1 00:00:00,000 --> 00:00:00,040 2 00:00:00,040 --> 00:00:01,940 ANNOUNCER: The following content is provided under a 3 00:00:01,940 --> 00:00:03,690 Creative Commons license. 4 00:00:03,690 --> 00:00:06,630 Your support will help MIT OpenCourseWare continue to 5 00:00:06,630 --> 00:00:09,990 offer high quality educational resources for free. 6 00:00:09,990 --> 00:00:12,830 To make a donation or to view additional materials from 7 00:00:12,830 --> 00:00:16,760 hundreds of MIT courses, visit MIT OpenCourseWare at 8 00:00:16,760 --> 00:00:18,010 ocw.mit.edu. 9 00:00:18,010 --> 00:00:31,180 10 00:00:31,180 --> 00:00:34,570 HERBERT GROSS: Hi, our lecture today, if we're looking at 11 00:00:34,570 --> 00:00:38,140 this from an analytical point view, should be called 'Maxima 12 00:00:38,140 --> 00:00:39,470 and Minima'. 13 00:00:39,470 --> 00:00:41,510 And if we're looking at it from a geometric point of 14 00:00:41,510 --> 00:00:44,470 view, 'High Points and Low Points'. 15 00:00:44,470 --> 00:00:47,530 And actually, whichever point of view we're looking at from, 16 00:00:47,530 --> 00:00:51,400 it's a very nice application off much of the theory that we 17 00:00:51,400 --> 00:00:53,260 have learned up until now. 18 00:00:53,260 --> 00:00:56,330 So without further ado, let's talk a little bit about what 19 00:00:56,330 --> 00:01:00,830 we mean by high points, low points, maxima or minima. 20 00:01:00,830 --> 00:01:03,250 Which is, as I say, I've called the lecture today. 21 00:01:03,250 --> 00:01:06,340 Now, as is always the case, we usually have to have some sort 22 00:01:06,340 --> 00:01:09,480 of a fundamental result from which all of our 23 00:01:09,480 --> 00:01:11,340 other results follow. 24 00:01:11,340 --> 00:01:14,620 And in this context what I call the fundamental theorem 25 00:01:14,620 --> 00:01:18,020 for a study of maxima minima is the following. 26 00:01:18,020 --> 00:01:23,765 Suppose that 'f of c' is at least as great as 'f of x' for 27 00:01:23,765 --> 00:01:26,540 all 'x' in a delta neighborhood of 'c'. 28 00:01:26,540 --> 00:01:31,090 In other words, we have some open interval with delta 29 00:01:31,090 --> 00:01:32,570 surrounding 'c'. 30 00:01:32,570 --> 00:01:36,290 And then for all 'x' in that neighborhood, 'f of c' is at 31 00:01:36,290 --> 00:01:38,110 least as great as 'f of x'. 32 00:01:38,110 --> 00:01:43,690 Or equivalently, it might be that 'f of c' is less than or 33 00:01:43,690 --> 00:01:47,270 equal to 'f of x' for all 'x' in this neighborhood. 34 00:01:47,270 --> 00:01:51,280 And suppose also, that 'f prime of c' exists. 35 00:01:51,280 --> 00:01:53,900 Then the theorem says, 'f prime of c' in this 36 00:01:53,900 --> 00:01:56,310 case, must be 0. 37 00:01:56,310 --> 00:01:59,530 And whereas this can be proven analytically, again the 38 00:01:59,530 --> 00:02:02,190 analytical proof is motivated by what's happening 39 00:02:02,190 --> 00:02:03,160 geometrically. 40 00:02:03,160 --> 00:02:05,700 And the geometric demonstration is particularly 41 00:02:05,700 --> 00:02:08,380 easy to visualize in this particular case. 42 00:02:08,380 --> 00:02:13,130 Let me run the risk of drawing this fairly freehand here. 43 00:02:13,130 --> 00:02:15,390 See what we're saying is this. 44 00:02:15,390 --> 00:02:23,470 Suppose you have at the point (c, 'f of c') on this 45 00:02:23,470 --> 00:02:24,840 particular curve. 46 00:02:24,840 --> 00:02:28,240 Suppose that this particular curve the derivative exists 47 00:02:28,240 --> 00:02:29,060 and is positive. 48 00:02:29,060 --> 00:02:30,070 See that's the way I've drawn this. 49 00:02:30,070 --> 00:02:32,490 In other words, notice that in the neighborhood of the point 50 00:02:32,490 --> 00:02:36,200 'c' here, for example, the derivative here is positive. 51 00:02:36,200 --> 00:02:38,450 The curve is always rising. 52 00:02:38,450 --> 00:02:41,540 Now, what we're saying over here is that if you look at 53 00:02:41,540 --> 00:02:45,660 this particular picture, observe that 'f of c' will not 54 00:02:45,660 --> 00:02:48,700 exceed 'f of x' for all 'x' in this neighborhood. 55 00:02:48,700 --> 00:02:50,710 In fact, I think you can see just by looking at this 56 00:02:50,710 --> 00:02:55,920 picture that as 'x' increases, 'f of x' increases. 57 00:02:55,920 --> 00:02:59,490 In other words, in terms of this picture, if 'x' is less 58 00:02:59,490 --> 00:03:04,140 than 'c', 'f of x' is less than 'f of c'. 59 00:03:04,140 --> 00:03:09,030 And if 'x' is greater than 'c', 'f of x' is 60 00:03:09,030 --> 00:03:11,730 greater than 'f of c'. 61 00:03:11,730 --> 00:03:14,750 In other words, what we're saying is if the derivative is 62 00:03:14,750 --> 00:03:17,800 positive, it means that the curve is always rising. 63 00:03:17,800 --> 00:03:20,840 And hence, the point in the middle of that neighborhood 64 00:03:20,840 --> 00:03:24,060 cannot be the highest point in that neighborhood. 65 00:03:24,060 --> 00:03:26,350 Nor, for example, can it be the lowest point. 66 00:03:26,350 --> 00:03:30,910 In fact, a similar argument holds if we want to illustrate 67 00:03:30,910 --> 00:03:33,950 that 'f of c' is less than or equal to 'f of x' in this 68 00:03:33,950 --> 00:03:35,640 particular neighborhood. 69 00:03:35,640 --> 00:03:39,470 And by the way, again, observe what we're saying. 70 00:03:39,470 --> 00:03:41,450 This is rather crucial over here. 71 00:03:41,450 --> 00:03:44,660 First of all, we're talking about a sufficiently small 72 00:03:44,660 --> 00:03:46,040 neighborhood of 'c'. 73 00:03:46,040 --> 00:03:50,360 What I mean by that is something like this. 74 00:03:50,360 --> 00:03:52,910 And by the way, this could cause a little bit of a 75 00:03:52,910 --> 00:03:55,935 problem if you're not careful with your language. 76 00:03:55,935 --> 00:03:58,930 77 00:03:58,930 --> 00:04:02,120 If we drew a picture like this and you say to somebody, where 78 00:04:02,120 --> 00:04:03,740 are the high-low points on this? 79 00:04:03,740 --> 00:04:06,080 I think it's quite natural that the person would say, 80 00:04:06,080 --> 00:04:07,820 well, this is the high point. 81 00:04:07,820 --> 00:04:09,350 I'll call that 'H'. 82 00:04:09,350 --> 00:04:10,630 And this is the low point. 83 00:04:10,630 --> 00:04:12,270 I'll call that 'L'. 84 00:04:12,270 --> 00:04:16,180 And he might tend to forget about this point here because 85 00:04:16,180 --> 00:04:19,360 even though it's fairly high, it's not nearly as high as 86 00:04:19,360 --> 00:04:20,160 this point. 87 00:04:20,160 --> 00:04:26,960 What I'd like you to see however is that our definition 88 00:04:26,960 --> 00:04:28,750 talks about what? 89 00:04:28,750 --> 00:04:31,430 In a neighborhood of the given point. 90 00:04:31,430 --> 00:04:34,050 You see what we're saying here is that if we pick a 91 00:04:34,050 --> 00:04:37,470 particular neighborhood, an appropriately chosen 92 00:04:37,470 --> 00:04:41,190 neighborhood surrounding 'c', then what is true is that at 93 00:04:41,190 --> 00:04:45,760 the value of 'x' corresponding to 'c', 'f of c' is the 94 00:04:45,760 --> 00:04:49,900 highest point in a suitable neighborhood of 'c'. 95 00:04:49,900 --> 00:04:53,140 In other words, if you knew that for some reason or other, 96 00:04:53,140 --> 00:04:57,210 you had to be working in this neighborhood here, you could 97 00:04:57,210 --> 00:05:00,980 say, well, in the domain in which I'm interested in, this 98 00:05:00,980 --> 00:05:02,730 is a high point. 99 00:05:02,730 --> 00:05:07,170 And this is why we talk about 'local' or 'relative' maxima 100 00:05:07,170 --> 00:05:13,080 or minima in addition to 'absolute' maxima and minima. 101 00:05:13,080 --> 00:05:15,980 In fact, you see, this happens quite frequently in practice. 102 00:05:15,980 --> 00:05:18,420 Suppose you were doing an experiment and you really 103 00:05:18,420 --> 00:05:21,500 wanted to produce the largest possible value of 'y'. 104 00:05:21,500 --> 00:05:25,000 Well, you see, utopianaly you would like to pick 'x' out 105 00:05:25,000 --> 00:05:26,140 here someplace. 106 00:05:26,140 --> 00:05:29,440 But suppose because of some constraint in the laboratory, 107 00:05:29,440 --> 00:05:31,930 the largest value of 'x' that you could choose 108 00:05:31,930 --> 00:05:33,430 might be over here. 109 00:05:33,430 --> 00:05:35,670 And then you see what the equivalent problem would be. 110 00:05:35,670 --> 00:05:38,410 And this is where the domain of the function has such a 111 00:05:38,410 --> 00:05:42,520 powerful meaning in terms of practical applications. 112 00:05:42,520 --> 00:05:45,520 What you're saying is look-it , if the domain of my function 113 00:05:45,520 --> 00:05:48,110 is limited to this interval over here, then this 114 00:05:48,110 --> 00:05:50,300 particular point as far as I'm concerned 115 00:05:50,300 --> 00:05:52,840 is the highest point. 116 00:05:52,840 --> 00:05:56,265 In other words then, notice that the labelling 'N sub 117 00:05:56,265 --> 00:05:59,950 'delta of c'' indicates that you're talking locally rather 118 00:05:59,950 --> 00:06:03,640 than globally in a neighborhood of 'c'. 119 00:06:03,640 --> 00:06:06,140 And this is the important issue here. 120 00:06:06,140 --> 00:06:10,560 Now, the hardest part about this particular result as I 121 00:06:10,560 --> 00:06:15,520 see it, is not understanding the result as much as it is of 122 00:06:15,520 --> 00:06:18,570 reading more into the result than what's really there. 123 00:06:18,570 --> 00:06:21,420 And so I have a few cautions for you. 124 00:06:21,420 --> 00:06:23,800 The three commandments they turn out to be. 125 00:06:23,800 --> 00:06:27,510 The first is, beware of false converses. 126 00:06:27,510 --> 00:06:30,550 And before you can be beware of false converses, you have 127 00:06:30,550 --> 00:06:32,210 to know what a converse is. 128 00:06:32,210 --> 00:06:36,790 Roughly speaking, a converse applies only to an if-then 129 00:06:36,790 --> 00:06:38,350 type of statement. 130 00:06:38,350 --> 00:06:41,120 And you get the converse of a given statement by 131 00:06:41,120 --> 00:06:45,540 interchanging the clauses that follow the if and the then. 132 00:06:45,540 --> 00:06:47,860 And the reason you have to beware is that a true 133 00:06:47,860 --> 00:06:50,370 statement can have a false converse. 134 00:06:50,370 --> 00:06:53,250 For example, consider the following true statement. 135 00:06:53,250 --> 00:06:58,390 If a person is listening to me lecture now, then he is alive. 136 00:06:58,390 --> 00:07:01,140 Hopefully, a true statement. 137 00:07:01,140 --> 00:07:04,590 If we now interchange the clauses to form the converse 138 00:07:04,590 --> 00:07:07,460 it says if a person is alive, then he is 139 00:07:07,460 --> 00:07:09,650 listening to me lecture. 140 00:07:09,650 --> 00:07:12,420 A tragically false statement. 141 00:07:12,420 --> 00:07:15,770 But notice the difference between inverting the clauses 142 00:07:15,770 --> 00:07:17,590 of an if-then statement. 143 00:07:17,590 --> 00:07:18,870 And the idea is this. 144 00:07:18,870 --> 00:07:25,010 Notice that our theorem says if 'f prime' exists, then-- or 145 00:07:25,010 --> 00:07:28,270 if there is a local maximum or a local minimum, 'f 146 00:07:28,270 --> 00:07:29,960 prime of c' is 0. 147 00:07:29,960 --> 00:07:34,020 It does not say that if 'f prime of c' is 0 we have a 148 00:07:34,020 --> 00:07:36,220 local maximum or a local minimum. 149 00:07:36,220 --> 00:07:38,560 Perhaps the easiest way to see that is 150 00:07:38,560 --> 00:07:40,720 in terms of an example. 151 00:07:40,720 --> 00:07:43,790 That's the nicest thing to show how a converse is false. 152 00:07:43,790 --> 00:07:46,550 To prove that something is true, you can't do it just by 153 00:07:46,550 --> 00:07:48,350 showing its true in certain cases. 154 00:07:48,350 --> 00:07:51,180 But to show that something is not true, all you have to do 155 00:07:51,180 --> 00:07:54,060 is exhibit one example in which the result is false. 156 00:07:54,060 --> 00:07:56,780 That's enough to prove that it can't always be true. 157 00:07:56,780 --> 00:07:59,660 For example, in this particular diagram, notice 158 00:07:59,660 --> 00:08:02,340 that in the curve 'y' equals 'f of x', which I've drawn 159 00:08:02,340 --> 00:08:06,900 here, the curve at the point 'c' comma 'f of c' has a 160 00:08:06,900 --> 00:08:08,230 horizontal tangent. 161 00:08:08,230 --> 00:08:10,860 In other words, 'f prime of c' is 0 here. 162 00:08:10,860 --> 00:08:14,520 But notice that in any neighborhood that surrounds 163 00:08:14,520 --> 00:08:18,780 'c', in any open interval that's around 'c', notice that 164 00:08:18,780 --> 00:08:22,650 if 'x' is less than 'c', 'f of x' will be less than 'f of c'. 165 00:08:22,650 --> 00:08:25,610 And if 'x' is greater than 'c', 'f of x' will be greater 166 00:08:25,610 --> 00:08:26,710 than 'f of c'. 167 00:08:26,710 --> 00:08:29,000 In other words, notice that except for the stationary 168 00:08:29,000 --> 00:08:32,370 value at which we have a horizontal tangent, the graph 169 00:08:32,370 --> 00:08:36,460 is always rising in any neighborhood of 'c'. 170 00:08:36,460 --> 00:08:39,409 That's the first point I want you to see. 171 00:08:39,409 --> 00:08:42,860 The second point says beware if 'f prime 172 00:08:42,860 --> 00:08:44,670 of c' doesn't exist. 173 00:08:44,670 --> 00:08:46,650 See all our theorem said is, look it. 174 00:08:46,650 --> 00:08:49,720 If you have a relative high point, a relative low point, 175 00:08:49,720 --> 00:08:53,900 relative max, or a relative min at 'x' equal 'c', and if 176 00:08:53,900 --> 00:09:00,050 'f prime of c' exists, then 'f prime of c' is 0. 177 00:09:00,050 --> 00:09:03,060 But who said that 'f prime of c' has to exist? 178 00:09:03,060 --> 00:09:05,440 Again, let's look in terms of an example. 179 00:09:05,440 --> 00:09:09,830 If we let 'f of x' equal the absolute value of 'x', recall 180 00:09:09,830 --> 00:09:15,160 from our previous assignments and the like that the 181 00:09:15,160 --> 00:09:18,050 derivative of 'f of x', in this case, does not 182 00:09:18,050 --> 00:09:19,570 exist when 'x' is 0. 183 00:09:19,570 --> 00:09:26,500 In other words, 'f prime of 0' doesn't exist. 184 00:09:26,500 --> 00:09:27,500 And why is that? 185 00:09:27,500 --> 00:09:30,480 Well, notice that the graph of 'y' equals the absolute value 186 00:09:30,480 --> 00:09:33,640 of 'x' is the straight line 'y' equals 'x' if 'x' is 187 00:09:33,640 --> 00:09:37,040 non-negative and the straight line 'y' equals 'minus x' if 188 00:09:37,040 --> 00:09:38,130 'x' is negative. 189 00:09:38,130 --> 00:09:41,960 In other words, 'f prime of x' is 1 if 'x' is positive. 190 00:09:41,960 --> 00:09:44,940 It's minus 1 if 'x' is negative. 191 00:09:44,940 --> 00:09:47,460 And hence, a jump discontinuity in the 192 00:09:47,460 --> 00:09:49,840 derivative at 0. 193 00:09:49,840 --> 00:09:51,030 But you see, the point is this. 194 00:09:51,030 --> 00:09:55,430 If you look at our graph, do we have a low point 195 00:09:55,430 --> 00:09:57,020 at 'x' equals 0? 196 00:09:57,020 --> 00:09:59,030 In other words, is this the lowest point in the 197 00:09:59,030 --> 00:10:01,180 neighborhood of 0? 198 00:10:01,180 --> 00:10:02,230 And the answer is yes. 199 00:10:02,230 --> 00:10:04,130 In fact, it's an 'absolute' low point. 200 00:10:04,130 --> 00:10:07,030 Meaning that no matter where you go, no point on our graph 201 00:10:07,030 --> 00:10:10,550 can be less or lower than this particular point here. 202 00:10:10,550 --> 00:10:11,440 But that's irrelevant. 203 00:10:11,440 --> 00:10:12,350 The point is what? 204 00:10:12,350 --> 00:10:16,010 If we looked for a place where 'f prime of c' was 0 in this 205 00:10:16,010 --> 00:10:18,620 example, we wouldn't find one. 206 00:10:18,620 --> 00:10:20,520 That would not mean that there wasn't a low point. 207 00:10:20,520 --> 00:10:21,880 What happened was what? 208 00:10:21,880 --> 00:10:24,220 The low point snuck in at a place where the 209 00:10:24,220 --> 00:10:26,300 derivative did not exist. 210 00:10:26,300 --> 00:10:28,490 And again, you have to be careful. 211 00:10:28,490 --> 00:10:29,750 All I said was what? 212 00:10:29,750 --> 00:10:34,210 Beware of points for which 'f prime of c' does not exist. 213 00:10:34,210 --> 00:10:37,030 It does not mean that at each point where 'f prime of c' 214 00:10:37,030 --> 00:10:39,030 does not exist that you're going to have a 215 00:10:39,030 --> 00:10:40,270 high or a low point. 216 00:10:40,270 --> 00:10:44,090 For example, remember that the derivative not existing 217 00:10:44,090 --> 00:10:46,440 loosely speaking means what? 218 00:10:46,440 --> 00:10:48,570 That there's a sharp corner to the curve. 219 00:10:48,570 --> 00:10:50,510 What I'm thinking of is something like this. 220 00:10:50,510 --> 00:10:52,520 Let's take a curve that's always rising, say 221 00:10:52,520 --> 00:10:53,900 something like this. 222 00:10:53,900 --> 00:10:57,300 Now, let's suppose we put a sharp corner in here, but in 223 00:10:57,300 --> 00:10:58,940 such a way that the curve still 224 00:10:58,940 --> 00:11:00,770 continues to always rise. 225 00:11:00,770 --> 00:11:03,180 Say like this. 226 00:11:03,180 --> 00:11:06,390 Now you see, at this particular value of 'c', 'f 227 00:11:06,390 --> 00:11:11,860 prime of c' doesn't exist. 228 00:11:11,860 --> 00:11:13,010 I think you can see intuitively 229 00:11:13,010 --> 00:11:14,580 what's happening here. 230 00:11:14,580 --> 00:11:17,800 The slope approaches one value as you approach 'c' from the 231 00:11:17,800 --> 00:11:22,040 left and another value if you approach 'c' from the right. 232 00:11:22,040 --> 00:11:25,970 Now even know 'f prime of c' doesn't exist at this point, 233 00:11:25,970 --> 00:11:27,550 notice that this point is neither a 234 00:11:27,550 --> 00:11:29,170 high nor a low point. 235 00:11:29,170 --> 00:11:30,830 Meaning that all points what? 236 00:11:30,830 --> 00:11:34,710 To the left of this point are below it and all points to the 237 00:11:34,710 --> 00:11:37,540 right of this point are above it. 238 00:11:37,540 --> 00:11:38,460 So again, what? 239 00:11:38,460 --> 00:11:42,190 Beware when 'f prime of c' equals 0, but don't jump to 240 00:11:42,190 --> 00:11:44,020 any false conclusions. 241 00:11:44,020 --> 00:11:48,450 And the third caution is an extremely important one. 242 00:11:48,450 --> 00:11:51,450 In fact, for the first time from a practical point of 243 00:11:51,450 --> 00:11:55,760 view, we are going to probably see analytically what is the 244 00:11:55,760 --> 00:11:57,490 difference between an open interval 245 00:11:57,490 --> 00:11:59,300 and a closed interval. 246 00:11:59,300 --> 00:12:03,190 Suppose I have the function 'f of x' equals 'x squared', but 247 00:12:03,190 --> 00:12:06,810 the domain of 'f' is now the open interval from 2 to 3. 248 00:12:06,810 --> 00:12:10,700 In other words, the inputs of the 'f' machine are restricted 249 00:12:10,700 --> 00:12:13,850 to all those numbers which are greater than 2, 250 00:12:13,850 --> 00:12:16,430 but less than 3. 251 00:12:16,430 --> 00:12:17,670 Now, let's take a look here. 252 00:12:17,670 --> 00:12:21,750 Let's first of all, see where 'f prime of x' is 0. 253 00:12:21,750 --> 00:12:26,610 First of all, 'f prime of x' is '2x' and that equals 0 if 254 00:12:26,610 --> 00:12:29,480 and only if 'x' equals 0. 255 00:12:29,480 --> 00:12:31,880 Now here's where the domain is very important. 256 00:12:31,880 --> 00:12:35,870 Is 'x' equal 0 in our domain? 257 00:12:35,870 --> 00:12:37,920 The answer is no. 258 00:12:37,920 --> 00:12:39,420 'x' equals 0 is not in our domain. 259 00:12:39,420 --> 00:12:42,480 Our domain is restricted to be the open interval from 2 to 3. 260 00:12:42,480 --> 00:12:45,660 Therefore, as far as our function 'f' is concerned-- 261 00:12:45,660 --> 00:12:48,790 and remember, way back in one of our early, I don't even 262 00:12:48,790 --> 00:12:49,270 think it was a lecture. 263 00:12:49,270 --> 00:12:50,620 It was in our supplementary notes. 264 00:12:50,620 --> 00:12:53,700 We pointed out that when you define a function, you need 265 00:12:53,700 --> 00:12:57,510 not only the rule, but you must specify the domain. 266 00:12:57,510 --> 00:13:00,110 Remember, two functions were equal not only if they were 267 00:13:00,110 --> 00:13:04,250 the same rule, but they had to be defined on the same domain. 268 00:13:04,250 --> 00:13:07,160 So f here is defined on a domain from 2 to 3. 269 00:13:07,160 --> 00:13:10,380 And on that particular domain of definition there is no 270 00:13:10,380 --> 00:13:12,750 place where the derivative is 0. 271 00:13:12,750 --> 00:13:16,900 By the same token, since the derivative is a polynomial and 272 00:13:16,900 --> 00:13:19,620 all polynomials are differentiable, and all 273 00:13:19,620 --> 00:13:23,270 differentiable functions are continuous, there certainly 274 00:13:23,270 --> 00:13:26,750 will be no places where 'f prime of x' doesn't exist. 275 00:13:26,750 --> 00:13:34,140 In other words, 'f prime' exists for all 'x' in the 276 00:13:34,140 --> 00:13:36,290 domain of 'f'. 277 00:13:36,290 --> 00:13:41,100 In other words, here is a particular example in which a 278 00:13:41,100 --> 00:13:45,350 particular function on its domain of definition does not 279 00:13:45,350 --> 00:13:48,500 have any high or low points on it. 280 00:13:48,500 --> 00:13:52,100 And I'll illustrate that graphically in a few moments. 281 00:13:52,100 --> 00:13:55,730 I simply wanted to put this on the board first to put it into 282 00:13:55,730 --> 00:13:59,970 sharp contrast with what we're going to say next. 283 00:13:59,970 --> 00:14:03,180 If we're not careful, in fact, the next problem looks exactly 284 00:14:03,180 --> 00:14:05,340 the same as the problem that we just solved. 285 00:14:05,340 --> 00:14:08,820 Namely, what we want to do now is investigate the function 'f 286 00:14:08,820 --> 00:14:10,940 of x' equals 'x squared'. 287 00:14:10,940 --> 00:14:14,200 But we want the domain of 'f' now to be what? 288 00:14:14,200 --> 00:14:16,680 The closed interval from 2 to 3. 289 00:14:16,680 --> 00:14:19,080 In other words, the only difference between this 290 00:14:19,080 --> 00:14:22,300 problem and the problem that we just solved is that now we 291 00:14:22,300 --> 00:14:26,190 want the endpoints included. 292 00:14:26,190 --> 00:14:28,720 Now, there's no sense repeating the 293 00:14:28,720 --> 00:14:30,210 part that we did before. 294 00:14:30,210 --> 00:14:36,030 First of all, will 'f prime of x' equal 0 in the domain of 295 00:14:36,030 --> 00:14:37,110 definition? 296 00:14:37,110 --> 00:14:38,990 As we saw before, no. 297 00:14:38,990 --> 00:14:41,650 298 00:14:41,650 --> 00:14:50,330 Is 'f prime' nonexistent any place in the interval? 299 00:14:50,330 --> 00:14:51,400 No, it's differentiable. 300 00:14:51,400 --> 00:14:53,150 It's a smooth polynomial curve. 301 00:14:53,150 --> 00:14:55,380 Answer is no. 302 00:14:55,380 --> 00:14:59,380 Now, here's where we come to two very important points that 303 00:14:59,380 --> 00:15:02,410 hopefully, will clarify certain conventions that were 304 00:15:02,410 --> 00:15:04,090 made in the textbook. 305 00:15:04,090 --> 00:15:07,420 In our section on continuity there was a little result that 306 00:15:07,420 --> 00:15:09,370 may have seemed a little bit obscure. 307 00:15:09,370 --> 00:15:10,300 It said what? 308 00:15:10,300 --> 00:15:14,190 That a continuous function defined on a closed interval 309 00:15:14,190 --> 00:15:17,760 must take on its maximum and minimum values someplace on 310 00:15:17,760 --> 00:15:19,360 that closed interval. 311 00:15:19,360 --> 00:15:22,630 You see, all we've proven over here is what? 312 00:15:22,630 --> 00:15:33,640 That if 'f' has a max or min, what we've proven is what? 313 00:15:33,640 --> 00:15:42,625 It does not occur in the open interval from 'a' to 'b'. 314 00:15:42,625 --> 00:15:43,700 Well, look it. 315 00:15:43,700 --> 00:15:48,260 If the high and low points have to occur some place on 316 00:15:48,260 --> 00:15:52,120 the closed interval and they can't appear, as we've seen, 317 00:15:52,120 --> 00:15:56,960 in the open interval, where must they occur? 318 00:15:56,960 --> 00:15:59,310 Well, if they can't be inside and they have to be on the 319 00:15:59,310 --> 00:16:02,850 interval, it must be that they take place at the endpoints. 320 00:16:02,850 --> 00:16:05,950 321 00:16:05,950 --> 00:16:07,590 Let's go back without referring 322 00:16:07,590 --> 00:16:08,920 back to another board. 323 00:16:08,920 --> 00:16:11,400 Remember we wrote down when we're talking about high and 324 00:16:11,400 --> 00:16:15,370 low points that we talked about 'x' being in a delta 325 00:16:15,370 --> 00:16:16,890 neighborhood of 'c'. 326 00:16:16,890 --> 00:16:18,040 It meant what? 327 00:16:18,040 --> 00:16:23,390 That you could surround 'c' by some bandwidth 'delta'. 328 00:16:23,390 --> 00:16:28,330 329 00:16:28,330 --> 00:16:30,870 And notice that our definition in the textbook of a 330 00:16:30,870 --> 00:16:33,750 neighborhood was always an open interval. 331 00:16:33,750 --> 00:16:36,440 And the reason that the definition is given to be open 332 00:16:36,440 --> 00:16:39,420 is that notice that what the definition now says is what? 333 00:16:39,420 --> 00:16:44,090 You know what's going on on either side of 'c'. 334 00:16:44,090 --> 00:16:46,830 We know what's going on either side of 'c'. 335 00:16:46,830 --> 00:16:52,050 Notice that in the case where you have a closed interval, by 336 00:16:52,050 --> 00:16:54,530 definition of a closed interval notice that what? 337 00:16:54,530 --> 00:17:00,220 We know what's happening as we come in on 'a' from the right. 338 00:17:00,220 --> 00:17:03,450 And we know what's happening to 'b' as we come in on it 339 00:17:03,450 --> 00:17:04,420 from the left. 340 00:17:04,420 --> 00:17:07,740 But allegedly, meaning that since the function is only 341 00:17:07,740 --> 00:17:10,470 defined on the closed interval from 'a' to 'b', we don't know 342 00:17:10,470 --> 00:17:13,150 what's happening before 'a', and we don't know what's 343 00:17:13,150 --> 00:17:15,520 happening before 'b'. 344 00:17:15,520 --> 00:17:19,510 In other words, this is why the test for 'f prime of c' 345 00:17:19,510 --> 00:17:25,250 equaling 0 applies only to 'c' being in the interior. 346 00:17:25,250 --> 00:17:28,040 In other words, in an open interval. 347 00:17:28,040 --> 00:17:31,030 So in other words then, you see if a function happens to 348 00:17:31,030 --> 00:17:36,080 be continuous on the closed interval and it doesn't have 349 00:17:36,080 --> 00:17:39,990 any high-low points in the interior of the interval, then 350 00:17:39,990 --> 00:17:42,670 it must have its high-low points where? 351 00:17:42,670 --> 00:17:44,720 It must be at the endpoints. 352 00:17:44,720 --> 00:17:47,460 And to show you quite simply what was going on in this 353 00:17:47,460 --> 00:17:51,650 particular problem, notice that if we graph 'f of x' 354 00:17:51,650 --> 00:17:55,830 equals 'x squared' and look at this say, first of all, on the 355 00:17:55,830 --> 00:17:59,490 closed interval from 2 to 3 what we're saying is look it, 356 00:17:59,490 --> 00:18:03,570 any point in which we look at a neighborhood that isn't 2 or 357 00:18:03,570 --> 00:18:07,540 3, the curve is rising on one side of the point. 358 00:18:07,540 --> 00:18:09,630 In other words, the curve is rising on both sides of the 359 00:18:09,630 --> 00:18:13,030 point, so that if you are to the left of the point, the 360 00:18:13,030 --> 00:18:15,700 height will be less than the point you're interested in. 361 00:18:15,700 --> 00:18:18,410 If you're to the right of it, the height will exceed the 362 00:18:18,410 --> 00:18:19,810 point that you're interested in. 363 00:18:19,810 --> 00:18:23,260 But notice that at the end points themselves, you do have 364 00:18:23,260 --> 00:18:24,420 what in this case? 365 00:18:24,420 --> 00:18:27,320 Not a relative high or low, but actually an 366 00:18:27,320 --> 00:18:29,220 absolute high or low. 367 00:18:29,220 --> 00:18:31,570 In other words, the point '2 comma 4'. 368 00:18:31,570 --> 00:18:34,890 See, what is the endpoint here? 369 00:18:34,890 --> 00:18:37,800 One endpoint is 4, the other endpoint going up on the 370 00:18:37,800 --> 00:18:39,440 y-direction here is 9. 371 00:18:39,440 --> 00:18:40,890 What you're saying is what? 372 00:18:40,890 --> 00:18:43,130 4 is less than 9. 373 00:18:43,130 --> 00:18:46,420 Every value of 'x squared' between 2 and 3 falls 374 00:18:46,420 --> 00:18:48,190 between 4 and 9. 375 00:18:48,190 --> 00:18:49,230 And what you're saying is what? 376 00:18:49,230 --> 00:18:51,800 That the lowest point occurs when 'x' is 2, the highest 377 00:18:51,800 --> 00:18:54,090 point occurs when 'y' is 3. 378 00:18:54,090 --> 00:18:57,410 The interesting point to note is that if you now look at the 379 00:18:57,410 --> 00:19:01,530 open interval and exclude the endpoints, you cannot get a 380 00:19:01,530 --> 00:19:03,600 lowest point or a highest point. 381 00:19:03,600 --> 00:19:06,960 Namely, notice that if you allow yourself to get as close 382 00:19:06,960 --> 00:19:10,150 to 2 as you want without ever getting there, it means you 383 00:19:10,150 --> 00:19:11,290 could have done what? 384 00:19:11,290 --> 00:19:12,880 Moved closer to 2. 385 00:19:12,880 --> 00:19:17,210 In other words, if 'x' is greater than 2, you can pick 386 00:19:17,210 --> 00:19:19,190 another value that's what? 387 00:19:19,190 --> 00:19:20,270 Between 'x' and 2. 388 00:19:20,270 --> 00:19:21,700 There's a space there. 389 00:19:21,700 --> 00:19:23,240 In other words, what you saying is that no matter how 390 00:19:23,240 --> 00:19:26,930 low you get here, as long as you're not exactly at 2, you 391 00:19:26,930 --> 00:19:28,820 can find the point that's lower. 392 00:19:28,820 --> 00:19:31,950 And in the same way as you move out this way, as you move 393 00:19:31,950 --> 00:19:34,610 closer and closer to this point, if you exclude this 394 00:19:34,610 --> 00:19:37,300 point itself, no matter where you stop, you could have 395 00:19:37,300 --> 00:19:40,220 always found the point that was a little bit higher. 396 00:19:40,220 --> 00:19:42,430 And again, you have to be where? 397 00:19:42,430 --> 00:19:46,340 When I say check the endpoints of a closed interval, it does 398 00:19:46,340 --> 00:19:48,070 not mean that the endpoints will give 399 00:19:48,070 --> 00:19:49,710 you high or low points. 400 00:19:49,710 --> 00:19:52,670 For example, look at the following curve. 401 00:19:52,670 --> 00:19:54,220 It looks something like this. 402 00:19:54,220 --> 00:19:55,250 It's continuous. 403 00:19:55,250 --> 00:19:58,010 It's defined on the closed interval from 'a' to 'b'. 404 00:19:58,010 --> 00:20:02,710 Notice that at a we do not get an absolute maximum. 405 00:20:02,710 --> 00:20:03,780 In fact, what? 406 00:20:03,780 --> 00:20:07,180 All of these points on the curve are higher than what's 407 00:20:07,180 --> 00:20:08,480 happening here. 408 00:20:08,480 --> 00:20:09,790 Same thing happens, what? 409 00:20:09,790 --> 00:20:12,670 All of these points are lower than what the height is 410 00:20:12,670 --> 00:20:14,500 corresponding to 'x' equals 'a'. 411 00:20:14,500 --> 00:20:17,250 And in the similar way, this is what happens at 'b'. 412 00:20:17,250 --> 00:20:20,520 You see, with the endpoints, obviously since you can't see 413 00:20:20,520 --> 00:20:23,710 what's happening before, this will be either the highest 414 00:20:23,710 --> 00:20:27,180 point or the lowest point near here depending on how the 415 00:20:27,180 --> 00:20:28,200 curve is sloped. 416 00:20:28,200 --> 00:20:29,550 But all we're saying is what? 417 00:20:29,550 --> 00:20:32,630 In terms of absolute high values and absolute low 418 00:20:32,630 --> 00:20:35,500 values, meaning the highest possible points and the lowest 419 00:20:35,500 --> 00:20:39,620 possible points, we must always check the endpoints. 420 00:20:39,620 --> 00:20:42,970 But we can't be positive that the endpoints 421 00:20:42,970 --> 00:20:45,450 are going to be chosen. 422 00:20:45,450 --> 00:20:47,070 In fact, let's summarize. 423 00:20:47,070 --> 00:20:48,750 And I'm going to summarize again at 424 00:20:48,750 --> 00:20:49,540 the end of the lecture. 425 00:20:49,540 --> 00:20:50,830 But the idea is this. 426 00:20:50,830 --> 00:20:54,680 If 'f of x' is continuous on the closed interval from 'a' 427 00:20:54,680 --> 00:20:56,800 to 'b'-- and here's the key word. 428 00:20:56,800 --> 00:20:59,310 To find candidates-- 429 00:20:59,310 --> 00:21:01,280 you know the old cliche about many are 430 00:21:01,280 --> 00:21:02,930 called, but few are chosen. 431 00:21:02,930 --> 00:21:07,860 In this case, few are called and even fewer are chosen. 432 00:21:07,860 --> 00:21:09,630 Namely, what we're saying is look it. 433 00:21:09,630 --> 00:21:12,580 When we're looking for high-low points, the 434 00:21:12,580 --> 00:21:13,700 candidates are what? 435 00:21:13,700 --> 00:21:17,410 Those points for which the derivative is 0. 436 00:21:17,410 --> 00:21:20,550 We can check those out because those are possibilities. 437 00:21:20,550 --> 00:21:23,030 Those points for which the derivative fails to exist. 438 00:21:23,030 --> 00:21:26,220 We can check those out because they're possibilities. 439 00:21:26,220 --> 00:21:27,470 And the endpoints. 440 00:21:27,470 --> 00:21:30,370 441 00:21:30,370 --> 00:21:32,030 Namely, if the function is continuous, 442 00:21:32,030 --> 00:21:32,810 we check the endpoints. 443 00:21:32,810 --> 00:21:35,970 By the way, if the function is not continuous, then there is 444 00:21:35,970 --> 00:21:37,780 no need to check-- well, let's put it this way. 445 00:21:37,780 --> 00:21:40,190 If you're on an open interval there's no need to check the 446 00:21:40,190 --> 00:21:42,440 endpoints because there aren't any. 447 00:21:42,440 --> 00:21:44,260 In other words, notice that I'm talking about what? 448 00:21:44,260 --> 00:21:48,190 That the function 'f' is not only continuous, but on a 449 00:21:48,190 --> 00:21:49,620 closed interval. 450 00:21:49,620 --> 00:21:50,530 That's all there is to this. 451 00:21:50,530 --> 00:21:53,230 In other words, these are all the possible candidates. 452 00:21:53,230 --> 00:21:54,920 Now you see the bigger question is, 453 00:21:54,920 --> 00:21:56,500 how do you use this? 454 00:21:56,500 --> 00:21:59,140 And I thought that I would make up a makeshift exercise, 455 00:21:59,140 --> 00:22:01,840 one that's rather easy to do at the blackboard. 456 00:22:01,840 --> 00:22:06,370 For deeper exercises, for more quantitative results, we have 457 00:22:06,370 --> 00:22:10,130 several exercises in the learning exercises. 458 00:22:10,130 --> 00:22:11,610 Several exercises worked out 459 00:22:11,610 --> 00:22:13,320 illustratively in the textbook. 460 00:22:13,320 --> 00:22:17,250 But let's pick a particularly straightforward example. 461 00:22:17,250 --> 00:22:24,430 Let's suppose that what I want to do is construct a cylinder. 462 00:22:24,430 --> 00:22:26,510 This is a cross sectional view of a cylinder. 463 00:22:26,510 --> 00:22:30,230 'x' represents the radius of the base and 'y' represents 464 00:22:30,230 --> 00:22:31,290 the height. 465 00:22:31,290 --> 00:22:33,860 This is a right circular cylinder. 466 00:22:33,860 --> 00:22:37,240 I'm given a constraint, namely I'm told that for some reason 467 00:22:37,240 --> 00:22:40,350 or other, and I don't know why anybody would ever want to 468 00:22:40,350 --> 00:22:43,140 impose this condition other than the fact that we want 469 00:22:43,140 --> 00:22:45,160 some condition imposed here to see what's happening. 470 00:22:45,160 --> 00:22:50,270 I'm told that I want the sum of the radius of the base and 471 00:22:50,270 --> 00:22:53,040 the altitude to be exactly 30. 472 00:22:53,040 --> 00:22:55,400 In other words, if the radius of the base is going to be 6 473 00:22:55,400 --> 00:22:58,440 inches, I want the altitude to be 24 inches. 474 00:22:58,440 --> 00:22:59,360 That's a constraint. 475 00:22:59,360 --> 00:23:02,090 And by the way, I'll come back to this later also. 476 00:23:02,090 --> 00:23:05,330 Notice how you're almost begging a related rates 477 00:23:05,330 --> 00:23:07,050 relationship here. 478 00:23:07,050 --> 00:23:09,440 Or an implicit relationship that 'x' and 'y' are not 479 00:23:09,440 --> 00:23:10,280 independent. 480 00:23:10,280 --> 00:23:12,275 But I've now put a constraint on here. 481 00:23:12,275 --> 00:23:17,990 At any rate the question is, how shall I use up my 30 482 00:23:17,990 --> 00:23:22,510 inches, say, if I want to make the volume of the resulting 483 00:23:22,510 --> 00:23:25,080 cylinder as large as possible? 484 00:23:25,080 --> 00:23:26,450 And notice how we work this thing. 485 00:23:26,450 --> 00:23:30,640 We say, well, the volume is equal to 'pi 'x squared' y'. 486 00:23:30,640 --> 00:23:34,410 In this particular case, it's easy to see explicitly that 487 00:23:34,410 --> 00:23:37,420 'y' is equal to '30 - x'. 488 00:23:37,420 --> 00:23:40,710 It's also easy to see physically that 'x' must be 489 00:23:40,710 --> 00:23:41,620 more than 0. 490 00:23:41,620 --> 00:23:44,120 You can't have a negative radius of the base. 491 00:23:44,120 --> 00:23:48,180 It must be less than 30 because if you used up more 492 00:23:48,180 --> 00:23:51,080 than 30 inches in the radius of your base, how can the 493 00:23:51,080 --> 00:23:54,970 radius of the base plus the altitude add up to exactly 30? 494 00:23:54,970 --> 00:23:57,180 Because physically, the constraint is that the 495 00:23:57,180 --> 00:23:59,100 altitude can't be negative. 496 00:23:59,100 --> 00:24:01,270 It's certainly a positive value. 497 00:24:01,270 --> 00:24:03,340 So our analytic relationship is what? 498 00:24:03,340 --> 00:24:06,060 That 'v' equals 'pi 'x squared' y', which can be 499 00:24:06,060 --> 00:24:08,770 written as 'pi 'x squared' times '30 - x''. 500 00:24:08,770 --> 00:24:12,290 Which in turn, can be written as the polynomial '30 pi 'x 501 00:24:12,290 --> 00:24:15,750 squared'' minus 'pi 'x cubed'', where 'x' is the open 502 00:24:15,750 --> 00:24:19,950 interval or defined on the open interval from 0 to 30. 503 00:24:19,950 --> 00:24:21,730 Now, how do I proceed? 504 00:24:21,730 --> 00:24:24,480 What was my test for membership? 505 00:24:24,480 --> 00:24:26,630 I have three ways of checking out where 506 00:24:26,630 --> 00:24:28,380 high-low points will occur. 507 00:24:28,380 --> 00:24:31,190 Or max-min points in this case, since notice that this 508 00:24:31,190 --> 00:24:34,750 function does not require a graph to understand it. 509 00:24:34,750 --> 00:24:38,240 I first check out to see where the derivative is 0. 510 00:24:38,240 --> 00:24:41,090 Well, the derivative is simply '60 pi x' 511 00:24:41,090 --> 00:24:43,210 minus '3 pi 'x squared''. 512 00:24:43,210 --> 00:24:48,250 If I set this thing equal to 0, I find upon factoring that 513 00:24:48,250 --> 00:24:52,090 either 'x' must be 0 or 'x' must be 20. 514 00:24:52,090 --> 00:24:58,520 I can immediately exclude 'x' equals 0 because notice that 515 00:24:58,520 --> 00:25:02,800 my function 'v' had its domain of definition on the open 516 00:25:02,800 --> 00:25:04,430 interval from 0 to 30. 517 00:25:04,430 --> 00:25:07,690 'x' equals 0 is not in the open interval from 0 to 30. 518 00:25:07,690 --> 00:25:10,190 Consequently, we can rule this thing out. 519 00:25:10,190 --> 00:25:13,390 And what we discover is that 'x' equals 20. 520 00:25:13,390 --> 00:25:15,980 So 'x' equals 20 is the only possible 521 00:25:15,980 --> 00:25:17,410 candidate that we have. 522 00:25:17,410 --> 00:25:20,650 And by the way, since the sum of 'x' and 'y' must be 20, if 523 00:25:20,650 --> 00:25:23,240 'x' equals 20, 'y' must be 10. 524 00:25:23,240 --> 00:25:25,715 So the only candidate that we have by setting the derivative 525 00:25:25,715 --> 00:25:29,330 equal to 0 is that 'x' should be 20 and 'y' should be 10. 526 00:25:29,330 --> 00:25:32,540 We do not get any candidates in the sense of where the 527 00:25:32,540 --> 00:25:33,890 derivative doesn't exist. 528 00:25:33,890 --> 00:25:37,250 Because if you look at '60 pi x' minus '3 pi 'x squared'', 529 00:25:37,250 --> 00:25:40,170 this certainly exists for all values of 'x'. 530 00:25:40,170 --> 00:25:43,410 And finally, we have no endpoints to check. 531 00:25:43,410 --> 00:25:46,800 Because again, our function 'v' is 532 00:25:46,800 --> 00:25:50,310 defined on an open interval. 533 00:25:50,310 --> 00:25:54,170 By the way, if I tried to draw this somewhat to scale, a very 534 00:25:54,170 --> 00:25:58,690 interesting result turns up, which may show the power of 535 00:25:58,690 --> 00:26:03,280 analytical methods versus more intuitive types of methods. 536 00:26:03,280 --> 00:26:05,220 You see, what we showed here was what? 537 00:26:05,220 --> 00:26:07,190 Without going into the details, that to get the 538 00:26:07,190 --> 00:26:11,760 largest volume cylinder, 'x' should be 20. 539 00:26:11,760 --> 00:26:14,210 In other words, the radius of your base should be 20 and the 540 00:26:14,210 --> 00:26:15,750 altitude should be 10. 541 00:26:15,750 --> 00:26:19,240 Let's take a look at that drawn roughly to scale. 542 00:26:19,240 --> 00:26:22,530 The radius of the base here is 20 and the altitude is 10. 543 00:26:22,530 --> 00:26:23,800 By the way, that means that the cross 544 00:26:23,800 --> 00:26:25,160 section will be what? 545 00:26:25,160 --> 00:26:30,430 A rectangle whose height is 10 and whose base is 40. 546 00:26:30,430 --> 00:26:33,670 Now intuitively, I think it's easy to see that for a 547 00:26:33,670 --> 00:26:39,100 rectangle, for a given perimeter the largest possible 548 00:26:39,100 --> 00:26:42,880 area rectangle is the one which is a square. 549 00:26:42,880 --> 00:26:45,690 Well, without even trying to prove that, let's go to this 550 00:26:45,690 --> 00:26:46,150 thing here. 551 00:26:46,150 --> 00:26:49,370 Instead of using up the 20 and the 10, let's draw a second 552 00:26:49,370 --> 00:26:52,940 rectangle, or a second cross section of a cylinder where 553 00:26:52,940 --> 00:26:56,350 the radius is 15 and the height is 15. 554 00:26:56,350 --> 00:26:58,860 Notice that this still satisfies the fact that the 555 00:26:58,860 --> 00:27:02,520 sum of the radius and the altitude are 30. 556 00:27:02,520 --> 00:27:03,890 Now, look at this. 557 00:27:03,890 --> 00:27:08,220 If we compute the area of this particular rectangle, 40 times 558 00:27:08,220 --> 00:27:10,350 10, it's 400. 559 00:27:10,350 --> 00:27:12,490 On the other hand, the volume is what? 560 00:27:12,490 --> 00:27:14,380 Pi times 20 squared. 561 00:27:14,380 --> 00:27:15,830 The radius of the base squared. 562 00:27:15,830 --> 00:27:17,320 Times the height, which is 10. 563 00:27:17,320 --> 00:27:20,890 And that yields the result of 4,000 pi. 564 00:27:20,890 --> 00:27:24,540 On the other hand, if we look at our second rectangle, its 565 00:27:24,540 --> 00:27:27,890 area is 450. 566 00:27:27,890 --> 00:27:29,240 But its volume is what? 567 00:27:29,240 --> 00:27:37,270 It's pi times 15 squared times 15, which is 3,375 pi. 568 00:27:37,270 --> 00:27:40,760 The interesting result here is what? 569 00:27:40,760 --> 00:27:45,050 The area of this rectangle is greater than the 570 00:27:45,050 --> 00:27:48,280 area of this rectangle. 571 00:27:48,280 --> 00:27:52,430 In fact, the area of the second rectangle is 450. 572 00:27:52,430 --> 00:27:54,800 The area of the first rectangle is only 400. 573 00:27:54,800 --> 00:27:58,840 But notice that when you revolve this thing to form the 574 00:27:58,840 --> 00:28:05,170 cylinder, the smaller cross sectional area generates the 575 00:28:05,170 --> 00:28:06,440 larger volume. 576 00:28:06,440 --> 00:28:09,340 And the reason, of course, for that is that the relationship 577 00:28:09,340 --> 00:28:11,840 in our variables was not linear. 578 00:28:11,840 --> 00:28:14,860 In other words notice that when 'x' is large, a 579 00:28:14,860 --> 00:28:18,340 relatively small change in 'x' produces a large 580 00:28:18,340 --> 00:28:20,760 change in 'x squared'. 581 00:28:20,760 --> 00:28:24,490 In other words, a relatively small value in 'x' can offset 582 00:28:24,490 --> 00:28:27,740 a relatively large value or increase in 'y'. 583 00:28:27,740 --> 00:28:29,840 And this is kind of interesting because try to 584 00:28:29,840 --> 00:28:33,040 figure out intuitively how you would figure out where these 585 00:28:33,040 --> 00:28:34,880 stop compensating for one another? 586 00:28:34,880 --> 00:28:37,350 Where does it turn out that finally you've taken so much 587 00:28:37,350 --> 00:28:41,100 away from 'x' that even though you square it, it can't 588 00:28:41,100 --> 00:28:42,940 compensate the change in 'y'? 589 00:28:42,940 --> 00:28:46,050 How would you intuitively pick off where the high-low points 590 00:28:46,050 --> 00:28:47,900 occur in a problem like this? 591 00:28:47,900 --> 00:28:51,080 And all I want you to see is again, the beautiful gentle 592 00:28:51,080 --> 00:28:53,840 balance between intuitive calculus 593 00:28:53,840 --> 00:28:55,570 and rigorous calculus. 594 00:28:55,570 --> 00:28:57,460 That we don't throw away our intuition. 595 00:28:57,460 --> 00:29:01,160 But notice how, in many cases, where our intuition fails us, 596 00:29:01,160 --> 00:29:04,190 the analytic recipes come to our rescue. 597 00:29:04,190 --> 00:29:07,550 But enough said about that, let me now again, highlight 598 00:29:07,550 --> 00:29:10,840 the difference or the relationship between functions 599 00:29:10,840 --> 00:29:11,850 and graphs. 600 00:29:11,850 --> 00:29:14,660 Namely, in all of this discussion that we've done on 601 00:29:14,660 --> 00:29:17,510 this particular board, we're talking about what? 602 00:29:17,510 --> 00:29:19,330 'v' being a function of 'x'. 603 00:29:19,330 --> 00:29:22,810 We do not have to visualize this thing pictorially. 604 00:29:22,810 --> 00:29:28,140 But if we wish, what we can say is let's graph 'v' as a 605 00:29:28,140 --> 00:29:30,560 function of 'x'. 606 00:29:30,560 --> 00:29:33,520 And you see, going back to the material of last time. 607 00:29:33,520 --> 00:29:36,560 And notice, you see how interrelated these things are. 608 00:29:36,560 --> 00:29:39,660 Notice how curve plotting ties in very nicely with 609 00:29:39,660 --> 00:29:40,740 derivatives and the like. 610 00:29:40,740 --> 00:29:42,120 All we're saying is what? 611 00:29:42,120 --> 00:29:45,920 Given this relationship, which is how 'v' is related to 'x', 612 00:29:45,920 --> 00:29:48,530 we can form the first derivative, we can form the 613 00:29:48,530 --> 00:29:51,350 second derivative, we can look to see where the first 614 00:29:51,350 --> 00:29:53,880 derivative is 0, we can look to see where the second 615 00:29:53,880 --> 00:29:54,960 derivative is 0. 616 00:29:54,960 --> 00:29:57,980 I leave these details to you because after the homework 617 00:29:57,980 --> 00:30:01,370 assignment you did last time these should be fairly trivial 618 00:30:01,370 --> 00:30:02,970 exercises to do. 619 00:30:02,970 --> 00:30:05,580 But the idea is if you now utilize all of this 620 00:30:05,580 --> 00:30:09,300 information, you find that if you plot 'v' verses 'x', you 621 00:30:09,300 --> 00:30:12,190 get a graph something like this. 622 00:30:12,190 --> 00:30:16,330 By the way, again notice that we were not talking about just 623 00:30:16,330 --> 00:30:18,410 'v' being a function of 'x'. 624 00:30:18,410 --> 00:30:22,010 The domain of 'v' was restricted to be the open 625 00:30:22,010 --> 00:30:25,420 interval from 0 to 30, and this is a very crucial thing 626 00:30:25,420 --> 00:30:26,580 to keep in mind. 627 00:30:26,580 --> 00:30:29,120 You can get into a whole bunch of trouble if you start 628 00:30:29,120 --> 00:30:31,210 looking to see what happens out here. 629 00:30:31,210 --> 00:30:33,595 For example, you say, hey, this curve is always going to 630 00:30:33,595 --> 00:30:34,240 keep going up. 631 00:30:34,240 --> 00:30:37,210 Won't this be greater than this maximum value over here 632 00:30:37,210 --> 00:30:38,040 eventually? 633 00:30:38,040 --> 00:30:39,600 The answer is yes, it will be. 634 00:30:39,600 --> 00:30:42,600 But what does it mean to say that 'x' is negative? 635 00:30:42,600 --> 00:30:44,500 'x' was the radius of our base. 636 00:30:44,500 --> 00:30:46,070 So in other words, what we should 637 00:30:46,070 --> 00:30:48,360 really say here is what? 638 00:30:48,360 --> 00:30:50,930 That the function that we're talking about is not this 639 00:30:50,930 --> 00:30:51,930 whole curve. 640 00:30:51,930 --> 00:30:54,470 Yikes, that wasn't a very good job of drawing. 641 00:30:54,470 --> 00:30:55,560 But rather what? 642 00:30:55,560 --> 00:30:59,370 Just this portion of the curve defined on the open interval 643 00:30:59,370 --> 00:31:01,420 from 0 to 30. 644 00:31:01,420 --> 00:31:05,120 By the way, let me point out something else. 645 00:31:05,120 --> 00:31:07,690 You recall in an earlier lecture we talked about 646 00:31:07,690 --> 00:31:08,960 related rates. 647 00:31:08,960 --> 00:31:11,130 We had an assignment with other lecture where you've 648 00:31:11,130 --> 00:31:13,420 solved some problems using related rates. 649 00:31:13,420 --> 00:31:16,690 Let me show you how related rates play a very important 650 00:31:16,690 --> 00:31:21,300 computational role in dealing with max-min problems. 651 00:31:21,300 --> 00:31:24,780 In this particular problem, we had that 'v' equals 'pi 'x 652 00:31:24,780 --> 00:31:28,240 squared' y', where 'y' happened to be a particular 653 00:31:28,240 --> 00:31:29,300 function of 'x'. 654 00:31:29,300 --> 00:31:33,270 In fact, implicitly it was given by the fact that 'x + y' 655 00:31:33,270 --> 00:31:34,970 happened to equal 30. 656 00:31:34,970 --> 00:31:36,730 Now let's keep track of something here. 657 00:31:36,730 --> 00:31:40,220 In this particular problem, notice that it was very easy 658 00:31:40,220 --> 00:31:42,550 to change this implicit relationship 659 00:31:42,550 --> 00:31:44,180 to an explicit one. 660 00:31:44,180 --> 00:31:48,520 It was also easy once you expressed 'y' explicitly in 661 00:31:48,520 --> 00:31:51,120 terms of 'x' that when you wanted to substitute into 662 00:31:51,120 --> 00:31:54,950 here, it was a very easy computational job to carry out 663 00:31:54,950 --> 00:31:56,080 the operations. 664 00:31:56,080 --> 00:31:59,430 But suppose there happened to be cube roots in here, or all 665 00:31:59,430 --> 00:32:02,380 sorts of nasty things, whatever they might be. 666 00:32:02,380 --> 00:32:05,910 And suppose instead of 'x + y' equals 30, you had our old 667 00:32:05,910 --> 00:32:08,870 friend something like 'x to the eighth' plus ''x to the 668 00:32:08,870 --> 00:32:13,560 sixth' 'y squared'' plus 'y to the sixth' equals 3. 669 00:32:13,560 --> 00:32:15,530 How would you solve for 'y' explicitly in 670 00:32:15,530 --> 00:32:16,870 terms of 'x' there? 671 00:32:16,870 --> 00:32:18,970 And the point that I'd like you to see is that we can 672 00:32:18,970 --> 00:32:23,570 solve this problem very nicely without having to resort to 673 00:32:23,570 --> 00:32:26,460 explicitly replacing 'y' as a function of 'x'. 674 00:32:26,460 --> 00:32:30,130 Namely, implicitly assuming that 'y' is a differentiable 675 00:32:30,130 --> 00:32:32,270 function of 'x', we can differentiate 676 00:32:32,270 --> 00:32:33,760 this thing as a product. 677 00:32:33,760 --> 00:32:35,320 'dv dx' will be what? 678 00:32:35,320 --> 00:32:38,280 The derivative of the first factor, which is '2 pi x', 679 00:32:38,280 --> 00:32:39,690 times the second. 680 00:32:39,690 --> 00:32:42,970 Plus the first factor, which is 'pi 'x squared', times the 681 00:32:42,970 --> 00:32:45,730 derivative of 'y' with respect to 'x'. 682 00:32:45,730 --> 00:32:47,440 So that's 'dv dx'. 683 00:32:47,440 --> 00:32:50,960 On the other hand, from this relationship here, we can 684 00:32:50,960 --> 00:32:54,380 conclude by differentiating implicitly that '1 685 00:32:54,380 --> 00:32:57,810 + 'dy dx'' is 0. 686 00:32:57,810 --> 00:33:03,550 And therefore, 'dy dx' is minus 1. 687 00:33:03,550 --> 00:33:08,130 And putting that value for 'dy dx' in here, we wind up with 688 00:33:08,130 --> 00:33:09,920 this explicit relationship. 689 00:33:09,920 --> 00:33:14,340 And we can now see that 'dv dx' is 0 if and only if just 690 00:33:14,340 --> 00:33:15,710 by solving this thing, setting it equal to 691 00:33:15,710 --> 00:33:17,870 0, 'x' equals '2y'. 692 00:33:17,870 --> 00:33:20,400 By the way, that's exactly what happened in our problem. 693 00:33:20,400 --> 00:33:23,340 You'll notice that 'x' turned out to be 20 and 'y' turned 694 00:33:23,340 --> 00:33:25,050 out to be 10. 695 00:33:25,050 --> 00:33:27,370 You see again, 'x' and 'y' are related. 696 00:33:27,370 --> 00:33:31,060 You can say, gee, couldn't 'x' be 60 and 'y' be 30? 697 00:33:31,060 --> 00:33:32,370 Answer, no. 698 00:33:32,370 --> 00:33:36,440 Because the constraint is that 'x + y' must be 30. 699 00:33:36,440 --> 00:33:37,250 Well, look it. 700 00:33:37,250 --> 00:33:40,820 I don't want us to get too wrapped up on the idea of 701 00:33:40,820 --> 00:33:42,330 computational differences now. 702 00:33:42,330 --> 00:33:47,320 What I do want to do is review our basic result. 703 00:33:47,320 --> 00:33:48,660 And in fact, let me come over here. 704 00:33:48,660 --> 00:33:50,180 I hope this doesn't spoil you. 705 00:33:50,180 --> 00:33:51,750 I'd like to come back to the board. 706 00:33:51,750 --> 00:33:54,330 And actually, since I've got this all written down, let's 707 00:33:54,330 --> 00:33:57,180 close on this particular result again. 708 00:33:57,180 --> 00:34:01,170 To find the high-low points of a function 'f of x', we first 709 00:34:01,170 --> 00:34:04,700 of all, check out when 'f prime of c' is 0 for 710 00:34:04,700 --> 00:34:05,350 candidates. 711 00:34:05,350 --> 00:34:08,790 We check out where 'f prime of c' fails to exist. 712 00:34:08,790 --> 00:34:12,460 And if it's a closed interval, we check the endpoints. 713 00:34:12,460 --> 00:34:15,480 This is the mechanism behind what we're doing from that 714 00:34:15,480 --> 00:34:19,370 point on, as the cliche goes, it's all engineering's baby. 715 00:34:19,370 --> 00:34:21,900 It's all computational know how. 716 00:34:21,900 --> 00:34:25,460 At any rate, I think this is enough in terms of emphasizing 717 00:34:25,460 --> 00:34:27,060 the points that we wanted to make. 718 00:34:27,060 --> 00:34:29,120 And so until next time, goodbye. 719 00:34:29,120 --> 00:34:32,199 720 00:34:32,199 --> 00:34:34,739 ANNOUNCER: Funding for the publication of this video was 721 00:34:34,739 --> 00:34:39,449 provided by the Gabriella and Paul Rosenbaum Foundation. 722 00:34:39,449 --> 00:34:43,630 Help OCW continue to provide free and open access to MIT 723 00:34:43,630 --> 00:34:47,820 courses by making a donation at ocw.mit.edu/donate. 724 00:34:47,820 --> 00:34:52,564