0:00:00.000,0:00:01.510 PROFESSOR: --here. 0:00:01.510,0:00:06.780 I have excuse two[br]people for being sick. 0:00:06.780,0:00:09.980 But I haven't[br]excused anybody else. 0:00:09.980,0:00:12.690 You are not the complete group. 0:00:12.690,0:00:15.360 I would like to take[br]attendance as soon as possible. 0:00:15.360,0:00:17.328 Would you mind starting[br]an attendance sheet? 0:00:17.328,0:00:17.820 STUDENT: We already got one. 0:00:17.820,0:00:18.312 STUDENT: We already got one. 0:00:18.312,0:00:18.990 STUDENT: Yeah, we[br]already got one. 0:00:18.990,0:00:19.980 PROFESSOR: Oh, you[br]already have one. 0:00:19.980,0:00:20.480 OK. 0:00:20.480,0:00:25.447 0:00:25.447,0:00:28.429 I understand having[br]to struggle with snow, 0:00:28.429,0:00:33.896 but you are expected[br]to come here. 0:00:33.896,0:00:39.650 And I don't want to punish[br]the people who don't make it. 0:00:39.650,0:00:42.490 I want to reward the people[br]who make it every time. 0:00:42.490,0:00:47.866 That's the principle behind[br]perfect attendance for this. 0:00:47.866,0:00:50.850 All right. 0:00:50.850,0:00:54.390 Today we are going to[br]cover something new. 0:00:54.390,0:00:58.000 It is new and it's not new. 0:00:58.000,0:01:06.902 It's an extension of[br]the ideas in 11.7, 0:01:06.902,0:01:16.730 which were finding extrema[br]of functions of the type 0:01:16.730,0:01:21.436 z equals f of xy, and[br]classifying those. 0:01:21.436,0:01:29.220 0:01:29.220,0:01:35.110 In 11.8, we provide a[br]very specific method. 0:01:35.110,0:01:37.580 That's Lagrange multipliers. 0:01:37.580,0:01:47.080 0:01:47.080,0:01:52.650 Of finding extrema,[br]you struggled. 0:01:52.650,0:01:56.253 Well, you didn't[br]struggle, but it 0:01:56.253,0:02:04.015 wasn't easy to find those[br]absolute extrema at every time. 0:02:04.015,0:02:08.380 The Lagrange multipliers[br]are going to help you. 0:02:08.380,0:02:12.220 So practically, what[br]should we assume 0:02:12.220,0:02:17.590 to know that the function of[br]two variables that we deal with 0:02:17.590,0:02:20.090 is c1. 0:02:20.090,0:02:22.380 Sometimes I assume it's smooth. 0:02:22.380,0:02:24.590 What do we need? 0:02:24.590,0:02:28.352 We need the derivatives. 0:02:28.352,0:02:30.336 Derivatives. 0:02:30.336,0:02:36.610 Derivative exist[br]and are continuous. 0:02:36.610,0:02:39.765 0:02:39.765,0:02:41.232 I assume differentiability. 0:02:41.232,0:02:48.570 0:02:48.570,0:02:49.550 OK? 0:02:49.550,0:02:52.200 And what else do I assume? 0:02:52.200,0:02:56.990 I assume that you have a[br]constraint that is also smooth. 0:02:56.990,0:03:00.200 0:03:00.200,0:03:04.093 Let's say g of xy equals c. 0:03:04.093,0:03:07.235 0:03:07.235,0:03:08.330 Do you remember? 0:03:08.330,0:03:14.810 We talked about constraints[br]last time on Tuesday as well. 0:03:14.810,0:03:17.610 So practically,[br]x and y are bound 0:03:17.610,0:03:23.320 to be together by some sort of[br]agreement, contract, marriage. 0:03:23.320,0:03:24.710 They depend on one another. 0:03:24.710,0:03:29.320 They cannot leave[br]this constraint. 0:03:29.320,0:03:33.090 And last time, I[br]really don't remember 0:03:33.090,0:03:37.505 what problem I took last time. 0:03:37.505,0:03:43.010 But we had something like, given[br]the function f of xy-- that 0:03:43.010,0:03:48.310 was nice and smooth--[br]find the absolute maximum 0:03:48.310,0:03:52.200 and the absolute[br]minimum of that function 0:03:52.200,0:03:57.350 inside the-- or on[br]the closed disk. 0:03:57.350,0:03:58.212 Remember that? 0:03:58.212,0:04:00.840 We had the closed[br]disk, x squared 0:04:00.840,0:04:03.120 plus y squared less[br]than or equal to 1. 0:04:03.120,0:04:05.570 And we said, let's[br]find-- somebody gives you 0:04:05.570,0:04:07.220 this very nice function. 0:04:07.220,0:04:10.450 We found the critical[br]point inside. 0:04:10.450,0:04:11.670 And we said, that's it. 0:04:11.670,0:04:13.860 Relative max or relative min. 0:04:13.860,0:04:17.300 Maybe we have more[br]depending on the function. 0:04:17.300,0:04:21.980 What was crucial for us to[br]do-- to study the extrema that 0:04:21.980,0:04:24.920 could come from the boundary. 0:04:24.920,0:04:27.255 And in order for them to[br]come from the boundary, 0:04:27.255,0:04:29.150 we played this little game. 0:04:29.150,0:04:30.420 We took the boundary. 0:04:30.420,0:04:31.715 We said, that's the circle. 0:04:31.715,0:04:34.440 X squared plus y[br]squared equals 1. 0:04:34.440,0:04:37.670 We pulled out the[br]y in terms of x 0:04:37.670,0:04:41.266 and brought it back in[br]the original expression, 0:04:41.266,0:04:43.730 z equals f of xy. 0:04:43.730,0:04:49.800 Since y would depend on x as y[br]squared is 1 minus x squared, 0:04:49.800,0:04:53.380 we plugged in and we got[br]a function of x only. 0:04:53.380,0:04:57.830 For that function of x only[br]on the boundary, we said, 0:04:57.830,0:05:01.235 we look for those critical[br]points for the function. 0:05:01.235,0:05:04.109 It was a little bit of[br]time-consuming stuff. 0:05:04.109,0:05:05.067 Critical points. 0:05:05.067,0:05:07.490 That would give you[br]relative max or min 0:05:07.490,0:05:09.570 for that function[br]on the boundary. 0:05:09.570,0:05:14.915 Plus, we said, but that function[br]has n points in the domain, 0:05:14.915,0:05:17.570 because the domain[br]would be for x 0:05:17.570,0:05:22.720 between minus 1 and 1--[br]inclusively minus 1 and 1. 0:05:22.720,0:05:27.900 So those minus 1 and 1's[br]as endpoints can also 0:05:27.900,0:05:31.590 generate absolute max and min. 0:05:31.590,0:05:34.630 So we made a table of[br]all the possible values, 0:05:34.630,0:05:36.855 including all the critical[br]points and the values 0:05:36.855,0:05:37.960 at the endpoints. 0:05:37.960,0:05:41.580 We said, whoever's the[br]tallest guy over here's 0:05:41.580,0:05:43.260 gonna be the maximum. 0:05:43.260,0:05:46.510 Whoever's the smallest[br]one will be the minimum. 0:05:46.510,0:05:50.410 And that was what the[br]philosophy was before. 0:05:50.410,0:05:53.820 Now we have to find a[br]different method, which 0:05:53.820,0:06:00.270 is providing the same solution,[br]but it's more systematic 0:06:00.270,0:06:06.300 in approach and is[br]based on a result that 0:06:06.300,0:06:11.670 was due to Lagrange, one[br]of the-- well, the fathers. 0:06:11.670,0:06:14.340 The fathers were[br]Euler and Leibniz. 0:06:14.340,0:06:16.946 Lagrange had lots[br]of contributions 0:06:16.946,0:06:20.960 to physics, mechanics[br]especially, and calculus. 0:06:20.960,0:06:24.730 So he's also a father. 0:06:24.730,0:06:27.840 As a father, he came up[br]with this beautiful theorem, 0:06:27.840,0:06:34.410 that says, if you have[br]these conditions satisfied 0:06:34.410,0:06:40.420 and if has-- if f has[br]an extrema already-- 0:06:40.420,0:06:42.180 we know that has an extrema. 0:06:42.180,0:06:45.201 0:06:45.201,0:06:54.010 At some point, P0 of x0 y0 along[br]the curve-- the boundary curve. 0:06:54.010,0:06:59.330 Let's call this boundary curve[br]as script C. Do you understand? 0:06:59.330,0:07:00.525 This is not an l. 0:07:00.525,0:07:02.340 I don't know how to denote. 0:07:02.340,0:07:08.990 Script C. Script C. How[br]do you draw a script C? 0:07:08.990,0:07:11.110 Let's draw it like that. 0:07:11.110,0:07:12.121 I'm not an l. 0:07:12.121,0:07:12.620 OK? 0:07:12.620,0:07:14.750 C from curve. 0:07:14.750,0:07:28.560 Then there exists-- I[br]taught you the sign. 0:07:28.560,0:07:33.370 There exists a lambda--[br]real number-- such 0:07:33.370,0:07:36.420 that the gradients are parallel. 0:07:36.420,0:07:37.430 What? 0:07:37.430,0:07:43.350 The gradient of f[br]of x0 y0 would be 0:07:43.350,0:07:47.510 parallel of a proportionality[br]factor, lambda, 0:07:47.510,0:07:52.720 to the gradient of g,[br]the constraint function. 0:07:52.720,0:07:56.670 So we have two[br]Musketeers here that 0:07:56.670,0:08:00.300 matter-- the gradient of[br]the original function, 0:08:00.300,0:08:03.735 the one you want to[br]optimize, and the gradient 0:08:03.735,0:08:07.535 of the constraint function[br]as a function of x 0:08:07.535,0:08:08.990 and y at the point. 0:08:08.990,0:08:11.220 And we claim that[br]at that point, we 0:08:11.220,0:08:13.740 have an extremum of some sort. 0:08:13.740,0:08:16.150 Then something magical happens. 0:08:16.150,0:08:20.120 There is a lambda-- there[br]is a proportionality 0:08:20.120,0:08:23.290 between those two gradients. 0:08:23.290,0:08:26.220 So you say something magical[br]happens-- that the gradients 0:08:26.220,0:08:28.250 will be in the same direction. 0:08:28.250,0:08:32.640 And the proportionality factor[br]is this beautiful lambda. 0:08:32.640,0:08:35.700 0:08:35.700,0:08:42.460 If Mr. g-- this is a tricky[br]thing you have to make sure 0:08:42.460,0:08:43.470 happens. 0:08:43.470,0:08:51.930 If Mr. g, let's say, is-- at[br]its 0y0 is different from 0. 0:08:51.930,0:08:57.203 Because if it is equal to 0,[br]well, then it's gonna be crazy. 0:08:57.203,0:09:01.487 We will have 0 equals 0 for[br]any lambda multiplication here. 0:09:01.487,0:09:05.085 So that would complicate[br]things, and you would 0:09:05.085,0:09:07.070 get something that's lost. 0:09:07.070,0:09:12.780 So how do I view the procedure? 0:09:12.780,0:09:14.660 How do I get the lambda? 0:09:14.660,0:09:18.940 Once I grab this lambda,[br]I think I would be done. 0:09:18.940,0:09:21.030 Because once I[br]grab the lambda, I 0:09:21.030,0:09:25.130 could figure out who the x0,[br]y0 are from the equations. 0:09:25.130,0:09:30.890 So I have this feeling I need a[br]procedure, I need an algorithm. 0:09:30.890,0:09:34.122 Engineers are more[br]algorithmically oriented 0:09:34.122,0:09:35.080 than us mathematicians. 0:09:35.080,0:09:39.042 And this is what I appreciate[br]mostly about engineers. 0:09:39.042,0:09:42.920 They have a very[br]organized, systematic mind. 0:09:42.920,0:09:45.720 So if I were to[br]write an algorithm, 0:09:45.720,0:09:50.360 a procedure for the[br]method, I would say, 0:09:50.360,0:09:53.349 assume that f and[br]g are nice to you. 0:09:53.349,0:09:54.140 You don't say that. 0:09:54.140,0:09:55.090 Don't write that. 0:09:55.090,0:10:01.990 Now assume that f and g[br]satisfy Lagrange's theorem. 0:10:01.990,0:10:04.912 Satisfy the conditions[br]of Lagrange's theorem. 0:10:04.912,0:10:14.180 0:10:14.180,0:10:15.760 OK? 0:10:15.760,0:10:19.380 The notion, by the way, has[br]nothing to do with Calc 3. 0:10:19.380,0:10:22.110 But the notion of[br]Lagrangian and Hamiltonian 0:10:22.110,0:10:25.120 are something you are[br]learning in engineering. 0:10:25.120,0:10:30.330 And the Lagrangian is a[br]product of Mr. Lagrange. 0:10:30.330,0:10:33.100 So he's done a lot for[br]science in general, not just 0:10:33.100,0:10:38.040 for calculus, for mathematics,[br]for pure mathematics. 0:10:38.040,0:10:38.540 OK. 0:10:38.540,0:10:41.100 0:10:41.100,0:10:42.810 In that case, what[br]do you need to do? 0:10:42.810,0:10:43.530 Step one. 0:10:43.530,0:10:46.190 0:10:46.190,0:10:47.840 You need to recover that. 0:10:47.840,0:10:56.220 So solve for x0, y0, and[br]lambda the following system. 0:10:56.220,0:10:58.540 What does it mean the two[br]gradients are parallel 0:10:58.540,0:11:00.450 to each other? 0:11:00.450,0:11:04.335 This fella over here is[br]going to be what vector? 0:11:04.335,0:11:07.250 f of xy-- f sub y. 0:11:07.250,0:11:11.340 This gal over here will[br]be g sub x, g sub y. 0:11:11.340,0:11:15.130 For them to be proportional,[br]you should have this, 0:11:15.130,0:11:21.026 then-- f sub x equals g[br]sub x times the lambda. 0:11:21.026,0:11:21.970 Right? 0:11:21.970,0:11:26.910 And f sub y equals g[br]sub y times the lambda 0:11:26.910,0:11:31.640 for the same lambda, your hero. 0:11:31.640,0:11:36.580 So both coordinates have to be[br]multiplied by the same lambda 0:11:36.580,0:11:39.940 to get you the other[br]partial velocities. 0:11:39.940,0:11:42.205 STUDENT: And it has to be[br]evaluated at that point? 0:11:42.205,0:11:42.830 PROFESSOR: Yes. 0:11:42.830,0:11:46.927 So you're gonna solve for--[br]you're gonna solve this system, 0:11:46.927,0:11:49.100 and you are going to[br]get-- and I'm sorry. 0:11:49.100,0:11:50.100 With a constraint. 0:11:50.100,0:11:54.860 And with the absolute constraint[br]that you have at g of xy 0:11:54.860,0:11:57.060 equals c, because[br]these guys are married. 0:11:57.060,0:12:01.020 They always are in[br]this relationship. 0:12:01.020,0:12:05.630 And from all the[br]information of the system, 0:12:05.630,0:12:13.270 you're going to get a-- not one,[br]maybe several values of lambda, 0:12:13.270,0:12:21.580 you get values of lambda, x0,[br]y0, that satisfy the system. 0:12:21.580,0:12:25.570 You have to satisfy all[br]the three constraints, all 0:12:25.570,0:12:27.065 the three equations. 0:12:27.065,0:12:30.460 I'm going to put them[br]in bullets, red bullets. 0:12:30.460,0:12:33.330 You don't have colors, but I do. 0:12:33.330,0:12:35.320 So. 0:12:35.320,0:12:38.670 And then at all these points[br]that we found in step one, 0:12:38.670,0:12:40.490 step two. 0:12:40.490,0:12:42.770 Step two I'm going[br]to erase here. 0:12:42.770,0:12:48.780 For all the points-- x0, y0,[br]and lambda 0-- you got step one. 0:12:48.780,0:12:57.318 You have to evaluate[br]the f function. 0:12:57.318,0:13:09.560 Evaluate f at those x0, y0,[br]lambda 0 we got 4 lambda 0. 0:13:09.560,0:13:19.910 And get to compare values[br]in a table just like before. 0:13:19.910,0:13:25.140 0:13:25.140,0:13:26.590 See all the points. 0:13:26.590,0:13:32.365 All points will[br]give you an idea who 0:13:32.365,0:13:35.104 is going to be the[br]absolute max, absolute min. 0:13:35.104,0:13:39.088 0:13:39.088,0:13:48.580 And I'm just going to go ahead[br]and solve one typical example 0:13:48.580,0:13:50.060 for your better understanding. 0:13:50.060,0:13:53.130 You know, it's not solved[br]in the book by both methods. 0:13:53.130,0:13:55.380 But I'm thinking[br]since I'm teaching you 0:13:55.380,0:13:59.820 how to apply the Lagrange[br]theorem today and do the step 0:13:59.820,0:14:02.180 one, step two[br]procedure for Lagrange 0:14:02.180,0:14:04.770 multipliers I'm going to solve[br]it with Lagrange multipliers 0:14:04.770,0:14:05.640 first. 0:14:05.640,0:14:09.236 And the same problem, I'm[br]going to solve it in the spirit 0:14:09.236,0:14:12.935 that we have employed[br]last time in 11.7. 0:14:12.935,0:14:15.970 And then I'm going[br]to ask you to vote 0:14:15.970,0:14:18.442 which method is easier for you. 0:14:18.442,0:14:21.820 And I'm really curious,[br]because of course, I 0:14:21.820,0:14:24.560 can predict what theorems[br]I'm going to cover. 0:14:24.560,0:14:27.080 And I can predict[br]the results I'm 0:14:27.080,0:14:29.255 going to get in the exercises. 0:14:29.255,0:14:35.440 But I cannot predict what you[br]perceive to be easier or more 0:14:35.440,0:14:40.360 difficult. And I'm[br]curious about it. 0:14:40.360,0:14:42.220 So let's see what you think. 0:14:42.220,0:14:44.760 Just keep an eye[br]on both of them. 0:14:44.760,0:14:48.360 Compare them, and then[br]tell me what you think 0:14:48.360,0:14:54.400 was more efficient or easier[br]to follow or understand. 0:14:54.400,0:14:55.050 OK. 0:14:55.050,0:14:58.450 I'll take this one[br]that's really pretty. 0:14:58.450,0:15:00.720 Example one. 0:15:00.720,0:15:05.190 It is practically[br]straight out of the book. 0:15:05.190,0:15:10.490 It appeared as an obsession[br]in several final exams 0:15:10.490,0:15:12.870 with little variations. 0:15:12.870,0:15:15.296 The constraint was a[br]little, pretty one. 0:15:15.296,0:15:18.840 It's a linear constraint that[br]you have on the variables. 0:15:18.840,0:15:21.390 0:15:21.390,0:15:25.756 The g function I was talking[br]about, the marriage constraint, 0:15:25.756,0:15:27.155 is x plus y. 0:15:27.155,0:15:30.200 And this is the c, little[br]c we were talking about. 0:15:30.200,0:15:32.960 0:15:32.960,0:15:38.020 So how do I know that there[br]exists an x0, y0 extreme? 0:15:38.020,0:15:45.660 How do I know there[br]is an x0, y0 extreme? 0:15:45.660,0:15:49.350 0:15:49.350,0:15:50.430 I need to look baffled. 0:15:50.430,0:15:51.376 How do I look? 0:15:51.376,0:15:54.690 0:15:54.690,0:15:55.560 I don't know. 0:15:55.560,0:15:59.430 I'm just thinking, well,[br]maybe I can find it. 0:15:59.430,0:16:01.660 And once it verifies[br]all the conditions 0:16:01.660,0:16:05.620 of Lagrange's theorem, that[br]I know I'm in business-- 0:16:05.620,0:16:09.450 and I would compute everything,[br]and compare the values, 0:16:09.450,0:16:11.940 and get my max and my min. 0:16:11.940,0:16:14.470 So what do I need[br]to do in step one? 0:16:14.470,0:16:16.490 Step one. 0:16:16.490,0:16:17.090 Oh, my god. 0:16:17.090,0:16:19.259 You guys, remind me,[br]because I forgot. 0:16:19.259,0:16:20.550 I'm just pretending, of course. 0:16:20.550,0:16:24.680 But I want to see if you[br]were able to remember. 0:16:24.680,0:16:28.840 The two gradients of[br]f and g, respectively, 0:16:28.840,0:16:30.210 have to be proportional. 0:16:30.210,0:16:31.940 That's kind of the idea. 0:16:31.940,0:16:34.760 And the proportionality[br]factor is lambda. 0:16:34.760,0:16:38.860 So I do f sub x[br]equals lambda g sub x. 0:16:38.860,0:16:44.525 f sub y equals lambda g sub y,[br]assuming that the gradient of g 0:16:44.525,0:16:49.710 is non-zero at that point where[br]I'm looking and assuming that g 0:16:49.710,0:16:55.570 of xy equals-- guys, I'm-- well,[br]OK, I'm going to write it now. 0:16:55.570,0:16:58.520 But then I have to say[br]who these guys are, 0:16:58.520,0:17:00.130 because that's the[br]important thing. 0:17:00.130,0:17:01.700 And this is where[br]I need your help. 0:17:01.700,0:17:05.180 0:17:05.180,0:17:06.665 So you tell me. 0:17:06.665,0:17:09.462 Who is this fellow, f sub x? 0:17:09.462,0:17:10.394 STUDENT: Negative 2x. 0:17:10.394,0:17:14.155 PROFESSOR: Minus[br]2x equals lambda. 0:17:14.155,0:17:14.910 Lambda. 0:17:14.910,0:17:16.500 Mr. Lambda is important. 0:17:16.500,0:17:20.300 I'm going to put it in red. 0:17:20.300,0:17:22.839 Know why I'm putting him[br]in red-- because he needs 0:17:22.839,0:17:26.608 to just jump into my eyes. 0:17:26.608,0:17:29.770 Maybe I can[br]eliminate the lambda. 0:17:29.770,0:17:31.920 This is the general philosophy. 0:17:31.920,0:17:35.061 Maybe to solve the system,[br]I can eliminate the lambda 0:17:35.061,0:17:36.310 between the equations somehow. 0:17:36.310,0:17:38.990 0:17:38.990,0:17:43.360 How about Mr. g sub x? 0:17:43.360,0:17:45.980 g sub x is 1, so[br]it's a blessing. 0:17:45.980,0:17:48.560 I shouldn't write[br]times 1, but I am silly 0:17:48.560,0:17:50.130 and you know me by now. 0:17:50.130,0:17:51.405 So I'm going to keep going. 0:17:51.405,0:17:54.590 And I say, minus two[br]more is the same way. 0:17:54.590,0:17:59.000 Mr. Lambda in red very[br]happy to be there. 0:17:59.000,0:18:01.307 And times-- 0:18:01.307,0:18:02.015 STUDENT: 1 again. 0:18:02.015,0:18:03.350 PROFESSOR: 1 again. 0:18:03.350,0:18:04.690 Thank god. 0:18:04.690,0:18:09.520 And then this easy condition,[br]that translates as x plus y 0:18:09.520,0:18:10.780 is 1. 0:18:10.780,0:18:12.410 And now what do we do? 0:18:12.410,0:18:20.160 Now we start staring at the[br]system, and we see patterns. 0:18:20.160,0:18:24.020 And we think, what[br]would be the easiest way 0:18:24.020,0:18:27.150 to deal with these patterns? 0:18:27.150,0:18:30.850 We see a pattern like[br]x plus y is known. 0:18:30.850,0:18:37.170 And if we were to sum[br]up the two equations, 0:18:37.170,0:18:40.980 like summing up the left-hand[br]side and right-hand side, 0:18:40.980,0:18:46.250 x plus y would be included as[br]in something in there as a unit. 0:18:46.250,0:18:49.790 So I'm just trying to[br]be creative and say, 0:18:49.790,0:18:51.800 there is no unique[br]way to solve it. 0:18:51.800,0:18:54.240 You can solve it in many ways. 0:18:54.240,0:18:56.830 But the easiest way[br]that comes to mind 0:18:56.830,0:19:02.494 is like, add up the left-hand[br]side and the right-hand side. 0:19:02.494,0:19:04.430 How much is that, the[br]lambda plus lambda? 0:19:04.430,0:19:05.400 STUDENT: 2 lambda. 0:19:05.400,0:19:07.790 PROFESSOR: 2 lambda, right? 0:19:07.790,0:19:13.065 And the x plus y is 1, because[br]God provided this to you. 0:19:13.065,0:19:16.390 You cannot change this. 0:19:16.390,0:19:17.280 OK? 0:19:17.280,0:19:19.110 It's an axiom. 0:19:19.110,0:19:21.150 So you replace it here. 0:19:21.150,0:19:23.060 1. 0:19:23.060,0:19:24.840 And you say, OK, I divide by 2. 0:19:24.840,0:19:25.340 Whatever. 0:19:25.340,0:19:31.260 Lambda has to be minus 1. 0:19:31.260,0:19:35.420 So if lambda is minus 1, do[br]I have other possibilities? 0:19:35.420,0:19:38.265 So first thing, when you[br]look at this algorithm, 0:19:38.265,0:19:42.080 you say, well, I know[br]what I have to do, 0:19:42.080,0:19:44.760 but are there any[br]other possibilities? 0:19:44.760,0:19:48.432 And then you say no,[br]that's the only one. 0:19:48.432,0:19:53.980 For lambda equals minus 1,[br]fortunately, you get what? 0:19:53.980,0:19:57.620 x equals 1/2. 0:19:57.620,0:20:02.880 Unless-- give it a name,[br]because this variable 0:20:02.880,0:20:04.135 means an arbitrary variable. 0:20:04.135,0:20:04.740 It's 0. 0:20:04.740,0:20:07.570 It's not-- and for[br]the same, you get 0:20:07.570,0:20:12.720 y0 equals 1/2 in the same way. 0:20:12.720,0:20:17.230 And then you say, OK,[br]for I know x0 y0 are now, 0:20:17.230,0:20:23.595 that's the only extremum that[br]I'm having to look at for now. 0:20:23.595,0:20:25.280 What is the 0? 0:20:25.280,0:20:26.950 So I'm going to[br]go ahead and say, 0:20:26.950,0:20:31.350 the point will be P0, 1/2, 1/2. 0:20:31.350,0:20:37.940 And then I plug in, and I[br]say, 1 minus 1/4 minus 1/4 0:20:37.940,0:20:38.810 is again 1/2. 0:20:38.810,0:20:41.540 0:20:41.540,0:20:45.795 When you compute problems,[br]when you computationally 0:20:45.795,0:20:49.270 solve problems,[br]many times you're 0:20:49.270,0:20:52.190 going to see that you[br]make algebra mistakes. 0:20:52.190,0:20:54.710 If you think I[br]don't make them, you 0:20:54.710,0:20:58.217 have proof that I make[br]them myself sometimes. 0:20:58.217,0:20:59.925 What is the best way[br]to protect yourself? 0:20:59.925,0:21:02.760 0:21:02.760,0:21:04.710 When you get numerical[br]answers a little bit, 0:21:04.710,0:21:07.311 see if they make sense. 0:21:07.311,0:21:08.300 Does that make sense? 0:21:08.300,0:21:08.799 Yes. 0:21:08.799,0:21:09.860 Is the sum 1? 0:21:09.860,0:21:10.550 Yes. 0:21:10.550,0:21:13.420 A little bit of double-checking[br]with your constraints, 0:21:13.420,0:21:18.040 your original data,[br]it looks good. 0:21:18.040,0:21:18.640 All right. 0:21:18.640,0:21:20.840 So the question[br]here is-- right now 0:21:20.840,0:21:24.770 the question is, are we done? 0:21:24.770,0:21:28.030 The answer is no,[br]we haven't quite 0:21:28.030,0:21:32.100 looked at what happens[br]with the constraint g, 0:21:32.100,0:21:36.230 because c-- oh, I forgot[br]to tell you that the book-- 0:21:36.230,0:21:39.272 if you look in the book--[br]that's why you should have 0:21:39.272,0:21:44.030 the book in electronic format,[br]so you can read it in Kindle. 0:21:44.030,0:21:49.940 Example one had the[br]additional requirement 0:21:49.940,0:21:52.780 that x and y are positive. 0:21:52.780,0:21:56.215 Is such a requirement[br]natural in applications 0:21:56.215,0:21:59.766 of calculus, because this is[br]Calculus 3 with applications. 0:21:59.766,0:22:05.630 Can you give me an example[br]where x and y, being positive, 0:22:05.630,0:22:07.599 would be a must? 0:22:07.599,0:22:09.140 STUDENT: When they're[br]both distances? 0:22:09.140,0:22:13.180 PROFESSOR: Distances[br]or some physical things 0:22:13.180,0:22:15.450 that are measurable. 0:22:15.450,0:22:20.590 Lengths, widths, the[br]girth around an object, 0:22:20.590,0:22:25.770 some positive numbers that-- OK. 0:22:25.770,0:22:26.450 All right. 0:22:26.450,0:22:30.250 So we will see an example[br]involving dimensions 0:22:30.250,0:22:33.890 of a box and volume of a[br]box, where, of course, x, 0:22:33.890,0:22:36.206 y, z will be the[br]length, the width, 0:22:36.206,0:22:39.200 and the height of the box. 0:22:39.200,0:22:43.125 So that would come naturally[br]as x, y, z positive. 0:22:43.125,0:22:45.850 Next we are going to do that. 0:22:45.850,0:22:50.150 Now, what's going to happen[br]for this kind of constraint? 0:22:50.150,0:22:54.210 So I want to see if x and y are[br]positive but at the same time, 0:22:54.210,0:22:57.105 they are married to[br]Px plus y equals 1, 0:22:57.105,0:23:00.130 I do not have just[br]all the possibilities. 0:23:00.130,0:23:05.440 I have to have in mind[br]their picture as a couple. 0:23:05.440,0:23:11.470 x plus y, as a couple,[br]must be 1, meaning you get 0:23:11.470,0:23:12.740 the segment, this segment. 0:23:12.740,0:23:14.480 Are you guys with me? 0:23:14.480,0:23:17.170 Why don't I expand[br]to the whole line? 0:23:17.170,0:23:19.940 I say, I want to expand[br]to the whole line, which 0:23:19.940,0:23:21.040 would be stupid. 0:23:21.040,0:23:23.370 Why would it be stupid? 0:23:23.370,0:23:27.020 I would get y equals[br]something negative here. 0:23:27.020,0:23:31.370 And if I expand in the other[br]direction, x would be negative. 0:23:31.370,0:23:34.160 So it's not a good thing. 0:23:34.160,0:23:39.170 So the only thing I[br]have is the segment, 0:23:39.170,0:23:40.500 which has two endpoints. 0:23:40.500,0:23:42.307 Those two endpoints[br]are boo-boos. 0:23:42.307,0:23:45.100 0:23:45.100,0:23:47.370 The endpoints can give[br]you extrema as well. 0:23:47.370,0:23:49.220 We talked about it last time. 0:23:49.220,0:23:51.500 So every time you do[br]this, you're fine, 0:23:51.500,0:23:55.300 but you have to compare the[br]results against the extrema. 0:23:55.300,0:23:57.460 These are artificial cuts. 0:23:57.460,0:23:58.540 In what sense artificial? 0:23:58.540,0:24:03.940 In the sense that you[br]don't let the whole thing 0:24:03.940,0:24:06.920 evolve over the[br]whole real domain. 0:24:06.920,0:24:09.190 Once you artificially[br]cut something-- 0:24:09.190,0:24:10.530 let me give you another example. 0:24:10.530,0:24:13.095 Don't put this in[br]the notes, because I 0:24:13.095,0:24:14.742 don't want it to confuse you. 0:24:14.742,0:24:17.646 You have some natural, so-called[br]relative max and minima here, 0:24:17.646,0:24:18.614 right? 0:24:18.614,0:24:21.870 That's a relative min, that's[br]a relative max, and so on. 0:24:21.870,0:24:27.460 If I make an artificial[br]cut anywhere-- let's 0:24:27.460,0:24:31.860 say this is not going[br]to be a minimum anymore. 0:24:31.860,0:24:35.720 I make an artificial cut here,[br]I make an artificial cut here. 0:24:35.720,0:24:39.100 These endpoints will[br]generate other possibilities 0:24:39.100,0:24:42.115 for my absolute max[br]and absolute min. 0:24:42.115,0:24:44.532 So those extrema are[br]extremely important. 0:24:44.532,0:24:46.890 I have one. 0:24:46.890,0:24:49.770 What is this guy's-- 1, 0. 0:24:49.770,0:24:51.350 Am I right? 0:24:51.350,0:24:54.100 And this is 0,1. 0:24:54.100,0:24:55.645 So I have to look[br]at the possibility. 0:24:55.645,0:24:59.220 When it's 1 by 1, it[br]goes-- say it again. 0:24:59.220,0:25:00.590 1,0. 0:25:00.590,0:25:03.174 And x2y2 was hmm? 0:25:03.174,0:25:04.578 0,1. 0:25:04.578,0:25:07.386 And of course, both[br]of them satisfy that. 0:25:07.386,0:25:13.068 In this case, f of 1, 0 has[br]to be evaluated as well. 0:25:13.068,0:25:18.000 That's going to be 1 minus[br]1 squared minus 0 equals 0. 0:25:18.000,0:25:22.390 And by the symmetry[br]of this polynomial, 0:25:22.390,0:25:28.620 you are going to have the[br]same answer, 0, in both cases. 0:25:28.620,0:25:30.800 You're going to draw the table. 0:25:30.800,0:25:35.490 And this is the perfect[br]place for the table. 0:25:35.490,0:25:38.230 Perfect place in the[br]sense that you have x, y 0:25:38.230,0:25:43.568 and you have-- who are your[br]notable, noticeable guys? 0:25:43.568,0:25:45.940 1/2, 1/2. 0:25:45.940,0:25:49.270 Who said 1,0? 0:25:49.270,0:25:52.700 And 0, 1. 0:25:52.700,0:25:56.600 And who was the zz[br]was the 1/2 here. 0:25:56.600,0:25:59.432 And here was 0, and here was 0. 0:25:59.432,0:26:03.106 And I'm going to start[br]making faces and drawing. 0:26:03.106,0:26:06.962 0:26:06.962,0:26:08.408 Did I get the answer? 0:26:08.408,0:26:10.818 Did I solve this[br]problem at home? 0:26:10.818,0:26:12.760 Yes, I did. 0:26:12.760,0:26:14.014 And I got the same answer. 0:26:14.014,0:26:14.850 All right. 0:26:14.850,0:26:18.585 So this is max. 0:26:18.585,0:26:20.700 This is min. 0:26:20.700,0:26:23.120 This is mean, the same. 0:26:23.120,0:26:25.540 So both of these are what? 0:26:25.540,0:26:28.220 Absolute minima. 0:26:28.220,0:26:31.740 And these are the[br]absolute extrema 0:26:31.740,0:26:39.231 for this problem[br]with constraint. 0:26:39.231,0:26:43.370 I'm going to go ahead[br]and erase and say, 0:26:43.370,0:26:46.280 remember in the[br]eyes of your mind 0:26:46.280,0:26:49.413 how much work it was to do this? 0:26:49.413,0:26:52.270 And I'm going to apply[br]the other method. 0:26:52.270,0:26:53.580 So how much space? 0:26:53.580,0:26:58.570 So we needed one[br]board largely written. 0:26:58.570,0:27:01.820 You want to go to follow[br]the steps one and two. 0:27:01.820,0:27:02.870 Should I erase that? 0:27:02.870,0:27:03.370 STUDENT: No. 0:27:03.370,0:27:06.375 0:27:06.375,0:27:07.750 PROFESSOR: You[br]are my note-taker. 0:27:07.750,0:27:10.070 Of course I will listen to you. 0:27:10.070,0:27:16.110 And then let's see what method[br]number two I had in mind 0:27:16.110,0:27:17.660 is the one from last time. 0:27:17.660,0:27:19.440 So this is a what? 0:27:19.440,0:27:23.990 It's a review of-- what[br]is the section time? 0:27:23.990,0:27:25.822 11.7. 0:27:25.822,0:27:31.250 And I'm going to make a face,[br]happy that I can have yet 0:27:31.250,0:27:34.755 another application for you. 0:27:34.755,0:27:39.100 When this problem appeared on[br]the final at least five times 0:27:39.100,0:27:46.004 in the last 10 finals or[br]more, different instructors 0:27:46.004,0:27:48.606 viewed it differently. 0:27:48.606,0:27:52.600 Practically, the[br]general instruction 0:27:52.600,0:27:55.010 given to the students--[br]solve it anyway you 0:27:55.010,0:27:57.220 find it easier for you. 0:27:57.220,0:27:59.230 Just don't make mistakes. 0:27:59.230,0:28:03.250 So we did not[br]encourage instructors 0:28:03.250,0:28:05.482 to say, do this by[br]Lagrange multipliers, 0:28:05.482,0:28:08.560 or do this by-- no, no, no, no. 0:28:08.560,0:28:10.150 Whatever is easier[br]for the student. 0:28:10.150,0:28:13.475 So what did we do last[br]time about the constraint? 0:28:13.475,0:28:17.380 Since x and y are[br]married, y depend on x. 0:28:17.380,0:28:19.970 So y is 1 times x. 0:28:19.970,0:28:22.210 And we say, this[br]is my guy that I 0:28:22.210,0:28:26.200 have to plug in[br]into the function, 0:28:26.200,0:28:28.120 into the original function. 0:28:28.120,0:28:30.240 And then f would[br]not be a function 0:28:30.240,0:28:32.410 of two independent[br]variables anymore. 0:28:32.410,0:28:35.930 But it's going to become a[br]function of one variable. 0:28:35.930,0:28:37.845 Thank god it's not hard. 0:28:37.845,0:28:40.920 It's no hard[br]because in this way, 0:28:40.920,0:28:47.100 you have just to pull[br]out the y1 minus x, 0:28:47.100,0:28:50.740 and square it, and[br]do the algebra. 0:28:50.740,0:28:54.040 So 1 minus x squared. 0:28:54.040,0:28:56.210 And I'm going to do[br]this really quickly. 0:28:56.210,0:28:59.450 Minus 1 minus x[br]squared and plus 2x. 0:28:59.450,0:29:03.508 0:29:03.508,0:29:04.300 And OK. 0:29:04.300,0:29:09.620 So we say, all right, all[br]right, so 1 and minus 1 go away. 0:29:09.620,0:29:13.980 0:29:13.980,0:29:19.040 It make our life easier,[br]because I have minus 2x squared 0:29:19.040,0:29:21.150 plus So of course,[br]I could do it fast, 0:29:21.150,0:29:24.000 but the whole idea[br]is not to amaze you 0:29:24.000,0:29:30.090 with my capability of working[br]fast, but be able to follow. 0:29:30.090,0:29:31.540 So you have minus what? 0:29:31.540,0:29:32.920 So tell me. 0:29:32.920,0:29:36.447 You can pull out a minus 2x. 0:29:36.447,0:29:39.620 And you get x. 0:29:39.620,0:29:40.550 And a minus 1. 0:29:40.550,0:29:44.440 0:29:44.440,0:29:50.740 And what is special about that? 0:29:50.740,0:29:53.810 Well, do I really[br]need to do that? 0:29:53.810,0:29:56.087 That's the question. 0:29:56.087,0:29:57.170 Could I have stopped here? 0:29:57.170,0:29:59.930 Is this the point[br]of factoring out? 0:29:59.930,0:30:03.080 Not really because factoring[br]out is not going to help you. 0:30:03.080,0:30:07.030 What I want is to chase[br]after Mr. f prime of x 0:30:07.030,0:30:12.660 and solve the critical point[br]equation f prime of x equals 0. 0:30:12.660,0:30:14.610 Right? 0:30:14.610,0:30:21.520 I need to find that x0 that will[br]satisfy f prime of x equals 0. 0:30:21.520,0:30:22.360 What do I get? 0:30:22.360,0:30:29.520 I get minus 4x plus 2 equals 0. 0:30:29.520,0:30:33.200 And I see I'm already relieved. 0:30:33.200,0:30:38.210 The moment I saw that, I[br]felt that I'm doing this 0:30:38.210,0:30:42.910 the right way, because I had[br]the previous method that led me 0:30:42.910,0:30:48.000 to a 1/2 that Alex provided for[br]somebody for the first time. 0:30:48.000,0:30:50.890 So now I feel I'm going[br]to get the same thing. 0:30:50.890,0:30:53.210 Let's see how much faster[br]or how much slower. 0:30:53.210,0:30:57.122 Why 0 corresponding[br]to it will be 1/2? 0:30:57.122,0:30:59.980 Because 1/2 plus 1/2 is a 1. 0:30:59.980,0:31:02.070 So what do I do? 0:31:02.070,0:31:05.190 Just as before, I start[br]my table and I say, 0:31:05.190,0:31:10.108 x and y must be 1/2 and 1/2[br]to give me the critical point 0:31:10.108,0:31:10.720 in the middle. 0:31:10.720,0:31:13.530 And I'm going to[br]get a 1/2 for that. 0:31:13.530,0:31:14.730 And I don't yet. 0:31:14.730,0:31:15.720 I pretend. 0:31:15.720,0:31:19.124 I don't know that's[br]gonna be a maximum. 0:31:19.124,0:31:25.490 What other points will provide[br]the books, the so-called-- 0:31:25.490,0:31:26.990 STUDENT: Endpoints. 0:31:26.990,0:31:28.406 PROFESSOR: The endpoints. 0:31:28.406,0:31:31.250 And for those endpoints,[br]I keep in mind 0:31:31.250,0:31:33.830 that x and y,[br]again, are positive. 0:31:33.830,0:31:38.150 I should keep this picture in[br]mind, because if I don't, well, 0:31:38.150,0:31:40.150 it's not going to be very good. 0:31:40.150,0:31:43.742 So x is not allowed to move. 0:31:43.742,0:31:50.115 See, x has limited freedom[br]from the constraint. 0:31:50.115,0:31:54.076 So he's not allowed[br]to leave this segment. 0:31:54.076,0:31:56.300 x is going to be[br]between 0 and 1. 0:31:56.300,0:32:03.030 So for the endpoint x equals[br]0-- will provide me with y 0:32:03.030,0:32:03.950 equals 1. 0:32:03.950,0:32:07.230 And I'll put it in the[br]table, and I'll say, 0:32:07.230,0:32:11.010 when x equals 0 and y[br]equals 1-- and in that case, 0:32:11.010,0:32:14.730 I plug back in here and I get 0. 0:32:14.730,0:32:18.490 And again, for the[br]same type of-- I mean, 0:32:18.490,0:32:24.120 the other endpoint, I[br]get 1,0, and I get 0. 0:32:24.120,0:32:25.990 And it's the same thing. 0:32:25.990,0:32:29.270 I got the same thing[br]through another method. 0:32:29.270,0:32:33.120 This is the max, and[br]these are the mins. 0:32:33.120,0:32:38.920 And one of my students asked me[br]in my office hour-- by the way, 0:32:38.920,0:32:46.220 if you cannot make it to[br]Tuesday's 3:00 to 5:00, 0:32:46.220,0:32:47.240 you can come today. 0:32:47.240,0:32:50.330 At 2:00 after we are done,[br]I'm going to be in my office 0:32:50.330,0:32:50.830 as well. 0:32:50.830,0:32:52.740 So just. 0:32:52.740,0:32:58.332 So I have Tuesdays and Thursdays[br]after class, right after class. 0:32:58.332,0:33:04.090 Now, no matter what, if you get[br]the same answer, what if you 0:33:04.090,0:33:07.320 forget about one of the values? 0:33:07.320,0:33:11.450 Like, this student asked me,[br]what if I got the right maximum 0:33:11.450,0:33:14.900 and I got the right minimum, and[br]I say those are your extrema, 0:33:14.900,0:33:18.650 and I don't prove, mind you,[br]both points when it happens, 0:33:18.650,0:33:21.825 only one? 0:33:21.825,0:33:23.589 I don't know. 0:33:23.589,0:33:25.380 It's different from a[br]problem to the other. 0:33:25.380,0:33:28.160 Maybe I'm subtracting[br]some credit. 0:33:28.160,0:33:32.250 But you get most of the[br]partial credit in that case. 0:33:32.250,0:33:34.090 There will be many[br]values in which 0:33:34.090,0:33:36.840 you get the same altitude. 0:33:36.840,0:33:38.143 This is the altitude. 0:33:38.143,0:33:38.642 My z. 0:33:38.642,0:33:41.890 0:33:41.890,0:33:43.430 Do you have questions of that? 0:33:43.430,0:33:43.930 OK. 0:33:43.930,0:33:48.010 Now it's my turn[br]to make you vote. 0:33:48.010,0:33:51.100 And if you cannot[br]vote, you abstain. 0:33:51.100,0:33:53.420 Which one was easier? 0:33:53.420,0:33:57.000 The first method, the[br]Lagrange multipliers? 0:33:57.000,0:34:01.890 Or the second one, the-- how[br]should I call the second one, 0:34:01.890,0:34:02.550 the-- 0:34:02.550,0:34:03.870 STUDENT: Integration. 0:34:03.870,0:34:07.820 PROFESSOR: The ray substitution[br]method then derivation, 0:34:07.820,0:34:11.110 count one type method? 0:34:11.110,0:34:14.590 So who is for-- OK. 0:34:14.590,0:34:18.530 You got this on the[br]midterm, say, or final. 0:34:18.530,0:34:21.975 How many of you would feel the[br]first method would be easier 0:34:21.975,0:34:23.875 to employ? 0:34:23.875,0:34:26.730 STUDENT: The second. 0:34:26.730,0:34:29.129 PROFESSOR: And how many of[br]you think the second method 0:34:29.129,0:34:31.650 is easier to employ? 0:34:31.650,0:34:34.554 And how many people[br]say that they 0:34:34.554,0:34:39.110 are equally long, or short,[br]or how many people abstain? 0:34:39.110,0:34:41.480 STUDENT: I would say it[br]depends on the problem. 0:34:41.480,0:34:42.800 PROFESSOR: Yeah, absolutely. 0:34:42.800,0:34:44.550 But I'm talking about[br]this particular one, 0:34:44.550,0:34:45.340 because I'm curious. 0:34:45.340,0:34:46.380 STUDENT: Oh, on this one. 0:34:46.380,0:34:47.500 Oh, OK. 0:34:47.500,0:34:49.960 PROFESSOR: I'm going to go[br]on and do another problem. 0:34:49.960,0:34:51.945 And for that, also, I will ask. 0:34:51.945,0:34:57.401 0:34:57.401,0:35:01.880 with other problems, it[br]may be that it's easier 0:35:01.880,0:35:05.350 to solve the system for[br]the Lagrange multipliers 0:35:05.350,0:35:11.260 than it is to pull out the y[br]explicitly from the constraint 0:35:11.260,0:35:12.840 and put it back. 0:35:12.840,0:35:19.132 0:35:19.132,0:35:21.730 What else have I[br]prepared for you? 0:35:21.730,0:35:23.850 I had cooked up something. 0:35:23.850,0:35:27.040 0:35:27.040,0:35:30.112 I had cooked up[br]some extra credit. 0:35:30.112,0:35:31.840 But I don't know[br]if you have time. 0:35:31.840,0:35:32.680 But write it anyway. 0:35:32.680,0:35:35.320 So please write[br]down, for one point 0:35:35.320,0:35:44.690 extra credit for[br]the next seven days, 0:35:44.690,0:35:54.174 read and summarize both[br]of the following methods-- 0:35:54.174,0:36:05.140 Lagrange multipliers with[br]one parameter lambda, 0:36:05.140,0:36:07.853 which is exactly the[br]same I taught you. 0:36:07.853,0:36:11.790 Same I taught. 0:36:11.790,0:36:17.420 And one that is not required[br]for the examinations, which 0:36:17.420,0:36:19.675 is Lagrange multipliers[br]with two parameters. 0:36:19.675,0:36:24.330 0:36:24.330,0:36:26.442 And that is a big[br]headache when you 0:36:26.442,0:36:29.690 do that, because you[br]have two parameters. 0:36:29.690,0:36:32.730 Let's call them lambda and mu. 0:36:32.730,0:36:36.020 I don't know what to call them. 0:36:36.020,0:36:40.290 When you have that kind[br]of method, it's longer. 0:36:40.290,0:36:45.200 So it may take you several[br]pages of computation 0:36:45.200,0:36:48.890 to get to the lambdas and to[br]the extrema and everything. 0:36:48.890,0:36:52.090 But I would like you to[br]at least read the theorem 0:36:52.090,0:36:56.245 and write down a short[br]paragraph about one of these. 0:36:56.245,0:36:58.640 So both of them are one point. 0:36:58.640,0:37:02.922 Both of them are one point[br]extra credit at the end. 0:37:02.922,0:37:04.005 STUDENT: Together or each? 0:37:04.005,0:37:04.515 PROFESSOR: Yes, sir? 0:37:04.515,0:37:06.230 STUDENT: One point each[br]or one point together? 0:37:06.230,0:37:07.646 PROFESSOR: No, one[br]both, together. 0:37:07.646,0:37:08.346 I'm sorry. 0:37:08.346,0:37:11.755 Because there will be other[br]chances to get extra credit. 0:37:11.755,0:37:15.960 And I'm cooking up something[br]I didn't say on the syllabus, 0:37:15.960,0:37:21.030 like a brownie point[br]or cake or something. 0:37:21.030,0:37:24.160 At the end of the[br]class, I would like 0:37:24.160,0:37:28.320 you to write me a statement[br]of two pages on how 0:37:28.320,0:37:32.290 you think Calculus 3[br]relates to your major. 0:37:32.290,0:37:34.815 And one question from[br]a previous student 0:37:34.815,0:37:38.800 was, I've changed[br]my major four times. 0:37:38.800,0:37:41.398 Which one shall I pick? 0:37:41.398,0:37:44.070 I said, whichever[br]you are in right now. 0:37:44.070,0:37:45.280 How does that relate? 0:37:45.280,0:37:48.600 How is Calculus 3[br]relevant to your major? 0:37:48.600,0:37:51.290 Give me some[br]examples and how you 0:37:51.290,0:37:55.055 think functions of two[br]variables or three variables-- 0:37:55.055,0:37:55.930 STUDENT: What's this? 0:37:55.930,0:38:00.370 PROFESSOR: Up here[br]in your main major. 0:38:00.370,0:38:03.458 STUDENT: So it's a two[br]parameter question? 0:38:03.458,0:38:05.702 Like, would there be any[br]question regarding that? 0:38:05.702,0:38:06.660 PROFESSOR: No, nothing. 0:38:06.660,0:38:07.990 Not in the homework. 0:38:07.990,0:38:11.740 We don't cover that, we[br]don't do that in the test. 0:38:11.740,0:38:14.060 Most instructors[br]don't even mention it, 0:38:14.060,0:38:16.705 but I said, mm, you[br]are our students, 0:38:16.705,0:38:20.400 so I want to let you do[br]a little bit of research. 0:38:20.400,0:38:23.460 It's about a page and[br]a half of reading. 0:38:23.460,0:38:24.370 Individual study. 0:38:24.370,0:38:25.832 STUDENT: Is that in the book? 0:38:25.832,0:38:27.040 PROFESSOR: It is in the book. 0:38:27.040,0:38:28.140 So individual study. 0:38:28.140,0:38:30.453 One page or one page and a half. 0:38:30.453,0:38:31.490 Something like that. 0:38:31.490,0:38:32.819 Maybe less. 0:38:32.819,0:38:33.318 OK. 0:38:33.318,0:38:36.060 0:38:36.060,0:38:43.400 One other one that I cooked[br]up-- it's not in the book. 0:38:43.400,0:38:47.190 But I liked it because it sounds[br]like a real-life application. 0:38:47.190,0:38:50.038 It is a real-life application. 0:38:50.038,0:38:54.540 And I was talking[br]to the mailman. 0:38:54.540,0:38:59.680 And he was saying, I wonder[br]how-- because a guy, poor guy, 0:38:59.680,0:39:01.675 was carrying these[br]Priority boxes. 0:39:01.675,0:39:05.870 And he said, I wonder[br]how they optimize? 0:39:05.870,0:39:09.280 When they say "flat[br]rate," how do they 0:39:09.280,0:39:11.672 come up with those dimensions? 0:39:11.672,0:39:15.116 And it's an[br]optimization problem, 0:39:15.116,0:39:18.685 and there are many like[br]that in the real world. 0:39:18.685,0:39:23.160 But for my case,[br]I would say, let's 0:39:23.160,0:39:27.255 assume that somebody says,[br]the sum of the lengths 0:39:27.255,0:39:32.510 plus widths plus height is[br]constrained to be some number. 0:39:32.510,0:39:36.890 x plus y plus z equals[br]the maximum possible. 0:39:36.890,0:39:38.600 Could be 50 inches. 0:39:38.600,0:39:40.550 But instead of 50[br]inches-- because I 0:39:40.550,0:39:42.905 don't want to work with[br]that kind of numbers, 0:39:42.905,0:39:48.380 I'm too lazy-- I put x[br]plus y plus equals 1. 0:39:48.380,0:39:50.110 That's my constraint, g. 0:39:50.110,0:39:57.690 0:39:57.690,0:40:03.094 I would like to[br]maximize the volume. 0:40:03.094,0:40:04.320 Say it again, Magdalena. 0:40:04.320,0:40:05.456 What is the problem? 0:40:05.456,0:40:06.700 What's your problem? 0:40:06.700,0:40:11.190 My problem is example three. 0:40:11.190,0:40:34.310 Maximize the volume of a box[br]of length, height, and width x, 0:40:34.310,0:40:41.360 y, z, just to make our[br]life easier in a way 0:40:41.360,0:40:46.464 that the girth cross[br]the-- well, OK. 0:40:46.464,0:40:48.929 Let me make this interesting. 0:40:48.929,0:40:53.860 The sum of the[br]dimensions equals 1. 0:40:53.860,0:40:56.235 And where can you[br]find this problem? 0:40:56.235,0:41:00.140 Well, this problem can be[br]found in several resources. 0:41:00.140,0:41:03.085 We haven't dealt very[br]much with functions 0:41:03.085,0:41:06.130 of three variables, x, y, z. 0:41:06.130,0:41:09.120 But the procedure[br]is exactly the same. 0:41:09.120,0:41:11.780 I stole that from[br]a library online 0:41:11.780,0:41:16.750 that's called Paul's[br]Online Calculus Notes. 0:41:16.750,0:41:19.580 And imagine that[br]the same thing I 0:41:19.580,0:41:24.020 taught you would be applied to[br]functions of three variables. 0:41:24.020,0:41:26.686 Tell me who the volume will be. 0:41:26.686,0:41:29.584 The volume would be a[br]function of three variables. 0:41:29.584,0:41:33.448 Let's call it f, which is what? 0:41:33.448,0:41:34.897 Who is telling me what? 0:41:34.897,0:41:36.126 STUDENT: x times y. 0:41:36.126,0:41:37.000 PROFESSOR: x times y. 0:41:37.000,0:41:37.920 Thanks. 0:41:37.920,0:41:40.130 And are we happy about it? 0:41:40.130,0:41:41.687 Ah, it's a beautiful function. 0:41:41.687,0:41:43.770 It's not going to give you[br]too much of a headache. 0:41:43.770,0:41:46.670 0:41:46.670,0:41:53.210 I would like you to cook up[br]step one and step two for me 0:41:53.210,0:41:55.680 by the Lagrange[br]multipliers I specify. 0:41:55.680,0:42:04.547 0:42:04.547,0:42:06.536 For functions of[br]three variables. 0:42:06.536,0:42:13.680 0:42:13.680,0:42:15.080 Maximize and minimize. 0:42:15.080,0:42:19.780 0:42:19.780,0:42:20.280 Yeah. 0:42:20.280,0:42:24.600 0:42:24.600,0:42:25.100 OK. 0:42:25.100,0:42:31.580 So the gradients are not[br]going to be in R2 anymore. 0:42:31.580,0:42:33.030 They will be in R3. 0:42:33.030,0:42:33.530 And so what? 0:42:33.530,0:42:34.880 It doesn't matter. 0:42:34.880,0:42:35.464 Step one. 0:42:35.464,0:42:39.550 0:42:39.550,0:42:40.740 Say it again, Magdalena. 0:42:40.740,0:42:41.470 What do you mean? 0:42:41.470,0:42:44.700 I mean that when you're going[br]to have something like that, 0:42:44.700,0:42:52.215 the system for nabla f of[br]x, y, z at the point x0, y0, 0:42:52.215,0:43:00.590 z0 will be lambda times[br]nabla g of x at x0, y0 is 0, 0:43:00.590,0:43:04.641 where both nablas are in R3. 0:43:04.641,0:43:05.140 Right? 0:43:05.140,0:43:12.420 They will be f sub x, f sub[br]y, f sub z angular brackets. 0:43:12.420,0:43:16.519 So instead of having just[br]two equations in the system, 0:43:16.519,0:43:18.060 you're going to have[br]three equations. 0:43:18.060,0:43:21.480 That's the only big difference. 0:43:21.480,0:43:22.130 Big deal. 0:43:22.130,0:43:23.692 Not a big deal. 0:43:23.692,0:43:26.500 So you tell me what[br]I'm going to write. 0:43:26.500,0:43:33.540 So I'm going to write f sub[br]x equals lambda g sub x. 0:43:33.540,0:43:37.050 f sub y equals lambda g sub y. 0:43:37.050,0:43:40.670 f sub z equals lambda g sub z. 0:43:40.670,0:43:44.730 Thank god I don't have[br]more than three variables. 0:43:44.730,0:43:48.680 Now we-- in fact, it's[br]how do you think engineers 0:43:48.680,0:43:51.010 solve this kind of system? 0:43:51.010,0:43:52.430 Do they do this by hand? 0:43:52.430,0:43:53.160 No. 0:43:53.160,0:43:54.890 Life is complicated. 0:43:54.890,0:43:59.230 When you do Lagrange multipliers[br]on a thermodynamical problem 0:43:59.230,0:44:01.730 or mechanics problem,[br]physics problem, 0:44:01.730,0:44:05.940 you have really ugly[br]data that are programs 0:44:05.940,0:44:07.960 based on Lagrange multipliers. 0:44:07.960,0:44:10.140 You can have a[br]Lagrange multiplier 0:44:10.140,0:44:13.350 of seven different parameters,[br]including pressure, time, 0:44:13.350,0:44:14.500 and temperature. 0:44:14.500,0:44:16.060 And it's really horrible. 0:44:16.060,0:44:18.800 And you don't do that by hand. 0:44:18.800,0:44:22.660 That's why we have to be[br]thankful to technology 0:44:22.660,0:44:27.180 and the software, the[br]scientific software methods. 0:44:27.180,0:44:30.430 You can do that in MATLAB, you[br]can do that in Mathematica. 0:44:30.430,0:44:32.420 MATLAB is mostly for engineers. 0:44:32.420,0:44:35.110 There are programs written[br]especially for MATLAB 0:44:35.110,0:44:39.750 to solve the problem of[br]Lagrange multipliers. 0:44:39.750,0:44:43.290 Now, this has not complete. 0:44:43.290,0:44:46.966 We are missing the most[br]important, the marriage thing, 0:44:46.966,0:44:53.010 the g of x, y, z constraint. 0:44:53.010,0:44:55.415 Now there are three[br]in the picture. 0:44:55.415,0:45:00.843 I don't know what that means.[br]x plus y plus z equals 1. 0:45:00.843,0:45:05.230 0:45:05.230,0:45:08.830 So if and only if, who's going[br]to tell me what those will be? 0:45:08.830,0:45:10.000 Are they going to be hard? 0:45:10.000,0:45:11.720 No. 0:45:11.720,0:45:15.680 It's a real-life problem, but[br]it's not a hard problem. f of x 0:45:15.680,0:45:18.560 will be yz equals lambda. 0:45:18.560,0:45:20.390 Who is g sub x? 0:45:20.390,0:45:21.040 STUDENT: 1. 0:45:21.040,0:45:21.791 PROFESSOR: 1. 0:45:21.791,0:45:22.653 Thank god. 0:45:22.653,0:45:24.380 So it's fine. 0:45:24.380,0:45:26.370 It's not that. 0:45:26.370,0:45:30.430 F sub y would be[br]xz equals lambda. 0:45:30.430,0:45:34.380 f sub z is xy equals lambda. 0:45:34.380,0:45:39.250 Ah, there is a lot[br]of symmetry in that. 0:45:39.250,0:45:41.330 I have some thinking to do. 0:45:41.330,0:45:43.040 Well, I'm a scientist. 0:45:43.040,0:45:46.720 I have to take into account[br]all the possibilities. 0:45:46.720,0:45:50.140 If I lose one, I'm dead[br]meat, because that one 0:45:50.140,0:45:52.050 may be essential. 0:45:52.050,0:45:55.525 So if I were a computer,[br]I would branch out 0:45:55.525,0:45:58.850 all the possibilities[br]in a certain order. 0:45:58.850,0:45:59.970 But I'm not a computer. 0:45:59.970,0:46:03.734 But I have to think in[br]the same organized way 0:46:03.734,0:46:09.200 to exhaust all the[br]possibilities for that. 0:46:09.200,0:46:12.700 And for that matter, I[br]have to pay attention. 0:46:12.700,0:46:17.438 So I have x plus[br]y plus z equals 1. 0:46:17.438,0:46:18.920 OK. 0:46:18.920,0:46:22.030 I'm going to give you[br]about-- we have time? 0:46:22.030,0:46:22.530 Yes. 0:46:22.530,0:46:24.325 I'll give you two[br]minutes to think 0:46:24.325,0:46:28.266 how to solve-- how[br]does one solve that? 0:46:28.266,0:46:33.020 How does one solve it? 0:46:33.020,0:46:34.830 Think how you would grab. 0:46:34.830,0:46:36.700 Where would you grab[br]the problem from? 0:46:36.700,0:46:40.180 But think it for yourself, and[br]then I'm gone for two minutes. 0:46:40.180,0:46:42.640 And then I'm going to[br]discuss things out loud, 0:46:42.640,0:46:47.174 and I'll share with[br]you how I did it. 0:46:47.174,0:46:51.390 STUDENT: It could[br]be 1, 1 minus 1. 0:46:51.390,0:46:53.015 PROFESSOR: You are[br]like an engineer. 0:46:53.015,0:46:59.780 You already see, oh, maybe[br]I could have some equality 0:46:59.780,0:47:02.090 between the coordinate. 0:47:02.090,0:47:05.155 We have to do it in[br]a mathematical way. 0:47:05.155,0:47:06.040 All right? 0:47:06.040,0:47:11.940 So would it help me if I[br]subtracted the second equation 0:47:11.940,0:47:13.200 from the first equation? 0:47:13.200,0:47:15.190 What kind of[br]information would I get? 0:47:15.190,0:47:17.150 STUDENT: But that[br]can be your ratio. 0:47:17.150,0:47:18.400 STUDENT: We can divide better. 0:47:18.400,0:47:20.200 0:47:20.200,0:47:21.280 PROFESSOR: I can divide. 0:47:21.280,0:47:22.930 That's another possibility. 0:47:22.930,0:47:28.380 I can divide and[br]do y/x equals 1. 0:47:28.380,0:47:32.780 And that would give[br]you x equals y. 0:47:32.780,0:47:34.450 And then you plug it back. 0:47:34.450,0:47:36.290 And then you say, wait a minute. 0:47:36.290,0:47:40.200 If x equals y, then[br]x times x is lambda. 0:47:40.200,0:47:43.560 So lambda would be x squared. 0:47:43.560,0:47:45.650 So then we plug it in here. 0:47:45.650,0:47:50.220 And we go, x plus x equals 2x. 0:47:50.220,0:47:54.358 And then we see what else we[br]can find that information. 0:47:54.358,0:47:57.820 As you can see, there is no[br]unique way of doing that. 0:47:57.820,0:48:00.000 But what's unique[br]should be our answer. 0:48:00.000,0:48:06.030 No matter how I do it, I should[br]overlap with Nitish's method. 0:48:06.030,0:48:08.990 At some point, I should[br]get the possibility 0:48:08.990,0:48:10.465 that x and y are the same. 0:48:10.465,0:48:13.210 If I don't, that means[br]I'm doing something wrong. 0:48:13.210,0:48:18.220 So the way I approach this[br]problem-- OK, one observation, 0:48:18.220,0:48:21.090 I could subtract the second[br]from the first, where 0:48:21.090,0:48:23.170 I would subtract the[br]third from the second. 0:48:23.170,0:48:26.960 Or I could subtract the[br]second from the first 0:48:26.960,0:48:29.360 and analyze all[br]the possibilities. 0:48:29.360,0:48:32.700 Let's do only one[br]and then by symmetry, 0:48:32.700,0:48:34.720 because this is a[br]symmetric problem. 0:48:34.720,0:48:37.470 By symmetry, I'm going to[br]see all the other problems. 0:48:37.470,0:48:41.100 So how do you think in symmetry? 0:48:41.100,0:48:45.960 x and y and z have-- it's a[br]democratic world for them. 0:48:45.960,0:48:48.270 They have the same roles. 0:48:48.270,0:48:51.380 So at some point when you[br]got some solutions for x, y, 0:48:51.380,0:48:55.072 z in a certain way,[br]you may swap them. 0:48:55.072,0:48:58.130 You may change the[br]rules of x, y, z, 0:48:58.130,0:49:00.286 and get all the solutions. 0:49:00.286,0:49:07.947 So the way I did it was I took[br]first xz minus yz equals 0. 0:49:07.947,0:49:11.426 0:49:11.426,0:49:16.120 But then let's interpret what[br]this-- and a mathematician 0:49:16.120,0:49:19.860 will go either by if and[br]only-- if/or it implies. 0:49:19.860,0:49:23.472 I don't know if[br]anybody taught you. 0:49:23.472,0:49:25.695 Depends where[br]you're coming from, 0:49:25.695,0:49:27.800 because different[br]schools, different states, 0:49:27.800,0:49:29.920 different customs[br]for this differently. 0:49:29.920,0:49:34.950 But in professional mathematics,[br]one should go with if and only 0:49:34.950,0:49:40.960 if, or implication,[br]x minus yz equals 0. 0:49:40.960,0:49:43.710 0:49:43.710,0:49:45.940 And then what[br]implication do I have? 0:49:45.940,0:49:48.410 Now I don't have an implication. 0:49:48.410,0:49:53.710 I have it in the sense[br]that I have either/or. 0:49:53.710,0:49:58.790 So this will go, like in[br]computer science, either/or. 0:49:58.790,0:50:09.765 Either-- I do the branching--[br]x equals y, or z equals 0. 0:50:09.765,0:50:14.900 And I have to study[br]these cases separately. 0:50:14.900,0:50:17.140 You see? 0:50:17.140,0:50:19.380 It's not so obvious. 0:50:19.380,0:50:22.000 Let me take this one, because[br]it's closer in my area 0:50:22.000,0:50:22.840 on [INAUDIBLE]. 0:50:22.840,0:50:26.330 It doesn't matter in[br]which order I start. 0:50:26.330,0:50:29.910 For z equals 0, if I plug in[br]z equals 0, what do I get? 0:50:29.910,0:50:31.500 Lambda equals 0, right? 0:50:31.500,0:50:34.770 0:50:34.770,0:50:38.470 But if lambda equals 0, I[br]get another ramification. 0:50:38.470,0:50:41.120 So you are going to say,[br]oh, I'm getting a headache. 0:50:41.120,0:50:42.080 Not yet. 0:50:42.080,0:50:48.880 So lambda equals 0 will again[br]lead you to two possibilities. 0:50:48.880,0:50:52.980 Either x equals 0 or-- 0:50:52.980,0:50:53.813 STUDENT: y equals 0. 0:50:53.813,0:50:55.196 PROFESSOR: --y equals 0. 0:50:55.196,0:50:57.970 0:50:57.970,0:51:01.470 Let's take the first one. 0:51:01.470,0:51:03.790 Like a computer,[br]just like a computer, 0:51:03.790,0:51:06.905 computer will say,[br]if-- so I'm here. 0:51:06.905,0:51:11.880 If 0 is 0 and x was[br]0, what would y be? 0:51:11.880,0:51:14.220 y will be 1. 0:51:14.220,0:51:16.230 That is the only case I got. 0:51:16.230,0:51:19.580 And I make a smile, because[br]why do I make a smile now? 0:51:19.580,0:51:25.070 Because I got all three of[br]them, and I can start my table 0:51:25.070,0:51:27.450 that's a pink table. 0:51:27.450,0:51:34.290 And here I have x, y,[br]z significant values. 0:51:34.290,0:51:36.940 Everything else doesn't matter. 0:51:36.940,0:51:41.650 And this is z, which was the[br]volume, which was x, y, z. 0:51:41.650,0:51:42.800 Was it, guys? 0:51:42.800,0:51:48.460 So I have to compare volumes[br]for this thinking box. 0:51:48.460,0:51:49.440 Right? 0:51:49.440,0:51:50.420 OK. 0:51:50.420,0:51:54.653 Now, in this case, I have 0, x. 0:51:54.653,0:51:56.800 y is 1. 0:51:56.800,0:51:58.470 z is 0. 0:51:58.470,0:52:02.980 The volume will be--[br]and do I have a box? 0:52:02.980,0:52:04.510 No, I don't have a box. 0:52:04.510,0:52:07.140 I make a face like that. 0:52:07.140,0:52:09.230 But the value is[br]still there to put. 0:52:09.230,0:52:12.496 As a mathematician, I[br]have to record everything. 0:52:12.496,0:52:14.370 STUDENT: Do you have to[br]put this on the exam? 0:52:14.370,0:52:16.010 Because it doesn't make sense. 0:52:16.010,0:52:16.900 This would not-- 0:52:16.900,0:52:17.150 PROFESSOR: No. 0:52:17.150,0:52:17.650 No. 0:52:17.650,0:52:22.945 Because I haven't said,[br]if the box cannot be used. 0:52:22.945,0:52:25.000 I didn't say I[br]would use it or not. 0:52:25.000,0:52:28.545 So the volume 0 is a possible[br]value for the function. 0:52:28.545,0:52:31.330 And that will give[br]us the minimum. 0:52:31.330,0:52:33.745 So what do we-- I expect[br]you to say in the exam, 0:52:33.745,0:52:37.320 I have the absolute minimum. 0:52:37.320,0:52:39.880 One of the points-- I'm[br]going to have more points 0:52:39.880,0:52:42.711 when I have minima. 0:52:42.711,0:52:43.210 OK. 0:52:43.210,0:52:44.230 And the other case. 0:52:44.230,0:52:46.290 I don't want to get[br]distracted. y is 0. 0:52:46.290,0:52:49.291 So I get x equals 1. 0:52:49.291,0:52:50.740 Are you guys with me? 0:52:50.740,0:52:53.190 From here and here[br]and here, I get 0:52:53.190,0:52:56.840 x equals 1, because the sum[br]of all three of them will be, 0:52:56.840,0:52:57.740 again, 1. 0:52:57.740,0:52:58.760 So I have another pair. 0:52:58.760,0:53:00.610 0, 0. 0:53:00.610,0:53:03.490 STUDENT: Wouldn't it be 1, 0? 0:53:03.490,0:53:05.990 PROFESSOR: 1, 0. 0:53:05.990,0:53:08.600 1, 0, 0. 0:53:08.600,0:53:11.580 And the volume will be the same. 0:53:11.580,0:53:15.060 And another absolute minima. 0:53:15.060,0:53:18.770 Remember that everything[br]is positive-- the x, y, z, 0:53:18.770,0:53:20.030 and the [INAUDIBLE]. 0:53:20.030,0:53:22.780 I keep going. 0:53:22.780,0:53:28.263 And I say, how do I get-- I[br]have the feeling I'm going 0:53:28.263,0:53:29.830 to get 0, 0, 1 at some point. 0:53:29.830,0:53:32.860 But how am I going[br]to get this thing? 0:53:32.860,0:53:34.260 I'm going to get[br]to it naturally. 0:53:34.260,0:53:35.655 So I should never anticipate. 0:53:35.655,0:53:38.510 0:53:38.510,0:53:42.490 The other case[br]will give it to me. 0:53:42.490,0:53:43.010 OK? 0:53:43.010,0:53:44.250 So let's see. 0:53:44.250,0:53:48.160 When x equals y, I[br]didn't say anything. 0:53:48.160,0:53:52.290 When x equals y, I have[br]to see what happens. 0:53:52.290,0:53:56.660 And I got here again two cases. 0:53:56.660,0:54:01.665 Either x equals-- either,[br]Magdalena, either x 0:54:01.665,0:54:09.226 equals y equals 0, or x[br]equals y equals non-zero. 0:54:09.226,0:54:10.480 So I'm a robot. 0:54:10.480,0:54:11.520 I'm an android. 0:54:11.520,0:54:18.030 I don't let any logical[br]piece escape me. 0:54:18.030,0:54:21.740 Everything goes in[br]the right place. 0:54:21.740,0:54:25.357 When x equals y equals[br]0, the only possibility I 0:54:25.357,0:54:28.640 have is z to the 1. 0:54:28.640,0:54:29.870 And I make another face. 0:54:29.870,0:54:30.740 I'm why? 0:54:30.740,0:54:32.110 Happy that I'm at the end. 0:54:32.110,0:54:35.010 But then I realize[br]that it is, of course, 0:54:35.010,0:54:37.780 not what I hoped for. 0:54:37.780,0:54:40.320 It's another minimum. 0:54:40.320,0:54:43.320 So I have minima[br]0 for the volume 0:54:43.320,0:54:47.900 attained at all these three[br]possibilities, all the three 0:54:47.900,0:54:48.400 points. 0:54:48.400,0:54:52.567 0:54:52.567,0:54:53.150 And then what? 0:54:53.150,0:54:56.280 Then finally something[br]more interesting. 0:54:56.280,0:54:57.522 Finally. 0:54:57.522,0:55:00.198 x equals y different from 0. 0:55:00.198,0:55:02.495 What am I doing to[br]do with that case? 0:55:02.495,0:55:04.780 Of course, you can[br]do this in many ways. 0:55:04.780,0:55:13.910 But if you want to know what I[br]did, just don't laugh too hard. 0:55:13.910,0:55:16.450 I said, look, I'm[br]changing everything 0:55:16.450,0:55:17.800 in the original thing. 0:55:17.800,0:55:22.580 I'll take it aside, and[br]I'll plug in and see 0:55:22.580,0:55:25.370 what the system becomes. 0:55:25.370,0:55:29.590 So we'll assume x equals[br]y different from 0, 0:55:29.590,0:55:31.250 and plug it back in the system. 0:55:31.250,0:55:34.680 In that case, xy equals[br]lambda will become x squared 0:55:34.680,0:55:38.300 equals lambda, right? 0:55:38.300,0:55:41.240 Mr. x plus y plus[br]z equals 1 would 0:55:41.240,0:55:45.900 become x plus x plus z, which[br]is 2x plus z, which is 1. 0:55:45.900,0:55:49.420 0:55:49.420,0:55:52.510 And finally, these[br]two equations, 0:55:52.510,0:55:56.575 since x equals y are one and[br]the same, they become one. 0:55:56.575,0:55:59.310 xz equals lambda. 0:55:59.310,0:56:00.980 And I stare at this guy. 0:56:00.980,0:56:04.640 And somebody tell[br]me, can I solve that? 0:56:04.640,0:56:06.890 Well, it's a system,[br]not a linear system. 0:56:06.890,0:56:11.250 But it's a system[br]of three variables. 0:56:11.250,0:56:14.725 Three equations-- I'm sorry--[br]with three unknowns-- x, z, 0:56:14.725,0:56:15.280 and lambda. 0:56:15.280,0:56:20.310 So it should be easy[br]for me to solve it. 0:56:20.310,0:56:22.370 How did I solve it? 0:56:22.370,0:56:25.660 I got-- it's a little bit funny. 0:56:25.660,0:56:28.760 I got x equals lambda over z. 0:56:28.760,0:56:33.272 And then I went-- but let[br]me square the whole thing. 0:56:33.272,0:56:35.930 And I'm going to get--[br]why do I square it? 0:56:35.930,0:56:39.230 Because I want to compare[br]it to what I have here. 0:56:39.230,0:56:42.305 If I compare, I go, if[br]and only if x squared 0:56:42.305,0:56:47.590 equals lambda squared[br]over z squared. 0:56:47.590,0:56:51.670 But Mr. x squared is[br]known as being lambda. 0:56:51.670,0:56:54.890 So I will replace[br]him. x squared is 0:56:54.890,0:56:56.390 lambda from the first equation. 0:56:56.390,0:57:02.850 So I get lambda equals lambda[br]squared over z squared. 0:57:02.850,0:57:07.350 So I got that-- what did I get? 0:57:07.350,0:57:08.260 Nitish, tell me. 0:57:08.260,0:57:09.730 Lambda equals? 0:57:09.730,0:57:10.522 STUDENT: x squared. 0:57:10.522,0:57:11.396 PROFESSOR: z squared. 0:57:11.396,0:57:13.050 STUDENT: Lambda is[br]equal to z squared. 0:57:13.050,0:57:19.120 PROFESSOR: So if and only[br]if lambda equals z squared. 0:57:19.120,0:57:22.790 But lambda was x[br]squared as well. 0:57:22.790,0:57:24.260 So lambda was what? 0:57:24.260,0:57:29.292 Lambda was z squared,[br]and lambda was x squared. 0:57:29.292,0:57:34.310 And it implies that x equals z. 0:57:34.310,0:57:36.150 x is equal to z. 0:57:36.150,0:57:38.010 But it's also equal to y. 0:57:38.010,0:57:39.600 Alex jump on me. 0:57:39.600,0:57:41.029 Why would that be? 0:57:41.029,0:57:42.570 STUDENT: Because[br]you just said that-- 0:57:42.570,0:57:44.840 PROFESSOR: Because x was[br]y from the assumption. 0:57:44.840,0:57:46.110 So equal to y. 0:57:46.110,0:57:49.770 So this is the beautiful thing,[br]where all the three dimensions 0:57:49.770,0:57:50.520 are the same. 0:57:50.520,0:57:54.980 0:57:54.980,0:57:59.442 So what do we know that[br]thingie-- x equals z equals y? 0:57:59.442,0:58:00.530 STUDENT: It's a box. 0:58:00.530,0:58:01.910 PROFESSOR: It's a box of a what? 0:58:01.910,0:58:02.780 STUDENT: It's a square. 0:58:02.780,0:58:03.070 STUDENT: Square. 0:58:03.070,0:58:03.410 PROFESSOR: It's a-- 0:58:03.410,0:58:04.145 ALL STUDENTS: Cube. 0:58:04.145,0:58:04.811 PROFESSOR: Cube. 0:58:04.811,0:58:05.401 OK. 0:58:05.401,0:58:06.343 So for the cube-- 0:58:06.343,0:58:07.285 STUDENT: Square box. 0:58:07.285,0:58:09.660 PROFESSOR: We get--[br]for z, they were 0:58:09.660,0:58:12.740 stingy about the[br]dimensions we can have. 0:58:12.740,0:58:17.490 So they said, x plus y plus[br]z should be, at most, 1. 0:58:17.490,0:58:23.190 But we managed to maximize[br]the volume by the cube. 0:58:23.190,0:58:27.300 The cube is the only one[br]that maximizes the volume. 0:58:27.300,0:58:29.020 How do I get it back? 0:58:29.020,0:58:34.310 So I get it back by saying,[br]x plus y plus z equals 1. 0:58:34.310,0:58:38.860 So the only possibility that[br]comes out from here is that-- 0:58:38.860,0:58:39.940 STUDENT: They're all 1/3. 0:58:39.940,0:58:45.050 PROFESSOR: That I[br]have 1/3, 1/3, 1/3. 0:58:45.050,0:58:47.530 And I have to take[br]this significant point. 0:58:47.530,0:58:51.260 This is the significant[br]point that I was praying for. 0:58:51.260,0:58:54.592 And the volume will be 1/27. 0:58:54.592,0:58:57.780 0:58:57.780,0:58:58.710 And I'm happy. 0:58:58.710,0:59:00.910 Why am I so happy? 0:59:00.910,0:59:04.400 Is this 1/27 the best I can get? 0:59:04.400,0:59:06.540 In this case, yes. 0:59:06.540,0:59:10.050 So I have the maximum. 0:59:10.050,0:59:11.912 Now, assume that[br]somebody would have-- 0:59:11.912,0:59:12.834 STUDENT: That's a[br]really small box. 0:59:12.834,0:59:14.110 PROFESSOR: It's a small box. 0:59:14.110,0:59:14.700 Exactly. 0:59:14.700,0:59:17.790 I'm switching to[br]something, so assume-- 0:59:17.790,0:59:22.390 I don't know why airlines[br]do that, but they do. 0:59:22.390,0:59:27.060 They say, the girth plus[br]the height will be this. 0:59:27.060,0:59:31.004 Girth meaning-- the[br]girth would be-- 0:59:31.004,0:59:35.090 so this is the height[br]of your-- can I get? 0:59:35.090,0:59:35.739 Or this one? 0:59:35.739,0:59:36.447 No, that's yours. 0:59:36.447,0:59:37.878 Oh. 0:59:37.878,0:59:38.593 It's heavy. 0:59:38.593,0:59:40.720 You shouldn't make[br]me carry this. 0:59:40.720,0:59:41.220 OK. 0:59:41.220,0:59:46.050 So x plus y plus x[br]plus y is the girth. 0:59:46.050,0:59:48.006 And some airlines[br]are really weird. 0:59:48.006,0:59:50.370 I've dealt with at least[br]12 different airlines. 0:59:50.370,0:59:55.210 And the low-cost airlines that[br]I've dealt with in Europe, 0:59:55.210,0:59:56.670 they don't tell you what. 0:59:56.670,0:59:59.520 They say, maximum, 10-kilo max. 0:59:59.520,1:00:00.680 That's about 20 pounds. 1:00:00.680,1:00:05.530 And the girth plus the length[br]has to be a certain thing. 1:00:05.530,1:00:08.720 And others say just the--[br]some of the three dimensions 1:00:08.720,1:00:11.324 should be something like that. 1:00:11.324,1:00:13.950 Whatever they give you. 1:00:13.950,1:00:17.930 So I know you don't think[br]in centimeters usually. 1:00:17.930,1:00:20.435 But imagine that[br]somebody gives you 1:00:20.435,1:00:23.110 the sum of the three[br]dimensions of your check-in bag 1:00:23.110,1:00:27.290 would be 100. 1:00:27.290,1:00:30.270 That is horrible. 1:00:30.270,1:00:34.790 What would be the maximum[br]volume in that case? 1:00:34.790,1:00:36.790 STUDENT: It would all be[br]33 and 1/3 centimeters. 1:00:36.790,1:00:37.415 PROFESSOR: Huh? 1:00:37.415,1:00:39.417 STUDENT: It would all be[br]33 and 1/3 centimeters. 1:00:39.417,1:00:40.250 PROFESSOR: You mean? 1:00:40.250,1:00:44.630 STUDENT: They'll all be 33 and[br]1/3 centimeters, x, y, and z. 1:00:44.630,1:00:47.522 PROFESSOR: Not the sum. x[br]plus y plus z would be 100. 1:00:47.522,1:00:48.730 STUDENT: So each one of them? 1:00:48.730,1:00:52.190 PROFESSOR: And then you[br]have 33.33 whatever. 1:00:52.190,1:00:56.080 And then you cube that,[br]and you get the volume. 1:00:56.080,1:00:58.188 Now, would that be practical? 1:00:58.188,1:00:59.389 STUDENT: No. 1:00:59.389,1:01:00.180 PROFESSOR: Why not? 1:01:00.180,1:01:03.666 1:01:03.666,1:01:05.160 STUDENT: It doesn't fit. 1:01:05.160,1:01:07.970 PROFESSOR: It doesn't the[br]head bin and whatever. 1:01:07.970,1:01:10.912 So we try to-- because[br]the head bin is already 1:01:10.912,1:01:14.410 sort of flattened out, we[br]have the flattened ones. 1:01:14.410,1:01:17.360 But in any case,[br]it's a hassle just 1:01:17.360,1:01:21.480 having to deal with[br]this kind of constraint. 1:01:21.480,1:01:24.440 And when you come back[br]to the United States, 1:01:24.440,1:01:27.570 you really feel-- I don't know[br]if you have this experience. 1:01:27.570,1:01:30.145 The problem is not in[br]between continents. 1:01:30.145,1:01:32.770 1:01:32.770,1:01:36.230 You have plenty of-- you[br]can check in a baggage. 1:01:36.230,1:01:39.390 But if you don't, which I[br]don't, because I'm really weird. 1:01:39.390,1:01:42.480 I get a big carry-on,[br]and I can fit that. 1:01:42.480,1:01:43.890 And I'm very happy. 1:01:43.890,1:01:48.050 I have everything I need for[br]three weeks to one month. 1:01:48.050,1:01:53.170 But if you deal with low-cost[br]airlines, on that kind of 70 1:01:53.170,1:01:58.775 euro or something between London[br]and Milan, or Paris, or London 1:01:58.775,1:02:04.870 and Athens, or something,[br]and you pay that little, they 1:02:04.870,1:02:07.360 have all sorts of weird[br]constraints like this one. 1:02:07.360,1:02:10.730 x plus y plus z has to[br]be no more than that. 1:02:10.730,1:02:13.580 And the weight should be[br]no more than 20 pounds. 1:02:13.580,1:02:15.840 And I'll see how[br]you deal with that. 1:02:15.840,1:02:16.630 It's not easy. 1:02:16.630,1:02:20.296 So yes, we complain about[br]American airlines all the time, 1:02:20.296,1:02:25.140 but compared to those airlines,[br]we are really spoiled. 1:02:25.140,1:02:27.170 In the ticket price,[br]we are paying, 1:02:27.170,1:02:32.810 let's say, $300 from[br]here to Memphis, 1:02:32.810,1:02:38.800 we have a lot of goodies[br]includes that we may not always 1:02:38.800,1:02:40.450 appreciate. 1:02:40.450,1:02:42.510 I'm not working for[br]American Airlines. 1:02:42.510,1:02:46.570 Actually, I prefer[br]Southwest a lot 1:02:46.570,1:02:50.150 by the way they treat[br]us customers and so on. 1:02:50.150,1:02:52.890 But I'm saying,[br]think of restrictions 1:02:52.890,1:02:57.280 when it comes to[br]volume and weight, 1:02:57.280,1:03:00.540 because they represent something[br]in real-life applications. 1:03:00.540,1:03:01.480 Yes, sir. 1:03:01.480,1:03:03.730 STUDENT: I have a question[br]about the previous problem. 1:03:03.730,1:03:05.905 I found the 1/3 just[br]by finding the ratio-- 1:03:05.905,1:03:08.510 the first and the second and[br]then the second and the third. 1:03:08.510,1:03:09.050 PROFESSOR: That's how Nitish-- 1:03:09.050,1:03:10.136 STUDENT: Yeah, that's[br]how I did it as well. 1:03:10.136,1:03:11.432 PROFESSOR: You were[br]napping a little bit. 1:03:11.432,1:03:11.931 But yeah. 1:03:11.931,1:03:13.052 But then you woke up. 1:03:13.052,1:03:15.482 [LAUGHTER] 1:03:15.482,1:03:18.390 While you were napping, he goes,[br]divide by the first equation 1:03:18.390,1:03:20.430 by the second one,[br]and you get 1. 1:03:20.430,1:03:21.775 x/y is 1. 1:03:21.775,1:03:26.545 And so you get the solution[br]of having all of them equal, 1:03:26.545,1:03:27.057 all three. 1:03:27.057,1:03:27.640 STUDENT: Yeah. 1:03:27.640,1:03:29.196 Just because I did that[br]out, and then I was like, 1:03:29.196,1:03:31.738 oh, it's y is equal to x,[br]y is equal to z, and then 1:03:31.738,1:03:32.738 just change it all to y. 1:03:32.738,1:03:33.446 PROFESSOR: Right. 1:03:33.446,1:03:37.310 So how do you think I'm going[br]to proceed about your exams? 1:03:37.310,1:03:38.050 Do I care? 1:03:38.050,1:03:38.740 No. 1:03:38.740,1:03:41.990 As long as you get the right[br]answer, the same answer, 1:03:41.990,1:03:45.170 I don't care which[br]method you were using. 1:03:45.170,1:03:51.044 The problem for me comes where[br]you have had the right idea. 1:03:51.044,1:03:53.640 You messed up in the[br]middle of the algebra, 1:03:53.640,1:03:57.560 and you gave me the[br]wrong algebraic solution. 1:03:57.560,1:04:01.020 That's where I have to ponder[br]how much partial credit 1:04:01.020,1:04:02.965 I want to give you[br]or not give you. 1:04:02.965,1:04:08.740 But I'm trying to be fair,[br]in those cases, to everybody. 1:04:08.740,1:04:11.560 I wanted to tell you-- I[br]don't know if you realize, 1:04:11.560,1:04:17.280 but I stole from you every[br]Tuesday about seven minutes 1:04:17.280,1:04:19.670 from your break. 1:04:19.670,1:04:23.020 It should be a little[br]bit cumulative. 1:04:23.020,1:04:25.240 I'm going to give you[br]back those minutes 1:04:25.240,1:04:28.390 right now, hoping that[br]those seven minutes, 1:04:28.390,1:04:31.890 you're going to use them doing[br]something useful for yourself. 1:04:31.890,1:04:35.050 1:04:35.050,1:04:38.550 At the same time, I'm[br]waiting for your questions 1:04:38.550,1:04:43.400 either now, either here,[br]or in my office upstairs. 1:04:43.400,1:04:46.830 And I know many of you[br]solved a lot of the homework. 1:04:46.830,1:04:48.790 I'm proud of you. 1:04:48.790,1:04:50.260 Some of you did not. 1:04:50.260,1:04:51.730 Some of you still struggle. 1:04:51.730,1:04:54.180 I'm there to help you. 1:04:54.180,1:04:58.270 Is it too early-- I[br]mean, somebody asked me, 1:04:58.270,1:05:02.660 if I read ahead Chapter 12,[br]can I have the homework early? 1:05:02.660,1:05:04.100 Is it too early? 1:05:04.100,1:05:05.540 I don't know what to do. 1:05:05.540,1:05:08.050 I mean, I feel it's[br]too early to give you 1:05:08.050,1:05:11.040 Chapter 12 [INAUDIBLE][br]and Chapter 1:05:11.040,1:05:13.740 12 problems ahead of time. 1:05:13.740,1:05:20.640 But if you feel it's OK, I can[br]send you the homework next. 1:05:20.640,1:05:21.840 Yeah? 1:05:21.840,1:05:22.440 All right. 1:05:22.440,1:05:23.640 Whatever you want. 1:05:23.640,1:05:26.040 We will start[br]Chapter 12 next week. 1:05:26.040,1:05:29.640 So I extended the deadline[br]for Chapter 11 already, 1:05:29.640,1:05:34.740 and I can go ahead and start[br]the homework for Chapter 12 1:05:34.740,1:05:36.240 already. 1:05:36.240,1:05:38.340 And keep it for a month or so. 1:05:38.340,1:05:43.140 I feel that as long as you[br]don't procrastinate, it's OK. 1:05:43.140,1:05:46.790 STUDENT: I solved that one,[br]because I've seen it before.