1 00:00:00,040 --> 00:00:02,400 The following content is provided under a Creative 2 00:00:02,400 --> 00:00:03,690 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 00:00:32,670 --> 00:00:33,470 PROFESSOR: Hi. 9 00:00:33,470 --> 00:00:36,680 Our lecture today is called 'Rolle's Theorem and Its 10 00:00:36,680 --> 00:00:37,930 Consequences'. 11 00:00:37,930 --> 00:00:40,960 And I suppose we could've made a take off on what goes up 12 00:00:40,960 --> 00:00:44,030 must come down, and say that what Rolle's theorem says 13 00:00:44,030 --> 00:00:47,970 intuitively is that what goes up smoothly and comes down 14 00:00:47,970 --> 00:00:51,480 smoothly must level off somewhere. 15 00:00:51,480 --> 00:00:51,880 OK? 16 00:00:51,880 --> 00:00:55,280 Now because that may sound too easy to understand, let's 17 00:00:55,280 --> 00:00:59,350 cloak that in the language of more formal mathematics. 18 00:00:59,350 --> 00:01:02,110 Rolle's theorem says this. 19 00:01:02,110 --> 00:01:05,760 Let 'f' be defined and continuous on the closed 20 00:01:05,760 --> 00:01:08,050 interval from 'a' to 'b'. 21 00:01:08,050 --> 00:01:08,620 In other words, what? 22 00:01:08,620 --> 00:01:09,930 The domain of 'f' is the closed 23 00:01:09,930 --> 00:01:11,140 interval from 'a' to 'b'. 24 00:01:11,140 --> 00:01:14,340 The graph of 'f' is unbroken on this interval. 25 00:01:14,340 --> 00:01:18,770 And differentiable in the open interval from 'a' to 'b'. 26 00:01:18,770 --> 00:01:21,840 In other words, you want the function to be continuous on 27 00:01:21,840 --> 00:01:25,270 the entire interval, but for differentiability, you only 28 00:01:25,270 --> 00:01:29,000 require that it be smooth, differentiable, in the 29 00:01:29,000 --> 00:01:32,010 interior of the interval, in the open interval. 30 00:01:32,010 --> 00:01:38,000 Suppose also that 'f of a' and 'f of b' are 0. 31 00:01:38,000 --> 00:01:41,990 Then, what Rolle's theorem says is that 'f prime of c', 32 00:01:41,990 --> 00:01:48,180 the derivative of 'f of x', must be 0 for some number 'c', 33 00:01:48,180 --> 00:01:51,680 at least one number 'c', in the open 34 00:01:51,680 --> 00:01:54,160 interval from 'a' to 'b'. 35 00:01:54,160 --> 00:01:58,420 Now what this thing says intuitively is simply this, 36 00:01:58,420 --> 00:02:04,440 suppose you have a curve that is unbroken for all values of 37 00:02:04,440 --> 00:02:07,130 'x' between 'a' and 'b' inclusively. 38 00:02:07,130 --> 00:02:09,580 Suppose the curve is smooth. 39 00:02:09,580 --> 00:02:14,780 Suppose the curve starts here and ends here, then what we're 40 00:02:14,780 --> 00:02:18,330 saying is there must be some point in here where the curve 41 00:02:18,330 --> 00:02:19,030 levels off. 42 00:02:19,030 --> 00:02:21,800 In other words, someplace where you have a horizontal 43 00:02:21,800 --> 00:02:24,800 tangent, which is what 'f prime of c' equals 0 means. 44 00:02:24,800 --> 00:02:29,000 In this particular diagram, this would be the value of 'c' 45 00:02:29,000 --> 00:02:30,510 that we're talking about. 46 00:02:30,510 --> 00:02:35,080 By the way, I think the proof is intuitively clear. 47 00:02:35,080 --> 00:02:39,510 Namely, if the curve never leaves the x-axis, then it's 48 00:02:39,510 --> 00:02:42,660 leveled off for the entire domain. 49 00:02:42,660 --> 00:02:45,937 And if the curve does leave the x-axis, for example, if 50 00:02:45,937 --> 00:02:50,400 the curve starts to rise, OK, since it must eventually get 51 00:02:50,400 --> 00:02:55,320 back to the x-axis when 'b' is 0, it must ultimately begin to 52 00:02:55,320 --> 00:02:56,240 fall again. 53 00:02:56,240 --> 00:02:59,860 Well, if the curve goes from rising to falling, 54 00:02:59,860 --> 00:03:01,040 it must have what? 55 00:03:01,040 --> 00:03:04,010 Since it's coming up and then going down, it must attain a 56 00:03:04,010 --> 00:03:05,520 maximum value. 57 00:03:05,520 --> 00:03:09,030 Because the curve is unbroken and smooth, as we saw in our 58 00:03:09,030 --> 00:03:12,690 previous lecture, the maximum value is characterized by the 59 00:03:12,690 --> 00:03:15,220 derivative at that point being 0. 60 00:03:15,220 --> 00:03:18,920 In fact, the analytic proof is precisely what we've just 61 00:03:18,920 --> 00:03:22,650 said, only translated into more mathematical language. 62 00:03:22,650 --> 00:03:25,280 By the way, I would like to make a slight aside here, 63 00:03:25,280 --> 00:03:28,980 because I think it sometimes gets confusing to students to 64 00:03:28,980 --> 00:03:33,600 see, why do you say that the function has to be continuous 65 00:03:33,600 --> 00:03:38,000 on the closed interval, but differentiable only in the 66 00:03:38,000 --> 00:03:39,220 open interval? 67 00:03:39,220 --> 00:03:42,580 I thought you might like to see a contrived example as to 68 00:03:42,580 --> 00:03:46,360 what goes wrong if you allow the curve to be broken at the 69 00:03:46,360 --> 00:03:47,220 end points. 70 00:03:47,220 --> 00:03:49,520 See, all I'm thinking of is something like this. 71 00:03:49,520 --> 00:03:52,880 Suppose I say, look, let's define a curve as follows. 72 00:03:52,880 --> 00:03:56,510 At 'x' equals 'a' and 'x' equals 'b', the 73 00:03:56,510 --> 00:03:59,040 curve will be 0. 74 00:03:59,040 --> 00:04:01,550 So in other words, it'll cross the x-axis. 75 00:04:01,550 --> 00:04:04,980 Then immediately, for 'x' greater than 'a', the curve 76 00:04:04,980 --> 00:04:09,090 jumps up to here, comes down along this line, and then, you 77 00:04:09,090 --> 00:04:13,480 see, when 'x' equals 'b', it jumps down here again. 78 00:04:13,480 --> 00:04:15,560 In other words, why not let the curve 79 00:04:15,560 --> 00:04:16,950 be defined as follows? 80 00:04:16,950 --> 00:04:20,170 It will be 0 at these two endpoints. 81 00:04:20,170 --> 00:04:23,650 It'll be this curve on the open interval. 82 00:04:23,650 --> 00:04:28,010 Notice in this contrived example that 'f of a' and 'f 83 00:04:28,010 --> 00:04:31,780 of b' are both equal to 0, but there is no place in the open 84 00:04:31,780 --> 00:04:35,880 interval where the curve has a horizontal tangent line. 85 00:04:35,880 --> 00:04:38,790 In other words, the significance is you've got to 86 00:04:38,790 --> 00:04:41,830 be sure that the curve doesn't get broken at the ends, 87 00:04:41,830 --> 00:04:43,940 because with these gaps, all sorts of 88 00:04:43,940 --> 00:04:45,780 crazy things can happen. 89 00:04:45,780 --> 00:04:49,180 Now, just as in our previous lecture, there are some rather 90 00:04:49,180 --> 00:04:53,960 important cautions that have to be taken in understanding 91 00:04:53,960 --> 00:04:55,210 Rolle's theorem. 92 00:04:55,210 --> 00:04:59,020 As simple as it is, we have to be sure that we understand 93 00:04:59,020 --> 00:05:02,070 exactly what's really happening here. 94 00:05:02,070 --> 00:05:06,260 The first caution is that Rolle's theorem is what we 95 00:05:06,260 --> 00:05:09,920 mathematicians call an 'existence theorem'. 96 00:05:09,920 --> 00:05:13,800 It says, under certain conditions, there exists at 97 00:05:13,800 --> 00:05:17,080 least one number 'c' that has a certain property. 98 00:05:17,080 --> 00:05:19,790 It doesn't tell us how many "c's" there are. 99 00:05:19,790 --> 00:05:21,870 It doesn't tell us where to find them. 100 00:05:21,870 --> 00:05:23,040 It just says, what? 101 00:05:23,040 --> 00:05:25,720 There exists at least one such 'c'. 102 00:05:25,720 --> 00:05:28,770 And the point is, you must be careful to remember-- 103 00:05:28,770 --> 00:05:29,930 so let's take an example. 104 00:05:29,930 --> 00:05:31,970 Here's 'a', here's 'b'. 105 00:05:31,970 --> 00:05:34,826 If the function is continuous and smooth, in other words, if 106 00:05:34,826 --> 00:05:37,950 the graph is continuous and smooth, all we're saying is 107 00:05:37,950 --> 00:05:40,710 that at least one number between 'a' and 'b', the 108 00:05:40,710 --> 00:05:43,520 curve, must possess a horizontal tangent. 109 00:05:43,520 --> 00:05:44,640 There may be more than one. 110 00:05:44,640 --> 00:05:46,480 You see, the curve, for example, could do 111 00:05:46,480 --> 00:05:48,250 something like this. 112 00:05:48,250 --> 00:05:51,640 See, in other words, here is one value of 'c', which we'll 113 00:05:51,640 --> 00:05:54,060 call 'c1', horizontal tangent here. 114 00:05:54,060 --> 00:05:57,730 Here's another value, which we'll call 'c2', horizontal 115 00:05:57,730 --> 00:05:59,280 tangent here. 116 00:05:59,280 --> 00:06:03,100 See, again, the meaning of at least one. 117 00:06:03,100 --> 00:06:11,370 Secondly, we must make sure that we remember that the 118 00:06:11,370 --> 00:06:12,290 curve is smooth. 119 00:06:12,290 --> 00:06:12,780 Meaning what? 120 00:06:12,780 --> 00:06:14,570 That the function is differentiable. 121 00:06:14,570 --> 00:06:15,980 Now I'm taking the liberty of drawing 122 00:06:15,980 --> 00:06:17,880 these things in freehand. 123 00:06:17,880 --> 00:06:21,760 There's some mixed emotions here, if I draw the diagrams 124 00:06:21,760 --> 00:06:24,260 too smoothly all the time, you lose the significance of 125 00:06:24,260 --> 00:06:26,200 what's going on because of the picture. 126 00:06:26,200 --> 00:06:28,020 And if I draw them freehand all the time, you won't 127 00:06:28,020 --> 00:06:29,230 understand what I'm doing, because I 128 00:06:29,230 --> 00:06:30,380 don't draw very well. 129 00:06:30,380 --> 00:06:32,270 But I think here we can get away with this. 130 00:06:32,270 --> 00:06:33,650 What I'm driving at is this. 131 00:06:33,650 --> 00:06:36,710 Let's suppose you have 'a' and 'b' here. 132 00:06:36,710 --> 00:06:38,810 Let's suppose that the function, the curve that we're 133 00:06:38,810 --> 00:06:41,230 drawing, passes through these two points. 134 00:06:41,230 --> 00:06:44,260 But suppose there happens to be a sharp corner in here. 135 00:06:44,260 --> 00:06:46,530 Maybe the curve goes like this, it goes up like this, 136 00:06:46,530 --> 00:06:50,010 then very abruptly comes down like this. 137 00:06:50,010 --> 00:06:51,550 Notice, what? 138 00:06:51,550 --> 00:06:54,270 That the curve is continuous. 139 00:06:54,270 --> 00:06:57,380 It does reach a local maximum. 140 00:06:57,380 --> 00:07:01,890 But the point is, for this particular value of 'c', 'f 141 00:07:01,890 --> 00:07:05,320 prime of c' is not 0 by default. 142 00:07:05,320 --> 00:07:10,320 Namely, 'f prime of c' doesn't even exist. 143 00:07:10,320 --> 00:07:15,230 So in other words, Rolle's theorem doesn't apply if you 144 00:07:15,230 --> 00:07:16,700 don't have differentiabilities. 145 00:07:16,700 --> 00:07:19,890 I want to make sure you see where each of the parts of the 146 00:07:19,890 --> 00:07:22,600 hypotheses for the theorem are used. 147 00:07:22,600 --> 00:07:25,470 By the way, here's another interesting result, which has 148 00:07:25,470 --> 00:07:27,200 nothing to do with the statement of the theorem, but 149 00:07:27,200 --> 00:07:31,140 again, another piece of evidence as to why we like to 150 00:07:31,140 --> 00:07:34,700 shy away from functions which are not single value. 151 00:07:34,700 --> 00:07:37,700 See, for example, suppose you allowed the function to be 152 00:07:37,700 --> 00:07:41,520 multivalued, and you say, OK, I want the curve to pass 153 00:07:41,520 --> 00:07:42,960 through here and here. 154 00:07:42,960 --> 00:07:44,890 And I want it to be smooth. 155 00:07:44,890 --> 00:07:48,180 But the curve does not have to be single value. 156 00:07:48,180 --> 00:07:49,660 Notice what you can do. 157 00:07:49,660 --> 00:07:52,070 You could have a curve that does something like this. 158 00:07:55,490 --> 00:07:57,210 I don't know. 159 00:07:57,210 --> 00:08:00,190 Now you see, there will be a point 'c' where the 160 00:08:00,190 --> 00:08:02,390 derivative will be 0. 161 00:08:02,390 --> 00:08:04,790 Even as badly as I've drawn this, I think roughly 162 00:08:04,790 --> 00:08:09,430 speaking, we can see that 'c' would be something like this. 163 00:08:09,430 --> 00:08:12,130 Notice, however, that in Rolle's theorem, the 164 00:08:12,130 --> 00:08:13,150 statement is what? 165 00:08:13,150 --> 00:08:16,490 That 'c' must be on the open interval from 'a' to 'b'. 166 00:08:16,490 --> 00:08:20,010 If the function is not single value, as long as the curve is 167 00:08:20,010 --> 00:08:23,330 smooth, there will be places where the curve levels off. 168 00:08:23,330 --> 00:08:27,130 But the x-coordinate of the point at which the curve 169 00:08:27,130 --> 00:08:30,530 levels off may not be in the interval-- 170 00:08:30,530 --> 00:08:34,179 may not be, it could be, but it might not be, I should 171 00:08:34,179 --> 00:08:37,220 say-- in the open interval from 'a' to 'b'. 172 00:08:37,220 --> 00:08:39,070 This is a very, very important result. 173 00:08:39,070 --> 00:08:43,070 In fact, I'll have reason to make reference to this in a 174 00:08:43,070 --> 00:08:45,700 little while later in the lecture. 175 00:08:45,700 --> 00:08:47,660 I couldn't make reference to it later earlier in the 176 00:08:47,660 --> 00:08:49,570 lecture, I guess. 177 00:08:49,570 --> 00:08:54,200 The fourth assumption here is also an aside, and it's one 178 00:08:54,200 --> 00:08:55,360 that's rather crucial. 179 00:08:55,360 --> 00:08:59,470 In most textbooks in which Rolle's theorem is stated, the 180 00:08:59,470 --> 00:09:00,350 condition is what? 181 00:09:00,350 --> 00:09:04,180 That 'f of a' equals 'f of b' equals 0. 182 00:09:04,180 --> 00:09:09,340 It turns out that this is too restrictive, that essentially, 183 00:09:09,340 --> 00:09:13,630 all you need is 'f of a' equals 'f of b'. 184 00:09:13,630 --> 00:09:17,200 What I mean by that is, let's suppose 'f of a' is not 0. 185 00:09:17,200 --> 00:09:19,810 Let's suppose this height represents 'f of a'. 186 00:09:19,810 --> 00:09:24,300 What I'm saying is suppose that 'f of a' and 187 00:09:24,300 --> 00:09:27,210 'f of b' are equal. 188 00:09:27,210 --> 00:09:29,760 What that means is, if I want to think of a new axis, which 189 00:09:29,760 --> 00:09:32,330 I call the 'x sub 1' axis-- 190 00:09:32,330 --> 00:09:35,980 see, with respect to the 'x sub 1' axis, the curve crosses 191 00:09:35,980 --> 00:09:37,550 the axis at these two points. 192 00:09:37,550 --> 00:09:40,235 In other words, notice that as long as these two points are 193 00:09:40,235 --> 00:09:43,160 at the same level, the same argument that we used to prove 194 00:09:43,160 --> 00:09:46,970 Rolle's theorem goes through unimpeded over here. 195 00:09:46,970 --> 00:09:48,170 Namely, we say what? 196 00:09:48,170 --> 00:09:52,490 What goes up smoothly and comes down smoothly-- 197 00:09:52,490 --> 00:09:54,670 because it has to come down, because it comes back to the 198 00:09:54,670 --> 00:10:00,540 same level here-- must reach a point someplace in here where 199 00:10:00,540 --> 00:10:02,280 it levels off. 200 00:10:02,280 --> 00:10:07,100 So these are the four cautions that I'd like you to look at 201 00:10:07,100 --> 00:10:09,200 when we view Rolle's theorem. 202 00:10:09,200 --> 00:10:12,110 Now, if somebody were to say to us, what's so important 203 00:10:12,110 --> 00:10:14,290 about Rolle's theorem? 204 00:10:14,290 --> 00:10:17,250 And this happens so often in mathematics that frequently, 205 00:10:17,250 --> 00:10:21,510 the most important thing about one particular theorem, is 206 00:10:21,510 --> 00:10:24,420 that it may be the building block by which a more 207 00:10:24,420 --> 00:10:26,670 important, or more useful, or more 208 00:10:26,670 --> 00:10:28,910 practical theorem is derived. 209 00:10:28,910 --> 00:10:32,680 And in this respect, I would say for my own opinion, that 210 00:10:32,680 --> 00:10:35,830 the most important application of Rolle's theorem is that it 211 00:10:35,830 --> 00:10:39,560 facilitates a very famous result known as the 'Mean 212 00:10:39,560 --> 00:10:40,590 Value theorem'. 213 00:10:40,590 --> 00:10:43,790 As our course proceeds, from time to time we will have 214 00:10:43,790 --> 00:10:47,330 ample reason to back track and make references to the mean 215 00:10:47,330 --> 00:10:48,390 value theorem. 216 00:10:48,390 --> 00:10:51,640 I intend not to make too deep references to the mean value 217 00:10:51,640 --> 00:10:54,190 theorem now, because what I would like to do is to 218 00:10:54,190 --> 00:10:57,220 establish the result, give you enough of an intuitive feeling 219 00:10:57,220 --> 00:11:00,960 so that you can tuck it under your belt without feeling too 220 00:11:00,960 --> 00:11:04,880 overwhelmed by it, and just enough applications of it so 221 00:11:04,880 --> 00:11:07,450 that we can get into the next phase of our calculus course. 222 00:11:07,450 --> 00:11:10,080 But the mean value theorem is another one of these things 223 00:11:10,080 --> 00:11:12,680 where if you state the thing analytically, and have no 224 00:11:12,680 --> 00:11:15,490 feeling for what's going on pictorially, the thing can 225 00:11:15,490 --> 00:11:17,180 become overwhelming. 226 00:11:17,180 --> 00:11:19,940 Let's, in fact, do it in an overwhelming way, and then 227 00:11:19,940 --> 00:11:22,040 show what the thing means pictorially. 228 00:11:22,040 --> 00:11:25,520 Notice again how this thing now starts off the same way as 229 00:11:25,520 --> 00:11:26,560 Rolle's theorem. 230 00:11:26,560 --> 00:11:29,560 Let 'f' be continuous on the closed interval from 'a' to 231 00:11:29,560 --> 00:11:34,150 'b', and differentiable in the open interval from 'a' to 'b'. 232 00:11:34,150 --> 00:11:38,110 By the way, this is just an idiosyncrasy of mine, I don't 233 00:11:38,110 --> 00:11:39,050 know if it's standard. 234 00:11:39,050 --> 00:11:42,070 When I talk about the closed interval, I have the habit of 235 00:11:42,070 --> 00:11:44,050 saying 'on' the interval. 236 00:11:44,050 --> 00:11:47,280 When I talk about the open interval, I like to say 'in' 237 00:11:47,280 --> 00:11:49,840 the interval, to sort of emphasize the interior. 238 00:11:49,840 --> 00:11:53,250 It's just a vocabulary trait, and don't read 239 00:11:53,250 --> 00:11:54,080 too much into this. 240 00:11:54,080 --> 00:11:55,160 Don't be upset by it. 241 00:11:55,160 --> 00:11:58,060 But it's continuous on the closed interval, 242 00:11:58,060 --> 00:12:00,050 differentiable in the open interval. 243 00:12:00,050 --> 00:12:02,360 Now it's again an existence theorem. 244 00:12:02,360 --> 00:12:04,750 It says then there exists a number 'c'. 245 00:12:04,750 --> 00:12:06,580 When I say there exists, it means what? 246 00:12:06,580 --> 00:12:10,190 There is at least one number 'c' in the open interval from 247 00:12:10,190 --> 00:12:11,080 'a' to 'b'-- 248 00:12:11,080 --> 00:12:13,240 and this is the part that looks kind of tough-- 249 00:12:13,240 --> 00:12:19,030 such that ''f of b' minus 'f of a'' divided by 'b - a' is 250 00:12:19,030 --> 00:12:20,600 'f prime of c'. 251 00:12:20,600 --> 00:12:24,040 And this somehow or other may seem at first glance to be 252 00:12:24,040 --> 00:12:27,290 more ominous than the intuitive feeling about 253 00:12:27,290 --> 00:12:28,520 Rolle's theorem. 254 00:12:28,520 --> 00:12:31,590 By the way, as the name implies, where by 'mean' we 255 00:12:31,590 --> 00:12:34,900 don't mean nasty, we mean average, if you'd like to see 256 00:12:34,900 --> 00:12:37,240 what this thing means, and I'll draw you a picture in a 257 00:12:37,240 --> 00:12:44,020 second, all it says is that if a particle is moving from 258 00:12:44,020 --> 00:12:48,790 point 'a' to point 'b', say, at at least one point during 259 00:12:48,790 --> 00:12:51,750 its trip, the instantaneous speed must 260 00:12:51,750 --> 00:12:54,980 equal the average speed. 261 00:12:54,980 --> 00:12:55,210 You see? 262 00:12:55,210 --> 00:12:57,380 Because after all, if you're always going less than your 263 00:12:57,380 --> 00:13:00,490 average speed, how could you have had an average speed as 264 00:13:00,490 --> 00:13:02,010 high as your average speed? 265 00:13:02,010 --> 00:13:04,250 And if you're always going less than your average speed, 266 00:13:04,250 --> 00:13:08,270 how could you have had, you see, an average speed equaling 267 00:13:08,270 --> 00:13:09,660 this, what it did? 268 00:13:09,660 --> 00:13:13,610 So that somehow or other, all you're saying is that somehow 269 00:13:13,610 --> 00:13:17,050 or other, the instantaneous speed at a particular instance 270 00:13:17,050 --> 00:13:21,160 must equal your average speed someplace along the path. 271 00:13:21,160 --> 00:13:24,550 Now what that means pictorially is this-- 272 00:13:24,550 --> 00:13:27,750 again, I'll chance a freehand diagram-- 273 00:13:27,750 --> 00:13:31,320 suppose our curve is 'y' equals 'f of x'. 274 00:13:31,320 --> 00:13:35,230 See, I've drawn it to be smooth. 275 00:13:35,230 --> 00:13:39,380 Suppose it's continuous and smooth on this open interval 276 00:13:39,380 --> 00:13:41,680 from 'a' to 'b'. 277 00:13:41,680 --> 00:13:45,530 Now what is, if I think of a particle moving from point 'p' 278 00:13:45,530 --> 00:13:50,270 to point 'q', how do I identify the average speed? 279 00:13:50,270 --> 00:13:53,200 The average speed is the slope of the straight line that 280 00:13:53,200 --> 00:13:55,530 joins 'p' to 'q'. 281 00:13:55,530 --> 00:13:57,880 On the other hand, what is the instantaneous speed? 282 00:13:57,880 --> 00:14:00,280 If we think of it in terms of the picture, it's the slope of 283 00:14:00,280 --> 00:14:03,620 the line tangent to the curve at a particular point. 284 00:14:03,620 --> 00:14:06,420 So in other words, what we're saying is this. 285 00:14:06,420 --> 00:14:11,270 You see, if we were to take the line 'PQ', and we shift it 286 00:14:11,270 --> 00:14:15,070 parallel to itself, I think you can sense that the points 287 00:14:15,070 --> 00:14:19,070 'P' and 'Q', if we labeled 'P' and 'Q' the points at which 288 00:14:19,070 --> 00:14:22,170 this chord intersects the curve, the points 'P' and 'Q' 289 00:14:22,170 --> 00:14:24,440 will roll in closer and closer together. 290 00:14:24,440 --> 00:14:29,760 Ultimately, the line will fail to intersect the curve, and at 291 00:14:29,760 --> 00:14:32,340 the transition point, if the curve is smooth, all we're 292 00:14:32,340 --> 00:14:36,780 saying is that the last point at which that line touches the 293 00:14:36,780 --> 00:14:41,460 curve as we move it out, OK, that the line would be tangent 294 00:14:41,460 --> 00:14:44,060 to the curve at that particular point. 295 00:14:44,060 --> 00:14:45,430 You see, all we're saying is what? 296 00:14:45,430 --> 00:14:48,650 That someplace between here and here there must be a point 297 00:14:48,650 --> 00:14:51,760 where the tangent line to the curve is parallel 298 00:14:51,760 --> 00:14:53,310 to the chord 'PQ'. 299 00:14:53,310 --> 00:14:56,320 And now, we have all the ingredients that we need to 300 00:14:56,320 --> 00:14:59,500 see what the mean value theorem says geometrically. 301 00:14:59,500 --> 00:15:01,550 Let's call the tangent line 'l'. 302 00:15:01,550 --> 00:15:04,820 First of all, what is the slope of the line 'PQ'? 303 00:15:04,820 --> 00:15:07,120 Well, it's a straight line. 304 00:15:07,120 --> 00:15:09,610 The slope of a straight line is 'delta y' 305 00:15:09,610 --> 00:15:11,090 divided by 'delta x'. 306 00:15:11,090 --> 00:15:13,840 Well, notice that this height here is by 307 00:15:13,840 --> 00:15:15,680 definition 'f of b'. 308 00:15:15,680 --> 00:15:17,880 This height here is 'f of a'. 309 00:15:17,880 --> 00:15:22,820 So this height here is just 'f of b' minus 'f of a'. 310 00:15:22,820 --> 00:15:26,770 This length here is just 'b - a'. 311 00:15:26,770 --> 00:15:30,710 So the slope, 'delta y' divided by 'delta x', is just 312 00:15:30,710 --> 00:15:37,180 ''f of b' minus 'f of a'', over 'b - a'. 313 00:15:37,180 --> 00:15:40,010 On the other hand, what is the slope of the line 'l'? 314 00:15:40,010 --> 00:15:44,340 By definition, it's 'f prime of x' evaluated at 'x' equals 315 00:15:44,340 --> 00:15:47,100 'c', that's 'f prime of c'. 316 00:15:47,100 --> 00:15:49,240 Now what does it mean in terms of slopes for 317 00:15:49,240 --> 00:15:50,700 two lines to be parallel? 318 00:15:50,700 --> 00:15:53,630 It means that their slopes are equal. 319 00:15:53,630 --> 00:15:55,170 And where is 'c'? 320 00:15:55,170 --> 00:16:02,240 'c' is someplace in the open interval from 'a' to 'b'. 321 00:16:02,240 --> 00:16:07,730 Now the reason I call this intuitively an extension of 322 00:16:07,730 --> 00:16:09,490 Rolle's theorem-- 323 00:16:09,490 --> 00:16:12,270 and by the way, you'll notice that what I have to say is a 324 00:16:12,270 --> 00:16:16,490 much simpler demonstration than what's given in the book. 325 00:16:16,490 --> 00:16:19,460 But before you think I'm being egotistic about this, let me 326 00:16:19,460 --> 00:16:22,610 point out, as is so often the case that wherever my 327 00:16:22,610 --> 00:16:25,930 demonstrations are easier than the one in the book, I'm 328 00:16:25,930 --> 00:16:28,520 losing something in my presentation. 329 00:16:28,520 --> 00:16:31,410 Either I haven't shown the most analytic representation, 330 00:16:31,410 --> 00:16:34,760 or I'm overlooking a particular complicated side 331 00:16:34,760 --> 00:16:36,760 effect that might occur. 332 00:16:36,760 --> 00:16:39,330 But disregarding that for the moment, you see what I'm 333 00:16:39,330 --> 00:16:42,790 saying is this, let's suppose, for the sake of argument, we 334 00:16:42,790 --> 00:16:47,720 visualize the line 'PQ' as being our new x-axis. 335 00:16:47,720 --> 00:16:49,710 I'll call that the x1-axis. 336 00:16:49,710 --> 00:16:53,110 And now let's take a line perpendicular to 'PQ' and call 337 00:16:53,110 --> 00:16:56,740 that our new y-axis, the y1-axis. 338 00:16:56,740 --> 00:16:58,870 Now look at the curve that we've drawn. 339 00:16:58,870 --> 00:17:05,079 With respect to the y1- x1-axis, notice that the curve 340 00:17:05,079 --> 00:17:06,819 is smooth, right? 341 00:17:06,819 --> 00:17:08,329 It's unbroken. 342 00:17:08,329 --> 00:17:13,050 And it cuts the x1-axis at two points. 343 00:17:13,050 --> 00:17:16,270 Now if we apply Rolle's theorem with respect to the 344 00:17:16,270 --> 00:17:19,970 x1- y1-axis, we say, look, here's a curve 345 00:17:19,970 --> 00:17:21,770 which cuts the x-axis-- 346 00:17:21,770 --> 00:17:22,660 the x1-axis-- 347 00:17:22,660 --> 00:17:23,790 at two points. 348 00:17:23,790 --> 00:17:26,560 It's smooth. 349 00:17:26,560 --> 00:17:28,740 Therefore, it must level off someplace. 350 00:17:28,740 --> 00:17:31,370 In other words, there must be some point on this curve where 351 00:17:31,370 --> 00:17:36,480 the tangent line to the curve is parallel to the x1-axis. 352 00:17:36,480 --> 00:17:39,670 That's exactly, you see, what this thing here says. 353 00:17:39,670 --> 00:17:42,660 That's another geometric interpretation that indicates 354 00:17:42,660 --> 00:17:44,620 how Rolle's theorem might be used. 355 00:17:44,620 --> 00:17:49,000 However, there is a very, very subtle flaw in what I've said. 356 00:17:49,000 --> 00:17:52,240 One that is so subtle that you may not even notice it until I 357 00:17:52,240 --> 00:17:54,830 point it out to you, and even after I point it out, there's 358 00:17:54,830 --> 00:17:56,760 a chance you may not realize what I've said. 359 00:17:56,760 --> 00:17:59,910 Because it's a point that I know took me a long, long time 360 00:17:59,910 --> 00:18:01,390 to discover for myself. 361 00:18:01,390 --> 00:18:03,470 And it all hinges on the concept of 362 00:18:03,470 --> 00:18:05,200 single valuedness again. 363 00:18:05,200 --> 00:18:07,840 The trouble with this interpretation is the 364 00:18:07,840 --> 00:18:09,380 following-- 365 00:18:09,380 --> 00:18:12,770 and by the way, let me point out, I'm not knocking my 366 00:18:12,770 --> 00:18:15,390 interpretation, I think it's still a tremendous way of 367 00:18:15,390 --> 00:18:18,110 visualizing the result, but from an analytical point of 368 00:18:18,110 --> 00:18:20,600 view, why we have to be careful. 369 00:18:20,600 --> 00:18:23,830 Let's suppose that my curve 'y' equals 'f of x' happens to 370 00:18:23,830 --> 00:18:27,110 look something like this, OK? 371 00:18:27,110 --> 00:18:30,200 Happens to look something like this. 372 00:18:30,200 --> 00:18:33,160 Notice, barring any bad drawing that I've done here, 373 00:18:33,160 --> 00:18:37,130 that this curve is single value, that no line parallel 374 00:18:37,130 --> 00:18:40,670 to the y-axis cuts this curve in more than one place. 375 00:18:40,670 --> 00:18:43,150 Now here's my 'a' and here's my 'b'. 376 00:18:45,670 --> 00:18:49,820 And so I say, OK, by Rolle's theorem, if I look at this as 377 00:18:49,820 --> 00:18:52,740 being the x-axis and this as being the y-axis-- 378 00:18:52,740 --> 00:18:55,500 in other words, the x1- y1-axis again-- 379 00:18:55,500 --> 00:18:58,930 I say to myself, look, here's a smooth curve, it cuts the 380 00:18:58,930 --> 00:19:04,620 x-axis in two points, therefore, someplace between 381 00:19:04,620 --> 00:19:07,210 these two points, there must be a place where the curve 382 00:19:07,210 --> 00:19:11,120 levels off, et cetera, et cetera, et cetera. 383 00:19:11,120 --> 00:19:15,370 And the interesting point is to notice that a given curve, 384 00:19:15,370 --> 00:19:18,850 as to whether it's single valued or not, is dependent 385 00:19:18,850 --> 00:19:21,830 upon the orientation of the axes. 386 00:19:21,830 --> 00:19:24,040 In other words, notice that I've drawn this particular 387 00:19:24,040 --> 00:19:28,100 curve so it is single valued with respect to the xy-plane. 388 00:19:28,100 --> 00:19:31,580 On the other hand, with respect to the x1- 389 00:19:31,580 --> 00:19:36,020 y1-coordinate system, this curve is not single valued. 390 00:19:36,020 --> 00:19:40,920 Namely, observe how a line parallel to the y1-axis can 391 00:19:40,920 --> 00:19:45,630 intersect this curve at more than one point. 392 00:19:45,630 --> 00:19:48,590 In other words, whether a curve is single valued or not 393 00:19:48,590 --> 00:19:51,570 is not an absolute property independent of 394 00:19:51,570 --> 00:19:52,970 the coordinate system. 395 00:19:52,970 --> 00:19:57,160 So again, if I could be sure that when I rotated my 396 00:19:57,160 --> 00:20:01,460 coordinate axes the original single valued curve was still 397 00:20:01,460 --> 00:20:04,010 single valued, then my above proof 398 00:20:04,010 --> 00:20:05,250 would have been rigorous. 399 00:20:05,250 --> 00:20:07,410 But of course, I can't be sure of that. 400 00:20:07,410 --> 00:20:11,700 By the way, the technique used in the book is quite standard, 401 00:20:11,700 --> 00:20:14,270 and what it does is the following, it still utilizes 402 00:20:14,270 --> 00:20:17,630 Rolle's theorem, but the technique behind the proof in 403 00:20:17,630 --> 00:20:18,780 the book is this. 404 00:20:18,780 --> 00:20:23,160 The function that we set up is the vertical distance between 405 00:20:23,160 --> 00:20:27,930 the chord and the curve, as we move along this way. 406 00:20:27,930 --> 00:20:32,090 And notice that that distance is 0 at these two endpoints. 407 00:20:32,090 --> 00:20:33,450 OK? 408 00:20:33,450 --> 00:20:37,510 And therefore, Rolle's theorem applies to that function. 409 00:20:37,510 --> 00:20:39,710 And the whole idea is something like this. 410 00:20:39,710 --> 00:20:40,950 All we say is-- 411 00:20:40,950 --> 00:20:43,780 and the analytic part proves this-- 412 00:20:43,780 --> 00:20:47,190 all we say is look, the point at which this chord would have 413 00:20:47,190 --> 00:20:50,960 been tangent to the curve is the place where the vertical 414 00:20:50,960 --> 00:20:55,360 distance between the chord and the curve is what? 415 00:20:55,360 --> 00:20:56,730 Maximum. 416 00:20:56,730 --> 00:20:58,420 And we won't go into that right now, that 417 00:20:58,420 --> 00:21:00,250 is done in the text. 418 00:21:00,250 --> 00:21:03,520 All I wanted to do, as I always will do when possible, 419 00:21:03,520 --> 00:21:06,580 is that whenever the rigorous proof seems far more 420 00:21:06,580 --> 00:21:10,520 complicated than proofs which are more intuitive, I will not 421 00:21:10,520 --> 00:21:13,830 take the time in general, in our lectures, to give the more 422 00:21:13,830 --> 00:21:14,700 rigorous proof. 423 00:21:14,700 --> 00:21:17,590 What I will take the time to do is to show why the less 424 00:21:17,590 --> 00:21:19,610 rigorous proof has pitfalls. 425 00:21:19,610 --> 00:21:21,640 Well, enough said about the statement of 426 00:21:21,640 --> 00:21:22,950 the mean value theorem. 427 00:21:22,950 --> 00:21:26,080 Time is getting very short, and we don't need much more 428 00:21:26,080 --> 00:21:29,260 time to make the home run ball pitch that we 429 00:21:29,260 --> 00:21:30,520 want to make now. 430 00:21:30,520 --> 00:21:35,040 And the idea is this, that the most important analytical 431 00:21:35,040 --> 00:21:39,740 reason for having the mean value theorem is, for those of 432 00:21:39,740 --> 00:21:43,560 us who like to use our geometric intuition, it turns 433 00:21:43,560 --> 00:21:47,080 out that almost every geometrically obvious fact 434 00:21:47,080 --> 00:21:52,200 that has a proper analytic counterpart has the property 435 00:21:52,200 --> 00:21:54,640 that the analytic counterpart is proven by 436 00:21:54,640 --> 00:21:56,130 the mean value theorem. 437 00:21:56,130 --> 00:21:58,320 See, let me give you a simple for instance. 438 00:21:58,320 --> 00:22:02,700 In fact, in the text book this is called the first corollary 439 00:22:02,700 --> 00:22:04,530 to the mean value theorem. 440 00:22:04,530 --> 00:22:07,810 Suppose we have a function capital 'F of x', and we know 441 00:22:07,810 --> 00:22:11,330 that the derivative is always equal to 0. 442 00:22:11,330 --> 00:22:14,730 The claim is that 'F of x' itself must be a constant. 443 00:22:14,730 --> 00:22:17,160 By the way, two cautions here. 444 00:22:17,160 --> 00:22:20,140 Don't say that we've had this result before. 445 00:22:20,140 --> 00:22:23,030 The result that we had before was actually 446 00:22:23,030 --> 00:22:24,660 the converse of this. 447 00:22:24,660 --> 00:22:27,790 The result that we had before was the one that said what? 448 00:22:27,790 --> 00:22:32,570 If 'F of x' is a constant, then its derivative is 0. 449 00:22:32,570 --> 00:22:34,800 Now we're saying the opposite--not the opposite, 450 00:22:34,800 --> 00:22:35,400 but the converse. 451 00:22:35,400 --> 00:22:38,000 Now we're saying, look, if the derivative is always 0-- 452 00:22:38,000 --> 00:22:39,980 notice the use of my identity symbol here-- 453 00:22:39,980 --> 00:22:44,680 if the derivative is 0 for all values of 'x', then the 454 00:22:44,680 --> 00:22:46,480 function must've been a constant. 455 00:22:46,480 --> 00:22:48,660 Now, you know, geometrically this is a very 456 00:22:48,660 --> 00:22:50,090 simple thing to visualize. 457 00:22:50,090 --> 00:22:53,160 You say, look, the derivative is the slope. 458 00:22:53,160 --> 00:22:56,150 And if you're saying that the slope of the curve is always 459 00:22:56,150 --> 00:23:00,300 horizontal, the curve itself must be a straight line. 460 00:23:00,300 --> 00:23:02,960 And if the curve is a straight line, that's exactly what you 461 00:23:02,960 --> 00:23:05,590 mean by saying that the function is a constant. 462 00:23:05,590 --> 00:23:08,500 How do we prove this using the mean value theorem? 463 00:23:08,500 --> 00:23:11,250 See, and I just wanted to go through a proof here once, 464 00:23:11,250 --> 00:23:14,970 just so to get the idea of what a proof means. 465 00:23:14,970 --> 00:23:16,350 You see, to show that something is a 466 00:23:16,350 --> 00:23:17,930 constant should mean what? 467 00:23:17,930 --> 00:23:23,430 That if 'a' is unequal to 'b' for any two values 'a' and 468 00:23:23,430 --> 00:23:25,850 'b', 'F of a'-- 469 00:23:25,850 --> 00:23:28,570 well, I'm using capital 'F' here--capital 'F of a' has to 470 00:23:28,570 --> 00:23:30,210 equal capital 'F of b'. 471 00:23:30,210 --> 00:23:32,500 That's what you mean for a function to be a constant. 472 00:23:32,500 --> 00:23:36,530 No matter what the input is, the outputs are always equal. 473 00:23:36,530 --> 00:23:39,310 By the way, if 'a' equals 'b', it's trivial that 'F of a' 474 00:23:39,310 --> 00:23:40,350 equals 'F of b'. 475 00:23:40,350 --> 00:23:43,500 But essentially, to prove that capital F is a constant, this 476 00:23:43,500 --> 00:23:44,810 is what I have to prove. 477 00:23:44,810 --> 00:23:47,740 That if 'a' is different from 'b', no matter what 'a' and 478 00:23:47,740 --> 00:23:50,960 'b' I use, that 'F of a' equals 'F of b'. 479 00:23:50,960 --> 00:23:53,570 And the idea is by the mean value theorem, we say, look, 480 00:23:53,570 --> 00:23:56,090 what does the mean value theorem say? 481 00:23:56,090 --> 00:23:59,440 We're assuming now that 'F' is a continuous and 482 00:23:59,440 --> 00:24:05,420 differentiable function on an interval, OK, from 'a' to 'b'. 483 00:24:05,420 --> 00:24:08,210 The mean value theorem says under these conditions, there 484 00:24:08,210 --> 00:24:10,600 exists a number 'c' between 'a' and 485 00:24:10,600 --> 00:24:12,730 'b' with what property? 486 00:24:12,730 --> 00:24:20,260 That ''F of b' minus 'F of a'' over 'b - a' is equal to 'F 487 00:24:20,260 --> 00:24:22,810 prime of c'. 488 00:24:22,810 --> 00:24:26,260 That's just a statement of the mean value theorem. 489 00:24:26,260 --> 00:24:29,060 This is always true if the conditions of the mean value 490 00:24:29,060 --> 00:24:30,150 theorem apply. 491 00:24:30,150 --> 00:24:33,480 Now all we're saying is, in this particular problem, what 492 00:24:33,480 --> 00:24:36,130 property that capital 'F' have? 493 00:24:36,130 --> 00:24:38,670 It had the property that its derivative for all 494 00:24:38,670 --> 00:24:40,250 values of 'x' was 0. 495 00:24:40,250 --> 00:24:43,970 In particular then, when 'c' is the value that we're 496 00:24:43,970 --> 00:24:47,890 talking about, if 'F prime of x' is 0 for all values of 'x', 497 00:24:47,890 --> 00:24:51,400 in particular, then, it's 0 when 'x' is equal to 'c'. 498 00:24:51,400 --> 00:24:55,920 In other words, by our given hypothesis, this is 0. 499 00:24:55,920 --> 00:25:00,380 But if a fraction is 0, its numerator must be 0. 500 00:25:03,120 --> 00:25:04,300 That says what? 501 00:25:04,300 --> 00:25:07,920 'F of b' minus 'F of a' is 0. 502 00:25:07,920 --> 00:25:11,450 See, the only way a quotient can be 0 is for the 503 00:25:11,450 --> 00:25:12,370 numerator-- 504 00:25:12,370 --> 00:25:15,860 or the dividend, the divisor, I don't know these formal 505 00:25:15,860 --> 00:25:20,210 names, they slipped my mind, but the top one has to be 0. 506 00:25:20,210 --> 00:25:24,110 And if 'F of b' minus 'F of a' is 0, that says 'F of b' 507 00:25:24,110 --> 00:25:27,730 equals 'F of a', and that's precisely what we had to show 508 00:25:27,730 --> 00:25:30,830 to show that 'F' was a constant. 509 00:25:30,830 --> 00:25:31,490 OK? 510 00:25:31,490 --> 00:25:33,800 So again, notice, it's not that we're saying that the 511 00:25:33,800 --> 00:25:37,190 mean value theorem is a harder way of proving what we already 512 00:25:37,190 --> 00:25:40,610 know to be intuitively true, what we're saying is what? 513 00:25:40,610 --> 00:25:44,890 That we know that many intuitively obvious results 514 00:25:44,890 --> 00:25:46,380 frequently turn out to be false. 515 00:25:46,380 --> 00:25:49,760 We would like some analytical way of knowing which of the 516 00:25:49,760 --> 00:25:52,240 intuitive results are actually correct. 517 00:25:52,240 --> 00:25:55,125 All I'm saying is the mean value theorem gives us a big 518 00:25:55,125 --> 00:25:56,080 hint that way. 519 00:25:56,080 --> 00:25:59,750 By the way, let me close by giving you one more important 520 00:25:59,750 --> 00:26:01,400 illustration of what we can prove by 521 00:26:01,400 --> 00:26:03,030 the mean value theorem. 522 00:26:03,030 --> 00:26:05,810 And this is called a corollary of a corollary, as I'll show 523 00:26:05,810 --> 00:26:07,240 you what I mean in a minute. 524 00:26:07,240 --> 00:26:11,900 The next example that I want to use is what it means to say 525 00:26:11,900 --> 00:26:15,560 suppose I'm given two functions 'f' and 'g', and all 526 00:26:15,560 --> 00:26:17,790 I know about these two functions is that the 527 00:26:17,790 --> 00:26:19,970 derivatives are identical. 528 00:26:19,970 --> 00:26:23,280 In other ways, that 'f' and 'g' have the property that for 529 00:26:23,280 --> 00:26:26,950 every value of 'x', 'f prime of x' is equal 530 00:26:26,950 --> 00:26:28,320 to 'g prime of x'. 531 00:26:28,320 --> 00:26:31,200 By the way, when I say every value of 'x', again, it's 532 00:26:31,200 --> 00:26:32,890 local versus global. 533 00:26:32,890 --> 00:26:35,850 It's not necessary that this happens for all 'x', what is 534 00:26:35,850 --> 00:26:37,240 important is what? 535 00:26:37,240 --> 00:26:41,385 That 'x' be defined on some interval. 536 00:26:41,385 --> 00:26:44,340 In other words, even if I know that this property is true for 537 00:26:44,340 --> 00:26:47,470 some interval, I don't really care what happens outside of 538 00:26:47,470 --> 00:26:47,760 that interval. 539 00:26:47,760 --> 00:26:50,660 In terms of local properties, all I'm saying is, all I know 540 00:26:50,660 --> 00:26:53,300 is that for some interval, maybe the whole axis, doesn't 541 00:26:53,300 --> 00:26:58,390 make any difference, 'f prime' is identical to 'g prime'. 542 00:26:58,390 --> 00:27:01,690 Now, you would like to be able to say, maybe, that if 'f 543 00:27:01,690 --> 00:27:04,300 prime' is equal to 'g prime', 'f' equals 'g'. 544 00:27:04,300 --> 00:27:05,920 But that's not the case. 545 00:27:05,920 --> 00:27:08,510 What is the case is that the difference between the two 546 00:27:08,510 --> 00:27:10,740 functions must be a constant. 547 00:27:10,740 --> 00:27:14,180 Again, geometrically, what you're saying is what? 548 00:27:14,180 --> 00:27:20,400 That if you have two curves, which point by point always 549 00:27:20,400 --> 00:27:21,810 have the same slope-- 550 00:27:21,810 --> 00:27:23,410 in other words, for each 'x' value, the 551 00:27:23,410 --> 00:27:24,540 slopes are the same-- 552 00:27:24,540 --> 00:27:26,590 is just essentially saying that the 553 00:27:26,590 --> 00:27:28,530 two curves are parallel. 554 00:27:28,530 --> 00:27:31,130 And if they're parallel curves, what's a way of 555 00:27:31,130 --> 00:27:32,900 stating that two curves are parallel? 556 00:27:32,900 --> 00:27:36,910 That one is a constant displacement of the other. 557 00:27:36,910 --> 00:27:41,300 In other words, the geometric impact of two curves having 558 00:27:41,300 --> 00:27:44,110 the same derivative is not that the curves are the same, 559 00:27:44,110 --> 00:27:45,780 but that they're parallel. 560 00:27:45,780 --> 00:27:50,200 And by the way, the proof of this result is again a 561 00:27:50,200 --> 00:27:52,290 corollary to the mean value theorem. 562 00:27:52,290 --> 00:27:54,600 Namely, let's look at the function 'f of 563 00:27:54,600 --> 00:27:55,970 x' minus 'g of x'. 564 00:27:55,970 --> 00:27:58,000 Call that capital 'F of x'. 565 00:27:58,000 --> 00:28:02,100 Let capital 'F of x' be ''f of x' minus 'g of x''. 566 00:28:02,100 --> 00:28:04,460 Since the derivative of a difference is the difference 567 00:28:04,460 --> 00:28:08,430 of the derivatives, that would say the derivative of capital 568 00:28:08,430 --> 00:28:11,970 'F' is the derivative little 'f' minus the 569 00:28:11,970 --> 00:28:14,360 derivative of 'g'. 570 00:28:14,360 --> 00:28:15,090 OK? 571 00:28:15,090 --> 00:28:18,470 Now what do we know about 'f prime' and 'g prime of x'? 572 00:28:18,470 --> 00:28:21,250 We know that 'f prime of x' equals 'g prime 573 00:28:21,250 --> 00:28:22,970 of x' for all 'x'. 574 00:28:22,970 --> 00:28:28,180 Consequently, the difference between these two must be 0. 575 00:28:28,180 --> 00:28:30,550 Remember, if two functions are identical, their 576 00:28:30,550 --> 00:28:32,000 difference is 0. 577 00:28:32,000 --> 00:28:35,510 That says, therefore, that capital 'F prime of x' is 578 00:28:35,510 --> 00:28:36,920 identically 0. 579 00:28:36,920 --> 00:28:38,870 And by our previous theorem-- 580 00:28:38,870 --> 00:28:40,430 notice the beautiful logic of this-- 581 00:28:40,430 --> 00:28:44,540 from the mean value theorem, we proved that if the 582 00:28:44,540 --> 00:28:47,400 derivative of a function is identically 0, the function 583 00:28:47,400 --> 00:28:49,660 must be a constant. 584 00:28:49,660 --> 00:28:52,260 So we apply that here. 585 00:28:52,260 --> 00:28:54,020 But what was capital 'F'? 586 00:28:54,020 --> 00:28:57,410 It was 'little 'f - g'. 587 00:28:57,410 --> 00:29:00,130 And that proves our desired result. 588 00:29:00,130 --> 00:29:04,510 Again, what I want you to see here is that we have not done 589 00:29:04,510 --> 00:29:07,530 anything different with the mean value theorem. 590 00:29:07,530 --> 00:29:10,140 We're not trying to say we're going to prove results we 591 00:29:10,140 --> 00:29:11,410 couldn't prove before. 592 00:29:11,410 --> 00:29:12,230 Rather, what? 593 00:29:12,230 --> 00:29:15,970 The mean value theorem is our way of showing that certain 594 00:29:15,970 --> 00:29:19,540 intuitive results hold true analytically, that we can talk 595 00:29:19,540 --> 00:29:21,920 about parallel curves, and things like this. 596 00:29:21,920 --> 00:29:25,250 Most important, in terms of summarizing this lecture from 597 00:29:25,250 --> 00:29:28,650 a point of view of what's coming next, it's crucial to 598 00:29:28,650 --> 00:29:33,050 observe that this last example is what is going to allow us 599 00:29:33,050 --> 00:29:35,080 to enter the study of something called the 600 00:29:35,080 --> 00:29:36,640 'indefinite integral'. 601 00:29:36,640 --> 00:29:39,820 Or in another manner of speaking, something called the 602 00:29:39,820 --> 00:29:42,600 inverse of taking a derivative. 603 00:29:42,600 --> 00:29:46,290 You see, the idea is, notice that in these two examples we 604 00:29:46,290 --> 00:29:49,490 start with information about the derivative and deduce 605 00:29:49,490 --> 00:29:51,820 what's true about the original function. 606 00:29:51,820 --> 00:29:54,820 That's inverting the emphasis of what we've been doing up 607 00:29:54,820 --> 00:29:57,740 until now, where we've started with the function and 608 00:29:57,740 --> 00:29:59,800 investigated its derivative. 609 00:29:59,800 --> 00:30:02,910 To see this in more detail, join me again next time. 610 00:30:02,910 --> 00:30:04,520 And until next time, goodbye. 611 00:30:07,500 --> 00:30:10,700 Funding for the publication of this video was provided by the 612 00:30:10,700 --> 00:30:14,750 Gabriella and Paul Rosenbaum Foundation. 613 00:30:14,750 --> 00:30:18,930 Help OCW continue to provide free and open access to MIT 614 00:30:18,930 --> 00:30:23,130 courses by making a donation at ocw.mit.edu/donate.