experience with. 175 00:08:40,390 --> 00:08:43,810 In economics, I told you they use logarithms. 176 00:08:43,810 --> 00:08:46,030 So sometimes they use log-linear functions. 177 00:08:46,030 --> 00:08:48,010 Sometimes they use log-quadratic functions 178 00:08:48,010 --> 00:08:49,860 when the log-linear ones don't work. 179 00:08:49,860 --> 00:08:53,472 So most modeling in economics is with log-quadratic functions. 180 00:08:53,472 --> 00:08:55,680 And if you've made it any more complicated than that, 181 00:08:55,680 --> 00:08:56,740 it's useless. 182 00:08:56,740 --> 00:08:57,850 And it's a mess. 183 00:08:57,850 --> 00:08:58,810 And people don't do it. 184 00:08:58,810 --> 00:09:02,790 So they stick with the quadratic ones, typically. 185 00:09:02,790 --> 00:09:07,010 So the basic formula here, and I'm going to take the base 186 00:09:07,010 --> 00:09:14,380 point to be 0, is that f(x) is approximately f(0) + f'(0) x. 187 00:09:14,380 --> 00:09:16,100 That's the linear part. 188 00:09:16,100 --> 00:09:17,174 Plus this extra term. 189 00:09:17,174 --> 00:09:18,090 Which is f''(0)/2 x^2. 190 00:09:21,440 --> 00:09:27,740 And this is supposed to work for x near 0. 191 00:09:27,740 --> 00:09:34,310 So I've chosen the base point as simply as possible. 192 00:09:34,310 --> 00:09:38,060 So here's more or less where we left off last time. 193 00:09:38,060 --> 00:09:41,710 And one thing that I said I was going to explain, 194 00:09:41,710 --> 00:09:48,390 which I will now, is why it's 1/2 f''(0). 195 00:09:48,390 --> 00:09:51,610 So we need to know that. 196 00:09:51,610 --> 00:09:54,770 So let's work that out here first of all. 197 00:09:54,770 --> 00:09:57,290 So I'm just going to do it by example. 198 00:09:57,290 --> 00:09:59,860 So if you like, the answer is just, well, 199 00:09:59,860 --> 00:10:03,600 what happens when you have a parabola? 200 00:10:03,600 --> 00:10:06,640 A parabola's a quadratic. 201 00:10:06,640 --> 00:10:09,250 It had better-- its quadratic approximation had better 202 00:10:09,250 --> 00:10:10,314 be itself. 203 00:10:10,314 --> 00:10:11,480 It's got to be the best one. 204 00:10:11,480 --> 00:10:13,010 So it's got to be itself. 205 00:10:13,010 --> 00:10:15,240 So this formula, if it's going to work, 206 00:10:15,240 --> 00:10:21,890 has to work on the nose, for quadratic functions. 207 00:10:21,890 --> 00:10:23,770 So, let's take a look. 208 00:10:23,770 --> 00:10:28,130 If I differentiate, I get b + 2cx. 209 00:10:28,130 --> 00:10:32,230 If I differentiate a second time, I get 2c. 210 00:10:32,230 --> 00:10:34,580 And now let's plug it in. 211 00:10:34,580 --> 00:10:39,450 Well, we can recover, what is it that we want to recover? 212 00:10:39,450 --> 00:10:42,150 We want to recover these numbers a, b and c 213 00:10:42,150 --> 00:10:45,930 using the derivatives evaluated at 0. 214 00:10:45,930 --> 00:10:49,500 So let's see. 215 00:10:49,500 --> 00:10:51,210 It's pretty easy, actually. 216 00:10:51,210 --> 00:10:52,569 f(0) = a. 217 00:10:52,569 --> 00:10:53,360 That's on the nose. 218 00:10:53,360 --> 00:10:58,070 If you plug in x = 0 here, these terms drop out and you get a. 219 00:10:58,070 --> 00:11:02,220 And now, f'(0), whoops that was wrong. 220 00:11:02,220 --> 00:11:05,100 So I wrote f' but what I meant was f. 221 00:11:05,100 --> 00:11:07,950 So f(0) is a. 222 00:11:07,950 --> 00:11:09,200 Let's back up. 223 00:11:09,200 --> 00:11:13,130 f(0) is a, so if I plug in x = 0 I get a. 224 00:11:13,130 --> 00:11:18,220 Now, f'(0), that's this next formula here, 225 00:11:18,220 --> 00:11:21,180 f'(0), I plug in 0 here, and I get b. 226 00:11:21,180 --> 00:11:22,220 That's also good. 227 00:11:22,220 --> 00:11:24,220 And that's exactly what the linear approximation 228 00:11:24,220 --> 00:11:25,540 is supposed to be. 229 00:11:25,540 --> 00:11:28,480 But now you notice, f'' is 2c. 230 00:11:28,480 --> 00:11:34,440 So to recover c, I better take half of it. 231 00:11:34,440 --> 00:11:35,120 And that's it. 232 00:11:35,120 --> 00:11:38,010 That's the reason. 233 00:11:38,010 --> 00:11:40,690 There's no chance that any other formula could work. 234 00:11:40,690 --> 00:11:45,050 And this one does. 235 00:11:45,050 --> 00:11:50,310 So that's the explanation for the formula. 236 00:11:50,310 --> 00:11:54,215 And now I remind you that I had a collection of basic formulas 237 00:11:54,215 --> 00:11:55,090 written on the board. 238 00:11:55,090 --> 00:12:00,700 And I want to just make sure we know all of them again. 239 00:12:00,700 --> 00:12:07,580 So, first of all, there was sine x is approximately x. 240 00:12:07,580 --> 00:12:12,610 Cosine x is approximately 1 - 1/2 x^2. 241 00:12:12,610 --> 00:12:19,230 And e^x is approximately 1 + x + 1/2 x^2. 242 00:12:19,230 --> 00:12:22,230 So those were three that I mentioned last time. 243 00:12:22,230 --> 00:12:28,980 And, again, this is all for x near 0. 244 00:12:28,980 --> 00:12:31,160 All for x near 0 only. 245 00:12:31,160 --> 00:12:35,360 These are wildly wrong far away, but near 0 they're 246 00:12:35,360 --> 00:12:37,730 nice, good, quadratic approximations. 247 00:12:37,730 --> 00:12:39,300 Now, the other two approximations 248 00:12:39,300 --> 00:12:44,960 that I want to mention are the logarithm, 249 00:12:44,960 --> 00:12:46,960 and we use the base point shifted. 250 00:12:46,960 --> 00:12:50,500 So we can put it at x near 0. 251 00:12:50,500 --> 00:12:54,545 And this one - sorry, this is an approximately equals sign there 252 00:12:54,545 --> 00:12:59,410 - turns out to be x - 1/2 x^2. 253 00:12:59,410 --> 00:13:08,979 And the last one is (1+x)^r, which turns out to be 1 + rx + 254 00:13:08,979 --> 00:13:09,520 r(r-1)/2 x^2. 255 00:13:14,480 --> 00:13:19,980 Now, eventually, your mind will converge on all of these 256 00:13:19,980 --> 00:13:23,380 and you'll find them relatively easy to memorize. 257 00:13:23,380 --> 00:13:25,820 But it'll take some getting used to. 258 00:13:25,820 --> 00:13:29,670 And I'm not claiming that you should recognize them 259 00:13:29,670 --> 00:13:32,370 and understand them all now. 260 00:13:32,370 --> 00:13:35,630 But I'm going to put a giant box around this. 261 00:13:35,630 --> 00:13:41,793 STUDENT: [INAUDIBLE] 262 00:13:41,793 --> 00:13:42,293 PROF. 263 00:13:42,293 --> 00:13:42,834 JERISON: Yes. 264 00:13:42,834 --> 00:13:44,543 So the question was, you get all of these 265 00:13:44,543 --> 00:13:45,834 if you use that equation there. 266 00:13:45,834 --> 00:13:47,500 That's exactly what I'm going to do. 267 00:13:47,500 --> 00:13:52,350 So I already did it actually for these three, last time. 268 00:13:52,350 --> 00:13:55,880 But I didn't do it yet for these two. 269 00:13:55,880 --> 00:13:58,490 But I will do it in about two minutes. 270 00:13:58,490 --> 00:14:00,405 Well, maybe five minutes. 271 00:14:00,405 --> 00:14:10,410 But first I want to explain just a few things about these. 272 00:14:10,410 --> 00:14:12,850 They all follow from the basic formula. 273 00:14:12,850 --> 00:14:16,780 In fact, that one deserves a pink box too, doesn't it. 274 00:14:16,780 --> 00:14:18,020 That one's pretty important. 275 00:14:18,020 --> 00:14:19,770 Alright. 276 00:14:19,770 --> 00:14:23,020 Yeah. 277 00:14:23,020 --> 00:14:25,100 Maybe even some little sparkles. 278 00:14:25,100 --> 00:14:32,830 Alright. 279 00:14:32,830 --> 00:14:33,430 OK. 280 00:14:33,430 --> 00:14:36,390 So that's pretty important. 281 00:14:36,390 --> 00:14:39,900 Almost as important as the more basic one without this term 282 00:14:39,900 --> 00:14:41,190 here. 283 00:14:41,190 --> 00:14:47,480 So now, let me just tell you a little bit more 284 00:14:47,480 --> 00:14:54,740 about the significance. 285 00:14:54,740 --> 00:14:56,641 Again, this is just to reinforce something 286 00:14:56,641 --> 00:14:57,640 that we've already done. 287 00:14:57,640 --> 00:14:59,264 But it's closely related to what you're 288 00:14:59,264 --> 00:15:00,880 doing on your problem set. 289 00:15:00,880 --> 00:15:05,230 So it's worth your while to recall this. 290 00:15:05,230 --> 00:15:10,870 So, there's this expression that we were dealing with. 291 00:15:10,870 --> 00:15:13,370 And we talked about it in lecture. 292 00:15:13,370 --> 00:15:19,610 And we showed that this tends to e as k goes to infinity. 293 00:15:19,610 --> 00:15:21,110 So that's what we showed in lecture. 294 00:15:21,110 --> 00:15:25,310 And the way that we did that was, we took the logarithm 295 00:15:25,310 --> 00:15:27,320 and we wrote it as k times, sorry, 296 00:15:27,320 --> 00:15:29,640 the logarithm of 1 + 1/k. 297 00:15:32,530 --> 00:15:35,780 And then we evaluated the limit of this. 298 00:15:35,780 --> 00:15:38,000 And I want to do this limit again, 299 00:15:38,000 --> 00:15:40,220 using linear approximation. 300 00:15:40,220 --> 00:15:42,660 To show you how easy it is if you just remember 301 00:15:42,660 --> 00:15:44,820 the linear approximation. 302 00:15:44,820 --> 00:15:47,720 And then we'll explain where the quadratic approximation comes 303 00:15:47,720 --> 00:15:48,390 in. 304 00:15:48,390 --> 00:15:51,770 So I claim that this is approximately 305 00:15:51,770 --> 00:15:56,470 equal to k times 1/k. 306 00:15:59,300 --> 00:16:00,750 Now, why is that? 307 00:16:00,750 --> 00:16:04,350 Well, that's just this linear approximation. 308 00:16:04,350 --> 00:16:05,500 So what did I use here? 309 00:16:05,500 --> 00:16:14,760 I used log of 1+x is approximately x, for x = 1/k. 310 00:16:14,760 --> 00:16:18,600 That's what I used in this approximation here. 311 00:16:18,600 --> 00:16:20,330 And that's the linear approximation 312 00:16:20,330 --> 00:16:24,010 to the natural logarithm. 313 00:16:24,010 --> 00:16:27,250 And this number is relatively easy to evaluate. 314 00:16:27,250 --> 00:16:28,150 I know how to do it. 315 00:16:28,150 --> 00:16:31,010 It's equal to 1. 316 00:16:31,010 --> 00:16:34,680 That's the same, well, so where does this work? 317 00:16:34,680 --> 00:16:37,450 This works where this thing is near 0. 318 00:16:37,450 --> 00:16:41,610 Which is when k is going to infinity. 319 00:16:41,610 --> 00:16:44,170 This thing is working only when k is going to infinity. 320 00:16:44,170 --> 00:16:46,660 So what it's really saying, this approximation formula, 321 00:16:46,660 --> 00:16:50,690 it's really saying that as we go to infinity, in k, 322 00:16:50,690 --> 00:16:54,720 this thing is going to 1. 323 00:16:54,720 --> 00:16:58,890 As k goes to infinity. 324 00:16:58,890 --> 00:17:00,410 So that's what it's saying. 325 00:17:00,410 --> 00:17:01,740 That's the substance there. 326 00:17:01,740 --> 00:17:05,200 And that's how we want to use it, in many instances. 327 00:17:05,200 --> 00:17:06,580 Just to evaluate limits. 328 00:17:06,580 --> 00:17:09,180 We also want to realize that it's nearby 329 00:17:09,180 --> 00:17:12,800 when k is pretty large, like 100 or something like that. 330 00:17:12,800 --> 00:17:16,660 Now, so that's the idea of the linear approximation. 331 00:17:16,660 --> 00:17:21,920 Now, if you want to get the rate of convergence here, 332 00:17:21,920 --> 00:17:27,950 so the rate of what's called convergence. 333 00:17:27,950 --> 00:17:34,260 So convergence means how fast this is going towards that. 334 00:17:34,260 --> 00:17:36,220 What I have to do is take the difference. 335 00:17:36,220 --> 00:17:40,670 I have to take ln a_k, and I have to subtract 1 from it. 336 00:17:40,670 --> 00:17:42,420 And I know that this is going to 0, 337 00:17:42,420 --> 00:17:47,870 and the question is how big is this. 338 00:17:47,870 --> 00:17:51,760 We want it to be very small. 339 00:17:51,760 --> 00:17:54,490 And the answer we're going to get, 340 00:17:54,490 --> 00:18:03,570 so the answer just uses the quadratic approximation. 341 00:18:03,570 --> 00:18:06,520 So if I just have a little bit more detail, then 342 00:18:06,520 --> 00:18:08,720 this expression here, in other words, 343 00:18:08,720 --> 00:18:11,210 I have the next higher-order term. 344 00:18:11,210 --> 00:18:15,420 This is like 1/k, this is like 1 / k^2. 345 00:18:15,420 --> 00:18:18,890 Then I can understand how big the difference 346 00:18:18,890 --> 00:18:24,260 is between the expression that I've got and its limit. 347 00:18:24,260 --> 00:18:26,200 And so that's what's on your homework. 348 00:18:26,200 --> 00:18:31,610 This is on your problem set. 349 00:18:31,610 --> 00:18:35,050 OK, so that is more or less an explanation for one 350 00:18:35,050 --> 00:18:38,860 of the things that quadratic approximations are good for. 351 00:18:38,860 --> 00:18:42,900 And I'm going to give you one more illustration. 352 00:18:42,900 --> 00:18:45,360 One more illustration. 353 00:18:45,360 --> 00:18:47,472 And then we'll actually check these formulas. 354 00:18:47,472 --> 00:18:48,430 Yeah, another question. 355 00:18:48,430 --> 00:18:55,520 STUDENT: [INAUDIBLE] 356 00:18:55,520 --> 00:18:56,020 PROF. 357 00:18:56,020 --> 00:18:58,470 JERISON: That's a very good question here. 358 00:18:58,470 --> 00:19:01,260 When they, which in this case means maybe, 359 00:19:01,260 --> 00:19:07,510 me, when I give you a question, does one 360 00:19:07,510 --> 00:19:13,030 specify whether you want to use a linear or a quadratic 361 00:19:13,030 --> 00:19:14,430 approximation. 362 00:19:14,430 --> 00:19:17,910 The answer is, in real life when you're 363 00:19:17,910 --> 00:19:22,470 faced with a problem like this, where some satellite is 364 00:19:22,470 --> 00:19:24,700 orbiting and you want to know the effects of gravity 365 00:19:24,700 --> 00:19:28,304 or something like that, nobody is going to tell you anything. 366 00:19:28,304 --> 00:19:29,720 They're not even going to tell you 367 00:19:29,720 --> 00:19:32,480 whether a linear approximation is relevant, or a quadratic 368 00:19:32,480 --> 00:19:33,740 or anything. 369 00:19:33,740 --> 00:19:36,170 So you're on your own. 370 00:19:36,170 --> 00:19:39,830 When I give you a question, at least for right now, 371 00:19:39,830 --> 00:19:42,640 I'm always going to tell you. 372 00:19:42,640 --> 00:19:44,640 But as time goes on I'd like you to get 373 00:19:44,640 --> 00:19:48,360 used to when it's enough to get away 374 00:19:48,360 --> 00:19:49,840 with a linear approximation. 375 00:19:49,840 --> 00:19:53,540 And you should only use a quadratic approximation 376 00:19:53,540 --> 00:19:55,690 if somebody forces you to. 377 00:19:55,690 --> 00:19:57,960 You should always start trying with a linear one. 378 00:19:57,960 --> 00:20:00,610 Because the quadratic ones are much more complicated as you'll 379 00:20:00,610 --> 00:20:03,390 see in this next example. 380 00:20:03,390 --> 00:20:05,320 OK, so the example that I want to use 381 00:20:05,320 --> 00:20:07,740 is, you're going to be stuck with it because I'm 382 00:20:07,740 --> 00:20:09,110 asking for the quadratic. 383 00:20:09,110 --> 00:20:16,190 So we're going to find the quadratic approximation 384 00:20:16,190 --> 00:20:23,740 near-- for x near 0. 385 00:20:23,740 --> 00:20:24,525 To what? 386 00:20:24,525 --> 00:20:26,360 Well, this is the same function that we 387 00:20:26,360 --> 00:20:30,840 used in the last lecture. 388 00:20:30,840 --> 00:20:32,880 I think this was it. 389 00:20:32,880 --> 00:20:34,090 e^(-3x) (1+x)^(-1/2). 390 00:20:37,460 --> 00:20:40,670 OK. 391 00:20:40,670 --> 00:20:45,060 So, unfortunately, I stuck it in the wrong place 392 00:20:45,060 --> 00:20:47,620 to be able to fit this very long formula here. 393 00:20:47,620 --> 00:20:51,100 So I'm going to switch it. 394 00:20:51,100 --> 00:20:57,070 I'm just going to write it here. 395 00:20:57,070 --> 00:21:00,140 And we're going to just do the approximation. 396 00:21:00,140 --> 00:21:03,830 So we're going to say quadratic, in parentheses. 397 00:21:03,830 --> 00:21:08,450 And we'll say x near 0. 398 00:21:08,450 --> 00:21:12,160 So that's what I want. 399 00:21:12,160 --> 00:21:15,424 So now, here's what I have to do. 400 00:21:15,424 --> 00:21:17,590 Well, I have to write in the quadratic approximation 401 00:21:17,590 --> 00:21:25,710 for e^(-3x), and I'm going to use this formula right here. 402 00:21:25,710 --> 00:21:33,970 And so that's 1 + (-3x) + (-3x)^2 / 2. 403 00:21:33,970 --> 00:21:36,530 And the other factor, I'm going to have 404 00:21:36,530 --> 00:21:40,910 to use this formula down here. 405 00:21:40,910 --> 00:21:43,460 Because r is - 1/2. 406 00:21:43,460 --> 00:21:50,300 And so that's 1 - 1/2 x + 1/2 (-1/2)(-3/2)x^2. 407 00:21:59,090 --> 00:22:09,970 So this is the r term, and this is the r - 1 term. 408 00:22:09,970 --> 00:22:11,630 And now I'm going to do something 409 00:22:11,630 --> 00:22:15,550 which is the only good thing about quadratic approximations. 410 00:22:15,550 --> 00:22:17,660 They're messy, they're long, there's nothing 411 00:22:17,660 --> 00:22:19,450 particularly good about them. 412 00:22:19,450 --> 00:22:21,670 But there is one good thing about them. 413 00:22:21,670 --> 00:22:25,610 Which is that you always get to ignore the higher order terms. 414 00:22:25,610 --> 00:22:29,420 So even though this looks like a very ugly multiplication, 415 00:22:29,420 --> 00:22:31,590 I can do it in my head. 416 00:22:31,590 --> 00:22:33,940 Just watching it. 417 00:22:33,940 --> 00:22:38,710 Because I get a 1 * 1, I'm forced with that term here. 418 00:22:38,710 --> 00:22:41,200 And then I get the cross terms which are linear, 419 00:22:41,200 --> 00:22:44,150 which is -3x - 1/2 x. 420 00:22:44,150 --> 00:22:46,460 We already did that when we calculated 421 00:22:46,460 --> 00:22:48,550 the linear approximation, so that's 422 00:22:48,550 --> 00:22:51,670 this times the 1 and this times that 1. 423 00:22:51,670 --> 00:22:55,230 And now I have three cross terms which are quadratic. 424 00:22:55,230 --> 00:22:58,880 So one of them is these two linear terms are multiplying. 425 00:22:58,880 --> 00:23:02,160 So that's plus 3/2 x^2. 426 00:23:02,160 --> 00:23:05,210 That's -3 times -1/2. 427 00:23:05,210 --> 00:23:08,160 And then there's this term, multiplying the 1, 428 00:23:08,160 --> 00:23:11,730 that's plus 9/2 x^2. 429 00:23:11,730 --> 00:23:15,080 And then there's one last term, which is this monster here. 430 00:23:15,080 --> 00:23:23,260 Multiplying 1, and that is -3/8. 431 00:23:23,260 --> 00:23:31,430 So the great thing is, we drop x^3, x^4, etc., terms. 432 00:23:31,430 --> 00:23:37,930 Yeah? 433 00:23:37,930 --> 00:23:39,656 STUDENT: [INAUDIBLE] 434 00:23:39,656 --> 00:23:40,156 PROF. 435 00:23:40,156 --> 00:23:42,500 JERISON: OK, well so copy it down. 436 00:23:42,500 --> 00:23:45,620 And you work it out as I'm doing it now. 437 00:23:45,620 --> 00:23:47,930 So what I did is, I multiplied 1 by 1. 438 00:23:47,930 --> 00:23:50,290 I'm using the distributive law here. 439 00:23:50,290 --> 00:23:51,560 That was this one. 440 00:23:51,560 --> 00:23:55,120 I multiplied this 3x by this one, that was that term. 441 00:23:55,120 --> 00:23:58,550 I multiplied this by this, that's that term. 442 00:23:58,550 --> 00:24:01,120 And then I multiplied this by this. 443 00:24:01,120 --> 00:24:03,860 In other words, two x terms that gave me an x^2 444 00:24:03,860 --> 00:24:06,940 and a (-3)(-1/2). 445 00:24:06,940 --> 00:24:08,499 And I'm going to stop at that point. 446 00:24:08,499 --> 00:24:10,290 Because the point is it's just all the rest 447 00:24:10,290 --> 00:24:12,230 of the terms that come up. 448 00:24:12,230 --> 00:24:14,320 Now, the reason, the only reason why it's easy, 449 00:24:14,320 --> 00:24:16,319 is that I only have to go up to x squared terms. 450 00:24:16,319 --> 00:24:21,462 I don't have to do the higher ones. 451 00:24:21,462 --> 00:24:22,795 Another question, way back here. 452 00:24:22,795 --> 00:24:23,545 Yeah, right there. 453 00:24:23,545 --> 00:24:28,696 STUDENT: [INAUDIBLE] 454 00:24:28,696 --> 00:24:29,196 PROF. 455 00:24:29,196 --> 00:24:29,696 JERISON: OK. 456 00:24:29,696 --> 00:24:38,690 So somebody can check my arithmetic, too. 457 00:24:38,690 --> 00:24:39,470 Good. 458 00:24:39,470 --> 00:24:40,361 STUDENT: [INAUDIBLE] 459 00:24:40,361 --> 00:24:40,861 PROF. 460 00:24:40,861 --> 00:24:43,466 JERISON: Why do I get to drop all the higher-order terms. 461 00:24:43,466 --> 00:24:46,210 So, that's because the situation where 462 00:24:46,210 --> 00:24:50,200 I'm going to apply this is the situation in which x is, 463 00:24:50,200 --> 00:24:52,610 say, 1/100. 464 00:24:52,610 --> 00:24:55,030 So here's about 1/100. 465 00:24:55,030 --> 00:24:57,210 Here's something which is on the order of 1/100. 466 00:24:57,210 --> 00:25:00,130 This is on the order of 1/100^2. 467 00:25:00,130 --> 00:25:02,450 1/100^2, all of these terms. 468 00:25:02,450 --> 00:25:07,400 Now, these cubic and quartic terms are of the order 469 00:25:07,400 --> 00:25:10,520 of 1/100^3. 470 00:25:10,520 --> 00:25:11,420 That's 10^(-6). 471 00:25:11,420 --> 00:25:12,420 6. 472 00:25:12,420 --> 00:25:14,730 And the point is that I'm not claiming 473 00:25:14,730 --> 00:25:16,230 that I have an exact answer. 474 00:25:16,230 --> 00:25:19,950 And I'm going to drop things of that order of magnitude. 475 00:25:19,950 --> 00:25:22,890 So I'm saving everything up to 4 decimal places. 476 00:25:22,890 --> 00:25:29,300 I'm throwing away things which are 6 decimal places out. 477 00:25:29,300 --> 00:25:30,870 Does that answer your question? 478 00:25:30,870 --> 00:25:35,211 STUDENT: [INAUDIBLE] 479 00:25:35,211 --> 00:25:35,710 PROF. 480 00:25:35,710 --> 00:25:36,370 JERISON: So. 481 00:25:36,370 --> 00:25:39,770 That's the situation, and now you can combine the terms. 482 00:25:39,770 --> 00:25:43,780 I mean, it's not very impressive here. 483 00:25:43,780 --> 00:25:54,300 This is equal to 1 - 7/2 x, maybe, plus 51/8 x^2. 484 00:25:54,300 --> 00:25:57,530 If I've made that-- if those minus signs hadn't canceled, 485 00:25:57,530 --> 00:25:59,560 I would have gotten the wrong answer here. 486 00:25:59,560 --> 00:26:00,140 Anyway. 487 00:26:00,140 --> 00:26:03,430 So, this is a 2 here, sorry. 488 00:26:03,430 --> 00:26:04,920 7/2. 489 00:26:04,920 --> 00:26:07,864 This is the linear approximation we got last time 490 00:26:07,864 --> 00:26:09,530 and here's the extra information that we 491 00:26:09,530 --> 00:26:11,540 got from this calculation. 492 00:26:11,540 --> 00:26:17,650 Which is this 51/8 term. 493 00:26:17,650 --> 00:26:19,620 Right, you have to accept that there's 494 00:26:19,620 --> 00:26:21,890 a certain degree of complexity to this problem 495 00:26:21,890 --> 00:26:23,840 and the answer is sufficiently complicated 496 00:26:23,840 --> 00:26:25,740 so it can't be less arithmetic because we 497 00:26:25,740 --> 00:26:29,380 get this peculiar 51/8 there, right. 498 00:26:29,380 --> 00:26:31,950 So one of the things to realize is 499 00:26:31,950 --> 00:26:35,120 that these kinds of problems, because they involve 500 00:26:35,120 --> 00:26:37,135 many, many terms are always going 501 00:26:37,135 --> 00:26:43,960 to involve a little bit of complicated arithmetic. 502 00:26:43,960 --> 00:26:46,910 Last little bit, I did promise you 503 00:26:46,910 --> 00:26:51,360 that I was going to derive these two relations, as I said. 504 00:26:51,360 --> 00:26:53,020 Did the ones in the left column. 505 00:26:53,020 --> 00:26:56,640 So let's carry that out. 506 00:26:56,640 --> 00:27:01,400 And as someone just pointed out, it all comes from this formula 507 00:27:01,400 --> 00:27:01,900 here. 508 00:27:01,900 --> 00:27:07,710 So let's just check it. 509 00:27:07,710 --> 00:27:12,160 So we'll start with the log function. 510 00:27:12,160 --> 00:27:18,810 This is the function, f, and then f' is 1/(1+x). 511 00:27:18,810 --> 00:27:24,620 And f'', so this is f', this is f'', is - -1 / (1+x)^2. 512 00:27:28,160 --> 00:27:31,730 And now I have to plug in x = 0. 513 00:27:31,730 --> 00:27:35,180 So at x = 0 this is ln 1, which is 0. 514 00:27:35,180 --> 00:27:37,860 So this is at x = 0. 515 00:27:37,860 --> 00:27:41,850 I'm getting 0 here, I plug in 0 and I get 1. 516 00:27:41,850 --> 00:27:45,720 And here, I plug in 0 and I get -1. 517 00:27:45,720 --> 00:27:48,050 So now I go and I look up at that formula, 518 00:27:48,050 --> 00:27:50,270 which is way in that upper corner there. 519 00:27:50,270 --> 00:27:53,420 And I see that the coefficient on the constant is 0. 520 00:27:53,420 --> 00:27:55,260 The coefficient on x is 1. 521 00:27:55,260 --> 00:27:59,290 And then the other coefficient, the very last one, is -1/2. 522 00:27:59,290 --> 00:28:01,010 So this is the -1 here. 523 00:28:01,010 --> 00:28:05,340 And then in the formula, there's a 2 in the denominator. 524 00:28:05,340 --> 00:28:08,400 So it's half of whatever I get for this second derivative, 525 00:28:08,400 --> 00:28:10,590 at 0. 526 00:28:10,590 --> 00:28:13,100 So this is the approximation formula, which 527 00:28:13,100 --> 00:28:17,260 is way up in that corner there. 528 00:28:17,260 --> 00:28:21,290 Similarly, if I do it for (1+x)^r, 529 00:28:21,290 --> 00:28:25,990 I have to differentiate that, I get r(1+x)^(r-1), 530 00:28:25,990 --> 00:28:27,300 and then r(r-1)(x+1)^(r-2). 531 00:28:31,270 --> 00:28:33,830 So here are the derivatives. 532 00:28:33,830 --> 00:28:41,110 And so if I evaluate them at x = 0, I get 1. 533 00:28:41,110 --> 00:28:43,240 That's 1^r is 1. 534 00:28:43,240 --> 00:28:46,910 And here I get r. 535 00:28:46,910 --> 00:28:49,580 1^(r-1) times r. 536 00:28:49,580 --> 00:28:54,010 And here, I plug in x = 0 and I get r(r-1). 537 00:28:59,790 --> 00:29:02,880 So again, the pattern is right above it here. 538 00:29:02,880 --> 00:29:05,300 The 1 is there, the r is there. 539 00:29:05,300 --> 00:29:08,920 And then instead of r(r-1), I have half that. 540 00:29:08,920 --> 00:29:21,480 For the coefficient. 541 00:29:21,480 --> 00:29:22,660 So these are just examples. 542 00:29:22,660 --> 00:29:24,700 Obviously if we had a more complicated function, 543 00:29:24,700 --> 00:29:26,050 we might carry this out. 544 00:29:26,050 --> 00:29:28,210 But as a practical matter, we try 545 00:29:28,210 --> 00:29:30,810 to stick with the ones in the pink box 546 00:29:30,810 --> 00:29:42,460 and just use algebra to get other formulas. 547 00:29:42,460 --> 00:29:46,430 So I want to shift gears now and treat the subject that was 548 00:29:46,430 --> 00:29:48,910 supposed to be this lecture. 549 00:29:48,910 --> 00:29:51,230 And we're not quite caught up, but we 550 00:29:51,230 --> 00:29:54,890 will try to do our best to do as much as we can today. 551 00:29:54,890 --> 00:29:59,880 So the next topic is curve sketching. 552 00:29:59,880 --> 00:30:18,120 And so let's get started with that. 553 00:30:18,120 --> 00:30:24,950 So now, happily in this subject, there are more pictures 554 00:30:24,950 --> 00:30:26,880 and it's a little bit more geometric. 555 00:30:26,880 --> 00:30:30,420 And there's relatively little computation. 556 00:30:30,420 --> 00:30:33,600 So let's hope we can do this. 557 00:30:33,600 --> 00:30:36,890 So I want to-- so here we go, we'll 558 00:30:36,890 --> 00:30:44,930 start with curve sketching. 559 00:30:44,930 --> 00:30:47,520 And the goal here-- 560 00:30:47,520 --> 00:30:56,410 STUDENT: [INAUDIBLE] 561 00:30:56,410 --> 00:30:56,910 PROF. 562 00:30:56,910 --> 00:31:00,880 JERISON: So that's like 'liner', the last time. 563 00:31:00,880 --> 00:31:09,600 That's kind of sketchy spelling, isn't it? 564 00:31:09,600 --> 00:31:12,800 Yeah, there are certain kinds of things which I can't spell. 565 00:31:12,800 --> 00:31:16,640 But, all right. 566 00:31:16,640 --> 00:31:18,741 Sketching. 567 00:31:18,741 --> 00:31:19,240 Alright. 568 00:31:19,240 --> 00:31:21,350 So here's our goal. 569 00:31:21,350 --> 00:31:38,280 Our goal is to draw the graph of f, using f' and f''. 570 00:31:42,380 --> 00:31:47,190 Whether they're positive or negative. 571 00:31:47,190 --> 00:31:48,440 So that's it. 572 00:31:48,440 --> 00:31:52,400 This is the goal here. 573 00:31:52,400 --> 00:31:57,020 However, there is a big warning that I want to give you. 574 00:31:57,020 --> 00:32:04,000 And this is one that unfortunately 575 00:32:04,000 --> 00:32:07,005 I now have to make you unlearn, especially 576 00:32:07,005 --> 00:32:08,380 those that you that have actually 577 00:32:08,380 --> 00:32:10,700 had a little bit of calculus before, I 578 00:32:10,700 --> 00:32:12,929 want to make you unlearn some of your instincts 579 00:32:12,929 --> 00:32:13,720 that you developed. 580 00:32:13,720 --> 00:32:15,511 So this will be harder for those of you who 581 00:32:15,511 --> 00:32:19,840 have actually done this before. 582 00:32:19,840 --> 00:32:22,650 But for the rest of you, it will be relatively easy. 583 00:32:22,650 --> 00:32:35,490 Which is, don't abandon your precalculus skills. 584 00:32:35,490 --> 00:32:42,380 And common sense. 585 00:32:42,380 --> 00:32:46,760 So there's a great deal of common sense in this. 586 00:32:46,760 --> 00:32:50,555 And it actually trumps some of the calculus. 587 00:32:50,555 --> 00:32:56,190 The calculus just fills in what you didn't quite know yet. 588 00:32:56,190 --> 00:32:59,660 So I will try to illustrate this. 589 00:32:59,660 --> 00:33:01,390 And because we're running a bit late, 590 00:33:01,390 --> 00:33:03,910 I won't get to the some of the main punchlines 591 00:33:03,910 --> 00:33:05,970 until next lecture. 592 00:33:05,970 --> 00:33:07,380 But I want you to do it. 593 00:33:07,380 --> 00:33:09,412 So for now, I'm just going to tell you 594 00:33:09,412 --> 00:33:10,620 about the general principles. 595 00:33:10,620 --> 00:33:14,100 And in the process I'm going to introduce the terminology. 596 00:33:14,100 --> 00:33:17,167 Just, the words that we need to use to describe what 597 00:33:17,167 --> 00:33:18,000 is that we're doing. 598 00:33:18,000 --> 00:33:19,375 And there's also a certain amount 599 00:33:19,375 --> 00:33:22,160 of carelessness with that in many of the treatments 600 00:33:22,160 --> 00:33:23,070 that you'll see. 601 00:33:23,070 --> 00:33:24,750 And a lot of hastiness. 602 00:33:24,750 --> 00:33:29,350 So just be a little patient and we will do this. 603 00:33:29,350 --> 00:33:33,740 So, the first principle is the following. 604 00:33:33,740 --> 00:33:40,670 If f' is positive, then f is increasing. 605 00:33:40,670 --> 00:33:44,530 That's a straightforward idea, and it's closely related 606 00:33:44,530 --> 00:33:46,730 to this tangent line approximation 607 00:33:46,730 --> 00:33:49,220 or the linear approximation that I just did. 608 00:33:49,220 --> 00:33:50,230 You can just imagine. 609 00:33:50,230 --> 00:33:53,050 Here's the tangent line, here's the function. 610 00:33:53,050 --> 00:33:55,680 And if the tangent line is pointing up, 611 00:33:55,680 --> 00:33:58,490 then the function is also going up, too. 612 00:33:58,490 --> 00:34:00,310 So that's all that's going on here. 613 00:34:00,310 --> 00:34:10,370 Similarly, if f' is negative, then f is decreasing. 614 00:34:10,370 --> 00:34:12,220 And that's the basic idea. 615 00:34:12,220 --> 00:34:17,170 Now, the second step is also fairly straightforward. 616 00:34:17,170 --> 00:34:21,130 It's just a second-order effect of the same type. 617 00:34:21,130 --> 00:34:29,190 If you have f'' as positive, then that means that f' is 618 00:34:29,190 --> 00:34:33,150 increasing. 619 00:34:33,150 --> 00:34:36,740 That's the same principle applied one step up. 620 00:34:36,740 --> 00:34:37,320 Right? 621 00:34:37,320 --> 00:34:41,880 Because if f'' is positive, that means it's the derivative 622 00:34:41,880 --> 00:34:42,710 of f'. 623 00:34:42,710 --> 00:34:45,970 So it's the same principle just repeated. 624 00:34:45,970 --> 00:34:49,710 And now I just want to draw a picture of this. 625 00:34:49,710 --> 00:34:52,470 Here's a picture of it, I claim. 626 00:34:52,470 --> 00:34:54,790 And it looks like something's going down. 627 00:34:54,790 --> 00:34:56,430 And I did that on purpose. 628 00:34:56,430 --> 00:34:58,650 But there is something that's increasing here. 629 00:34:58,650 --> 00:35:02,470 Which is, the slope is very steep negative here. 630 00:35:02,470 --> 00:35:06,330 And it's less steep negative over here. 631 00:35:06,330 --> 00:35:09,790 So we have the slope which is some negative number, say, -4. 632 00:35:09,790 --> 00:35:13,500 And here it's -3. 633 00:35:13,500 --> 00:35:14,800 So it's increasing. 634 00:35:14,800 --> 00:35:18,250 It's getting less negative, and maybe eventually it'll 635 00:35:18,250 --> 00:35:19,680 curve up the other way. 636 00:35:19,680 --> 00:35:23,180 And this is a picture of what I'm talking about here. 637 00:35:23,180 --> 00:35:25,630 That's what it means to say that f' is increasing. 638 00:35:25,630 --> 00:35:28,020 The slope is getting larger. 639 00:35:28,020 --> 00:35:34,930 And the way to describe a curve like this is that it's concave. 640 00:35:34,930 --> 00:35:41,620 So f is concave up. 641 00:35:41,620 --> 00:35:49,320 And similarly, f'' negative is going to be the same thing as f 642 00:35:49,320 --> 00:35:59,770 concave-- or implies f concave down. 643 00:35:59,770 --> 00:36:04,850 So those are the ways in which derivatives will help us 644 00:36:04,850 --> 00:36:08,270 qualitatively to draw graphs. 645 00:36:08,270 --> 00:36:09,880 But as I said before, we still have 646 00:36:09,880 --> 00:36:13,360 to use a little bit of common sense when we draw the graphs. 647 00:36:13,360 --> 00:36:15,710 These are just the additional bits of help 648 00:36:15,710 --> 00:36:17,850 that we have from calculus. 649 00:36:17,850 --> 00:36:25,410 In drawing pictures. 650 00:36:25,410 --> 00:36:30,650 So I'm going to go through one example 651 00:36:30,650 --> 00:36:36,190 to introduce all the notations. 652 00:36:36,190 --> 00:36:42,210 And then eventually, so probably at the beginning of next time, 653 00:36:42,210 --> 00:36:45,340 I'll give you a systematic strategy 654 00:36:45,340 --> 00:36:48,600 that's going to work when what I'm describing now 655 00:36:48,600 --> 00:36:52,360 goes wrong, or a little bit wrong. 656 00:36:52,360 --> 00:36:58,730 So let's begin with a straightforward example. 657 00:36:58,730 --> 00:37:03,280 So, the first example that I'll give you is the function f(x) = 658 00:37:03,280 --> 00:37:04,690 3x - x^3. 659 00:37:07,520 --> 00:37:11,130 Just, as I said, to be able to introduce all the notations. 660 00:37:11,130 --> 00:37:17,640 Now, if you differentiate it, you get 3 - 3x^2. 661 00:37:17,640 --> 00:37:20,884 And I can factor that. 662 00:37:20,884 --> 00:37:21,800 This is 3 (1-x) (1+x). 663 00:37:26,440 --> 00:37:27,750 OK? 664 00:37:27,750 --> 00:37:33,140 And so, I can decide whether the derivative 665 00:37:33,140 --> 00:37:37,390 is positive or negative. 666 00:37:37,390 --> 00:37:38,830 Easily enough. 667 00:37:38,830 --> 00:37:50,120 Namely, just staring at this, I can see that when -1 < x < 1, 668 00:37:50,120 --> 00:37:53,550 in that range there, both these numbers, both these factors, 669 00:37:53,550 --> 00:37:55,550 are positive. 670 00:37:55,550 --> 00:37:59,290 1-x is a positive number and 1 1+x is a positive number. 671 00:37:59,290 --> 00:38:04,710 So, in this range, f'(x) is positive. 672 00:38:04,710 --> 00:38:09,700 So this thing is, so f is increasing. 673 00:38:09,700 --> 00:38:13,310 And similarly, in the other ranges, 674 00:38:13,310 --> 00:38:15,770 if x is very, very large, this becomes, 675 00:38:15,770 --> 00:38:18,190 if it crosses 1, in fact, this becomes, 676 00:38:18,190 --> 00:38:21,310 this factor becomes negative and this one stays positive. 677 00:38:21,310 --> 00:38:29,680 So when x > 1, we have that f'(x) is negative. 678 00:38:29,680 --> 00:38:35,510 And so f is decreasing. 679 00:38:35,510 --> 00:38:39,540 And the same thing goes for the other side. 680 00:38:39,540 --> 00:38:42,500 When it's less than -1, that also works this way. 681 00:38:42,500 --> 00:38:46,420 Because when it's less than -1, this factor is positive. 682 00:38:46,420 --> 00:38:50,300 But the other one is negative. 683 00:38:50,300 --> 00:38:56,290 So in both of these cases, we get that it's decreasing. 684 00:38:56,290 --> 00:39:07,300 So now, here's the schematic picture of this function. 685 00:39:07,300 --> 00:39:12,950 So here's -1, here's 1. 686 00:39:12,950 --> 00:39:19,840 It's going to go down, up, down. 687 00:39:19,840 --> 00:39:21,590 That's what it's doing. 688 00:39:21,590 --> 00:39:23,710 Maybe I'll just leave it alone like this. 689 00:39:23,710 --> 00:39:28,120 That's what it looks like. 690 00:39:28,120 --> 00:39:31,400 So, this is the kind of information 691 00:39:31,400 --> 00:39:32,980 we can get right off the bat. 692 00:39:32,980 --> 00:39:38,089 And you notice immediately that it's very important, 693 00:39:38,089 --> 00:39:39,880 from the features of the function, the sort 694 00:39:39,880 --> 00:39:42,060 of key features of the function that we see here, 695 00:39:42,060 --> 00:39:44,690 are these two places. 696 00:39:44,690 --> 00:39:49,270 Maybe I'll even mark them in a, like this. 697 00:39:49,270 --> 00:39:59,090 And these things are turning points. 698 00:39:59,090 --> 00:40:00,430 So what are they? 699 00:40:00,430 --> 00:40:03,920 Well, they're just the points where 700 00:40:03,920 --> 00:40:05,749 the derivative changes sign. 701 00:40:05,749 --> 00:40:07,790 Where it's negative here and it's positive there, 702 00:40:07,790 --> 00:40:09,850 so there it must be 0. 703 00:40:09,850 --> 00:40:12,830 So we have a definition, and this is the most important 704 00:40:12,830 --> 00:40:21,110 definition in this subject, which is that is if f'(x_0) = 705 00:40:21,110 --> 00:40:33,370 0, we call x_0 a critical point. 706 00:40:33,370 --> 00:40:36,620 The word 'turning point' is not used just because, in fact, it 707 00:40:36,620 --> 00:40:38,880 doesn't have to turn around at those points. 708 00:40:38,880 --> 00:40:42,710 But certainly, if it turns around then this will happen. 709 00:40:42,710 --> 00:40:44,980 And we also have another notation, 710 00:40:44,980 --> 00:40:50,580 which is the number y_0 which is f(x_0) is 711 00:40:50,580 --> 00:40:59,100 called a critical value. 712 00:40:59,100 --> 00:41:02,230 So these are the key numbers that we're 713 00:41:02,230 --> 00:41:05,340 going to have to work out in order to understand 714 00:41:05,340 --> 00:41:18,750 what the function looks like. 715 00:41:18,750 --> 00:41:27,990 So what I'm going to do is just plot them. 716 00:41:27,990 --> 00:41:30,440 We're going to plot the critical points and the values. 717 00:41:30,440 --> 00:41:34,650 Well, we found the critical points relatively easily. 718 00:41:34,650 --> 00:41:37,240 I didn't write it down here but it's pretty obvious. 719 00:41:37,240 --> 00:41:42,353 If you set f(x) = 0, that implies that (1 - 720 00:41:42,353 --> 00:41:50,570 x)(1 + x) = 0, which implies that x is plus or minus 1. 721 00:41:50,570 --> 00:41:52,710 So those are known as the critical points. 722 00:41:52,710 --> 00:41:56,420 And now, in order to get the critical values here, 723 00:41:56,420 --> 00:42:02,280 I have to plug in f(1), for instance, the function is 3x - 724 00:42:02,280 --> 00:42:07,480 x^2, so there's this 3 * 1 - 1^3, which is 2. 725 00:42:07,480 --> 00:42:17,180 And f(-1), which is 3(-1) - (-1)^3, which is -2. 726 00:42:17,180 --> 00:42:21,780 And so I can plot the function here. 727 00:42:21,780 --> 00:42:26,160 So here's the point -1 and here's, up here, is 2. 728 00:42:26,160 --> 00:42:28,620 So this is-- whoops, which one is it? 729 00:42:28,620 --> 00:42:29,160 Yeah. 730 00:42:29,160 --> 00:42:32,560 This is -1, so it's down here. 731 00:42:32,560 --> 00:42:35,160 So it's (-1, -2). 732 00:42:35,160 --> 00:42:41,570 And then over here, I have the point (1, 2). 733 00:42:41,570 --> 00:42:44,570 Alright, now, what information do 734 00:42:44,570 --> 00:42:48,930 I get from - so I've now plotted two, I claim, 735 00:42:48,930 --> 00:42:50,630 very interesting points. 736 00:42:50,630 --> 00:42:55,000 What information do I get from this? 737 00:42:55,000 --> 00:42:59,680 The answer is, I know something very nearby. 738 00:42:59,680 --> 00:43:01,830 Because I've already checked that the thing 739 00:43:01,830 --> 00:43:04,770 is coming down from the left, and coming back up. 740 00:43:04,770 --> 00:43:07,370 And so it must be shaped like this. 741 00:43:07,370 --> 00:43:08,330 Over here. 742 00:43:08,330 --> 00:43:11,620 The tangent line is 0, it's going to be level there. 743 00:43:11,620 --> 00:43:14,490 And similarly over here, it's going to do that. 744 00:43:14,490 --> 00:43:22,211 So this is what we know so far, based on what we've computed. 745 00:43:22,211 --> 00:43:22,710 Question. 746 00:43:22,710 --> 00:43:36,916 STUDENT: [INAUDIBLE] 747 00:43:36,916 --> 00:43:37,416 PROF. 748 00:43:37,416 --> 00:43:38,150 JERISON: The question is, what happens 749 00:43:38,150 --> 00:43:39,233 if there's a sharp corner. 750 00:43:39,233 --> 00:43:43,350 The answer is, calculus is-- it's not 751 00:43:43,350 --> 00:43:45,050 called a critical point. 752 00:43:45,050 --> 00:43:47,090 It's a something else. 753 00:43:47,090 --> 00:43:50,440 And it's a very important point, too. 754 00:43:50,440 --> 00:43:52,570 And we will be discussing those kinds of points. 755 00:43:52,570 --> 00:43:54,540 There are much more dramatic instances of that. 756 00:43:54,540 --> 00:43:56,640 That's part of what we're going to say. 757 00:43:56,640 --> 00:43:58,670 But I just want to save that, all right. 758 00:43:58,670 --> 00:44:02,490 We will be discussing. 759 00:44:02,490 --> 00:44:03,010 Yeah. 760 00:44:03,010 --> 00:44:03,540 Question. 761 00:44:03,540 --> 00:44:08,516 STUDENT: [INAUDIBLE] PROF. 762 00:44:08,516 --> 00:44:10,316 JERISON: The question that was asked 763 00:44:10,316 --> 00:44:12,070 was, how did I know at the critical point 764 00:44:12,070 --> 00:44:17,190 that it's concave down over here and concave up over here. 765 00:44:17,190 --> 00:44:21,590 The answer is that I actually did not 766 00:44:21,590 --> 00:44:24,300 use the second derivative yet. 767 00:44:24,300 --> 00:44:26,510 What I used is another piece of information. 768 00:44:26,510 --> 00:44:28,760 I used the information that I derived over here. 769 00:44:28,760 --> 00:44:32,500 That f' is positive, where f' is positive 770 00:44:32,500 --> 00:44:33,670 and where it's negative. 771 00:44:33,670 --> 00:44:36,980 So what I know is that the graph is going down 772 00:44:36,980 --> 00:44:39,470 to the left of -1. 773 00:44:39,470 --> 00:44:41,530 It's going up to the right, here. 774 00:44:41,530 --> 00:44:44,920 It's going up here and it's going down there. 775 00:44:44,920 --> 00:44:47,060 I did not use the second derivative. 776 00:44:47,060 --> 00:44:49,960 I used the first derivative. 777 00:44:49,960 --> 00:44:51,840 OK, but I didn't just use the fact 778 00:44:51,840 --> 00:44:55,834 that there was a turning point here. 779 00:44:55,834 --> 00:44:57,250 So, actually, I was using the fact 780 00:44:57,250 --> 00:44:58,416 that it was a turning point. 781 00:44:58,416 --> 00:45:01,080 I wasn't using the fact that it had the second derivative, 782 00:45:01,080 --> 00:45:01,730 though. 783 00:45:01,730 --> 00:45:02,230 OK. 784 00:45:02,230 --> 00:45:03,120 For now. 785 00:45:03,120 --> 00:45:09,500 You can also see it by the second derivative as well. 786 00:45:09,500 --> 00:45:13,710 So now, the next thing that I'd like to do, 787 00:45:13,710 --> 00:45:16,170 I need to finish off this graph. 788 00:45:16,170 --> 00:45:19,950 And I just want to do it a little bit carefully here. 789 00:45:19,950 --> 00:45:22,710 In the order that is reasonable. 790 00:45:22,710 --> 00:45:26,830 Now, you might happen to notice, and there's 791 00:45:26,830 --> 00:45:31,660 nothing wrong with this. 792 00:45:31,660 --> 00:45:34,826 So let's even fill in a guess. 793 00:45:34,826 --> 00:45:36,700 In order to fill in a guess, though, and have 794 00:45:36,700 --> 00:45:38,190 it be even vaguely right, I do have 795 00:45:38,190 --> 00:45:39,606 to notice that this thing crosses, 796 00:45:39,606 --> 00:45:41,800 this function crosses the origin. 797 00:45:41,800 --> 00:45:48,120 The function f(x) = 3x - x^3 happens to also have 798 00:45:48,120 --> 00:45:50,780 the property that f(0) = 0. 799 00:45:50,780 --> 00:45:52,560 Again, common sense. 800 00:45:52,560 --> 00:45:54,590 You're allowed to use your common sense. 801 00:45:54,590 --> 00:45:58,320 You're allowed to notice a value of the function and put it in. 802 00:45:58,320 --> 00:46:00,270 So there's nothing wrong with that. 803 00:46:00,270 --> 00:46:03,960 If you happen to have such a value. 804 00:46:03,960 --> 00:46:06,650 So, now we can guess what our function is going to look like. 805 00:46:06,650 --> 00:46:10,010 It's going to maybe come down like this. 806 00:46:10,010 --> 00:46:11,670 Come up like this. 807 00:46:11,670 --> 00:46:13,130 And come down like this. 808 00:46:13,130 --> 00:46:15,230 That could be what it looks like. 809 00:46:15,230 --> 00:46:16,800 But, you know, another possibility 810 00:46:16,800 --> 00:46:19,820 is it sort of comes along here and goes out that way. 811 00:46:19,820 --> 00:46:22,260 Comes along here and goes out that way, who knows? 812 00:46:22,260 --> 00:46:25,070 It happens, by the way, that it's an odd function. 813 00:46:25,070 --> 00:46:25,570 Right? 814 00:46:25,570 --> 00:46:26,640 Those are all odd powers. 815 00:46:26,640 --> 00:46:28,710 So, actually, it's symmetric on the right half 816 00:46:28,710 --> 00:46:29,860 and the left half. 817 00:46:29,860 --> 00:46:31,289 And crosses at 0. 818 00:46:31,289 --> 00:46:32,830 So everything that we do on the right 819 00:46:32,830 --> 00:46:34,996 is going to be the same as what happens on the left. 820 00:46:34,996 --> 00:46:36,620 That's another piece of common sense. 821 00:46:36,620 --> 00:46:38,120 You want to make use of that as much 822 00:46:38,120 --> 00:46:40,060 as possible, whenever you're drawing anything. 823 00:46:40,060 --> 00:46:42,340 Don't want to throw out information. 824 00:46:42,340 --> 00:46:46,540 So this function happens to be odd. 825 00:46:46,540 --> 00:46:48,700 Odd, and f(0) = 0. 826 00:46:48,700 --> 00:46:52,520 I'm considering those to be kinds of precalculus skills 827 00:46:52,520 --> 00:47:00,610 that I want you to use as much as you can. 828 00:47:00,610 --> 00:47:02,880 So now, here's the first feature which 829 00:47:02,880 --> 00:47:08,880 is unfortunately ignored in most discussions of functions. 830 00:47:08,880 --> 00:47:11,340 And it's strange, because nowadays we 831 00:47:11,340 --> 00:47:13,380 have graphing things. 832 00:47:13,380 --> 00:47:19,080 And it's really the only part of the exercise 833 00:47:19,080 --> 00:47:22,480 that you couldn't do, at least on this relatively simpleminded 834 00:47:22,480 --> 00:47:28,180 level, with a graphing calculator. 835 00:47:28,180 --> 00:47:33,790 And that is what I would call the ends of the problem. 836 00:47:33,790 --> 00:47:37,000 So what happens off the screen, is the question. 837 00:47:37,000 --> 00:47:39,880 And that basically is the theoretical part of the problem 838 00:47:39,880 --> 00:47:41,430 that you have to address. 839 00:47:41,430 --> 00:47:42,920 You can program this. 840 00:47:42,920 --> 00:47:45,410 You can draw all the pictures that you want. 841 00:47:45,410 --> 00:47:48,190 But what you won't see is what's off the screen. 842 00:47:48,190 --> 00:47:50,260 You need to know something to figure out 843 00:47:50,260 --> 00:47:51,780 what's off the screen. 844 00:47:51,780 --> 00:47:54,560 So, in this case, I'm talking about what's off the screen 845 00:47:54,560 --> 00:48:01,930 going to the right, or going to the left. 846 00:48:01,930 --> 00:48:06,100 So let's check the ends. 847 00:48:06,100 --> 00:48:07,680 So here, let's just take a look. 848 00:48:07,680 --> 00:48:12,280 We have the function f(x), which is, sorry, 3x - x^3. 849 00:48:12,280 --> 00:48:14,620 Again this is a precalculus sort of thing. 850 00:48:14,620 --> 00:48:16,230 And we're imagining now, let's just 851 00:48:16,230 --> 00:48:18,360 do x goes to plus infinity. 852 00:48:18,360 --> 00:48:19,670 So what happens here. 853 00:48:19,670 --> 00:48:25,460 When x is gigantic, this term is completely negligible. 854 00:48:25,460 --> 00:48:30,480 And it just behaves like -x^3, which goes to minus infinity 855 00:48:30,480 --> 00:48:32,600 as x goes to plus infinity. 856 00:48:32,600 --> 00:48:40,330 And similarly, f(x) goes to plus infinity 857 00:48:40,330 --> 00:48:46,150 if x goes to minus infinity. 858 00:48:46,150 --> 00:48:50,060 Now let me pull down this picture again, and show you 859 00:48:50,060 --> 00:48:53,140 what piece of the information we've got. 860 00:48:53,140 --> 00:48:55,610 We now know that it is heading up this way. 861 00:48:55,610 --> 00:48:58,560 It doesn't go like this, it goes up like that. 862 00:48:58,560 --> 00:49:00,220 And I'm going to put an arrow for it, 863 00:49:00,220 --> 00:49:02,580 And it's going down like this. 864 00:49:02,580 --> 00:49:06,500 Heading down to minus infinity as x goes out farther 865 00:49:06,500 --> 00:49:07,310 to the right. 866 00:49:07,310 --> 00:49:15,120 And going out to plus infinity as x goes farther to the left. 867 00:49:15,120 --> 00:49:19,070 So now there's hardly anything left 868 00:49:19,070 --> 00:49:21,910 of this function to describe. 869 00:49:21,910 --> 00:49:27,120 There's really nothing left except maybe decoration. 870 00:49:27,120 --> 00:49:29,650 And we kind of like that decoration, 871 00:49:29,650 --> 00:49:32,560 so we will pay attention to it. 872 00:49:32,560 --> 00:49:35,580 And to do that, we'll have to check the second derivative. 873 00:49:35,580 --> 00:49:40,150 So if we differentiate a second time, the first derivative was, 874 00:49:40,150 --> 00:49:42,680 remember, 3 - 3x^2. 875 00:49:42,680 --> 00:49:53,540 So the second derivative is -6x. 876 00:49:53,540 --> 00:50:02,470 So now we notice that f''(x) is negative if x is positive. 877 00:50:02,470 --> 00:50:08,310 And f''(x) is positive if x is negative. 878 00:50:08,310 --> 00:50:13,170 And so in this part it's concave down. 879 00:50:13,170 --> 00:50:18,990 And in this part it's concave up. 880 00:50:18,990 --> 00:50:22,390 And now I'm going to switch the boards so that you'll, and draw 881 00:50:22,390 --> 00:50:24,860 it. 882 00:50:24,860 --> 00:50:30,630 And you see that it was begging to be this way. 883 00:50:30,630 --> 00:50:32,960 So we'll fill in the rest of it here. 884 00:50:32,960 --> 00:50:35,560 Maybe in a nice color here. 885 00:50:35,560 --> 00:50:38,100 So this is the whole graph and this is the correct graph. 886 00:50:38,100 --> 00:50:41,450 It comes down in one swoop down here, and comes up here. 887 00:50:41,450 --> 00:50:46,620 And then it changes to concave down right at the origin. 888 00:50:46,620 --> 00:50:49,250 So this point is of interest, not only 889 00:50:49,250 --> 00:50:51,750 because it's the place where it crosses the axis, 890 00:50:51,750 --> 00:51:00,690 but it's also what's called an inflection point. 891 00:51:00,690 --> 00:51:02,970 Inflection point, that's a point where-- 892 00:51:02,970 --> 00:51:07,620 because f'' at that place is equal to 0. 893 00:51:07,620 --> 00:51:10,970 So it's a place where the second derivative is 0. 894 00:51:10,970 --> 00:51:15,160 We also consider those to be interesting points. 895 00:51:15,160 --> 00:51:21,010 Now, so let me just making one closing remark here. 896 00:51:21,010 --> 00:51:26,590 Which is that all of this information fits together. 897 00:51:26,590 --> 00:51:29,610 And we're going to have much, much harder examples of this 898 00:51:29,610 --> 00:51:32,570 where you'll actually have to think about what's going on. 899 00:51:32,570 --> 00:51:35,190 But there's a lot of stuff protecting you. 900 00:51:35,190 --> 00:51:39,050 And functions will behave themselves 901 00:51:39,050 --> 00:51:40,920 and turn around appropriately. 902 00:51:40,920 --> 00:51:43,473 Anyway, we'll talk about it next time.