WEBVTT 00:00:00.000 --> 00:00:06.508 The following content is provided under a Creative 00:00:06.508 --> 00:00:07.174 Commons license. 00:00:07.174 --> 00:00:07.438 Your support will help MIT OpenCourseWare 00:00:07.438 --> 00:00:09.940 continue to offer high quality educational resources for free. 00:00:09.940 --> 00:00:12.065 To make a donation, or to view additional materials 00:00:12.065 --> 00:00:18.970 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.970 --> 00:00:22.780 at ocw.mit.edu. 00:00:22.780 --> 00:00:24.390 PROFESSOR: In the twelfth lecture, 00:00:24.390 --> 00:00:31.110 we're going to talk about maxima and minima. 00:00:31.110 --> 00:00:33.110 Let's finish up what we did last time. 00:00:33.110 --> 00:00:35.710 We really only just started with maxima and minima. 00:00:35.710 --> 00:00:38.100 And then we're going to talk about related rates. 00:00:38.100 --> 00:00:48.180 So, right now I want to give you some examples 00:00:48.180 --> 00:00:51.430 of max-min problems. 00:00:51.430 --> 00:00:55.540 And we're going to start with a fairly basic one. 00:00:55.540 --> 00:00:58.540 So what's the thing about max-min problems? 00:00:58.540 --> 00:01:01.240 The main thing is that we're asking 00:01:01.240 --> 00:01:05.560 you to do a little bit more of the interpretation of word 00:01:05.560 --> 00:01:06.060 problems. 00:01:06.060 --> 00:01:09.970 So many of the problems are expressed in terms of words. 00:01:09.970 --> 00:01:18.650 And so, in this case, we have a wire which is length 1. 00:01:18.650 --> 00:01:29.610 Cut into two pieces. 00:01:29.610 --> 00:01:38.150 And then each piece encloses a square. 00:01:38.150 --> 00:01:44.400 Sorry, encloses a square. 00:01:44.400 --> 00:01:47.170 And the problem - so this is the setup. 00:01:47.170 --> 00:02:02.650 And the problem is to find the largest area enclosed. 00:02:02.650 --> 00:02:03.910 So here's the problem. 00:02:03.910 --> 00:02:10.340 Now, in all of these cases, in all these cases, 00:02:10.340 --> 00:02:11.600 there's a bunch of words. 00:02:11.600 --> 00:02:18.599 And your job is typically to draw a diagram. 00:02:18.599 --> 00:02:20.890 So the first thing you want to do is to draw a diagram. 00:02:20.890 --> 00:02:23.340 In this case, it can be fairly schematic. 00:02:23.340 --> 00:02:25.400 Here's your unit length. 00:02:25.400 --> 00:02:27.440 And when you draw the diagram, you're 00:02:27.440 --> 00:02:29.370 going to have to pick variables. 00:02:29.370 --> 00:02:34.700 So those are really the two main tasks. 00:02:34.700 --> 00:02:35.665 To set up the problem. 00:02:35.665 --> 00:02:37.170 So you're drawing a diagram. 00:02:37.170 --> 00:02:40.860 This is like word problems of old, in grade school 00:02:40.860 --> 00:02:42.410 through high school. 00:02:42.410 --> 00:02:50.020 Draw a diagram and name the variables. 00:02:50.020 --> 00:02:52.900 So we'll be doing a lot of that today. 00:02:52.900 --> 00:02:54.950 So here's my unit length. 00:02:54.950 --> 00:02:58.160 And I'm going to choose the variable x 00:02:58.160 --> 00:03:00.280 to be the length of one of the pieces of wire. 00:03:00.280 --> 00:03:03.240 And that makes the other piece 1 - x. 00:03:03.240 --> 00:03:05.579 And that's pretty much the whole diagram, 00:03:05.579 --> 00:03:07.870 except that there's something that we did with the wire 00:03:07.870 --> 00:03:09.710 after we cut it in half. 00:03:09.710 --> 00:03:13.370 Namely, we built two little boxes out of it. 00:03:13.370 --> 00:03:15.520 Like this, these are our squares. 00:03:15.520 --> 00:03:19.900 And their side lengths are x/4 and (1-x)/4. 00:03:25.020 --> 00:03:26.710 So, so far, so good. 00:03:26.710 --> 00:03:28.880 And now we have to think, well, we 00:03:28.880 --> 00:03:30.450 want to find the largest area. 00:03:30.450 --> 00:03:33.050 So I need a formula for area in terms of variables 00:03:33.050 --> 00:03:34.490 that I've described. 00:03:34.490 --> 00:03:35.790 And so that's the last thing. 00:03:35.790 --> 00:03:39.400 I'll give the letter A as the label for the area. 00:03:39.400 --> 00:03:43.760 And then the area is just the square of x/4 00:03:43.760 --> 00:03:51.805 plus the square of 1 minus x - whoops, that strange 2 got 00:03:51.805 --> 00:03:57.780 in here - over 4. 00:03:57.780 --> 00:03:59.110 So far, so good. 00:03:59.110 --> 00:04:01.750 Now, the instinct that you'll have, 00:04:01.750 --> 00:04:04.020 and I'm going to yield to that instinct, 00:04:04.020 --> 00:04:07.370 is we should charge ahead and just differentiate. 00:04:07.370 --> 00:04:07.870 All right? 00:04:07.870 --> 00:04:08.578 That's all right. 00:04:08.578 --> 00:04:10.430 We'll find the critical points. 00:04:10.430 --> 00:04:12.995 So we know that those are important points. 00:04:12.995 --> 00:04:17.840 So we're going to find the critical points. 00:04:17.840 --> 00:04:21.460 In other words, we take the derivative, 00:04:21.460 --> 00:04:27.390 we set the derivative of A with respect to x = 0. 00:04:27.390 --> 00:04:33.020 So if I do that differentiation, I get the, well, 00:04:33.020 --> 00:04:37.240 so the first one, x^2 / 16, that's 8. 00:04:37.240 --> 00:04:40.870 Sorry. 00:04:40.870 --> 00:04:44.060 That's x/8, right? 00:04:44.060 --> 00:04:45.570 That's the derivative of this. 00:04:45.570 --> 00:04:51.290 And if I differentiate this, I get well, the derivative of 1 - 00:04:51.290 --> 00:04:56.930 x^2 is 2(1-x)(a-1). 00:04:56.930 --> 00:04:59.620 So it's -(1-x)/8. 00:05:03.170 --> 00:05:05.010 So there are two minus signs in there, 00:05:05.010 --> 00:05:07.520 I'll let you ponder that differentiation, which 00:05:07.520 --> 00:05:10.900 I did by the chain rule. 00:05:10.900 --> 00:05:12.730 Hang on a sec, OK? 00:05:12.730 --> 00:05:14.960 Just wait until we're done. 00:05:14.960 --> 00:05:17.480 So here's the derivative. 00:05:17.480 --> 00:05:19.890 Is there a problem? 00:05:19.890 --> 00:05:29.060 STUDENT: [INAUDIBLE] 00:05:29.060 --> 00:05:31.320 PROFESSOR: Right, so there's a 1/16 here. 00:05:31.320 --> 00:05:32.850 This is x^2 / 16. 00:05:32.850 --> 00:05:36.870 And so it's 2x over 8, over 16, sorry. 00:05:36.870 --> 00:05:40.360 Which has an 8. 00:05:40.360 --> 00:05:41.790 That's OK. 00:05:41.790 --> 00:05:56.080 All right, so now, This is equal to 0 if and only if x = 1 - x. 00:05:56.080 --> 00:06:02.170 That's 2x = 1, or in other words x = 1/2. 00:06:02.170 --> 00:06:03.280 All right? 00:06:03.280 --> 00:06:06.530 So there's our critical point. 00:06:06.530 --> 00:06:11.710 So x = 1/2 is the critical point. 00:06:11.710 --> 00:06:17.640 And the critical value, which is what 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. 00:06:26.260 --> 00:06:38.030 So that's (1/8)^2 + (1/8)^2 which is 1/32. 00:06:38.030 --> 00:06:45.480 So, so far, so good. 00:06:45.480 --> 00:06:48.050 But we're not done yet. 00:06:48.050 --> 00:06:56.850 We're not done. 00:06:56.850 --> 00:06:59.070 So why aren't we done? 00:06:59.070 --> 00:07:04.590 Because we haven't checked the end points. 00:07:04.590 --> 00:07:08.370 So let's check the end points. 00:07:08.370 --> 00:07:10.450 Now, in this problem, the end points 00:07:10.450 --> 00:07:12.650 are really sort of excluded. 00:07:12.650 --> 00:07:17.530 The ends are between 0 and 1 here. 00:07:17.530 --> 00:07:22.740 That's the possible lengths of the cut. 00:07:22.740 --> 00:07:25.410 And so what we should really be doing is evaluating 00:07:25.410 --> 00:07:29.230 in the limit, so that would be the right-hand limit as x goes 00:07:29.230 --> 00:07:33.780 to 0 of A. And if you plug in x = 0, 00:07:33.780 --> 00:07:42.000 what you get here is 0 + (1/4)^2, which is 1/16. 00:07:42.000 --> 00:07:49.710 And similarly, at the other end, that's 1-, from the left, 00:07:49.710 --> 00:07:56.650 we get (1/4)^2 + 0, which is also 1/16. 00:07:56.650 --> 00:08:01.750 So, what you see is that the schematic picture 00:08:01.750 --> 00:08:05.090 of this function, and this isn't even so far 00:08:05.090 --> 00:08:08.100 off from being the right picture here, 00:08:08.100 --> 00:08:12.120 is that its level here is 1/16 and then it 00:08:12.120 --> 00:08:14.520 dips down and goes up. 00:08:14.520 --> 00:08:15.020 Right? 00:08:15.020 --> 00:08:19.960 This is 1/2, this is 1, and this level here is a half that. 00:08:19.960 --> 00:08:23.070 This is 1/32. 00:08:23.070 --> 00:08:25.900 So we did not find, when we found 00:08:25.900 --> 00:08:29.740 the critical point we did not find the largest area enclosed. 00:08:29.740 --> 00:08:33.060 We found the least area enclosed. 00:08:33.060 --> 00:08:36.850 So if you don't pay attention to what the function looks like, 00:08:36.850 --> 00:08:40.240 not only will you about half the time get the wrong answer, 00:08:40.240 --> 00:08:43.980 you'll get the absolute worst answer. 00:08:43.980 --> 00:08:47.770 You'll get the one which is the polar opposite from what 00:08:47.770 --> 00:08:49.680 you want. 00:08:49.680 --> 00:08:52.250 So you have to pay a little bit of attention to the function 00:08:52.250 --> 00:08:53.440 that you've got. 00:08:53.440 --> 00:08:55.530 And in this case it's just very schematic. 00:08:55.530 --> 00:08:57.320 It dips down and goes up, and that's 00:08:57.320 --> 00:08:59.514 true of pretty much most functions. 00:08:59.514 --> 00:09:00.430 They're fairly simple. 00:09:00.430 --> 00:09:02.096 They maybe only have one critical point. 00:09:02.096 --> 00:09:03.550 They only turn around once. 00:09:03.550 --> 00:09:06.310 But then, maybe the critical point is the maximum 00:09:06.310 --> 00:09:07.760 or maybe it's the minimum. 00:09:07.760 --> 00:09:09.670 Or maybe it's neither, in fact. 00:09:09.670 --> 00:09:15.820 So we'll be discussing that maybe some other time. 00:09:15.820 --> 00:09:26.260 So what we find here is that we have the least area enclosed 00:09:26.260 --> 00:09:30.420 is 1/32. 00:09:30.420 --> 00:09:35.880 And this is true when x = 1/2. 00:09:35.880 --> 00:09:44.190 So these are equal squares. 00:09:44.190 --> 00:09:55.970 And most when there's only one square. 00:09:55.970 --> 00:10:00.480 Which is more or less the limiting situation, 00:10:00.480 --> 00:10:05.490 if one of the pieces disappears. 00:10:05.490 --> 00:10:09.840 Now, so that's the first kind of example. 00:10:09.840 --> 00:10:14.560 And I just want to make one more comment about terminology 00:10:14.560 --> 00:10:16.600 before we go on. 00:10:16.600 --> 00:10:20.690 And I will introduce it with the following question. 00:10:20.690 --> 00:10:31.510 What is the minimum? 00:10:31.510 --> 00:10:39.620 So, what is the minimum? 00:10:39.620 --> 00:10:40.120 Yeah. 00:10:40.120 --> 00:10:44.032 STUDENT: [INAUDIBLE] 00:10:44.032 --> 00:10:44.740 PROFESSOR: Right. 00:10:44.740 --> 00:10:46.640 The lowest value of the function. 00:10:46.640 --> 00:10:52.600 So the answer to that question is 1/32. 00:10:52.600 --> 00:10:56.830 Now, the problem with this question and you 00:10:56.830 --> 00:11:07.100 will-- so that refers to the minimum value. 00:11:07.100 --> 00:11:08.900 But then there's this other question 00:11:08.900 --> 00:11:15.950 which is where is the minimum. 00:11:15.950 --> 00:11:22.390 And the answer to that is x = 1/2. 00:11:22.390 --> 00:11:30.310 So one of them is the minimum point, 00:11:30.310 --> 00:11:33.640 and the other one is the minimum value. 00:11:33.640 --> 00:11:35.840 So they're two separate things. 00:11:35.840 --> 00:11:39.210 Now, the problem is that people are sloppy. 00:11:39.210 --> 00:11:43.550 And especially since you usually find the critical point 00:11:43.550 --> 00:11:48.620 first, and the value, that is plugging in for A, 00:11:48.620 --> 00:11:51.430 second, people will stop short and they'll 00:11:51.430 --> 00:11:54.010 give the wrong answer to the question, for instance. 00:11:54.010 --> 00:11:57.760 Now, both questions are important to answer. 00:11:57.760 --> 00:12:01.040 You just need to have a word to put there. 00:12:01.040 --> 00:12:02.490 So this is a little bit careless. 00:12:02.490 --> 00:12:06.060 When we say what is the minimum, some people will say 1/2. 00:12:06.060 --> 00:12:08.030 And that's literally wrong. 00:12:08.030 --> 00:12:09.800 They know what they mean. 00:12:09.800 --> 00:12:10.890 But it's just wrong. 00:12:10.890 --> 00:12:13.520 And when people ask this question, they're being sloppy. 00:12:13.520 --> 00:12:14.210 Anyway. 00:12:14.210 --> 00:12:16.010 They should maybe be a little clearer 00:12:16.010 --> 00:12:17.850 and say what's the minimum value. 00:12:17.850 --> 00:12:20.320 Or, where is the value achieved. 00:12:20.320 --> 00:12:27.080 It's achieved at, or where is the minimum value achieved. 00:12:27.080 --> 00:12:31.970 "Where is min achieved?", would be a better way of phrasing 00:12:31.970 --> 00:12:34.050 this second question. 00:12:34.050 --> 00:12:37.900 So that it has an unambiguous answer. 00:12:37.900 --> 00:12:41.670 And when people ask you for the minimum point, 00:12:41.670 --> 00:12:45.490 they're also-- so why is it that we call it the minimum point? 00:12:45.490 --> 00:12:48.510 We have this word, critical point, which is what x = 1/2 00:12:48.510 --> 00:12:50.330 is, and critical value. 00:12:50.330 --> 00:12:52.840 And so I'm making those same distinctions here. 00:12:52.840 --> 00:12:59.680 But there's another notion of a minimum point, 00:12:59.680 --> 00:13:09.340 and this is an alternative if you like. 00:13:09.340 --> 00:13:19.540 The minimum point is the point (1/2, 1/32). 00:13:19.540 --> 00:13:25.350 Right, that's a point on the graph. 00:13:25.350 --> 00:13:29.490 It's the point - well, so that graph is way up there, 00:13:29.490 --> 00:13:30.740 but I'll just put it on there. 00:13:30.740 --> 00:13:33.780 That's this point. 00:13:33.780 --> 00:13:35.792 And you might say min there. 00:13:35.792 --> 00:13:38.166 And you might point to this point, and you might say max. 00:13:38.166 --> 00:13:42.540 And similarly, this one might be a max. 00:13:42.540 --> 00:13:44.840 So in other words, what this means 00:13:44.840 --> 00:13:49.375 is simply that people are a little sloppy. 00:13:49.375 --> 00:13:50.750 And sometimes they mean one thing 00:13:50.750 --> 00:13:53.770 and sometimes they mean another. 00:13:53.770 --> 00:13:56.470 And you're just stuck with this, because there'll 00:13:56.470 --> 00:13:58.200 be some authors who will say one thing 00:13:58.200 --> 00:13:59.575 and some people will mean another 00:13:59.575 --> 00:14:02.770 and you just have to live with this little bit of annoying 00:14:02.770 --> 00:14:05.150 ambiguity. 00:14:05.150 --> 00:14:05.690 Yeah? 00:14:05.690 --> 00:14:08.280 STUDENT: [INAUDIBLE] 00:14:08.280 --> 00:14:12.735 PROFESSOR: OK, so that's a good - very good. 00:14:12.735 --> 00:14:16.710 So here we go, find the largest area enclosed. 00:14:16.710 --> 00:14:22.290 So that's sort of a trick question, isn't it? 00:14:22.290 --> 00:14:28.410 So there are various - that's a good thing to ask. 00:14:28.410 --> 00:14:30.530 That's sort of a trick question, why? 00:14:30.530 --> 00:14:34.840 Because according to the rules, we're 00:14:34.840 --> 00:14:38.280 trapped between the two maxima at something 00:14:38.280 --> 00:14:40.320 which is strictly below. 00:14:40.320 --> 00:14:44.140 So in other words, one answer to this question would be, 00:14:44.140 --> 00:14:48.000 and this is the answer that I would probably give, is 1/16. 00:14:48.000 --> 00:14:50.640 But that's not really true. 00:14:50.640 --> 00:14:56.560 Because that's only in the limit. 00:14:56.560 --> 00:15:01.346 As x goes to 0, or as x goes to 1-. 00:15:01.346 --> 00:15:02.720 And if you like, the most is when 00:15:02.720 --> 00:15:04.830 you've only got one square. 00:15:04.830 --> 00:15:07.550 Which breaks the rules of the problem. 00:15:07.550 --> 00:15:11.290 So, essentially, it's a trick question. 00:15:11.290 --> 00:15:13.670 But I would answer it this way. 00:15:13.670 --> 00:15:16.220 Because that's the most interesting part of the answer, 00:15:16.220 --> 00:15:18.780 which is that it's 1/16 and it occurs really 00:15:18.780 --> 00:15:28.920 when one of the squares disappears to nothing. 00:15:28.920 --> 00:15:34.180 So now, let's do another example here. 00:15:34.180 --> 00:15:41.450 And I just want to illustrate the second style, 00:15:41.450 --> 00:15:42.980 or the second type of question. 00:15:42.980 --> 00:15:43.480 Yeah. 00:15:43.480 --> 00:15:51.750 STUDENT: [INAUDIBLE] 00:15:51.750 --> 00:15:56.320 PROFESSOR: The question is, since the question was, 00:15:56.320 --> 00:16:00.090 what was the largest area, why did we find the least area. 00:16:00.090 --> 00:16:04.050 The reason is that when we go about our procedure of looking 00:16:04.050 --> 00:16:10.116 for the least, or the most, we'll automatically find both. 00:16:10.116 --> 00:16:12.490 Because we don't know which one is which until we compare 00:16:12.490 --> 00:16:14.730 values. 00:16:14.730 --> 00:16:17.170 And actually, it's much more to your advantage 00:16:17.170 --> 00:16:20.540 to figure out both the maximum and minimum 00:16:20.540 --> 00:16:22.316 whenever you answer such a question. 00:16:22.316 --> 00:16:24.440 Because otherwise you won't understand the behavior 00:16:24.440 --> 00:16:26.590 of the function very well. 00:16:26.590 --> 00:16:27.850 So, the question. 00:16:27.850 --> 00:16:30.500 We started out with one question, we answered both. 00:16:30.500 --> 00:16:31.750 We answered two questions. 00:16:31.750 --> 00:16:36.480 We answered the question of what the largest and the smallest 00:16:36.480 --> 00:16:37.290 value was. 00:16:37.290 --> 00:16:39.505 STUDENT: Also, I'm wondering if you 00:16:39.505 --> 00:16:42.680 can check both the minimum [INAUDIBLE] approaches 00:16:42.680 --> 00:16:47.505 [INAUDIBLE]. 00:16:47.505 --> 00:16:48.130 PROFESSOR: Yes. 00:16:48.130 --> 00:16:49.880 One can also use-- the question is, can we 00:16:49.880 --> 00:16:51.490 use the second derivative test. 00:16:51.490 --> 00:16:53.480 And the answer is, yes we can. 00:16:53.480 --> 00:16:55.610 In fact, you can actually also stare at this 00:16:55.610 --> 00:16:57.590 and see that it's a sum of squares. 00:16:57.590 --> 00:17:00.850 So it's always curving up. 00:17:00.850 --> 00:17:04.710 It's a parabola with a positive second coefficient. 00:17:04.710 --> 00:17:06.670 So you can differentiate this twice. 00:17:06.670 --> 00:17:09.670 If you do you'll get 1/8 plus another 1/8 00:17:09.670 --> 00:17:11.450 and you'll get 1/16. 00:17:11.450 --> 00:17:14.540 So the second derivative is 1/16. 00:17:14.540 --> 00:17:17.730 Is 1/4. 00:17:17.730 --> 00:17:21.444 And that's an acceptable way of figuring it out. 00:17:21.444 --> 00:17:23.360 I'll mention the second derivative test again, 00:17:23.360 --> 00:17:24.950 in this second example. 00:17:24.950 --> 00:17:32.580 So let me talk about a second example. 00:17:32.580 --> 00:17:35.070 So again, this is going to be another question. 00:17:35.070 --> 00:17:37.950 STUDENT: [INAUDIBLE] 00:17:37.950 --> 00:17:43.660 PROFESSOR: The question is, when I say minimum or maximum 00:17:43.660 --> 00:17:44.880 point which will I mean. 00:17:44.880 --> 00:17:50.570 STUDENT: [INAUDIBLE] 00:17:50.570 --> 00:17:53.100 PROFESSOR: So I just repeated the question. 00:17:53.100 --> 00:17:56.650 So the question is, when I say minimum point, 00:17:56.650 --> 00:17:58.580 what will I mean? 00:17:58.580 --> 00:18:00.100 OK? 00:18:00.100 --> 00:18:05.490 And the answer is that for the purposes of this class 00:18:05.490 --> 00:18:09.090 I will probably avoid saying that. 00:18:09.090 --> 00:18:13.340 But I will say, probably, where is the minimum achieved. 00:18:13.340 --> 00:18:14.680 Just in order to avoid that. 00:18:14.680 --> 00:18:17.980 If I actually said it, I often am referring to the graph, 00:18:17.980 --> 00:18:19.300 and I mean this. 00:18:19.300 --> 00:18:21.280 And in fact, when you get your little review 00:18:21.280 --> 00:18:23.090 for the second exam, I'll say exactly 00:18:23.090 --> 00:18:25.860 that on the review sheet. 00:18:25.860 --> 00:18:28.700 And I'll make this very clear when we were doing this. 00:18:28.700 --> 00:18:30.700 However, I just want to prepare you for the fact 00:18:30.700 --> 00:18:35.480 that in real life, and even me when I'm talking colloquially, 00:18:35.480 --> 00:18:37.750 when I say what's the minimum point of something, 00:18:37.750 --> 00:18:48.730 I might actually be mixing it up with this other notion here. 00:18:48.730 --> 00:18:54.380 So let's do another example. 00:18:54.380 --> 00:18:57.440 So this is an example to get us used 00:18:57.440 --> 00:18:59.880 to the notion of constraints. 00:18:59.880 --> 00:19:08.250 So we have, so consider a box without a top. 00:19:08.250 --> 00:19:16.860 Or, if you like, we're going to find the box without a top. 00:19:16.860 --> 00:19:35.380 With least surface area for a fixed volume. 00:19:35.380 --> 00:19:38.950 Find the box without a top with least 00:19:38.950 --> 00:19:42.360 surface area for a fixed volume. 00:19:42.360 --> 00:19:47.940 The procedure for working this out is the following. 00:19:47.940 --> 00:19:51.480 You make this diagram. 00:19:51.480 --> 00:19:56.220 And you set up the variables. 00:19:56.220 --> 00:20:00.390 In this case, we're going to have four names of variables. 00:20:00.390 --> 00:20:02.890 We have four letters that we have to choose. 00:20:02.890 --> 00:20:05.680 And we'll choose them in a kind of a standard way, alright? 00:20:05.680 --> 00:20:08.260 So first I have to tell you one more thing. 00:20:08.260 --> 00:20:12.880 Which is something that we could calculate separately 00:20:12.880 --> 00:20:15.080 but I'm just going to give it to you in advance. 00:20:15.080 --> 00:20:16.746 Which is that it turns out that the best 00:20:16.746 --> 00:20:21.420 box has a square bottom. 00:20:21.420 --> 00:20:24.710 And that's going to get rid of one of our variables for us. 00:20:24.710 --> 00:20:28.500 So it's got a square bottom, and so let's draw a picture of it. 00:20:28.500 --> 00:20:36.860 So here's our box. 00:20:36.860 --> 00:20:40.680 Well, that goes down like this, almost. 00:20:40.680 --> 00:20:49.620 Maybe I should get it a little farther down. 00:20:49.620 --> 00:20:52.070 So here's our box. 00:20:52.070 --> 00:20:54.900 Let's correct that just a bit. 00:20:54.900 --> 00:20:57.760 So now, what about the dimensions of this box? 00:20:57.760 --> 00:21:02.050 Well, this is going to be x, and this is very foreshortened, 00:21:02.050 --> 00:21:03.180 but it's also x. 00:21:03.180 --> 00:21:06.130 The bottom is x by x, it's the same dimensions. 00:21:06.130 --> 00:21:12.190 And then the vertical dimension is y. 00:21:12.190 --> 00:21:13.750 So far, so good. 00:21:13.750 --> 00:21:16.620 Now, I promised you two more letter names. 00:21:16.620 --> 00:21:21.040 I want to compute the volume. 00:21:21.040 --> 00:21:24.440 The volume is, the base is x^2, and the height is y. 00:21:24.440 --> 00:21:26.780 So there's the volume. 00:21:26.780 --> 00:21:32.870 And then the area, the area is the area of the bottom, 00:21:32.870 --> 00:21:37.940 which is x^2, that's the bottom. 00:21:37.940 --> 00:21:40.510 And then there are the four sides. 00:21:40.510 --> 00:21:44.230 And the four sides are rectangles of dimensions x, y. 00:21:44.230 --> 00:21:49.500 So it's 4xy. 00:21:49.500 --> 00:21:53.630 So these are the sides. 00:21:53.630 --> 00:21:56.680 And remember, there's no top. 00:21:56.680 --> 00:21:58.610 So that's our setup. 00:21:58.610 --> 00:22:02.670 So now, the difference between this problem 00:22:02.670 --> 00:22:05.230 and the last problem is that there 00:22:05.230 --> 00:22:06.690 are two variables floating around, 00:22:06.690 --> 00:22:09.710 namely x and y, which are not determined. 00:22:09.710 --> 00:22:17.670 But there's what's called a constraint here. 00:22:17.670 --> 00:22:22.920 Namely, we've fixed the relationship between x and y. 00:22:22.920 --> 00:22:30.880 And so, that means that we can solve for y in terms of x. 00:22:30.880 --> 00:22:33.310 So y = V/x^2. 00:22:40.650 --> 00:22:43.660 And then, we can plug that into the formula for A. 00:22:43.660 --> 00:22:50.280 So here we have A which is x^2 + 4x v/x^2. 00:23:00.911 --> 00:23:01.410 Question. 00:23:01.410 --> 00:23:14.600 STUDENT: [INAUDIBLE] 00:23:14.600 --> 00:23:16.100 PROFESSOR: The question is, will you 00:23:16.100 --> 00:23:17.350 need to know this intuitively? 00:23:17.350 --> 00:23:17.849 No. 00:23:17.849 --> 00:23:20.830 That's something that I would have to give to you. 00:23:20.830 --> 00:23:24.870 I mean, it's actually true that a lot 00:23:24.870 --> 00:23:28.040 of things, the correct answer is something symmetric. 00:23:28.040 --> 00:23:30.240 In this last problem, the minimum 00:23:30.240 --> 00:23:33.100 turned out to be exactly halfway in between because there 00:23:33.100 --> 00:23:35.900 were sort of equal demands from the two sides. 00:23:35.900 --> 00:23:37.480 And similarly, here, what happens 00:23:37.480 --> 00:23:40.130 is if you elongate one side, you get less 00:23:40.130 --> 00:23:46.050 - it actually is involved with a two variable problem. 00:23:46.050 --> 00:23:49.440 Namely, if you have a rectangle and you have a certain amount 00:23:49.440 --> 00:23:52.610 of length associated with it, what's the optimal thing 00:23:52.610 --> 00:23:53.720 you can do with that. 00:23:53.720 --> 00:23:58.280 But I won't-- in other words, the optimal rectangle, 00:23:58.280 --> 00:24:01.397 the least perimeter rectangle, turns out to be a square. 00:24:01.397 --> 00:24:03.230 That's the little sub-problem that leads you 00:24:03.230 --> 00:24:05.960 to this square bottom. 00:24:05.960 --> 00:24:09.270 But so that would have been a separate max-min problem. 00:24:09.270 --> 00:24:11.900 Which I'm skipping, because I want to do this slightly more 00:24:11.900 --> 00:24:16.240 interesting one. 00:24:16.240 --> 00:24:20.540 So now, here's our formula for A, 00:24:20.540 --> 00:24:27.340 and now I want to follow the same procedure as before. 00:24:27.340 --> 00:24:29.320 Namely, we look for the critical point. 00:24:29.320 --> 00:24:35.530 Or points. 00:24:35.530 --> 00:24:37.260 So let's take a look. 00:24:37.260 --> 00:24:40.020 So again, A is (x^2 + 4v) / x. 00:24:43.430 --> 00:24:49.800 And A' is 2x - 4v/x^2. 00:24:49.800 --> 00:24:59.510 So if we set that equal to 0, we get 2x = 2v/x^2. 00:24:59.510 --> 00:25:01.620 So 2x^3. 00:25:04.690 --> 00:25:07.270 How did that happen to change into 2? 00:25:07.270 --> 00:25:09.590 Interesting, guess that's wrong. 00:25:09.590 --> 00:25:12.950 OK. 00:25:12.950 --> 00:25:20.480 So this is x^3 2v. 00:25:20.480 --> 00:25:23.730 And so x = 2^(1/3) v^(1/3). 00:25:28.010 --> 00:25:36.230 So this is the critical point. 00:25:36.230 --> 00:25:38.040 So we are not done. 00:25:38.040 --> 00:25:38.619 Right? 00:25:38.619 --> 00:25:40.160 We're not done, because we don't even 00:25:40.160 --> 00:25:41.701 know whether this is going to give us 00:25:41.701 --> 00:25:44.410 the worst box or the best box, from this point of view. 00:25:44.410 --> 00:25:48.100 The one that uses the most surface area or the least. 00:25:48.100 --> 00:25:51.990 So let's check the ends, right away. 00:25:51.990 --> 00:25:54.030 To see what's happening. 00:25:54.030 --> 00:25:56.280 So can somebody tell me what the ends, what 00:25:56.280 --> 00:25:58.060 the end values of x are? 00:25:58.060 --> 00:25:59.520 Where does x range from? 00:25:59.520 --> 00:26:05.124 STUDENT: [INAUDIBLE] 00:26:05.124 --> 00:26:07.040 PROFESSOR: What's the smallest x can be, yeah. 00:26:07.040 --> 00:26:16.520 STUDENT: [INAUDIBLE] 00:26:16.520 --> 00:26:19.400 PROFESSOR: OK, the claim was that the largest x could be 00:26:19.400 --> 00:26:24.040 root A, because somehow there's this x^2 here and you can't get 00:26:24.040 --> 00:26:25.685 any further past than that. 00:26:25.685 --> 00:26:28.700 But there's a key feature here of this problem. 00:26:28.700 --> 00:26:32.050 Which is that A is variable. 00:26:32.050 --> 00:26:34.390 The only thing that's fixed in the problem 00:26:34.390 --> 00:26:47.440 is V. So if V is fixed, what do you know about x? 00:26:47.440 --> 00:26:49.810 STUDENT: [INAUDIBLE] 00:26:49.810 --> 00:26:51.160 PROFESSOR: x > 0, yeah. 00:26:51.160 --> 00:26:53.060 The lower end point, that's safe. 00:26:53.060 --> 00:26:55.440 Because that has to do geometrically with the fact 00:26:55.440 --> 00:26:59.430 that we don't have any boxes with negative dimensions. 00:26:59.430 --> 00:27:02.360 That would be refused by the Post Office, definitely. 00:27:02.360 --> 00:27:04.190 Over and above the empty top, which 00:27:04.190 --> 00:27:05.510 they wouldn't accept either. 00:27:05.510 --> 00:27:13.330 STUDENT: [INAUDIBLE] 00:27:13.330 --> 00:27:17.260 PROFESSOR: It's true that x is less than square root of V / y. 00:27:17.260 --> 00:27:19.250 So that's using this relationship. 00:27:19.250 --> 00:27:23.750 But notice that y = V/x^2. 00:27:26.850 --> 00:27:30.060 So 0 to infinity, I just got a guess there 00:27:30.060 --> 00:27:32.080 over here, that's right. 00:27:32.080 --> 00:27:33.350 Here's the upper limit. 00:27:33.350 --> 00:27:35.750 So this is really important to realize. 00:27:35.750 --> 00:27:37.420 This is most problems. 00:27:37.420 --> 00:27:40.340 Most problems, the variable, if it doesn't have a limitation, 00:27:40.340 --> 00:27:42.370 usually just goes out to infinity. 00:27:42.370 --> 00:27:45.790 And infinity is a very important end for the problem. 00:27:45.790 --> 00:27:51.516 It's usually an easy end to the problem, too. 00:27:51.516 --> 00:27:53.890 So there's a possibility that if we push all the way down 00:27:53.890 --> 00:27:56.240 to x = 0, we'll get a better box. 00:27:56.240 --> 00:27:58.230 It would be very strange box. 00:27:58.230 --> 00:28:01.620 A little bit like our vanishing enclosure. 00:28:01.620 --> 00:28:04.920 And maybe an infinitely long box, 00:28:04.920 --> 00:28:06.840 also very inconvenient one. 00:28:06.840 --> 00:28:07.990 Might be the best box. 00:28:07.990 --> 00:28:10.390 We'll have to see. 00:28:10.390 --> 00:28:12.810 So let's just take a look at what happens. 00:28:12.810 --> 00:28:18.740 So we're looking at A, at 0+. 00:28:18.740 --> 00:28:25.760 And that's x^2 + 4V / x with x at 0+. 00:28:25.760 --> 00:28:26.760 So what happens to that? 00:28:26.760 --> 00:28:35.220 Notice right here, this is going to infinity. 00:28:35.220 --> 00:28:38.500 So this is infinite. 00:28:38.500 --> 00:28:42.250 So that turns out to be a bad box. 00:28:42.250 --> 00:28:45.730 Let's take a look at the other end. 00:28:45.730 --> 00:28:51.930 So this is x^2 + 4V / x, x going to infinity. 00:28:51.930 --> 00:28:59.930 And again, this term here means that this thing is infinite. 00:28:59.930 --> 00:29:01.870 So the shape of this thing, I'll draw 00:29:01.870 --> 00:29:05.000 this tiny little schematic diagram over here. 00:29:05.000 --> 00:29:11.110 The shape of this thing is like this, right? 00:29:11.110 --> 00:29:13.950 And so, when we find that one turnaround point, 00:29:13.950 --> 00:29:19.000 which happened to be at this strange point 2/3-- sorry, 00:29:19.000 --> 00:29:24.440 2^(1/3) V^(1/3), that is going to be the minimum. 00:29:24.440 --> 00:29:30.090 So we've just discovered that it's the minimum. 00:29:30.090 --> 00:29:31.690 Which is just what we were hoping for. 00:29:31.690 --> 00:29:38.200 This is going to be the optimal box. 00:29:38.200 --> 00:29:44.660 Now, since you asked earlier and since it's worth 00:29:44.660 --> 00:29:49.360 checking this as well, let's also 00:29:49.360 --> 00:29:51.740 check an alternative justification. 00:29:51.740 --> 00:30:02.540 So an alternative to checking ends 00:30:02.540 --> 00:30:11.540 is the second derivative test. 00:30:11.540 --> 00:30:14.200 I do not recommend the second derivative test. 00:30:14.200 --> 00:30:16.230 I try my best, when I give you problems, 00:30:16.230 --> 00:30:19.290 to make it really hard to apply the second derivative test. 00:30:19.290 --> 00:30:21.320 But in this example, the function 00:30:21.320 --> 00:30:24.570 is simple enough so that it's perfectly OK. 00:30:24.570 --> 00:30:28.110 If you take the derivative here, remember, 00:30:28.110 --> 00:30:34.760 this was whatever it was, 2x - 4V / x^2. 00:30:34.760 --> 00:30:43.610 If I take the second derivative, it's 2 + 8V / x^3. 00:30:43.610 --> 00:30:45.310 And that's positive. 00:30:45.310 --> 00:30:49.580 So this thing is concave up. 00:30:49.580 --> 00:30:52.180 And that's consistent with its being-- 00:30:52.180 --> 00:30:59.840 the critical point is a min. 00:30:59.840 --> 00:31:00.840 Is a minimum point. 00:31:00.840 --> 00:31:05.180 See how I almost said, is a min, as opposed to minimum point. 00:31:05.180 --> 00:31:05.780 So watch out. 00:31:05.780 --> 00:31:06.280 Yes. 00:31:06.280 --> 00:31:12.235 STUDENT: [INAUDIBLE] 00:31:12.235 --> 00:31:13.860 PROFESSOR: You're one step ahead of me. 00:31:13.860 --> 00:31:15.985 The question is, is this the answer to the question 00:31:15.985 --> 00:31:20.090 or would we have to give y and A and so on and so forth. 00:31:20.090 --> 00:31:23.120 So, again, this is something that I 00:31:23.120 --> 00:31:26.220 want to emphasize and take my time with right now. 00:31:26.220 --> 00:31:29.800 Because it depends, what kind of real-life problem 00:31:29.800 --> 00:31:33.360 you're answering, what kind of answer is appropriate. 00:31:33.360 --> 00:31:36.010 So, so far we've found the critical point. 00:31:36.010 --> 00:31:38.080 We haven't found the critical value. 00:31:38.080 --> 00:31:42.190 We haven't found the dimensions of the box. 00:31:42.190 --> 00:31:46.400 So we're going to spend a little bit more time on this, exactly 00:31:46.400 --> 00:31:48.920 in order to address these questions. 00:31:48.920 --> 00:31:50.170 So, first of all. 00:31:50.170 --> 00:31:51.470 The value of y. 00:31:51.470 --> 00:31:55.120 So, so far we have x = 2^(1/3) V^(1/3). 00:31:55.120 --> 00:31:57.170 And certainly if you're going to build the box, 00:31:57.170 --> 00:32:00.410 you also want to know what the y-value is. 00:32:00.410 --> 00:32:04.970 The y-value is going to be, let's see. 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, 00:32:12.110 --> 00:32:19.770 which comes out to be 2^(2/3) V^(1/3). 00:32:19.770 --> 00:32:22.580 So there's the y-value. 00:32:22.580 --> 00:32:29.505 On top of that, we could figure out the value of A. 00:32:29.505 --> 00:32:31.880 So that's also a perfectly reasonable part of the answer. 00:32:31.880 --> 00:32:33.780 Depending on what one is interested in, 00:32:33.780 --> 00:32:35.675 you might care how much money it's going 00:32:35.675 --> 00:32:38.200 to cost you to build this box. 00:32:38.200 --> 00:32:39.330 This optimal box. 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. 00:32:43.370 --> 00:32:47.740 It's x x^2 + 4V/x. 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) 00:32:57.810 --> 00:32:58.310 V^(1/3) ). 00:33:02.330 --> 00:33:07.300 And if you work that all out, what you get turns out to be 3 00:33:07.300 --> 00:33:08.170 * 2^(1/3) V^(2/3). 00:33:13.730 --> 00:33:17.540 So if you like, one way of answering this question 00:33:17.540 --> 00:33:23.160 is these three things. 00:33:23.160 --> 00:33:26.490 That would be the minimum point corresponding to the graph. 00:33:26.490 --> 00:33:28.950 That would be the answer to this question. 00:33:28.950 --> 00:33:31.710 But the reason why I'm carrying it out in such detail 00:33:31.710 --> 00:33:35.370 is I want to show you that there are much more meaningful ways 00:33:35.370 --> 00:33:37.150 of answering this question. 00:33:37.150 --> 00:34:02.810 So let me explain that. 00:34:02.810 --> 00:34:07.290 So let me go through some more meaningful answers here. 00:34:07.290 --> 00:34:14.770 The first more meaningful answer is the following idea. 00:34:14.770 --> 00:34:29.780 Simply, what are known as dimensionless variables. 00:34:29.780 --> 00:34:33.050 So the first thing that you notice is the scaling law. 00:34:33.050 --> 00:34:36.580 That A / V^(2/3) is the thing that's a dimensionless 00:34:36.580 --> 00:34:37.260 quantity. 00:34:37.260 --> 00:34:39.360 That happens to be 3 * 2^(1/3). 00:34:42.150 --> 00:34:43.970 So that's one thing. 00:34:43.970 --> 00:34:45.470 If you want to expand the volume, 00:34:45.470 --> 00:34:49.220 you'll have to expand the area by the 2/3 power of the volume. 00:34:49.220 --> 00:34:55.430 And if you think of the area as being in, say, square inches, 00:34:55.430 --> 00:35:00.000 and the volume of the box as being in cubic inches, 00:35:00.000 --> 00:35:02.360 then you can see that this is a dimensionless quantity 00:35:02.360 --> 00:35:04.070 and you have a dimensionless number here, 00:35:04.070 --> 00:35:06.730 which is a characteristic independent of what 00:35:06.730 --> 00:35:09.310 A and V were. 00:35:09.310 --> 00:35:15.710 The other dimensionless quantity is the ratio of x to y. 00:35:15.710 --> 00:35:19.640 Or x to y. 00:35:19.640 --> 00:35:23.680 So, again, that's inches divided by inches. 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), 00:35:32.950 --> 00:35:36.150 which happens to be 2. 00:35:36.150 --> 00:35:41.390 So this is actually the best answer to the question. 00:35:41.390 --> 00:35:46.120 And it shows you that the box is a 2:1 box. 00:35:46.120 --> 00:35:50.200 If this is 2 and this is 1, that's the good box. 00:35:50.200 --> 00:35:56.630 And that is just the shape, if you like, 00:35:56.630 --> 00:36:02.670 and it's the optimal shape. 00:36:02.670 --> 00:36:04.490 And certainly that, aesthetically, that's 00:36:04.490 --> 00:36:12.187 the cleanest answer to the question. 00:36:12.187 --> 00:36:13.520 There was a question right here. 00:36:13.520 --> 00:36:14.020 Yes. 00:36:14.020 --> 00:36:20.537 STUDENT: [INAUDIBLE] 00:36:20.537 --> 00:36:22.620 PROFESSOR: Could you repeat that, I couldn't hear. 00:36:22.620 --> 00:36:24.036 STUDENT: I'm wondering if you'd be 00:36:24.036 --> 00:36:29.850 able to get that answer if you [INAUDIBLE] square. 00:36:29.850 --> 00:36:31.350 PROFESSOR: The question is, could we 00:36:31.350 --> 00:36:33.058 have gotten the answer if we weren't told 00:36:33.058 --> 00:36:34.380 that the bottom was square. 00:36:34.380 --> 00:36:39.014 The answer is, yes in 18.02 with multivariable. 00:36:39.014 --> 00:36:41.222 You would have to have three letters here, an x, a y, 00:36:41.222 --> 00:36:43.470 and a z, if you like. 00:36:43.470 --> 00:36:48.170 And then you'd have to work with all three of them. 00:36:48.170 --> 00:36:51.890 So I separated out into one, there's 00:36:51.890 --> 00:36:55.550 a separate one variable problem that you can do for the base. 00:36:55.550 --> 00:36:57.510 And then this is a second one variable problem 00:36:57.510 --> 00:36:58.510 for this other thing. 00:36:58.510 --> 00:37:00.910 And it's just two consecutive one variable problems 00:37:00.910 --> 00:37:03.170 that solve the multivariable problem. 00:37:03.170 --> 00:37:05.650 Or, as I say in multivariable calculus, 00:37:05.650 --> 00:37:08.740 you can just do it all in one step. 00:37:08.740 --> 00:37:09.240 Yeah? 00:37:09.240 --> 00:37:11.820 STUDENT: [INAUDIBLE] 00:37:11.820 --> 00:37:14.550 PROFESSOR: Why did I divide x by y, 00:37:14.550 --> 00:37:17.830 rather than y by x, or in any? 00:37:17.830 --> 00:37:22.320 So, again, what I was aiming for was dimensionless quantities. 00:37:22.320 --> 00:37:26.640 So x and y are measured in the same units. 00:37:26.640 --> 00:37:29.065 And also the proportions of the box 00:37:29.065 --> 00:37:34.310 - so that's another word for this is proportions - 00:37:34.310 --> 00:37:38.430 are something that's universal, independent of the volume 00:37:38.430 --> 00:37:42.020 V. It's something you can say about any box, at any scale. 00:37:42.020 --> 00:37:45.040 Whether it be, you know, something 00:37:45.040 --> 00:37:48.781 by Cristo in the Common. 00:37:48.781 --> 00:37:50.530 Maybe we'll get in here to do some fancy-- 00:37:50.530 --> 00:37:55.540 STUDENT: [INAUDIBLE] 00:37:55.540 --> 00:37:58.630 PROFESSOR: The proportions is - with geometric problems 00:37:58.630 --> 00:38:01.930 typically, when there's a scaling to the problem, where 00:38:01.930 --> 00:38:05.020 the answer is the same at small scales and at large scales 00:38:05.020 --> 00:38:07.000 - this is capturing that. 00:38:07.000 --> 00:38:10.070 So that's why, the ratios are what's capturing that. 00:38:10.070 --> 00:38:12.410 And that's why it's aesthetically the nicest thing 00:38:12.410 --> 00:38:13.860 to ask. 00:38:13.860 --> 00:38:17.905 STUDENT: So, what exactly does the ratio 00:38:17.905 --> 00:38:21.639 of the area to the volume ratio [INAUDIBLE] tell us? 00:38:21.639 --> 00:38:23.180 PROFESSOR: Unfortunately, this number 00:38:23.180 --> 00:38:25.490 is a really obscure number. 00:38:25.490 --> 00:38:28.006 So the question is what does this tell us. 00:38:28.006 --> 00:38:29.630 The only thing that I want to emphasize 00:38:29.630 --> 00:38:31.800 is what's on the left-hand side here. 00:38:31.800 --> 00:38:34.870 Which is, it's the area to the 2/3 power over the volume, 00:38:34.870 --> 00:38:38.550 so it's a dimensionless quantity that happens to be this. 00:38:38.550 --> 00:38:43.700 If you do this, for example, in general with planar diagrams, 00:38:43.700 --> 00:38:47.200 circumference to area is a bad ratio to take. 00:38:47.200 --> 00:38:48.810 What you want to take is the square 00:38:48.810 --> 00:38:50.932 of circumference to area. 00:38:50.932 --> 00:38:52.390 Because the square of circumference 00:38:52.390 --> 00:38:54.690 has the same dimensions, that is, say, 00:38:54.690 --> 00:38:56.880 inches squared to area. 00:38:56.880 --> 00:38:58.840 Which is in square inches. 00:38:58.840 --> 00:39:01.360 So, again, it's these dimensionless quantities 00:39:01.360 --> 00:39:03.440 that you want to cook up. 00:39:03.440 --> 00:39:06.810 And those are the ones that will have universal properties. 00:39:06.810 --> 00:39:09.990 The most famous of these is the circle 00:39:09.990 --> 00:39:13.880 that encloses the most area for its circumference. 00:39:13.880 --> 00:39:17.876 And, again, that's only true if you take the square 00:39:17.876 --> 00:39:18.880 of the circumference. 00:39:18.880 --> 00:39:25.620 You do the units correctly. 00:39:25.620 --> 00:39:26.290 Anyway. 00:39:26.290 --> 00:39:29.760 So we're here, we've got a shape. 00:39:29.760 --> 00:39:31.560 We've got an answer to this question. 00:39:31.560 --> 00:39:36.940 And I now want to do this problem. 00:39:36.940 --> 00:39:38.267 Well, let's put it this way. 00:39:38.267 --> 00:39:40.350 I wanted to do this problem by a different method. 00:39:40.350 --> 00:39:43.430 I think I'll take the time to do it. 00:39:43.430 --> 00:39:46.790 So I want to do this problem by a slightly different method 00:39:46.790 --> 00:39:48.880 here. 00:39:48.880 --> 00:39:59.160 So, here's Example 2 by implicit differentiation. 00:39:59.160 --> 00:40:01.950 So the same example, but now I'm going to do it 00:40:01.950 --> 00:40:03.720 by implicit differentiation. 00:40:03.720 --> 00:40:07.050 Well, I'll tell you the advantages and disadvantages 00:40:07.050 --> 00:40:08.880 to this method here. 00:40:08.880 --> 00:40:20.130 So the situation is, you have to start the same way. 00:40:20.130 --> 00:40:24.230 So here is the starting place of the problem. 00:40:24.230 --> 00:40:34.980 And the goal was the minimum of A with V constant. 00:40:34.980 --> 00:40:38.540 So this was the situation that we were in. 00:40:38.540 --> 00:40:45.130 And now, what I want to do is just differentiate. 00:40:45.130 --> 00:40:46.990 The function y is implicitly a function 00:40:46.990 --> 00:40:54.550 of x, so I can differentiate the first expression. 00:40:54.550 --> 00:41:00.380 And that yields 0 = 2xy + x^2 y'. 00:41:03.330 --> 00:41:11.590 So this is giving me my implicit formula for y', So y' = -2xy / 00:41:11.590 --> 00:41:13.080 x^2. 00:41:13.080 --> 00:41:14.950 Or in other words, -2y/x. 00:41:19.660 --> 00:41:24.990 And then I also have the dA/dx. 00:41:24.990 --> 00:41:28.750 Now, you may notice I'm not using primes quite as much. 00:41:28.750 --> 00:41:31.510 Because all of the variables are varying, 00:41:31.510 --> 00:41:34.300 and so here I'm emphasizing that it's a differentiation 00:41:34.300 --> 00:41:36.790 with respect to the variable x. 00:41:36.790 --> 00:41:48.130 And this becomes 2x + 4y + 4xy'. 00:41:48.130 --> 00:41:53.140 So again, this is using the product rule. 00:41:53.140 --> 00:41:57.450 And now I can plug in for what y' is, which is right above it. 00:41:57.450 --> 00:42:04.890 So this is 2x + 4y + 4x (-2y/x). 00:42:09.330 --> 00:42:14.150 And that's equal to 0. 00:42:14.150 --> 00:42:24.770 And so let's gather that together. 00:42:24.770 --> 00:42:25.960 So what do we have? 00:42:25.960 --> 00:42:34.290 We have 2x + 4y, and then, altogether, this is 8 - 8y, 00:42:34.290 --> 00:42:36.550 equals 0. 00:42:36.550 --> 00:42:41.600 So that's the same thing as 2x = 4y. 00:42:41.600 --> 00:42:44.100 The - 4y goes to the other side. 00:42:44.100 --> 00:42:54.420 And so, x/y = 2. 00:42:54.420 --> 00:42:59.260 So this, I claim, so you have to decide for yourself. 00:42:59.260 --> 00:43:03.630 But I claim that this is faster. 00:43:03.630 --> 00:43:07.350 It's faster, and also it gets to the heart 00:43:07.350 --> 00:43:10.870 of the matter, which is this scale-invariant proportions. 00:43:10.870 --> 00:43:13.360 Which is basically also nicer. 00:43:13.360 --> 00:43:16.720 So it gets to the nicer answer, also. 00:43:16.720 --> 00:43:19.970 So those are the advantages that this has. 00:43:19.970 --> 00:43:23.190 So it's faster, and it gets to this, 00:43:23.190 --> 00:43:26.260 I'm going to call it nicer. 00:43:26.260 --> 00:43:38.640 And the disadvantage is it did not 00:43:38.640 --> 00:43:55.620 check whether this critical point is a max, min, 00:43:55.620 --> 00:43:59.540 or neither. 00:43:59.540 --> 00:44:02.280 So we didn't quite finish the problem. 00:44:02.280 --> 00:44:10.820 But we got to the answer very fast. 00:44:10.820 --> 00:44:11.640 Yeah, question. 00:44:11.640 --> 00:44:13.480 STUDENT: [INAUDIBLE] 00:44:13.480 --> 00:44:17.437 PROFESSOR: How would you check it? 00:44:17.437 --> 00:44:18.270 STUDENT: [INAUDIBLE] 00:44:18.270 --> 00:44:20.510 PROFESSOR: Well, so it gives you a candidate. 00:44:20.510 --> 00:44:23.390 The answer is-- so the question is, how would you check it? 00:44:23.390 --> 00:44:26.720 The answer is that for this particular problem, 00:44:26.720 --> 00:44:31.300 the only way to do it is to do something like this. 00:44:31.300 --> 00:44:34.480 So in other words, it doesn't save you that much time. 00:44:34.480 --> 00:44:37.730 But with many, many, examples, you actually 00:44:37.730 --> 00:44:42.010 can tell immediately that if the two ends, the thing is, say, 0, 00:44:42.010 --> 00:44:43.650 and inside it's positive. 00:44:43.650 --> 00:44:44.370 Things like that. 00:44:44.370 --> 00:44:53.640 So in many, many, cases this is just as good. 00:44:53.640 --> 00:44:58.510 So now I'm going to change subjects here. 00:44:58.510 --> 00:45:02.220 But the subject that I'm going to talk about next 00:45:02.220 --> 00:45:07.550 is almost-- is very, very closely linked. 00:45:07.550 --> 00:45:10.850 Namely, I talked about implicit differentiation. 00:45:10.850 --> 00:45:12.770 Now, we're going to just talk about dealing 00:45:12.770 --> 00:45:13.920 with lots of variables. 00:45:13.920 --> 00:45:15.360 And rates of change. 00:45:15.360 --> 00:45:16.960 So, essentially, we're going to talk 00:45:16.960 --> 00:45:19.680 about the same type of thing. 00:45:19.680 --> 00:45:23.470 So, I'm going to tell you about a subject which 00:45:23.470 --> 00:45:25.220 is called related rates. 00:45:25.220 --> 00:45:26.980 Which is really just another excuse 00:45:26.980 --> 00:45:32.510 for getting used to setting up variables and equations. 00:45:32.510 --> 00:45:34.320 So, here we go. 00:45:34.320 --> 00:45:36.000 Related rates. 00:45:36.000 --> 00:45:41.970 And I'm going to illustrate this with one example today, one 00:45:41.970 --> 00:45:44.350 tomorrow. 00:45:44.350 --> 00:45:47.600 So here's my example for today. 00:45:47.600 --> 00:45:50.810 So, again, this is going to be a police problem. 00:45:50.810 --> 00:45:53.630 But this is going to be a word problem and-- sorry, 00:45:53.630 --> 00:45:56.910 I'm don't want to scare you, no police. 00:45:56.910 --> 00:45:59.940 Well, there are police in the story but they're not present. 00:45:59.940 --> 00:46:05.070 So, but I'm going to draw it immediately with the diagram 00:46:05.070 --> 00:46:08.410 because I'm going to save us the trouble. 00:46:08.410 --> 00:46:11.580 Although, you know, the point here is to get from the words 00:46:11.580 --> 00:46:15.940 to the diagram. 00:46:15.940 --> 00:46:21.990 So you have the police, and they're 30 feet from the road. 00:46:21.990 --> 00:46:25.750 And here's the road. 00:46:25.750 --> 00:46:36.610 And you're coming along, here, in your, let's see, 00:46:36.610 --> 00:46:39.420 in your car going in this direction here. 00:46:39.420 --> 00:46:43.760 And the police have radar. 00:46:43.760 --> 00:46:46.490 Which is bouncing off of your car. 00:46:46.490 --> 00:46:53.880 And what they read off is that you're 50 feet away. 00:46:53.880 --> 00:46:57.100 They also know that you're approaching 00:46:57.100 --> 00:47:12.370 along the line of the radar at a rate of 80 feet per second. 00:47:12.370 --> 00:47:19.260 Now, the question is, are you speeding. 00:47:19.260 --> 00:47:20.730 That's the question. 00:47:20.730 --> 00:47:29.030 So when you're speeding, by the way, 95 feet per second 00:47:29.030 --> 00:47:32.090 is about 65 miles per hour. 00:47:32.090 --> 00:47:35.890 So that's the threshold here. 00:47:35.890 --> 00:47:40.080 So what I want to do now is show you 00:47:40.080 --> 00:47:43.860 how you set up a problem like this. 00:47:43.860 --> 00:47:46.150 This distance is 50. 00:47:46.150 --> 00:47:50.460 This is 30, and because it's the distance to a straight line 00:47:50.460 --> 00:47:52.030 you know that this is a right angle. 00:47:52.030 --> 00:47:54.580 So we know that this is a right triangle. 00:47:54.580 --> 00:47:56.810 And this is set out to be a right triangle, which 00:47:56.810 --> 00:48:00.300 is an easy one, a 3, 4, 5 right triangle 00:48:00.300 --> 00:48:05.640 just so that we can do the computations easily. 00:48:05.640 --> 00:48:09.180 So now, the question is, how do we put the letters in 00:48:09.180 --> 00:48:10.970 to make this problem work, to figure out 00:48:10.970 --> 00:48:12.310 what the rate of change is. 00:48:12.310 --> 00:48:15.930 So now, let me explain that right now. 00:48:15.930 --> 00:48:18.590 And we will actually do the computation next time. 00:48:18.590 --> 00:48:22.940 So the first thing is, you have to understand what's changing 00:48:22.940 --> 00:48:24.850 and what's not. 00:48:24.850 --> 00:48:30.880 And we're going to use t for time, in seconds. 00:48:30.880 --> 00:48:35.210 And now, an important distance here 00:48:35.210 --> 00:48:37.820 is the distance to this foot of this perpendicular. 00:48:37.820 --> 00:48:41.310 So I'm going to name that x. 00:48:41.310 --> 00:48:42.710 I'm going to give that letter x. 00:48:42.710 --> 00:48:44.480 Now, x is varying. 00:48:44.480 --> 00:48:47.359 The reason why I need a letter for it as opposed to this 40 00:48:47.359 --> 00:48:49.150 is that it's going to have a rate of change 00:48:49.150 --> 00:48:50.220 with respect to t. 00:48:50.220 --> 00:48:54.520 And, in fact, it's related to-- the question 00:48:54.520 --> 00:49:00.432 is whether dx/dt is faster or slower than 95. 00:49:00.432 --> 00:49:01.890 So that's the thing that's varying. 00:49:01.890 --> 00:49:04.010 Now, there's something else that's varying. 00:49:04.010 --> 00:49:06.000 This distance here is also varying. 00:49:06.000 --> 00:49:08.760 So we need a letter for that. 00:49:08.760 --> 00:49:11.620 We do not need a letter for this. 00:49:11.620 --> 00:49:12.920 Because it's never changing. 00:49:12.920 --> 00:49:15.010 We're assuming the police are parked. 00:49:15.010 --> 00:49:17.382 They're not ready to roar out and catch you just yet, 00:49:17.382 --> 00:49:18.840 and they're certainly not in motion 00:49:18.840 --> 00:49:20.880 when they've got the radar guns aimed at you. 00:49:20.880 --> 00:49:24.110 So you need to know something about the sociology 00:49:24.110 --> 00:49:28.160 and style of police. 00:49:28.160 --> 00:49:30.950 So you need to know things about the real world. 00:49:30.950 --> 00:49:36.145 Now, the last bit is, what about this 80 here. 00:49:36.145 --> 00:49:37.770 So this is how fast you're approaching. 00:49:37.770 --> 00:49:40.290 Now, that's measured along the radar gun. 00:49:40.290 --> 00:49:44.950 I claim that that's d/dt of this quantity here. 00:49:44.950 --> 00:49:46.910 So this is D is also changing. 00:49:46.910 --> 00:49:49.200 That's why we needed a letter for it, too. 00:49:49.200 --> 00:49:51.730 So, next time, we'll just put that all together 00:49:51.730 --> 00:49:55.010 and compute dx / dt.