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 --> 00:00:09 offer high quality educational resources for free. 6 00:00:09 --> 00:00:13 To make a donation, or to view additional materials from 7 00:00:13 --> 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 --> 00:00:24 PROFESSOR: To begin today I want to remind you, I need to 10 00:00:24 --> 00:00:28 write it down on the board at least twice, of the fundamental 11 00:00:28 --> 00:00:33 theorem of calculus. 12 00:00:33 --> 00:00:38 We called it FTC 1 because it's the first version of 13 00:00:38 --> 00:00:39 the fundamental theorem. 14 00:00:39 --> 00:00:42 We'll be talking about another version, called 15 00:00:42 --> 00:00:44 the second version, today. 16 00:00:44 --> 00:00:58 And what it says is this: If F' = f, then the integral from a 17 00:00:58 --> 00:01:06 to b of f (x) dx = F ( b) - F ( a). 18 00:01:06 --> 00:01:10 So that's the fundamental theorem of calculus. 19 00:01:10 --> 00:01:17 And the way we used it last time was, this was used 20 00:01:17 --> 00:01:22 to evaluate integrals. 21 00:01:22 --> 00:01:28 Not surprisingly, that's how we used it. 22 00:01:28 --> 00:01:35 But today, I want to reverse that point of view. 23 00:01:35 --> 00:01:39 We're going to read the equation backwards, and we're 24 00:01:39 --> 00:01:49 going to write it this way. 25 00:01:49 --> 00:02:01 And we're going to use f to understand capital F. 26 00:02:01 --> 00:02:04 Or in other words, the derivative. 27 00:02:04 --> 00:02:07 To understand the function. 28 00:02:07 --> 00:02:13 So that's the reversal of point of view that I'd like to make. 29 00:02:13 --> 00:02:16 And we'll make this point in various ways. 30 00:02:16 --> 00:02:28 So information about F, about F', gives us 31 00:02:28 --> 00:02:36 information about F. 32 00:02:36 --> 00:02:38 Now, since there were questions about the mean value theorem, 33 00:02:38 --> 00:02:43 I'm going to illustrate this first by making a comparison 34 00:02:43 --> 00:02:46 between the fundamental theorem of calculus and the 35 00:02:46 --> 00:02:50 mean value theorem. 36 00:02:50 --> 00:02:56 So we're going to compare this fundamental theorem of calculus 37 00:02:56 --> 00:03:01 with what we call the mean value theorem. 38 00:03:01 --> 00:03:03 And in order to do that, I'm going to introduce 39 00:03:03 --> 00:03:05 a couple of notations. 40 00:03:05 --> 00:03:11 I'll write delta F as F ( b ) - F ( a). 41 00:03:11 --> 00:03:17 And another highly imaginative notation, delta x = b - a. 42 00:03:17 --> 00:03:21 So here's the change in f, there's the change in x. 43 00:03:21 --> 00:03:27 And then, this fundamental theorem can be written, of 44 00:03:27 --> 00:03:30 course, right up above there is the formula. 45 00:03:30 --> 00:03:36 And it's the formula for delta F. 46 00:03:36 --> 00:03:38 So this is what we call the fundamental 47 00:03:38 --> 00:03:44 theorem of calculus. 48 00:03:44 --> 00:03:51 I'm going to divide by delta x, now. 49 00:03:51 --> 00:03:56 And If I divide by delta x, that's the same thing as 1 50 00:03:56 --> 00:04:02 / b - a times the integral from a to b of f ( x) dx. 51 00:04:02 --> 00:04:05 So I've just rewritten the formula here. 52 00:04:05 --> 00:04:11 And this expression here, on the right-hand side, is 53 00:04:11 --> 00:04:13 a fairly important one. 54 00:04:13 --> 00:04:23 This is the average of f. 55 00:04:23 --> 00:04:29 That's the average value of f. 56 00:04:29 --> 00:04:34 Now, so this is going to permit me to make the comparison 57 00:04:34 --> 00:04:37 between the mean value theorem, which we don't have 58 00:04:37 --> 00:04:38 stated yet here. 59 00:04:38 --> 00:04:41 And the fundamental theorem. 60 00:04:41 --> 00:04:49 And I'll do it in the form of inequalities. 61 00:04:49 --> 00:04:51 So right in the middle here, I'm going to put 62 00:04:51 --> 00:04:52 the fundamental theorem. 63 00:04:52 --> 00:04:57 It says that delta F in this notation is equal to, well if 64 00:04:57 --> 00:04:59 I multiply by delta x again, I can write it as 65 00:04:59 --> 00:05:01 the average of F. 66 00:05:01 --> 00:05:04 So I'm going to write it as the average of F' here. 67 00:05:04 --> 00:05:07 Times delta x. 68 00:05:07 --> 00:05:10 So we have this factor here, which is the average of F', 69 00:05:10 --> 00:05:13 or the average of little f, it's the same thing. 70 00:05:13 --> 00:05:15 And then I multiplied through again. 71 00:05:15 --> 00:05:20 So I put the thing in the red box. 72 00:05:20 --> 00:05:29 Here. 73 00:05:29 --> 00:05:37 Isn't what the average of big F. 74 00:05:37 --> 00:05:41 So the question is, why is this the average of little f rather 75 00:05:41 --> 00:05:44 than the average of big F. 76 00:05:44 --> 00:05:50 So the average of a function is the typical value. 77 00:05:50 --> 00:05:56 If, for example, little f were constant, little f were 78 00:05:56 --> 00:06:01 constant, then this integral would be, so the question is 79 00:06:01 --> 00:06:03 why is this the average. 80 00:06:03 --> 00:06:08 And I'll take a little second to explain that. 81 00:06:08 --> 00:06:13 But I think I'll explain it over here. 82 00:06:13 --> 00:06:19 Because I'm going to erase it. 83 00:06:19 --> 00:06:27 So the idea of an average is the following. 84 00:06:27 --> 00:06:32 For example, imagine that a = 0 and b = n, let's 85 00:06:32 --> 00:06:33 say for example. 86 00:06:33 --> 00:06:46 And so we might sum the function from 1 to n. 87 00:06:46 --> 00:06:49 Now, that would be the sum of the values from 1 to n. 88 00:06:49 --> 00:06:54 But the average is, we divide by n here. 89 00:06:54 --> 00:06:56 So this is the average. 90 00:06:56 --> 00:07:01 And this is a kind of Riemann sum, representing the integral 91 00:07:01 --> 00:07:06 from 0 to n, of f (x) dx. 92 00:07:06 --> 00:07:10 Where the increment, delta x, is 1. 93 00:07:10 --> 00:07:13 So this is the notion of an average value here. 94 00:07:13 --> 00:07:15 But in the continuum setting as opposed to 95 00:07:15 --> 00:07:20 the discrete setting. 96 00:07:20 --> 00:07:26 Whereas what's on the left-hand side is the change in f. 97 00:07:26 --> 00:07:28 The capital F. 98 00:07:28 --> 00:07:31 And this is the average of the little f. 99 00:07:31 --> 00:07:33 So an average is a sum. 100 00:07:33 --> 00:07:39 And it's like an integral. 101 00:07:39 --> 00:07:43 So, in other words what I have here is that the change in f is 102 00:07:43 --> 00:07:45 the average of its infinitesimal change 103 00:07:45 --> 00:07:50 times the amount of time elapsed, if you like. 104 00:07:50 --> 00:07:56 So this is the statement of the fundamental theorem. 105 00:07:56 --> 00:07:56 Just rewritten. 106 00:07:56 --> 00:07:58 Exactly what I wrote there. 107 00:07:58 --> 00:08:01 But I multiplied back by delta x. 108 00:08:01 --> 00:08:07 Now, let me compare this with the mean value theorem. 109 00:08:07 --> 00:08:13 The mean values theorem also is an equation. 110 00:08:13 --> 00:08:15 The mean value theorem says that this is equal 111 00:08:15 --> 00:08:21 to F' (c) delta x. 112 00:08:21 --> 00:08:23 Now, I pulled a fast one on you. 113 00:08:23 --> 00:08:26 I used capital F's here to make the analogy clear. 114 00:08:26 --> 00:08:30 But the role of the letter is important to make the 115 00:08:30 --> 00:08:32 transition to this comparison. 116 00:08:32 --> 00:08:35 We're talking about the function capital F here. 117 00:08:35 --> 00:08:36 And its derivative. 118 00:08:36 --> 00:08:38 Now, this is true. 119 00:08:38 --> 00:08:42 So now I claim that this thing is fairly specific. 120 00:08:42 --> 00:08:47 Whereas this, unfortunately, is a little bit vague. 121 00:08:47 --> 00:08:50 And the reason why it's vague is that c is just 122 00:08:50 --> 00:08:52 somewhere in the interval. 123 00:08:52 --> 00:09:01 So some c - sorry, this is some c, in between a and b. 124 00:09:01 --> 00:09:04 So really, since we don't know where this thing is, we don't 125 00:09:04 --> 00:09:07 know which of the values it is, we can't say what it is. 126 00:09:07 --> 00:09:12 All we can do is say well for sure it's less than 127 00:09:12 --> 00:09:13 the largest value. 128 00:09:13 --> 00:09:18 Say, (the maximum of F' ) delta x. 129 00:09:18 --> 00:09:21 And the only thing we can say for sure on the other end is 130 00:09:21 --> 00:09:24 that it's less than or equal to - sorry, it's greater than or 131 00:09:24 --> 00:09:27 equal to, (the minimum of F' ) delta x. 132 00:09:27 --> 00:09:29 Over that same interval. 133 00:09:29 --> 00:09:39 This is over 0 less than - sorry, a < x < b. 134 00:09:39 --> 00:09:43 So that means that the fundamental theorem of calculus 135 00:09:43 --> 00:09:45 is a much more specific thing. 136 00:09:45 --> 00:09:48 And indeed it gives the same conclusion. 137 00:09:48 --> 00:09:50 It's much stronger than the mean value theorem. 138 00:09:50 --> 00:09:52 It's way better than the mean value theorem. 139 00:09:52 --> 00:09:56 In fact, as soon as we have integrals, we can abandon 140 00:09:56 --> 00:09:57 the mean value theorem. 141 00:09:57 --> 00:09:58 We don't want it. 142 00:09:58 --> 00:10:00 It's too simple-minded. 143 00:10:00 --> 00:10:03 And what we have is something much more sophisticated, 144 00:10:03 --> 00:10:04 which we can use. 145 00:10:04 --> 00:10:05 Which is this. 146 00:10:05 --> 00:10:08 So it's obvious that if this is the average, the average 147 00:10:08 --> 00:10:09 is less than the maximum. 148 00:10:09 --> 00:10:14 So it's obvious that it works just as well to 149 00:10:14 --> 00:10:15 draw this conclusion. 150 00:10:15 --> 00:10:20 And similarly over here with the minimum. 151 00:10:20 --> 00:10:22 OK, the average is always bigger than the minimum 152 00:10:22 --> 00:10:25 and smaller than the max. 153 00:10:25 --> 00:10:28 So this is the connection, if you like. 154 00:10:28 --> 00:10:33 And I'm going to elaborate just one step further by talking 155 00:10:33 --> 00:10:36 about the problem that you had on the exam. 156 00:10:36 --> 00:10:39 So there was an Exam 2 problem. 157 00:10:39 --> 00:10:43 And I'll show you how it works using the mean value theorem 158 00:10:43 --> 00:10:45 and how it works using integrals. 159 00:10:45 --> 00:10:47 But I'm going to have to use this notation capital F. 160 00:10:47 --> 00:10:52 So capital F', as opposed to the little f, which was what 161 00:10:52 --> 00:10:55 was the notation that was on your exam. 162 00:10:55 --> 00:10:57 So we had this situation here. 163 00:10:57 --> 00:11:01 These were the givens of the problem. 164 00:11:01 --> 00:11:11 And then the question was, the mean value theorem says, or 165 00:11:11 --> 00:11:14 implies, if you like, it doesn't say it, but 166 00:11:14 --> 00:11:20 it implies it. 167 00:11:20 --> 00:11:38 Implies a < capital F ( 4) < b, for which a and b? 168 00:11:38 --> 00:11:43 So let's take a look at what it says. 169 00:11:43 --> 00:11:49 Well, the mean value theorem says that F( 4) - F 170 00:11:49 --> 00:11:57 (0) = F' ( c)( 4 - 0). 171 00:11:57 --> 00:12:03 This is this (F') delta x, this is the change in x. 172 00:12:03 --> 00:12:12 And that's the same thing as (1 / 1 + c )( 4). 173 00:12:12 --> 00:12:20 And so the range of values of this number here is from (1 174 00:12:20 --> 00:12:24 / 1 plus 0)( 4), that's 4. 175 00:12:24 --> 00:12:28 To, that's the largest value, to the smallest that it gets, 176 00:12:28 --> 00:12:32 which is (1 / 1 + 4)( 4). 177 00:12:32 --> 00:12:41 That's the range. 178 00:12:41 --> 00:12:52 And so the conclusion is that F (4) - f ( 0 ) is 179 00:12:52 --> 00:12:55 between, well, let's see. 180 00:12:55 --> 00:12:59 It's between 4 and 4/5. 181 00:12:59 --> 00:13:01 Which are those two numbers down there. 182 00:13:01 --> 00:13:04 And if you remember that F ( 0) was 1, this is the same as F 183 00:13:04 --> 00:13:15 ( 4), is between 5 and 9/5. 184 00:13:15 --> 00:13:19 So that's the way that you were supposed to solve 185 00:13:19 --> 00:13:22 the problem on the exam. 186 00:13:22 --> 00:13:26 On the other hand, let's compare to what you would 187 00:13:26 --> 00:13:31 do with the fundamental theorem of calculus. 188 00:13:31 --> 00:13:33 With the fundamentals theorem of calculus, we have 189 00:13:33 --> 00:13:34 the following formula. 190 00:13:34 --> 00:13:46 F( 4) - F ( 0) = the integral from 0 to 4 of dx / 1 + x. 191 00:13:46 --> 00:13:52 That's what the fundamental theorem says. 192 00:13:52 --> 00:13:58 And now I claim that we can get these same types of results by 193 00:13:58 --> 00:14:00 a very elementary observation. 194 00:14:00 --> 00:14:02 It's really the same observation that I made up 195 00:14:02 --> 00:14:06 here, that the average is less than or equal to the maximum. 196 00:14:06 --> 00:14:11 Which is that the biggest this can ever be is, let's see. 197 00:14:11 --> 00:14:15 The biggest it is when x is 0, that's 1. 198 00:14:15 --> 00:14:20 So the biggest it ever gets is this. 199 00:14:20 --> 00:14:25 And that's equal to 4. 200 00:14:25 --> 00:14:25 Right? 201 00:14:25 --> 00:14:27 On the other hand, the smallest it ever gets to 202 00:14:27 --> 00:14:36 be, it's equal to this. 203 00:14:36 --> 00:14:39 The smallest it ever gets to be is the integral 204 00:14:39 --> 00:14:42 from 0 to 4 of 1/5 dx. 205 00:14:42 --> 00:14:46 Because that's the lowest value that the integrand takes. 206 00:14:46 --> 00:14:48 When x = 4, it's 1/5. 207 00:14:48 --> 00:14:54 And that's equal to 4/5. 208 00:14:54 --> 00:14:57 Now, there's a little tiny detail which is that really we 209 00:14:57 --> 00:15:00 know that this is the area of some rectangle and this 210 00:15:00 --> 00:15:01 is strictly smaller. 211 00:15:01 --> 00:15:03 And we know that these inequalities are 212 00:15:03 --> 00:15:05 actually strict. 213 00:15:05 --> 00:15:07 But that's a minor point. 214 00:15:07 --> 00:15:12 And certainly not one that we'll pay close attention to. 215 00:15:12 --> 00:15:17 But now, let me show you what this looks like geometrically. 216 00:15:17 --> 00:15:22 So geometrically, we interpret this as the area under a curve. 217 00:15:22 --> 00:15:31 Here's a piece of the curve y = 1/1 + x. 218 00:15:31 --> 00:15:37 And it's going up to 4 and starting at 0 here. 219 00:15:37 --> 00:15:43 And the first estimate that we made; that is, the upper bound, 220 00:15:43 --> 00:15:53 was by trapping this in this big rectangle here. 221 00:15:53 --> 00:15:56 We compared it to the constant function, which was 1 222 00:15:56 --> 00:15:57 all the way across. 223 00:15:57 --> 00:16:00 This is y = 1. 224 00:16:00 --> 00:16:06 And then we also trapped it from underneath by the function 225 00:16:06 --> 00:16:08 which was at the bottom. 226 00:16:08 --> 00:16:14 And this was y = 1/5. 227 00:16:14 --> 00:16:19 And so what this really is is, these things are the simplest 228 00:16:19 --> 00:16:21 possible Riemann sum. 229 00:16:21 --> 00:16:22 Sort of a silly Riemann sum. 230 00:16:22 --> 00:16:34 This is a Riemann sum with one rectangle. 231 00:16:34 --> 00:16:36 This is the simplest possible one. 232 00:16:36 --> 00:16:38 And so this is a very, very crude estimate. 233 00:16:38 --> 00:16:40 You can see it misses by a mile. 234 00:16:40 --> 00:16:43 The larger and the smaller values are off by 235 00:16:43 --> 00:16:46 a factor of 5. 236 00:16:46 --> 00:16:49 But this one is called the, this one is the 237 00:16:49 --> 00:16:53 lower Riemann sum. 238 00:16:53 --> 00:16:59 And that one is less than our actual integral. 239 00:16:59 --> 00:17:14 Which is less than the upper Riemann sum. 240 00:17:14 --> 00:17:18 And you should, by now, have looked at those upper and 241 00:17:18 --> 00:17:20 lower sums on your homework. 242 00:17:20 --> 00:17:22 So it's just the rectangles underneath and the 243 00:17:22 --> 00:17:25 rectangles on top. 244 00:17:25 --> 00:17:27 So at this point, we can literally abandon the 245 00:17:27 --> 00:17:28 mean value theorem. 246 00:17:28 --> 00:17:30 Because we have a much better way of getting at things. 247 00:17:30 --> 00:17:33 If we chop things up into more rectangles, we'll get much 248 00:17:33 --> 00:17:35 better numerical approximations. 249 00:17:35 --> 00:17:38 And if we use simpleminded expressions with integrals, 250 00:17:38 --> 00:17:40 we'll be able to figure out any bound we could get using 251 00:17:40 --> 00:17:42 the mean value theorem. 252 00:17:42 --> 00:17:45 So that's not the relevance of the mean value theorem. 253 00:17:45 --> 00:17:48 I'll explain to you why we talked about it, even, 254 00:17:48 --> 00:17:51 in a few minutes. 255 00:17:51 --> 00:18:00 OK, are there any questions before we go on? 256 00:18:00 --> 00:18:00 Yeah. 257 00:18:00 --> 00:18:07 STUDENT: [INAUDIBLE] 258 00:18:07 --> 00:18:09 PROFESSOR: I knew that the range of c was from 0 to 4, 259 00:18:09 --> 00:18:11 I should have said that right here. 260 00:18:11 --> 00:18:13 This is true for this theorem. 261 00:18:13 --> 00:18:16 The mean value theorem comes with an extra statement, 262 00:18:16 --> 00:18:17 which I missed. 263 00:18:17 --> 00:18:21 Which is that this is for some c between 0 and 4. 264 00:18:21 --> 00:18:23 So I know the range is between 0 and 4. 265 00:18:23 --> 00:18:25 The reason why it's between 0 and 4 is that 's part of 266 00:18:25 --> 00:18:27 the mean value theorem. 267 00:18:27 --> 00:18:29 We started at 0, we ended at 4. 268 00:18:29 --> 00:18:32 So the c has to be somewhere in between. 269 00:18:32 --> 00:18:42 That's part of the mean value theorem. 270 00:18:42 --> 00:18:42 STUDENT: [INAUDIBLE] 271 00:18:42 --> 00:18:44 PROFESSOR: The question is, do you exclude any values that 272 00:18:44 --> 00:18:45 are above 4 and below 0. 273 00:18:45 --> 00:18:47 Yes, absolutely. 274 00:18:47 --> 00:18:49 The point is that in order to figure out how F changes, 275 00:18:49 --> 00:18:52 capital F changes, between 0 and 4, you need only 276 00:18:52 --> 00:18:54 pay attention to the values in between. 277 00:18:54 --> 00:18:56 You don't have to pay any attention to what the function 278 00:18:56 --> 00:18:59 is doing below 0 or above 4. 279 00:18:59 --> 00:19:07 Those things are strictly irrelevant. 280 00:19:07 --> 00:19:16 STUDENT: [INAUDIBLE] 281 00:19:16 --> 00:19:18 PROFESSOR: Yeah, I mean it's strictly in between 282 00:19:18 --> 00:19:19 these two numbers. 283 00:19:19 --> 00:19:23 I have to understand what the lowest and the highest one is. 284 00:19:23 --> 00:19:24 STUDENT: [INAUDIBLE] 285 00:19:24 --> 00:19:35 PROFESSOR: It's approaching that, so. 286 00:19:35 --> 00:19:35 OK. 287 00:19:35 --> 00:19:40 So now, the next thing that we're going to talk about is, 288 00:19:40 --> 00:19:42 since I've got that 1 up there, that Fundamental Theorem of 289 00:19:42 --> 00:20:05 Calculus 1, I need to talk about version 2. 290 00:20:05 --> 00:20:15 So here is the Fundamental Theorem of Calculus version 2. 291 00:20:15 --> 00:20:20 I'm going to start out with a function little f, and I'm 292 00:20:20 --> 00:20:28 going to assume that it's continuous. 293 00:20:28 --> 00:20:30 And then I'm going to define a new function, which is defined 294 00:20:30 --> 00:20:32 as a definite integral. 295 00:20:32 --> 00:20:40 G ( x ) is the integral from a to x of f ( t ) dt. 296 00:20:40 --> 00:20:42 Now, I want to emphasize here because it's the first time 297 00:20:42 --> 00:20:44 that I'm writing something like this, that this is a 298 00:20:44 --> 00:20:47 fairly complicated gadget. 299 00:20:47 --> 00:20:52 It plays a very basic and very fundamental, but simple role 300 00:20:52 --> 00:20:54 but it nevertheless is a little complicated. 301 00:20:54 --> 00:20:58 What's happening here is that the upper limit I've now called 302 00:20:58 --> 00:21:06 x, and the variable t is ranging between a and x, and 303 00:21:06 --> 00:21:12 that the a and the x are fixed when I calculate the integral. 304 00:21:12 --> 00:21:14 And the t is what's called the dummy variable. 305 00:21:14 --> 00:21:16 It's the variable of integration. 306 00:21:16 --> 00:21:21 You'll see a lot of people who will mix this x with this t. 307 00:21:21 --> 00:21:25 And if you do that, you will get confused, potentially 308 00:21:25 --> 00:21:28 hopelessly confused, in this class. 309 00:21:28 --> 00:21:32 In 18.02 you will be completely lost if you do that. 310 00:21:32 --> 00:21:33 So don't do it. 311 00:21:33 --> 00:21:38 Don't mix these two guys up. 312 00:21:38 --> 00:21:42 It's actually done by many people in textbooks, and 313 00:21:42 --> 00:21:43 it's fairly careless. 314 00:21:43 --> 00:21:45 Especially in old-fashioned textbooks. 315 00:21:45 --> 00:21:48 But don't do it. 316 00:21:48 --> 00:21:50 So here we have this G ( x). 317 00:21:50 --> 00:21:56 Now, remember, this G ( x) really does make sense. 318 00:21:56 --> 00:22:00 If you give me an a, and you give me an x, I can figure out 319 00:22:00 --> 00:22:02 what this is, because I can figure out the Riemann sum. 320 00:22:02 --> 00:22:05 So of course I need to know what the function is, too. 321 00:22:05 --> 00:22:07 But anyway, we have a numerical procedure for figuring out 322 00:22:07 --> 00:22:09 what the function g is. 323 00:22:09 --> 00:22:13 Now, as is suggested by this mysterious letter x being in 324 00:22:13 --> 00:22:16 the place where it is, I'm actually going to vary x. 325 00:22:16 --> 00:22:19 So the conclusion is that if this is true, and this is 326 00:22:19 --> 00:22:21 just a parenthesism not part of the theorem. 327 00:22:21 --> 00:22:25 It's just an indication of what the notation means. 328 00:22:25 --> 00:22:40 Then G' = f. 329 00:22:40 --> 00:22:43 Let me first explain what the significance of this theorem 330 00:22:43 --> 00:22:47 is, from the point of view of differential equations. 331 00:22:47 --> 00:23:04 G ( x) solves the differential equation y' = f ( x). 332 00:23:04 --> 00:23:09 So y' = f, you put the x in if I got it here, with 333 00:23:09 --> 00:23:13 the condition y ( a) = 0. 334 00:23:13 --> 00:23:19 So it solves this pair of conditions here. 335 00:23:19 --> 00:23:21 The rate of change, and the initial position 336 00:23:21 --> 00:23:23 is specified here. 337 00:23:23 --> 00:23:29 Because when you integrate from a to a, you get 0 always. 338 00:23:29 --> 00:23:34 And what this theorem says is you can always 339 00:23:34 --> 00:23:35 solve that equation. 340 00:23:35 --> 00:23:38 When we did differential equations, I said that already. 341 00:23:38 --> 00:23:39 I said we'll treat these as always solved. 342 00:23:39 --> 00:23:41 Well, here's the reason. 343 00:23:41 --> 00:23:45 We have a numerical procedure for computing things like this. 344 00:23:45 --> 00:23:49 We could always solve this equation. 345 00:23:49 --> 00:23:54 And the formula is a fairly complicated gadget, but so 346 00:23:54 --> 00:23:58 far just associated with Riemann sums. 347 00:23:58 --> 00:24:01 Alright, now. 348 00:24:01 --> 00:24:13 Let's just do one example. 349 00:24:13 --> 00:24:17 Unfortunately, not a complicated example and maybe 350 00:24:17 --> 00:24:21 not persuasive as to why you would care about this just yet. 351 00:24:21 --> 00:24:23 But nevertheless very important. 352 00:24:23 --> 00:24:26 Because this is the quiz question which everybody gets 353 00:24:26 --> 00:24:29 wrong until they practice it. 354 00:24:29 --> 00:24:38 So the integral from, say 1 to x of dt / t ^2. 355 00:24:38 --> 00:24:45 Let's try this one here. 356 00:24:45 --> 00:24:56 So here's an example of this theorem, I claim. 357 00:24:56 --> 00:25:01 Now, this is a question which challenges your ability to 358 00:25:01 --> 00:25:04 understand what the question means. 359 00:25:04 --> 00:25:06 Because it's got a lot of symbols. 360 00:25:06 --> 00:25:09 It's got the integration and it's got the differentiation. 361 00:25:09 --> 00:25:15 However, what it really is an exercise in recopying. 362 00:25:15 --> 00:25:20 You look at it and you write down the answer. 363 00:25:20 --> 00:25:25 And the reason is that, by definition, this function in 364 00:25:25 --> 00:25:30 here is a function of the form G ( x) of the 365 00:25:30 --> 00:25:34 theorem over here. 366 00:25:34 --> 00:25:35 So this is the G ( x). 367 00:25:35 --> 00:25:42 And by definition, we said that G' ( x) = f(x). 368 00:25:42 --> 00:25:46 Well, what's the f ( x)? 369 00:25:46 --> 00:25:47 Look inside here. 370 00:25:47 --> 00:25:48 It's what's called the integrand. 371 00:25:48 --> 00:25:53 This is the integral from 0 to x of f ( t ) dt, right? 372 00:25:53 --> 00:25:59 Where f ( t) = 1 / t^2. 373 00:26:00 --> 00:26:02 So your ability is challenged. 374 00:26:02 --> 00:26:06 You have to take that 1 / t ^2 and you have to plug in the 375 00:26:06 --> 00:26:09 letter x, instead of t for it. 376 00:26:09 --> 00:26:11 And then write it down. 377 00:26:11 --> 00:26:18 As I say, this is an exercise in recopying what's there. 378 00:26:18 --> 00:26:19 So this is quite easy to do, right? 379 00:26:19 --> 00:26:21 I mean, you just look and you write it down. 380 00:26:21 --> 00:26:28 But nevertheless, it looks like a long, elaborate object here. 381 00:26:28 --> 00:26:28 Pardon me? 382 00:26:28 --> 00:26:30 STUDENT: [INAUDIBLE] 383 00:26:30 --> 00:26:32 PROFESSOR: So the question was, why did I integrate. 384 00:26:32 --> 00:26:34 STUDENT: [INAUDIBLE] 385 00:26:34 --> 00:26:36 PROFESSOR: Why did I not integrate? 386 00:26:36 --> 00:26:37 Ah. 387 00:26:37 --> 00:26:38 Very good question. 388 00:26:38 --> 00:26:41 Why did I not integrate. 389 00:26:41 --> 00:26:45 The reason why I didn't integrate is I didn't need to. 390 00:26:45 --> 00:26:48 Just as when you take the - sorry, the derivative of 391 00:26:48 --> 00:26:50 something, you take the antiderivative, you get 392 00:26:50 --> 00:26:51 back to the thing. 393 00:26:51 --> 00:26:54 So, in this case, we're taking the antiderivative of something 394 00:26:54 --> 00:26:56 and we're differentiating. 395 00:26:56 --> 00:26:58 So we end back in the same place where we started. 396 00:26:58 --> 00:27:01 We started with f ( t), we're ending with f. 397 00:27:01 --> 00:27:04 Little f. 398 00:27:04 --> 00:27:06 So you integrate, and then differentiate. 399 00:27:06 --> 00:27:09 And you get back to the same place. 400 00:27:09 --> 00:27:11 Now, the only difference between this and the other 401 00:27:11 --> 00:27:15 version is, in this case when you differentiate and integrate 402 00:27:15 --> 00:27:18 you could be off by a constant. 403 00:27:18 --> 00:27:19 That's what that shift, why there are two 404 00:27:19 --> 00:27:21 pieces to this one. 405 00:27:21 --> 00:27:23 But there's never an extra piece here. 406 00:27:23 --> 00:27:25 There's no + c here. 407 00:27:25 --> 00:27:26 When you integrate and differentiate, you kill 408 00:27:26 --> 00:27:28 whatever the constant is. 409 00:27:28 --> 00:27:31 Because the derivative of a constant is 0. 410 00:27:31 --> 00:27:36 So no matter what the constant is hiding inside of g, you're 411 00:27:36 --> 00:27:37 getting the same result. 412 00:27:37 --> 00:27:41 So this is the basic idea. 413 00:27:41 --> 00:27:46 Now, I just want to double-check it, using 414 00:27:46 --> 00:27:52 the Fundamental Theorem of Calculus 1 here. 415 00:27:52 --> 00:27:53 So let's actually evaluate the integral. 416 00:27:53 --> 00:27:55 So now I'm going to do what you've suggested, which is 417 00:27:55 --> 00:27:57 I'm just going to check whether it's true. 418 00:27:57 --> 00:27:59 No, no I am because I'm going just double-check 419 00:27:59 --> 00:28:01 that it's consistent. 420 00:28:01 --> 00:28:03 It certainly is slower this way, and we're not going to 421 00:28:03 --> 00:28:06 want to do this all the time, but we might as well check one. 422 00:28:06 --> 00:28:09 So this is our integral. 423 00:28:09 --> 00:28:10 And we know how to do it. 424 00:28:10 --> 00:28:13 No, I need to do it. 425 00:28:13 --> 00:28:17 And this is - t ^ - 1, evaluated at 1 and x. 426 00:28:17 --> 00:28:22 Again, there's something subliminally here for 427 00:28:22 --> 00:28:23 you to think about. 428 00:28:23 --> 00:28:28 Which is that, remember, its t is ranging between 1 and t = x. 429 00:28:28 --> 00:28:31 And this is one of the big reasons why this letter t 430 00:28:31 --> 00:28:32 has to be different from x. 431 00:28:32 --> 00:28:35 Because here it's 1 and there it's x. 432 00:28:35 --> 00:28:37 It's not x. 433 00:28:37 --> 00:28:38 So you can't put an x here. 434 00:28:38 --> 00:28:44 Again, this is t = 1 and this is t = x over there. 435 00:28:44 --> 00:28:48 And now if I plug that in, I get what? 436 00:28:48 --> 00:28:55 I get - 1 / x, and then I get - ( - 1). 437 00:28:55 --> 00:28:59 So this is, let me get rid of those little t's there. 438 00:28:59 --> 00:29:05 This is a little easier to read. 439 00:29:05 --> 00:29:07 And so now let's check it. 440 00:29:07 --> 00:29:07 It's d / dx. 441 00:29:07 --> 00:29:09 So here's what G ( x) is. 442 00:29:09 --> 00:29:12 G ( x) = 1 - 1 / x. 443 00:29:12 --> 00:29:14 That's what G( x) is. 444 00:29:14 --> 00:29:20 And if I differentiate that, I get + 1 / x ^2. 445 00:29:20 --> 00:29:26 That's it. 446 00:29:26 --> 00:29:40 You see the constant washed away. 447 00:29:40 --> 00:29:42 So now, here's my job. 448 00:29:42 --> 00:29:44 My job is to prove these theorems. 449 00:29:44 --> 00:29:46 I never did prove them for you. 450 00:29:46 --> 00:29:47 So, I'm going to prove the Fundamental 451 00:29:47 --> 00:29:49 Theorem of Calculus. 452 00:29:49 --> 00:29:51 But I'm going to do 2 first. 453 00:29:51 --> 00:29:53 And then I'm going to do 1. 454 00:29:53 --> 00:29:56 And it's just going to take me just one blackboard. 455 00:29:56 --> 00:30:00 It's not that hard. 456 00:30:00 --> 00:30:03 The proof is by picture. 457 00:30:03 --> 00:30:08 And, using the interpretation as area under the curve. 458 00:30:08 --> 00:30:13 So if here's the value of a, and this is the graph of the 459 00:30:13 --> 00:30:22 function y equals f of x. 460 00:30:22 --> 00:30:26 Then I want to draw three vertical lines. 461 00:30:26 --> 00:30:29 One of them is going to be at x. 462 00:30:29 --> 00:30:33 And one of them is going to be at x + delta x. 463 00:30:33 --> 00:30:37 So here I have the interval from 0 to x, and next I have 464 00:30:37 --> 00:30:42 the interval from x to delta x more than that. 465 00:30:42 --> 00:30:50 And now the pieces that I've got are the area of this part. 466 00:30:50 --> 00:30:53 So this has area which has a name. 467 00:30:53 --> 00:30:55 It's called G ( x ). 468 00:30:55 --> 00:31:00 By definition, G( x ), which is sitting right over here in the 469 00:31:00 --> 00:31:04 fundamental theorem, is the integral from a to x 470 00:31:04 --> 00:31:06 of this function. 471 00:31:06 --> 00:31:07 So it's the area under the curve. 472 00:31:07 --> 00:31:10 So that area is G ( x ). 473 00:31:10 --> 00:31:21 Now this other chunk here, I claim that this is delta G. 474 00:31:21 --> 00:31:23 This is the change in G. 475 00:31:23 --> 00:31:25 It's the value of G ( x ) that is the area of the whole 476 00:31:25 --> 00:31:29 business all the way up to x + delta x - the first 477 00:31:29 --> 00:31:30 part, G ( x ). 478 00:31:30 --> 00:31:31 So it's what's left over. 479 00:31:31 --> 00:31:39 It's the incremental amount of area there. 480 00:31:39 --> 00:31:45 And now I am going to carry out a pretty standard 481 00:31:45 --> 00:31:46 estimation here. 482 00:31:46 --> 00:31:48 This is practically a rectangle. 483 00:31:48 --> 00:31:51 And it's got a base of delta x, and so we need to figure 484 00:31:51 --> 00:31:55 out what its height is. 485 00:31:55 --> 00:32:02 This is delta G, and it's approximately its base 486 00:32:02 --> 00:32:05 times its height. 487 00:32:05 --> 00:32:06 But what is the height? 488 00:32:06 --> 00:32:10 Well, the height is maybe either this segment 489 00:32:10 --> 00:32:11 or this segment or something in between. 490 00:32:11 --> 00:32:13 But they're all about the same. 491 00:32:13 --> 00:32:17 So I'm just going to put in the value at the first point. 492 00:32:17 --> 00:32:19 That's the left end there. 493 00:32:19 --> 00:32:25 So that's this height here, is f ( x ). 494 00:32:25 --> 00:32:28 So this is f (x), and so really I approximate it 495 00:32:28 --> 00:32:30 by that rectangle there. 496 00:32:30 --> 00:32:34 And now if I divide and take the limit, as delta x goes to 497 00:32:34 --> 00:32:43 0 of delta G / delta x, it's going to equal f ( x ). 498 00:32:43 --> 00:32:48 And this is where I'm using the fact that f is continuous. 499 00:32:48 --> 00:32:51 Because I need the values nearby to be similar to 500 00:32:51 --> 00:32:59 the value in the limit. 501 00:32:59 --> 00:33:00 OK, that's the end. 502 00:33:00 --> 00:33:03 This the end of the proof, so I'll put a nice 503 00:33:03 --> 00:33:10 little q.e.d. here. 504 00:33:10 --> 00:33:14 So we've done Fundamental Theorem of Calculus 2, and now 505 00:33:14 --> 00:33:38 we're ready for Fundamental Theorem of Calculus 1. 506 00:33:38 --> 00:33:44 So now I still have it on the blackboard to remind you. 507 00:33:44 --> 00:33:49 It says that the integral of the derivative is the function, 508 00:33:49 --> 00:33:51 at least the difference between the values of the 509 00:33:51 --> 00:33:54 function at two places. 510 00:33:54 --> 00:34:08 So the place where we start is with this property that F' = f. 511 00:34:08 --> 00:34:10 That's the starting, that's the hypothesis. 512 00:34:10 --> 00:34:13 Now, unfortunately, I'm going to have to assume something 513 00:34:13 --> 00:34:18 extra in order to use the Fundamental Theorem of Calculus 514 00:34:18 --> 00:34:27 2, which is I'm going to assume that f is continuous. 515 00:34:27 --> 00:34:31 That's not really necessary, but that's just a very minor 516 00:34:31 --> 00:34:34 technical point, which I'm just going to ignore. 517 00:34:34 --> 00:34:40 So we're going to start with F' = f. 518 00:34:40 --> 00:34:46 And then I'm going to go somewhere else. 519 00:34:46 --> 00:34:53 I'm going to define a new function, G (x), which is 520 00:34:53 --> 00:35:00 the integral from a to x of f ( t ) dt. 521 00:35:00 --> 00:35:04 This is where we needed all of the labor of Riemann's sums. 522 00:35:04 --> 00:35:07 Because otherwise we don't have a way of making sense out 523 00:35:07 --> 00:35:10 of what this even means. 524 00:35:10 --> 00:35:14 So hiding behind this one sentence is the fact that 525 00:35:14 --> 00:35:16 we actually have a number. 526 00:35:16 --> 00:35:18 We have a formula for such functions. 527 00:35:18 --> 00:35:21 So there is a function g (x) which, once you've produced a 528 00:35:21 --> 00:35:27 little f for me, I can cook up a function capital G for you. 529 00:35:27 --> 00:35:31 Now, we're going to apply this Fundamental Theorem of Calculus 530 00:35:31 --> 00:35:34 2, the one that we've already checked. 531 00:35:34 --> 00:35:36 So what does it say? 532 00:35:36 --> 00:35:46 It says that G' = f. 533 00:35:46 --> 00:35:49 And so now we're in the following situation. 534 00:35:49 --> 00:35:58 We know that F' ( x ) = G' ( x). 535 00:35:58 --> 00:36:00 That's what we've got so far. 536 00:36:00 --> 00:36:06 And now we have one last step to get a good connection 537 00:36:06 --> 00:36:07 between F and G. 538 00:36:07 --> 00:36:12 Which is that we can conclude that F ( x ) = G ( x) + c. 539 00:36:12 --> 00:36:20 540 00:36:20 --> 00:36:29 Now, this little step may seem innocuous but I remind you that 541 00:36:29 --> 00:36:36 this is the spot that requires the mean value theorem. 542 00:36:36 --> 00:36:40 So in order not too lie to you, we actually tell you what the 543 00:36:40 --> 00:36:42 underpinnings of all of calculus are. 544 00:36:42 --> 00:36:46 And they're this: the fact, if you like, that if two functions 545 00:36:46 --> 00:36:48 have the same derivative, they differ by a constant. 546 00:36:48 --> 00:36:51 Or that if a function has derivative 0, it's 547 00:36:51 --> 00:36:53 a constant itself. 548 00:36:53 --> 00:36:58 Now, that is the fundamental step that's needed, the 549 00:36:58 --> 00:36:59 underlying step that's needed. 550 00:36:59 --> 00:37:02 And, unfortunately, there aren't any proofs of it that 551 00:37:02 --> 00:37:06 are less complicated than using the mean value theorem. 552 00:37:06 --> 00:37:08 And so that's why we talk a little bit about the mean value 553 00:37:08 --> 00:37:10 theorem, because we don't want to lie to you about 554 00:37:10 --> 00:37:11 what's really going on. 555 00:37:11 --> 00:37:12 Yes. 556 00:37:12 --> 00:37:19 STUDENT: [INAUDIBLE] 557 00:37:19 --> 00:37:24 PROFESSOR: The question is how did I get from here, to here. 558 00:37:24 --> 00:37:28 And the answer is that if G' is little f, and we also know that 559 00:37:28 --> 00:37:32 F' is little f, then F' is G'. 560 00:37:32 --> 00:37:37 OK. 561 00:37:37 --> 00:37:50 Other questions? 562 00:37:50 --> 00:37:52 Alright, so we're almost done. 563 00:37:52 --> 00:37:57 I just have to work out the arithmetic here. 564 00:37:57 --> 00:38:04 So I start with F(b) - F ( a). 565 00:38:04 --> 00:38:18 And that's equal to (G ( b) + c) - (G (a) + c). 566 00:38:18 --> 00:38:20 And then I cancel the c's. 567 00:38:20 --> 00:38:23 So I have here G(b) - G(a). 568 00:38:23 --> 00:38:29 569 00:38:29 --> 00:38:32 And now I just have to check what each of these is. 570 00:38:32 --> 00:38:35 So Remember the definition of G here. 571 00:38:35 --> 00:38:38 G ( b) is just what we want. 572 00:38:38 --> 00:38:42 The integral from a to b of f(x) dx. 573 00:38:42 --> 00:38:46 Well I called it f ( t) dt, that's the same as f(x) dx now, 574 00:38:46 --> 00:38:49 because I have the limit being b and I'm allowed to use 575 00:38:49 --> 00:38:52 x as the dummy variable. 576 00:38:52 --> 00:38:55 Now the other one, I claim, is 0. 577 00:38:55 --> 00:38:59 Because it's the integral from a to a. 578 00:38:59 --> 00:39:03 This one is the integral from a to a. 579 00:39:03 --> 00:39:06 Which gives us 0. 580 00:39:06 --> 00:39:09 So this is just this - 0, and that's the end. 581 00:39:09 --> 00:39:13 That's it. 582 00:39:13 --> 00:39:20 I started with F( b) - F( a), I got to the integral. 583 00:39:20 --> 00:39:20 Question? 584 00:39:20 --> 00:39:27 STUDENT: [INAUDIBLE] 585 00:39:27 --> 00:39:32 PROFESSOR: How did I get from F ( b) - F (a), is (G ( b) + c) - 586 00:39:32 --> 00:39:35 (G( a) + c), that's the question. 587 00:39:35 --> 00:39:40 STUDENT: [INAUDIBLE] 588 00:39:40 --> 00:39:44 PROFESSOR: Oh, sorry this is an equals sign. 589 00:39:44 --> 00:39:47 Sorry, the second line didn't draw. 590 00:39:47 --> 00:39:48 OK, equals. 591 00:39:48 --> 00:39:53 Because we're plugging in for f (x) the formula for it. 592 00:39:53 --> 00:39:53 Yes. 593 00:39:53 --> 00:39:57 STUDENT: [INAUDIBLE] 594 00:39:57 --> 00:39:59 PROFESSOR: This step here? 595 00:39:59 --> 00:40:04 Or this one? there's 596 00:40:04 --> 00:40:09 STUDENT: [INAUDIBLE] 597 00:40:09 --> 00:40:10 PROFESSOR: Right. 598 00:40:10 --> 00:40:12 So that was a good question. 599 00:40:12 --> 00:40:15 But the answer is that that's the statement 600 00:40:15 --> 00:40:16 that we're aiming for. 601 00:40:16 --> 00:40:18 That's the Fundamental Theorem of Calculus 1, 602 00:40:18 --> 00:40:19 which we don't know yet. 603 00:40:19 --> 00:40:21 So we're trying to prove it, and that's why we haven't, 604 00:40:21 --> 00:40:25 we can't assume it. 605 00:40:25 --> 00:40:30 OK, so let me just notice that in the example that 606 00:40:30 --> 00:40:36 we had, before we go on to something else here. 607 00:40:36 --> 00:40:48 In the example above, what we had was the following thing. 608 00:40:48 --> 00:40:58 We had, say, F ( x ) = - 1 / x. 609 00:40:58 --> 00:41:01 So F' (x) = 1 / x ^2. 610 00:41:01 --> 00:41:08 And, say, G ( x) = 1 - (1 / x). 611 00:41:08 --> 00:41:12 And you can see that either way you do that, if you integrate 612 00:41:12 --> 00:41:16 from 1 to 2, let's say, which is what we had over there, dt 613 00:41:16 --> 00:41:26 / t ^2, you're going to get either - 1 / t, 1 to 2 or, if 614 00:41:26 --> 00:41:31 you like, 1 - (1 / t), 1 to 2. 615 00:41:31 --> 00:41:33 So this is the F version, this is the G version. 616 00:41:33 --> 00:41:36 And that's what plays itself out here, in 617 00:41:36 --> 00:41:45 this general proof. 618 00:41:45 --> 00:41:49 Alright. 619 00:41:49 --> 00:41:56 So now I want to go back to the theme for today, which is using 620 00:41:56 --> 00:42:00 little f to understand capital F. 621 00:42:00 --> 00:42:03 In other words, using the derivative of f to 622 00:42:03 --> 00:42:05 understand capital F. 623 00:42:05 --> 00:42:22 And I want to illustrate it by some more complicated examples. 624 00:42:22 --> 00:42:26 So I guess I just erased it, but we just took the 625 00:42:26 --> 00:42:29 antiderivative of 1 / t ^2. 626 00:42:29 --> 00:42:32 And there's all of the powers work easily. 627 00:42:32 --> 00:42:39 But one, and the tricky one is the power 1 / x. 628 00:42:39 --> 00:42:41 So let's consider the differential equation 629 00:42:41 --> 00:42:44 L' ( x) = 1 / x. 630 00:42:44 --> 00:42:54 And say, with the initial value L (1) = 0. 631 00:42:54 --> 00:42:56 The solution, so the Fundamental Theorem of Calculus 632 00:42:56 --> 00:43:07 2 tells us the solution is this function here. 633 00:43:07 --> 00:43:14 L( x) equals the integral from 1 to x, dt / t. 634 00:43:14 --> 00:43:16 That's how we solve all such equations. 635 00:43:16 --> 00:43:19 We just integrate, take the definite integral. 636 00:43:19 --> 00:43:27 And I'm starting at 1 because I insisted that L( 1 ) be 0. 637 00:43:27 --> 00:43:31 So that's the solution to the problem. 638 00:43:31 --> 00:43:34 And now the thing that's interesting here is that we 639 00:43:34 --> 00:43:35 started from a polynomial. 640 00:43:35 --> 00:43:37 Or we started from a rational, a ratio of polynomials; 641 00:43:37 --> 00:43:40 that is, 1 / t or 1 / x. 642 00:43:40 --> 00:43:43 And we get to a function which is actually what's known as 643 00:43:43 --> 00:43:44 a transcendental function. 644 00:43:44 --> 00:43:46 It's not an algebraic function. 645 00:43:46 --> 00:43:47 Yeah, question. 646 00:43:47 --> 00:43:57 STUDENT: [INAUDIBLE] 647 00:43:57 --> 00:44:04 PROFESSOR: The question is why is this equal to that. 648 00:44:04 --> 00:44:08 And the answer is, it's for the same reason that 649 00:44:08 --> 00:44:10 this is equal to that. 650 00:44:10 --> 00:44:14 It's the same reason as this. 651 00:44:14 --> 00:44:16 It's that the 1's cancel. 652 00:44:16 --> 00:44:19 We're taken the value of something at 2 minus 653 00:44:19 --> 00:44:20 the value at 1. 654 00:44:20 --> 00:44:22 The value at 2 minus the value at 1. 655 00:44:22 --> 00:44:24 And you'll get a 1 in the one case, and you get 656 00:44:24 --> 00:44:25 a 1 in the other case. 657 00:44:25 --> 00:44:27 And you subtract them and they will cancel. 658 00:44:27 --> 00:44:28 They'll give you 0. 659 00:44:28 --> 00:44:31 These two things really are equal. 660 00:44:31 --> 00:44:33 This is not a function evaluated at one place, it's 661 00:44:33 --> 00:44:35 the difference between the function evaluated at 662 00:44:35 --> 00:44:37 2 and the value at 1. 663 00:44:37 --> 00:44:39 And whenever you subtract two things like that, 664 00:44:39 --> 00:44:40 constants drop out. 665 00:44:40 --> 00:44:43 STUDENT: [INAUDIBLE] 666 00:44:43 --> 00:44:44 PROFESSOR: That's right. 667 00:44:44 --> 00:44:46 If I put 2, here if I put c here, it would 668 00:44:46 --> 00:44:47 have been the same. 669 00:44:47 --> 00:44:49 It would just have dropped out. 670 00:44:49 --> 00:44:50 It's not there. 671 00:44:50 --> 00:44:53 And that's exactly this arithmetic right here. 672 00:44:53 --> 00:44:55 It doesn't matter which antiderivative you take. 673 00:44:55 --> 00:44:57 When you take the differences, the c's will cancel. 674 00:44:57 --> 00:45:03 You always get the same answer in the end. 675 00:45:03 --> 00:45:05 That's exactly why I wrote this down, so that 676 00:45:05 --> 00:45:06 you would see that. 677 00:45:06 --> 00:45:12 It doesn't matter which one you do. 678 00:45:12 --> 00:45:21 So, we still have a couple of minutes left here. 679 00:45:21 --> 00:45:23 This is actually, so let me go back. 680 00:45:23 --> 00:45:29 So here's the antiderivative of 1 / x, with value 1 at 0. 681 00:45:29 --> 00:45:32 Now, in disguise, we know what this function is. 682 00:45:32 --> 00:45:35 We know this function is the logarithm function. 683 00:45:35 --> 00:45:40 But this is actually a better way of deriving all of the 684 00:45:40 --> 00:45:42 formulas for the logarithm. 685 00:45:42 --> 00:45:44 This is a much quicker and more efficient way of doing it. 686 00:45:44 --> 00:45:47 We had to do it by very laborious processes. 687 00:45:47 --> 00:45:51 This will allow us to do it very easily. 688 00:45:51 --> 00:45:56 And so, I'm going to do that next time. 689 00:45:56 --> 00:45:59 But rather than do that now, I'm going to point out to you 690 00:45:59 --> 00:46:10 that we can also get truly new functions. 691 00:46:10 --> 00:46:12 OK, so there are all kinds of new functions. 692 00:46:12 --> 00:46:15 So this is the first example of this kind would be, for 693 00:46:15 --> 00:46:21 example, to solve the equation y' = e ^ - x^2 with y( 694 00:46:21 --> 00:46:25 0 ) = 0, let's say. 695 00:46:25 --> 00:46:28 Now, the solution to that is a function which again I 696 00:46:28 --> 00:46:30 can write down by the fundamental theorem. 697 00:46:30 --> 00:46:48 It's the integral from 0 to x of e ^ - t^2 dt. 698 00:46:48 --> 00:46:52 This is a very famous function. 699 00:46:52 --> 00:46:55 This shape here is known as the bell curve. 700 00:46:55 --> 00:46:59 And it's the thing that comes up in probability all the time. 701 00:46:59 --> 00:47:01 This shape e ^ - x^2. 702 00:47:01 --> 00:47:04 And our function is geometrically just the area 703 00:47:04 --> 00:47:06 under the curve here. 704 00:47:06 --> 00:47:09 This is F (x). 705 00:47:09 --> 00:47:12 If this place is x. 706 00:47:12 --> 00:47:14 So I have a geometric definition, I have a way 707 00:47:14 --> 00:47:16 of constructing what it is by Riemann's sums. 708 00:47:16 --> 00:47:18 And I have a function here. 709 00:47:18 --> 00:47:26 But the curious thing about F ( x ) is that F ( x ) cannot be 710 00:47:26 --> 00:47:34 expressed in terms of any function you've 711 00:47:34 --> 00:47:35 seen previously. 712 00:47:35 --> 00:47:44 So logs, exponentials, trig functions, cannot be. 713 00:47:44 --> 00:47:51 It's a totally new function. 714 00:47:51 --> 00:47:54 Nevertheless, we'll be able to get any possible piece of 715 00:47:54 --> 00:47:56 information we would want to, out of this function. 716 00:47:56 --> 00:47:59 It's perfectly acceptable function, it will work 717 00:47:59 --> 00:48:00 just great for us. 718 00:48:00 --> 00:48:01 Just like any other function. 719 00:48:01 --> 00:48:03 Just like the ln. 720 00:48:03 --> 00:48:08 And what this is analogous to is the following kind of thing. 721 00:48:08 --> 00:48:12 If you take the circle, the ancient Greeks, if you like, 722 00:48:12 --> 00:48:15 already understood that if you have a circle of radius 723 00:48:15 --> 00:48:23 1, then its area is pi. 724 00:48:23 --> 00:48:25 So that's a geometric construction of what you 725 00:48:25 --> 00:48:31 could call a new number. 726 00:48:31 --> 00:48:34 Which is outside of the realm of what you might expect. 727 00:48:34 --> 00:48:41 And the weird thing about this number pi is that it is not the 728 00:48:41 --> 00:48:57 root of an algebraic equation with rational coefficients. 729 00:48:57 --> 00:49:00 It's what's called transcendental. 730 00:49:00 --> 00:49:02 Meaning, it's just completely outside of 731 00:49:02 --> 00:49:04 the realm of algebra. 732 00:49:04 --> 00:49:07 And, indeed, the logarithm function is called a 733 00:49:07 --> 00:49:09 transcendental function, because it's completely out 734 00:49:09 --> 00:49:11 of the realm of algebra. 735 00:49:11 --> 00:49:14 It's only in calculus that you come up with 736 00:49:14 --> 00:49:16 this kind of thing. 737 00:49:16 --> 00:49:21 So these kinds of functions will have access to a huge 738 00:49:21 --> 00:49:24 class of new functions here, all of which are important 739 00:49:24 --> 00:49:26 tools in science and engineering. 740 00:49:26 --> 00:49:29 So, see you next time. 741 00:49:29 --> 00:49:30