1 00:00:00,000 --> 00:00:06,508 The following content is provided under a Creative 2 00:00:06,508 --> 00:00:07,174 Commons license. 3 00:00:07,174 --> 00:00:07,438 Your support will help MIT OpenCourseWare 4 00:00:07,438 --> 00:00:09,940 continue to offer high quality educational resources for free. 5 00:00:09,940 --> 00:00:12,065 To make a donation, or to view additional materials 6 00:00:12,065 --> 00:00:18,970 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:18,970 --> 00:00:22,780 at ocw.mit.edu. 8 00:00:22,780 --> 00:00:24,390 PROFESSOR: In the twelfth lecture, 9 00:00:24,390 --> 00:00:31,110 we're going to talk about maxima and minima. 10 00:00:31,110 --> 00:00:33,110 Let's finish up what we did last time. 11 00:00:33,110 --> 00:00:35,710 We really only just started with maxima and minima. 12 00:00:35,710 --> 00:00:38,100 And then we're going to talk about related rates. 13 00:00:38,100 --> 00:00:48,180 So, right now I want to give you some examples 14 00:00:48,180 --> 00:00:51,430 of max-min problems. 15 00:00:51,430 --> 00:00:55,540 And we're going to start with a fairly basic one. 16 00:00:55,540 --> 00:00:58,540 So what's the thing about max-min problems? 17 00:00:58,540 --> 00:01:01,240 The main thing is that we're asking 18 00:01:01,240 --> 00:01:05,560 you to do a little bit more of the interpretation of word 19 00:01:05,560 --> 00:01:06,060 problems. 20 00:01:06,060 --> 00:01:09,970 So many of the problems are expressed in terms of words. 21 00:01:09,970 --> 00:01:18,650 And so, in this case, we have a wire which is length 1. 22 00:01:18,650 --> 00:01:29,610 Cut into two pieces. 23 00:01:29,610 --> 00:01:38,150 And then each piece encloses a square. 24 00:01:38,150 --> 00:01:44,400 Sorry, encloses a square. 25 00:01:44,400 --> 00:01:47,170 And the problem - so this is the setup. 26 00:01:47,170 --> 00:02:02,650 And the problem is to find the largest area enclosed. 27 00:02:02,650 --> 00:02:03,910 So here's the problem. 28 00:02:03,910 --> 00:02:10,340 Now, in all of these cases, in all these cases, 29 00:02:10,340 --> 00:02:11,600 there's a bunch of words. 30 00:02:11,600 --> 00:02:18,599 And your job is typically to draw a diagram. 31 00:02:18,599 --> 00:02:20,890 So the first thing you want to do is to draw a diagram. 32 00:02:20,890 --> 00:02:23,340 In this case, it can be fairly schematic. 33 00:02:23,340 --> 00:02:25,400 Here's your unit length. 34 00:02:25,400 --> 00:02:27,440 And when you draw the diagram, you're 35 00:02:27,440 --> 00:02:29,370 going to have to pick variables. 36 00:02:29,370 --> 00:02:34,700 So those are really the two main tasks. 37 00:02:34,700 --> 00:02:35,665 To set up the problem. 38 00:02:35,665 --> 00:02:37,170 So you're drawing a diagram. 39 00:02:37,170 --> 00:02:40,860 This is like word problems of old, in grade school 40 00:02:40,860 --> 00:02:42,410 through high school. 41 00:02:42,410 --> 00:02:50,020 Draw a diagram and name the variables. 42 00:02:50,020 --> 00:02:52,900 So we'll be doing a lot of that today. 43 00:02:52,900 --> 00:02:54,950 So here's my unit length. 44 00:02:54,950 --> 00:02:58,160 And I'm going to choose the variable x 45 00:02:58,160 --> 00:03:00,280 to be the length of one of the pieces of wire. 46 00:03:00,280 --> 00:03:03,240 And that makes the other piece 1 - x. 47 00:03:03,240 --> 00:03:05,579 And that's pretty much the whole diagram, 48 00:03:05,579 --> 00:03:07,870 except that there's something that we did with the wire 49 00:03:07,870 --> 00:03:09,710 after we cut it in half. 50 00:03:09,710 --> 00:03:13,370 Namely, we built two little boxes out of it. 51 00:03:13,370 --> 00:03:15,520 Like this, these are our squares. 52 00:03:15,520 --> 00:03:19,900 And their side lengths are x/4 and (1-x)/4. 53 00:03:25,020 --> 00:03:26,710 So, so far, so good. 54 00:03:26,710 --> 00:03:28,880 And now we have to think, well, we 55 00:03:28,880 --> 00:03:30,450 want to find the largest area. 56 00:03:30,450 --> 00:03:33,050 So I need a formula for area in terms of variables 57 00:03:33,050 --> 00:03:34,490 that I've described. 58 00:03:34,490 --> 00:03:35,790 And so that's the last thing. 59 00:03:35,790 --> 00:03:39,400 I'll give the letter A as the label for the area. 60 00:03:39,400 --> 00:03:43,760 And then the area is just the square of x/4 61 00:03:43,760 --> 00:03:51,805 plus the square of 1 minus x - whoops, that strange 2 got 62 00:03:51,805 --> 00:03:57,780 in here - over 4. 63 00:03:57,780 --> 00:03:59,110 So far, so good. 64 00:03:59,110 --> 00:04:01,750 Now, the instinct that you'll have, 65 00:04:01,750 --> 00:04:04,020 and I'm going to yield to that instinct, 66 00:04:04,020 --> 00:04:07,370 is we should charge ahead and just differentiate. 67 00:04:07,370 --> 00:04:07,870 All right? 68 00:04:07,870 --> 00:04:08,578 That's all right. 69 00:04:08,578 --> 00:04:10,430 We'll find the critical points. 70 00:04:10,430 --> 00:04:12,995 So we know that those are important points. 71 00:04:12,995 --> 00:04:17,840 So we're going to find the critical points. 72 00:04:17,840 --> 00:04:21,460 In other words, we take the derivative, 73 00:04:21,460 --> 00:04:27,390 we set the derivative of A with respect to x = 0. 74 00:04:27,390 --> 00:04:33,020 So if I do that differentiation, I get the, well, 75 00:04:33,020 --> 00:04:37,240 so the first one, x^2 / 16, that's 8. 76 00:04:37,240 --> 00:04:40,870 Sorry. 77 00:04:40,870 --> 00:04:44,060 That's x/8, right? 78 00:04:44,060 --> 00:04:45,570 That's the derivative of this. 79 00:04:45,570 --> 00:04:51,290 And if I differentiate this, I get well, the derivative of 1 - 80 00:04:51,290 --> 00:04:56,930 x^2 is 2(1-x)(a-1). 81 00:04:56,930 --> 00:04:59,620 So it's -(1-x)/8. 82 00:05:03,170 --> 00:05:05,010 So there are two minus signs in there, 83 00:05:05,010 --> 00:05:07,520 I'll let you ponder that differentiation, which 84 00:05:07,520 --> 00:05:10,900 I did by the chain rule. 85 00:05:10,900 --> 00:05:12,730 Hang on a sec, OK? 86 00:05:12,730 --> 00:05:14,960 Just wait until we're done. 87 00:05:14,960 --> 00:05:17,480 So here's the derivative. 88 00:05:17,480 --> 00:05:19,890 Is there a problem? 89 00:05:19,890 --> 00:05:29,060 STUDENT: [INAUDIBLE] 90 00:05:29,060 --> 00:05:31,320 PROFESSOR: Right, so there's a 1/16 here. 91 00:05:31,320 --> 00:05:32,850 This is x^2 / 16. 92 00:05:32,850 --> 00:05:36,870 And so it's 2x over 8, over 16, sorry. 93 00:05:36,870 --> 00:05:40,360 Which has an 8. 94 00:05:40,360 --> 00:05:41,790 That's OK. 95 00:05:41,790 --> 00:05:56,080 All right, so now, This is equal to 0 if and only if x = 1 - x. 96 00:05:56,080 --> 00:06:02,170 That's 2x = 1, or in other words x = 1/2. 97 00:06:02,170 --> 00:06:03,280 All right? 98 00:06:03,280 --> 00:06:06,530 So there's our critical point. 99 00:06:06,530 --> 00:06:11,710 So x = 1/2 is the critical point. 100 00:06:11,710 --> 00:06:17,640 And the critical value, which is what 101 00:06:17,640 --> 00:06:26,260 you get when you evaluate A at 1/2, is (1/2) / 4, that's 1/8. 102 00:06:26,260 --> 00:06:38,030 So that's (1/8)^2 + (1/8)^2 which is 1/32. 103 00:06:38,030 --> 00:06:45,480 So, so far, so good. 104 00:06:45,480 --> 00:06:48,050 But we're not done yet. 105 00:06:48,050 --> 00:06:56,850 We're not done. 106 00:06:56,850 --> 00:06:59,070 So why aren't we done? 107 00:06:59,070 --> 00:07:04,590 Because we haven't checked the end points. 108 00:07:04,590 --> 00:07:08,370 So let's check the end points. 109 00:07:08,370 --> 00:07:10,450 Now, in this problem, the end points 110 00:07:10,450 --> 00:07:12,650 are really sort of excluded. 111 00:07:12,650 --> 00:07:17,530 The ends are between 0 and 1 here. 112 00:07:17,530 --> 00:07:22,740 That's the possible lengths of the cut. 113 00:07:22,740 --> 00:07:25,410 And so what we should really be doing is evaluating 114 00:07:25,410 --> 00:07:29,230 in the limit, so that would be the right-hand limit as x goes 115 00:07:29,230 --> 00:07:33,780 to 0 of A. And if you plug in x = 0, 116 00:07:33,780 --> 00:07:42,000 what you get here is 0 + (1/4)^2, which is 1/16. 117 00:07:42,000 --> 00:07:49,710 And similarly, at the other end, that's 1-, from the left, 118 00:07:49,710 --> 00:07:56,650 we get (1/4)^2 + 0, which is also 1/16. 119 00:07:56,650 --> 00:08:01,750 So, what you see is that the schematic picture 120 00:08:01,750 --> 00:08:05,090 of this function, and this isn't even so far 121 00:08:05,090 --> 00:08:08,100 off from being the right picture here, 122 00:08:08,100 --> 00:08:12,120 is that its level here is 1/16 and then it 123 00:08:12,120 --> 00:08:14,520 dips down and goes up. 124 00:08:14,520 --> 00:08:15,020 Right? 125 00:08:15,020 --> 00:08:19,960 This is 1/2, this is 1, and this level here is a half that. 126 00:08:19,960 --> 00:08:23,070 This is 1/32. 127 00:08:23,070 --> 00:08:25,900 So we did not find, when we found 128 00:08:25,900 --> 00:08:29,740 the critical point we did not find the largest area enclosed. 129 00:08:29,740 --> 00:08:33,060 We found the least area enclosed. 130 00:08:33,060 --> 00:08:36,850 So if you don't pay attention to what the function looks like, 131 00:08:36,850 --> 00:08:40,240 not only will you about half the time get the wrong answer, 132 00:08:40,240 --> 00:08:43,980 you'll get the absolute worst answer. 133 00:08:43,980 --> 00:08:47,770 You'll get the one which is the polar opposite from what 134 00:08:47,770 --> 00:08:49,680 you want. 135 00:08:49,680 --> 00:08:52,250 So you have to pay a little bit of attention to the function 136 00:08:52,250 --> 00:08:53,440 that you've got. 137 00:08:53,440 --> 00:08:55,530 And in this case it's just very schematic. 138 00:08:55,530 --> 00:08:57,320 It dips down and goes up, and that's 139 00:08:57,320 --> 00:08:59,514 true of pretty much most functions. 140 00:08:59,514 --> 00:09:00,430 They're fairly simple. 141 00:09:00,430 --> 00:09:02,096 They maybe only have one critical point. 142 00:09:02,096 --> 00:09:03,550 They only turn around once. 143 00:09:03,550 --> 00:09:06,310 But then, maybe the critical point is the maximum 144 00:09:06,310 --> 00:09:07,760 or maybe it's the minimum. 145 00:09:07,760 --> 00:09:09,670 Or maybe it's neither, in fact. 146 00:09:09,670 --> 00:09:15,820 So we'll be discussing that maybe some other time. 147 00:09:15,820 --> 00:09:26,260 So what we find here is that we have the least area enclosed 148 00:09:26,260 --> 00:09:30,420 is 1/32. 149 00:09:30,420 --> 00:09:35,880 And this is true when x = 1/2. 150 00:09:35,880 --> 00:09:44,190 So these are equal squares. 151 00:09:44,190 --> 00:09:55,970 And most when there's only one square. 152 00:09:55,970 --> 00:10:00,480 Which is more or less the limiting situation, 153 00:10:00,480 --> 00:10:05,490 if one of the pieces disappears. 154 00:10:05,490 --> 00:10:09,840 Now, so that's the first kind of example. 155 00:10:09,840 --> 00:10:14,560 And I just want to make one more comment about terminology 156 00:10:14,560 --> 00:10:16,600 before we go on. 157 00:10:16,600 --> 00:10:20,690 And I will introduce it with the following question. 158 00:10:20,690 --> 00:10:31,510 What is the minimum? 159 00:10:31,510 --> 00:10:39,620 So, what is the minimum? 160 00:10:39,620 --> 00:10:40,120 Yeah. 161 00:10:40,120 --> 00:10:44,032 STUDENT: [INAUDIBLE] 162 00:10:44,032 --> 00:10:44,740 PROFESSOR: Right. 163 00:10:44,740 --> 00:10:46,640 The lowest value of the function. 164 00:10:46,640 --> 00:10:52,600 So the answer to that question is 1/32. 165 00:10:52,600 --> 00:10:56,830 Now, the problem with this question and you 166 00:10:56,830 --> 00:11:07,100 will-- so that refers to the minimum value. 167 00:11:07,100 --> 00:11:08,900 But then there's this other question 168 00:11:08,900 --> 00:11:15,950 which is where is the minimum. 169 00:11:15,950 --> 00:11:22,390 And the answer to that is x = 1/2. 170 00:11:22,390 --> 00:11:30,310 So one of them is the minimum point, 171 00:11:30,310 --> 00:11:33,640 and the other one is the minimum value. 172 00:11:33,640 --> 00:11:35,840 So they're two separate things. 173 00:11:35,840 --> 00:11:39,210 Now, the problem is that people are sloppy. 174 00:11:39,210 --> 00:11:43,550 And especially since you usually find the critical point 175 00:11:43,550 --> 00:11:48,620 first, and the value, that is plugging in for A, 176 00:11:48,620 --> 00:11:51,430 second, people will stop short and they'll 177 00:11:51,430 --> 00:11:54,010 give the wrong answer to the question, for instance. 178 00:11:54,010 --> 00:11:57,760 Now, both questions are important to answer. 179 00:11:57,760 --> 00:12:01,040 You just need to have a word to put there. 180 00:12:01,040 --> 00:12:02,490 So this is a little bit careless. 181 00:12:02,490 --> 00:12:06,060 When we say what is the minimum, some people will say 1/2. 182 00:12:06,060 --> 00:12:08,030 And that's literally wrong. 183 00:12:08,030 --> 00:12:09,800 They know what they mean. 184 00:12:09,800 --> 00:12:10,890 But it's just wrong. 185 00:12:10,890 --> 00:12:13,520 And when people ask this question, they're being sloppy. 186 00:12:13,520 --> 00:12:14,210 Anyway. 187 00:12:14,210 --> 00:12:16,010 They should maybe be a little clearer 188 00:12:16,010 --> 00:12:17,850 and say what's the minimum value. 189 00:12:17,850 --> 00:12:20,320 Or, where is the value achieved. 190 00:12:20,320 --> 00:12:27,080 It's achieved at, or where is the minimum value achieved. 191 00:12:27,080 --> 00:12:31,970 "Where is min achieved?", would be a better way of phrasing 192 00:12:31,970 --> 00:12:34,050 this second question. 193 00:12:34,050 --> 00:12:37,900 So that it has an unambiguous answer. 194 00:12:37,900 --> 00:12:41,670 And when people ask you for the minimum point, 195 00:12:41,670 --> 00:12:45,490 they're also-- so why is it that we call it the minimum point? 196 00:12:45,490 --> 00:12:48,510 We have this word, critical point, which is what x = 1/2 197 00:12:48,510 --> 00:12:50,330 is, and critical value. 198 00:12:50,330 --> 00:12:52,840 And so I'm making those same distinctions here. 199 00:12:52,840 --> 00:12:59,680 But there's another notion of a minimum point, 200 00:12:59,680 --> 00:13:09,340 and this is an alternative if you like. 201 00:13:09,340 --> 00:13:19,540 The minimum point is the point (1/2, 1/32). 202 00:13:19,540 --> 00:13:25,350 Right, that's a point on the graph. 203 00:13:25,350 --> 00:13:29,490 It's the point - well, so that graph is way up there, 204 00:13:29,490 --> 00:13:30,740 but I'll just put it on there. 205 00:13:30,740 --> 00:13:33,780 That's this point. 206 00:13:33,780 --> 00:13:35,792 And you might say min there. 207 00:13:35,792 --> 00:13:38,166 And you might point to this point, and you might say max. 208 00:13:38,166 --> 00:13:42,540 And similarly, this one might be a max. 209 00:13:42,540 --> 00:13:44,840 So in other words, what this means 210 00:13:44,840 --> 00:13:49,375 is simply that people are a little sloppy. 211 00:13:49,375 --> 00:13:50,750 And sometimes they mean one thing 212 00:13:50,750 --> 00:13:53,770 and sometimes they mean another. 213 00:13:53,770 --> 00:13:56,470 And you're just stuck with this, because there'll 214 00:13:56,470 --> 00:13:58,200 be some authors who will say one thing 215 00:13:58,200 --> 00:13:59,575 and some people will mean another 216 00:13:59,575 --> 00:14:02,770 and you just have to live with this little bit of annoying 217 00:14:02,770 --> 00:14:05,150 ambiguity. 218 00:14:05,150 --> 00:14:05,690 Yeah? 219 00:14:05,690 --> 00:14:08,280 STUDENT: [INAUDIBLE] 220 00:14:08,280 --> 00:14:12,735 PROFESSOR: OK, so that's a good - very good. 221 00:14:12,735 --> 00:14:16,710 So here we go, find the largest area enclosed. 222 00:14:16,710 --> 00:14:22,290 So that's sort of a trick question, isn't it? 223 00:14:22,290 --> 00:14:28,410 So there are various - that's a good thing to ask. 224 00:14:28,410 --> 00:14:30,530 That's sort of a trick question, why? 225 00:14:30,530 --> 00:14:34,840 Because according to the rules, we're 226 00:14:34,840 --> 00:14:38,280 trapped between the two maxima at something 227 00:14:38,280 --> 00:14:40,320 which is strictly below. 228 00:14:40,320 --> 00:14:44,140 So in other words, one answer to this question would be, 229 00:14:44,140 --> 00:14:48,000 and this is the answer that I would probably give, is 1/16. 230 00:14:48,000 --> 00:14:50,640 But that's not really true. 231 00:14:50,640 --> 00:14:56,560 Because that's only in the limit. 232 00:14:56,560 --> 00:15:01,346 As x goes to 0, or as x goes to 1-. 233 00:15:01,346 --> 00:15:02,720 And if you like, the most is when 234 00:15:02,720 --> 00:15:04,830 you've only got one square. 235 00:15:04,830 --> 00:15:07,550 Which breaks the rules of the problem. 236 00:15:07,550 --> 00:15:11,290 So, essentially, it's a trick question. 237 00:15:11,290 --> 00:15:13,670 But I would answer it this way. 238 00:15:13,670 --> 00:15:16,220 Because that's the most interesting part of the answer, 239 00:15:16,220 --> 00:15:18,780 which is that it's 1/16 and it occurs really 240 00:15:18,780 --> 00:15:28,920 when one of the squares disappears to nothing. 241 00:15:28,920 --> 00:15:34,180 So now, let's do another example here. 242 00:15:34,180 --> 00:15:41,450 And I just want to illustrate the second style, 243 00:15:41,450 --> 00:15:42,980 or the second type of question. 244 00:15:42,980 --> 00:15:43,480 Yeah. 245 00:15:43,480 --> 00:15:51,750 STUDENT: [INAUDIBLE] 246 00:15:51,750 --> 00:15:56,320 PROFESSOR: The question is, since the question was, 247 00:15:56,320 --> 00:16:00,090 what was the largest area, why did we find the least area. 248 00:16:00,090 --> 00:16:04,050 The reason is that when we go about our procedure of looking 249 00:16:04,050 --> 00:16:10,116 for the least, or the most, we'll automatically find both. 250 00:16:10,116 --> 00:16:12,490 Because we don't know which one is which until we compare 251 00:16:12,490 --> 00:16:14,730 values. 252 00:16:14,730 --> 00:16:17,170 And actually, it's much more to your advantage 253 00:16:17,170 --> 00:16:20,540 to figure out both the maximum and minimum 254 00:16:20,540 --> 00:16:22,316 whenever you answer such a question. 255 00:16:22,316 --> 00:16:24,440 Because otherwise you won't understand the behavior 256 00:16:24,440 --> 00:16:26,590 of the function very well. 257 00:16:26,590 --> 00:16:27,850 So, the question. 258 00:16:27,850 --> 00:16:30,500 We started out with one question, we answered both. 259 00:16:30,500 --> 00:16:31,750 We answered two questions. 260 00:16:31,750 --> 00:16:36,480 We answered the question of what the largest and the smallest 261 00:16:36,480 --> 00:16:37,290 value was. 262 00:16:37,290 --> 00:16:39,505 STUDENT: Also, I'm wondering if you 263 00:16:39,505 --> 00:16:42,680 can check both the minimum [INAUDIBLE] approaches 264 00:16:42,680 --> 00:16:47,505 [INAUDIBLE]. 265 00:16:47,505 --> 00:16:48,130 PROFESSOR: Yes. 266 00:16:48,130 --> 00:16:49,880 One can also use-- the question is, can we 267 00:16:49,880 --> 00:16:51,490 use the second derivative test. 268 00:16:51,490 --> 00:16:53,480 And the answer is, yes we can. 269 00:16:53,480 --> 00:16:55,610 In fact, you can actually also stare at this 270 00:16:55,610 --> 00:16:57,590 and see that it's a sum of squares. 271 00:16:57,590 --> 00:17:00,850 So it's always curving up. 272 00:17:00,850 --> 00:17:04,710 It's a parabola with a positive second coefficient. 273 00:17:04,710 --> 00:17:06,670 So you can differentiate this twice. 274 00:17:06,670 --> 00:17:09,670 If you do you'll get 1/8 plus another 1/8 275 00:17:09,670 --> 00:17:11,450 and you'll get 1/16. 276 00:17:11,450 --> 00:17:14,540 So the second derivative is 1/16. 277 00:17:14,540 --> 00:17:17,730 Is 1/4. 278 00:17:17,730 --> 00:17:21,444 And that's an acceptable way of figuring it out. 279 00:17:21,444 --> 00:17:23,360 I'll mention the second derivative test again, 280 00:17:23,360 --> 00:17:24,950 in this second example. 281 00:17:24,950 --> 00:17:32,580 So let me talk about a second example. 282 00:17:32,580 --> 00:17:35,070 So again, this is going to be another question. 283 00:17:35,070 --> 00:17:37,950 STUDENT: [INAUDIBLE] 284 00:17:37,950 --> 00:17:43,660 PROFESSOR: The question is, when I say minimum or maximum 285 00:17:43,660 --> 00:17:44,880 point which will I mean. 286 00:17:44,880 --> 00:17:50,570 STUDENT: [INAUDIBLE] 287 00:17:50,570 --> 00:17:53,100 PROFESSOR: So I just repeated the question. 288 00:17:53,100 --> 00:17:56,650 So the question is, when I say minimum point, 289 00:17:56,650 --> 00:17:58,580 what will I mean? 290 00:17:58,580 --> 00:18:00,100 OK? 291 00:18:00,100 --> 00:18:05,490 And the answer is that for the purposes of this class 292 00:18:05,490 --> 00:18:09,090 I will probably avoid saying that. 293 00:18:09,090 --> 00:18:13,340 But I will say, probably, where is the minimum achieved. 294 00:18:13,340 --> 00:18:14,680 Just in order to avoid that. 295 00:18:14,680 --> 00:18:17,980 If I actually said it, I often am referring to the graph, 296 00:18:17,980 --> 00:18:19,300 and I mean this. 297 00:18:19,300 --> 00:18:21,280 And in fact, when you get your little review 298 00:18:21,280 --> 00:18:23,090 for the second exam, I'll say exactly 299 00:18:23,090 --> 00:18:25,860 that on the review sheet. 300 00:18:25,860 --> 00:18:28,700 And I'll make this very clear when we were doing this. 301 00:18:28,700 --> 00:18:30,700 However, I just want to prepare you for the fact 302 00:18:30,700 --> 00:18:35,480 that in real life, and even me when I'm talking colloquially, 303 00:18:35,480 --> 00:18:37,750 when I say what's the minimum point of something, 304 00:18:37,750 --> 00:18:48,730 I might actually be mixing it up with this other notion here. 305 00:18:48,730 --> 00:18:54,380 So let's do another example. 306 00:18:54,380 --> 00:18:57,440 So this is an example to get us used 307 00:18:57,440 --> 00:18:59,880 to the notion of constraints. 308 00:18:59,880 --> 00:19:08,250 So we have, so consider a box without a top. 309 00:19:08,250 --> 00:19:16,860 Or, if you like, we're going to find the box without a top. 310 00:19:16,860 --> 00:19:35,380 With least surface area for a fixed volume. 311 00:19:35,380 --> 00:19:38,950 Find the box without a top with least 312 00:19:38,950 --> 00:19:42,360 surface area for a fixed volume. 313 00:19:42,360 --> 00:19:47,940 The procedure for working this out is the following. 314 00:19:47,940 --> 00:19:51,480 You make this diagram. 315 00:19:51,480 --> 00:19:56,220 And you set up the variables. 316 00:19:56,220 --> 00:20:00,390 In this case, we're going to have four names of variables. 317 00:20:00,390 --> 00:20:02,890 We have four letters that we have to choose. 318 00:20:02,890 --> 00:20:05,680 And we'll choose them in a kind of a standard way, alright? 319 00:20:05,680 --> 00:20:08,260 So first I have to tell you one more thing. 320 00:20:08,260 --> 00:20:12,880 Which is something that we could calculate separately 321 00:20:12,880 --> 00:20:15,080 but I'm just going to give it to you in advance. 322 00:20:15,080 --> 00:20:16,746 Which is that it turns out that the best 323 00:20:16,746 --> 00:20:21,420 box has a square bottom. 324 00:20:21,420 --> 00:20:24,710 And that's going to get rid of one of our variables for us. 325 00:20:24,710 --> 00:20:28,500 So it's got a square bottom, and so let's draw a picture of it. 326 00:20:28,500 --> 00:20:36,860 So here's our box. 327 00:20:36,860 --> 00:20:40,680 Well, that goes down like this, almost. 328 00:20:40,680 --> 00:20:49,620 Maybe I should get it a little farther down. 329 00:20:49,620 --> 00:20:52,070 So here's our box. 330 00:20:52,070 --> 00:20:54,900 Let's correct that just a bit. 331 00:20:54,900 --> 00:20:57,760 So now, what about the dimensions of this box? 332 00:20:57,760 --> 00:21:02,050 Well, this is going to be x, and this is very foreshortened, 333 00:21:02,050 --> 00:21:03,180 but it's also x. 334 00:21:03,180 --> 00:21:06,130 The bottom is x by x, it's the same dimensions. 335 00:21:06,130 --> 00:21:12,190 And then the vertical dimension is y. 336 00:21:12,190 --> 00:21:13,750 So far, so good. 337 00:21:13,750 --> 00:21:16,620 Now, I promised you two more letter names. 338 00:21:16,620 --> 00:21:21,040 I want to compute the volume. 339 00:21:21,040 --> 00:21:24,440 The volume is, the base is x^2, and the height is y. 340 00:21:24,440 --> 00:21:26,780 So there's the volume. 341 00:21:26,780 --> 00:21:32,870 And then the area, the area is the area of the bottom, 342 00:21:32,870 --> 00:21:37,940 which is x^2, that's the bottom. 343 00:21:37,940 --> 00:21:40,510 And then there are the four sides. 344 00:21:40,510 --> 00:21:44,230 And the four sides are rectangles of dimensions x, y. 345 00:21:44,230 --> 00:21:49,500 So it's 4xy. 346 00:21:49,500 --> 00:21:53,630 So these are the sides. 347 00:21:53,630 --> 00:21:56,680 And remember, there's no top. 348 00:21:56,680 --> 00:21:58,610 So that's our setup. 349 00:21:58,610 --> 00:22:02,670 So now, the difference between this problem 350 00:22:02,670 --> 00:22:05,230 and the last problem is that there 351 00:22:05,230 --> 00:22:06,690 are two variables floating around, 352 00:22:06,690 --> 00:22:09,710 namely x and y, which are not determined. 353 00:22:09,710 --> 00:22:17,670 But there's what's called a constraint here. 354 00:22:17,670 --> 00:22:22,920 Namely, we've fixed the relationship between x and y. 355 00:22:22,920 --> 00:22:30,880 And so, that means that we can solve for y in terms of x. 356 00:22:30,880 --> 00:22:33,310 So y = V/x^2. 357 00:22:40,650 --> 00:22:43,660 And then, we can plug that into the formula for A. 358 00:22:43,660 --> 00:22:50,280 So here we have A which is x^2 + 4x v/x^2. 359 00:23:00,911 --> 00:23:01,410 Question. 360 00:23:01,410 --> 00:23:14,600 STUDENT: [INAUDIBLE] 361 00:23:14,600 --> 00:23:16,100 PROFESSOR: The question is, will you 362 00:23:16,100 --> 00:23:17,350 need to know this intuitively? 363 00:23:17,350 --> 00:23:17,849 No. 364 00:23:17,849 --> 00:23:20,830 That's something that I would have to give to you. 365 00:23:20,830 --> 00:23:24,870 I mean, it's actually true that a lot 366 00:23:24,870 --> 00:23:28,040 of things, the correct answer is something symmetric. 367 00:23:28,040 --> 00:23:30,240 In this last problem, the minimum 368 00:23:30,240 --> 00:23:33,100 turned out to be exactly halfway in between because there 369 00:23:33,100 --> 00:23:35,900 were sort of equal demands from the two sides. 370 00:23:35,900 --> 00:23:37,480 And similarly, here, what happens 371 00:23:37,480 --> 00:23:40,130 is if you elongate one side, you get less 372 00:23:40,130 --> 00:23:46,050 - it actually is involved with a two variable problem. 373 00:23:46,050 --> 00:23:49,440 Namely, if you have a rectangle and you have a certain amount 374 00:23:49,440 --> 00:23:52,610 of length associated with it, what's the optimal thing 375 00:23:52,610 --> 00:23:53,720 you can do with that. 376 00:23:53,720 --> 00:23:58,280 But I won't-- in other words, the optimal rectangle, 377 00:23:58,280 --> 00:24:01,397 the least perimeter rectangle, turns out to be a square. 378 00:24:01,397 --> 00:24:03,230 That's the little sub-problem that leads you 379 00:24:03,230 --> 00:24:05,960 to this square bottom. 380 00:24:05,960 --> 00:24:09,270 But so that would have been a separate max-min problem. 381 00:24:09,270 --> 00:24:11,900 Which I'm skipping, because I want to do this slightly more 382 00:24:11,900 --> 00:24:16,240 interesting one. 383 00:24:16,240 --> 00:24:20,540 So now, here's our formula for A, 384 00:24:20,540 --> 00:24:27,340 and now I want to follow the same procedure as before. 385 00:24:27,340 --> 00:24:29,320 Namely, we look for the critical point. 386 00:24:29,320 --> 00:24:35,530 Or points. 387 00:24:35,530 --> 00:24:37,260 So let's take a look. 388 00:24:37,260 --> 00:24:40,020 So again, A is (x^2 + 4v) / x. 389 00:24:43,430 --> 00:24:49,800 And A' is 2x - 4v/x^2. 390 00:24:49,800 --> 00:24:59,510 So if we set that equal to 0, we get 2x = 2v/x^2. 391 00:24:59,510 --> 00:25:01,620 So 2x^3. 392 00:25:04,690 --> 00:25:07,270 How did that happen to change into 2? 393 00:25:07,270 --> 00:25:09,590 Interesting, guess that's wrong. 394 00:25:09,590 --> 00:25:12,950 OK. 395 00:25:12,950 --> 00:25:20,480 So this is x^3 2v. 396 00:25:20,480 --> 00:25:23,730 And so x = 2^(1/3) v^(1/3). 397 00:25:28,010 --> 00:25:36,230 So this is the critical point. 398 00:25:36,230 --> 00:25:38,040 So we are not done. 399 00:25:38,040 --> 00:25:38,619 Right? 400 00:25:38,619 --> 00:25:40,160 We're not done, because we don't even 401 00:25:40,160 --> 00:25:41,701 know whether this is going to give us 402 00:25:41,701 --> 00:25:44,410 the worst box or the best box, from this point of view. 403 00:25:44,410 --> 00:25:48,100 The one that uses the most surface area or the least. 404 00:25:48,100 --> 00:25:51,990 So let's check the ends, right away. 405 00:25:51,990 --> 00:25:54,030 To see what's happening. 406 00:25:54,030 --> 00:25:56,280 So can somebody tell me what the ends, what 407 00:25:56,280 --> 00:25:58,060 the end values of x are? 408 00:25:58,060 --> 00:25:59,520 Where does x range from? 409 00:25:59,520 --> 00:26:05,124 STUDENT: [INAUDIBLE] 410 00:26:05,124 --> 00:26:07,040 PROFESSOR: What's the smallest x can be, yeah. 411 00:26:07,040 --> 00:26:16,520 STUDENT: [INAUDIBLE] 412 00:26:16,520 --> 00:26:19,400 PROFESSOR: OK, the claim was that the largest x could be 413 00:26:19,400 --> 00:26:24,040 root A, because somehow there's this x^2 here and you can't get 414 00:26:24,040 --> 00:26:25,685 any further past than that. 415 00:26:25,685 --> 00:26:28,700 But there's a key feature here of this problem. 416 00:26:28,700 --> 00:26:32,050 Which is that A is variable. 417 00:26:32,050 --> 00:26:34,390 The only thing that's fixed in the problem 418 00:26:34,390 --> 00:26:47,440 is V. So if V is fixed, what do you know about x? 419 00:26:47,440 --> 00:26:49,810 STUDENT: [INAUDIBLE] 420 00:26:49,810 --> 00:26:51,160 PROFESSOR: x > 0, yeah. 421 00:26:51,160 --> 00:26:53,060 The lower end point, that's safe. 422 00:26:53,060 --> 00:26:55,440 Because that has to do geometrically with the fact 423 00:26:55,440 --> 00:26:59,430 that we don't have any boxes with negative dimensions. 424 00:26:59,430 --> 00:27:02,360 That would be refused by the Post Office, definitely. 425 00:27:02,360 --> 00:27:04,190 Over and above the empty top, which 426 00:27:04,190 --> 00:27:05,510 they wouldn't accept either. 427 00:27:05,510 --> 00:27:13,330 STUDENT: [INAUDIBLE] 428 00:27:13,330 --> 00:27:17,260 PROFESSOR: It's true that x is less than square root of V / y. 429 00:27:17,260 --> 00:27:19,250 So that's using this relationship. 430 00:27:19,250 --> 00:27:23,750 But notice that y = V/x^2. 431 00:27:26,850 --> 00:27:30,060 So 0 to infinity, I just got a guess there 432 00:27:30,060 --> 00:27:32,080 over here, that's right. 433 00:27:32,080 --> 00:27:33,350 Here's the upper limit. 434 00:27:33,350 --> 00:27:35,750 So this is really important to realize. 435 00:27:35,750 --> 00:27:37,420 This is most problems. 436 00:27:37,420 --> 00:27:40,340 Most problems, the variable, if it doesn't have a limitation, 437 00:27:40,340 --> 00:27:42,370 usually just goes out to infinity. 438 00:27:42,370 --> 00:27:45,790 And infinity is a very important end for the problem. 439 00:27:45,790 --> 00:27:51,516 It's usually an easy end to the problem, too. 440 00:27:51,516 --> 00:27:53,890 So there's a possibility that if we push all the way down 441 00:27:53,890 --> 00:27:56,240 to x = 0, we'll get a better box. 442 00:27:56,240 --> 00:27:58,230 It would be very strange box. 443 00:27:58,230 --> 00:28:01,620 A little bit like our vanishing enclosure. 444 00:28:01,620 --> 00:28:04,920 And maybe an infinitely long box, 445 00:28:04,920 --> 00:28:06,840 also very inconvenient one. 446 00:28:06,840 --> 00:28:07,990 Might be the best box. 447 00:28:07,990 --> 00:28:10,390 We'll have to see. 448 00:28:10,390 --> 00:28:12,810 So let's just take a look at what happens. 449 00:28:12,810 --> 00:28:18,740 So we're looking at A, at 0+. 450 00:28:18,740 --> 00:28:25,760 And that's x^2 + 4V / x with x at 0+. 451 00:28:25,760 --> 00:28:26,760 So what happens to that? 452 00:28:26,760 --> 00:28:35,220 Notice right here, this is going to infinity. 453 00:28:35,220 --> 00:28:38,500 So this is infinite. 454 00:28:38,500 --> 00:28:42,250 So that turns out to be a bad box. 455 00:28:42,250 --> 00:28:45,730 Let's take a look at the other end. 456 00:28:45,730 --> 00:28:51,930 So this is x^2 + 4V / x, x going to infinity. 457 00:28:51,930 --> 00:28:59,930 And again, this term here means that this thing is infinite. 458 00:28:59,930 --> 00:29:01,870 So the shape of this thing, I'll draw 459 00:29:01,870 --> 00:29:05,000 this tiny little schematic diagram over here. 460 00:29:05,000 --> 00:29:11,110 The shape of this thing is like this, right? 461 00:29:11,110 --> 00:29:13,950 And so, when we find that one turnaround point, 462 00:29:13,950 --> 00:29:19,000 which happened to be at this strange point 2/3-- sorry, 463 00:29:19,000 --> 00:29:24,440 2^(1/3) V^(1/3), that is going to be the minimum. 464 00:29:24,440 --> 00:29:30,090 So we've just discovered that it's the minimum. 465 00:29:30,090 --> 00:29:31,690 Which is just what we were hoping for. 466 00:29:31,690 --> 00:29:38,200 This is going to be the optimal box. 467 00:29:38,200 --> 00:29:44,660 Now, since you asked earlier and since it's worth 468 00:29:44,660 --> 00:29:49,360 checking this as well, let's also 469 00:29:49,360 --> 00:29:51,740 check an alternative justification. 470 00:29:51,740 --> 00:30:02,540 So an alternative to checking ends 471 00:30:02,540 --> 00:30:11,540 is the second derivative test. 472 00:30:11,540 --> 00:30:14,200 I do not recommend the second derivative test. 473 00:30:14,200 --> 00:30:16,230 I try my best, when I give you problems, 474 00:30:16,230 --> 00:30:19,290 to make it really hard to apply the second derivative test. 475 00:30:19,290 --> 00:30:21,320 But in this example, the function 476 00:30:21,320 --> 00:30:24,570 is simple enough so that it's perfectly OK. 477 00:30:24,570 --> 00:30:28,110 If you take the derivative here, remember, 478 00:30:28,110 --> 00:30:34,760 this was whatever it was, 2x - 4V / x^2. 479 00:30:34,760 --> 00:30:43,610 If I take the second derivative, it's 2 + 8V / x^3. 480 00:30:43,610 --> 00:30:45,310 And that's positive. 481 00:30:45,310 --> 00:30:49,580 So this thing is concave up. 482 00:30:49,580 --> 00:30:52,180 And that's consistent with its being-- 483 00:30:52,180 --> 00:30:59,840 the critical point is a min. 484 00:30:59,840 --> 00:31:00,840 Is a minimum point. 485 00:31:00,840 --> 00:31:05,180 See how I almost said, is a min, as opposed to minimum point. 486 00:31:05,180 --> 00:31:05,780 So watch out. 487 00:31:05,780 --> 00:31:06,280 Yes. 488 00:31:06,280 --> 00:31:12,235 STUDENT: [INAUDIBLE] 489 00:31:12,235 --> 00:31:13,860 PROFESSOR: You're one step ahead of me. 490 00:31:13,860 --> 00:31:15,985 The question is, is this the answer to the question 491 00:31:15,985 --> 00:31:20,090 or would we have to give y and A and so on and so forth. 492 00:31:20,090 --> 00:31:23,120 So, again, this is something that I 493 00:31:23,120 --> 00:31:26,220 want to emphasize and take my time with right now. 494 00:31:26,220 --> 00:31:29,800 Because it depends, what kind of real-life problem 495 00:31:29,800 --> 00:31:33,360 you're answering, what kind of answer is appropriate. 496 00:31:33,360 --> 00:31:36,010 So, so far we've found the critical point. 497 00:31:36,010 --> 00:31:38,080 We haven't found the critical value. 498 00:31:38,080 --> 00:31:42,190 We haven't found the dimensions of the box. 499 00:31:42,190 --> 00:31:46,400 So we're going to spend a little bit more time on this, exactly 500 00:31:46,400 --> 00:31:48,920 in order to address these questions. 501 00:31:48,920 --> 00:31:50,170 So, first of all. 502 00:31:50,170 --> 00:31:51,470 The value of y. 503 00:31:51,470 --> 00:31:55,120 So, so far we have x = 2^(1/3) V^(1/3). 504 00:31:55,120 --> 00:31:57,170 And certainly if you're going to build the box, 505 00:31:57,170 --> 00:32:00,410 you also want to know what the y-value is. 506 00:32:00,410 --> 00:32:04,970 The y-value is going to be, let's see. 507 00:32:04,970 --> 00:32:12,110 Well, it's V / x^2, so that's V / (2^(1/3) V^(1/3) )^2, 508 00:32:12,110 --> 00:32:19,770 which comes out to be 2^(2/3) V^(1/3). 509 00:32:19,770 --> 00:32:22,580 So there's the y-value. 510 00:32:22,580 --> 00:32:29,505 On top of that, we could figure out the value of A. 511 00:32:29,505 --> 00:32:31,880 So that's also a perfectly reasonable part of the answer. 512 00:32:31,880 --> 00:32:33,780 Depending on what one is interested in, 513 00:32:33,780 --> 00:32:35,675 you might care how much money it's going 514 00:32:35,675 --> 00:32:38,200 to cost you to build this box. 515 00:32:38,200 --> 00:32:39,330 This optimal box. 516 00:32:39,330 --> 00:32:43,370 And so you plug in the value of A. So A, let's see, is up here. 517 00:32:43,370 --> 00:32:47,740 It's x x^2 + 4V/x. 518 00:32:47,740 --> 00:32:57,810 So that's going to be (2^(1/3) V^(1/3) )^2 plus 4V / (2^(1/3) 519 00:32:57,810 --> 00:32:58,310 V^(1/3) ). 520 00:33:02,330 --> 00:33:07,300 And if you work that all out, what you get turns out to be 3 521 00:33:07,300 --> 00:33:08,170 * 2^(1/3) V^(2/3). 522 00:33:13,730 --> 00:33:17,540 So if you like, one way of answering this question 523 00:33:17,540 --> 00:33:23,160 is these three things. 524 00:33:23,160 --> 00:33:26,490 That would be the minimum point corresponding to the graph. 525 00:33:26,490 --> 00:33:28,950 That would be the answer to this question. 526 00:33:28,950 --> 00:33:31,710 But the reason why I'm carrying it out in such detail 527 00:33:31,710 --> 00:33:35,370 is I want to show you that there are much more meaningful ways 528 00:33:35,370 --> 00:33:37,150 of answering this question. 529 00:33:37,150 --> 00:34:02,810 So let me explain that. 530 00:34:02,810 --> 00:34:07,290 So let me go through some more meaningful answers here. 531 00:34:07,290 --> 00:34:14,770 The first more meaningful answer is the following idea. 532 00:34:14,770 --> 00:34:29,780 Simply, what are known as dimensionless variables. 533 00:34:29,780 --> 00:34:33,050 So the first thing that you notice is the scaling law. 534 00:34:33,050 --> 00:34:36,580 That A / V^(2/3) is the thing that's a dimensionless 535 00:34:36,580 --> 00:34:37,260 quantity. 536 00:34:37,260 --> 00:34:39,360 That happens to be 3 * 2^(1/3). 537 00:34:42,150 --> 00:34:43,970 So that's one thing. 538 00:34:43,970 --> 00:34:45,470 If you want to expand the volume, 539 00:34:45,470 --> 00:34:49,220 you'll have to expand the area by the 2/3 power of the volume. 540 00:34:49,220 --> 00:34:55,430 And if you think of the area as being in, say, square inches, 541 00:34:55,430 --> 00:35:00,000 and the volume of the box as being in cubic inches, 542 00:35:00,000 --> 00:35:02,360 then you can see that this is a dimensionless quantity 543 00:35:02,360 --> 00:35:04,070 and you have a dimensionless number here, 544 00:35:04,070 --> 00:35:06,730 which is a characteristic independent of what 545 00:35:06,730 --> 00:35:09,310 A and V were. 546 00:35:09,310 --> 00:35:15,710 The other dimensionless quantity is the ratio of x to y. 547 00:35:15,710 --> 00:35:19,640 Or x to y. 548 00:35:19,640 --> 00:35:23,680 So, again, that's inches divided by inches. 549 00:35:23,680 --> 00:35:32,950 And it's 2^(1/3) V^(1/3) divided by 2^(-2/3) V^(1/3), 550 00:35:32,950 --> 00:35:36,150 which happens to be 2. 551 00:35:36,150 --> 00:35:41,390 So this is actually the best answer to the question. 552 00:35:41,390 --> 00:35:46,120 And it shows you that the box is a 2:1 box. 553 00:35:46,120 --> 00:35:50,200 If this is 2 and this is 1, that's the good box. 554 00:35:50,200 --> 00:35:56,630 And that is just the shape, if you like, 555 00:35:56,630 --> 00:36:02,670 and it's the optimal shape. 556 00:36:02,670 --> 00:36:04,490 And certainly that, aesthetically, that's 557 00:36:04,490 --> 00:36:12,187 the cleanest answer to the question. 558 00:36:12,187 --> 00:36:13,520 There was a question right here. 559 00:36:13,520 --> 00:36:14,020 Yes. 560 00:36:14,020 --> 00:36:20,537 STUDENT: [INAUDIBLE] 561 00:36:20,537 --> 00:36:22,620 PROFESSOR: Could you repeat that, I couldn't hear. 562 00:36:22,620 --> 00:36:24,036 STUDENT: I'm wondering if you'd be 563 00:36:24,036 --> 00:36:29,850 able to get that answer if you [INAUDIBLE] square. 564 00:36:29,850 --> 00:36:31,350 PROFESSOR: The question is, could we 565 00:36:31,350 --> 00:36:33,058 have gotten the answer if we weren't told 566 00:36:33,058 --> 00:36:34,380 that the bottom was square. 567 00:36:34,380 --> 00:36:39,014 The answer is, yes in 18.02 with multivariable. 568 00:36:39,014 --> 00:36:41,222 You would have to have three letters here, an x, a y, 569 00:36:41,222 --> 00:36:43,470 and a z, if you like. 570 00:36:43,470 --> 00:36:48,170 And then you'd have to work with all three of them. 571 00:36:48,170 --> 00:36:51,890 So I separated out into one, there's 572 00:36:51,890 --> 00:36:55,550 a separate one variable problem that you can do for the base. 573 00:36:55,550 --> 00:36:57,510 And then this is a second one variable problem 574 00:36:57,510 --> 00:36:58,510 for this other thing. 575 00:36:58,510 --> 00:37:00,910 And it's just two consecutive one variable problems 576 00:37:00,910 --> 00:37:03,170 that solve the multivariable problem. 577 00:37:03,170 --> 00:37:05,650 Or, as I say in multivariable calculus, 578 00:37:05,650 --> 00:37:08,740 you can just do it all in one step. 579 00:37:08,740 --> 00:37:09,240 Yeah? 580 00:37:09,240 --> 00:37:11,820 STUDENT: [INAUDIBLE] 581 00:37:11,820 --> 00:37:14,550 PROFESSOR: Why did I divide x by y, 582 00:37:14,550 --> 00:37:17,830 rather than y by x, or in any? 583 00:37:17,830 --> 00:37:22,320 So, again, what I was aiming for was dimensionless quantities. 584 00:37:22,320 --> 00:37:26,640 So x and y are measured in the same units. 585 00:37:26,640 --> 00:37:29,065 And also the proportions of the box 586 00:37:29,065 --> 00:37:34,310 - so that's another word for this is proportions - 587 00:37:34,310 --> 00:37:38,430 are something that's universal, independent of the volume 588 00:37:38,430 --> 00:37:42,020 V. It's something you can say about any box, at any scale. 589 00:37:42,020 --> 00:37:45,040 Whether it be, you know, something 590 00:37:45,040 --> 00:37:48,781 by Cristo in the Common. 591 00:37:48,781 --> 00:37:50,530 Maybe we'll get in here to do some fancy-- 592 00:37:50,530 --> 00:37:55,540 STUDENT: [INAUDIBLE] 593 00:37:55,540 --> 00:37:58,630 PROFESSOR: The proportions is - with geometric problems 594 00:37:58,630 --> 00:38:01,930 typically, when there's a scaling to the problem, where 595 00:38:01,930 --> 00:38:05,020 the answer is the same at small scales and at large scales 596 00:38:05,020 --> 00:38:07,000 - this is capturing that. 597 00:38:07,000 --> 00:38:10,070 So that's why, the ratios are what's capturing that. 598 00:38:10,070 --> 00:38:12,410 And that's why it's aesthetically the nicest thing 599 00:38:12,410 --> 00:38:13,860 to ask. 600 00:38:13,860 --> 00:38:17,905 STUDENT: So, what exactly does the ratio 601 00:38:17,905 --> 00:38:21,639 of the area to the volume ratio [INAUDIBLE] tell us? 602 00:38:21,639 --> 00:38:23,180 PROFESSOR: Unfortunately, this number 603 00:38:23,180 --> 00:38:25,490 is a really obscure number. 604 00:38:25,490 --> 00:38:28,006 So the question is what does this tell us. 605 00:38:28,006 --> 00:38:29,630 The only thing that I want to emphasize 606 00:38:29,630 --> 00:38:31,800 is what's on the left-hand side here. 607 00:38:31,800 --> 00:38:34,870 Which is, it's the area to the 2/3 power over the volume, 608 00:38:34,870 --> 00:38:38,550 so it's a dimensionless quantity that happens to be this. 609 00:38:38,550 --> 00:38:43,700 If you do this, for example, in general with planar diagrams, 610 00:38:43,700 --> 00:38:47,200 circumference to area is a bad ratio to take. 611 00:38:47,200 --> 00:38:48,810 What you want to take is the square 612 00:38:48,810 --> 00:38:50,932 of circumference to area. 613 00:38:50,932 --> 00:38:52,390 Because the square of circumference 614 00:38:52,390 --> 00:38:54,690 has the same dimensions, that is, say, 615 00:38:54,690 --> 00:38:56,880 inches squared to area. 616 00:38:56,880 --> 00:38:58,840 Which is in square inches. 617 00:38:58,840 --> 00:39:01,360 So, again, it's these dimensionless quantities 618 00:39:01,360 --> 00:39:03,440 that you want to cook up. 619 00:39:03,440 --> 00:39:06,810 And those are the ones that will have universal properties. 620 00:39:06,810 --> 00:39:09,990 The most famous of these is the circle 621 00:39:09,990 --> 00:39:13,880 that encloses the most area for its circumference. 622 00:39:13,880 --> 00:39:17,876 And, again, that's only true if you take the square 623 00:39:17,876 --> 00:39:18,880 of the circumference. 624 00:39:18,880 --> 00:39:25,620 You do the units correctly. 625 00:39:25,620 --> 00:39:26,290 Anyway. 626 00:39:26,290 --> 00:39:29,760 So we're here, we've got a shape. 627 00:39:29,760 --> 00:39:31,560 We've got an answer to this question. 628 00:39:31,560 --> 00:39:36,940 And I now want to do this problem. 629 00:39:36,940 --> 00:39:38,267 Well, let's put it this way. 630 00:39:38,267 --> 00:39:40,350 I wanted to do this problem by a different method. 631 00:39:40,350 --> 00:39:43,430 I think I'll take the time to do it. 632 00:39:43,430 --> 00:39:46,790 So I want to do this problem by a slightly different method 633 00:39:46,790 --> 00:39:48,880 here. 634 00:39:48,880 --> 00:39:59,160 So, here's Example 2 by implicit differentiation. 635 00:39:59,160 --> 00:40:01,950 So the same example, but now I'm going to do it 636 00:40:01,950 --> 00:40:03,720 by implicit differentiation. 637 00:40:03,720 --> 00:40:07,050 Well, I'll tell you the advantages and disadvantages 638 00:40:07,050 --> 00:40:08,880 to this method here. 639 00:40:08,880 --> 00:40:20,130 So the situation is, you have to start the same way. 640 00:40:20,130 --> 00:40:24,230 So here is the starting place of the problem. 641 00:40:24,230 --> 00:40:34,980 And the goal was the minimum of A with V constant. 642 00:40:34,980 --> 00:40:38,540 So this was the situation that we were in. 643 00:40:38,540 --> 00:40:45,130 And now, what I want to do is just differentiate. 644 00:40:45,130 --> 00:40:46,990 The function y is implicitly a function 645 00:40:46,990 --> 00:40:54,550 of x, so I can differentiate the first expression. 646 00:40:54,550 --> 00:41:00,380 And that yields 0 = 2xy + x^2 y'. 647 00:41:03,330 --> 00:41:11,590 So this is giving me my implicit formula for y', So y' = -2xy / 648 00:41:11,590 --> 00:41:13,080 x^2. 649 00:41:13,080 --> 00:41:14,950 Or in other words, -2y/x. 650 00:41:19,660 --> 00:41:24,990 And then I also have the dA/dx. 651 00:41:24,990 --> 00:41:28,750 Now, you may notice I'm not using primes quite as much. 652 00:41:28,750 --> 00:41:31,510 Because all of the variables are varying, 653 00:41:31,510 --> 00:41:34,300 and so here I'm emphasizing that it's a differentiation 654 00:41:34,300 --> 00:41:36,790 with respect to the variable x. 655 00:41:36,790 --> 00:41:48,130 And this becomes 2x + 4y + 4xy'. 656 00:41:48,130 --> 00:41:53,140 So again, this is using the product rule. 657 00:41:53,140 --> 00:41:57,450 And now I can plug in for what y' is, which is right above it. 658 00:41:57,450 --> 00:42:04,890 So this is 2x + 4y + 4x (-2y/x). 659 00:42:09,330 --> 00:42:14,150 And that's equal to 0. 660 00:42:14,150 --> 00:42:24,770 And so let's gather that together. 661 00:42:24,770 --> 00:42:25,960 So what do we have? 662 00:42:25,960 --> 00:42:34,290 We have 2x + 4y, and then, altogether, this is 8 - 8y, 663 00:42:34,290 --> 00:42:36,550 equals 0. 664 00:42:36,550 --> 00:42:41,600 So that's the same thing as 2x = 4y. 665 00:42:41,600 --> 00:42:44,100 The - 4y goes to the other side. 666 00:42:44,100 --> 00:42:54,420 And so, x/y = 2. 667 00:42:54,420 --> 00:42:59,260 So this, I claim, so you have to decide for yourself. 668 00:42:59,260 --> 00:43:03,630 But I claim that this is faster. 669 00:43:03,630 --> 00:43:07,350 It's faster, and also it gets to the heart 670 00:43:07,350 --> 00:43:10,870 of the matter, which is this scale-invariant proportions. 671 00:43:10,870 --> 00:43:13,360 Which is basically also nicer. 672 00:43:13,360 --> 00:43:16,720 So it gets to the nicer answer, also. 673 00:43:16,720 --> 00:43:19,970 So those are the advantages that this has. 674 00:43:19,970 --> 00:43:23,190 So it's faster, and it gets to this, 675 00:43:23,190 --> 00:43:26,260 I'm going to call it nicer. 676 00:43:26,260 --> 00:43:38,640 And the disadvantage is it did not 677 00:43:38,640 --> 00:43:55,620 check whether this critical point is a max, min, 678 00:43:55,620 --> 00:43:59,540 or neither. 679 00:43:59,540 --> 00:44:02,280 So we didn't quite finish the problem. 680 00:44:02,280 --> 00:44:10,820 But we got to the answer very fast. 681 00:44:10,820 --> 00:44:11,640 Yeah, question. 682 00:44:11,640 --> 00:44:13,480 STUDENT: [INAUDIBLE] 683 00:44:13,480 --> 00:44:17,437 PROFESSOR: How would you check it? 684 00:44:17,437 --> 00:44:18,270 STUDENT: [INAUDIBLE] 685 00:44:18,270 --> 00:44:20,510 PROFESSOR: Well, so it gives you a candidate. 686 00:44:20,510 --> 00:44:23,390 The answer is-- so the question is, how would you check it? 687 00:44:23,390 --> 00:44:26,720 The answer is that for this particular problem, 688 00:44:26,720 --> 00:44:31,300 the only way to do it is to do something like this. 689 00:44:31,300 --> 00:44:34,480 So in other words, it doesn't save you that much time. 690 00:44:34,480 --> 00:44:37,730 But with many, many, examples, you actually 691 00:44:37,730 --> 00:44:42,010 can tell immediately that if the two ends, the thing is, say, 0, 692 00:44:42,010 --> 00:44:43,650 and inside it's positive. 693 00:44:43,650 --> 00:44:44,370 Things like that. 694 00:44:44,370 --> 00:44:53,640 So in many, many, cases this is just as good. 695 00:44:53,640 --> 00:44:58,510 So now I'm going to change subjects here. 696 00:44:58,510 --> 00:45:02,220 But the subject that I'm going to talk about next 697 00:45:02,220 --> 00:45:07,550 is almost-- is very, very closely linked. 698 00:45:07,550 --> 00:45:10,850 Namely, I talked about implicit differentiation. 699 00:45:10,850 --> 00:45:12,770 Now, we're going to just talk about dealing 700 00:45:12,770 --> 00:45:13,920 with lots of variables. 701 00:45:13,920 --> 00:45:15,360 And rates of change. 702 00:45:15,360 --> 00:45:16,960 So, essentially, we're going to talk 703 00:45:16,960 --> 00:45:19,680 about the same type of thing. 704 00:45:19,680 --> 00:45:23,470 So, I'm going to tell you about a subject which 705 00:45:23,470 --> 00:45:25,220 is called related rates. 706 00:45:25,220 --> 00:45:26,980 Which is really just another excuse 707 00:45:26,980 --> 00:45:32,510 for getting used to setting up variables and equations. 708 00:45:32,510 --> 00:45:34,320 So, here we go. 709 00:45:34,320 --> 00:45:36,000 Related rates. 710 00:45:36,000 --> 00:45:41,970 And I'm going to illustrate this with one example today, one 711 00:45:41,970 --> 00:45:44,350 tomorrow. 712 00:45:44,350 --> 00:45:47,600 So here's my example for today. 713 00:45:47,600 --> 00:45:50,810 So, again, this is going to be a police problem. 714 00:45:50,810 --> 00:45:53,630 But this is going to be a word problem and-- sorry, 715 00:45:53,630 --> 00:45:56,910 I'm don't want to scare you, no police. 716 00:45:56,910 --> 00:45:59,940 Well, there are police in the story but they're not present. 717 00:45:59,940 --> 00:46:05,070 So, but I'm going to draw it immediately with the diagram 718 00:46:05,070 --> 00:46:08,410 because I'm going to save us the trouble. 719 00:46:08,410 --> 00:46:11,580 Although, you know, the point here is to get from the words 720 00:46:11,580 --> 00:46:15,940 to the diagram. 721 00:46:15,940 --> 00:46:21,990 So you have the police, and they're 30 feet from the road. 722 00:46:21,990 --> 00:46:25,750 And here's the road. 723 00:46:25,750 --> 00:46:36,610 And you're coming along, here, in your, let's see, 724 00:46:36,610 --> 00:46:39,420 in your car going in this direction here. 725 00:46:39,420 --> 00:46:43,760 And the police have radar. 726 00:46:43,760 --> 00:46:46,490 Which is bouncing off of your car. 727 00:46:46,490 --> 00:46:53,880 And what they read off is that you're 50 feet away. 728 00:46:53,880 --> 00:46:57,100 They also know that you're approaching 729 00:46:57,100 --> 00:47:12,370 along the line of the radar at a rate of 80 feet per second. 730 00:47:12,370 --> 00:47:19,260 Now, the question is, are you speeding. 731 00:47:19,260 --> 00:47:20,730 That's the question. 732 00:47:20,730 --> 00:47:29,030 So when you're speeding, by the way, 95 feet per second 733 00:47:29,030 --> 00:47:32,090 is about 65 miles per hour. 734 00:47:32,090 --> 00:47:35,890 So that's the threshold here. 735 00:47:35,890 --> 00:47:40,080 So what I want to do now is show you 736 00:47:40,080 --> 00:47:43,860 how you set up a problem like this. 737 00:47:43,860 --> 00:47:46,150 This distance is 50. 738 00:47:46,150 --> 00:47:50,460 This is 30, and because it's the distance to a straight line 739 00:47:50,460 --> 00:47:52,030 you know that this is a right angle. 740 00:47:52,030 --> 00:47:54,580 So we know that this is a right triangle. 741 00:47:54,580 --> 00:47:56,810 And this is set out to be a right triangle, which 742 00:47:56,810 --> 00:48:00,300 is an easy one, a 3, 4, 5 right triangle 743 00:48:00,300 --> 00:48:05,640 just so that we can do the computations easily. 744 00:48:05,640 --> 00:48:09,180 So now, the question is, how do we put the letters in 745 00:48:09,180 --> 00:48:10,970 to make this problem work, to figure out 746 00:48:10,970 --> 00:48:12,310 what the rate of change is. 747 00:48:12,310 --> 00:48:15,930 So now, let me explain that right now. 748 00:48:15,930 --> 00:48:18,590 And we will actually do the computation next time. 749 00:48:18,590 --> 00:48:22,940 So the first thing is, you have to understand what's changing 750 00:48:22,940 --> 00:48:24,850 and what's not. 751 00:48:24,850 --> 00:48:30,880 And we're going to use t for time, in seconds. 752 00:48:30,880 --> 00:48:35,210 And now, an important distance here 753 00:48:35,210 --> 00:48:37,820 is the distance to this foot of this perpendicular. 754 00:48:37,820 --> 00:48:41,310 So I'm going to name that x. 755 00:48:41,310 --> 00:48:42,710 I'm going to give that letter x. 756 00:48:42,710 --> 00:48:44,480 Now, x is varying. 757 00:48:44,480 --> 00:48:47,359 The reason why I need a letter for it as opposed to this 40 758 00:48:47,359 --> 00:48:49,150 is that it's going to have a rate of change 759 00:48:49,150 --> 00:48:50,220 with respect to t. 760 00:48:50,220 --> 00:48:54,520 And, in fact, it's related to-- the question 761 00:48:54,520 --> 00:49:00,432 is whether dx/dt is faster or slower than 95. 762 00:49:00,432 --> 00:49:01,890 So that's the thing that's varying. 763 00:49:01,890 --> 00:49:04,010 Now, there's something else that's varying. 764 00:49:04,010 --> 00:49:06,000 This distance here is also varying. 765 00:49:06,000 --> 00:49:08,760 So we need a letter for that. 766 00:49:08,760 --> 00:49:11,620 We do not need a letter for this. 767 00:49:11,620 --> 00:49:12,920 Because it's never changing. 768 00:49:12,920 --> 00:49:15,010 We're assuming the police are parked. 769 00:49:15,010 --> 00:49:17,382 They're not ready to roar out and catch you just yet, 770 00:49:17,382 --> 00:49:18,840 and they're certainly not in motion 771 00:49:18,840 --> 00:49:20,880 when they've got the radar guns aimed at you. 772 00:49:20,880 --> 00:49:24,110 So you need to know something about the sociology 773 00:49:24,110 --> 00:49:28,160 and style of police. 774 00:49:28,160 --> 00:49:30,950 So you need to know things about the real world. 775 00:49:30,950 --> 00:49:36,145 Now, the last bit is, what about this 80 here. 776 00:49:36,145 --> 00:49:37,770 So this is how fast you're approaching. 777 00:49:37,770 --> 00:49:40,290 Now, that's measured along the radar gun. 778 00:49:40,290 --> 00:49:44,950 I claim that that's d/dt of this quantity here. 779 00:49:44,950 --> 00:49:46,910 So this is D is also changing. 780 00:49:46,910 --> 00:49:49,200 That's why we needed a letter for it, too. 781 00:49:49,200 --> 00:49:51,730 So, next time, we'll just put that all together 782 00:49:51,730 --> 00:49:55,010 and compute dx / dt.