1 00:00:00,000 --> 00:00:02,330 The following content is provided under a Creative 2 00:00:02,330 --> 00:00:03,620 Commons license. 3 00:00:03,620 --> 00:00:05,960 Your support will help MIT OpenCourseWare 4 00:00:05,960 --> 00:00:09,950 continue to offer high quality educational resources for free. 5 00:00:09,950 --> 00:00:12,540 To make a donation, or to view additional materials 6 00:00:12,540 --> 00:00:16,150 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,150 --> 00:00:22,162 at ocw.mit.edu. 8 00:00:22,162 --> 00:00:24,120 PROFESSOR: To begin today I want to remind you, 9 00:00:24,120 --> 00:00:27,170 I need to write it down on the board at least twice, 10 00:00:27,170 --> 00:00:33,230 of the fundamental theorem of calculus. 11 00:00:33,230 --> 00:00:38,496 We called it FTC 1 because it's the first version 12 00:00:38,496 --> 00:00:39,620 of the fundamental theorem. 13 00:00:39,620 --> 00:00:42,340 We'll be talking about another version, called 14 00:00:42,340 --> 00:00:44,940 the second version, today. 15 00:00:44,940 --> 00:00:53,610 And what it says is this: If F' = f, 16 00:00:53,610 --> 00:01:03,330 then the integral from a to b of f(x) dx is equal to F(b) - 17 00:01:03,330 --> 00:01:03,830 F(a). 18 00:01:06,410 --> 00:01:10,750 So that's the fundamental theorem of calculus. 19 00:01:10,750 --> 00:01:15,600 And the way we used it last time was, 20 00:01:15,600 --> 00:01:22,090 this was used to evaluate integrals. 21 00:01:22,090 --> 00:01:28,110 Not surprisingly, that's how we used it. 22 00:01:28,110 --> 00:01:35,670 But today, I want to reverse that point of view. 23 00:01:35,670 --> 00:01:38,900 We're going to read the equation backwards, 24 00:01:38,900 --> 00:01:49,240 and we're going to write it this way. 25 00:01:49,240 --> 00:02:01,010 And we're going to use f to understand capital F. 26 00:02:01,010 --> 00:02:04,370 Or in other words, the derivative 27 00:02:04,370 --> 00:02:07,700 to understand the function. 28 00:02:07,700 --> 00:02:13,380 So that's the reversal of point of view that I'd like to make. 29 00:02:13,380 --> 00:02:16,310 And we'll make this point in various ways. 30 00:02:16,310 --> 00:02:28,230 So information about f, about F', gives us 31 00:02:28,230 --> 00:02:37,750 information about F. Now, since there were questions 32 00:02:37,750 --> 00:02:39,740 about the mean value theorem, I'm 33 00:02:39,740 --> 00:02:42,380 going to illustrate this first by making 34 00:02:42,380 --> 00:02:46,710 a comparison between the fundamental theorem of calculus 35 00:02:46,710 --> 00:02:50,600 and the mean value theorem. 36 00:02:50,600 --> 00:02:56,460 So we're going to compare this fundamental theorem of calculus 37 00:02:56,460 --> 00:03:01,489 with what we call the mean value theorem. 38 00:03:01,489 --> 00:03:02,905 And in order to do that, I'm going 39 00:03:02,905 --> 00:03:05,070 to introduce a couple of notations. 40 00:03:05,070 --> 00:03:08,870 I'll write delta F as F(b) - F(a). 41 00:03:11,820 --> 00:03:17,770 And another highly imaginative notation, delta x = b - a. 42 00:03:17,770 --> 00:03:21,760 So here's the change in F, there's the change in x. 43 00:03:21,760 --> 00:03:25,690 And then, this fundamental theorem 44 00:03:25,690 --> 00:03:30,800 can be written, of course, right up above there is the formula. 45 00:03:30,800 --> 00:03:36,430 And it's the formula for delta F. 46 00:03:36,430 --> 00:03:39,240 So this is what we call the fundamental theorem 47 00:03:39,240 --> 00:03:44,090 of calculus. 48 00:03:44,090 --> 00:03:51,810 I'm going to divide by delta x, now. 49 00:03:51,810 --> 00:03:56,510 And If I divide by delta x, that's the same thing as 1 / 50 00:03:56,510 --> 00:04:02,880 (b-a) times the integral from a to b of f(x) dx. 51 00:04:02,880 --> 00:04:05,110 So I've just rewritten the formula here. 52 00:04:05,110 --> 00:04:11,750 And this expression here, on the right-hand side, 53 00:04:11,750 --> 00:04:13,650 is a fairly important one. 54 00:04:13,650 --> 00:04:23,370 This is the average of f. 55 00:04:23,370 --> 00:04:29,490 That's the average value of f. 56 00:04:29,490 --> 00:04:32,560 Now, so this is going to permit me 57 00:04:32,560 --> 00:04:35,980 to make the comparison between the mean value theorem, which 58 00:04:35,980 --> 00:04:38,630 we don't have stated yet here. 59 00:04:38,630 --> 00:04:41,010 And the fundamental theorem. 60 00:04:41,010 --> 00:04:49,130 And I'll do it in the form of inequalities. 61 00:04:49,130 --> 00:04:51,079 So right in the middle here, I'm going 62 00:04:51,079 --> 00:04:52,370 to put the fundamental theorem. 63 00:04:52,370 --> 00:04:56,580 It says that delta F in this notation is equal to, 64 00:04:56,580 --> 00:04:59,150 well if I multiply by delta x again, I 65 00:04:59,150 --> 00:05:01,850 can write it as the average of f-- 66 00:05:01,850 --> 00:05:04,870 So I'm going to write it as the average of F' here. 67 00:05:04,870 --> 00:05:07,100 Times delta x. 68 00:05:07,100 --> 00:05:09,120 So we have this factor here, which 69 00:05:09,120 --> 00:05:11,870 is the average of F', or the average of little f, 70 00:05:11,870 --> 00:05:13,810 it's the same thing. 71 00:05:13,810 --> 00:05:15,430 And then I multiplied through again. 72 00:05:15,430 --> 00:05:20,810 So I put the thing in the red box, here. 73 00:05:20,810 --> 00:05:29,970 STUDENT: [INAUDIBLE] 74 00:05:29,970 --> 00:05:37,190 PROFESSOR: Isn't what the average of big F? 75 00:05:37,190 --> 00:05:39,990 So the question is, why is this the average 76 00:05:39,990 --> 00:05:44,990 of little f rather than the average of big F. 77 00:05:44,990 --> 00:05:50,130 So the average of a function is the typical value. 78 00:05:50,130 --> 00:05:54,680 If, for example, little f were constant, 79 00:05:54,680 --> 00:05:59,770 little f were constant, then this integral would be-- 80 00:05:59,770 --> 00:06:03,920 So the question is why is this the average. 81 00:06:03,920 --> 00:06:08,970 And I'll take a little second to explain that. 82 00:06:08,970 --> 00:06:13,050 But I think I'll explain it over here. 83 00:06:13,050 --> 00:06:19,010 Because I'm going to erase it. 84 00:06:19,010 --> 00:06:27,270 So the idea of an average is the following. 85 00:06:27,270 --> 00:06:32,281 For example, imagine that a = 0 and b = n, 86 00:06:32,281 --> 00:06:33,600 let's say for example. 87 00:06:33,600 --> 00:06:46,850 And so we might sum the function from 1 to n. 88 00:06:46,850 --> 00:06:49,520 Now, that would be the sum of the values from 1 to n. 89 00:06:49,520 --> 00:06:54,960 But the average is, we divide by n here. 90 00:06:54,960 --> 00:06:56,140 So this is the average. 91 00:06:56,140 --> 00:07:00,180 And this is a kind of Riemann sum, 92 00:07:00,180 --> 00:07:06,290 representing the integral from 0 to n, of f(x) dx. 93 00:07:06,290 --> 00:07:10,700 Where the increment, delta x, is 1. 94 00:07:10,700 --> 00:07:13,470 So this is the notion of an average value here, 95 00:07:13,470 --> 00:07:15,760 but in the continuum setting, as opposed 96 00:07:15,760 --> 00:07:20,570 to the discrete setting. 97 00:07:20,570 --> 00:07:24,270 Whereas what's on the left-hand side 98 00:07:24,270 --> 00:07:28,250 is the change in F. The capital F. 99 00:07:28,250 --> 00:07:31,790 And this is the average of the little f. 100 00:07:31,790 --> 00:07:33,860 So an average is a sum. 101 00:07:33,860 --> 00:07:39,520 And it's like an integral. 102 00:07:39,520 --> 00:07:42,720 So, in other words what I have here is that the change in F 103 00:07:42,720 --> 00:07:45,620 is the average of its infinitesimal change 104 00:07:45,620 --> 00:07:50,000 times the amount of time elapsed, if you like. 105 00:07:50,000 --> 00:07:56,100 So this is the statement of the fundamental theorem. 106 00:07:56,100 --> 00:07:56,960 Just rewritten. 107 00:07:56,960 --> 00:07:58,450 Exactly what I wrote there. 108 00:07:58,450 --> 00:08:01,780 But I multiplied back by delta x. 109 00:08:01,780 --> 00:08:07,530 Now, let me compare this with the mean value theorem. 110 00:08:07,530 --> 00:08:13,590 The mean value theorem also is an equation. 111 00:08:13,590 --> 00:08:17,790 The mean value theorem says that this is equal to F'(c) delta x. 112 00:08:21,560 --> 00:08:23,270 Now, I pulled a fast one on you. 113 00:08:23,270 --> 00:08:26,560 I used capital F's here to make the analogy clear. 114 00:08:26,560 --> 00:08:30,710 But the role of the letter is important to make 115 00:08:30,710 --> 00:08:32,200 the transition to this comparison. 116 00:08:32,200 --> 00:08:35,010 We're talking about the function capital F here. 117 00:08:35,010 --> 00:08:36,620 And its derivative. 118 00:08:36,620 --> 00:08:38,510 Now, this is true. 119 00:08:38,510 --> 00:08:42,510 So now I claim that this thing is fairly specific. 120 00:08:42,510 --> 00:08:47,750 Whereas this, unfortunately, is a little bit vague. 121 00:08:47,750 --> 00:08:51,240 And the reason why it's vague is that c is just somewhere 122 00:08:51,240 --> 00:08:52,790 in the interval. 123 00:08:52,790 --> 00:09:01,010 So some c-- Sorry, this is some c, in between a and b. 124 00:09:01,010 --> 00:09:04,475 So really, since we don't know where this thing is, 125 00:09:04,475 --> 00:09:05,850 we don't know which of the values 126 00:09:05,850 --> 00:09:07,690 it is, we can't say what it is. 127 00:09:07,690 --> 00:09:11,700 All we can do is say, well for sure 128 00:09:11,700 --> 00:09:13,280 it's less than the largest value, 129 00:09:13,280 --> 00:09:18,090 say, the maximum of F', times delta x. 130 00:09:18,090 --> 00:09:20,720 And the only thing we can say for sure on the other end 131 00:09:20,720 --> 00:09:23,720 is that it's less than or equal to-- sorry, 132 00:09:23,720 --> 00:09:25,700 it's greater than or equal to the minimum 133 00:09:25,700 --> 00:09:27,410 of F' times delta x. 134 00:09:27,410 --> 00:09:29,090 Over that same interval. 135 00:09:29,090 --> 00:09:39,650 This is over 0 less than-- sorry, a < x < b. 136 00:09:39,650 --> 00:09:43,220 So that means that the fundamental theorem of calculus 137 00:09:43,220 --> 00:09:45,950 is a much more specific thing. 138 00:09:45,950 --> 00:09:48,272 And indeed it gives the same conclusion. 139 00:09:48,272 --> 00:09:50,230 It's much stronger than the mean value theorem. 140 00:09:50,230 --> 00:09:52,330 It's way better than the mean value theorem. 141 00:09:52,330 --> 00:09:55,100 In fact, as soon as we have integrals, 142 00:09:55,100 --> 00:09:57,280 we can abandon the mean value theorem. 143 00:09:57,280 --> 00:09:58,550 We don't want it. 144 00:09:58,550 --> 00:10:00,810 It's too simple-minded. 145 00:10:00,810 --> 00:10:03,700 And what we have is something much more sophisticated, 146 00:10:03,700 --> 00:10:04,940 which we can use. 147 00:10:04,940 --> 00:10:05,760 Which is this. 148 00:10:05,760 --> 00:10:08,400 So it's obvious that if this is the average, 149 00:10:08,400 --> 00:10:09,960 the average is less than the maximum. 150 00:10:09,960 --> 00:10:14,610 So it's obvious that it works just as well 151 00:10:14,610 --> 00:10:15,950 to draw this conclusion. 152 00:10:15,950 --> 00:10:20,010 And similarly over here with the minimum. 153 00:10:20,010 --> 00:10:22,240 OK, the average is always bigger than the minimum 154 00:10:22,240 --> 00:10:25,430 and smaller than the max. 155 00:10:25,430 --> 00:10:28,680 So this is the connection, if you like. 156 00:10:28,680 --> 00:10:33,050 And I'm going to elaborate just one step further by talking 157 00:10:33,050 --> 00:10:36,340 about the problem that you had on the exam. 158 00:10:36,340 --> 00:10:39,640 So there was an Exam 2 problem. 159 00:10:39,640 --> 00:10:43,590 And I'll show you how it works using the mean value theorem 160 00:10:43,590 --> 00:10:45,602 and how it works using integrals. 161 00:10:45,602 --> 00:10:47,810 But I'm going to have to use this notation capital F. 162 00:10:47,810 --> 00:10:51,850 So capital F', as opposed to the little f, 163 00:10:51,850 --> 00:10:55,270 which was what was the notation that was on your exam. 164 00:10:55,270 --> 00:10:57,620 So we had this situation here. 165 00:10:57,620 --> 00:11:01,380 These were the givens of the problem. 166 00:11:01,380 --> 00:11:11,620 And then the question was, the mean value theorem says, 167 00:11:11,620 --> 00:11:14,160 or implies, if you like, it doesn't say it, 168 00:11:14,160 --> 00:11:25,370 but it implies it - implies A is less than capital F of 4 169 00:11:25,370 --> 00:11:38,690 is less than B, for which A and B? 170 00:11:38,690 --> 00:11:43,330 So let's take a look at what it says. 171 00:11:43,330 --> 00:11:48,840 Well, the mean value theorem says that F( F(4) - 172 00:11:48,840 --> 00:11:53,620 F(0) = F'(c) (4 - 0). 173 00:11:57,240 --> 00:12:03,770 This is this F' times delta x, this is the change in x. 174 00:12:03,770 --> 00:12:08,100 And that's the same thing as 1/(1+c) times 4. 175 00:12:12,750 --> 00:12:21,780 And so the range of values of this number here is from / 176 00:12:21,780 --> 00:12:24,630 1/(1+0) times 4, that's 4. 177 00:12:24,630 --> 00:12:28,000 To, that's the largest value, to the smallest that it gets, 178 00:12:28,000 --> 00:12:32,510 which is 1/(1+4) times 4. 179 00:12:32,510 --> 00:12:41,020 That's the range. 180 00:12:41,020 --> 00:12:54,410 And so the conclusion is that F(4) - f(0) is between, well, 181 00:12:54,410 --> 00:12:55,090 let's see. 182 00:12:55,090 --> 00:12:59,040 It's between 4 and 4/5. 183 00:12:59,040 --> 00:13:01,850 Which are those two numbers down there. 184 00:13:01,850 --> 00:13:03,920 And if you remember that F(0) was 1, 185 00:13:03,920 --> 00:13:15,680 this is the same F(4) is between 5 and 9/5. 186 00:13:15,680 --> 00:13:19,650 So that's the way that you were supposed to solve 187 00:13:19,650 --> 00:13:22,570 the problem on the exam. 188 00:13:22,570 --> 00:13:25,760 On the other hand, let's compare to what 189 00:13:25,760 --> 00:13:27,940 you would do with the fundamental theorem 190 00:13:27,940 --> 00:13:31,140 of calculus. 191 00:13:31,140 --> 00:13:33,090 With the fundamentals theorem of calculus, 192 00:13:33,090 --> 00:13:35,090 we have the following formula. 193 00:13:35,090 --> 00:13:41,710 F(4) - F(0) is equal to the integral from 0 to 4 of dx / 194 00:13:41,710 --> 00:13:42,210 (1+x). 195 00:13:46,100 --> 00:13:52,700 That's what the fundamental theorem says. 196 00:13:52,700 --> 00:13:58,550 And now I claim that we can get these same types of results 197 00:13:58,550 --> 00:14:00,780 by a very elementary observation. 198 00:14:00,780 --> 00:14:03,470 It's really the same observation that I made up here, 199 00:14:03,470 --> 00:14:06,110 that the average is less than or equal to the maximum. 200 00:14:06,110 --> 00:14:11,810 Which is that the biggest this can ever be is, let's see. 201 00:14:11,810 --> 00:14:15,280 The biggest it is when x is 0, that's 1. 202 00:14:15,280 --> 00:14:20,360 So the biggest it ever gets is this. 203 00:14:20,360 --> 00:14:25,110 And that's equal to 4. 204 00:14:25,110 --> 00:14:25,660 Right? 205 00:14:25,660 --> 00:14:30,910 On the other hand, the smallest it ever gets to be, 206 00:14:30,910 --> 00:14:36,770 it's equal to this. 207 00:14:36,770 --> 00:14:39,620 The smallest it ever gets to be is the integral 208 00:14:39,620 --> 00:14:42,400 from 0 to 4 of 1/5 dx. 209 00:14:42,400 --> 00:14:46,110 Because that's the lowest value that the integrand takes. 210 00:14:46,110 --> 00:14:48,390 When x = 4, it's 1/5. 211 00:14:48,390 --> 00:14:54,270 And that's equal to 4/5. 212 00:14:54,270 --> 00:14:56,440 Now, there's a little tiny detail 213 00:14:56,440 --> 00:14:58,350 which is that really we know that this 214 00:14:58,350 --> 00:15:01,560 is the area of some rectangle and this is strictly smaller. 215 00:15:01,560 --> 00:15:03,990 And we know that these inequalities are actually 216 00:15:03,990 --> 00:15:05,210 strict. 217 00:15:05,210 --> 00:15:07,620 But that's a minor point. 218 00:15:07,620 --> 00:15:12,270 And certainly not one that we'll pay close attention to. 219 00:15:12,270 --> 00:15:17,910 But now, let me show you what this looks like geometrically. 220 00:15:17,910 --> 00:15:22,270 So geometrically, we interpret this as the area under a curve. 221 00:15:22,270 --> 00:15:31,850 Here's a piece of the curve y = 1/(1+x). 222 00:15:31,850 --> 00:15:37,130 And it's going up to 4 and starting at 0 here. 223 00:15:37,130 --> 00:15:41,510 And the first estimate that we made 224 00:15:41,510 --> 00:15:46,030 - that is, the upper bound - was by trapping this 225 00:15:46,030 --> 00:15:53,200 in this big rectangle here. 226 00:15:53,200 --> 00:15:55,610 We compared it to the constant function, 227 00:15:55,610 --> 00:15:57,990 which was 1 all the way across. 228 00:15:57,990 --> 00:16:00,300 This is y = 1. 229 00:16:00,300 --> 00:16:05,780 And then we also trapped it from underneath 230 00:16:05,780 --> 00:16:08,600 by the function which was at the bottom. 231 00:16:08,600 --> 00:16:14,430 And this was y = 1/5. 232 00:16:14,430 --> 00:16:18,410 And so what this really is is, these things 233 00:16:18,410 --> 00:16:21,060 are the simplest possible Riemann sum. 234 00:16:21,060 --> 00:16:22,860 Sort of a silly Riemann sum. 235 00:16:22,860 --> 00:16:34,390 This is a Riemann sum with one rectangle. 236 00:16:34,390 --> 00:16:36,420 This is the simplest possible one. 237 00:16:36,420 --> 00:16:38,580 And so this is a very, very crude estimate. 238 00:16:38,580 --> 00:16:40,970 You can see it misses by a mile. 239 00:16:40,970 --> 00:16:42,840 The larger and the smaller values 240 00:16:42,840 --> 00:16:46,370 are off by a factor of 5. 241 00:16:46,370 --> 00:16:50,660 But this one is called the-- this one is the lower 242 00:16:50,660 --> 00:16:53,390 Riemann sum. 243 00:16:53,390 --> 00:16:59,560 And that one is less than our actual integral. 244 00:16:59,560 --> 00:17:14,100 Which is less than the upper Riemann sum. 245 00:17:14,100 --> 00:17:19,070 And you should, by now, have looked at those upper and lower 246 00:17:19,070 --> 00:17:20,670 sums on your homework. 247 00:17:20,670 --> 00:17:22,460 So it's just the rectangles underneath 248 00:17:22,460 --> 00:17:25,390 and the rectangles on top. 249 00:17:25,390 --> 00:17:27,370 So at this point, we can literally 250 00:17:27,370 --> 00:17:28,662 abandon the mean value theorem. 251 00:17:28,662 --> 00:17:30,953 Because we have a much better way of getting at things. 252 00:17:30,953 --> 00:17:32,770 If we chop things up into more rectangles, 253 00:17:32,770 --> 00:17:35,550 we'll get much better numerical approximations. 254 00:17:35,550 --> 00:17:38,587 And if we use simpleminded expressions with integrals, 255 00:17:38,587 --> 00:17:40,670 we'll be able to figure out any bound we could get 256 00:17:40,670 --> 00:17:42,610 using the mean value theorem. 257 00:17:42,610 --> 00:17:45,790 So that's not the relevance of the mean value theorem. 258 00:17:45,790 --> 00:17:48,810 I'll explain to you why we talked about it, even, 259 00:17:48,810 --> 00:17:51,310 in a few minutes. 260 00:17:51,310 --> 00:17:59,790 OK, are there any questions before we go on? 261 00:17:59,790 --> 00:18:00,290 Yeah. 262 00:18:00,290 --> 00:18:07,000 STUDENT: [INAUDIBLE] 263 00:18:07,000 --> 00:18:09,860 PROFESSOR: I knew that the range of c was from 0 to 4, 264 00:18:09,860 --> 00:18:11,980 I should have said that right here. 265 00:18:11,980 --> 00:18:13,715 This is true for this theorem. 266 00:18:13,715 --> 00:18:16,720 The mean value theorem comes with an extra statement, 267 00:18:16,720 --> 00:18:17,950 which I missed. 268 00:18:17,950 --> 00:18:21,960 Which is that this is for some c between 0 and 4. 269 00:18:21,960 --> 00:18:23,690 So I know the range is between 0 and 4. 270 00:18:23,690 --> 00:18:25,148 The reason why it's between 0 and 4 271 00:18:25,148 --> 00:18:27,560 is that's part of the mean value theorem. 272 00:18:27,560 --> 00:18:29,610 We started at 0, we ended at 4. 273 00:18:29,610 --> 00:18:32,420 So the c has to be somewhere in between. 274 00:18:32,420 --> 00:18:42,057 That's part of the mean value theorem. 275 00:18:42,057 --> 00:18:42,890 STUDENT: [INAUDIBLE] 276 00:18:42,890 --> 00:18:43,730 PROFESSOR: The question is, do you 277 00:18:43,730 --> 00:18:45,820 exclude any values that are above 4 and below 0. 278 00:18:45,820 --> 00:18:46,884 Yes, absolutely. 279 00:18:46,884 --> 00:18:48,550 The point is that in order to figure out 280 00:18:48,550 --> 00:18:51,942 how F changes, capital F changes, between 0 and 4, 281 00:18:51,942 --> 00:18:54,150 you need only pay attention to the values in between. 282 00:18:54,150 --> 00:18:55,608 You don't have to pay any attention 283 00:18:55,608 --> 00:18:59,740 to what the function is doing below 0 or above 4. 284 00:18:59,740 --> 00:19:07,090 Those things are strictly irrelevant. 285 00:19:07,090 --> 00:19:16,569 STUDENT: [INAUDIBLE] 286 00:19:16,569 --> 00:19:18,110 PROFESSOR: Yeah, I mean it's strictly 287 00:19:18,110 --> 00:19:19,950 in between these two numbers. 288 00:19:19,950 --> 00:19:23,420 I have to understand what the lowest and the highest one is. 289 00:19:23,420 --> 00:19:24,700 STUDENT: [INAUDIBLE] 290 00:19:24,700 --> 00:19:35,031 PROFESSOR: It's approaching that, so. 291 00:19:35,031 --> 00:19:35,530 OK. 292 00:19:35,530 --> 00:19:39,650 So now, the next thing that we're going to talk about 293 00:19:39,650 --> 00:19:41,820 is, since I've got that 1 up there, 294 00:19:41,820 --> 00:19:44,180 that Fundamental Theorem of Calculus 1, I 295 00:19:44,180 --> 00:20:05,070 need to talk about version 2. 296 00:20:05,070 --> 00:20:15,530 So here is the Fundamental Theorem of Calculus version 2. 297 00:20:15,530 --> 00:20:20,660 I'm going to start out with a function little f, 298 00:20:20,660 --> 00:20:28,002 and I'm going to assume that it's continuous. 299 00:20:28,002 --> 00:20:30,210 And then I'm going to define a new function, which is 300 00:20:30,210 --> 00:20:33,030 defined as a definite integral. 301 00:20:33,030 --> 00:20:40,310 G(x) is the integral from a to x of f(t) dt. 302 00:20:40,310 --> 00:20:42,829 Now, I want to emphasize here because it's the first time 303 00:20:42,829 --> 00:20:44,370 that I'm writing something like this, 304 00:20:44,370 --> 00:20:47,300 that this is a fairly complicated gadget. 305 00:20:47,300 --> 00:20:52,110 It plays a very basic and very fundamental but simple role, 306 00:20:52,110 --> 00:20:54,490 but it nevertheless is a little complicated. 307 00:20:54,490 --> 00:20:58,370 What's happening here is that the upper limit I've now 308 00:20:58,370 --> 00:21:05,840 called x, and the variable t is ranging between a and x, 309 00:21:05,840 --> 00:21:08,030 and that the a and the x are fixed 310 00:21:08,030 --> 00:21:12,290 when I calculate the integral. 311 00:21:12,290 --> 00:21:14,730 And the t is what's called the dummy variable. 312 00:21:14,730 --> 00:21:16,290 It's the variable of integration. 313 00:21:16,290 --> 00:21:21,420 You'll see a lot of people who will mix this x with this t. 314 00:21:21,420 --> 00:21:25,190 And if you do that, you will get confused, 315 00:21:25,190 --> 00:21:28,150 potentially hopelessly confused, in this class. 316 00:21:28,150 --> 00:21:32,310 In 18.02 you will be completely lost if you do that. 317 00:21:32,310 --> 00:21:33,720 So don't do it. 318 00:21:33,720 --> 00:21:38,410 Don't mix these two guys up. 319 00:21:38,410 --> 00:21:42,089 It's actually done by many people in textbooks, 320 00:21:42,089 --> 00:21:43,130 and it's fairly careless. 321 00:21:43,130 --> 00:21:45,230 Especially in old-fashioned textbooks. 322 00:21:45,230 --> 00:21:48,610 But don't do it. 323 00:21:48,610 --> 00:21:50,570 So here we have this G(x). 324 00:21:50,570 --> 00:21:56,250 Now, remember, this G(x) really does make sense. 325 00:21:56,250 --> 00:21:59,650 If you give me an a, and you give me an x, 326 00:21:59,650 --> 00:22:01,960 I can figure out what this is, because I can figure out 327 00:22:01,960 --> 00:22:02,830 the Riemann sum. 328 00:22:02,830 --> 00:22:05,090 So of course I need to know what the function is, too. 329 00:22:05,090 --> 00:22:07,960 But anyway, we have a numerical procedure for figuring out 330 00:22:07,960 --> 00:22:09,920 what the function G is. 331 00:22:09,920 --> 00:22:13,010 Now, as is suggested by this mysterious letter x being 332 00:22:13,010 --> 00:22:16,350 in the place where it is, I'm actually going to vary x. 333 00:22:16,350 --> 00:22:19,240 So the conclusion is that if this is true, 334 00:22:19,240 --> 00:22:21,870 and this is just a parenthesis, not part of the theorem. 335 00:22:21,870 --> 00:22:25,770 It's just an indication of what the notation means. 336 00:22:25,770 --> 00:22:40,430 Then G' = f. 337 00:22:40,430 --> 00:22:43,030 Let me first explain what the significance of this theorem 338 00:22:43,030 --> 00:22:48,280 is, from the point of view of differential equations. 339 00:22:48,280 --> 00:23:04,420 G(x) solves the differential equation y' = f(x). 340 00:23:04,420 --> 00:23:09,390 So y' = f, I shouldn't put the x in if I got it here, 341 00:23:09,390 --> 00:23:13,240 with the condition y(a) = 0. 342 00:23:13,240 --> 00:23:19,000 So it solves this pair of conditions here. 343 00:23:19,000 --> 00:23:21,860 The rate of change, and the initial position 344 00:23:21,860 --> 00:23:23,280 is specified here. 345 00:23:23,280 --> 00:23:29,660 Because when you integrate from a to a, you get 0 always. 346 00:23:29,660 --> 00:23:34,100 And what this theorem says is you can always 347 00:23:34,100 --> 00:23:35,700 solve that equation. 348 00:23:35,700 --> 00:23:38,130 When we did differential equations, I said that already. 349 00:23:38,130 --> 00:23:39,990 I said we'll treat these as always solved. 350 00:23:39,990 --> 00:23:41,530 Well, here's the reason. 351 00:23:41,530 --> 00:23:45,080 We have a numerical procedure for computing things like this. 352 00:23:45,080 --> 00:23:49,330 We could always solve this equation. 353 00:23:49,330 --> 00:23:52,290 And the formula is a fairly complicated gadget, 354 00:23:52,290 --> 00:23:58,400 but so far just associated with Riemann sums. 355 00:23:58,400 --> 00:24:01,000 Alright, now. 356 00:24:01,000 --> 00:24:13,820 Let's just do one example. 357 00:24:13,820 --> 00:24:17,900 Unfortunately, not a complicated example and maybe not 358 00:24:17,900 --> 00:24:21,230 persuasive as to why you would care about this just yet. 359 00:24:21,230 --> 00:24:23,670 But nevertheless very important. 360 00:24:23,670 --> 00:24:26,470 Because this is the quiz question which everybody gets 361 00:24:26,470 --> 00:24:29,530 wrong until they practice it. 362 00:24:29,530 --> 00:24:35,960 So the integral from, say 1 to x, of dt / t^2. 363 00:24:38,710 --> 00:24:45,420 Let's try this one here. 364 00:24:45,420 --> 00:24:56,650 So here's an example of this theorem, I claim. 365 00:24:56,650 --> 00:25:00,850 Now, this is a question which challenges your ability 366 00:25:00,850 --> 00:25:04,740 to understand what the question means. 367 00:25:04,740 --> 00:25:06,800 Because it's got a lot of symbols. 368 00:25:06,800 --> 00:25:09,730 It's got the integration and it's got the differentiation. 369 00:25:09,730 --> 00:25:15,700 However, what it really is is an exercise in recopying. 370 00:25:15,700 --> 00:25:20,660 You look at it and you write down the answer. 371 00:25:20,660 --> 00:25:24,570 And the reason is that, by definition, 372 00:25:24,570 --> 00:25:29,190 this function in here is a function of the form 373 00:25:29,190 --> 00:25:34,320 G(x) of the theorem over here. 374 00:25:34,320 --> 00:25:35,920 So this is the G(x). 375 00:25:35,920 --> 00:25:42,980 And by definition, we said that G'(x) = f(x). 376 00:25:42,980 --> 00:25:46,170 Well, what's the f(x)? 377 00:25:46,170 --> 00:25:47,010 Look inside here. 378 00:25:47,010 --> 00:25:48,760 It's what's called the integrand. 379 00:25:48,760 --> 00:25:53,640 This is the integral from 0 to x of f(t) dt, right? 380 00:25:53,640 --> 00:26:00,710 Where the f(t) is equal to 1 / t^2. 381 00:26:00,710 --> 00:26:02,910 So your ability is challenged. 382 00:26:02,910 --> 00:26:06,060 You have to take that 1 / t^2 and you have to plug 383 00:26:06,060 --> 00:26:09,960 in the letter x, instead of t, for it. 384 00:26:09,960 --> 00:26:11,480 And then write it down. 385 00:26:11,480 --> 00:26:18,120 As I say, this is an exercise in recopying what's there. 386 00:26:18,120 --> 00:26:19,917 So this is quite easy to do, right? 387 00:26:19,917 --> 00:26:21,750 I mean, you just look and you write it down. 388 00:26:21,750 --> 00:26:28,260 But nevertheless, it looks like a long, elaborate object here. 389 00:26:28,260 --> 00:26:28,760 Pardon me? 390 00:26:28,760 --> 00:26:30,590 STUDENT: [INAUDIBLE] 391 00:26:30,590 --> 00:26:32,940 PROFESSOR: So the question was, why did I integrate. 392 00:26:32,940 --> 00:26:34,000 STUDENT: [INAUDIBLE] 393 00:26:34,000 --> 00:26:36,990 PROFESSOR: Why did I not integrate? 394 00:26:36,990 --> 00:26:37,720 Ah. 395 00:26:37,720 --> 00:26:38,870 Very good question. 396 00:26:38,870 --> 00:26:41,010 Why did I not integrate. 397 00:26:41,010 --> 00:26:45,300 The reason why I didn't integrate is I didn't need to. 398 00:26:45,300 --> 00:26:47,665 Just as when you take the antiderivative-- sorry, 399 00:26:47,665 --> 00:26:50,040 the derivative of something, you take the antiderivative, 400 00:26:50,040 --> 00:26:51,750 you get back to the thing. 401 00:26:51,750 --> 00:26:54,947 So, in this case, we're taking the antiderivative of something 402 00:26:54,947 --> 00:26:56,030 and we're differentiating. 403 00:26:56,030 --> 00:26:58,380 So we end back in the same place where we started. 404 00:26:58,380 --> 00:27:01,780 We started with f(t), we're ending with f. 405 00:27:01,780 --> 00:27:04,990 Little f. 406 00:27:04,990 --> 00:27:06,950 So you integrate, and then differentiate. 407 00:27:06,950 --> 00:27:09,410 And you get back to the same place. 408 00:27:09,410 --> 00:27:12,330 Now, the only difference between this and the other version 409 00:27:12,330 --> 00:27:15,610 is, in this case when you differentiate and integrate 410 00:27:15,610 --> 00:27:18,195 you could be off by a constant. 411 00:27:18,195 --> 00:27:19,570 That's what that shift, why there 412 00:27:19,570 --> 00:27:21,420 are two pieces to this one. 413 00:27:21,420 --> 00:27:23,020 But there's never an extra piece here. 414 00:27:23,020 --> 00:27:24,919 There's no plus c here. 415 00:27:24,919 --> 00:27:26,460 When you integrate and differentiate, 416 00:27:26,460 --> 00:27:28,080 you kill whatever the constant is. 417 00:27:28,080 --> 00:27:31,920 Because the derivative of a constant is 0. 418 00:27:31,920 --> 00:27:36,220 So no matter what the constant is, hiding inside of G, 419 00:27:36,220 --> 00:27:41,140 you're getting the same result. So this is the basic idea. 420 00:27:41,140 --> 00:27:46,220 Now, I just want to double-check it, 421 00:27:46,220 --> 00:27:52,124 using the Fundamental Theorem of Calculus 1 here. 422 00:27:52,124 --> 00:27:53,790 So let's actually evaluate the integral. 423 00:27:53,790 --> 00:27:54,880 So now I'm going to do what you've 424 00:27:54,880 --> 00:27:56,185 suggested, which is I'm just going 425 00:27:56,185 --> 00:27:57,370 to check whether it's true. 426 00:27:57,370 --> 00:27:59,560 No, no I am because I'm going just double-check 427 00:27:59,560 --> 00:28:01,120 that it's consistent. 428 00:28:01,120 --> 00:28:03,447 It certainly is slower this way, and we're not 429 00:28:03,447 --> 00:28:05,030 going to want to do this all the time, 430 00:28:05,030 --> 00:28:06,810 but we might as well check one. 431 00:28:06,810 --> 00:28:09,370 So this is our integral. 432 00:28:09,370 --> 00:28:10,860 And we know how to do it. 433 00:28:10,860 --> 00:28:13,260 No, I need to do it. 434 00:28:13,260 --> 00:28:17,820 And this is -t^(-1), evaluated at 1 and x. 435 00:28:17,820 --> 00:28:21,670 Again, there's something subliminally here 436 00:28:21,670 --> 00:28:23,300 for you to think about. 437 00:28:23,300 --> 00:28:28,737 Which is that, remember, it's t is ranging between 1 and t = x. 438 00:28:28,737 --> 00:28:30,820 And this is one of the big reasons why this letter 439 00:28:30,820 --> 00:28:32,860 t has to be different from x. 440 00:28:32,860 --> 00:28:35,370 Because here it's 1 and there it's x. 441 00:28:35,370 --> 00:28:37,210 It's not x. 442 00:28:37,210 --> 00:28:38,890 So you can't put an x here. 443 00:28:38,890 --> 00:28:44,130 Again, this is t = 1 and this is t = x over there. 444 00:28:44,130 --> 00:28:48,630 And now if I plug that in, I get what? 445 00:28:48,630 --> 00:28:55,860 I get -1/x, and then I get -(-1). 446 00:28:55,860 --> 00:28:59,090 So this is, let me get rid of those little t's there. 447 00:28:59,090 --> 00:29:05,690 This is a little easier to read. 448 00:29:05,690 --> 00:29:07,030 And so now let's check it. 449 00:29:07,030 --> 00:29:07,730 It's d/dx. 450 00:29:07,730 --> 00:29:09,500 So here's what G(x) is. 451 00:29:09,500 --> 00:29:12,260 G(x) = 1 - 1/x. 452 00:29:12,260 --> 00:29:14,860 That's what G(x) is. 453 00:29:14,860 --> 00:29:20,500 And if I differentiate that, I get +1 / x^2. 454 00:29:20,500 --> 00:29:26,170 That's it. 455 00:29:26,170 --> 00:29:40,810 You see the constant washed away. 456 00:29:40,810 --> 00:29:42,070 So now, here's my job. 457 00:29:42,070 --> 00:29:44,440 My job is to prove these theorems. 458 00:29:44,440 --> 00:29:45,809 I never did prove them for you. 459 00:29:45,809 --> 00:29:47,725 So, I'm going to prove the Fundamental Theorem 460 00:29:47,725 --> 00:29:49,340 of Calculus. 461 00:29:49,340 --> 00:29:51,860 But I'm going to do 2 first. 462 00:29:51,860 --> 00:29:53,420 And then I'm going to do 1. 463 00:29:53,420 --> 00:29:56,490 And it's just going to take me just one blackboard. 464 00:29:56,490 --> 00:30:00,310 It's not that hard. 465 00:30:00,310 --> 00:30:03,150 The proof is by picture. 466 00:30:03,150 --> 00:30:08,310 And, using the interpretation as area under the curve. 467 00:30:08,310 --> 00:30:12,170 So if here's the value of a, and this 468 00:30:12,170 --> 00:30:22,550 is the graph of the function y equals f of x. 469 00:30:22,550 --> 00:30:26,380 Then I want to draw three vertical lines. 470 00:30:26,380 --> 00:30:29,330 One of them is going to be at x. 471 00:30:29,330 --> 00:30:33,790 And one of them is going to be at x + delta x. 472 00:30:33,790 --> 00:30:35,950 So here I have the interval from 0 473 00:30:35,950 --> 00:30:39,510 to x, and next I have the interval from x to delta 474 00:30:39,510 --> 00:30:42,900 x more than that. 475 00:30:42,900 --> 00:30:50,360 And now the pieces that I've got are the area of this part. 476 00:30:50,360 --> 00:30:53,430 So this has area which has a name. 477 00:30:53,430 --> 00:30:55,860 It's called G(x). 478 00:30:55,860 --> 00:31:00,790 By definition, G(x), which is sitting right over here 479 00:31:00,790 --> 00:31:03,940 in the fundamental theorem, is the integral from a 480 00:31:03,940 --> 00:31:06,100 to x of this function. 481 00:31:06,100 --> 00:31:07,890 So it's the area under the curve. 482 00:31:07,890 --> 00:31:10,400 So that area is G(x). 483 00:31:10,400 --> 00:31:17,770 Now this other chunk here, I claim 484 00:31:17,770 --> 00:31:23,430 that this is delta G. This is the change in G. It's 485 00:31:23,430 --> 00:31:26,260 the value of G(x) that is the area of the whole business all 486 00:31:26,260 --> 00:31:30,530 the way up to x + delta x minus the first part, G(x). 487 00:31:30,530 --> 00:31:31,640 So it's what's left over. 488 00:31:31,640 --> 00:31:39,780 It's the incremental amount of area there. 489 00:31:39,780 --> 00:31:45,830 And now I am going to carry out a pretty standard estimation 490 00:31:45,830 --> 00:31:46,520 here. 491 00:31:46,520 --> 00:31:48,850 This is practically a rectangle. 492 00:31:48,850 --> 00:31:51,700 And it's got a base of delta x, and so we need to figure out 493 00:31:51,700 --> 00:31:55,320 what its height is. 494 00:31:55,320 --> 00:32:02,120 This is delta G, and it's approximately its base 495 00:32:02,120 --> 00:32:05,340 times its height. 496 00:32:05,340 --> 00:32:06,820 But what is the height? 497 00:32:06,820 --> 00:32:10,740 Well, the height is maybe either this segment or this segment 498 00:32:10,740 --> 00:32:11,920 or something in between. 499 00:32:11,920 --> 00:32:13,720 But they're all about the same. 500 00:32:13,720 --> 00:32:17,070 So I'm just going to put in the value at the first point. 501 00:32:17,070 --> 00:32:19,430 That's the left end there. 502 00:32:19,430 --> 00:32:25,550 So that's this height here, is f(x). 503 00:32:25,550 --> 00:32:28,200 So this is f(x), and so really I approximate it 504 00:32:28,200 --> 00:32:30,050 by that rectangle there. 505 00:32:30,050 --> 00:32:33,760 And now if I divide and take the limit, 506 00:32:33,760 --> 00:32:38,510 as delta x goes to 0, of delta G / delta x, 507 00:32:38,510 --> 00:32:40,270 it's going to equal f(x). 508 00:32:43,120 --> 00:32:48,920 And this is where I'm using the fact that f is continuous. 509 00:32:48,920 --> 00:32:51,270 Because I need the values nearby to be 510 00:32:51,270 --> 00:32:59,690 similar to the value in the limit. 511 00:32:59,690 --> 00:33:00,650 OK, that's the end. 512 00:33:00,650 --> 00:33:05,020 This the end of the proof, so I'll put a nice little Q.E.D. 513 00:33:05,020 --> 00:33:10,800 here. 514 00:33:10,800 --> 00:33:14,360 So we've done Fundamental Theorem of Calculus 2, 515 00:33:14,360 --> 00:33:19,440 and now we're ready for Fundamental Theorem of Calculus 516 00:33:19,440 --> 00:33:38,420 1. 517 00:33:38,420 --> 00:33:44,580 So now I still have it on the blackboard to remind you. 518 00:33:44,580 --> 00:33:48,200 It says that the integral of the derivative 519 00:33:48,200 --> 00:33:50,980 is the function, at least the difference between the values 520 00:33:50,980 --> 00:33:54,830 of the function at two places. 521 00:33:54,830 --> 00:34:08,040 So the place where we start is with this property that F' = f. 522 00:34:08,040 --> 00:34:10,600 That's the starting-- that's the hypothesis. 523 00:34:10,600 --> 00:34:12,710 Now, unfortunately, I'm going to have 524 00:34:12,710 --> 00:34:14,420 to assume something extra in order 525 00:34:14,420 --> 00:34:18,660 to use the Fundamental Theorem of Calculus 2, 526 00:34:18,660 --> 00:34:27,980 which is I'm going to assume that f is continuous. 527 00:34:27,980 --> 00:34:30,490 That's not really necessary, but that's 528 00:34:30,490 --> 00:34:32,170 just a very minor technical point, 529 00:34:32,170 --> 00:34:34,280 which I'm just going to ignore. 530 00:34:34,280 --> 00:34:40,610 So we're going to start with F' = f. 531 00:34:40,610 --> 00:34:46,530 And then I'm going to go somewhere else. 532 00:34:46,530 --> 00:34:52,620 I'm going to define a new function, G(x), 533 00:34:52,620 --> 00:35:00,130 which is the integral from a to x of f(t) dt. 534 00:35:00,130 --> 00:35:04,560 This is where we needed all of the labor of Riemann sums. 535 00:35:04,560 --> 00:35:07,170 Because otherwise we don't have a way of making sense out 536 00:35:07,170 --> 00:35:10,340 of what this even means. 537 00:35:10,340 --> 00:35:13,350 So hiding behind this one sentence 538 00:35:13,350 --> 00:35:16,050 is the fact that we actually have a number. 539 00:35:16,050 --> 00:35:18,430 We have a formula for such functions. 540 00:35:18,430 --> 00:35:20,550 So there is a function G(x) which, 541 00:35:20,550 --> 00:35:22,220 once you've produced a little f for me, 542 00:35:22,220 --> 00:35:27,420 I can cook up a function capital G for you. 543 00:35:27,420 --> 00:35:31,390 Now, we're going to apply this Fundamental Theorem of Calculus 544 00:35:31,390 --> 00:35:34,650 2, the one that we've already checked. 545 00:35:34,650 --> 00:35:36,030 So what does it say? 546 00:35:36,030 --> 00:35:46,510 It says that G' = f. 547 00:35:46,510 --> 00:35:49,820 And so now we're in the following situation. 548 00:35:49,820 --> 00:35:54,030 We know that F'(x) = G'(x). 549 00:35:58,080 --> 00:36:00,360 That's what we've got so far. 550 00:36:00,360 --> 00:36:07,210 And now we have one last step to get a good connection between F 551 00:36:07,210 --> 00:36:10,120 and G. Which is that we can conclude that F(x) 552 00:36:10,120 --> 00:36:12,060 is G(x) plus a constant. 553 00:36:20,440 --> 00:36:26,520 Now, this little step may seem innocuous 554 00:36:26,520 --> 00:36:32,610 but I remind you that this is the spot that requires 555 00:36:32,610 --> 00:36:36,030 the mean value theorem. 556 00:36:36,030 --> 00:36:39,650 So in order not to lie to you, we actually 557 00:36:39,650 --> 00:36:42,680 tell you what the underpinnings of all of calculus are. 558 00:36:42,680 --> 00:36:45,152 And they're this: the fact, if you like, 559 00:36:45,152 --> 00:36:47,110 that if two functions have the same derivative, 560 00:36:47,110 --> 00:36:48,430 they differ by a constant. 561 00:36:48,430 --> 00:36:50,750 Or that if a function has derivative 0, 562 00:36:50,750 --> 00:36:53,960 it's a constant itself. 563 00:36:53,960 --> 00:36:57,850 Now, that is the fundamental step that's needed, 564 00:36:57,850 --> 00:36:59,740 the underlying step that's needed. 565 00:36:59,740 --> 00:37:02,500 And, unfortunately, there aren't any proofs of it 566 00:37:02,500 --> 00:37:06,050 that are less complicated than using the mean value theorem. 567 00:37:06,050 --> 00:37:08,560 And so that's why we talk a little bit about the mean value 568 00:37:08,560 --> 00:37:10,934 theorem, because we don't want to lie to you about what's 569 00:37:10,934 --> 00:37:11,731 really going on. 570 00:37:11,731 --> 00:37:12,230 Yes. 571 00:37:12,230 --> 00:37:19,070 STUDENT: [INAUDIBLE] 572 00:37:19,070 --> 00:37:24,210 PROFESSOR: The question is how did I get from here, to here. 573 00:37:24,210 --> 00:37:27,440 And the answer is that if G' is little f, 574 00:37:27,440 --> 00:37:32,950 and we also know that F' is little f, then F' is G'. 575 00:37:32,950 --> 00:37:37,090 OK. 576 00:37:37,090 --> 00:37:50,710 Other questions? 577 00:37:50,710 --> 00:37:52,750 Alright, so we're almost done. 578 00:37:52,750 --> 00:37:57,400 I just have to work out the arithmetic here. 579 00:37:57,400 --> 00:38:02,030 So I start with F(b) - F(a). 580 00:38:04,810 --> 00:38:12,330 And that's equal to (G(b) + c) - (G(a) + c). 581 00:38:18,130 --> 00:38:20,970 And then I cancel the c's. 582 00:38:20,970 --> 00:38:23,310 So I have here G(b) - G(a). 583 00:38:29,960 --> 00:38:32,665 And now I just have to check what each of these is. 584 00:38:32,665 --> 00:38:35,470 So remember the definition of G here. 585 00:38:35,470 --> 00:38:38,100 G(b) is just what we want. 586 00:38:38,100 --> 00:38:42,460 The integral from a to b of f(x) dx. 587 00:38:42,460 --> 00:38:46,430 Well I called it f(t) dt, that's the same as f(x) dx 588 00:38:46,430 --> 00:38:48,665 now, because I have the limit being b 589 00:38:48,665 --> 00:38:52,900 and I'm allowed to use x as the dummy variable. 590 00:38:52,900 --> 00:38:55,780 Now the other one, I claim, is 0. 591 00:38:55,780 --> 00:38:59,680 Because it's the integral from a to a. 592 00:38:59,680 --> 00:39:03,490 This one is the integral from a to a. 593 00:39:03,490 --> 00:39:06,510 Which gives us 0. 594 00:39:06,510 --> 00:39:09,370 So this is just this minus 0, and that's the end. 595 00:39:09,370 --> 00:39:13,460 That's it. 596 00:39:13,460 --> 00:39:20,271 I started with F(b) - F(a), I got to the integral. 597 00:39:20,271 --> 00:39:20,770 Question? 598 00:39:20,770 --> 00:39:27,370 STUDENT: [INAUDIBLE] 599 00:39:27,370 --> 00:39:32,610 PROFESSOR: How did I get from F(b) - F(a), is (G(b) + c) - 600 00:39:32,610 --> 00:39:35,430 (G(a) + c), that's the question. 601 00:39:35,430 --> 00:39:40,190 STUDENT: [INAUDIBLE] 602 00:39:40,190 --> 00:39:44,160 PROFESSOR: Oh, sorry this is an equals sign. 603 00:39:44,160 --> 00:39:47,090 Sorry, the second line didn't draw. 604 00:39:47,090 --> 00:39:48,640 OK, equals. 605 00:39:48,640 --> 00:39:53,141 Because we're plugging in, for F(x), the formula for it. 606 00:39:53,141 --> 00:39:53,640 Yes. 607 00:39:53,640 --> 00:39:57,310 STUDENT: [INAUDIBLE] 608 00:39:57,310 --> 00:39:59,630 PROFESSOR: This step here? 609 00:39:59,630 --> 00:40:04,730 Or this one? 610 00:40:04,730 --> 00:40:09,392 STUDENT: [INAUDIBLE] 611 00:40:09,392 --> 00:40:10,100 PROFESSOR: Right. 612 00:40:10,100 --> 00:40:12,090 So that was a good question. 613 00:40:12,090 --> 00:40:15,120 But the answer is that that's the statement 614 00:40:15,120 --> 00:40:16,246 that we're aiming for. 615 00:40:16,246 --> 00:40:18,120 That's the Fundamental Theorem of Calculus 1, 616 00:40:18,120 --> 00:40:19,614 which we don't know yet. 617 00:40:19,614 --> 00:40:22,030 So we're trying to prove it, and that's why we haven't, we 618 00:40:22,030 --> 00:40:25,950 can't assume it. 619 00:40:25,950 --> 00:40:33,270 OK, so let me just notice that in the example that we had, 620 00:40:33,270 --> 00:40:36,410 before we go on to something else here. 621 00:40:36,410 --> 00:40:48,530 In the example above, what we had was the following thing. 622 00:40:48,530 --> 00:40:58,070 We had, say, F(x) = -1/x. 623 00:40:58,070 --> 00:41:01,890 So F'(x) = 1 / x^2. 624 00:41:01,890 --> 00:41:08,850 And, say, G(x) = 1 - 1/x. 625 00:41:08,850 --> 00:41:11,970 And you can see that either way you do that, 626 00:41:11,970 --> 00:41:14,310 if you integrate from 1 to 2, let's say, 627 00:41:14,310 --> 00:41:18,670 which is what we had over there, dt / t^2, 628 00:41:18,670 --> 00:41:28,140 you're going to get either -1/t, 1 to 2, or, if you like, 1 - 629 00:41:28,140 --> 00:41:31,200 1/t, 1 to 2. 630 00:41:31,200 --> 00:41:33,930 So this is the F version, this is the G version. 631 00:41:33,930 --> 00:41:45,410 And that's what plays itself out here, in this general proof. 632 00:41:45,410 --> 00:41:49,640 Alright. 633 00:41:49,640 --> 00:41:54,910 So now I want to go back to the theme for today, which 634 00:41:54,910 --> 00:42:01,290 is using little f to understand capital F. In other words, 635 00:42:01,290 --> 00:42:05,890 using the derivative of F to understand capital F. 636 00:42:05,890 --> 00:42:22,630 And I want to illustrate it by some more complicated examples. 637 00:42:22,630 --> 00:42:27,510 So I guess I just erased it, but we just took the antiderivative 638 00:42:27,510 --> 00:42:29,740 of 1 / t^2. 639 00:42:29,740 --> 00:42:34,340 And there's-- all of the powers work easily, but one. 640 00:42:34,340 --> 00:42:39,190 And the tricky one is the power 1 / x. 641 00:42:39,190 --> 00:42:43,280 So let's consider the differential equation L' (x) = 642 00:42:43,280 --> 00:42:44,550 1 / x. 643 00:42:44,550 --> 00:42:54,490 And say, with the initial value L(1) = 0. 644 00:42:54,490 --> 00:42:58,840 The solution, so the Fundamental Theorem of Calculus 2 645 00:42:58,840 --> 00:43:07,330 tells us the solution is this function here. 646 00:43:07,330 --> 00:43:14,070 L(x) equals the integral from 1 to x, dt / t. 647 00:43:14,070 --> 00:43:16,490 That's how we solve all such equations. 648 00:43:16,490 --> 00:43:19,290 We just integrate, take the definite integral. 649 00:43:19,290 --> 00:43:27,400 And I'm starting at 1 because I insisted that L(1) be 0. 650 00:43:27,400 --> 00:43:31,100 So that's the solution to the problem. 651 00:43:31,100 --> 00:43:33,900 And now the thing that's interesting here 652 00:43:33,900 --> 00:43:35,720 is that we started from a polynomial. 653 00:43:35,720 --> 00:43:37,990 Or we started from a rational, a ratio of polynomials; 654 00:43:37,990 --> 00:43:40,460 that is, 1 / t or 1 / x. 655 00:43:40,460 --> 00:43:42,490 And we get to a function which is actually 656 00:43:42,490 --> 00:43:44,990 what's known as a transcendental function. 657 00:43:44,990 --> 00:43:46,455 It's not an algebraic function. 658 00:43:46,455 --> 00:43:47,080 Yeah, question. 659 00:43:47,080 --> 00:43:57,000 STUDENT: [INAUDIBLE] 660 00:43:57,000 --> 00:44:04,970 PROFESSOR: The question is why is this equal to that. 661 00:44:04,970 --> 00:44:08,210 And the answer is, it's for the same reason 662 00:44:08,210 --> 00:44:10,200 that this is equal to that. 663 00:44:10,200 --> 00:44:14,030 It's the same reason as this. 664 00:44:14,030 --> 00:44:16,760 It's that the 1's cancel. 665 00:44:16,760 --> 00:44:19,800 We've taken the value of something at 2 minus the value 666 00:44:19,800 --> 00:44:20,300 at 1. 667 00:44:20,300 --> 00:44:22,500 The value at 2 minus the value at 1. 668 00:44:22,500 --> 00:44:24,720 And you'll get a 1 in the one case, and you get a 1 669 00:44:24,720 --> 00:44:25,470 in the other case. 670 00:44:25,470 --> 00:44:27,580 And you subtract them and they will cancel. 671 00:44:27,580 --> 00:44:28,620 They'll give you 0. 672 00:44:28,620 --> 00:44:31,380 These two things really are equal. 673 00:44:31,380 --> 00:44:33,507 This is not a function evaluated at one place, 674 00:44:33,507 --> 00:44:35,590 it's the difference between the function evaluated 675 00:44:35,590 --> 00:44:37,025 at 2 and the value at 1. 676 00:44:37,025 --> 00:44:39,570 And whenever you subtract two things like that, 677 00:44:39,570 --> 00:44:40,900 constants drop out. 678 00:44:40,900 --> 00:44:44,370 STUDENT: [INAUDIBLE] PROFESSOR: That's right. 679 00:44:44,370 --> 00:44:46,030 If I put 2 here, if I put c here, 680 00:44:46,030 --> 00:44:47,370 it would have been the same. 681 00:44:47,370 --> 00:44:49,010 It would just have dropped out. 682 00:44:49,010 --> 00:44:50,640 It's not there. 683 00:44:50,640 --> 00:44:53,260 And that's exactly this arithmetic right here. 684 00:44:53,260 --> 00:44:55,520 It doesn't matter which antiderivative you take. 685 00:44:55,520 --> 00:44:57,850 When you take the differences, the c's will cancel. 686 00:44:57,850 --> 00:45:03,130 You always get the same answer in the end. 687 00:45:03,130 --> 00:45:04,850 That's exactly why I wrote this down, 688 00:45:04,850 --> 00:45:06,290 so that you would see that. 689 00:45:06,290 --> 00:45:12,890 It doesn't matter which one you do. 690 00:45:12,890 --> 00:45:21,540 So, we still have a couple of minutes left here. 691 00:45:21,540 --> 00:45:23,630 This is actually-- So let me go back. 692 00:45:23,630 --> 00:45:29,720 So here's the antiderivative of 1 / x, with value 1 at 0. 693 00:45:29,720 --> 00:45:32,180 Now, in disguise, we know what this function is. 694 00:45:32,180 --> 00:45:35,410 We know this function is the logarithm function. 695 00:45:35,410 --> 00:45:37,580 But this is actually a better way 696 00:45:37,580 --> 00:45:42,030 of deriving all of the formulas for the logarithm. 697 00:45:42,030 --> 00:45:44,850 This is a much quicker and more efficient way of doing it. 698 00:45:44,850 --> 00:45:47,280 We had to do it by very laborious processes. 699 00:45:47,280 --> 00:45:51,690 This will allow us to do it very easily. 700 00:45:51,690 --> 00:45:56,040 And so, I'm going to do that next time. 701 00:45:56,040 --> 00:45:58,395 But rather than do that now, I'm going 702 00:45:58,395 --> 00:46:10,280 to point out to you that we can also get truly new functions. 703 00:46:10,280 --> 00:46:12,130 OK, so there are all kinds of new functions. 704 00:46:12,130 --> 00:46:15,690 So this is the first example of this kind would be, 705 00:46:15,690 --> 00:46:21,630 for example, to solve the equation y' = e^(-x^2) with 706 00:46:21,630 --> 00:46:25,140 y(0) = 0, let's say. 707 00:46:25,140 --> 00:46:28,047 Now, the solution to that is a function 708 00:46:28,047 --> 00:46:30,380 which again I can write down by the fundamental theorem. 709 00:46:30,380 --> 00:46:48,160 It's the integral from 0 to x of e^(-t^2) dt. 710 00:46:48,160 --> 00:46:52,370 This is a very famous function. 711 00:46:52,370 --> 00:46:55,820 This shape here is known as the bell curve. 712 00:46:55,820 --> 00:46:59,170 And it's the thing that comes up in probability all the time. 713 00:46:59,170 --> 00:47:01,880 This shape e^(-x^2). 714 00:47:01,880 --> 00:47:04,590 And our function is geometrically just the area 715 00:47:04,590 --> 00:47:06,310 under the curve here. 716 00:47:06,310 --> 00:47:09,080 This is F(x). 717 00:47:09,080 --> 00:47:12,350 If this place is x. 718 00:47:12,350 --> 00:47:13,860 So I have a geometric definition, 719 00:47:13,860 --> 00:47:16,620 I have a way of constructing what it is by Riemann sums. 720 00:47:16,620 --> 00:47:18,810 And I have a function here. 721 00:47:18,810 --> 00:47:26,850 But the curious thing about F(x) is that F(x) cannot be 722 00:47:26,850 --> 00:47:34,470 expressed in terms of any function you've seen 723 00:47:34,470 --> 00:47:35,360 previously. 724 00:47:35,360 --> 00:47:44,210 So logs, exponentials, trig functions, cannot be. 725 00:47:44,210 --> 00:47:51,600 It's a totally new function. 726 00:47:51,600 --> 00:47:53,730 Nevertheless, we'll be able to get 727 00:47:53,730 --> 00:47:56,030 any possible piece of information we would want to, 728 00:47:56,030 --> 00:47:56,910 out of this function. 729 00:47:56,910 --> 00:47:58,770 It's perfectly acceptable function, 730 00:47:58,770 --> 00:48:00,250 it will work just great for us. 731 00:48:00,250 --> 00:48:01,700 Just like any other function. 732 00:48:01,700 --> 00:48:03,540 Just like the log. 733 00:48:03,540 --> 00:48:08,890 And what this is analogous to is the following kind of thing. 734 00:48:08,890 --> 00:48:12,650 If you take the circle, the ancient Greeks, if you like, 735 00:48:12,650 --> 00:48:18,030 already understood that if you have a circle of radius 1, then 736 00:48:18,030 --> 00:48:23,290 its area is pi. 737 00:48:23,290 --> 00:48:24,820 So that's a geometric construction 738 00:48:24,820 --> 00:48:31,430 of what you could call a new number. 739 00:48:31,430 --> 00:48:34,280 Which is outside of the realm of what you might expect. 740 00:48:34,280 --> 00:48:37,840 And the weird thing about this number pi 741 00:48:37,840 --> 00:48:49,780 is that it is not the root of an algebraic equation 742 00:48:49,780 --> 00:48:57,260 with rational coefficients. 743 00:48:57,260 --> 00:49:00,540 It's what's called transcendental. 744 00:49:00,540 --> 00:49:02,680 Meaning, it's just completely outside of the realm 745 00:49:02,680 --> 00:49:04,050 of algebra. 746 00:49:04,050 --> 00:49:06,660 And, indeed, the logarithm function 747 00:49:06,660 --> 00:49:08,790 is called a transcendental function, 748 00:49:08,790 --> 00:49:11,350 because it's completely out of the realm of algebra. 749 00:49:11,350 --> 00:49:14,710 It's only in calculus that you come up 750 00:49:14,710 --> 00:49:16,190 with this kind of thing. 751 00:49:16,190 --> 00:49:20,030 So these kinds of functions will have access 752 00:49:20,030 --> 00:49:23,130 to a huge class of new functions here, all of which 753 00:49:23,130 --> 00:49:26,730 are important tools in science and engineering. 754 00:49:26,730 --> 00:49:29,537 So, see you next time.