1 00:00:00,000 --> 00:00:01,940 FEMALE SPEAKER: The following content is provided under a 2 00:00:01,940 --> 00:00:03,690 Creative Commons license. 3 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 4 00:00:06,630 --> 00:00:09,990 offer high-quality educational resources for free. 5 00:00:09,990 --> 00:00:12,830 To make a donation or to view additional materials from 6 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 8 00:00:18,010 --> 00:00:27,675 9 00:00:27,675 --> 00:00:28,340 PROFESSOR: Hi. 10 00:00:28,340 --> 00:00:31,370 Our subject matter today concerns that phase of 11 00:00:31,370 --> 00:00:36,080 calculus known as the 'indefinite integral', or the 12 00:00:36,080 --> 00:00:38,010 'antiderivative'. 13 00:00:38,010 --> 00:00:41,260 In terms of the concepts that we've talked about so far in 14 00:00:41,260 --> 00:00:45,930 our course, and keeping in mind that one concept with 15 00:00:45,930 --> 00:00:50,050 many applications is educationally more meaningful 16 00:00:50,050 --> 00:00:54,320 than many concepts each with one application, I prefer to 17 00:00:54,320 --> 00:00:59,810 call today's lesson the inverse derivative, or inverse 18 00:00:59,810 --> 00:01:01,340 differentiation. 19 00:01:01,340 --> 00:01:05,530 And again, bring into play a very well-known feature of our 20 00:01:05,530 --> 00:01:08,830 course, namely, our discussion of inverse functions. 21 00:01:08,830 --> 00:01:12,750 Now, you see, the only problem that comes up in this context 22 00:01:12,750 --> 00:01:16,240 is a rather simple one and that is that all the time that 23 00:01:16,240 --> 00:01:18,820 we've been taking a derivative, we may not have 24 00:01:18,820 --> 00:01:23,010 realized that we were using the idea of a function in a 25 00:01:23,010 --> 00:01:24,990 rather different way. 26 00:01:24,990 --> 00:01:28,730 Namely, noticed that by taking a derivative, we had a rule 27 00:01:28,730 --> 00:01:33,620 which told us how, given a particular function, to assign 28 00:01:33,620 --> 00:01:37,260 to it a new function called its 'derivative'. 29 00:01:37,260 --> 00:01:40,130 In other words, what we could have done is to have 30 00:01:40,130 --> 00:01:43,620 visualized a different function machine, which I will 31 00:01:43,620 --> 00:01:45,490 call the 'D-machine'. 32 00:01:45,490 --> 00:01:49,050 Hopefully, the 'D' will suggest differentiation, where 33 00:01:49,050 --> 00:01:51,130 the domain of my the machine-- 34 00:01:51,130 --> 00:01:54,410 in other words, the input of my 'D-machine' will be 35 00:01:54,410 --> 00:01:59,620 differentiable functions and the output, you see, the image 36 00:01:59,620 --> 00:02:03,190 of my 'D-machine' will be the derivative of 37 00:02:03,190 --> 00:02:05,290 that particular function. 38 00:02:05,290 --> 00:02:08,310 So in other words then, if we want to use our typical 39 00:02:08,310 --> 00:02:11,500 notation, what we're saying is that the 40 00:02:11,500 --> 00:02:13,080 'D-machine' does what? 41 00:02:13,080 --> 00:02:16,480 Given a differentiable function as its input, the 42 00:02:16,480 --> 00:02:19,830 output will be the derivative 'f prime'. 43 00:02:19,830 --> 00:02:22,710 Now again, we're used to talking about the derivative 44 00:02:22,710 --> 00:02:25,080 with respect to a given variable. 45 00:02:25,080 --> 00:02:28,950 Unless otherwise specified, the variable is 'x'. 46 00:02:28,950 --> 00:02:32,150 And so, perhaps with that in mind, maybe it would be more 47 00:02:32,150 --> 00:02:37,310 suggestive if I were to do something like this to 48 00:02:37,310 --> 00:02:41,540 indicate, you see, that the input is 'f of x' and the 49 00:02:41,540 --> 00:02:44,250 output is 'f prime of x'. 50 00:02:44,250 --> 00:02:48,230 Now, you may recall also that long before we started dealing 51 00:02:48,230 --> 00:02:51,190 with functions of real variables, we discussed 52 00:02:51,190 --> 00:02:53,840 functions in general in terms of circle 53 00:02:53,840 --> 00:02:55,470 diagrams and the like. 54 00:02:55,470 --> 00:02:59,020 In this respect, notice that we can think of our 55 00:02:59,020 --> 00:03:03,740 'D-machine' as operating on a particular domain, the domain 56 00:03:03,740 --> 00:03:04,650 being what? 57 00:03:04,650 --> 00:03:08,070 The set of all differentiable functions. 58 00:03:08,070 --> 00:03:12,600 And the range, or the image, will also be what? 59 00:03:12,600 --> 00:03:15,640 All functions which are derivatives of 60 00:03:15,640 --> 00:03:17,780 differentiable functions. 61 00:03:17,780 --> 00:03:20,420 Now, what kind of a function is 'D'? 62 00:03:20,420 --> 00:03:22,960 Don't lose track of the fact that things like 63 00:03:22,960 --> 00:03:27,800 one-to-oneness and ontoness are concepts that transcend 64 00:03:27,800 --> 00:03:29,570 dealing with numbers. 65 00:03:29,570 --> 00:03:31,440 They apply whenever we're dealing with 66 00:03:31,440 --> 00:03:32,850 any kind of a function. 67 00:03:32,850 --> 00:03:34,590 The idea is something like this. 68 00:03:34,590 --> 00:03:38,010 If 'x squared' were to go into the 'D-machine', the output 69 00:03:38,010 --> 00:03:40,020 would be '2x'. 70 00:03:40,020 --> 00:03:43,060 See? 'x squared' is mapped into '2x'. 71 00:03:43,060 --> 00:03:46,920 'x cubed' is mapped into '3 x squared'. 72 00:03:46,920 --> 00:03:49,230 See, the derivative of 'x cubed' with respect to 'x' is 73 00:03:49,230 --> 00:03:50,710 '3 x squared'. 74 00:03:50,710 --> 00:03:55,590 'x squared plus 7', also mapped into '2x'. 75 00:03:55,590 --> 00:03:58,350 In other words, without going any further, notice that 76 00:03:58,350 --> 00:04:06,350 whatever 'D' is, 'D' is not 1 to 1. 77 00:04:06,350 --> 00:04:09,950 You see, two different functions can have the same 78 00:04:09,950 --> 00:04:10,930 derivative. 79 00:04:10,930 --> 00:04:14,860 In fact, many different functions can have the same 80 00:04:14,860 --> 00:04:15,730 derivative. 81 00:04:15,730 --> 00:04:18,620 For example, once we know that the derivative of 'x squared' 82 00:04:18,620 --> 00:04:22,180 is '2x', we certainly know that the derivative of 'x 83 00:04:22,180 --> 00:04:26,080 squared' plus any constant is also '2x'. 84 00:04:26,080 --> 00:04:29,050 In other words then, we cannot make an inverse function 85 00:04:29,050 --> 00:04:31,400 machine as things stand here. 86 00:04:31,400 --> 00:04:32,530 And why is that? 87 00:04:32,530 --> 00:04:36,360 To have an inverse function, what you must be able to do is 88 00:04:36,360 --> 00:04:39,930 to reverse arrowheads and still have a function. 89 00:04:39,930 --> 00:04:44,420 Notice that, in a way, 'D inverse' would be a little bit 90 00:04:44,420 --> 00:04:46,320 of a tricky thing to talk about here. 91 00:04:46,320 --> 00:04:47,940 But let's pretend that we could anyway. 92 00:04:47,940 --> 00:04:50,960 In other words, what would go wrong if we tried to build a 93 00:04:50,960 --> 00:04:53,230 'D' inverse machine? 94 00:04:53,230 --> 00:04:56,340 Well, in terms of the example that we're talking about, if 95 00:04:56,340 --> 00:05:01,780 the input to the D inverse machine happened to be '2x', 96 00:05:01,780 --> 00:05:04,390 then there would be infinitely many different outputs that 97 00:05:04,390 --> 00:05:06,050 we're sure of. 98 00:05:06,050 --> 00:05:06,880 Namely what? 99 00:05:06,880 --> 00:05:09,830 Every function of the form 'x squared plus c' 100 00:05:09,830 --> 00:05:11,410 where 'c' is a constant. 101 00:05:11,410 --> 00:05:14,450 And what do I mean, we can be sure of that many? 102 00:05:14,450 --> 00:05:17,880 You see, all I know going back to this diagram here is that 103 00:05:17,880 --> 00:05:21,980 every function of the form 'x squared plus c' will map into 104 00:05:21,980 --> 00:05:24,550 '2x' under the derivative function. 105 00:05:24,550 --> 00:05:28,170 The other question that comes up is, how do you know that 106 00:05:28,170 --> 00:05:32,690 you can find a function whose derivative is '2x' that comes 107 00:05:32,690 --> 00:05:35,100 from a function which doesn't have the form 'x 108 00:05:35,100 --> 00:05:36,210 squared plus c'? 109 00:05:36,210 --> 00:05:39,480 How do you know there isn't some other function 'f of x' 110 00:05:39,480 --> 00:05:42,580 which has the property that 'f of x' maps into '2x'? 111 00:05:42,580 --> 00:05:45,640 In other words, the derivative of 'f of x' is '2x' even 112 00:05:45,640 --> 00:05:50,040 though 'f of x' does not have the form 'x squared plus c'. 113 00:05:50,040 --> 00:05:53,810 And this is exactly where the mean value theorem comes in to 114 00:05:53,810 --> 00:05:55,330 help us over here. 115 00:05:55,330 --> 00:05:59,040 You see, what we're saying here is, suppose that 'D' of 116 00:05:59,040 --> 00:06:00,590 'f of x' is '2x'. 117 00:06:00,590 --> 00:06:03,300 All we know is that when you run 'f of x' through the 118 00:06:03,300 --> 00:06:06,970 'D-machine', you wind up with '2x'. 119 00:06:06,970 --> 00:06:08,050 What does that mean? 120 00:06:08,050 --> 00:06:11,010 'f prime of x' is '2x'. 121 00:06:11,010 --> 00:06:14,700 Well, we know the derivative of 'x squared' is also '2x'. 122 00:06:14,700 --> 00:06:19,360 Therefore, whatever 'f of x' is by the corollary to the 123 00:06:19,360 --> 00:06:22,690 mean value theorem that we've studied before, since 'f of x' 124 00:06:22,690 --> 00:06:27,200 and 'x squared' have identical derivatives, namely, they're 125 00:06:27,200 --> 00:06:31,930 both '2x', it means that they must differ by a constant. 126 00:06:31,930 --> 00:06:35,580 In other words, 'f of x' minus 'x squared' is a constant. 127 00:06:35,580 --> 00:06:38,700 But to say that 'f of x' minus 'x squared' is a constant is 128 00:06:38,700 --> 00:06:43,000 the same as saying that 'f of x' belongs to the family 'x 129 00:06:43,000 --> 00:06:44,580 squared plus c'. 130 00:06:44,580 --> 00:06:46,920 In other words, 'f of x' equals 'x 131 00:06:46,920 --> 00:06:49,420 squared' plus a constant. 132 00:06:49,420 --> 00:06:52,280 What the mean value theorem tells us is not only does 133 00:06:52,280 --> 00:06:55,430 every function of the form 'x squared plus c' have its 134 00:06:55,430 --> 00:06:58,120 derivative equal to '2x' but that every function whose 135 00:06:58,120 --> 00:07:01,980 derivative is '2x' has the form 'x squared plus c'. 136 00:07:01,980 --> 00:07:04,560 137 00:07:04,560 --> 00:07:06,960 To generalize this, consider the following. 138 00:07:06,960 --> 00:07:10,570 Suppose we're given 'f of x', OK? 139 00:07:10,570 --> 00:07:16,040 And we now say, look, every time you differentiate a 140 00:07:16,040 --> 00:07:19,650 function, if you add on a constant, you don't change 141 00:07:19,650 --> 00:07:21,200 anything, meaning what? 142 00:07:21,200 --> 00:07:22,960 The derivative of a constant is 0. 143 00:07:22,960 --> 00:07:25,480 In other words, what we're saying is, if the derivative 144 00:07:25,480 --> 00:07:30,100 of 'f of x' is 'f prime of x', any function of the form 'f of 145 00:07:30,100 --> 00:07:33,720 x' plus 'c' will have the same derivative. 146 00:07:33,720 --> 00:07:37,190 And not only that, the only function whose derivative is 147 00:07:37,190 --> 00:07:40,560 'f prime of x' must be one of these. 148 00:07:40,560 --> 00:07:43,070 That's what this 'e sub f' stands for. 149 00:07:43,070 --> 00:07:45,730 You see, what I'm really saying here is that certainly 150 00:07:45,730 --> 00:07:49,730 when you change the constant, you change the function. 151 00:07:49,730 --> 00:07:52,220 The point is, with respect to the thing called the 152 00:07:52,220 --> 00:07:55,670 derivative, you cannot tell the difference between two 153 00:07:55,670 --> 00:07:58,830 functions just by looking at their derivatives if they 154 00:07:58,830 --> 00:08:00,290 differ by a constant. 155 00:08:00,290 --> 00:08:03,600 In other words, with respect to differentiation, two 156 00:08:03,600 --> 00:08:06,750 functions which differ by a constant are equivalent and 157 00:08:06,750 --> 00:08:10,260 that's why I invented this notation, 'e of f'. 158 00:08:10,260 --> 00:08:14,020 All I'm saying is that if we visualize now that the input 159 00:08:14,020 --> 00:08:17,850 of the 'D-machine' is not individual functions but whole 160 00:08:17,850 --> 00:08:19,890 classes of functions-- 161 00:08:19,890 --> 00:08:22,780 in other words, such that they differ only by a constant, 162 00:08:22,780 --> 00:08:25,780 then you see there is a one-to-one correspondence 163 00:08:25,780 --> 00:08:28,840 between the input and the output. 164 00:08:28,840 --> 00:08:30,610 Now that's a subtle point. 165 00:08:30,610 --> 00:08:33,049 It's a point which I'm sure most of you will grasp. 166 00:08:33,049 --> 00:08:38,240 But more importantly, the important thing is simply that 167 00:08:38,240 --> 00:08:42,370 once we've seen one function with a given derivative, in a 168 00:08:42,370 --> 00:08:44,960 manner of speaking, we've seen them all. 169 00:08:44,960 --> 00:08:47,550 In other words, we only have enough leeway as to fool 170 00:08:47,550 --> 00:08:48,825 around with an arbitrary constant. 171 00:08:48,825 --> 00:08:52,410 172 00:08:52,410 --> 00:08:54,790 So let's generalize again. 173 00:08:54,790 --> 00:08:59,250 What do we mean by 'D inverse' of 'f of x'? 174 00:08:59,250 --> 00:09:00,550 We mean what? 175 00:09:00,550 --> 00:09:04,830 The set of all function 'g of x' whose derivative with 176 00:09:04,830 --> 00:09:07,955 respect to 'x' is the given 'f of x'. 177 00:09:07,955 --> 00:09:10,780 See, that's exactly what the inverse function means. 178 00:09:10,780 --> 00:09:14,530 And by the way, notice that this tells me my set 179 00:09:14,530 --> 00:09:15,430 implicitly. 180 00:09:15,430 --> 00:09:17,610 Let me put that in parentheses over here. 181 00:09:17,610 --> 00:09:22,520 Namely, suppose somebody says, does 'g of x' belong to this 182 00:09:22,520 --> 00:09:24,690 particular set called 'D inverse'? 183 00:09:24,690 --> 00:09:28,880 All I have to do is differentiate the given 'g' 184 00:09:28,880 --> 00:09:30,530 and see if I get 'f'. 185 00:09:30,530 --> 00:09:33,560 If I do, 'G' belongs to the set. 186 00:09:33,560 --> 00:09:36,060 If I don't, it doesn't belong to the set. 187 00:09:36,060 --> 00:09:39,050 But as we've so often stressed about inverse functions, 188 00:09:39,050 --> 00:09:41,520 notice that the test to see whether something belongs to 189 00:09:41,520 --> 00:09:45,710 'D inverse', it's sufficient to know how to differentiate. 190 00:09:45,710 --> 00:09:53,320 By the way, to summarize what we did before, to list this 191 00:09:53,320 --> 00:09:57,310 thing explicitly, notice that what we're saying is, to find 192 00:09:57,310 --> 00:10:01,400 the set of all functions whose derivative is little 'f of x', 193 00:10:01,400 --> 00:10:03,420 all we have to do is what? 194 00:10:03,420 --> 00:10:07,030 Find one function, capital 'F', whose derivative is 195 00:10:07,030 --> 00:10:08,280 little 'f'. 196 00:10:08,280 --> 00:10:11,380 And then our set, explicitly, is what? 197 00:10:11,380 --> 00:10:14,990 The set of all functions of the form capital 'F of x' plus 198 00:10:14,990 --> 00:10:18,540 'c' where 'c' is an arbitrary constant. 199 00:10:18,540 --> 00:10:20,580 Now, I'm sure this concept is not 200 00:10:20,580 --> 00:10:22,230 difficult for you to grasp. 201 00:10:22,230 --> 00:10:26,140 For those of you who have been through calculus before, as 202 00:10:26,140 --> 00:10:29,170 our course is intended to be for this type of person too, 203 00:10:29,170 --> 00:10:32,820 you see, you may not be familiar with the notation, 204 00:10:32,820 --> 00:10:35,440 the 'D inverse' notation, which I want to stress here. 205 00:10:35,440 --> 00:10:39,380 But the concept, I hope, is clear in its own right. 206 00:10:39,380 --> 00:10:44,030 Now of course, you see, the problem that comes up is that 207 00:10:44,030 --> 00:10:47,840 it's much easier to differentiate a function than 208 00:10:47,840 --> 00:10:52,020 it is to be given a function and then have to try to find 209 00:10:52,020 --> 00:10:54,690 what you have to differentiate to get it. 210 00:10:54,690 --> 00:10:57,310 Without meaning it as facetiously as it may sound, I 211 00:10:57,310 --> 00:11:01,120 use this in on many occasions, I prefer to say is much 212 00:11:01,120 --> 00:11:04,910 easier, you see, to scramble an egg than to unscramble one. 213 00:11:04,910 --> 00:11:07,950 Told what to differentiate, that's easy enough to do. 214 00:11:07,950 --> 00:11:10,730 Given the derivative, that may not be so easy. 215 00:11:10,730 --> 00:11:13,780 Let's look in terms of an example. 216 00:11:13,780 --> 00:11:18,200 But this is, again, a very nice teachers trick. 217 00:11:18,200 --> 00:11:20,250 To get the right answer, you start with the answer and then 218 00:11:20,250 --> 00:11:21,120 work to the problem. 219 00:11:21,120 --> 00:11:25,200 I'll start with the function 'h of x' equals 'x' times the 220 00:11:25,200 --> 00:11:26,970 'square root of 'x squared plus 1''. 221 00:11:26,970 --> 00:11:31,300 Or written more conveniently in exponential notation, 'x' 222 00:11:31,300 --> 00:11:34,050 times ''x squared plus 1' to the 1/2'. 223 00:11:34,050 --> 00:11:35,670 Let me differentiate that. 224 00:11:35,670 --> 00:11:36,990 Remember, this is a product. 225 00:11:36,990 --> 00:11:39,430 The derivative of a product is what? 226 00:11:39,430 --> 00:11:42,690 It's the first factor which is 'x' times the derivative of 227 00:11:42,690 --> 00:11:43,560 the second. 228 00:11:43,560 --> 00:11:44,270 That means what? 229 00:11:44,270 --> 00:11:48,490 I bring the 1/2 down to a power 1 less, times the 230 00:11:48,490 --> 00:11:51,620 derivative of what's inside with respect to 'x'. 231 00:11:51,620 --> 00:11:53,110 See, the some old chain rule again. 232 00:11:53,110 --> 00:11:54,410 That's '2x'. 233 00:11:54,410 --> 00:11:55,160 Then what? 234 00:11:55,160 --> 00:11:59,610 Plus the second factor, 'x squared plus 1' to the 1/2 235 00:11:59,610 --> 00:12:01,770 times the derivative of the first with respect to 'x', 236 00:12:01,770 --> 00:12:03,500 which is 1. 237 00:12:03,500 --> 00:12:08,530 At any rate, simplifying, bringing the minus 1/2 power 238 00:12:08,530 --> 00:12:12,360 into the denominator, then putting everything over a 239 00:12:12,360 --> 00:12:16,580 common denominator, I wind up with the fact that 'h prime of 240 00:12:16,580 --> 00:12:21,340 x' is '2 x squared plus 1' over the 'square root of 'x 241 00:12:21,340 --> 00:12:22,890 squared plus 1''. 242 00:12:22,890 --> 00:12:24,870 I hope I haven't made a careless error here. 243 00:12:24,870 --> 00:12:27,675 But again, one of the beauties of the new mathematics is that 244 00:12:27,675 --> 00:12:30,240 it's the method that's important, OK? 245 00:12:30,240 --> 00:12:32,250 Now, at any rate, what do I know? 246 00:12:32,250 --> 00:12:35,640 Starting with 'h of x' equaling 'x' times the 'square 247 00:12:35,640 --> 00:12:39,330 root of 'x squared plus 1'', I now know that its derivative 248 00:12:39,330 --> 00:12:43,160 is '2 x squared plus 1' over the 'square root of 'x 249 00:12:43,160 --> 00:12:44,480 squared plus 1''. 250 00:12:44,480 --> 00:12:49,190 Let's write that in terms of our 'D inverse' notation. 251 00:12:49,190 --> 00:12:53,960 Namely, what we're saying is that 'x' times the 'square 252 00:12:53,960 --> 00:12:57,280 root of 'x squared plus 1'' has the property that its 253 00:12:57,280 --> 00:13:00,620 derivative is '2 x squared plus 1' over the 'square root 254 00:13:00,620 --> 00:13:02,180 of 'x squared plus 1''. 255 00:13:02,180 --> 00:13:05,970 Consequently, every function which has the property that 256 00:13:05,970 --> 00:13:09,400 its derivative is '2 x squared plus 1' over the 'square root 257 00:13:09,400 --> 00:13:13,520 of 'x squared plus 1'' must come from this family. 258 00:13:13,520 --> 00:13:14,440 You see? 259 00:13:14,440 --> 00:13:16,820 Everything of this form we'll have its 260 00:13:16,820 --> 00:13:18,000 derivative equal to this. 261 00:13:18,000 --> 00:13:21,110 And secondly, any other function cannot have its 262 00:13:21,110 --> 00:13:23,530 derivative equal to this by our corollary to the mean 263 00:13:23,530 --> 00:13:24,670 value theorem. 264 00:13:24,670 --> 00:13:27,080 Now you see, what I'm saying is-- 265 00:13:27,080 --> 00:13:30,270 and here's the beauty of what we mean by inverse operations 266 00:13:30,270 --> 00:13:31,330 and the like. 267 00:13:31,330 --> 00:13:34,310 It's conceivable that you might not have been 268 00:13:34,310 --> 00:13:37,380 sophisticated enough at this stage in the game to have been 269 00:13:37,380 --> 00:13:40,650 able to deduce this had we been given this. 270 00:13:40,650 --> 00:13:44,330 Notice that my cute trick was I started with this, found out 271 00:13:44,330 --> 00:13:46,160 what the derivative was, and then just 272 00:13:46,160 --> 00:13:47,800 inverted the emphasis. 273 00:13:47,800 --> 00:13:49,650 A change in emphasis again. 274 00:13:49,650 --> 00:13:52,500 However, notice the following. 275 00:13:52,500 --> 00:13:54,810 Suppose you weren't able to find this. 276 00:13:54,810 --> 00:13:58,810 And somebody said to you, I wonder if 'x' times the 277 00:13:58,810 --> 00:14:01,900 'square root of 'x squared plus 1'' is a function whose 278 00:14:01,900 --> 00:14:03,990 derivative is equal to this. 279 00:14:03,990 --> 00:14:06,600 And all I'm saying, without going through the work again 280 00:14:06,600 --> 00:14:09,670 because I've already done that, is to simply observe 281 00:14:09,670 --> 00:14:13,520 that even if you did not know this explicit representation, 282 00:14:13,520 --> 00:14:18,400 by definition, 'D inverse' of '2 x squared plus 1' over the 283 00:14:18,400 --> 00:14:21,310 'square root of 'x squared plus 1'' is simply what? 284 00:14:21,310 --> 00:14:25,530 The set of all functions 'g of x' such that 'g prime of x' is 285 00:14:25,530 --> 00:14:28,510 equal to '2 x squared plus 1' over the 'square root of 'x 286 00:14:28,510 --> 00:14:29,620 squared plus 1''. 287 00:14:29,620 --> 00:14:32,090 In other words, given any function at all, I could 288 00:14:32,090 --> 00:14:33,210 differentiate it. 289 00:14:33,210 --> 00:14:35,180 If the derivative came out to be this, 290 00:14:35,180 --> 00:14:36,700 then I have a solution. 291 00:14:36,700 --> 00:14:38,350 It belongs to the solution set. 292 00:14:38,350 --> 00:14:39,720 Otherwise, it doesn't. 293 00:14:39,720 --> 00:14:44,380 But again, notice that's to solve any 'D inverse' problem, 294 00:14:44,380 --> 00:14:46,270 it's sufficient to understand a 295 00:14:46,270 --> 00:14:49,170 corresponding derivative property. 296 00:14:49,170 --> 00:14:53,060 In fact, maybe now is a good time to show how we get 297 00:14:53,060 --> 00:14:57,530 certain recipes for 'D inverse' type things. 298 00:14:57,530 --> 00:15:00,490 Let me just write down a typical one. 299 00:15:00,490 --> 00:15:05,060 You see, 'D inverse of 'x to the n'' is 'x to the 'n + 1'' 300 00:15:05,060 --> 00:15:07,770 over 'n + 1' plus a constant. 301 00:15:07,770 --> 00:15:10,920 And of course, observe that as soon as you see something like 302 00:15:10,920 --> 00:15:14,120 this, you have to beware of 'n' equals negative 1. 303 00:15:14,120 --> 00:15:16,950 Otherwise we have a 0 denominator. 304 00:15:16,950 --> 00:15:19,700 Now, a person says, this doesn't look familiar to me. 305 00:15:19,700 --> 00:15:23,870 Again, keep in mind what D inverse means. 306 00:15:23,870 --> 00:15:28,390 Essentially, to say this is just a switch in emphasis from 307 00:15:28,390 --> 00:15:30,020 saying what? 308 00:15:30,020 --> 00:15:33,990 That if you run the family of functions 'x to the 'n + 1'' 309 00:15:33,990 --> 00:15:38,670 over 'n + 1' plus a constant through your 'D-machine', you 310 00:15:38,670 --> 00:15:39,770 get 'x to the n'. 311 00:15:39,770 --> 00:15:41,930 Or more familiarly, what? 312 00:15:41,930 --> 00:15:46,490 The derivative of any member in this family is 'x to the n' 313 00:15:46,490 --> 00:15:49,750 provided that 'n' is not equal to minus 1. 314 00:15:49,750 --> 00:15:52,760 Now again, this may look a little bit abstract to you. 315 00:15:52,760 --> 00:15:56,590 So to avoid this problem, let's just do a concrete 316 00:15:56,590 --> 00:15:57,670 illustration. 317 00:15:57,670 --> 00:15:59,740 Let's pick a particular value of 'n' and 318 00:15:59,740 --> 00:16:01,490 work with this thing. 319 00:16:01,490 --> 00:16:04,430 Let's suppose we're told to determine 'D inverse' 320 00:16:04,430 --> 00:16:05,660 of 'x to the 7th'. 321 00:16:05,660 --> 00:16:07,050 What does that mean? 322 00:16:07,050 --> 00:16:11,420 What it really means is, let's find a function whose 323 00:16:11,420 --> 00:16:13,660 derivative is 'x to the 7th'. 324 00:16:13,660 --> 00:16:16,150 And why do I say let's find a function? 325 00:16:16,150 --> 00:16:19,550 Because once I find a function, all I have to do is 326 00:16:19,550 --> 00:16:22,840 tack on arbitrary constants and the family that I get that 327 00:16:22,840 --> 00:16:26,460 way is the unique family of functions which have this 328 00:16:26,460 --> 00:16:27,870 particular derivative. 329 00:16:27,870 --> 00:16:29,730 So I play the detective game. 330 00:16:29,730 --> 00:16:32,680 I know from differential calculus that if I 331 00:16:32,680 --> 00:16:36,400 differentiate 'x' to the 8th power, I'll wind up with the 332 00:16:36,400 --> 00:16:37,620 exponent 7, at least. 333 00:16:37,620 --> 00:16:40,950 In other words, the derivative of 'x' to the 8th power. 334 00:16:40,950 --> 00:16:45,250 'D of 'x to the 8th'' is '8 'x to the 7th''. 335 00:16:45,250 --> 00:16:47,340 Well, what answer did I want to get? 336 00:16:47,340 --> 00:16:51,010 I wanted to get 'x to the 7th', not '8 'x to the 7th''. 337 00:16:51,010 --> 00:16:53,230 So I fudge this thing a little bit. 338 00:16:53,230 --> 00:16:56,220 I say, evidently what I should have done was to have started 339 00:16:56,220 --> 00:16:57,890 with 1/8 as much. 340 00:16:57,890 --> 00:17:01,290 In other words, multiplying equals by equals, I multiply 341 00:17:01,290 --> 00:17:05,950 both sides of this equation by 1/8 and I wind up with '1/8 D 342 00:17:05,950 --> 00:17:08,730 'x to the 8th'' equals 'x to the 7th'. 343 00:17:08,730 --> 00:17:11,240 And now comes a very crucial step. 344 00:17:11,240 --> 00:17:14,089 And let me write that down because I think it's something 345 00:17:14,089 --> 00:17:17,329 that we should pay very close attention to. 346 00:17:17,329 --> 00:17:20,480 And that is that the derivative has the property 347 00:17:20,480 --> 00:17:23,599 that if you want to differentiate a constant times 348 00:17:23,599 --> 00:17:28,420 a function, you can take the constant out and differentiate 349 00:17:28,420 --> 00:17:29,730 just the function. 350 00:17:29,730 --> 00:17:32,210 This is a very crucial point because you see-- 351 00:17:32,210 --> 00:17:35,180 and by the way, notice that in general, not all functions 352 00:17:35,180 --> 00:17:37,690 have this property. 353 00:17:37,690 --> 00:17:42,100 For example, if you're squaring something, if you 354 00:17:42,100 --> 00:17:46,810 double the number that's being squared, the output is 4 times 355 00:17:46,810 --> 00:17:49,290 as much because twice something squared 356 00:17:49,290 --> 00:17:50,320 is 4 times as much. 357 00:17:50,320 --> 00:17:54,050 In other words, in general, you do not say that if you 358 00:17:54,050 --> 00:17:56,840 double the input, you're going to double the output. 359 00:17:56,840 --> 00:17:58,620 Not every function has that property. 360 00:17:58,620 --> 00:18:01,420 But the function called 'D', the derivative, does have this 361 00:18:01,420 --> 00:18:02,620 particular property. 362 00:18:02,620 --> 00:18:04,600 And you see, with that in mind, I can 363 00:18:04,600 --> 00:18:06,460 bring this 1/8 inside. 364 00:18:06,460 --> 00:18:12,330 This is the key step, that 1/8 times the derivative of 'x to 365 00:18:12,330 --> 00:18:16,190 the 8th' is a derivative of ''1/8' x to the 8th'. 366 00:18:16,190 --> 00:18:17,850 That's this key step over here. 367 00:18:17,850 --> 00:18:20,900 And now, you see, putting all this together, I find what? 368 00:18:20,900 --> 00:18:23,940 That the derivative of ''1/8' x to the 8th' 369 00:18:23,940 --> 00:18:25,840 is 'x to the 7th'. 370 00:18:25,840 --> 00:18:28,950 And therefore, since I found one function whose derivative 371 00:18:28,950 --> 00:18:32,560 is 'x to the 7th', I have, in a sense, found them all, 372 00:18:32,560 --> 00:18:35,830 namely, ''1/8' x to the 8th' plus 'c'. 373 00:18:35,830 --> 00:18:39,180 In other words, that's what I call the equivalent class of 374 00:18:39,180 --> 00:18:40,270 ''1/8' x to the 8th'. 375 00:18:40,270 --> 00:18:43,030 All the functions that differ from ''1/8' x to 376 00:18:43,030 --> 00:18:45,350 the 8th' by a constant. 377 00:18:45,350 --> 00:18:48,720 And again, to emphasize this very important point, let me 378 00:18:48,720 --> 00:18:54,010 again mention, beware of non-constant factors. 379 00:18:54,010 --> 00:18:57,180 Let me give you a for instance. 380 00:18:57,180 --> 00:19:00,800 Let's suppose I take almost the same problem. 381 00:19:00,800 --> 00:19:02,650 And that almost as a big almost. 382 00:19:02,650 --> 00:19:05,910 Let's suppose I say, let me find all functions whose 383 00:19:05,910 --> 00:19:09,320 derivative, say, is 'x squared plus 1' to the 7th power. 384 00:19:09,320 --> 00:19:12,310 In other words, still something to the 7th power. 385 00:19:12,310 --> 00:19:14,930 So I argue something like, well, since whenever I 386 00:19:14,930 --> 00:19:18,460 differentiate I reduce the exponent by 1, to wind up with 387 00:19:18,460 --> 00:19:20,330 a 7th power, maybe I should have 388 00:19:20,330 --> 00:19:22,490 started with an 8th power. 389 00:19:22,490 --> 00:19:24,850 So I say, OK, that's what I'll do. 390 00:19:24,850 --> 00:19:26,290 I'll start with an 8th power. 391 00:19:26,290 --> 00:19:28,910 So I say, OK, what is the derivative of 'x squared plus 392 00:19:28,910 --> 00:19:30,490 1' to the 8th power? 393 00:19:30,490 --> 00:19:33,560 Now notice I know how to differentiate, hopefully. 394 00:19:33,560 --> 00:19:36,040 And again, let me make this point very strongly. 395 00:19:36,040 --> 00:19:39,280 There is no sense studying inverse functions if we don't 396 00:19:39,280 --> 00:19:41,180 know the original function itself. 397 00:19:41,180 --> 00:19:43,710 Because the whole purpose of the inverse function, or the 398 00:19:43,710 --> 00:19:47,410 whole strategy behind it, is to reduce it to the original 399 00:19:47,410 --> 00:19:50,020 function, namely, to switch the emphasis. 400 00:19:50,020 --> 00:19:52,510 So at any rate, I differentiate 'x squared plus 401 00:19:52,510 --> 00:19:53,760 1' to the 8th power. 402 00:19:53,760 --> 00:19:56,970 I get '8 'x squared plus 1' to the 7th'. 403 00:19:56,970 --> 00:20:01,410 But now, by the chain rule, I must remember that this part 404 00:20:01,410 --> 00:20:04,330 was only the derivative with respect to 'x squared plus 1'. 405 00:20:04,330 --> 00:20:05,740 The correction factor is what? 406 00:20:05,740 --> 00:20:07,960 The derivative of what's inside with respect to 'x'. 407 00:20:07,960 --> 00:20:09,000 That's '2x'. 408 00:20:09,000 --> 00:20:12,180 And so I wind up with that if I differentiate ''x squared 409 00:20:12,180 --> 00:20:15,960 plus 1' to the 8th', I get '16x' times ''x squared plus 410 00:20:15,960 --> 00:20:17,100 1' to the 7th'. 411 00:20:17,100 --> 00:20:18,590 Now, how much did I want to get? 412 00:20:18,590 --> 00:20:21,980 I wanted to get just ''x squared plus 1' to the 7th'. 413 00:20:21,980 --> 00:20:24,990 That put me off by a factor of '16x'. 414 00:20:24,990 --> 00:20:28,070 Now I say, OK, I'll fix that up. 415 00:20:28,070 --> 00:20:32,200 Namely, I'll divide both sides by '16x', assuming, of course, 416 00:20:32,200 --> 00:20:33,430 that 'x' is not 0. 417 00:20:33,430 --> 00:20:36,450 And this, by the way, is perfectly valid. 418 00:20:36,450 --> 00:20:41,360 I can now go from here to here and say, look, '1 over 16x' 419 00:20:41,360 --> 00:20:44,720 times the derivative of ''x squared plus 1' to the 8th' 420 00:20:44,720 --> 00:20:48,230 with respect to 'x' is ''x squared plus 1' to the 7th'. 421 00:20:48,230 --> 00:20:52,470 However, notice that I cannot take this factor and bring it 422 00:20:52,470 --> 00:20:53,560 inside here. 423 00:20:53,560 --> 00:20:57,220 And again, as I so often have said, also, of course I can 424 00:20:57,220 --> 00:20:58,100 bring it inside here. 425 00:20:58,100 --> 00:20:58,860 I just did. 426 00:20:58,860 --> 00:21:02,260 What I mean is, I don't get the right answer. 427 00:21:02,260 --> 00:21:04,650 And what's the best proof that I don't get the right answer? 428 00:21:04,650 --> 00:21:06,600 Very, very simple. 429 00:21:06,600 --> 00:21:13,400 Take this function, differentiate it, and see if 430 00:21:13,400 --> 00:21:15,980 you get ''x squared plus 1' to the 7th'. 431 00:21:15,980 --> 00:21:18,990 You won't, unless you differentiate incorrectly. 432 00:21:18,990 --> 00:21:21,520 Don't be like the person who just differentiates, brings 433 00:21:21,520 --> 00:21:26,050 the 8 down, replaces this by 1 less, multiplies by a 434 00:21:26,050 --> 00:21:29,180 derivative of what's inside, and cancels everything out. 435 00:21:29,180 --> 00:21:31,930 Notice that the expression inside the brackets that I've 436 00:21:31,930 --> 00:21:34,490 just circled is a quotient. 437 00:21:34,490 --> 00:21:37,060 And the derivative of a quotient is obtained in a very 438 00:21:37,060 --> 00:21:37,830 special way. 439 00:21:37,830 --> 00:21:40,750 The denominator times the derivative of the numerator 440 00:21:40,750 --> 00:21:43,700 minus the numerator times the derivative of the denominator 441 00:21:43,700 --> 00:21:45,480 over the square of the denominator. 442 00:21:45,480 --> 00:21:48,970 And all I'm saying is, you won't get ''x squared plus 1' 443 00:21:48,970 --> 00:21:50,750 to the 7th' power if you do that. 444 00:21:50,750 --> 00:21:54,820 Again, notice that you do not have to know the right answer 445 00:21:54,820 --> 00:21:59,150 in order to see what answer is wrong, OK? 446 00:21:59,150 --> 00:22:00,840 So this would be a wrong answer. 447 00:22:00,840 --> 00:22:04,350 By the way, this would also be a wrong answer because we've 448 00:22:04,350 --> 00:22:08,010 already seen that the derivative of ''1/8' x squared 449 00:22:08,010 --> 00:22:10,130 plus 1' to the 8th power is what? 450 00:22:10,130 --> 00:22:13,530 You bring the 8 down, which kills off the 1/8. 451 00:22:13,530 --> 00:22:16,620 You replace this to a power of one less, which gives you ''x 452 00:22:16,620 --> 00:22:18,390 squared plus 1' to the 7th'. 453 00:22:18,390 --> 00:22:21,960 But the correction factor here is you must also multiply by a 454 00:22:21,960 --> 00:22:24,220 derivative of what's inside with respect to 'x'. 455 00:22:24,220 --> 00:22:25,780 And that's to '2x'. 456 00:22:25,780 --> 00:22:27,350 In other words, you do not get ''x squared 457 00:22:27,350 --> 00:22:29,780 plus 1' to the 7th'. 458 00:22:29,780 --> 00:22:32,970 What you do get is what? ''x squared plus 1' to the 7th' 459 00:22:32,970 --> 00:22:34,470 times '2x'. 460 00:22:34,470 --> 00:22:37,540 And the question that may now come up is, how come this 461 00:22:37,540 --> 00:22:41,030 worked when you were raising 'x' to the 8th power but it 462 00:22:41,030 --> 00:22:43,600 didn't work when you were raising 'x squared plus 1' to 463 00:22:43,600 --> 00:22:46,050 the 8th power? 464 00:22:46,050 --> 00:22:49,580 Again, as always in these cases, the answer is 465 00:22:49,580 --> 00:22:53,120 immediately available in terms of derivatives. 466 00:22:53,120 --> 00:22:56,580 We can talk, in fact, about the inverse chain rule. 467 00:22:56,580 --> 00:22:59,470 That when we really talked about 'D'-- 468 00:22:59,470 --> 00:23:00,880 remember, we mentioned this at the very beginning of the 469 00:23:00,880 --> 00:23:04,900 lecture, that the variable inside the parentheses was the 470 00:23:04,900 --> 00:23:08,270 one with respect to which you were differentiating. 471 00:23:08,270 --> 00:23:16,480 In other words, what we saw was that if you wanted to get 472 00:23:16,480 --> 00:23:20,110 something to the 7th power, what you had to differentiate 473 00:23:20,110 --> 00:23:24,690 was 1/8 that same something to the 8th power if you were 474 00:23:24,690 --> 00:23:27,670 differentiating with respect to that same variable. 475 00:23:27,670 --> 00:23:31,110 In other words, what would have been ''x squared plus 1' 476 00:23:31,110 --> 00:23:33,520 to the 7th' would have been what? 477 00:23:33,520 --> 00:23:38,060 If you would differentiating not with respect to 'x' but 478 00:23:38,060 --> 00:23:42,070 with respect to 'x squared plus 1'. 479 00:23:42,070 --> 00:23:45,010 See, what we really wanted when we wrote this down was 480 00:23:45,010 --> 00:23:47,220 the derivative with respect to 'x'. 481 00:23:47,220 --> 00:23:50,920 And even though this notation may look a little bit strange 482 00:23:50,920 --> 00:23:55,130 to you, observe that once you get used to the notation, this 483 00:23:55,130 --> 00:23:58,160 is just another way of talking about the chain rule. 484 00:23:58,160 --> 00:24:01,830 Namely, to find the derivative of this with respect to 'x', 485 00:24:01,830 --> 00:24:05,840 you first take the derivative with respect to 'x squared 486 00:24:05,840 --> 00:24:10,180 plus 1' and then multiply that by the derivative of 'x 487 00:24:10,180 --> 00:24:13,150 squared plus 1' with respect to 'x'. 488 00:24:13,150 --> 00:24:16,140 And if we do that, you see, we get the answer that we've 489 00:24:16,140 --> 00:24:17,670 talked about before. 490 00:24:17,670 --> 00:24:21,710 In other words, what we could say is that the function that 491 00:24:21,710 --> 00:24:25,150 you have to differentiate to get ''x squared plus 1' to the 492 00:24:25,150 --> 00:24:31,060 7th' times '2x' is ''1/8' x squared plus 1' to the 8th' 493 00:24:31,060 --> 00:24:32,250 plus a constant. 494 00:24:32,250 --> 00:24:33,540 And how do I know that? 495 00:24:33,540 --> 00:24:36,830 Well, the way I know that is simply what? 496 00:24:36,830 --> 00:24:40,740 In terms of inverse functions, I started with ''1/8' x 497 00:24:40,740 --> 00:24:44,540 squared plus 1' to the 8th', differentiated it and found 498 00:24:44,540 --> 00:24:48,190 out I got ''x squared plus 1' to the 7th' times '2x'. 499 00:24:48,190 --> 00:24:51,050 And so this became the recipe. 500 00:24:51,050 --> 00:24:54,170 And again, a rather interesting aside, if you look 501 00:24:54,170 --> 00:25:04,620 at this, and look at this, it would appear at first glance 502 00:25:04,620 --> 00:25:07,810 that the top one should be a more difficult problem than 503 00:25:07,810 --> 00:25:08,810 the bottom one. 504 00:25:08,810 --> 00:25:13,500 The reason being that the input seems more simple in the 505 00:25:13,500 --> 00:25:14,200 bottom one. 506 00:25:14,200 --> 00:25:17,900 Yet, the interesting point is that the '2x', which seems to 507 00:25:17,900 --> 00:25:20,770 make this thing more complicated, is precisely the 508 00:25:20,770 --> 00:25:24,510 factor you need by the chain rule to make this thing work. 509 00:25:24,510 --> 00:25:27,400 Because when you differentiate 'x squared plus 1' to the 8th 510 00:25:27,400 --> 00:25:30,120 power, you're going to get a factor of '2x' 511 00:25:30,120 --> 00:25:31,360 by the chain rule. 512 00:25:31,360 --> 00:25:34,570 Now again, the main aim of the lectures is not to take the 513 00:25:34,570 --> 00:25:37,460 place of the computational drill supplied in our 514 00:25:37,460 --> 00:25:39,670 exercises and in the text but to give 515 00:25:39,670 --> 00:25:40,810 you sort of an insight. 516 00:25:40,810 --> 00:25:44,170 And they'll be plenty of drill on the mechanics of this in 517 00:25:44,170 --> 00:25:46,300 our exercises on this section. 518 00:25:46,300 --> 00:25:49,090 Let me just at least continue on with the 519 00:25:49,090 --> 00:25:51,000 concept of our recipes. 520 00:25:51,000 --> 00:25:54,270 For example, here's another one. 521 00:25:54,270 --> 00:25:55,920 And this one says what? 522 00:25:55,920 --> 00:25:59,165 That if you run the sum of two functions through the 'D 523 00:25:59,165 --> 00:26:03,160 inverse' machine, the output is the same as if you sent the 524 00:26:03,160 --> 00:26:08,130 functions through separately and then added them up, OK? 525 00:26:08,130 --> 00:26:11,290 I'll talk about that in more detail later. 526 00:26:11,290 --> 00:26:15,070 Again, all I want to see is that this is the analogous 527 00:26:15,070 --> 00:26:18,940 result of, again, a beautiful property of the derivative. 528 00:26:18,940 --> 00:26:25,870 And that is that the derivative of a sum is the sum 529 00:26:25,870 --> 00:26:27,660 of the derivatives. 530 00:26:27,660 --> 00:26:29,000 And how does that work over here? 531 00:26:29,000 --> 00:26:32,440 Again, a very interesting property throughout advanced 532 00:26:32,440 --> 00:26:34,920 calculus, linear algebra and the like. 533 00:26:34,920 --> 00:26:37,750 These properties are very, very special. 534 00:26:37,750 --> 00:26:40,200 And we'll have occasion, as the course goes on, to talk 535 00:26:40,200 --> 00:26:41,220 about them more. 536 00:26:41,220 --> 00:26:43,840 For the time being, rather than have you get lost in the 537 00:26:43,840 --> 00:26:47,440 maze of details, let me work a specific illustration. 538 00:26:47,440 --> 00:26:50,980 Let's suppose I would like to find the family of functions 539 00:26:50,980 --> 00:26:55,410 whose derivative is 'x to the 5th' plus 'x cubed', OK? 540 00:26:55,410 --> 00:26:57,050 Now, what I'm saying is this. 541 00:26:57,050 --> 00:27:00,030 By my previous result, I certainly know how to find the 542 00:27:00,030 --> 00:27:03,315 function whose derivative is 'x to the 5th', namely, ''1/6' 543 00:27:03,315 --> 00:27:06,450 x to the 6th'. 544 00:27:06,450 --> 00:27:09,160 I also know how to find the function whose derivative is 545 00:27:09,160 --> 00:27:13,200 'x cubed', namely, '1/4' x to the 4th'. 546 00:27:13,200 --> 00:27:16,810 Putting these two steps together and say equals added 547 00:27:16,810 --> 00:27:21,250 to equals are equal, I can conclude that 'D of ''1/6' x 548 00:27:21,250 --> 00:27:25,080 to the sixth'' plus 'D of ''1/4' x to the fourth'' is 'x 549 00:27:25,080 --> 00:27:27,080 to the 5th' plus 'x cubed'. 550 00:27:27,080 --> 00:27:31,030 Now, the key step is that since the derivative of a sum 551 00:27:31,030 --> 00:27:34,670 is the sum of the derivatives, I can say that the sum of 552 00:27:34,670 --> 00:27:37,800 these two derivatives is the derivative of the sum of the 553 00:27:37,800 --> 00:27:38,480 two functions. 554 00:27:38,480 --> 00:27:40,250 Namely, that this is what? 555 00:27:40,250 --> 00:27:45,470 'D of ''1/6' x to the 6th'' plus ''1/4' x to the 4th', a 556 00:27:45,470 --> 00:27:48,540 very important power and property of the derivative. 557 00:27:48,540 --> 00:27:52,030 Therefore, have I found one function whose derivative is 558 00:27:52,030 --> 00:27:53,830 'x to the 5th' plus 'x cubed'? 559 00:27:53,830 --> 00:27:56,750 The answer is yes. ''1/6' x to the 6th' plus 560 00:27:56,750 --> 00:27:58,420 ''1/4' x to the 4th'. 561 00:27:58,420 --> 00:28:02,040 Therefore, what is the family of all functions whose 562 00:28:02,040 --> 00:28:04,690 derivative is 'x 5th' plus 'x cubed'? 563 00:28:04,690 --> 00:28:08,030 And again, as before, the answer is, take your one 564 00:28:08,030 --> 00:28:11,610 solution that you found, tack on an arbitrary constant, and 565 00:28:11,610 --> 00:28:14,060 I suppose, technically speaking, I should put the 566 00:28:14,060 --> 00:28:17,490 braces in here to indicate that my solution is an 567 00:28:17,490 --> 00:28:22,060 infinite set, all belonging to one family called an 568 00:28:22,060 --> 00:28:24,590 equivalent set of functions because they have the same 569 00:28:24,590 --> 00:28:25,730 derivative. 570 00:28:25,730 --> 00:28:29,840 Well, let's continue on and do a little 571 00:28:29,840 --> 00:28:31,490 bit more harder stuff. 572 00:28:31,490 --> 00:28:34,370 Remember, we talked about implicit differentiation. 573 00:28:34,370 --> 00:28:39,330 Well, is there an analogue to implicit 'D inverses'? 574 00:28:39,330 --> 00:28:43,470 You see, notice that in every problem so far, when I wrote 575 00:28:43,470 --> 00:28:47,160 'D inverse', I was essentially telling you explicitly what 576 00:28:47,160 --> 00:28:49,040 came out of the 'D-machine'. 577 00:28:49,040 --> 00:28:51,480 Now, suppose I twist the emphasis a little bit. 578 00:28:51,480 --> 00:28:54,670 Suppose I tell you, look, I run a certain function through 579 00:28:54,670 --> 00:28:55,780 the 'D-machine'. 580 00:28:55,780 --> 00:28:57,650 In other words, I form its derivative. 581 00:28:57,650 --> 00:29:01,300 What the output is, is the square of the reciprocal of 582 00:29:01,300 --> 00:29:02,340 the function. 583 00:29:02,340 --> 00:29:06,940 In other words, if 'g of x' comes in, '1 over 'g of x'' 584 00:29:06,940 --> 00:29:08,650 squared comes out. 585 00:29:08,650 --> 00:29:12,050 And now the question is, what is the function 'g of x'? 586 00:29:12,050 --> 00:29:14,970 And again, notice that if we don't know what the right 'g 587 00:29:14,970 --> 00:29:18,690 of x', is you can certainly test a given 'g of x' to see 588 00:29:18,690 --> 00:29:19,580 whether it's right or not. 589 00:29:19,580 --> 00:29:20,930 Namely, what could you do? 590 00:29:20,930 --> 00:29:24,860 You differentiate 'g of x', see what you get, and if what 591 00:29:24,860 --> 00:29:30,130 you get isn't 1 over the square of 'g of x', you've got 592 00:29:30,130 --> 00:29:31,580 the wrong answer. 593 00:29:31,580 --> 00:29:34,030 But the question that comes up is, given this type of a 594 00:29:34,030 --> 00:29:38,000 problem, how do we handle it? 595 00:29:38,000 --> 00:29:38,200 See? 596 00:29:38,200 --> 00:29:42,810 In other words, where does the inverse idea come in here? 597 00:29:42,810 --> 00:29:46,710 The implicit relationship that 'g of x' is determined by this 598 00:29:46,710 --> 00:29:48,100 particular property. 599 00:29:48,100 --> 00:29:51,410 And again, notice how we use properties of derivatives. 600 00:29:51,410 --> 00:29:55,370 We're given that 'g prime of x' is a synonym, identity, for 601 00:29:55,370 --> 00:29:58,310 '1 over 'g squared of x'. 602 00:29:58,310 --> 00:30:02,300 We can cross-multiply and we get 'g squared of x' times 'g 603 00:30:02,300 --> 00:30:05,060 prime of x' is identically one. 604 00:30:05,060 --> 00:30:08,060 Now, if you're clever about this-- and remember, notice 605 00:30:08,060 --> 00:30:09,630 this very, very importantly. 606 00:30:09,630 --> 00:30:13,250 For example, ordinary division is the inverse of ordinary 607 00:30:13,250 --> 00:30:14,440 multiplication. 608 00:30:14,440 --> 00:30:17,410 Notice that to be really cute in division, you have to be 609 00:30:17,410 --> 00:30:19,320 pretty cute in multiplication. 610 00:30:19,320 --> 00:30:22,510 Since all you're doing is changing the emphasis, notice 611 00:30:22,510 --> 00:30:28,300 that to handle hard problems in antiderivatives, you have 612 00:30:28,300 --> 00:30:30,720 to be able to handle tough derivative problems. 613 00:30:30,720 --> 00:30:33,780 What I'm driving at is, you look at something like this 614 00:30:33,780 --> 00:30:36,850 and begin to wonder, do you know a function whose 615 00:30:36,850 --> 00:30:38,630 derivative is this? 616 00:30:38,630 --> 00:30:39,620 See? 617 00:30:39,620 --> 00:30:43,600 The idea is, if you're familiar with your chain rule, 618 00:30:43,600 --> 00:30:47,740 what is the derivative of 'g cubed of x'? 619 00:30:47,740 --> 00:30:50,140 To differentiate 'g cubed', what do you do? 620 00:30:50,140 --> 00:30:54,300 You bring the 3 down to a power 1 less times the 621 00:30:54,300 --> 00:30:56,500 derivative of 'g of x' with respect to 'x'. 622 00:30:56,500 --> 00:30:57,610 That's your chain rule. 623 00:30:57,610 --> 00:30:58,920 That's 'g prime of x'. 624 00:30:58,920 --> 00:31:02,140 In other words, if you're clever enough to see this, 625 00:31:02,140 --> 00:31:07,950 what you say here is, OK, now I multiply both sides by 3. 626 00:31:07,950 --> 00:31:13,100 The left hand side is just the derivative of 'g cubed of x'. 627 00:31:13,100 --> 00:31:17,720 The right hand side is the derivative off '3x'. 628 00:31:17,720 --> 00:31:22,210 Therefore, whatever g of x is, its cube has the same 629 00:31:22,210 --> 00:31:25,750 derivative with respect to 'x' as '3x'. 630 00:31:25,750 --> 00:31:28,600 And we've already learned that if two functions have 631 00:31:28,600 --> 00:31:30,520 identical derivatives, they differ 632 00:31:30,520 --> 00:31:32,730 by, at most, a constant. 633 00:31:32,730 --> 00:31:38,120 Consequently, 'g cubed of x' must equal 634 00:31:38,120 --> 00:31:40,590 '3x' plus some constant. 635 00:31:40,590 --> 00:31:47,880 In other words, 'g of x' is the cube root of '3x plus c'. 636 00:31:47,880 --> 00:31:51,190 Now, time is running short in terms of other things that I 637 00:31:51,190 --> 00:31:52,990 want to teach you in today's lesson. 638 00:31:52,990 --> 00:31:56,170 Let me leave this, then, just for you to check. 639 00:31:56,170 --> 00:32:01,710 Simply differentiate the cube root of '3x plus c'. 640 00:32:01,710 --> 00:32:05,420 And make sure that when you get that derivative, it does 641 00:32:05,420 --> 00:32:08,770 turn out to be '1 over 'g of x' squared'. 642 00:32:08,770 --> 00:32:11,310 As I say, it's a straightforward demonstration. 643 00:32:11,310 --> 00:32:12,940 I leave the details to you. 644 00:32:12,940 --> 00:32:15,980 But the point that's really important is that whenever you 645 00:32:15,980 --> 00:32:17,060 do get an answer-- 646 00:32:17,060 --> 00:32:18,720 the hard part is to get the answer. 647 00:32:18,720 --> 00:32:22,420 Whenever you do get the answer, you can check by means 648 00:32:22,420 --> 00:32:25,260 of just taking a derivative. 649 00:32:25,260 --> 00:32:29,090 Now, with all of this talk about 'D inverse' in mind, let 650 00:32:29,090 --> 00:32:32,930 me now go back to the more traditional notation, the 651 00:32:32,930 --> 00:32:36,510 notation that you'll find in most textbooks, the notation 652 00:32:36,510 --> 00:32:39,720 that, as I say, if you've had calculus before, most likely, 653 00:32:39,720 --> 00:32:41,930 you're more familiar with. 654 00:32:41,930 --> 00:32:46,410 And that is the following, that when we write 'D inverse' 655 00:32:46,410 --> 00:32:50,440 of 'f of x', the average textbook writes 656 00:32:50,440 --> 00:32:51,880 a symbol like this. 657 00:32:51,880 --> 00:32:54,740 It's called the 'integral' of 'f of x'. 658 00:32:54,740 --> 00:32:59,690 I will have later lectures to bemoan this choice of notation 659 00:32:59,690 --> 00:33:00,740 from a different point of view. 660 00:33:00,740 --> 00:33:03,410 But for the time being, all we're saying is, instead of 661 00:33:03,410 --> 00:33:04,630 writing 'D inverse'-- 662 00:33:04,630 --> 00:33:06,140 again, what's in the name? 663 00:33:06,140 --> 00:33:09,150 Just use this particular notation. 664 00:33:09,150 --> 00:33:12,830 That when you see this particular thing, perhaps read 665 00:33:12,830 --> 00:33:14,210 this as what? 666 00:33:14,210 --> 00:33:18,370 That this particular symbol is a code to tell you to find all 667 00:33:18,370 --> 00:33:21,490 functions whose derivative is 'f of x'. 668 00:33:21,490 --> 00:33:24,610 And by the way, to summarize the results that we've 669 00:33:24,610 --> 00:33:28,980 obtained so far, let me just rewrite some of these basic 670 00:33:28,980 --> 00:33:33,240 results in terms of the more traditional notation. 671 00:33:33,240 --> 00:33:35,520 When you write this, this is called the 'indefinite 672 00:33:35,520 --> 00:33:37,780 integral' and what we're saying is the indefinite 673 00:33:37,780 --> 00:33:41,860 integral of ''x to the n' dx' is 'x to the 'n + 1'' over 'n 674 00:33:41,860 --> 00:33:47,440 + 1' plus a constant when it is not equal to minus 1. 675 00:33:47,440 --> 00:33:51,180 The integral of a sum is the sum of the integrals. 676 00:33:51,180 --> 00:33:55,440 And the integral of a constant times a function is a constant 677 00:33:55,440 --> 00:33:56,680 times the integral. 678 00:33:56,680 --> 00:33:58,330 All this says is what? 679 00:33:58,330 --> 00:34:02,880 That this is a consequent of the fact that the derivative 680 00:34:02,880 --> 00:34:04,890 of the sum is the sum of the derivatives. 681 00:34:04,890 --> 00:34:08,090 This is a consequent of the fact that the derivative of a 682 00:34:08,090 --> 00:34:11,000 constant times a function is the constant times the 683 00:34:11,000 --> 00:34:12,590 derivative of the function. 684 00:34:12,590 --> 00:34:15,870 And again, all this is, same thing we were talking about 685 00:34:15,870 --> 00:34:20,739 before only with the 'D inverse' notation replaced by 686 00:34:20,739 --> 00:34:25,300 the more common indefinite integral. 687 00:34:25,300 --> 00:34:28,179 Now, we come to one more problem which will finish us 688 00:34:28,179 --> 00:34:31,800 up for the day, once we get through talking about it. 689 00:34:31,800 --> 00:34:32,820 And that's this. 690 00:34:32,820 --> 00:34:37,110 If all this means is find the functions whose derivative is 691 00:34:37,110 --> 00:34:41,239 'f of x', why write this notation here? 692 00:34:41,239 --> 00:34:44,000 Why couldn't we have just written, for example, the 693 00:34:44,000 --> 00:34:47,739 so-called integral sign with a little 'x' underneath? 694 00:34:47,739 --> 00:34:51,730 You see, we've talked about misleading notations before. 695 00:34:51,730 --> 00:34:54,460 You see, in terms of our differential notation, when 696 00:34:54,460 --> 00:34:58,510 you see ''f of x' dx', you have every right to think of a 697 00:34:58,510 --> 00:35:00,010 differential. 698 00:35:00,010 --> 00:35:04,380 Now, in all fairness, the chances are that this notation 699 00:35:04,380 --> 00:35:06,710 would not have been invented if there weren't some 700 00:35:06,710 --> 00:35:10,490 connection between derivatives and differentials. 701 00:35:10,490 --> 00:35:12,570 So let me mention this point. 702 00:35:12,570 --> 00:35:15,240 Going back as if we were starting the lecture all over 703 00:35:15,240 --> 00:35:18,900 again, could I have invented a different machine, which I'll 704 00:35:18,900 --> 00:35:21,910 call the 'script D- machine'? 705 00:35:21,910 --> 00:35:24,410 In other words, I don't want to call it the same 706 00:35:24,410 --> 00:35:27,230 'D-machine' as before because now it's going to have a 707 00:35:27,230 --> 00:35:28,660 different set of outputs. 708 00:35:28,660 --> 00:35:30,530 You see, now the input will still be 709 00:35:30,530 --> 00:35:32,100 differentiable functions. 710 00:35:32,100 --> 00:35:34,780 But the output, instead of being the derivative of the 711 00:35:34,780 --> 00:35:38,440 function, will be the differential of the function. 712 00:35:38,440 --> 00:35:41,420 For example, before, we said what? 713 00:35:41,420 --> 00:35:45,320 If 'x squared' goes in, the output would be '2x'. 714 00:35:45,320 --> 00:35:50,420 With the script 'D-machine', the output would be '2x dx'. 715 00:35:50,420 --> 00:35:52,950 Now, even though these machines are different because 716 00:35:52,950 --> 00:35:55,710 one machine gives you a differential as an output and 717 00:35:55,710 --> 00:35:57,900 the other gives you a derivative as an output, 718 00:35:57,900 --> 00:35:59,410 notice that they are equivalent. 719 00:35:59,410 --> 00:36:04,980 Namely, knowing the differential, we can pin down 720 00:36:04,980 --> 00:36:08,390 the function the same as when we knew the derivative. 721 00:36:08,390 --> 00:36:10,630 Now, the question that comes up is, why should we use this 722 00:36:10,630 --> 00:36:12,440 particular type of notation? 723 00:36:12,440 --> 00:36:16,410 And to use the examples that are most prevalent in the text 724 00:36:16,410 --> 00:36:19,510 and also in our notes, let me give you the same problem that 725 00:36:19,510 --> 00:36:21,350 we've done before, but now from a 726 00:36:21,350 --> 00:36:22,850 different point of view. 727 00:36:22,850 --> 00:36:26,210 Suppose we're still given the problem 'g prime of x' equals 728 00:36:26,210 --> 00:36:28,480 '1 over 'g of x' squared'. 729 00:36:28,480 --> 00:36:31,670 We say, OK, let 'y' equal 'g of x'. 730 00:36:31,670 --> 00:36:35,270 When we do that, this problem now translates into what? 731 00:36:35,270 --> 00:36:40,480 'dy dx' equals '1 over 'y squared''. 732 00:36:40,480 --> 00:36:44,190 Now, if we allow ourselves to use the differential notation, 733 00:36:44,190 --> 00:36:47,810 which we justified in previous lectures, this says what? 734 00:36:47,810 --> 00:36:52,000 Cross-multiplying ''y squared' dy' equals 'dx'. 735 00:36:52,000 --> 00:36:56,050 Notice, by the way, that 'y' is implicitly a differentiable 736 00:36:56,050 --> 00:36:57,840 function of 'x'. 737 00:36:57,840 --> 00:37:00,780 So what we're saying over here is, look, here are two 738 00:37:00,780 --> 00:37:05,850 functions which have the same differential, therefore, if we 739 00:37:05,850 --> 00:37:08,630 integrate them, they differ by a constant. 740 00:37:08,630 --> 00:37:13,520 Well, to mimic what we were doing before, let's just say-- 741 00:37:13,520 --> 00:37:14,530 let's see, this, I think, I have a 742 00:37:14,530 --> 00:37:15,070 little bit twisted here. 743 00:37:15,070 --> 00:37:19,300 If ''y squared' dy' equals '3x', '3 'y squared' dy' 744 00:37:19,300 --> 00:37:20,900 equals '3dx'. 745 00:37:20,900 --> 00:37:24,200 If the differentials are equal, then the functions 746 00:37:24,200 --> 00:37:25,540 differ by a constant. 747 00:37:25,540 --> 00:37:28,730 But 'y cubed' is the function whose differential is '3 'y 748 00:37:28,730 --> 00:37:31,240 squared' dy'. 749 00:37:31,240 --> 00:37:34,840 '3x' is the function whose differential is '3dx'. 750 00:37:34,840 --> 00:37:40,590 And we wind up with 'y cubed' equals '3x + c', or, 'y' 751 00:37:40,590 --> 00:37:45,710 equals the 'cube root of '3x + c''. 752 00:37:45,710 --> 00:37:49,120 By the way, let me just pull this board down so that we can 753 00:37:49,120 --> 00:37:51,440 make a little bit of a comparison here. 754 00:37:51,440 --> 00:37:55,330 You see, notice that in using this differential notation, 755 00:37:55,330 --> 00:37:58,770 which allowed us to use some nice algebraic devices and 756 00:37:58,770 --> 00:38:01,870 didn't seem quite as difficult for us to recognize certain 757 00:38:01,870 --> 00:38:04,820 things, I hope you notice that there seems to be a 758 00:38:04,820 --> 00:38:09,200 correspondence between the steps that we had here and the 759 00:38:09,200 --> 00:38:11,690 steps that took place over here. 760 00:38:11,690 --> 00:38:15,250 In other words, having done at the so-called rigorous way, 761 00:38:15,250 --> 00:38:21,620 notice that differentials give us a very nice, 762 00:38:21,620 --> 00:38:26,140 intuitively-simpler technique for solving certain types of 763 00:38:26,140 --> 00:38:30,720 problems when we have implicit relationships 764 00:38:30,720 --> 00:38:33,080 between 'y' and 'x'. 765 00:38:33,080 --> 00:38:35,540 And so, just to illustrate this in terms of one more 766 00:38:35,540 --> 00:38:38,710 problem, let's take one that's fairly geometric. 767 00:38:38,710 --> 00:38:41,880 Let's suppose we're given the following problem. 768 00:38:41,880 --> 00:38:45,790 'dy dx' is 'minus x' over 'y' and we know that when 'x' 769 00:38:45,790 --> 00:38:48,570 equals 3, 'y' equals 4. 770 00:38:48,570 --> 00:38:51,810 Using the language of differentials, what we would 771 00:38:51,810 --> 00:38:53,610 do over here is we would cross-multiply. 772 00:38:53,610 --> 00:38:56,890 773 00:38:56,890 --> 00:38:58,040 Whatever you want to do. 774 00:38:58,040 --> 00:39:03,370 we recognize that, to get '2y', you must differentiate 775 00:39:03,370 --> 00:39:04,100 'y squared'. 776 00:39:04,100 --> 00:39:05,990 This is a step you don't really need. 777 00:39:05,990 --> 00:39:07,230 I don't care how you want to do this. 778 00:39:07,230 --> 00:39:10,550 All I'm saying is that from this step, I can get to here. 779 00:39:10,550 --> 00:39:15,390 Recognizing that y squared has '2y dy' as its differential 780 00:39:15,390 --> 00:39:18,550 and that ''minus 'x squared'' has 'minus 2x dx' as its 781 00:39:18,550 --> 00:39:22,520 differential, I know that 'y squared' is equal to ''minus 782 00:39:22,520 --> 00:39:24,260 'x squared'' plus a constant. 783 00:39:24,260 --> 00:39:25,600 I transpose. 784 00:39:25,600 --> 00:39:28,800 I get that 'x squared' plus 'y squared' is a constant. 785 00:39:28,800 --> 00:39:32,380 Knowing that when 'x' equals 3, 'y' equals 4, I can 786 00:39:32,380 --> 00:39:37,030 determine that the constant must be 25 when 'x' equals 3 787 00:39:37,030 --> 00:39:38,250 and 'y' is 4. 788 00:39:38,250 --> 00:39:41,870 But since it's a constant, if it's 25 when 'x' equals 3 and 789 00:39:41,870 --> 00:39:46,050 'y' equals 4, it's 25 everyplace. 790 00:39:46,050 --> 00:39:48,780 And again, if you want a geometric interpretation of 791 00:39:48,780 --> 00:39:53,550 this, the circle centered at the origin with radius equal 792 00:39:53,550 --> 00:39:59,550 to 5 is the only curve in the whole world whose derivative 793 00:39:59,550 --> 00:40:04,070 at any given point is the negative of the x-coordinate 794 00:40:04,070 --> 00:40:07,050 over the y-coordinate and passes through 795 00:40:07,050 --> 00:40:09,140 the point (3 , 4). 796 00:40:09,140 --> 00:40:11,690 And if you want to see that geometrically, let me just 797 00:40:11,690 --> 00:40:13,300 take a second here. 798 00:40:13,300 --> 00:40:19,160 You see, notice that any point on the circle (x , y), notice 799 00:40:19,160 --> 00:40:22,560 that the tangent line to the curve, which has slope 'dy 800 00:40:22,560 --> 00:40:24,130 dx', is what? 801 00:40:24,130 --> 00:40:28,370 Perpendicular to the radius, again. 802 00:40:28,370 --> 00:40:31,610 The radius has slope 'y' over 'x'. 803 00:40:31,610 --> 00:40:32,940 And to be perpendicular-- 804 00:40:32,940 --> 00:40:35,000 if two lines are perpendicular, their slopes 805 00:40:35,000 --> 00:40:37,610 are negative reciprocals. 806 00:40:37,610 --> 00:40:40,200 This is this particular problem. 807 00:40:40,200 --> 00:40:42,580 And now, all I wanted to show you is that if you're nervous 808 00:40:42,580 --> 00:40:45,880 about differentials and you don't like to use them, notice 809 00:40:45,880 --> 00:40:48,150 that the same problem could have been stated without 810 00:40:48,150 --> 00:40:49,280 differentials. 811 00:40:49,280 --> 00:40:51,600 Namely, we are thinking of a function which we'll call 'g 812 00:40:51,600 --> 00:40:55,310 of x' such that the derivative of 'g of x' is 'minus 813 00:40:55,310 --> 00:40:57,370 x' over 'g of x'. 814 00:40:57,370 --> 00:41:00,670 In other words, if the input of the 'D-machine' is 'g', the 815 00:41:00,670 --> 00:41:03,210 output is 'minus x' over 'g'. 816 00:41:03,210 --> 00:41:06,990 And we also know that when 'x' equals 3, the output is 4. 817 00:41:06,990 --> 00:41:09,330 So to 'g' of 3 equals 4. 818 00:41:09,330 --> 00:41:12,800 And without going through the great details here, let's just 819 00:41:12,800 --> 00:41:13,420 notice that we could 820 00:41:13,420 --> 00:41:16,260 cross-multiply the same as before. 821 00:41:16,260 --> 00:41:20,580 We could multiply both sides by 2, the same as before. 822 00:41:20,580 --> 00:41:23,500 We could recognize that the left-hand side was the 823 00:41:23,500 --> 00:41:26,770 derivative of ''g of x' squared', that the right-hand 824 00:41:26,770 --> 00:41:29,350 side is the derivative of 'minus 'x squared''. 825 00:41:29,350 --> 00:41:32,830 Therefore, since they have the same identical derivatives, 826 00:41:32,830 --> 00:41:35,270 they must differ by a constant. 827 00:41:35,270 --> 00:41:37,070 OK. 828 00:41:37,070 --> 00:41:41,190 We actually have plus or minus here, but notice the fact that 829 00:41:41,190 --> 00:41:45,450 when the input is 3, the output is 4 means that we're 830 00:41:45,450 --> 00:41:48,820 on the positive branch of the curve, et cetera. 831 00:41:48,820 --> 00:41:52,910 I say et cetera not because these points aren't important 832 00:41:52,910 --> 00:41:56,410 but because every point that comes up now has already been 833 00:41:56,410 --> 00:42:00,640 discussed under the heading of differential calculus. 834 00:42:00,640 --> 00:42:04,370 In other words, that the inverse of differentiation can 835 00:42:04,370 --> 00:42:07,490 be handled very, very neatly just by knowing 836 00:42:07,490 --> 00:42:10,190 differentiation with a switch in emphasis. 837 00:42:10,190 --> 00:42:12,580 Why do we want to know this particular topic? 838 00:42:12,580 --> 00:42:16,650 Well, because in many cases, to look at it geometrically, 839 00:42:16,650 --> 00:42:19,280 in the past, we were given the curve. 840 00:42:19,280 --> 00:42:21,310 We wanted to find out what the slope was. 841 00:42:21,310 --> 00:42:24,700 In many physical applications, we are, in a sense, told what 842 00:42:24,700 --> 00:42:27,700 the slope is and have to figure out what the curve is. 843 00:42:27,700 --> 00:42:29,060 Hence the motivation. 844 00:42:29,060 --> 00:42:32,440 Well, at any rate, there'll be more about this in the text 845 00:42:32,440 --> 00:42:33,790 and in our exercises. 846 00:42:33,790 --> 00:42:37,890 So until next time, good bye. 847 00:42:37,890 --> 00:42:40,890 MALE SPEAKER: Funding for the publication of this video was 848 00:42:40,890 --> 00:42:45,610 provided by the Gabriella and Paul Rosenbaum Foundation. 849 00:42:45,610 --> 00:42:49,780 Help OCW continue to provide free and open access to MIT 850 00:42:49,780 --> 00:42:53,980 courses by making a donation at ocw.mit.edu/donate. 851 00:42:53,980 --> 00:42:58,730