0:00:00.000,0:00:00.500 0:00:00.500,0:00:03.087 People asked me if I'm[br]going to go over homework. 0:00:03.087,0:00:04.280 Of course I will. 0:00:04.280,0:00:05.300 Let me explain. 0:00:05.300,0:00:08.200 Out of the four[br]hours you have, three 0:00:08.200,0:00:11.090 should be more or[br]less lecture time. 0:00:11.090,0:00:14.470 And the fourth hour, which[br]is the instructor's latitude, 0:00:14.470,0:00:17.764 where they put it-- it's[br]applications, problems, 0:00:17.764,0:00:20.600 homework like problems, all[br]sorts of practice for exams 0:00:20.600,0:00:22.000 and so on. 0:00:22.000,0:00:23.410 It's not a recitation. 0:00:23.410,0:00:31.111 It's some sort of workshop that[br]the instructor conducts himself 0:00:31.111,0:00:33.083 personally. 0:00:33.083,0:00:36.041 All right. 0:00:36.041,0:00:38.834 If you don't have[br]questions, I'm just 0:00:38.834,0:00:42.030 going to go ahead and[br]review a little bit of what 0:00:42.030,0:00:44.580 we discussed last time. 0:00:44.580,0:00:53.190 Something new and exciting[br]was chapter 11, section 11.1. 0:00:53.190,0:00:55.174 And we did 11.2. 0:00:55.174,0:00:57.160 And what was that about? 0:00:57.160,0:00:59.480 That was about functions[br]of several variables. 0:00:59.480,0:01:07.970 0:01:07.970,0:01:10.430 And we discussed[br]several examples, 0:01:10.430,0:01:13.520 but then we focused[br]our attention mainly 0:01:13.520,0:01:19.280 to explicit functions, which[br]means z equals f of x, y, 0:01:19.280,0:01:21.383 of two variables. 0:01:21.383,0:01:25.330 And we call this a graph[br]because it is a graph. 0:01:25.330,0:01:33.350 In 3D, it's a surface whose[br]domain is on the floor. 0:01:33.350,0:01:38.670 And the altitude is z, and[br]that is the-- this is the-- OK. 0:01:38.670,0:01:40.970 How many of you are[br]non-math majors? 0:01:40.970,0:01:43.390 Can you raise hands? 0:01:43.390,0:01:44.500 Oh, OK. 0:01:44.500,0:01:47.340 So you know a little[br]bit about research 0:01:47.340,0:01:49.910 from your own classes,[br]science classes 0:01:49.910,0:01:51.870 or from science[br]fairs from school. 0:01:51.870,0:01:55.990 These are the independent[br]variables, x, y. 0:01:55.990,0:01:58.430 And z is the dependent variable. 0:01:58.430,0:02:01.400 We don't use this kind of[br]terminology in this class. 0:02:01.400,0:02:06.640 But so that you know-- we[br]discussed domain last time. 0:02:06.640,0:02:07.790 This was about what? 0:02:07.790,0:02:10.550 Domain, range. 0:02:10.550,0:02:12.435 After range, what did we do? 0:02:12.435,0:02:14.790 We talked about level curves. 0:02:14.790,0:02:17.616 What is the level curve? 0:02:17.616,0:02:22.220 Level curves are curves x,[br]y in the plane corresponding 0:02:22.220,0:02:24.885 to f of x, y equals constant. 0:02:24.885,0:02:27.555 0:02:27.555,0:02:29.970 These are called[br]level curves in plane, 0:02:29.970,0:02:32.860 in the plane called x, y plane. 0:02:32.860,0:02:36.180 0:02:36.180,0:02:37.850 What else have we discussed? 0:02:37.850,0:02:41.980 We went straight into 11.2. 0:02:41.980,0:02:44.650 In 11.2, we were very[br]happy to remember 0:02:44.650,0:02:49.450 a little bit of Calculus[br]1, which was practically 0:02:49.450,0:02:53.030 a review of limits from Calc 1. 0:02:53.030,0:02:54.420 And what did we do? 0:02:54.420,0:02:59.240 We did epsilon delta, which[br]was not covered in Calculus 1. 0:02:59.240,0:03:01.410 And where is Aaron? 0:03:01.410,0:03:01.910 OK. 0:03:01.910,0:03:04.680 Thank you, Aaron. 0:03:04.680,0:03:07.460 And today, I was thinking,[br]I want to show you actually 0:03:07.460,0:03:12.640 an example that is quite[br]easy of how you use epsilon 0:03:12.640,0:03:20.328 delta for continuity, to show[br]if the function is continuous, 0:03:20.328,0:03:23.744 but for a function[br]of true variables. 0:03:23.744,0:03:25.208 And that's not hard. 0:03:25.208,0:03:26.672 You may think, oh, my god. 0:03:26.672,0:03:27.648 That must be hard. 0:03:27.648,0:03:29.112 That's not hard at all. 0:03:29.112,0:03:32.700 I'm going to move on to the[br]second part of 11.2, which 0:03:32.700,0:03:34.770 is continuity. 0:03:34.770,0:03:38.240 11.2, second part. 0:03:38.240,0:03:39.650 The first part was what? 0:03:39.650,0:03:41.590 It was limits of[br]functions, right, guys? 0:03:41.590,0:03:45.160 We discussed[br]properties of limits, 0:03:45.160,0:03:49.810 algebraic properties of[br]adding sums and taking a limit 0:03:49.810,0:03:53.750 of a sum, taking a limit[br]of a product of functions, 0:03:53.750,0:03:58.500 taking the limit of a quotient[br]of function, when it exists, 0:03:58.500,0:04:00.290 when it doesn't. 0:04:00.290,0:04:06.165 Now the second part of[br]11.2 is called continuity. 0:04:06.165,0:04:08.095 Continuity of what? 0:04:08.095,0:04:09.470 Well, I'm too lazy[br]to right down, 0:04:09.470,0:04:16.450 but it's continuity of functions[br]of two variables, right? 0:04:16.450,0:04:20.170 Now in Calc 1-- you[br]reminded me last time. 0:04:20.170,0:04:21.839 I tried to remind you. 0:04:21.839,0:04:22.990 You tried to remind me. 0:04:22.990,0:04:24.640 Let's remind each other. 0:04:24.640,0:04:27.084 This is like a discussion. 0:04:27.084,0:04:43.010 What was the meaning of f of x[br]being a continuous function x0, 0:04:43.010,0:04:46.640 which is part of the domain? 0:04:46.640,0:04:48.305 x0 has to be in the domain. 0:04:48.305,0:04:55.952 0:04:55.952,0:04:58.045 This is if and only if what? 0:04:58.045,0:04:59.650 Well, what kind of[br]function is that? 0:04:59.650,0:05:01.925 A one variable[br]function, real value. 0:05:01.925,0:05:05.770 It takes values on, let's say,[br]an interval on the real line. 0:05:05.770,0:05:11.058 What was the group[br]of properties that 0:05:11.058,0:05:14.080 have to be[br]simultaneously satisfied, 0:05:14.080,0:05:15.690 satisfied at the same time? 0:05:15.690,0:05:18.350 0:05:18.350,0:05:21.198 And you told me it has[br]to be at the same time. 0:05:21.198,0:05:24.530 And I was very happy because[br]if one of the three conditions 0:05:24.530,0:05:28.254 is missing, then[br]goodbye, continuity. 0:05:28.254,0:05:30.192 One? 0:05:30.192,0:05:31.691 STUDENT: It's defined[br]at that point. 0:05:31.691,0:05:33.655 MAGDALENA TODA: Yes,[br]sir. f of x0 is defined. 0:05:33.655,0:05:36.620 0:05:36.620,0:05:39.850 Actually, I said that[br]here in the domain. 0:05:39.850,0:05:43.070 I'll remove it because[br]now I said it better. 0:05:43.070,0:05:44.943 Two? 0:05:44.943,0:05:46.215 STUDENT: The limit exists. 0:05:46.215,0:05:47.298 MAGDALENA TODA: Very good. 0:05:47.298,0:05:51.650 The limit, as I approach[br]x0 with any kind of value 0:05:51.650,0:05:59.855 closer and closer,[br]exists and is finite. 0:05:59.855,0:06:03.280 Let's give it a name. 0:06:03.280,0:06:09.634 Let's call it L. 0:06:09.634,0:06:11.092 STUDENT:[br][? The following value ?] 0:06:11.092,0:06:12.100 equals the limit. 0:06:12.100,0:06:13.141 MAGDALENA TODA: Yes, sir. 0:06:13.141,0:06:14.085 That's the last thing. 0:06:14.085,0:06:17.490 And I'm glad I didn't[br]have to pull the truth out 0:06:17.490,0:06:18.510 of your mouth. 0:06:18.510,0:06:27.410 So the limit will-- the limit[br]of f of x when x goes to x0 0:06:27.410,0:06:28.890 equals f of x0. 0:06:28.890,0:06:31.870 0:06:31.870,0:06:33.977 No examples. 0:06:33.977,0:06:38.510 You should know[br]Calc 1, and you do. 0:06:38.510,0:06:43.920 I'm just going to[br]move on to Calc 3. 0:06:43.920,0:06:48.500 And let's see what the[br]definition of continuity 0:06:48.500,0:06:52.075 would mean for us in Calc 3. 0:06:52.075,0:07:00.470 Can anybody mimic the properties[br]that-- well, f of x, y 0:07:00.470,0:07:15.262 is said to be[br]continuous at x0, y0 0:07:15.262,0:07:34.320 if and only if the following[br]conditions are-- my arm hurts. 0:07:34.320,0:07:36.055 Are simultaneously satisfied. 0:07:36.055,0:07:48.980 0:07:48.980,0:07:53.240 I don't like professors who[br]use PDF files or slides. 0:07:53.240,0:07:53.740 Shh. 0:07:53.740,0:07:54.240 OK. 0:07:54.240,0:07:56.150 I don't want anything premade. 0:07:56.150,0:08:00.810 The class is a[br]construction, is working, 0:08:00.810,0:08:04.630 is something like[br]a work in progress. 0:08:04.630,0:08:07.360 We are building things together. 0:08:07.360,0:08:08.680 This is teamwork. 0:08:08.680,0:08:11.110 If I come up with[br]some slides that were 0:08:11.110,0:08:13.080 made at home or a PDF file. 0:08:13.080,0:08:14.420 First of all, it means I'm lazy. 0:08:14.420,0:08:17.035 Second of all, it[br]means that I'm not 0:08:17.035,0:08:20.214 willing to take it[br]one step at a time 0:08:20.214,0:08:24.584 and show you how[br]the idea's revealed. 0:08:24.584,0:08:25.084 One. 0:08:25.084,0:08:33.470 0:08:33.470,0:08:34.230 Who is telling me? 0:08:34.230,0:08:35.320 I'm not going to say it. 0:08:35.320,0:08:36.332 It's a work in progress. 0:08:36.332,0:08:39.171 0:08:39.171,0:08:41.070 STUDENT: [INAUDIBLE] 0:08:41.070,0:08:43.014 MAGDALENA TODA: f of-- 0:08:43.014,0:08:44.450 STUDENT: [INAUDIBLE] 0:08:44.450,0:08:48.180 MAGDALENA TODA: Of[br]x0, y0 is defined. 0:08:48.180,0:08:49.856 And why not? 0:08:49.856,0:08:52.140 Well, just to have[br]a silly [? pun ?]. 0:08:52.140,0:08:52.640 Two. 0:08:52.640,0:08:55.260 0:08:55.260,0:09:02.922 Limit as the pair x, y[br]approaches x0, x0-- and guys, 0:09:02.922,0:09:05.840 when you close your eyes--[br]no you close your eyes-- 0:09:05.840,0:09:09.410 and you imagine[br]x, y going to x0, 0:09:09.410,0:09:16.030 y0 by any possible paths[br]in any possible way, 0:09:16.030,0:09:21.570 it's not that you have a[br]predetermined path to x0, y0, 0:09:21.570,0:09:23.650 because you may be trapped. 0:09:23.650,0:09:26.170 You may have-- as you've[br]seen last time, you may have, 0:09:26.170,0:09:28.540 coming from this direction,[br]the limit will exist, 0:09:28.540,0:09:29.990 will be this one. 0:09:29.990,0:09:32.600 Coming from that direction,[br]the limit will exist, 0:09:32.600,0:09:34.600 would be another one. 0:09:34.600,0:09:36.540 And then you don't[br]have overall limits. 0:09:36.540,0:09:41.450 So the limit-- when I call that,[br]that means the overall limit 0:09:41.450,0:09:51.250 exists, exists and[br]equals L. It's finite. 0:09:51.250,0:09:53.430 That's what I mean. 0:09:53.430,0:09:57.880 And three, the value[br]of the function at x0, 0:09:57.880,0:10:04.510 y0 must be equal to the limit[br]of the function that value 0:10:04.510,0:10:08.382 as you approach it, x0, y0. 0:10:08.382,0:10:11.550 And equals L, of course. 0:10:11.550,0:10:12.650 So great. 0:10:12.650,0:10:16.810 So it's so obvious[br]that we are following 0:10:16.810,0:10:19.358 exactly the same[br]type of definition, 0:10:19.358,0:10:22.214 the same type of pattern. 0:10:22.214,0:10:28.660 I'm going to ask you[br]to help me, to help 0:10:28.660,0:10:34.720 me solve a harder problem[br]that involves continuity. 0:10:34.720,0:10:38.558 And I'm asking you, if I[br]have the following function-- 0:10:38.558,0:10:40.950 I'm going to erase the[br]definition of continuity 0:10:40.950,0:10:43.308 from Calc 1. 0:10:43.308,0:10:45.720 I'm going to ask you, what if[br]I have this funny function? 0:10:45.720,0:10:49.082 You've seen it before, and[br]I gave you a little bit 0:10:49.082,0:10:50.384 of a warning about it. 0:10:50.384,0:10:53.320 0:10:53.320,0:11:00.860 Limit as x, y goes[br]to 0, 0 of x squared 0:11:00.860,0:11:07.170 plus y squared times sine of 1[br]over x squared plus y squared. 0:11:07.170,0:11:10.188 0:11:10.188,0:11:11.176 Does that exist? 0:11:11.176,0:11:16.120 0:11:16.120,0:11:16.920 And also-- 0:11:16.920,0:11:20.320 STUDENT: It's actually--[br]so the limit is actually 0:11:20.320,0:11:23.640 approaching a plane rather[br]than a set of [INAUDIBLE]. 0:11:23.640,0:11:26.850 MAGDALENA TODA: So[br]well, actually, it's 0:11:26.850,0:11:28.080 not approaching a plane. 0:11:28.080,0:11:29.800 Let's see what's[br]happening when-- 0:11:29.800,0:11:30.740 STUDENT: Sorry, sorry. 0:11:30.740,0:11:32.417 Not a plane, a [? line. ?] 0:11:32.417,0:11:33.250 MAGDALENA TODA: Yes. 0:11:33.250,0:11:36.037 STUDENT: And is the z-axis--[br]the entire z-axis is 0, 0? 0:11:36.037,0:11:37.620 MAGDALENA TODA: So[br]this is the z-axis. 0:11:37.620,0:11:43.930 And that means exactly that[br]x and y-- it will be 0. 0:11:43.930,0:11:47.030 Now I am just looking[br]at what happens 0:11:47.030,0:11:50.530 in the plane, in the[br]floor plane x, y. 0:11:50.530,0:11:53.730 The pairs x, y are wiggly. 0:11:53.730,0:11:56.220 They are like[br]little wormy worms. 0:11:56.220,0:12:00.680 And they float on the[br]water on the floor. 0:12:00.680,0:12:03.340 And these squiggly[br]things approach 0:12:03.340,0:12:05.870 x, y from any possible path. 0:12:05.870,0:12:06.995 They go like this. 0:12:06.995,0:12:09.040 They go like that. 0:12:09.040,0:12:11.192 They go in every possible way. 0:12:11.192,0:12:12.150 Let's see what happens. 0:12:12.150,0:12:15.050 0:12:15.050,0:12:17.550 Continuity-- is this continuous? 0:12:17.550,0:12:19.840 Well, you say,[br]Magdalena, come on. 0:12:19.840,0:12:21.960 You cannot have this[br]continuous at 0, 0, 0:12:21.960,0:12:24.480 because it's undefined at 0, 0. 0:12:24.480,0:12:24.980 Yes. 0:12:24.980,0:12:27.410 But maybe I can extend[br]it by continuity. 0:12:27.410,0:12:31.670 So let me introduce-- this[br]is my favorite, f of x, y. 0:12:31.670,0:12:35.790 But I'll say, f of x, y[br]is not defined at 0, 0. 0:12:35.790,0:12:46.775 But how about g of x, y as[br]being my f of x, y for any x, 0:12:46.775,0:12:50.380 y different from 0, 0. 0:12:50.380,0:12:55.540 And at the origin, at the[br]very origin, I will say, 0:12:55.540,0:12:59.105 I want to have--[br]when x, y equals 0, 0:12:59.105,0:13:00.430 0, I want to have a value. 0:13:00.430,0:13:05.460 Which value do you[br]think might extend 0:13:05.460,0:13:07.420 this function by continuity? 0:13:07.420,0:13:08.890 STUDENT: The limit. 0:13:08.890,0:13:10.650 MAGDALENA TODA: The[br]limit if it exists 0:13:10.650,0:13:15.960 and if-- well, you know already,[br]I think, what the limit is 0:13:15.960,0:13:18.780 because some of you[br]thought about this at home 0:13:18.780,0:13:20.160 for extra credit. 0:13:20.160,0:13:21.550 So it's not fair, right? 0:13:21.550,0:13:22.640 No, I'm just kidding. 0:13:22.640,0:13:26.690 So I claim that maybe--[br]if I put a 0 here, 0:13:26.690,0:13:28.894 will this be continuous? 0:13:28.894,0:13:31.284 Will g be continuous? 0:13:31.284,0:13:35.600 0:13:35.600,0:13:42.060 So prove, prove either way,[br]prove, justify your answer 0:13:42.060,0:13:45.665 by a proof, a complete[br]proof with epsilon delta. 0:13:45.665,0:13:46.425 Proof. 0:13:46.425,0:13:48.300 OK. 0:13:48.300,0:13:48.800 OK. 0:13:48.800,0:13:51.330 So now is a worried face. 0:13:51.330,0:13:52.660 Like, oh, my god. 0:13:52.660,0:13:54.965 This guy is worried[br]because, oh, my god. 0:13:54.965,0:13:55.786 Epsilon delta. 0:13:55.786,0:13:57.530 Oh, my god. 0:13:57.530,0:13:59.800 But the principle--[br]the intuition 0:13:59.800,0:14:03.920 tells us that we should look[br]first at some sort of a graph, 0:14:03.920,0:14:05.290 just like Ryan pointed out. 0:14:05.290,0:14:09.240 One should close their eyes and[br]imagine a graph of a function 0:14:09.240,0:14:16.800 with-- it's hard to visualize in[br]3D the graph of a function that 0:14:16.800,0:14:18.740 is a surface. 0:14:18.740,0:14:23.790 This is a surface. z[br]equals the whole shebang. 0:14:23.790,0:14:29.650 But when I'm going to look[br]at the one dimensional case 0:14:29.650,0:14:33.680 from last time, we[br]remember the sine of 1/x 0:14:33.680,0:14:35.210 was a crazy function. 0:14:35.210,0:14:39.320 We called it the harmonica,[br]well, 20-something years ago 0:14:39.320,0:14:40.710 when I was in high school. 0:14:40.710,0:14:42.736 I was in an advanced[br]calculus class. 0:14:42.736,0:14:46.170 And our teacher was[br]not funny at all. 0:14:46.170,0:14:49.260 He was also not teaching much,[br]gave us a lot of homework, 0:14:49.260,0:14:50.580 very challenging. 0:14:50.580,0:14:54.432 So in order to make our[br]life a little bit easier, 0:14:54.432,0:14:57.228 we always worked in[br]groups, which was allowed. 0:14:57.228,0:15:00.960 So we called it a harmonica[br]because it was oscillating 0:15:00.960,0:15:02.800 like that to the point[br]that-- you've seen 0:15:02.800,0:15:06.186 the harmonica-- the accordion. 0:15:06.186,0:15:12.940 When you bring it back to[br]the-- harmonica came to my mind 0:15:12.940,0:15:15.720 from the harmonic function. 0:15:15.720,0:15:19.260 So the accordion is--[br]when you actually 0:15:19.260,0:15:25.920 squeeze it, all that oscillation[br]things, the cusps are 0:15:25.920,0:15:28.490 closer and closer to a line. 0:15:28.490,0:15:33.920 So what you have here is[br]this kind of oscillation, 0:15:33.920,0:15:37.630 very, very rapid[br]oscillation for sine of 1/x. 0:15:37.630,0:15:41.806 When we want to multiply by[br]an x, what's going to happen? 0:15:41.806,0:15:47.510 Well, this has not limit at 0[br]because it takes all the values 0:15:47.510,0:15:49.600 infinitesimally close to 0. 0:15:49.600,0:15:52.480 It keeps going through all the[br]values between minus 1 and 1, 0:15:52.480,0:15:53.230 closer and closer. 0:15:53.230,0:15:55.560 So that was no good. 0:15:55.560,0:16:03.887 But if we take this guy--[br]that's going to go to-- well, 0:16:03.887,0:16:05.348 I cannot do better. 0:16:05.348,0:16:07.296 MATLAB can do better than me. 0:16:07.296,0:16:09.244 Mathematica can do better. 0:16:09.244,0:16:10.090 You can do that. 0:16:10.090,0:16:12.600 In most engineering[br]classes, if you are-- 0:16:12.600,0:16:15.690 who is an electrical[br]engineering major? 0:16:15.690,0:16:18.960 But even if you are[br]not, you are going 0:16:18.960,0:16:21.250 to see this type[br]of function a lot. 0:16:21.250,0:16:24.560 And you're going to see it[br]again in differential equations. 0:16:24.560,0:16:27.150 0:16:27.150,0:16:30.745 How can I imagine-- this[br]graph is hard to draw. 0:16:30.745,0:16:34.060 Don't ask me to draw that. 0:16:34.060,0:16:41.060 But ask me if I can use epsilon[br]delta to prove continuity. 0:16:41.060,0:16:44.965 So what would it mean,[br]proving continuity? 0:16:44.965,0:16:45.895 I have a feeling-- 0:16:45.895,0:16:47.290 STUDENT: Well, actually, if this[br]is-- going back to that graph, 0:16:47.290,0:16:49.010 doesn't that graph look like-- 0:16:49.010,0:16:50.301 MAGDALENA TODA: This goes to 0. 0:16:50.301,0:16:53.840 The limit exists for x[br]sine of 1/x, and it is 0. 0:16:53.840,0:16:55.580 Why? 0:16:55.580,0:16:57.170 Ryan? 0:16:57.170,0:17:01.050 RYAN: Wouldn't the graph[br]with the x squared plus 0:17:01.050,0:17:02.900 y squared times that[br]side-- wouldn't that 0:17:02.900,0:17:06.040 just look like a ripple[br]in a circle going out 0:17:06.040,0:17:07.372 from the center? 0:17:07.372,0:17:09.079 MAGDALENA TODA: Yeah,[br]it will be ripples. 0:17:09.079,0:17:10.680 STUDENT: Just like a[br][INAUDIBLE] from an epicenter 0:17:10.680,0:17:11.960 going outwards [INAUDIBLE]. 0:17:11.960,0:17:16.050 MAGDALENA TODA: And I think--[br]yes, we managed to-- you 0:17:16.050,0:17:19.896 have a concentric image, right? 0:17:19.896,0:17:20.480 STUDENT: Yeah. 0:17:20.480,0:17:22.369 MAGDALENA TODA: Like those[br]ripples, exactly like-- 0:17:22.369,0:17:24.036 STUDENT: So that's[br]what that looks like? 0:17:24.036,0:17:26.550 MAGDALENA TODA: --when you[br]throw a stone into the water, 0:17:26.550,0:17:27.680 this kind of wave. 0:17:27.680,0:17:30.730 But it's infinitesimally close. 0:17:30.730,0:17:32.910 It's like acting weird. 0:17:32.910,0:17:37.240 But then it sort[br]of shrinks here. 0:17:37.240,0:17:40.786 And that-- it[br]imposes the limit 0. 0:17:40.786,0:17:43.230 How come this goes[br]to 0, you say? 0:17:43.230,0:17:46.150 Well, Magdalena, this[br]guy is crazy, right? 0:17:46.150,0:17:49.155 Sine of 1/x goes[br]between minus 1 and 1 0:17:49.155,0:17:51.500 infinitely many times[br]as I go close, close, 0:17:51.500,0:17:57.930 closer and closer, more rapidly,[br]more and more rapidly close 0:17:57.930,0:17:58.890 to 0. 0:17:58.890,0:18:00.535 This will oscillate[br]more rapidly, 0:18:00.535,0:18:03.190 more rapidly, and more rapidly. 0:18:03.190,0:18:04.630 This is crazy, right? 0:18:04.630,0:18:07.690 How does this guy, x-- how[br]is this guy taming this guy? 0:18:07.690,0:18:10.445 STUDENT: Because[br]as 0 [INAUDIBLE]. 0:18:10.445,0:18:12.570 Something really small[br]times something [INAUDIBLE]. 0:18:12.570,0:18:14.069 MAGDALENA TODA:[br]Something very small 0:18:14.069,0:18:17.900 that shrinks to 0 times[br]something bounded. 0:18:17.900,0:18:20.880 Ryan brought the main idea. 0:18:20.880,0:18:25.160 If something goes strongly to[br]0, and that multiplies something 0:18:25.160,0:18:28.490 that's bounded, bounded[br]by a finite number, 0:18:28.490,0:18:31.069 the whole problem will go to 0. 0:18:31.069,0:18:32.961 Actually, you can prove[br]that as a theorem. 0:18:32.961,0:18:34.860 And some of you did. 0:18:34.860,0:18:36.630 In most honors[br]classes unfortunately, 0:18:36.630,0:18:39.100 epsilon delta was not covered. 0:18:39.100,0:18:43.350 So let's see how we prove[br]this with epsilon delta. 0:18:43.350,0:18:45.120 And, oh, my god. 0:18:45.120,0:18:52.825 Many of you read from the book[br]and may be able to help me. 0:18:52.825,0:19:00.100 So what am I supposed to[br]show with epsilon delta? 0:19:00.100,0:19:09.860 The limit of x squared plus[br]y squared sine of 1 over x 0:19:09.860,0:19:14.560 squared plus y squared is[br]0 as I approach the origin 0:19:14.560,0:19:19.735 with my pair, couple, x, y,[br]which can go any one path that 0:19:19.735,0:19:20.340 approaches 0. 0:19:20.340,0:19:23.780 0:19:23.780,0:19:27.575 So you say, oh, well, Magdalena,[br]the Ryan principle-- this 0:19:27.575,0:19:29.070 is the Ryan theorem. 0:19:29.070,0:19:32.180 It's the same because[br]this guy will be 0:19:32.180,0:19:34.100 bounded between minus 1 and 1. 0:19:34.100,0:19:37.630 I multiplied with a guy[br]that very determinedly goes 0:19:37.630,0:19:39.540 to 0 very strongly. 0:19:39.540,0:19:41.160 And he knows where he's going. 0:19:41.160,0:19:44.020 x squared plus y squared[br]says, I know what I'm doing. 0:19:44.020,0:19:45.880 I'm not going to change my mind. 0:19:45.880,0:19:49.480 This is like the guy who changes[br]his major too many times. 0:19:49.480,0:19:52.210 And this guy knows[br]what he's doing. 0:19:52.210,0:19:54.920 He's going there, and he's[br]a polynomial, goes to 0, 0:19:54.920,0:19:56.290 0 very rapidly. 0:19:56.290,0:20:00.520 Now it's clear what[br]happens intuitively. 0:20:00.520,0:20:02.910 But I'm a mathematician. 0:20:02.910,0:20:07.040 And if I don't publish[br]my proof, my article 0:20:07.040,0:20:12.675 will be very nicely rejected[br]by all the serious journals 0:20:12.675,0:20:13.970 on the market. 0:20:13.970,0:20:17.470 This is how it goes[br]in mathematics. 0:20:17.470,0:20:19.390 Even before journals[br]existed, mathematicians 0:20:19.390,0:20:23.110 had to show a rigorous[br]proof of their work, 0:20:23.110,0:20:25.720 of their conjecture. 0:20:25.720,0:20:26.920 OK. 0:20:26.920,0:20:35.080 So I go, for every epsilon[br]positive, no matter how small, 0:20:35.080,0:20:41.040 there must exist a[br]delta positive, which 0:20:41.040,0:20:51.990 depends on epsilon-- that[br]depends on epsilon-- such that 0:20:51.990,0:20:58.550 as soon as-- how did[br]we write the distance? 0:20:58.550,0:21:01.820 I'll write the distance[br]again because I'm lazy. 0:21:01.820,0:21:05.590 The distance between the[br]point x, y and the origin 0:21:05.590,0:21:07.931 is less than delta. 0:21:07.931,0:21:16.720 It follows that the[br]absolute value-- 0:21:16.720,0:21:24.330 these are all real numbers--[br]of f of x, y or g of x, 0:21:24.330,0:21:27.456 y-- g of x, y is the extension. 0:21:27.456,0:21:32.150 0:21:32.150,0:21:36.350 f of x, y minus 0, which[br]I claim to be the limit, 0:21:36.350,0:21:39.135 will be less than epsilon. 0:21:39.135,0:21:40.500 So you go, oh, my god. 0:21:40.500,0:21:42.830 What is this woman doing? 0:21:42.830,0:21:43.820 It's not hard. 0:21:43.820,0:21:45.980 I need your help though. 0:21:45.980,0:21:48.540 I need your help to do that. 0:21:48.540,0:21:53.100 So it's hard to see how you[br]should-- you take any epsilon. 0:21:53.100,0:21:58.300 You pick your favorite[br]epsilon, infinitesimally small, 0:21:58.300,0:22:01.120 any small number, but[br]then you go, but then I 0:22:01.120,0:22:03.400 have to show this delta exists. 0:22:03.400,0:22:06.590 You have to grab that delta[br]and say, you are my delta. 0:22:06.590,0:22:08.990 You cannot escape me. 0:22:08.990,0:22:10.926 I tell you who you are. 0:22:10.926,0:22:13.840 And that's the[br]hardest part in here, 0:22:13.840,0:22:18.240 figuring out who that delta must[br]be as a function of epsilon. 0:22:18.240,0:22:19.060 Is that hard? 0:22:19.060,0:22:21.320 How do you build[br]such a construction? 0:22:21.320,0:22:26.690 First of all,[br]understand what proof. 0:22:26.690,0:22:30.385 "Choose any positive epsilon." 0:22:30.385,0:22:32.885 Then forget about him,[br]because he's your friend, 0:22:32.885,0:22:36.400 and he's going to do whatever[br]you want to do with him. 0:22:36.400,0:22:40.120 Delta, chasing[br]after delta is going 0:22:40.120,0:22:42.390 to be a little bit harder. 0:22:42.390,0:22:56.010 "Chasing after delta[br]with that property." 0:22:56.010,0:22:58.430 Dot, dot, dot, dot, dot. 0:22:58.430,0:22:59.700 What is this distance? 0:22:59.700,0:23:01.950 You guys have[br]helped me last time, 0:23:01.950,0:23:04.670 you cannot let me down now. 0:23:04.670,0:23:08.350 So as soon as this distance,[br]your gradient distance 0:23:08.350,0:23:10.540 is less than delta,[br]you must have 0:23:10.540,0:23:13.312 that f of x, y [INAUDIBLE]. 0:23:13.312,0:23:15.170 Could you tell me[br]what that would be? 0:23:15.170,0:23:16.170 It was Euclidean, right? 0:23:16.170,0:23:21.670 So I had squared[br]root of-- did I? 0:23:21.670,0:23:30.396 Square root of x minus 0[br]squared plus y minus 0 squared. 0:23:30.396,0:23:33.170 You say, but that's[br]silly, Magdalena. 0:23:33.170,0:23:37.536 So you have to write[br]it down like that? 0:23:37.536,0:23:38.970 STUDENT: It's the [INAUDIBLE]. 0:23:38.970,0:23:40.410 MAGDALENA TODA: Huh? 0:23:40.410,0:23:42.290 Yeah. 0:23:42.290,0:23:47.174 So square root of this[br]plus square root of that 0:23:47.174,0:23:53.030 plus then delta,[br]that means what? 0:23:53.030,0:24:00.351 If and only if x squared plus[br]y squared is less than delta 0:24:00.351,0:24:00.850 squared. 0:24:00.850,0:24:08.160 0:24:08.160,0:24:11.110 And what do I want to do,[br]what do I want to build? 0:24:11.110,0:24:15.090 0:24:15.090,0:24:19.030 So we are thinking how[br]to set up all this thing. 0:24:19.030,0:24:21.209 How to choose the delta. 0:24:21.209,0:24:23.061 How to choose the delta. 0:24:23.061,0:24:25.830 0:24:25.830,0:24:28.350 OK, so what do I--[br]what am I after? 0:24:28.350,0:24:34.253 "I am after having" double dot. 0:24:34.253,0:24:39.830 F of x, y must be Mr. Ugly. 0:24:39.830,0:24:40.820 This one. 0:24:40.820,0:24:46.480 So absolute value of x squared[br]plus y squared, sine of 1 0:24:46.480,0:24:51.241 over x squared plus[br]y squared minus 0. 0:24:51.241,0:24:51.740 Duh. 0:24:51.740,0:24:55.170 I'm not going to write it. 0:24:55.170,0:24:59.260 We all know what that means. 0:24:59.260,0:25:00.090 Less than epsilon. 0:25:00.090,0:25:05.680 This is what must[br]follow as a conclusion. 0:25:05.680,0:25:12.360 This is what must[br]follow, must happen. 0:25:12.360,0:25:13.334 Must happen. 0:25:13.334,0:25:16.260 0:25:16.260,0:25:17.630 Now I'm getting excited. 0:25:17.630,0:25:18.130 Why? 0:25:18.130,0:25:21.050 Because I am thinking. 0:25:21.050,0:25:23.040 I started thinking. 0:25:23.040,0:25:26.420 Once I started thinking,[br]I'm dangerous, man. 0:25:26.420,0:25:31.530 So here sine of 1 over x squared[br]plus y squared is your friend. 0:25:31.530,0:25:34.210 Why is that your friend? 0:25:34.210,0:25:37.250 Sine of 1 over x squared[br]plus y squared, this 0:25:37.250,0:25:39.115 is always an absolute value. 0:25:39.115,0:25:42.977 The absolute value of that[br]is always less than 1. 0:25:42.977,0:25:43.476 OK? 0:25:43.476,0:25:45.290 STUDENT: Can't it be 4? 0:25:45.290,0:25:50.280 MAGDALENA TODA:[br]So-- so-- so what 0:25:50.280,0:25:54.940 shall I take in terms of[br]delta-- this is my question. 0:25:54.940,0:25:57.400 What shall I take[br]in terms of delta? 0:25:57.400,0:26:03.590 "Delta equals 1 as a[br]function of epsilon 0:26:03.590,0:26:20.160 in order to have the[br]conclusion satisfied." 0:26:20.160,0:26:20.920 You say, OK. 0:26:20.920,0:26:24.610 It's enough to choose delta[br]like that function of epsilon, 0:26:24.610,0:26:28.840 and I'm done, because then[br]everything will be fine. 0:26:28.840,0:26:33.510 So you chose your own epsilon,[br]positive, small, or God 0:26:33.510,0:26:34.480 gave you an epsilon. 0:26:34.480,0:26:37.040 You don't care how[br]you got the epsilon. 0:26:37.040,0:26:38.423 The epsilon is arbitrary. 0:26:38.423,0:26:40.910 You pick positive and small. 0:26:40.910,0:26:44.610 Now, it's up to[br]you to find delta. 0:26:44.610,0:26:48.820 So what delta would[br]satisfy everything? 0:26:48.820,0:26:50.940 What delta would[br]be good enough-- 0:26:50.940,0:26:52.830 you don't care[br]for all the good-- 0:26:52.830,0:26:54.550 it's like when you get married. 0:26:54.550,0:26:57.740 Do you care for all the[br]people who'd match you? 0:26:57.740,0:27:00.596 Hopefully not, because[br]then you would probably 0:27:00.596,0:27:05.416 have too large of a pool,[br]and it's hard to choose. 0:27:05.416,0:27:13.285 You only need one that satisfies[br]that assumption, that satisfies 0:27:13.285,0:27:14.974 all the conditions you have. 0:27:14.974,0:27:18.780 So what is the delta that[br]satisfies all the conditions 0:27:18.780,0:27:20.047 that I have? 0:27:20.047,0:27:20.880 [INTERPOSING VOICES] 0:27:20.880,0:27:22.270 MAGDALENA TODA: [INAUDIBLE]. 0:27:22.270,0:27:22.960 Who? 0:27:22.960,0:27:25.350 [INTERPOSING VOICES] 0:27:25.350,0:27:27.855 MAGDALENA TODA: For example,[br]delta equals epsilon. 0:27:27.855,0:27:28.785 Would that satisfy? 0:27:28.785,0:27:31.984 0:27:31.984,0:27:33.860 Well, let's see. 0:27:33.860,0:27:37.410 If I take delta to[br]be epsilon, then x 0:27:37.410,0:27:40.410 squared plus y squared would[br]be less than epsilon squared. 0:27:40.410,0:27:47.331 Now the question is is epsilon[br]squared less than epsilon? 0:27:47.331,0:27:48.510 Not always. 0:27:48.510,0:27:52.920 If epsilon is between 0[br]and 1, then epsilon squared 0:27:52.920,0:27:54.200 is less then epsilon. 0:27:54.200,0:27:59.200 But if I choose epsilon[br]to be greater than 1, 0:27:59.200,0:28:00.220 then oh, my God. 0:28:00.220,0:28:02.740 Then if it's greater than[br]1, then epsilon squared 0:28:02.740,0:28:06.790 is greater than 1--[br]greater than it. 0:28:06.790,0:28:14.700 So what if I choose[br]delta to be what? 0:28:14.700,0:28:18.652 0:28:18.652,0:28:19.595 STUDENT: 0? 0:28:19.595,0:28:20.720 MAGDALENA TODA: No, no, no. 0:28:20.720,0:28:22.090 Delta cannot be 0. 0:28:22.090,0:28:26.330 So delta-- look, there exists[br]delta strictly bigger than 0, 0:28:26.330,0:28:28.690 that depends on epsilon. 0:28:28.690,0:28:33.673 Maybe if epsilon is very small,[br]in a way Alexander was right. 0:28:33.673,0:28:37.350 But the delta [INAUDIBLE],[br]we don't go with epsilon 0:28:37.350,0:28:38.280 greater than 1. 0:28:38.280,0:28:39.000 Come on. 0:28:39.000,0:28:39.500 Be serious. 0:28:39.500,0:28:42.300 Epsilon is always[br]between 0 and 1. 0:28:42.300,0:28:44.603 I mean, it's a lot[br]smaller than that. 0:28:44.603,0:28:46.640 It's infinitesimal small. 0:28:46.640,0:28:49.480 So in the end, yes, in[br]that case epsilon squared 0:28:49.480,0:28:52.610 would be less than epsilon,[br]which would be OK for us 0:28:52.610,0:28:54.590 and that would be fine. 0:28:54.590,0:28:56.150 OK? 0:28:56.150,0:28:58.380 So that would be a[br]possibility to say, hey, 0:28:58.380,0:29:01.080 since epsilon-- Alexander,[br]if you write that as a proof 0:29:01.080,0:29:01.815 I'll be OK. 0:29:01.815,0:29:04.900 You say, I took my epsilon[br]to be a very small number, 0:29:04.900,0:29:07.020 so anyway it's going[br]to be less than 1. 0:29:07.020,0:29:09.190 So epsilon squared[br]is less than epsilon. 0:29:09.190,0:29:14.090 So when I take[br]delta to be epsilon, 0:29:14.090,0:29:18.210 for sure this guy will be less[br]than epsilon squared, which 0:29:18.210,0:29:21.148 is less than epsilon,[br]so I'm satisfied. 0:29:21.148,0:29:22.615 I'll give you a 100%. 0:29:22.615,0:29:24.082 I'm happy. 0:29:24.082,0:29:25.152 Is that the only way? 0:29:25.152,0:29:26.527 STUDENT: But what[br]about the sine? 0:29:26.527,0:29:27.505 What about [INAUDIBLE]. 0:29:27.505,0:29:28.483 STUDENT: Yeah. 0:29:28.483,0:29:30.108 MAGDALENA TODA: So[br]this doesn't matter. 0:29:30.108,0:29:32.395 Let me write it down. 0:29:32.395,0:29:39.730 So note that x squared[br]plus y squared sine of 1 0:29:39.730,0:29:42.650 over x squared plus[br]y square would always 0:29:42.650,0:29:46.380 be less than absolute[br]value of x squared 0:29:46.380,0:29:49.970 plus y, which is positive. 0:29:49.970,0:29:52.145 Why is that? 0:29:52.145,0:29:53.200 Is this true? 0:29:53.200,0:29:54.080 Yeah. 0:29:54.080,0:29:55.441 Why is that? 0:29:55.441,0:29:58.150 STUDENT: Because the sine can[br]only be one of these negatives. 0:29:58.150,0:30:00.445 MAGDALENA TODA: So[br]in absolute value, 0:30:00.445,0:30:05.780 sine of 1 over x squared plus y[br]squared is always less than 1. 0:30:05.780,0:30:08.525 STUDENT: Can't it equal 1? 0:30:08.525,0:30:11.885 MAGDALENA TODA: Well,[br]when does it equal 1? 0:30:11.885,0:30:14.310 STUDENT: Wouldn't it be x[br]squared plus y squared equals 1 0:30:14.310,0:30:15.647 [INAUDIBLE]? 0:30:15.647,0:30:17.230 MAGDALENA TODA: Less[br]than or equal to. 0:30:17.230,0:30:18.480 For some values it will. 0:30:18.480,0:30:19.160 STUDENT: Yeah. 0:30:19.160,0:30:19.659 OK. 0:30:19.659,0:30:21.870 MAGDALENA TODA: Now, will[br]that be a problem with us? 0:30:21.870,0:30:22.120 No. 0:30:22.120,0:30:23.060 Let's put it here. 0:30:23.060,0:30:27.370 Less than or equal to x[br]squared plus y squared, which 0:30:27.370,0:30:35.465 has to be less than epsilon[br]if and only if-- well, 0:30:35.465,0:30:38.780 if delta is what? 0:30:38.780,0:30:41.240 So, again, Alexander said,[br]well, but if I take delta 0:30:41.240,0:30:42.820 to be epsilon, I'm done. 0:30:42.820,0:30:45.760 0:30:45.760,0:30:46.740 STUDENT: [INAUDIBLE]. 0:30:46.740,0:30:49.690 MAGDALENA TODA: How[br]about square root? 0:30:49.690,0:30:52.150 Can I take delta to be[br]square root of epsilon. 0:30:52.150,0:30:53.541 STUDENT: That's what I said. 0:30:53.541,0:30:54.332 MAGDALENA TODA: No. 0:30:54.332,0:30:55.694 You said epsilon. 0:30:55.694,0:30:57.319 STUDENT: I said square[br]root of epsilon. 0:30:57.319,0:30:58.270 MAGDALENA TODA: OK. 0:30:58.270,0:31:01.290 If delta is square[br]root of epsilon, 0:31:01.290,0:31:05.280 then everything will be perfect[br]and it will be a perfect match. 0:31:05.280,0:31:05.997 In what case? 0:31:05.997,0:31:07.705 STUDENT: If epsilon[br]is in between 0 and 1 0:31:07.705,0:31:10.130 and if delta is equal[br]to bigger than epsilon. 0:31:10.130,0:31:13.060 0:31:13.060,0:31:17.740 MAGDALENA TODA: So that's[br]exactly the same assumption. 0:31:17.740,0:31:22.480 Epsilon should be[br]made in less than. 0:31:22.480,0:31:24.327 STUDENT: But I thought[br]delta was supposed 0:31:24.327,0:31:25.910 to be less than[br]epsilon in every case. 0:31:25.910,0:31:29.340 So if epsilon is between 0 and[br]1, the square root of epsilon 0:31:29.340,0:31:31.800 is going to be [INAUDIBLE]. 0:31:31.800,0:31:38.330 MAGDALENA TODA: So when[br]both of them are small, 0:31:38.330,0:31:45.040 delta squared will be-- if[br]I take delta-- so take delta 0:31:45.040,0:31:47.698 to be square root of epsilon. 0:31:47.698,0:31:50.118 STUDENT: Then anything less[br]than 1 and greater than 0, 0:31:50.118,0:31:51.784 epsilon would be great[br]than [INAUDIBLE]. 0:31:51.784,0:31:54.958 MAGDALENA TODA: "Delta to[br]be square root of epsilon, 0:31:54.958,0:32:01.570 then x squared plus y squared[br]less than delta squared equals 0:32:01.570,0:32:03.780 epsilon." 0:32:03.780,0:32:11.805 Then x squared plus[br]y squared sine of 1 0:32:11.805,0:32:14.990 over x squared plus[br]y squared less than 0:32:14.990,0:32:17.440 or equal to x squared[br]plus y squared. 0:32:17.440,0:32:19.155 I dont' need the absolute value. 0:32:19.155,0:32:20.380 I can [INAUDIBLE]. 0:32:20.380,0:32:23.320 Less than epsilon [INAUDIBLE]. 0:32:23.320,0:32:24.110 Qed. 0:32:24.110,0:32:26.050 STUDENT: Well, but[br]you told us delta 0:32:26.050,0:32:27.505 has to be less than epsilon. 0:32:27.505,0:32:28.475 Well, if-- 0:32:28.475,0:32:31.390 MAGDALENA TODA: No,[br]I didn't say that. 0:32:31.390,0:32:35.325 I didn't say that delta has[br]to be less than epsilon. 0:32:35.325,0:32:35.825 Absolutely-- 0:32:35.825,0:32:36.408 STUDENT: Yeah. 0:32:36.408,0:32:38.759 You said for all the values[br]of epsilon greater than 0, 0:32:38.759,0:32:42.280 there's a value of delta that is[br]greater than 0 that [INAUDIBLE] 0:32:42.280,0:32:45.648 such that as soon as the[br]distance between is less than 0:32:45.648,0:32:46.959 delta-- I don't remember what-- 0:32:46.959,0:32:48.250 MAGDALENA TODA: OK, so, again-- 0:32:48.250,0:32:50.166 STUDENT: Such that the[br]distance is less than-- 0:32:50.166,0:32:52.255 MAGDALENA TODA: So again,[br]for epsilon positive, 0:32:52.255,0:32:56.600 there is a delta[br]positive, very small. 0:32:56.600,0:32:58.740 Very small means very small, OK? 0:32:58.740,0:33:01.440 I'm not threatened by-- what? 0:33:01.440,0:33:04.688 For epsilon greater[br]than 0, very small, 0:33:04.688,0:33:07.128 there is a delta greater[br]than 0, very small, 0:33:07.128,0:33:10.544 which depends on epsilon-- I[br]didn't say it cannot be equal 0:33:10.544,0:33:21.100 to epsilon-- that depends on[br]epsilon such that whenever x, 0:33:21.100,0:33:30.012 y is within delta[br]distance from origin, 0:33:30.012,0:33:45.170 [INAUDIBLE] that f of x, y[br]is within epsilon of from l. 0:33:45.170,0:33:47.840 0:33:47.840,0:33:48.340 All right? 0:33:48.340,0:33:52.720 And now I will actually give[br]you another example where 0:33:52.720,0:33:55.970 maybe delta will be epsilon. 0:33:55.970,0:33:59.430 And let me challenge you[br]with another problem that's 0:33:59.430,0:34:00.530 not hard. 0:34:00.530,0:34:01.295 OK? 0:34:01.295,0:34:03.650 So let me give[br]you the function g 0:34:03.650,0:34:16.460 of x, y equals x sine[br]of 1 over y as x, y. 0:34:16.460,0:34:19.150 0:34:19.150,0:34:29.782 y is equal [? to delta 0. ?][br]And let's say 0 for the rest. 0:34:29.782,0:34:35.500 0:34:35.500,0:34:48.510 Can you show-- can you check[br]if g is continuous at 0, 0? 0:34:48.510,0:34:55.510 0:34:55.510,0:34:58.510 This is one of the[br]problems in your book. 0:34:58.510,0:35:02.300 So how do you check[br]that with epsilon delta? 0:35:02.300,0:35:04.130 Again, we recite the poetry. 0:35:04.130,0:35:05.555 We have to say that. 0:35:05.555,0:35:11.950 "For every epsilon[br]positive, small, very small, 0:35:11.950,0:35:16.000 there is a delta[br]positive that depends 0:35:16.000,0:35:33.830 on epsilon, such that as soon[br]as--" how is the distance? 0:35:33.830,0:35:42.424 Square root of x squared plus[br]y squared is less than delta. 0:35:42.424,0:35:46.861 This is the distance[br]between point and origin. 0:35:46.861,0:36:09.260 "It follows that absolute value[br]of x sine of 1 over y minus--" 0:36:09.260,0:36:12.247 so practically x, y no 0. 0:36:12.247,0:36:16.223 x, y different from 0. 0:36:16.223,0:36:17.720 OK? 0:36:17.720,0:36:21.720 I"m careful here, because[br]if y is 0, then I blow up. 0:36:21.720,0:36:23.080 And I don't want to blow up. 0:36:23.080,0:36:25.850 So x sine of 1 over y minus who? 0:36:25.850,0:36:30.810 Minus 0 is less than epsilon. 0:36:30.810,0:36:32.850 So now you're thinking,[br]OK, you want me 0:36:32.850,0:36:34.760 to prove there is such a delta? 0:36:34.760,0:36:35.830 Yes. 0:36:35.830,0:36:37.060 That depends on epsilon? 0:36:37.060,0:36:38.850 Yes. 0:36:38.850,0:36:40.430 And what would that delta be? 0:36:40.430,0:36:43.710 The simplest choice you[br]can have in this case. 0:36:43.710,0:36:45.080 So you go, oh, my God. 0:36:45.080,0:36:46.070 How do I do that? 0:36:46.070,0:36:48.470 You have to always[br]think backwards. 0:36:48.470,0:36:58.790 So "we need to satisfy[br]absolute value of x sine of 1 0:36:58.790,0:37:02.430 over y less than epsilon." 0:37:02.430,0:37:05.730 Is this hard? 0:37:05.730,0:37:10.295 What is your advantage here? 0:37:10.295,0:37:13.570 Do you have any advantage? 0:37:13.570,0:37:19.860 Remark absolute value[br]of x sine of 1 over y 0:37:19.860,0:37:22.690 is smaller than who? 0:37:22.690,0:37:26.670 Smaller than the product[br]of absolute values. 0:37:26.670,0:37:27.580 Say it again? 0:37:27.580,0:37:28.430 Yes? 0:37:28.430,0:37:32.005 STUDENT: But, like, for[br]example, the only condition 0:37:32.005,0:37:35.290 for that equation is that[br]y must not be equal to 0. 0:37:35.290,0:37:38.520 What if you used[br]another point for x? 0:37:38.520,0:37:43.170 Would the answer for[br]delta be different? 0:37:43.170,0:37:45.240 MAGDALENA TODA:[br]Well, x is-- you can 0:37:45.240,0:37:49.000 choose-- you were right here. 0:37:49.000,0:37:52.990 You can say, OK, can you be[br]more restrictive, Magdelena, 0:37:52.990,0:37:58.710 and say, for every point[br]of the type x equals 0 0:37:58.710,0:38:01.470 and y not 0, it's still OK? 0:38:01.470,0:38:03.490 Yes. 0:38:03.490,0:38:07.000 So you could be a[br]professional mathematician. 0:38:07.000,0:38:14.322 So practically all I care[br]about is x, y in the disk. 0:38:14.322,0:38:15.410 What disk? 0:38:15.410,0:38:16.810 What is this disk? 0:38:16.810,0:38:24.150 Disk of radius 0 when--[br]what is the radius? 0:38:24.150,0:38:31.930 Delta-- such that your[br]y should not be 0. 0:38:31.930,0:38:35.990 So a more rigorous[br]point would be 0:38:35.990,0:38:38.630 like take all the[br]couples that are 0:38:38.630,0:38:43.600 in this small disk[br]of radius delta, 0:38:43.600,0:38:46.020 except for those where y is 0. 0:38:46.020,0:38:48.650 So what do you actually remove? 0:38:48.650,0:38:54.860 You remove this stinking line. 0:38:54.860,0:39:01.080 But everybody else in this[br]disk, every couple in this disk 0:39:01.080,0:39:03.865 should be happy,[br]should be analyzed 0:39:03.865,0:39:06.080 as part of this thread. 0:39:06.080,0:39:08.100 Right? 0:39:08.100,0:39:09.070 OK. 0:39:09.070,0:39:13.180 x sine of 1 over y less[br]than-- is that true? 0:39:13.180,0:39:16.120 Is that less than the[br]absolute value of x? 0:39:16.120,0:39:16.930 STUDENT: Yeah. 0:39:16.930,0:39:17.846 MAGDALENA TODA: Right. 0:39:17.846,0:39:20.950 So it should be-- less[br]than should be made 0:39:20.950,0:39:23.570 should be less than epsilon. 0:39:23.570,0:39:27.040 When is this happening[br]on that occasion? 0:39:27.040,0:39:28.320 If I take delta-- meh? 0:39:28.320,0:39:29.570 STUDENT: When delta's epsilon. 0:39:29.570,0:39:31.310 MAGDALENA TODA: So if[br]I take-- very good. 0:39:31.310,0:39:35.570 So Alex saw that, hey,[br]Magdelena, your proof is over. 0:39:35.570,0:39:37.700 And I mean it's over. 0:39:37.700,0:39:42.990 Take delta, which is delta[br]of epsilon, to be epsilon. 0:39:42.990,0:39:44.350 You're done. 0:39:44.350,0:39:45.520 Why? 0:39:45.520,0:39:47.590 Let me explain what[br]Alex wants, because he 0:39:47.590,0:39:50.280 doesn't want to explain[br]much, but it's not his job. 0:39:50.280,0:39:51.260 He's not your teacher. 0:39:51.260,0:39:51.920 Right? 0:39:51.920,0:39:54.346 So why is this working? 0:39:54.346,0:40:02.850 Because in this case,[br]note that if I take delta 0:40:02.850,0:40:05.650 to be exactly epsilon,[br]what's going to happen? 0:40:05.650,0:40:08.620 0:40:08.620,0:40:13.760 x, Mr. x, could be[br]positive or negative. 0:40:13.760,0:40:15.950 See, x could be[br]positive or negative. 0:40:15.950,0:40:18.810 Let's take this guy and[br]protect him in absolute value. 0:40:18.810,0:40:23.350 He's always less than square[br]root of x square plus y 0:40:23.350,0:40:25.650 squared. 0:40:25.650,0:40:27.200 Why is that, guys? 0:40:27.200,0:40:30.730 STUDENT: Because y can't be 0. 0:40:30.730,0:40:34.482 MAGDALENA TODA: So this[br]is-- square it in your mind. 0:40:34.482,0:40:36.690 You got x squared less than[br]x squared plus y squared. 0:40:36.690,0:40:39.000 So this is always true. 0:40:39.000,0:40:40.640 Always satisfied. 0:40:40.640,0:40:44.940 But we chose this to[br]be less than delta, 0:40:44.940,0:40:49.450 and if we choose delta to be[br]epsilon, that's our choice. 0:40:49.450,0:40:54.310 So God gave us the epsilon,[br]but delta is our choice, 0:40:54.310,0:40:57.090 because you have to prove[br]you can do something 0:40:57.090,0:40:57.910 with your life. 0:40:57.910,0:40:58.410 Right? 0:40:58.410,0:41:00.700 So delta equals epsilon. 0:41:00.700,0:41:02.670 If you take delta[br]equals epsilon, 0:41:02.670,0:41:06.440 then you're done, because[br]in that case absolute value 0:41:06.440,0:41:11.980 of x is less than epsilon, and[br]your conclusion, which is this, 0:41:11.980,0:41:13.690 was satisfied. 0:41:13.690,0:41:16.680 Now, if a student[br]is really smart-- 0:41:16.680,0:41:20.650 one time I had a student,[br]I gave him this proof. 0:41:20.650,0:41:22.380 That was several[br]years ago in honors, 0:41:22.380,0:41:24.620 because we don't do epsilon[br]delta in non-honors. 0:41:24.620,0:41:28.200 And we very rarely do[br]it in honors as well. 0:41:28.200,0:41:31.140 His proof consisted of this. 0:41:31.140,0:41:34.060 Considering the fact that[br]absolute value of sine 0:41:34.060,0:41:38.300 is less than 1, if I[br]take delta to be epsilon, 0:41:38.300,0:41:39.870 that is sufficient. 0:41:39.870,0:41:41.775 I'm done. 0:41:41.775,0:41:44.470 And of course I gave[br]him 100%, because this 0:41:44.470,0:41:46.030 is the essence of the proof. 0:41:46.030,0:41:48.090 He didn't show any details. 0:41:48.090,0:41:52.180 And I thought, this is the[br]kind of guy who is great. 0:41:52.180,0:41:55.920 He's very smart, but he's not[br]going to make a good teacher. 0:41:55.920,0:41:59.380 So he's probably going to[br]be the next researcher, 0:41:59.380,0:42:04.810 the next astronaut, the next[br]something else, but not-- 0:42:04.810,0:42:11.040 And then, years later, he[br]took advanced calculus. 0:42:11.040,0:42:13.635 He graduated with[br]a graduate degree 0:42:13.635,0:42:17.600 in three years sponsored[br]by the Air Force. 0:42:17.600,0:42:20.750 And he works right[br]now for the Air Force. 0:42:20.750,0:42:24.420 He came out dressed[br]as a captain. 0:42:24.420,0:42:28.840 He came and gave a talk this[br]year at Tech in a conference-- 0:42:28.840,0:42:29.770 he was rushed. 0:42:29.770,0:42:32.150 I mean, if I talk[br]like that, my student 0:42:32.150,0:42:33.810 wouldn't be able to follow me. 0:42:33.810,0:42:38.320 But he was the same brilliant[br]student that I remember. 0:42:38.320,0:42:46.280 So he's working on some very[br]important top secret projects. 0:42:46.280,0:42:48.900 Very intelligent guy. 0:42:48.900,0:42:52.520 And every now and than going[br]to give talks at conferences. 0:42:52.520,0:42:58.170 Like, research talks[br]about what he's doing. 0:42:58.170,0:43:01.958 In his class-- he took[br]advanced calculus with me, 0:43:01.958,0:43:04.090 which was actually graduate[br]level [INAUDIBLE]-- 0:43:04.090,0:43:09.180 I explained epsilon delta, and[br]he had it very well understood. 0:43:09.180,0:43:13.270 And after I left the classroom[br]he explained it to his peers, 0:43:13.270,0:43:15.050 to his classmates. 0:43:15.050,0:43:16.885 And he explained[br]it better than me. 0:43:16.885,0:43:21.010 And I was there listening,[br]and I remember being jealous, 0:43:21.010,0:43:23.050 because although[br]he was very rushed, 0:43:23.050,0:43:27.430 he had a very clear[br]understanding of how 0:43:27.430,0:43:31.140 you take an epsilon, no[br]matter how small, and then 0:43:31.140,0:43:34.200 you take a little ball[br]here, radius delta. 0:43:34.200,0:43:38.680 So the image of that little[br]ball will fit in that ball 0:43:38.680,0:43:40.110 that you take here. 0:43:40.110,0:43:43.510 So even if you[br]shrink on the image, 0:43:43.510,0:43:46.020 you can take this[br]ball even smaller 0:43:46.020,0:43:48.670 so the image will[br]still fit inside. 0:43:48.670,0:43:51.490 And I was going, gosh,[br]this is the essence, 0:43:51.490,0:43:54.660 but I wish I could convey[br]it, because no book 0:43:54.660,0:43:58.550 will say it just-- or show you[br]how to do it with your hands. 0:43:58.550,0:43:59.202 0:43:59.202,0:44:00.035 STUDENT: [INAUDIBLE] 0:44:00.035,0:44:00.951 MAGDALENA TODA: Right. 0:44:00.951,0:44:04.985 So he was rushed, but he[br]had a very clear picture 0:44:04.985,0:44:06.965 of what is going on. 0:44:06.965,0:44:07.955 OK. 0:44:07.955,0:44:10.925 11.3 is a completely new start. 0:44:10.925,0:44:13.895 And you are gonna read[br]and be happy about that 0:44:13.895,0:44:16.370 because that's[br]partial derivatives. 0:44:16.370,0:44:20.360 And you say, Magdalena,[br]finally, this is piece of cake. 0:44:20.360,0:44:22.700 You see, I know these things. 0:44:22.700,0:44:25.760 I can do them in[br]my-- in my sleep. 0:44:25.760,0:44:29.960 So f of x and y[br]is still a graph. 0:44:29.960,0:44:33.115 And then you say,[br]how do we introduce 0:44:33.115,0:44:37.740 the partial derivative with[br]respect to one variable only. 0:44:37.740,0:44:39.768 You think, I draw the graph. 0:44:39.768,0:44:41.640 OK. 0:44:41.640,0:44:44.748 On this graph, I[br]pick a point x0, y0. 0:44:44.748,0:44:53.950 And if I were to take x to[br]be 0, what is-- what is the z 0:44:53.950,0:44:56.155 equals f of x0, y? 0:44:56.155,0:45:02.940 0:45:02.940,0:45:04.120 So I'll try to draw it. 0:45:04.120,0:45:05.120 It's not easy. 0:45:05.120,0:45:10.580 0:45:10.580,0:45:16.410 This is x and y and z, and you[br]want your x0 to be a constant. 0:45:16.410,0:45:17.457 STUDENT: [INAUDIBLE] 0:45:17.457,0:45:19.540 MAGDALENA TODA: It's a[br]so-called coordinate curve. 0:45:19.540,0:45:20.130 Very good. 0:45:20.130,0:45:23.085 It's a curve, but I want to[br]be good enough to draw it. 0:45:23.085,0:45:25.140 So you guys have[br]to wish me luck, 0:45:25.140,0:45:28.296 because I don't-- didn't have[br]enough coffee and I don't feel 0:45:28.296,0:45:30.060 like I can draw very well. 0:45:30.060,0:45:33.530 x0 is here. 0:45:33.530,0:45:39.580 So x is there, so you[br]cut with this board-- are 0:45:39.580,0:45:40.880 you guys with me? 0:45:40.880,0:45:44.115 You cut with this board[br]at the level x0 over here. 0:45:44.115,0:45:45.920 You cut. 0:45:45.920,0:45:49.290 When you cut with[br]this board-- you 0:45:49.290,0:45:52.563 cut your surface[br]with this board-- 0:45:52.563,0:45:54.882 you get a curve like that. 0:45:54.882,0:46:00.350 And we call that a[br]curve f of x0, y. 0:46:00.350,0:46:05.838 Some people who are a little bit[br]in a hurry and smarter than me, 0:46:05.838,0:46:07.782 they say x equals x0. 0:46:07.782,0:46:09.726 That's called coordinate curve. 0:46:09.726,0:46:16.550 0:46:16.550,0:46:20.270 So, the thing is, this--[br]it's a curve in plane. 0:46:20.270,0:46:21.324 This is the blue plane. 0:46:21.324,0:46:22.490 I don't know how to call it. 0:46:22.490,0:46:23.420 Pi. 0:46:23.420,0:46:25.840 You know I love to call it pi. 0:46:25.840,0:46:28.180 Since I'm in plane with[br]a point in a curve-- 0:46:28.180,0:46:33.570 a plane curve-- this curve[br]has a slope at x0, y0. 0:46:33.570,0:46:35.180 Can I draw that slope? 0:46:35.180,0:46:36.730 I'll try. 0:46:36.730,0:46:38.540 The slope of the[br]blue line, though. 0:46:38.540,0:46:39.976 Let me make it red. 0:46:39.976,0:46:43.950 The slope of the red line--[br]now, if you don't have colors 0:46:43.950,0:46:47.206 you can make it a dotted line. 0:46:47.206,0:46:57.810 The slope of the dotted line[br]is-- who the heck is that? 0:46:57.810,0:47:07.140 The derivative of f with respect[br]to y, because x0 is a constant. 0:47:07.140,0:47:09.490 So how do we write that? 0:47:09.490,0:47:12.800 Because x0 is sort of in[br]our way, driving us crazy. 0:47:12.800,0:47:15.000 Although he was fixed. 0:47:15.000,0:47:18.410 We keep him fixed by[br]keeping him in this plane. 0:47:18.410,0:47:20.130 x0 is fixed. 0:47:20.130,0:47:21.980 We have to write[br]another notation. 0:47:21.980,0:47:24.210 We cannot say f prime. 0:47:24.210,0:47:27.490 Because f depends[br]on two variables. 0:47:27.490,0:47:31.600 f prime were for when we[br]were babies in calculus 1. 0:47:31.600,0:47:32.934 We cannot use f prime anymore. 0:47:32.934,0:47:33.850 We have two variables. 0:47:33.850,0:47:36.310 Life became too complicated. 0:47:36.310,0:47:37.393 So we have to say-- 0:47:37.393,0:47:38.184 STUDENT: Professor? 0:47:38.184,0:47:40.701 MAGDALENA TODA: --instead[br]of df dy-- yes, sir. 0:47:40.701,0:47:41.700 May you use a subscript? 0:47:41.700,0:47:45.340 MAGDALENA TODA: You use--[br]yeah, you can do that as well. 0:47:45.340,0:47:47.300 That's what I do. 0:47:47.300,0:47:49.070 Let me do both. 0:47:49.070,0:47:55.751 f sub y at-- who[br]was fixed? x0 and y. 0:47:55.751,0:47:58.750 But this is my[br]favorite notation. 0:47:58.750,0:48:01.190 I'm going to make a[br]face because I love it. 0:48:01.190,0:48:02.760 This is what engineers love. 0:48:02.760,0:48:04.820 This is what we physicists love. 0:48:04.820,0:48:07.325 Mathematicians, though,[br]are crazy people. 0:48:07.325,0:48:08.240 They are. 0:48:08.240,0:48:09.330 All of them. 0:48:09.330,0:48:12.930 And they invented[br]another notation. 0:48:12.930,0:48:15.060 Do you remember[br]that Mr. Leibniz, 0:48:15.060,0:48:19.030 because he had nothing better to[br]do, when he invented calculus, 0:48:19.030,0:48:23.250 he did df dy, or df dx? 0:48:23.250,0:48:24.250 What is that? 0:48:24.250,0:48:27.140 That was the limit of[br]delta f, delta y, right? 0:48:27.140,0:48:28.550 That's what Leibniz did. 0:48:28.550,0:48:30.840 He introduced this[br]delta notation, 0:48:30.840,0:48:34.510 and then he said if you have[br]delta space over delta time, 0:48:34.510,0:48:38.080 then shrink both, and you[br]make a ratio in the limit, 0:48:38.080,0:48:40.620 you should read-- you[br]should write it df dy. 0:48:40.620,0:48:44.106 And that's the so-called[br]Leibniz notation, right? 0:48:44.106,0:48:46.756 That was in calc 1. 0:48:46.756,0:48:49.630 But I erased it because[br]that was calc 1. 0:48:49.630,0:48:54.180 Now, mathematicians, to[br]imitate the Leibniz notation, 0:48:54.180,0:48:57.870 they said, I cannot use df dy. 0:48:57.870,0:49:00.610 So what the heck shall I use? 0:49:00.610,0:49:02.650 After they thought[br]for about a year, 0:49:02.650,0:49:05.268 and I was reading through[br]the history about how 0:49:05.268,0:49:07.140 they invented this,[br]they said, let's take 0:49:07.140,0:49:09.860 the Greek-- the Greek d. 0:49:09.860,0:49:12.140 Which is the del. 0:49:12.140,0:49:13.990 That's partial. 0:49:13.990,0:49:19.400 The del f, del y, at x0, y. 0:49:19.400,0:49:22.400 When I was 20--[br]no, I was 18 when 0:49:22.400,0:49:27.100 I saw this the first time--[br]I had the hardest time making 0:49:27.100,0:49:27.925 this sign. 0:49:27.925,0:49:29.590 It's all in the wrist. 0:49:29.590,0:49:32.160 It's very-- OK. 0:49:32.160,0:49:32.950 Now. 0:49:32.950,0:49:33.590 df dy. 0:49:33.590,0:49:35.900 If you don't like it,[br]then what do you do? 0:49:35.900,0:49:38.858 You can adopt this notation. 0:49:38.858,0:49:41.710 And what is the meaning[br]of this by definition? 0:49:41.710,0:49:45.222 You say, you haven't even[br]defined it, Magdalena. 0:49:45.222,0:49:47.985 It has to be limit of[br]a difference quotient, 0:49:47.985,0:49:49.066 just like here. 0:49:49.066,0:49:53.320 But we have to be happy[br]and think of that. 0:49:53.320,0:49:57.360 What is the delta f[br]versus the delta y? 0:49:57.360,0:49:59.320 It has to be like that. 0:49:59.320,0:50:02.710 f of Mr. x0 is fixed. 0:50:02.710,0:50:07.060 x0, comma, y. 0:50:07.060,0:50:09.850 We have an increment in y. 0:50:09.850,0:50:16.310 y plus delta y. y plus[br]delta y minus-- that's 0:50:16.310,0:50:18.220 the difference quotient. 0:50:18.220,0:50:22.956 f of what-- the original[br]point was, well-- 0:50:22.956,0:50:24.450 STUDENT: x0, y0. 0:50:24.450,0:50:26.800 MAGDALENA TODA:[br]x0-- let me put y0 0:50:26.800,0:50:29.850 because our original[br]point was x0, y0. 0:50:29.850,0:50:37.810 x0, y0 over-- over delta y. 0:50:37.810,0:50:43.385 But if I am at x0, y0, I better[br]put x0, y0 fixed point here. 0:50:43.385,0:50:46.800 0:50:46.800,0:50:51.580 And I would like you to[br]photograph or put this thing-- 0:50:51.580,0:50:54.525 STUDENT: So is that a delta[br]that's in front of the f? 0:50:54.525,0:50:56.400 MAGDALENA TODA: Let me[br]review the whole thing 0:50:56.400,0:50:58.830 because it's very important. 0:50:58.830,0:51:00.810 Where shall I start,[br]here, or here? 0:51:00.810,0:51:01.860 It doesn't matter. 0:51:01.860,0:51:02.960 So the limit-- 0:51:02.960,0:51:05.330 STUDENT: [INAUDIBLE] start at m. 0:51:05.330,0:51:06.205 MAGDALENA TODA: At m? 0:51:06.205,0:51:06.704 At m. 0:51:06.704,0:51:07.930 OK, I'll start at m. 0:51:07.930,0:51:13.070 The slopes of this line at[br]x0, y0, right at my point, 0:51:13.070,0:51:18.640 will be, my favorite[br]notation is f sub y at x0, 0:51:18.640,0:51:22.030 y0, which means partial[br]derivative of f with respect 0:51:22.030,0:51:26.190 to y at the point--[br]fixed point x0, y0. 0:51:26.190,0:51:30.670 Or, for most mathematicians,[br]df-- of del-- del f, 0:51:30.670,0:51:34.310 del y at x0, y0. 0:51:34.310,0:51:38.880 Which is by definition the limit[br]of this difference quotient. 0:51:38.880,0:51:42.270 So x0 is held fixed[br]in both cases. 0:51:42.270,0:51:45.480 y0 is allowed to[br]deviate a little bit. 0:51:45.480,0:51:50.400 So y0 is fixed, but you[br]displace it by a little delta, 0:51:50.400,0:51:53.626 or by a little-- how did we[br]denote that in calc 1, h? 0:51:53.626,0:51:54.367 Little h? 0:51:54.367,0:51:54.950 STUDENT: Yeah. 0:51:54.950,0:51:56.616 MAGDALENA TODA: So[br]delta y, sometimes it 0:51:56.616,0:51:58.440 was called little h. 0:51:58.440,0:52:00.950 And this is the[br]same as little h. 0:52:00.950,0:52:03.840 Over that h. 0:52:03.840,0:52:07.370 Now you, without my[br]help, because you 0:52:07.370,0:52:10.870 have all the knowledge[br]and you're smart, 0:52:10.870,0:52:17.430 you should tell me how I[br]define f sub x at x0, y0, 0:52:17.430,0:52:22.192 and shut up, Magdalena,[br]let people talk. 0:52:22.192,0:52:23.947 This is hard. 0:52:23.947,0:52:24.780 [INTERPOSING VOICES] 0:52:24.780,0:52:26.090 MAGDALENA TODA: No. 0:52:26.090,0:52:28.250 I hope not. 0:52:28.250,0:52:31.515 As a limit of a[br]difference quotient, 0:52:31.515,0:52:34.570 so it's gonna be an[br]instantaneous rate of change. 0:52:34.570,0:52:37.062 That's the limit of a[br]difference quotient. 0:52:37.062,0:52:38.771 Limit of what? 0:52:38.771,0:52:39.271 Shut up. 0:52:39.271,0:52:40.534 I will zip my lips. 0:52:40.534,0:52:41.380 STUDENT: Delta x 0:52:41.380,0:52:42.860 MAGDALENA TODA:[br]Delta x, excellent. 0:52:42.860,0:52:44.492 Delta x going to 0. 0:52:44.492,0:52:47.990 So you shrink-- you displace[br]by a small displacement 0:52:47.990,0:52:50.254 only in the direction of x. 0:52:50.254,0:52:52.220 STUDENT: So f. 0:52:52.220,0:52:53.452 MAGDALENA TODA: f. 0:52:53.452,0:52:57.260 STUDENT: [INAUDIBLE] this[br]time, x is changing, so-- 0:52:57.260,0:52:58.580 [INTERPOSING VOICES] 0:52:58.580,0:53:04.890 MAGDALENA TODA: X0 plus[br]delta x, y0 is still fixed, 0:53:04.890,0:53:11.183 minus f of x0, y0. 0:53:11.183,0:53:13.115 Thank God this is always fixed. 0:53:13.115,0:53:14.081 I love this guy. 0:53:14.081,0:53:16.500 STUDENT: Delta-- 0:53:16.500,0:53:19.910 MAGDALENA TODA:[br]Delta x, which is 0:53:19.910,0:53:23.200 like the h we were[br]talking about. 0:53:23.200,0:53:24.680 Now in reality,[br]you never do that. 0:53:24.680,0:53:28.770 You would die if for every[br]exercise, derivation exercise, 0:53:28.770,0:53:31.500 you would have to compute a[br]limit of a difference quotient. 0:53:31.500,0:53:33.050 You will go bananas. 0:53:33.050,0:53:34.510 What we do? 0:53:34.510,0:53:37.400 We do exactly the same thing. 0:53:37.400,0:53:38.710 How can I draw? 0:53:38.710,0:53:40.620 Can anybody help me draw? 0:53:40.620,0:53:46.280 For y0, I would need to take[br]this other plane through y0. 0:53:46.280,0:53:47.332 Where is y0? 0:53:47.332,0:53:48.940 Here. 0:53:48.940,0:53:50.200 Is my drawing good enough? 0:53:50.200,0:53:51.598 I hope so. 0:53:51.598,0:53:56.470 So it's something like[br]I have this plane with, 0:53:56.470,0:53:57.920 oh, do you see that, guys? 0:53:57.920,0:53:58.570 OK. 0:53:58.570,0:54:02.842 So what is that, the other[br]curve, coordinate curve, look 0:54:02.842,0:54:03.342 like? 0:54:03.342,0:54:06.680 0:54:06.680,0:54:08.702 Oh my God. 0:54:08.702,0:54:10.260 Looks like that. 0:54:10.260,0:54:13.070 Through the same point,[br]and then the slope 0:54:13.070,0:54:18.370 of the line will be a[br]blue slope and the slope 0:54:18.370,0:54:23.710 will be f sub-- well OK. 0:54:23.710,0:54:28.130 So here I have in the red[br]one, which was the blue one, 0:54:28.130,0:54:33.670 this is f sub y, and for[br]this one, this is f sub x. 0:54:33.670,0:54:34.890 Right? 0:54:34.890,0:54:42.240 So guys, don't look[br]at the picture. 0:54:42.240,0:54:43.240 The picture's confusing. 0:54:43.240,0:54:46.510 This is x coming[br]towards me, right? 0:54:46.510,0:54:49.630 And y going there[br]and z is going up. 0:54:49.630,0:54:52.150 This is the graph. 0:54:52.150,0:54:54.650 When I do the[br]derivative with respect 0:54:54.650,0:55:00.830 to what is this, y, the[br]derivative with respect to y, 0:55:00.830,0:55:04.070 with respect to y, y[br]is my only variable, 0:55:04.070,0:55:06.640 so the curve will be like that. 0:55:06.640,0:55:11.480 And the slope will be for a[br]curve that depends on y only. 0:55:11.480,0:55:14.355 When I do derivative[br]with respect to x, 0:55:14.355,0:55:19.460 it's like I'm on top of a hill[br]and I decide to go skiing. 0:55:19.460,0:55:22.310 And I'm-- and I point[br]my skis like that, 0:55:22.310,0:55:27.100 and the slope is going down,[br]and that's the x direction. 0:55:27.100,0:55:27.750 OK? 0:55:27.750,0:55:30.910 And what I'm going to[br]describe as a skier 0:55:30.910,0:55:34.130 will be a plane curve going[br]down in this direction. 0:55:34.130,0:55:35.854 Zzzzsssshh, like that. 0:55:35.854,0:55:40.900 And the slope at every[br]point, the slope of the line, 0:55:40.900,0:55:44.720 of y trajectory, will[br]be the derivative. 0:55:44.720,0:55:47.460 So I have a curve like[br]that, and a curve like this. 0:55:47.460,0:55:50.140 And they're called[br]coordinate curves. 0:55:50.140,0:55:51.480 Now this is hard. 0:55:51.480,0:55:53.260 You'll see how[br]beautiful and easy 0:55:53.260,0:55:57.160 it is when you actually[br]compute the partial derivatives 0:55:57.160,0:55:59.890 of functions by hand. 0:55:59.890,0:56:01.930 Examples? 0:56:01.930,0:56:08.100 Let's take f of x, y to be[br]x squared plus y squared. 0:56:08.100,0:56:12.550 I'm asking you, who[br]is f sub x at x, y? 0:56:12.550,0:56:17.560 Who is f sub x at 1[br]minus 1, 1, 0, OK. 0:56:17.560,0:56:20.630 Who is f sub y at x, y? 0:56:20.630,0:56:26.130 And who is f sub y at 3 and 2. 0:56:26.130,0:56:28.245 Since I make up my[br]example-- I don't 0:56:28.245,0:56:30.460 want to copy the[br]examples from the book, 0:56:30.460,0:56:35.020 because you are supposedly[br]going to read the book. 0:56:35.020,0:56:40.230 This is-- should be another[br]example, just for you. 0:56:40.230,0:56:44.370 0:56:44.370,0:56:49.080 So who's gonna help me-- I'm[br]pausing a little bit-- who's 0:56:49.080,0:56:50.760 gonna help me here? 0:56:50.760,0:56:53.884 What's the answer here? 0:56:53.884,0:56:55.810 So how do I think? 0:56:55.810,0:57:00.060 I think I got-- when I[br]prime with respect to x, y 0:57:00.060,0:57:01.465 is like a held constant. 0:57:01.465,0:57:02.950 He's held prisoner. 0:57:02.950,0:57:05.205 Poor guy cannot leave his cell. 0:57:05.205,0:57:06.240 That's awful. 0:57:06.240,0:57:09.450 So you prime with respect to x. 0:57:09.450,0:57:11.350 Because x is the only variable. 0:57:11.350,0:57:12.176 And he is-- 0:57:12.176,0:57:14.515 STUDENT: So then it's 2x plus y? 0:57:14.515,0:57:17.068 MAGDALENA TODA: 2x plus 0. 0:57:17.068,0:57:17.764 Plus 0. 0:57:17.764,0:57:20.316 Because y is a constant and[br]when you prime a constant, 0:57:20.316,0:57:22.074 you get 0. 0:57:22.074,0:57:23.990 STUDENT: So when you[br]take partial derivatives, 0:57:23.990,0:57:25.656 you-- when you're[br]taking it with respect 0:57:25.656,0:57:28.610 to the first derivative, the[br]first variable [INAUDIBLE] 0:57:28.610,0:57:30.110 MAGDALENA TODA: You[br]don't completely 0:57:30.110,0:57:31.960 know because it[br]might be multiplied. 0:57:31.960,0:57:33.440 But you view it as a constant. 0:57:33.440,0:57:35.320 So for you-- very good, Ryan. 0:57:35.320,0:57:38.090 So for you, it's like,[br]as if y would be 7. 0:57:38.090,0:57:39.970 Imagine that y would be 7. 0:57:39.970,0:57:44.154 And then you have x squared plus[br]7 squared prime is u, right? 0:57:44.154,0:57:47.740 STUDENT: So then that means[br]f of 1-- or f x of 1,0 0:57:47.740,0:57:48.479 is [INAUDIBLE] 0:57:48.479,0:57:49.562 MAGDALENA TODA: Very good. 0:57:49.562,0:57:50.870 STUDENT: OK. 0:57:50.870,0:57:54.668 And in this case, f sub y,[br]what do you think it is? 0:57:54.668,0:57:56.439 STUDENT: 2y. 0:57:56.439,0:57:57.230 MAGDALENA TODA: 2y. 0:57:57.230,0:57:59.955 And what is f y of 3, 2? 0:57:59.955,0:58:01.092 STUDENT: 4. 0:58:01.092,0:58:02.050 MAGDALENA TODA: It's 4. 0:58:02.050,0:58:04.560 And you say, OK, that[br]makes sense, that was easy. 0:58:04.560,0:58:06.580 Let's try something hard. 0:58:06.580,0:58:09.385 I'm going to build them[br]on so many examples 0:58:09.385,0:58:12.670 that you say, stop,[br]Magdalena, because I became 0:58:12.670,0:58:15.580 an expert in partial[br]differentiation 0:58:15.580,0:58:19.290 and I-- now everything is so[br]trivial that you have to stop. 0:58:19.290,0:58:38.380 So example A, example B. A was f[br]of x, y [INAUDIBLE] x, y plus y 0:58:38.380,0:58:39.929 sine x. 0:58:39.929,0:58:41.470 And you say, wait,[br]wait, wait, you're 0:58:41.470,0:58:44.240 giving me a little[br]bit of trouble. 0:58:44.240,0:58:45.620 No, I don't mean to. 0:58:45.620,0:58:47.020 It's very easy. 0:58:47.020,0:58:50.260 Believe me guys,[br]very, very easy. 0:58:50.260,0:58:55.354 We just have to[br]think how we do this. 0:58:55.354,0:59:02.170 f sub x at 1 and 2, f[br]sub y at x, y in general, 0:59:02.170,0:59:06.980 f sub y at 1 and[br]2, for God's sake. 0:59:06.980,0:59:08.270 OK. 0:59:08.270,0:59:09.885 All right. 0:59:09.885,0:59:19.380 And now, while you're[br]staring at that, 0:59:19.380,0:59:23.595 I take out my beautiful[br]colors that I paid $6 for. 0:59:23.595,0:59:26.145 0:59:26.145,0:59:31.620 The department told me that[br]they don't buy different colors, 0:59:31.620,0:59:35.490 just two or three basic ones. 0:59:35.490,0:59:35.990 All right? 0:59:35.990,0:59:38.170 So what do we do? 0:59:38.170,0:59:40.620 STUDENT: First[br]one will be the y. 0:59:40.620,0:59:43.320 MAGDALENA TODA: It's like y[br]would be a constant 7, right, 0:59:43.320,0:59:46.710 but you have to keep in[br]mind it's mister called y. 0:59:46.710,0:59:48.580 Which for you is a constant. 0:59:48.580,0:59:52.730 So you go, I'm priming this[br]with respect to x only-- 0:59:52.730,0:59:54.514 STUDENT: Then you get y. 0:59:54.514,0:59:56.325 MAGDALENA TODA: Very good. 0:59:56.325,0:59:56.824 Plus-- 0:59:56.824,0:59:59.910 0:59:59.910,1:00:01.114 STUDENT: y cosine x. 1:00:01.114,1:00:01.596 MAGDALENA TODA: y cosine x. 1:00:01.596,1:00:02.096 Excellent. 1:00:02.096,1:00:03.524 And stop. 1:00:03.524,1:00:04.970 And stop. 1:00:04.970,1:00:06.185 Because that's all I have. 1:00:06.185,1:00:08.763 You see, it's not hard. 1:00:08.763,1:00:11.710 Let me put here a y. 1:00:11.710,1:00:13.443 OK. 1:00:13.443,1:00:19.040 And then, I plug[br]a different color. 1:00:19.040,1:00:21.520 I'm a girl, of course I[br]like different colors. 1:00:21.520,1:00:26.700 So 1, 2. x is 1, and y is 2. 1:00:26.700,1:00:30.270 2 plus 2 cosine 1. 1:00:30.270,1:00:33.314 And you say, oh, wait a minute,[br]what is that cosine of 1? 1:00:33.314,1:00:33.814 Never mind. 1:00:33.814,1:00:34.790 Don't worry about it. 1:00:34.790,1:00:37.230 It's like cosine[br]of 1, [INAUDIBLE] 1:00:37.230,1:00:41.134 plug it in the[br]calculator, nobody cares. 1:00:41.134,1:00:44.550 Well, in the final, you[br]don't have a calculator, 1:00:44.550,1:00:47.966 so you leave it like that. 1:00:47.966,1:00:49.430 Who cares? 1:00:49.430,1:00:52.750 It's just the perfect--[br]I would actually hate it 1:00:52.750,1:00:54.230 that you gave me--[br]because all you 1:00:54.230,1:00:56.480 could give me would be an[br]approximation, a truncation, 1:00:56.480,1:00:58.220 with two decimals. 1:00:58.220,1:01:01.330 I prefer you give me the[br]precise answer, which 1:01:01.330,1:01:03.960 is an exact answer like that. 1:01:03.960,1:01:04.750 f sub y. 1:01:04.750,1:01:07.500 Now, Mr. x is held prisoner. 1:01:07.500,1:01:09.006 He is a constant. 1:01:09.006,1:01:10.620 He cannot move. 1:01:10.620,1:01:11.800 Mr. y can move. 1:01:11.800,1:01:13.280 He has all the freedom. 1:01:13.280,1:01:16.603 So prime with respect[br]to y, what do you have? 1:01:16.603,1:01:17.102 STUDENT: x-- 1:01:17.102,1:01:18.026 [INTERPOSING VOICES] 1:01:18.026,1:01:21.952 MAGDALENA TODA: x plus[br]sine x is a constant. 1:01:21.952,1:01:25.200 So for God's sake,[br]I'll write it. 1:01:25.200,1:01:30.680 So then I get 1,[br]plug in x equals 1. y 1:01:30.680,1:01:31.960 doesn't appear in the picture. 1:01:31.960,1:01:33.110 I don't care. 1:01:33.110,1:01:35.132 1 plus sine 1. 1:01:35.132,1:01:38.340 1:01:38.340,1:01:39.820 And now comes-- don't erase. 1:01:39.820,1:01:42.240 Now comes the-- I mean,[br]you cannot erase it. 1:01:42.240,1:01:44.750 I can erase it. 1:01:44.750,1:01:48.768 Comes this mean professor[br]who says, wait a minute, 1:01:48.768,1:01:50.950 I want more. 1:01:50.950,1:01:53.601 Mathematicians always want more. 1:01:53.601,1:01:57.330 He goes, I want the[br]second derivative. 1:01:57.330,1:02:01.040 f sub x x of x, y. 1:02:01.040,1:02:03.530 And you say, what in[br]the world is that? 1:02:03.530,1:02:06.290 Even some mathematicians,[br]they denote it 1:02:06.290,1:02:13.295 as del 2 f dx 2, which[br]is d of-- d with respect 1:02:13.295,1:02:16.520 to x sub d u with respect to x. 1:02:16.520,1:02:17.811 What does it mean? 1:02:17.811,1:02:20.640 You take the first derivative[br]and you derive it again. 1:02:20.640,1:02:23.181 And don't drink and derive[br]because you'll be in trouble. 1:02:23.181,1:02:23.680 Right? 1:02:23.680,1:02:28.039 So you have d of dx primed[br]again, with-- differentiated 1:02:28.039,1:02:30.410 again with respect to x. 1:02:30.410,1:02:31.370 Is that hard? 1:02:31.370,1:02:31.870 Uh-uh. 1:02:31.870,1:02:32.950 What you do? 1:02:32.950,1:02:36.065 In the-- don't do it here. 1:02:36.065,1:02:37.430 You do it in general, right? 1:02:37.430,1:02:43.000 With respect to x as a variable,[br]y is again held as a prisoner, 1:02:43.000,1:02:44.570 constant. 1:02:44.570,1:02:47.680 So when you prime[br]that y goes away. 1:02:47.680,1:02:50.710 You're gonna get 0. 1:02:50.710,1:02:54.580 I'll write 0 like a silly[br]because we are just starters. 1:02:54.580,1:02:56.207 And what else? 1:02:56.207,1:02:57.700 STUDENT: Negative y sine of x. 1:02:57.700,1:02:59.900 MAGDALENA TODA:[br]Minus y sine of x. 1:02:59.900,1:03:02.340 And I know you've gonna[br]love this process. 1:03:02.340,1:03:04.940 You are becoming[br]experts in that. 1:03:04.940,1:03:10.050 And in a way I'm a little[br]bit sorry it's so easy, 1:03:10.050,1:03:13.180 but I guess not[br]everybody gets it. 1:03:13.180,1:03:16.290 There are students who[br]don't get it the first time. 1:03:16.290,1:03:17.980 So what do we get here? 1:03:17.980,1:03:18.907 Minus-- 1:03:18.907,1:03:21.770 STUDENT: 0. 1:03:21.770,1:03:25.821 MAGDALENA TODA: Please[br]tell me-- sine 1, 0. 1:03:25.821,1:03:26.320 Good. 1:03:26.320,1:03:30.002 I could do the same[br]thing for f y y. 1:03:30.002,1:03:34.800 I could do this thing--[br]what is f sub x y? 1:03:34.800,1:03:37.250 By definition f sub x y-- 1:03:37.250,1:03:39.975 STUDENT: Is that taking the[br]derivative of the derivative 1:03:39.975,1:03:42.100 with respect-- is that[br]taking the second derivative 1:03:42.100,1:03:44.058 with respect to y after[br]you take the derivative 1:03:44.058,1:03:46.110 of the-- first derivative[br]with respect to x? 1:03:46.110,1:03:47.026 MAGDALENA TODA: Right. 1:03:47.026,1:03:49.510 So when I write like that,[br]because that's a little bit 1:03:49.510,1:03:54.202 confusing, when students[br]ask me, which one is first? 1:03:54.202,1:03:57.830 First you do f sub[br]x, and then you do y. 1:03:57.830,1:04:02.766 And then f sub y x would be the[br]derivative with respect to y 1:04:02.766,1:04:04.650 primed again with respect to x. 1:04:04.650,1:04:07.480 Now, let me tell[br]you the good news. 1:04:07.480,1:04:13.390 They-- the book doesn't call[br]it any name, because we don't 1:04:13.390,1:04:14.886 like to call anybody names. 1:04:14.886,1:04:15.840 I'm just kidding. 1:04:15.840,1:04:23.710 It's called the[br]Schwartz principle, 1:04:23.710,1:04:27.470 or the theorem of Schwartz. 1:04:27.470,1:04:30.974 When I told my co-authors,[br]they said, who cares? 1:04:30.974,1:04:34.950 Well I care, because I was a[br]student when my professors told 1:04:34.950,1:04:38.290 me that this German[br]mathematician made 1:04:38.290,1:04:41.040 this discovery, which[br]is so beautiful. 1:04:41.040,1:04:55.380 If f is twice differentiable[br]with respect to x and y, 1:04:55.380,1:04:58.820 and the partial derivatives--[br]the second partial 1:04:58.820,1:05:14.750 derivatives-- are continuous,[br]then, now in English 1:05:14.750,1:05:17.480 it would say it doesn't[br]matter in which order 1:05:17.480,1:05:18.530 you differentiate. 1:05:18.530,1:05:20.920 The mixed ones are[br]always the same. 1:05:20.920,1:05:22.190 Say what? 1:05:22.190,1:05:26.760 f sub x y equals f sub[br]y x for every point. 1:05:26.760,1:05:31.590 For every-- do you remember[br]what I taught you for every x, y 1:05:31.590,1:05:32.360 in the domain. 1:05:32.360,1:05:36.110 Or for every x, y[br]where this happens. 1:05:36.110,1:05:38.410 So what does this mean? 1:05:38.410,1:05:41.219 That means that whether[br]you differentiate 1:05:41.219,1:05:43.760 first with respect to x and then[br]with respect to, y, or first 1:05:43.760,1:05:46.100 with respect to y and[br]then with respect to x, 1:05:46.100,1:05:48.250 it doesn't matter. 1:05:48.250,1:05:50.900 The mixed partial[br]derivatives are the same. 1:05:50.900,1:05:52.140 Which is wonderful. 1:05:52.140,1:05:55.110 I mean, this is one[br]of the best things 1:05:55.110,1:05:58.100 that ever happened to us. 1:05:58.100,1:06:01.306 Let's see if this[br]is true in our case. 1:06:01.306,1:06:03.864 I mean, of course it's true[br]because it's a theorem, 1:06:03.864,1:06:06.154 if it weren't true[br]I wouldn't teach it, 1:06:06.154,1:06:11.050 but let's verify it on a baby. 1:06:11.050,1:06:14.343 Not on a real baby,[br]on a baby example. 1:06:14.343,1:06:15.230 Right? 1:06:15.230,1:06:21.040 So, f sub x is y plus y[br]equals sine x primed again 1:06:21.040,1:06:22.782 with respect to y. 1:06:22.782,1:06:27.630 And what do we get out of it? 1:06:27.630,1:06:29.250 Cosine of x. 1:06:29.250,1:06:31.081 Are you guys with me? 1:06:31.081,1:06:35.130 So f sub x was y[br]plus y equals sine x. 1:06:35.130,1:06:39.078 Take this guy[br]again, put it here, 1:06:39.078,1:06:42.667 squeeze them up a little[br]bit, divide by-- no. 1:06:42.667,1:06:47.780 Time with respect to y, x is[br]a constant, what do you think? 1:06:47.780,1:06:48.794 Cosine of x, am I right? 1:06:48.794,1:06:49.960 STUDENT: 1 plus [INAUDIBLE]. 1:06:49.960,1:06:52.389 1:06:52.389,1:06:54.180 MAGDALENA TODA: That's[br]what it starts with. 1:06:54.180,1:06:56.700 Plus [INAUDIBLE]. 1:06:56.700,1:07:02.030 So cosine of x, [INAUDIBLE][br]a constant, plus 1. 1:07:02.030,1:07:04.965 Another way to have done[br]it is, like, wait a minute, 1:07:04.965,1:07:10.810 at this point I go, constant[br]out-- are you with me?-- 1:07:10.810,1:07:14.630 constant out, prime with respect[br]to y, equals sine x plus 1. 1:07:14.630,1:07:16.885 Thank you. 1:07:16.885,1:07:17.385 All right. 1:07:17.385,1:07:20.822 1:07:20.822,1:07:26.370 F sub yx is going to be f sub y. 1:07:26.370,1:07:32.336 x plus sine x, but I have[br]to take it from here, 1:07:32.336,1:07:38.416 and I prime again with respect[br]to x, and I get the same thing. 1:07:38.416,1:07:39.790 I don't know,[br]maybe I'm dyslexic, 1:07:39.790,1:07:43.450 I go from the right to the[br]left, what's the matter with me. 1:07:43.450,1:07:47.305 Instead of saying 1 plus,[br]I go cosine of x plus 1. 1:07:47.305,1:07:53.024 1:07:53.024,1:07:54.476 So it's the same thing. 1:07:54.476,1:07:55.444 Yes, sir. 1:07:55.444,1:07:58.611 STUDENT:I'm looking at[br]the f of xy from the-- 1:07:58.611,1:08:00.486 MAGDALENA TODA: Which[br]one are you looking at? 1:08:00.486,1:08:01.460 Show me. 1:08:01.460,1:08:03.518 STUDENT: It's in the purple. 1:08:03.518,1:08:05.150 MAGDALENA TODA: It[br]is in the purple. 1:08:05.150,1:08:05.680 STUDENT: It's that[br]one right there. 1:08:05.680,1:08:06.140 So-- 1:08:06.140,1:08:06.725 MAGDALENA TODA: This one? 1:08:06.725,1:08:07.308 STUDENT: Mmhm. 1:08:07.308,1:08:10.550 So, I'm looking at[br]the y plus y cosine x. 1:08:10.550,1:08:12.503 You got that from f of x. 1:08:12.503,1:08:14.196 MAGDALENA TODA: I[br]got this from f of x, 1:08:14.196,1:08:16.640 and I prime it again,[br]with respect to y. 1:08:16.640,1:08:18.920 The whole thing. 1:08:18.920,1:08:21.510 STUDENT: OK, so you're not[br]writing that as a derivative? 1:08:21.510,1:08:25.274 You're just substituting[br]that in for f of x? 1:08:25.274,1:08:27.475 MAGDALENA TODA: So,[br]let me write it better, 1:08:27.475,1:08:30.783 because I was a little bit[br]rushed, and I don't know, 1:08:30.783,1:08:32.167 silly or something. 1:08:32.167,1:08:35.135 When I prime this[br]with respect to y-- 1:08:35.135,1:08:38.274 STUDENT: Then you get[br]the cosine of x plus 1. 1:08:38.274,1:08:39.149 MAGDALENA TODA: Yeah. 1:08:39.149,1:08:42.426 I could say, I can take[br]out all the constants. 1:08:42.426,1:08:43.160 STUDENT: OK. 1:08:43.160,1:08:46.210 MAGDALENA TODA: And that[br]constant is this plus 1. 1:08:46.210,1:08:47.455 And that's all I'm left with. 1:08:47.455,1:08:47.955 Right? 1:08:47.955,1:08:51.609 It's the same thing[br]as 1 plus cosine x, 1:08:51.609,1:08:53.879 which is a constant times y. 1:08:53.879,1:08:57.238 Prime this with respect[br]to y, I get the constant. 1:08:57.238,1:09:04.020 It's the same principal as when[br]you have bdy of 7y equals 7. 1:09:04.020,1:09:06.752 Right? 1:09:06.752,1:09:08.703 OK. 1:09:08.703,1:09:10.130 Is this too easy? 1:09:10.130,1:09:13.420 I'll give you a nicer function. 1:09:13.420,1:09:28.760 I'm imitating the one[br]in WeBWorK [INAUDIBLE] 1:09:28.760,1:09:31.444 To make it harder for you. 1:09:31.444,1:09:34.354 Nothing I can make at this[br]point is hard for you, 1:09:34.354,1:09:39.250 because you're becoming experts[br]in partial differentiation, 1:09:39.250,1:09:41.720 and I cannot[br]challenge you on that. 1:09:41.720,1:09:54.113 1:09:54.113,1:09:57.048 I'm just trying to[br]make it harder for you. 1:09:57.048,1:09:59.004 And I'm trying to[br]look up something. 1:09:59.004,1:10:02.930 1:10:02.930,1:10:03.970 OK, how about that? 1:10:03.970,1:10:06.670 1:10:06.670,1:10:09.110 This is harder than the[br]ones you have in WeBWorK. 1:10:09.110,1:10:11.970 But that was kind of[br]the idea-- that when 1:10:11.970,1:10:15.620 you go home, and open[br]those WeBWorK problem sets, 1:10:15.620,1:10:17.430 that's a piece of cake. 1:10:17.430,1:10:20.600 What we did in class was harder. 1:10:20.600,1:10:23.960 When I was a graduate[br]student, one professor said, 1:10:23.960,1:10:27.320 the easy examples are the[br]ones that the professor's 1:10:27.320,1:10:29.740 supposed to write in[br]class, on the board. 1:10:29.740,1:10:31.440 The hard examples[br]are the ones that 1:10:31.440,1:10:34.290 are left for the[br]students' homework. 1:10:34.290,1:10:35.750 I disagree. 1:10:35.750,1:10:37.760 I think it should be[br]the other way around. 1:10:37.760,1:10:40.260 So f sub x. 1:10:40.260,1:10:43.380 1:10:43.380,1:10:50.552 That means bfdx for[br]the pair xy, any xy. 1:10:50.552,1:10:53.560 I'm not specifying an x and a y. 1:10:53.560,1:10:56.180 I'm not making them a constant. 1:10:56.180,1:10:58.970 What am I going to[br]have in this case? 1:10:58.970,1:11:03.850 Chain -- if I catch you[br]not knowing the chain rule, 1:11:03.850,1:11:05.370 you fail the final. 1:11:05.370,1:11:12.590 Not really, but, OK,[br]you get some penalty. 1:11:12.590,1:11:13.730 You know it. 1:11:13.730,1:11:16.110 Just pay attention[br]to what you do. 1:11:16.110,1:11:18.340 I make my own[br]mistakes sometimes. 1:11:18.340,1:11:21.160 So 1 over. 1:11:21.160,1:11:23.590 What do you do here[br]when you differentiate 1:11:23.590,1:11:24.350 with respect to x? 1:11:24.350,1:11:31.600 You think, OK, from the outside[br]to the inside, one at a time. 1:11:31.600,1:11:36.130 1 over the variable[br]squared plus 1, right? 1:11:36.130,1:11:42.215 Whatever that variable,[br]it's like you call variable 1:11:42.215,1:11:44.890 of the argument xy, right? 1:11:44.890,1:11:47.240 STUDENT: [INAUDIBLE] 1:11:47.240,1:11:49.558 MAGDALENA TODA: Square plus 1. 1:11:49.558,1:11:56.710 Times-- cover it with your[br]hand-- prime with respect to x. 1:11:56.710,1:11:59.000 y, right? 1:11:59.000,1:12:00.395 Good! 1:12:00.395,1:12:01.325 And you're done. 1:12:01.325,1:12:02.720 You see how easy it was. 1:12:02.720,1:12:07.510 Just don't forget something[br]because it can cost you points. 1:12:07.510,1:12:09.640 Are you guys with me? 1:12:09.640,1:12:13.380 So, once we are done with[br]saying, 1 over argument 1:12:13.380,1:12:16.130 squared plus 1, I cover[br]this with my hand, 1:12:16.130,1:12:20.110 xy prime with[br]respect to 2x is y. 1:12:20.110,1:12:22.380 And I'm done. 1:12:22.380,1:12:23.250 And I'm done. 1:12:23.250,1:12:26.250 And here, pause. 1:12:26.250,1:12:29.680 What's the easiest[br]way to do that? 1:12:29.680,1:12:32.010 You look at it like,[br]she wants me to get 1:12:32.010,1:12:34.310 caught in the quotient rule. 1:12:34.310,1:12:37.310 She wants to catch me[br]not knowing this rule, 1:12:37.310,1:12:40.250 while I can do better. 1:12:40.250,1:12:43.460 One way to do it would[br]be numerator prime plus 1:12:43.460,1:12:47.740 denominator, minus[br]numerator [INAUDIBLE] What's 1:12:47.740,1:12:50.410 the easier way to do it? 1:12:50.410,1:12:52.870 STUDENT: x squared plus[br]y squared, all of it 1:12:52.870,1:12:53.855 to the negative one. 1:12:53.855,1:12:54.771 MAGDALENA TODA: Right. 1:12:54.771,1:12:56.890 So you say, hey,[br]you cannot catch me, 1:12:56.890,1:13:00.680 I'm the gingerbread man. 1:13:00.680,1:13:01.280 Good! 1:13:01.280,1:13:03.100 That was a good idea. 1:13:03.100,1:13:10.330 Chain rule, and[br]minus 1/2, times-- 1:13:10.330,1:13:11.760 who tells me what's next? 1:13:11.760,1:13:13.210 I'm not going to say a word. 1:13:13.210,1:13:15.258 STUDENT: 2x plus y squared. 1:13:15.258,1:13:19.170 No, it's 2x. 1:13:19.170,1:13:20.637 x squared plus y squared. 1:13:20.637,1:13:22.595 MAGDALENA TODA: From the[br]outside to the inside. 1:13:22.595,1:13:25.340 From the outside-- to the what? 1:13:25.340,1:13:27.186 STUDENT: [INAUDIBLE] 1:13:27.186,1:13:28.061 MAGDALENA TODA: Good. 1:13:28.061,1:13:28.970 And now I'm done. 1:13:28.970,1:13:31.220 I don't see that anymore. 1:13:31.220,1:13:33.860 I focus to the core. 1:13:33.860,1:13:35.650 2x. 1:13:35.650,1:13:38.640 Times 2x. 1:13:38.640,1:13:42.390 And that is plenty. 1:13:42.390,1:13:45.250 OK, now, let me[br]ask you a question. 1:13:45.250,1:13:51.340 What if you would[br]ask a smart kid, 1:13:51.340,1:13:56.760 I don't know, somebody[br]who knows that, 1:13:56.760,1:14:01.620 can you pose the f sub y of xy[br]without doing the whole thing 1:14:01.620,1:14:03.370 all over again? 1:14:03.370,1:14:06.350 Can you sort of figure[br]out what it would be? 1:14:06.350,1:14:08.700 The beautiful[br]thing about x and y 1:14:08.700,1:14:11.042 is that these are[br]symmetric polynomials. 1:14:11.042,1:14:12.750 What does it mean,[br]symmetric polynomials? 1:14:12.750,1:14:19.260 That means, if you swap x[br]and y, and you swap x and y, 1:14:19.260,1:14:20.810 it's the same thing. 1:14:20.810,1:14:23.300 Just think of that--[br]swapping x and y. 1:14:23.300,1:14:25.230 Swapping the roles of x and y. 1:14:25.230,1:14:28.250 So what do you think[br]you're going to get? 1:14:28.250,1:14:31.130 OK, one student said,[br]this is for smart people, 1:14:31.130,1:14:32.564 not for people like me. 1:14:32.564,1:14:34.890 And I said, well, OK,[br]what's the matter with you? 1:14:34.890,1:14:36.240 I'm a hard worker. 1:14:36.240,1:14:39.910 I'm the kind of guy who takes[br]the whole thing again, and does 1:14:39.910,1:14:42.180 the derivation from scratch. 1:14:42.180,1:14:45.325 And thinking back in high[br]school, I think, even 1:14:45.325,1:14:47.810 for symmetric polynomials, 1:14:47.810,1:14:49.570 I'm sure that being[br]smart and being 1:14:49.570,1:14:53.510 able to guess the[br]whole thing-- but I 1:14:53.510,1:14:56.000 did the computation[br]many times mechanically, 1:14:56.000,1:14:59.440 just in the same way,[br]because I was a hard worker. 1:14:59.440,1:15:01.220 So what do you[br]have in that case? 1:15:01.220,1:15:09.810 1/xy squared plus 1 times x[br]plus-- the same kind of thing. 1:15:09.810,1:15:14.338 Attention, this is the symmetric[br]polynomial, and I go to that. 1:15:14.338,1:15:17.280 And then times 2y. 1:15:17.280,1:15:20.610 So, see-- that kind[br]of easy, fast thing. 1:15:20.610,1:15:24.340 Why is this a good[br]observation when 1:15:24.340,1:15:26.015 you have symmetric polynomials? 1:15:26.015,1:15:28.710 If you are on the final and[br]you don't have that much time, 1:15:28.710,1:15:33.540 or on any kind of exam when[br]you are in a time-crunch. 1:15:33.540,1:15:36.120 Now, we want those[br]exams so you are not 1:15:36.120,1:15:38.024 going to be in a time-crunch. 1:15:38.024,1:15:41.665 If there is something I hate,[br]I hate a final of 2 hours 1:15:41.665,1:15:44.640 and a half with 25[br]serious problems, 1:15:44.640,1:15:48.260 and you know nobody can do that. 1:15:48.260,1:15:50.710 So, it happens a lot. 1:15:50.710,1:15:55.880 I see that-- one of my jobs[br]is also to look at the finals 1:15:55.880,1:15:58.586 after people wrote[br]them, and I still 1:15:58.586,1:16:05.256 do that every semester-- I see[br]too many people making finals. 1:16:05.256,1:16:06.880 The finals are not[br]supposed to be long. 1:16:06.880,1:16:10.990 The finals are supposed[br]to be comprehensive, cover 1:16:10.990,1:16:16.300 everything, but not extensive. 1:16:16.300,1:16:21.060 So maybe you'll have 15[br]problems that cover practically 1:16:21.060,1:16:22.911 the material entirely. 1:16:22.911,1:16:23.410 Why? 1:16:23.410,1:16:29.080 Because every little problem[br]can have two short questions. 1:16:29.080,1:16:30.530 You were done with[br]a section, you 1:16:30.530,1:16:34.690 shot half of a chapter[br]only one question. 1:16:34.690,1:16:39.660 This is one example just--[br]not involving [INAUDIBLE] 1:16:39.660,1:16:41.210 of an expression like that, no. 1:16:41.210,1:16:43.310 That's too time-consuming. 1:16:43.310,1:16:47.600 But maybe just tangent of[br]x-squared plus y-squared, 1:16:47.600,1:16:49.964 find the partial derivatives. 1:16:49.964,1:16:53.380 That's a good exam[br]question, and that's enough 1:16:53.380,1:16:55.895 when it comes to[br]testing partials. 1:16:55.895,1:16:58.130 By the way, how[br]much-- what is that? 1:16:58.130,1:17:00.832 And I'm going to let[br]you go right now. 1:17:00.832,1:17:01.816 Use the bathroom. 1:17:01.816,1:17:05.240 And when you come back from the[br]bathroom, we'll fill in this. 1:17:05.240,1:17:10.515 You know I am horrible in the[br]sense that I want-- I'm greedy. 1:17:10.515,1:17:12.055 I need extra time. 1:17:12.055,1:17:15.340 I want to use more time. 1:17:15.340,1:17:17.850 I will do your[br]problems from now on, 1:17:17.850,1:17:22.061 and you can use the bathroom,[br]eat something, wash your hands. 1:17:22.061,1:17:26.498 1:17:26.498,1:17:28.470 I'll start in[br]about five minutes. 1:17:28.470,1:17:29.456 Don't worry. 1:17:29.456,1:17:32.907 1:17:32.907,1:17:33.893 Alexander? 1:17:33.893,1:17:35.372 Are you here? 1:17:35.372,1:17:37.837 Come get this. 1:17:37.837,1:17:40.350 I apologize. 1:17:40.350,1:17:42.426 This is long due back to you. 1:17:42.426,1:17:43.420 STUDENT: Oh. 1:17:43.420,1:17:43.920 Thank you. 1:17:43.920,1:17:47.406 1:17:47.406,1:17:49.900 STUDENT: Is there an[br]attendance sheet today? 1:17:49.900,1:17:53.110 MAGDALENA TODA: I will--[br]I'm making up one. 1:17:53.110,1:17:56.750 There is already on[br]one side attendance. 1:17:56.750,1:17:58.650 Let's use the other side. 1:17:58.650,1:18:01.566 Put today's date. 1:18:01.566,1:18:02.066 [INAUDIBLE] 1:18:02.066,1:18:44.396 1:18:44.396,1:18:48.380 [SIDE CONVERSATIONS] 1:18:48.380,1:18:56.348 1:18:56.348,1:18:58.340 MAGDALENA TODA: They[br]are spoiling me. 1:18:58.340,1:19:03.334 They give me new[br]sprays every week. 1:19:03.334,1:19:05.298 [INAUDIBLE] take care of this. 1:19:05.298,1:19:09.226 [SIDE CONVERSATIONS] 1:19:09.226,1:19:14.136 1:19:14.136,1:19:17.082 MAGDALENA TODA: So I'm[br]going to ask you something. 1:19:17.082,1:19:20.530 And you respond honestly. 1:19:20.530,1:19:24.810 Which chapter-- we already[br]browsed through three chapters. 1:19:24.810,1:19:26.840 I mean, Chapter 9[br]was vector spaces, 1:19:26.840,1:19:29.540 and it was all review[br]from-- from what? 1:19:29.540,1:19:30.540 From Calc 2. 1:19:30.540,1:19:34.610 Chapter 10 was curves in[br][INAUDIBLE] and curves 1:19:34.610,1:19:36.602 in space, practically. 1:19:36.602,1:19:40.540 1:19:40.540,1:19:46.626 And Chapter 11 is functions[br]of several variables. 1:19:46.626,1:19:48.640 Now you have a flavor[br]of all of them, 1:19:48.640,1:19:50.475 which one was hardest for you? 1:19:50.475,1:19:51.476 STUDENT: 9 and 10, both. 1:19:51.476,1:19:52.725 MAGDALENA TODA: 9 and 10 both. 1:19:52.725,1:19:53.710 STUDENT: [INAUDIBLE]. 1:19:53.710,1:19:56.420 MAGDALENA TODA: This is so[br]much better than the other. 1:19:56.420,1:20:00.660 No, I think you guys[br]actually-- it looks better, 1:20:00.660,1:20:06.696 because you've seen a lot more[br]vectors and vector functions. 1:20:06.696,1:20:08.592 STUDENT: I didn't[br]understand any of 9 or 10. 1:20:08.592,1:20:09.540 STUDENT: [INAUDIBLE]. 1:20:09.540,1:20:10.370 MAGDALENA TODA: Yes, ma'am. 1:20:10.370,1:20:12.370 STUDENT: Could you go over[br]parametrization [INAUDIBLE]? 1:20:12.370,1:20:14.370 MAGDALENA TODA: I will[br]go over that again. 1:20:14.370,1:20:18.370 And I will go over some[br]other parametrizations today. 1:20:18.370,1:20:24.500 And I promised that at the[br]end, in those 20 minutes, 1:20:24.500,1:20:28.380 I will do that problem that[br]gave a few of you trouble. 1:20:28.380,1:20:29.181 Yes, sir? 1:20:29.181,1:20:30.805 STUDENT: Do we take[br]the same final exam 1:20:30.805,1:20:33.230 as all the other [INAUDIBLE][br]classes? [INAUDIBLE]? 1:20:33.230,1:20:36.500 MAGDALENA TODA: Well, that's[br]what I was asked yesterday. 1:20:36.500,1:20:43.150 So practically, it's at the[br]latitude of the instructor who 1:20:43.150,1:20:45.320 teaches honors if they[br]write their own final, 1:20:45.320,1:20:48.620 and in general make[br]it harder, or they 1:20:48.620,1:20:51.060 take the general final[br]like everybody else. 1:20:51.060,1:20:53.988 For your formative[br]purposes, and as a study, 1:20:53.988,1:20:57.892 I would like you to[br]take the general final, 1:20:57.892,1:21:01.350 because I want to see[br]where you stand compared 1:21:01.350,1:21:02.780 to the rest of the population. 1:21:02.780,1:21:06.848 So you are my sample, and[br]they are the entire student 1:21:06.848,1:21:08.394 population of Calc[br]3, I want to make 1:21:08.394,1:21:14.494 the statistical analysis of your[br]performance compared to them. 1:21:14.494,1:21:16.446 STUDENT: So we'll[br]take the regular one? 1:21:16.446,1:21:17.422 MAGDALENA TODA: Yeah. 1:21:17.422,1:21:19.005 For this one, I just[br]have to make sure 1:21:19.005,1:21:22.302 that they also have that[br]extra credit added in. 1:21:22.302,1:21:25.840 Because if I have too much[br]extra credit in there, 1:21:25.840,1:21:27.310 well they also count that. 1:21:27.310,1:21:28.780 So that's what that means. 1:21:28.780,1:21:30.250 So we can [INAUDIBLE]. 1:21:30.250,1:21:34.180 1:21:34.180,1:21:35.690 All right. 1:21:35.690,1:21:37.015 Let me finish this exercise. 1:21:37.015,1:21:41.000 And then [? stop ?][br][INAUDIBLE] and go 1:21:41.000,1:21:45.635 over some homework problems and[br]some parametrization problems. 1:21:45.635,1:21:48.605 And I will see what else. 1:21:48.605,1:21:55.535 So tangent of [INAUDIBLE]. 1:21:55.535,1:21:59.495 1:21:59.495,1:22:00.485 Is this hard? 1:22:00.485,1:22:01.970 No, it's [INAUDIBLE]. 1:22:01.970,1:22:05.930 But you have to[br]remind me, because I 1:22:05.930,1:22:09.420 pretend that I[br]forgot-- let me pretend 1:22:09.420,1:22:14.250 that I forgot what the[br]derivative [INAUDIBLE] notation 1:22:14.250,1:22:17.722 of tangent of t was. 1:22:17.722,1:22:19.786 STUDENT: Secant squared. 1:22:19.786,1:22:23.237 MAGDALENA TODA: You guys love[br]that secant squared thingy. 1:22:23.237,1:22:26.195 1:22:26.195,1:22:30.632 Why do you like secant squared? 1:22:30.632,1:22:34.083 I, as a student, I didn't[br]like expressing it like that. 1:22:34.083,1:22:35.562 I liked [INAUDIBLE]. 1:22:35.562,1:22:37.041 Of course, it's the same thing. 1:22:37.041,1:22:40.492 But I always like it like[br]1 over cosine [INAUDIBLE]. 1:22:40.492,1:22:45.422 1:22:45.422,1:22:47.720 And of course, I have[br]to ask you something, 1:22:47.720,1:22:52.224 because I'm curious to[br]see what you remember. 1:22:52.224,1:22:55.212 And you say yeah,[br]curiosity killed the cat. 1:22:55.212,1:23:00.192 But where did the[br]derivative exist? 1:23:00.192,1:23:06.720 Because maybe was[br]that tangent of T-- 1:23:06.720,1:23:08.300 STUDENT: Wasn't[br]it a quotient rule 1:23:08.300,1:23:10.260 of sine and [? cosine x? ?] 1:23:10.260,1:23:11.730 MAGDALENA TODA: Good. 1:23:11.730,1:23:15.170 I'm proud of you. 1:23:15.170,1:23:17.430 That is the answer. 1:23:17.430,1:23:23.170 So [? my ?] [? have ?] this[br]blowing up, this blows up-- 1:23:23.170,1:23:29.520 blows up where cosine[br]T was zero, right? 1:23:29.520,1:23:32.380 So where did that blow up? 1:23:32.380,1:23:36.780 [INAUDIBLE] blow up of[br]cosine and zero [INAUDIBLE]. 1:23:36.780,1:23:40.740 The cosine was the[br]shadow on the x-axis. 1:23:40.740,1:23:43.992 So here you blow up here, you[br]blow up here, you blow up here, 1:23:43.992,1:23:44.700 you blow up here. 1:23:44.700,1:23:49.155 1:23:49.155,1:23:51.630 So [? what does ?] [INAUDIBLE]. 1:23:51.630,1:23:53.115 It should not be what? 1:23:53.115,1:23:55.496 STUDENT: Pi over 2. 1:23:55.496,1:23:56.370 MAGDALENA TODA: Yeah. 1:23:56.370,1:23:59.630 And can we express[br]that OK, among 0pi, 1:23:59.630,1:24:03.320 let's say you go in[br]between 0 and 2pi only. 1:24:03.320,1:24:08.300 I get rid of pi over[br]2 and 3pi over 2. 1:24:08.300,1:24:11.940 But if I express that in[br]general for [INAUDIBLE] T 1:24:11.940,1:24:15.000 not restricted to 0[br]to T, what do I say? 1:24:15.000,1:24:15.960 STUDENT: It's k. 1:24:15.960,1:24:18.820 STUDENT: So it can[br][? never be ?] pi over 2 1:24:18.820,1:24:19.320 plus pi? 1:24:19.320,1:24:21.240 MAGDALENA TODA: 2k plus 1. 1:24:21.240,1:24:23.952 2k plus 1. 1:24:23.952,1:24:25.398 Odd number over-- 1:24:25.398,1:24:26.243 STUDENT: Pi over 2. 1:24:26.243,1:24:27.326 MAGDALENA TODA: Pi over 2. 1:24:27.326,1:24:28.290 Odd number, pi over 2. 1:24:28.290,1:24:30.200 And all the odd[br]numbers are 2k plus 1. 1:24:30.200,1:24:30.700 Right? 1:24:30.700,1:24:32.146 All right. 1:24:32.146,1:24:38.990 So you have a not[br]existence and-- OK. 1:24:38.990,1:24:39.640 Coming back. 1:24:39.640,1:24:42.410 I'm just playing, because[br]we are still in the break. 1:24:42.410,1:24:44.290 Now we are ready. 1:24:44.290,1:24:50.195 What is dfdx, del f, del x, xy. 1:24:50.195,1:24:51.920 And what is del f, del y? 1:24:51.920,1:24:55.600 I'm not going to ask you for[br]the second partial derivative. 1:24:55.600,1:24:57.380 We've had enough of that. 1:24:57.380,1:25:05.230 We also agreed that we have[br]important results in that. 1:25:05.230,1:25:08.980 What is the final answer here? 1:25:08.980,1:25:13.940 STUDENT: [INAUDIBLE] plus[br]x-squared [INAUDIBLE]. 1:25:13.940,1:25:15.490 MAGDALENA TODA: 1[br]over [INAUDIBLE]. 1:25:15.490,1:25:17.980 I love this one, OK? 1:25:17.980,1:25:20.480 Don't tell me what I[br]want to [INAUDIBLE]. 1:25:20.480,1:25:22.376 I'm just kidding. 1:25:22.376,1:25:23.940 [INAUDIBLE] squared times-- 1:25:23.940,1:25:24.805 STUDENT: 2x. 1:25:24.805,1:25:26.450 MAGDALENA TODA: 2x, good. 1:25:26.450,1:25:27.700 How about the other one? 1:25:27.700,1:25:28.440 The same thing. 1:25:28.440,1:25:34.320 1:25:34.320,1:25:36.670 Times 2y. 1:25:36.670,1:25:41.840 1:25:41.840,1:25:43.300 OK. 1:25:43.300,1:25:46.280 I want to tell you something[br]that I will repeat. 1:25:46.280,1:25:49.365 But you will see it[br]all through the course. 1:25:49.365,1:25:52.180 There is a certain[br]notion that Alexander, 1:25:52.180,1:25:54.435 who is not talking--[br]I'm just kidding, 1:25:54.435,1:25:58.400 you can talk-- he[br]reminded me of gradient. 1:25:58.400,1:26:02.960 We don't talk about gradient[br]until a few sections from now. 1:26:02.960,1:26:05.120 But I'd like to[br]anticipate a little bit. 1:26:05.120,1:26:08.440 So the gradient of[br]a function, wherever 1:26:08.440,1:26:15.080 the partial derivatives exist,[br]with the partial derivative-- 1:26:15.080,1:26:21.215 that is, f sub x[br]and f sub y exist-- 1:26:21.215,1:26:26.572 I'm going to have that[br]delta f-- nabla f. 1:26:26.572,1:26:29.494 nabla is a [INAUDIBLE]. 1:26:29.494,1:26:34.340 Nable f at xy represents what? 1:26:34.340,1:26:34.850 The vector. 1:26:34.850,1:26:37.400 1:26:37.400,1:26:39.020 And I know you love vectors. 1:26:39.020,1:26:45.580 And that's why I'm going back[br]to the vector notation f sub x 1:26:45.580,1:26:51.508 at xy times i, i being[br]the standard vector i 1:26:51.508,1:26:59.130 unit along the x axis,[br]f sub y at xy times j. 1:26:59.130,1:27:03.450 STUDENT: So it's just like[br]the notation of [INAUDIBLE]? 1:27:03.450,1:27:05.322 MAGDALENA TODA: Just[br]the vector notation. 1:27:05.322,1:27:08.010 How else could I write it? 1:27:08.010,1:27:13.280 Angular bracket, f sub x x[br]at xy, comma, f sub y at xy. 1:27:13.280,1:27:16.935 And you know-- people who[br]saw my videos, colleagues 1:27:16.935,1:27:19.770 who teach Calc 3[br]at the same time 1:27:19.770,1:27:25.110 said I have a tendency of not[br]going by the book notations 1:27:25.110,1:27:28.260 all the time, and just give you[br]the [? round ?] parentheses. 1:27:28.260,1:27:28.960 It's OK. 1:27:28.960,1:27:31.340 I mean, different books,[br]different notations. 1:27:31.340,1:27:35.460 But what I mean is to represent[br]the vector in the standard way 1:27:35.460,1:27:37.430 [INAUDIBLE]. 1:27:37.430,1:27:38.010 All right. 1:27:38.010,1:27:38.510 OK. 1:27:38.510,1:27:41.760 Can you have this[br]notion for something 1:27:41.760,1:27:44.840 like a function of[br]three variables? 1:27:44.840,1:27:45.870 Absolutely. 1:27:45.870,1:27:48.430 Now I'll give you an easy one. 1:27:48.430,1:27:50.920 Suppose that you have[br]x-squared plus y-squared 1:27:50.920,1:27:54.402 plus z-squared equals 1. 1:27:54.402,1:28:00.130 And that is called-- let's[br]call it names-- f of x, y, z. 1:28:00.130,1:28:16.800 Compute the gradient nabla f[br]at any point x, y, z for f. 1:28:16.800,1:28:20.705 Find the meaning of that[br]gradient-- of that-- find 1:28:20.705,1:28:29.460 the geometric meaning of it. 1:28:29.460,1:28:34.210 For this case, not in[br]general, for this case. 1:28:34.210,1:28:35.860 So you say, wait,[br]wait, Magdalena. 1:28:35.860,1:28:38.846 A-dah-dah, you're confusing me. 1:28:38.846,1:28:39.941 This is the gradient. 1:28:39.941,1:28:40.440 Hmm. 1:28:40.440,1:28:43.360 Depends on how many[br]variables you have. 1:28:43.360,1:28:47.680 So you have to show a vector[br]whose coordinates represent 1:28:47.680,1:28:50.800 the partial derivatives with[br]respect to all the variables. 1:28:50.800,1:28:56.065 If I have n variables, I have[br]f sub x1 comma f sub x2 comma 1:28:56.065,1:28:58.770 f sub x3 comma f[br]sub xn, and stop. 1:28:58.770,1:28:59.570 Yes, sir. 1:28:59.570,1:29:03.610 STUDENT: If the formula[br]was just f of xy, 1:29:03.610,1:29:05.160 wouldn't that be implicit? 1:29:05.160,1:29:06.640 MAGDALENA TODA:[br]That is implicit. 1:29:06.640,1:29:08.510 That's exactly what I meant. 1:29:08.510,1:29:12.220 What's the geometric[br]meaning of this animal? 1:29:12.220,1:29:13.920 Forget about the left hand side. 1:29:13.920,1:29:15.510 I'm going to clean it quickly. 1:29:15.510,1:29:16.740 What is that animal? 1:29:16.740,1:29:19.640 That is a hippopotamus. 1:29:19.640,1:29:20.500 What is that? 1:29:20.500,1:29:22.148 STUDENT: It's a sphere. 1:29:22.148,1:29:23.398 MAGDALENA TODA: It's a sphere. 1:29:23.398,1:29:24.847 But what kind of sphere? 1:29:24.847,1:29:27.950 Center 0, 0, 0 with radius 1. 1:29:27.950,1:29:30.150 What do we call that? 1:29:30.150,1:29:30.670 Unit sphere. 1:29:30.670,1:29:33.594 Do you know what notation[br]that mathematicians 1:29:33.594,1:29:36.516 use for that object? 1:29:36.516,1:29:40.380 You don't know but I'll[br]tell you. s1 is the sphere. 1:29:40.380,1:29:42.340 We have s2, I'm[br]sorry, the sphere 1:29:42.340,1:29:45.190 of dimension 2, which[br]means the surface. 1:29:45.190,1:29:47.127 s1 is the circle. 1:29:47.127,1:29:49.115 s1 is a circle. 1:29:49.115,1:29:51.580 s2 is a sphere. 1:29:51.580,1:29:54.860 So what is this number[br]here for a mathematician? 1:29:54.860,1:29:59.050 That's the dimension of[br]that kind of manifold. 1:29:59.050,1:30:02.310 So if I have just a[br]circle, we call it s1 1:30:02.310,1:30:05.510 because there is only a one[br]independent variable, which 1:30:05.510,1:30:08.000 is time, and we parameterize. 1:30:08.000,1:30:09.135 Why go clockwise? 1:30:09.135,1:30:09.745 Shame on me. 1:30:09.745,1:30:12.190 Go counterclockwise. 1:30:12.190,1:30:13.030 All right. 1:30:13.030,1:30:14.110 That's s1. 1:30:14.110,1:30:16.260 For s2, I have two[br]degrees of freedom. 1:30:16.260,1:30:18.770 It's a surface. 1:30:18.770,1:30:23.005 On earth, what are those[br]two degrees of freedom? 1:30:23.005,1:30:25.980 It's a riddle. 1:30:25.980,1:30:27.065 No extra credit. 1:30:27.065,1:30:30.320 STUDENT: The latitude[br]and longitude? 1:30:30.320,1:30:31.882 MAGDALENA TODA: Who said it? 1:30:31.882,1:30:33.532 Who said it first? 1:30:33.532,1:30:35.300 STUDENT: [INAUDIBLE]. 1:30:35.300,1:30:40.185 MAGDALENA TODA: How many of[br]you said it at the same time? 1:30:40.185,1:30:40.935 Alexander said it. 1:30:40.935,1:30:42.726 STUDENT: I know there[br]was one other person. 1:30:42.726,1:30:43.910 I wasn't the only one. 1:30:43.910,1:30:44.660 STUDENT: I didn't. 1:30:44.660,1:30:47.490 1:30:47.490,1:30:48.882 STUDENT: [INAUDIBLE], sorry. 1:30:48.882,1:30:49.715 [INTERPOSING VOICES] 1:30:49.715,1:30:52.420 MAGDALENA TODA: I[br]don't have enough. 1:30:52.420,1:30:54.340 STUDENT: I'll take[br]the credit for it. 1:30:54.340,1:30:56.170 MAGDALENA TODA:[br][INAUDIBLE] extra credit. 1:30:56.170,1:30:58.620 OK, you choose. 1:30:58.620,1:30:59.560 These are good. 1:30:59.560,1:31:01.950 They are Valentine's hearts,[br]chocolate [INAUDIBLE]. 1:31:01.950,1:31:04.791 1:31:04.791,1:31:05.290 Wilson. 1:31:05.290,1:31:09.385 1:31:09.385,1:31:12.040 I heard you saying Wilson. 1:31:12.040,1:31:13.008 I have more. 1:31:13.008,1:31:13.976 I have more. 1:31:13.976,1:31:17.364 These are cough drops,[br]so I'm [INAUDIBLE]. 1:31:17.364,1:31:20.489 You set it right[br]next time, Alexander. 1:31:20.489,1:31:21.968 STUDENT: [INAUDIBLE]. 1:31:21.968,1:31:22.954 MAGDALENA TODA: OK. 1:31:22.954,1:31:23.940 Anybody else? 1:31:23.940,1:31:26.398 Anybody needing cough drops? 1:31:26.398,1:31:26.898 OK. 1:31:26.898,1:31:27.884 I'll leave them here. 1:31:27.884,1:31:29.363 Just let me see. 1:31:29.363,1:31:31.830 Do I have more chocolate? 1:31:31.830,1:31:32.986 Eh, next time. 1:31:32.986,1:31:35.160 I'm going to get some[br]before-- we have-- we 1:31:35.160,1:31:37.090 need before Valentine's, right? 1:31:37.090,1:31:37.970 So it's Thursday. 1:31:37.970,1:31:41.060 I'm going to bring[br]you a lot more. 1:31:41.060,1:31:46.650 So in that case, what[br]is the gradient of f? 1:31:46.650,1:31:47.970 An x, y, z. 1:31:47.970,1:31:48.470 Aha. 1:31:48.470,1:31:50.170 I have three variables. 1:31:50.170,1:31:52.520 What's the gradient? 1:31:52.520,1:31:56.050 I can write it as a[br]bracket, angular notation. 1:31:56.050,1:31:58.070 Am I right? 1:31:58.070,1:32:02.790 Or I can write it 2xi[br]plus 2ij plus 2zk. 1:32:02.790,1:32:06.590 Can anybody tell me why? 1:32:06.590,1:32:09.640 What in the world are[br]these, 2x, 2y, 2z? 1:32:09.640,1:32:11.697 STUDENT: Those are the[br]partial derivatives. 1:32:11.697,1:32:14.030 MAGDALENA TODA: They are[br]exactly the partial derivatives 1:32:14.030,1:32:17.810 with respect to x, with respect[br]to y, with respect to z. 1:32:17.810,1:32:19.410 Does this have a[br]geometric meaning? 1:32:19.410,1:32:20.540 I don't know. 1:32:20.540,1:32:21.870 I have to draw. 1:32:21.870,1:32:24.210 And maybe when I[br]draw, I get an idea. 1:32:24.210,1:32:29.105 1:32:29.105,1:32:31.526 Is this a unit vector? 1:32:31.526,1:32:32.482 Uh-uh. 1:32:32.482,1:32:33.980 It's not. 1:32:33.980,1:32:35.770 Nabla s, right. 1:32:35.770,1:32:36.490 In a way it is. 1:32:36.490,1:32:37.800 It's not a unit vector. 1:32:37.800,1:32:41.040 But if I were to[br][? uniterize ?] it-- 1:32:41.040,1:32:43.710 and you know very well what it[br]means to [? uniterize it ?]. 1:32:43.710,1:32:44.732 It means to-- 1:32:44.732,1:32:45.690 STUDENT: Divide it by-- 1:32:45.690,1:32:47.440 MAGDALENA TODA: Divide[br]it by its magnitude 1:32:47.440,1:32:51.075 and make it a unit vector[br]that would have a meaning. 1:32:51.075,1:32:52.180 This is the sphere. 1:32:52.180,1:32:56.016 1:32:56.016,1:32:57.670 What if I make like this? 1:32:57.670,1:33:04.145 n equals nabla f over[br]a magnitude of f. 1:33:04.145,1:33:10.050 And what is the meaning[br]of that going to be? 1:33:10.050,1:33:12.066 Can you tell me what[br]I'm going to get here? 1:33:12.066,1:33:18.830 1:33:18.830,1:33:24.870 In your head,[br]compute the magnitude 1:33:24.870,1:33:29.430 and divide by the magnitude,[br]and you have exactly 15 seconds 1:33:29.430,1:33:31.420 to tell me what it is. 1:33:31.420,1:33:32.845 STUDENT: [INAUDIBLE]. 1:33:32.845,1:33:34.470 MAGDALENA TODA:[br][? Ryan, ?] [? Ryan, ?] 1:33:34.470,1:33:36.170 you are in a Twilight Zone. 1:33:36.170,1:33:39.798 But I'm sure once I tell you,[br]once I tell you, [INAUDIBLE]. 1:33:39.798,1:33:41.548 STUDENT: 1 divided by[br]the square root of 2 1:33:41.548,1:33:42.922 for the [? i controller. ?] 1:33:42.922,1:33:43.797 STUDENT: [INAUDIBLE]. 1:33:43.797,1:33:47.610 1:33:47.610,1:33:49.730 MAGDALENA TODA: Well, OK. 1:33:49.730,1:33:51.125 Say it again, somebody. 1:33:51.125,1:33:52.990 STUDENT: x plus y plus z. 1:33:52.990,1:33:58.112 MAGDALENA TODA: xi plus yj[br]plus zk, not x plus x, y, 1:33:58.112,1:33:59.820 z because that[br]would be a mistake. 1:33:59.820,1:34:03.325 It would be a scalar function.[br][INAUDIBLE] has to be a vector. 1:34:03.325,1:34:07.190 If I am to draw this vector,[br]how am I going to draw it? 1:34:07.190,1:34:10.030 Well, this is the[br]position vector. 1:34:10.030,1:34:11.220 Say it again. 1:34:11.220,1:34:12.720 This is the position vector. 1:34:12.720,1:34:15.881 When I have a point on this[br]stinking earth, whatever 1:34:15.881,1:34:21.100 it is, x, y, z, the[br]position vector is x, y, z. 1:34:21.100,1:34:26.260 It's xi plus yj plus zk. 1:34:26.260,1:34:28.690 I have this identification[br]between the point 1:34:28.690,1:34:29.639 and the vector. 1:34:29.639,1:34:30.430 This is our vector. 1:34:30.430,1:34:33.300 So I'm going to draw these[br]needles, all these needles, 1:34:33.300,1:34:41.880 all these vectors whose tips[br]are exactly on the sphere. 1:34:41.880,1:34:42.960 So why? 1:34:42.960,1:34:43.870 You say, OK. 1:34:43.870,1:34:46.470 I understand that is[br]the position vector, 1:34:46.470,1:34:48.735 but why did you put an n here? 1:34:48.735,1:34:52.900 And anybody who answers[br]that gets a cough drops. 1:34:52.900,1:34:54.737 STUDENT: [INAUDIBLE]. 1:34:54.737,1:34:56.070 MAGDALENA TODA: Because that is? 1:34:56.070,1:34:58.235 STUDENT: The normal[br]to the surface. 1:34:58.235,1:34:59.360 MAGDALENA TODA: You get a-- 1:34:59.360,1:35:00.824 STUDENT: Yeah, cough drop. 1:35:00.824,1:35:02.288 MAGDALENA TODA: Two of them. 1:35:02.288,1:35:03.157 STUDENT: Aw, yeah. 1:35:03.157,1:35:04.240 MAGDALENA TODA: All right. 1:35:04.240,1:35:07.840 So that's the normal[br]to the surface, which 1:35:07.840,1:35:11.087 would be a continuation[br]of the position vector. 1:35:11.087,1:35:11.670 You see, guys? 1:35:11.670,1:35:14.450 So imagine you take[br]your position vector. 1:35:14.450,1:35:15.840 This is the sphere. 1:35:15.840,1:35:17.580 It's like an egg. 1:35:17.580,1:35:20.550 And these tips[br]are on the sphere. 1:35:20.550,1:35:24.950 If you continue from[br]sitting on the sphere, 1:35:24.950,1:35:29.240 another radius vector[br]colinear to that, 1:35:29.240,1:35:31.280 that would be the[br]normal to the sphere. 1:35:31.280,1:35:36.480 So in topology, we[br]have a name for that. 1:35:36.480,1:35:38.914 We call that the hairy ball. 1:35:38.914,1:35:41.970 The hairy ball in[br]mathematics, I'm not kidding, 1:35:41.970,1:35:44.410 it's a concentrated notations. 1:35:44.410,1:35:47.620 You see it in graduate[br]courses, if you're 1:35:47.620,1:35:50.157 going to become a graduate[br]student in mathematics, 1:35:50.157,1:35:51.990 or you want to do a[br]dual degree or whatever, 1:35:51.990,1:35:55.690 you're going to see the hairy[br]ball, all those normal vectors 1:35:55.690,1:35:58.820 of length 1. 1:35:58.820,1:36:01.620 It's also called[br]the normal field. 1:36:01.620,1:36:04.526 So if you ask Dr.[br]Ibragimov, because he 1:36:04.526,1:36:08.900 is in this kind of field[br]theory, [INAUDIBLE] normal field 1:36:08.900,1:36:10.080 to a surface. 1:36:10.080,1:36:13.010 But for the topologists[br]or geometers, 1:36:13.010,1:36:15.300 they say, oh, that's[br]the hairy ball. 1:36:15.300,1:36:18.860 So if you ask him what the[br]hairy ball is, he will say, 1:36:18.860,1:36:21.800 why are you talking[br]nonsense to me? 1:36:21.800,1:36:22.780 Right. 1:36:22.780,1:36:24.250 Exactly. 1:36:24.250,1:36:30.745 So here's where we stopped[br]our intrusion in chapter 11. 1:36:30.745,1:36:33.129 It's going to be as[br]fun as it was today 1:36:33.129,1:36:34.420 with these partial derivatives. 1:36:34.420,1:36:35.590 You're going to love them. 1:36:35.590,1:36:39.860 You have a lot of computations[br]like the ones we did today. 1:36:39.860,1:36:42.590 Let's go back to[br]something you hated, 1:36:42.590,1:36:45.630 which is the parameterizations. 1:36:45.630,1:36:48.590 So one of you--[br]no, three of you-- 1:36:48.590,1:36:51.606 asked me to redo one[br]problem like the one 1:36:51.606,1:36:54.366 with the parameterization[br]of a circle. 1:36:54.366,1:36:58.280 But now I have to pay[br]attention to the data 1:36:58.280,1:36:59.920 that I come up with. 1:36:59.920,1:37:14.258 So write the parameterization[br]of a circle of radius. 1:37:14.258,1:37:17.240 1:37:17.240,1:37:20.888 Do you want specific[br]data or you want letters? 1:37:20.888,1:37:21.763 STUDENT: [INAUDIBLE]. 1:37:21.763,1:37:25.561 1:37:25.561,1:37:26.352 MAGDALENA TODA: OK. 1:37:26.352,1:37:30.480 Let's do it [INAUDIBLE] r,[br]and then I'll give an example. 1:37:30.480,1:37:43.230 And center x0, y0 in plane[br]where-- what is the point? 1:37:43.230,1:37:57.200 Where is the particle[br]moving for time t equals 0? 1:37:57.200,1:37:59.260 Where is it located? 1:37:59.260,1:38:00.310 All right. 1:38:00.310,1:38:02.746 So review. 1:38:02.746,1:38:15.624 We had frame that we always[br]picked at the origin. 1:38:15.624,1:38:23.490 That was bad because we could[br]pick x0, y0 as a center, 1:38:23.490,1:38:25.115 and that has a separate radius. 1:38:25.115,1:38:31.905 1:38:31.905,1:38:39.050 And now, they want me to write[br]a parameterization of a circle. 1:38:39.050,1:38:41.020 How do you achieve it? 1:38:41.020,1:38:49.400 You say the circle is x minus x0[br]squared plus y minus y0 squared 1:38:49.400,1:38:50.980 equals r squared. 1:38:50.980,1:38:53.740 And one of you[br]asked me by email-- 1:38:53.740,1:38:56.650 and that was a good[br]question-- you said, come on. 1:38:56.650,1:38:58.920 Look, it was [INAUDIBLE]. 1:38:58.920,1:39:02.760 So you said, I was[br]quite good in math. 1:39:02.760,1:39:04.050 I was smart. 1:39:04.050,1:39:09.550 Why didn't I know the equations,[br]the parametric equations, 1:39:09.550,1:39:11.490 or even this? 1:39:11.490,1:39:13.730 I'll tell you why. 1:39:13.730,1:39:15.850 This used to be[br]covered in high school. 1:39:15.850,1:39:18.056 It's something called[br]college algebra. 1:39:18.056,1:39:21.460 We had a chapter,[br]either trigonometry 1:39:21.460,1:39:22.272 or college algebra. 1:39:22.272,1:39:24.520 We had a chapter called[br]analytic geometry. 1:39:24.520,1:39:26.340 This is analytic geometry. 1:39:26.340,1:39:28.530 It's the same chapter[br]in which you guys 1:39:28.530,1:39:33.510 covered conics, [INAUDIBLE],[br]ellipse, [INAUDIBLE], parabola. 1:39:33.510,1:39:36.120 It's no longer covered[br]in most high schools. 1:39:36.120,1:39:37.030 I asked around. 1:39:37.030,1:39:39.920 The teachers told[br]me that we reduced 1:39:39.920,1:39:41.810 the geometric[br]applications a lot, 1:39:41.810,1:39:47.920 according to the general[br]standards that are imposed. 1:39:47.920,1:39:51.603 That's a pity, because you[br]really need this in college. 1:39:51.603,1:39:52.590 All right. 1:39:52.590,1:39:55.520 So how do you come up[br]with a parameterization? 1:39:55.520,1:40:01.060 You say, I would like to[br]parameterize in such way 1:40:01.060,1:40:03.490 that this would be[br]easy to understand 1:40:03.490,1:40:06.300 this for Pythagorean theorem. 1:40:06.300,1:40:07.450 Oh, OK. 1:40:07.450,1:40:10.395 So what is the Pythagorean[br]theorem telling me? 1:40:10.395,1:40:14.240 It's telling you that if you are[br]in a unit circle practically, 1:40:14.240,1:40:19.005 then this is cosine and[br]theta and this is sine theta, 1:40:19.005,1:40:21.637 and the sum of[br]cosine theta squared 1:40:21.637,1:40:24.050 plus sine theta squared is 1. 1:40:24.050,1:40:26.778 This is 1, so that is[br]the Pythagorean theorem 1:40:26.778,1:40:28.722 [INAUDIBLE]. 1:40:28.722,1:40:38.230 So xy plus x0 should be cosine[br]of theta times an R. Why an R? 1:40:38.230,1:40:41.920 Because I want, when I square,[br]I want the R squared up. 1:40:41.920,1:40:46.230 And here, this guy inside[br]will be our sine [? thing. ?] 1:40:46.230,1:40:47.550 Am I going to be in good shape? 1:40:47.550,1:40:51.450 Yes, because when I[br]square this fellow squared 1:40:51.450,1:40:54.660 plus this fellow squared will[br]give me exactly R squared. 1:40:54.660,1:40:58.300 And here is my[br][INAUDIBLE] smiley face. 1:40:58.300,1:41:01.260 So I want to understand[br]what I'm doing. 1:41:01.260,1:41:05.440 x minus x0 must[br]be R cosine theta. 1:41:05.440,1:41:09.106 y minus y0 is R sine theta. 1:41:09.106,1:41:13.860 Theta in general is an[br]angular velocity, [INAUDIBLE]. 1:41:13.860,1:41:17.250 But it's also time, right? 1:41:17.250,1:41:19.290 It has the meaning[br]of time parameter. 1:41:19.290,1:41:22.980 So when we wrote those--[br]and some of you are bored, 1:41:22.980,1:41:25.620 but I think it's not[br]going to harm anybody 1:41:25.620,1:41:27.240 that I do this again. 1:41:27.240,1:41:36.406 R cosine of t plus x0 y is R[br]sine t plus x0, or plus y0. 1:41:36.406,1:41:41.490 Now note, all those[br]examples in web work, 1:41:41.490,1:41:43.840 they were not very imaginative. 1:41:43.840,1:41:47.580 They didn't mean for[br]you to try other things. 1:41:47.580,1:41:53.630 Like if one would put here[br]cosine of 5t or sine of 5t, 1:41:53.630,1:41:56.830 that person would move five[br]times faster on the circle. 1:41:56.830,1:42:00.240 And instead of being back[br]at 2 pi, in time 2 pi, 1:42:00.240,1:42:02.970 they would be there[br]in time 2 pi over 5. 1:42:02.970,1:42:06.860 All the examples-- and each of[br]you, it was randomized somehow. 1:42:06.860,1:42:09.730 Each of you has a[br]different data set. 1:42:09.730,1:42:11.970 Different R,[br]different x0 with 0, 1:42:11.970,1:42:15.570 and a different place where[br]the particle is moving. 1:42:15.570,1:42:18.580 But no matter what[br]they gave you, 1:42:18.580,1:42:21.910 it's a response to[br]the same problem. 1:42:21.910,1:42:26.930 And at time t equals[br]0, you have M. Do 1:42:26.930,1:42:28.650 you want me to call it M0? 1:42:28.650,1:42:33.090 Yes, from my initial-- M0. 1:42:33.090,1:42:41.040 For t equals 0, you're[br]going to have R plus x0. 1:42:41.040,1:42:44.670 And for t equals 0, you have y0. 1:42:44.670,1:42:50.425 So for example, Ryan had-- Ryan,[br]I don't remember what you had. 1:42:50.425,1:42:54.232 You had some where theta R was-- 1:42:54.232,1:42:54.940 STUDENT: 4 and 8. 1:42:54.940,1:42:57.057 MAGDALENA TODA: 7. 1:42:57.057,1:42:58.015 You, what did you have? 1:42:58.015,1:43:00.400 STUDENT: No, R was 7[br]and x was 3, y was 1. 1:43:00.400,1:43:03.510 MAGDALENA TODA: R[br]was 7 and x0 was-- 1:43:03.510,1:43:05.740 STUDENT: 3, 1. 1:43:05.740,1:43:11.570 MAGDALENA TODA: 3, 1 was x0, y0[br]so in that case, the point they 1:43:11.570,1:43:15.820 gave here was 7 plus 3. 1:43:15.820,1:43:16.825 Am I right, Ryan? 1:43:16.825,1:43:17.700 You can always check. 1:43:17.700,1:43:18.200 I remember. 1:43:18.200,1:43:22.010 It was 10 and God[br]knows, and 10 and 1. 1:43:22.010,1:43:25.810 So all of the data that[br]you had in that problem 1:43:25.810,1:43:30.480 was created so that you[br]have these equations. 1:43:30.480,1:43:36.392 And at time 0, you were exactly[br]at the time t equals 0 replaced 1:43:36.392,1:43:37.181 the t. 1:43:37.181,1:43:38.130 All right. 1:43:38.130,1:43:39.230 OK. 1:43:39.230,1:43:40.315 STUDENT: What's the M0? 1:43:40.315,1:43:42.280 What is-- 1:43:42.280,1:43:45.240 MAGDALENA TODA: M0[br]is Magdalena times 0. 1:43:45.240,1:43:46.820 I don't know. 1:43:46.820,1:43:51.045 I mean, it's the[br]point where you are. 1:43:51.045,1:43:55.140 I couldn't come up[br]with a better name. 1:43:55.140,1:44:01.569 So I'm going to[br]erase here and I'll 1:44:01.569,1:44:08.280 get to another problem, which[br]gave you guys a big headache. 1:44:08.280,1:44:16.570 And it's not so hard, but this[br]is the computational problem, 1:44:16.570,1:44:18.046 very pretty in itself. 1:44:18.046,1:44:24.934 1:44:24.934,1:44:35.706 [INAUDIBLE] cosine t i plus[br]e to the 3t sine t j plus e 1:44:35.706,1:44:36.540 to the 3tk. 1:44:36.540,1:44:40.040 1:44:40.040,1:44:43.850 And I think this was more or[br]less in everybody's homework 1:44:43.850,1:44:45.480 the same. 1:44:45.480,1:44:51.680 There's a position vector[br]given as parameterized form. 1:44:51.680,1:44:54.330 So since you love[br]parameterization so much, 1:44:54.330,1:45:00.523 I'm going to remind you what[br]that means for x and y and zr. 1:45:00.523,1:45:03.481 And what did they want from you? 1:45:03.481,1:45:07.920 I forget what number of[br]the problem that was. 1:45:07.920,1:45:16.270 They wanted the length of[br]the arc of a curve from t 1:45:16.270,1:45:18.604 equals-- I don't know. 1:45:18.604,1:45:19.580 STUDENT: 2 to 5. 1:45:19.580,1:45:21.044 MAGDALENA TODA: 2 to 5. 1:45:21.044,1:45:22.020 Thank you. 1:45:22.020,1:45:23.972 [INAUDIBLE] t equals 5. 1:45:23.972,1:45:30.316 So this is the beginning and the[br]end of the curve, the beginning 1:45:30.316,1:45:32.268 and the end of a curve. 1:45:32.268,1:45:35.630 So what is that going[br]to be [INAUDIBLE]? 1:45:35.630,1:45:40.500 How does [INAUDIBLE],[br]which we have 1:45:40.500,1:45:46.828 to write down 2 to 5[br]magnitude of r prime at t, dt. 1:45:46.828,1:45:50.140 1:45:50.140,1:45:53.000 And I don't know. 1:45:53.000,1:45:56.980 But I want to review this[br]because-- so what in the world? 1:45:56.980,1:45:59.540 Maybe I put this[br]on the midterm or I 1:45:59.540,1:46:03.530 make it a little bit easier,[br]but the same what I don't like, 1:46:03.530,1:46:05.070 it's time consuming. 1:46:05.070,1:46:07.780 But I can give you[br]something a lot easier 1:46:07.780,1:46:10.570 that tests the[br]concept, the idea, not 1:46:10.570,1:46:13.090 the computational power. 1:46:13.090,1:46:20.210 So r prime of t here with[br]a little bit of attention, 1:46:20.210,1:46:25.100 of course, most of you[br]computing this correctly. 1:46:25.100,1:46:28.010 You are just a little bit scared[br]of what happened after that, 1:46:28.010,1:46:30.467 and you should not be scared[br]because now I'll tell you 1:46:30.467,1:46:32.860 why you shouldn't be scared. 1:46:32.860,1:46:34.850 Chain rule, product rule. 1:46:34.850,1:46:37.890 So I have first prime-- 1:46:37.890,1:46:38.900 STUDENT: 3. 1:46:38.900,1:46:42.490 MAGDALENA TODA: 3 into the[br]3e second and [? time ?] 1:46:42.490,1:46:46.730 cosine t plus-- I'm[br]going to do that later. 1:46:46.730,1:46:48.320 I know what you're thinking. 1:46:48.320,1:46:49.792 STUDENT: e 3t. 1:46:49.792,1:46:53.264 MAGDALENA TODA: e to[br]the 3t minus sine. 1:46:53.264,1:46:55.600 I'm not worried[br]about this minus now. 1:46:55.600,1:46:57.430 I'll take care of that later. 1:46:57.430,1:46:58.010 Times i. 1:46:58.010,1:47:00.710 1:47:00.710,1:47:03.390 Now with your[br]permission-- when you 1:47:03.390,1:47:08.150 say, why is she not writing the[br]whole thing in continuation? 1:47:08.150,1:47:09.470 Because I don't want to. 1:47:09.470,1:47:09.970 No. 1:47:09.970,1:47:13.170 Because I want to help[br]you see what's going on. 1:47:13.170,1:47:16.310 You do the same kind of stuff[br]for this individual one. 1:47:16.310,1:47:17.804 I want to put it[br]right underneath. 1:47:17.804,1:47:21.290 If I put it right underneath,[br]it's going to [? agree ?]. 1:47:21.290,1:47:23.966 Otherwise it's not[br]going to [? agree ?]. 1:47:23.966,1:47:32.371 E to the 3t times sine t[br]plus e to the 3t cosine t. 1:47:32.371,1:47:34.111 You didn't have a[br]problem because you 1:47:34.111,1:47:36.000 know how to differentiate. 1:47:36.000,1:47:40.780 You started having the[br]problem from this point on. 1:47:40.780,1:47:44.412 3 into the 3tk. 1:47:44.412,1:47:47.010 The problem came when[br]you were supposed 1:47:47.010,1:47:55.510 to identify the coordinates and[br]square them and squeeze them 1:47:55.510,1:47:57.280 under the same square root. 1:47:57.280,1:48:01.250 And that drove you crazy[br]when you have enough. 1:48:01.250,1:48:04.210 Let me put the minus here to[br]make it more obvious what's 1:48:04.210,1:48:06.360 going to happen. 1:48:06.360,1:48:08.120 When you're going[br]to have problems 1:48:08.120,1:48:09.770 like that in[br]differential equations, 1:48:09.770,1:48:14.670 you better have the eye[br]for it, [INAUDIBLE]. 1:48:14.670,1:48:18.870 You should be able to recognize[br]this is like a pattern. 1:48:18.870,1:48:26.585 Have you seen the[br]movie A Beautiful Mind? 1:48:26.585,1:48:27.210 STUDENT: Yeah. 1:48:27.210,1:48:28.620 MAGDALENA TODA:[br]OK, so Nash, when 1:48:28.620,1:48:34.180 he was writing with the finger[br]on everything, on the walls 1:48:34.180,1:48:39.835 at Princeton, on the window,[br]he was thinking of patterns. 1:48:39.835,1:48:42.270 He's actually[br]trying to-- and it's 1:48:42.270,1:48:44.130 hard to visualize[br]without drawing, 1:48:44.130,1:48:48.374 but this is what most of us[br]recognize all the time when 1:48:48.374,1:48:51.212 a mathematician writes[br]down some computations 1:48:51.212,1:48:52.905 in a different way. 1:48:52.905,1:48:58.420 All we hope for is to get a[br]few steps behind that board 1:48:58.420,1:48:59.850 and see a pattern. 1:48:59.850,1:49:02.340 And when you do that,[br]you see the pattern. 1:49:02.340,1:49:05.580 This is an a minus b[br]and that's an a plus b. 1:49:05.580,1:49:08.685 And then you say, OK, if[br]I'm going to square them, 1:49:08.685,1:49:10.630 what's going to happen? 1:49:10.630,1:49:15.330 When you square an a minus[br]b and you square an a plus b 1:49:15.330,1:49:18.810 and you have this giggly[br]guy there-- leave him there. 1:49:18.810,1:49:21.980 He's having too much fun. 1:49:21.980,1:49:28.194 You actually develop these[br]guys and you put them one 1:49:28.194,1:49:31.530 under the other[br]and say wow, what 1:49:31.530,1:49:34.280 a beautiful simplification. 1:49:34.280,1:49:36.760 When I'm going to[br]add these guys, 1:49:36.760,1:49:40.340 this thing in the middle[br]will simply will cancel out, 1:49:40.340,1:49:44.700 but the a squared will double[br]and the b squared will double. 1:49:44.700,1:49:46.670 And that's the beauty[br]of seeing pattern. 1:49:46.670,1:49:50.970 You see how there is[br]something symmetric and magic 1:49:50.970,1:49:56.470 in mathematics that make[br]the answer simplified. 1:49:56.470,1:50:01.360 And that allows you to compress[br]your equations that originally 1:50:01.360,1:50:05.740 seemed to be a mess[br]into something that's 1:50:05.740,1:50:08.595 more easily expressed. 1:50:08.595,1:50:11.450 So when you're going[br]to compute this r 1:50:11.450,1:50:17.792 prime of t magic absolute[br]value of the magnitude, that's 1:50:17.792,1:50:21.664 going to be square root of--[br]instead of writing all the 1:50:21.664,1:50:25.190 [INAUDIBLE], I hate writing[br]and rewriting the whole thing 1:50:25.190,1:50:28.804 squared plus the whole thing[br]squared plus this squared. 1:50:28.804,1:50:32.970 If I love to write so much,[br]I'd be in humanities and not 1:50:32.970,1:50:34.780 in mathematics. 1:50:34.780,1:50:41.175 So as a mathematician, how[br]am I going to write that? 1:50:41.175,1:50:44.379 As a mathematician, I'm going[br]to use some sort of-- like the U 1:50:44.379,1:50:44.920 substitution. 1:50:44.920,1:50:48.910 So I say, I call this Mr.[br]A, and I call this Mr. B. 1:50:48.910,1:50:50.966 And that's A minus B,[br]and that's A plus B. 1:50:50.966,1:50:53.810 And that's somebody else. 1:50:53.810,1:50:57.470 So when I square[br]the first guy, and I 1:50:57.470,1:51:00.668 square the second component, and[br]I square the third component, 1:51:00.668,1:51:09.780 and I add them together,[br]I'm going to get what? 1:51:09.780,1:51:15.810 Square root of 2A[br]squared plus 2B squared. 1:51:15.810,1:51:19.010 Because I know that[br]these are the first two. 1:51:19.010,1:51:21.290 This guy squared[br]plus this guy squared 1:51:21.290,1:51:23.430 is going to be[br]exactly 2A squared 1:51:23.430,1:51:25.974 plus 2B squared,[br]nothing in the middle. 1:51:25.974,1:51:28.830 These guys cancel out. 1:51:28.830,1:51:30.636 STUDENT: A and B[br]are not the same. 1:51:30.636,1:51:34.210 1:51:34.210,1:51:42.030 MAGDALENA TODA: Well,[br]yeah, you're right. 1:51:42.030,1:51:45.869 Let me call-- you're[br]right, this is the same, 1:51:45.869,1:51:47.630 but these are different. 1:51:47.630,1:51:51.995 So let me call them[br]A prime plus B prime. 1:51:51.995,1:51:53.680 No, that's derivative. 1:51:53.680,1:51:56.244 Let me call them C[br]and D-- very good, 1:51:56.244,1:52:03.810 thank you-- C squared[br]plus 2CD plus D squared. 1:52:03.810,1:52:06.370 1:52:06.370,1:52:08.130 But the principle is the same. 1:52:08.130,1:52:11.492 So I'm going to have A[br]squared plus C squared. 1:52:11.492,1:52:12.940 This goes away. 1:52:12.940,1:52:14.660 Why? 1:52:14.660,1:52:18.530 Because this times that is[br]the same as this times that. 1:52:18.530,1:52:19.836 Say it again. 1:52:19.836,1:52:22.692 If we look in the[br]middle, the middle term 1:52:22.692,1:52:28.298 will have 3e to the 3t cosine[br]t times e to the 3t sine t. 1:52:28.298,1:52:33.030 Middle term here is 3e to the[br]3t e to the 3t sine and cosine. 1:52:33.030,1:52:36.440 So they will cancel[br]out, this and that. 1:52:36.440,1:52:40.191 So here I have the[br]sum of the square of A 1:52:40.191,1:52:45.907 plus the square[br]of C. And here I'm 1:52:45.907,1:52:50.680 going to have the square[br]of B plus the square of D. 1:52:50.680,1:52:54.388 OK, now when I square this[br]and that, what do I get? 1:52:54.388,1:52:57.220 1:52:57.220,1:53:00.970 The beauty of that-- let me[br]write it down then explicitly. 1:53:00.970,1:53:06.910 9e to the 3t cosine squared[br]t remains from this guy. 1:53:06.910,1:53:08.790 Plus from the square[br]of that, we'll 1:53:08.790,1:53:19.588 have 9e to the 3t-- no, just 3,[br]9 to the 6t, 9 to the 6t sine 1:53:19.588,1:53:21.540 squared. 1:53:21.540,1:53:22.870 So I take this guy. 1:53:22.870,1:53:23.545 I square it. 1:53:23.545,1:53:24.712 I take this guy. 1:53:24.712,1:53:26.920 I square it. 1:53:26.920,1:53:30.050 The middle terms will[br]disappear, thank god. 1:53:30.050,1:53:33.490 Then I have this guy, I square[br]it, that guy, I square it, 1:53:33.490,1:53:34.840 good. 1:53:34.840,1:53:41.290 Plus another parenthesis-- e[br]to the 6t sine squared t plus e 1:53:41.290,1:53:44.266 to the 6t cosine squared t. 1:53:44.266,1:53:47.110 1:53:47.110,1:53:50.340 So even if they don't[br]double because they're not 1:53:50.340,1:53:52.820 the same thing, what[br]is the principle 1:53:52.820,1:53:54.390 that will make my life easier? 1:53:54.390,1:53:58.840 The same pattern[br]of simplification. 1:53:58.840,1:54:00.730 What is that same pattern[br]of simplification? 1:54:00.730,1:54:03.720 Look at the beauty[br]of this guy and look 1:54:03.720,1:54:05.110 at the beauty of this guy. 1:54:05.110,1:54:06.790 And then there is[br]something missing, 1:54:06.790,1:54:12.560 the happy guy that was quiet[br]because I told him to be quiet. 1:54:12.560,1:54:17.162 That's 9e to the 6t. 1:54:17.162,1:54:18.488 He was there in the corner. 1:54:18.488,1:54:22.330 And you had to square this[br]guy and square this guy 1:54:22.330,1:54:26.188 and square this guy and[br]add them on top together. 1:54:26.188,1:54:27.649 Now what is the pattern? 1:54:27.649,1:54:35.441 The pattern is 9e to the 6t[br]with 9e to the 6t, same guy. 1:54:35.441,1:54:38.190 The orange guys-- that's[br]why I love the colors. 1:54:38.190,1:54:40.595 Cosine squared cosine[br]squared will be 1. 1:54:40.595,1:54:47.400 Another pattern like that, I[br]have e to the 6t, to the 6t, 1:54:47.400,1:54:52.260 and the same happy guys sine[br]squared t, sine squared t, 1:54:52.260,1:54:54.660 add them together is 1. 1:54:54.660,1:55:00.505 So all in all, this mess[br]is not a mess anymore. 1:55:00.505,1:55:11.290 So it becomes 9e to the 6t plus[br]e to the 6t plus 9e to the 6t. 1:55:11.290,1:55:12.630 Are you guys with me? 1:55:12.630,1:55:17.960 All right, now how many[br]e to the 6t's do we have? 1:55:17.960,1:55:25.850 9 plus 9 plus 1, 19, square[br]root of 19 e to the 6t. 1:55:25.850,1:55:29.900 So when we integrate,[br]we go integral 1:55:29.900,1:55:33.410 from 2 to 5 square root of 19. 1:55:33.410,1:55:34.850 Kick him out of your life. 1:55:34.850,1:55:36.990 He's just making[br]your life harder. 1:55:36.990,1:55:40.065 And then you have square root[br]of e to the 6t e to the 3t. 1:55:40.065,1:55:42.910 1:55:42.910,1:55:47.930 So after you kick[br]the guy out, you 1:55:47.930,1:55:55.060 have e to the 3t divided[br]by 3 between t equals 2 1:55:55.060,1:55:58.170 and t equals 5. 1:55:58.170,1:56:03.230 Actually, I took it right off[br]the WeBWorK problem you had. 1:56:03.230,1:56:06.104 So if you type this[br]in your WeBWorK-- 1:56:06.104,1:56:12.000 you probably already did-- you[br]should get exactly the answer 1:56:12.000,1:56:13.202 as being correct. 1:56:13.202,1:56:17.800 1:56:17.800,1:56:24.160 On the exam, do not[br]expect anything that long. 1:56:24.160,1:56:26.720 The idea of simplifying[br]these patterns 1:56:26.720,1:56:31.780 by finding the sine cosine, sine[br]squared plus cosine squared is 1:56:31.780,1:56:33.110 1, is still going to be there. 1:56:33.110,1:56:35.690 But don't expect[br]anything that long. 1:56:35.690,1:56:43.369 Also, don't expect-- once[br]you get to this state, 1:56:43.369,1:56:44.818 I don't want an answer. 1:56:44.818,1:56:46.267 This is the answer. 1:56:46.267,1:56:48.199 That's the precise answer. 1:56:48.199,1:56:52.560 I don't want any approximation[br]or anything like that. 1:56:52.560,1:56:54.272 A few of you did this[br]with a calculator. 1:56:54.272,1:56:57.655 Well, you will not have[br]calculators in the final. 1:56:57.655,1:56:59.285 You are going to[br]have easy problems. 1:56:59.285,1:57:03.170 If you did that[br]with a calculator, 1:57:03.170,1:57:05.230 and you truncated[br]your answer later, 1:57:05.230,1:57:11.270 and if you were within[br]0.01 of the correct answer, 1:57:11.270,1:57:12.340 you were fine. 1:57:12.340,1:57:14.861 But some people[br]approximated too much. 1:57:14.861,1:57:16.785 And that's always a problem. 1:57:16.785,1:57:19.490 So it's always a good[br]idea to enter something 1:57:19.490,1:57:23.860 like that in WeBWorK. 1:57:23.860,1:57:27.470 I said I wouldn't do it[br]except in the last 20 minutes. 1:57:27.470,1:57:31.190 But I wanted to do[br]something like that. 1:57:31.190,1:57:34.500 I want to give you another[br]example, because you love 1:57:34.500,1:57:39.216 parametrization so much it just[br]occurred to me that it would 1:57:39.216,1:57:41.940 be very, very helpful--[br]maybe, I don't 1:57:41.940,1:57:47.060 know-- to give you another[br]problem similar to this one. 1:57:47.060,1:57:50.250 It's not in the book,[br]but it was cooked up 1:57:50.250,1:57:53.698 by one of my colleagues[br]for his homework. 1:57:53.698,1:58:02.554 So I'd like to show it to you. 1:58:02.554,1:58:06.490 1:58:06.490,1:58:09.584 e to the t i is[br]a parametrization 1:58:09.584,1:58:13.240 of a [INAUDIBLE] space. 1:58:13.240,1:58:28.139 Plus e to the minus t j[br]plus square root of 2 tk. 1:58:28.139,1:58:36.030 1:58:36.030,1:58:37.470 And how do I know? 1:58:37.470,1:58:41.102 Well, one of his[br]students came to me 1:58:41.102,1:58:43.656 and asked for help[br]with homework. 1:58:43.656,1:58:51.450 Well, we don't give help when[br]it comes from another colleague. 1:58:51.450,1:58:55.790 So in the end, the student[br]went to the tutoring center. 1:58:55.790,1:58:58.711 And the tutoring center[br]helped only in parts. 1:58:58.711,1:59:00.520 She came back to me. 1:59:00.520,1:59:03.860 So what was the deal here? 1:59:03.860,1:59:13.662 Find f prime of t in[br]the most simplified form 1:59:13.662,1:59:16.440 and find the absolute[br]value r prime of t 1:59:16.440,1:59:17.830 in the most simplified form. 1:59:17.830,1:59:22.830 1:59:22.830,1:59:31.830 And find the length of the[br]arc of this curve between t 1:59:31.830,1:59:33.824 equals 0 and t equals 1. 1:59:33.824,1:59:36.632 If this were given[br]by a physicist, 1:59:36.632,1:59:39.760 how would that physicist[br]reformulate the problem? 1:59:39.760,1:59:47.895 He would say-- he or she--[br]what is the distance travelled 1:59:47.895,1:59:54.450 by the particle between[br]0 seconds and 1 second? 1:59:54.450,1:59:56.125 So how do you write that? 1:59:56.125,2:00:03.550 Integral from 0 to 1 of[br]r prime of t [INAUDIBLE]. 2:00:03.550,2:00:05.530 And you have to do the rest. 2:00:05.530,2:00:08.510 2:00:08.510,2:00:13.040 So arguably, this is[br]the Chapter 10 review. 2:00:13.040,2:00:15.070 It's very useful for[br]the midterm exam. 2:00:15.070,2:00:17.570 So although we are[br]just doing this review, 2:00:17.570,2:00:20.690 you should not erase[br]it from your memory. 2:00:20.690,2:00:24.380 Because I don't like to[br]put surprise problems 2:00:24.380,2:00:25.250 on the midterm. 2:00:25.250,2:00:28.950 But if you worked a[br]certain type of problem, 2:00:28.950,2:00:31.320 you may expect[br]something like that. 2:00:31.320,2:00:33.720 Maybe it's different[br]but in the same spirit. 2:00:33.720,2:00:37.690 r prime of t, who's going to[br]help me with r prime of t? 2:00:37.690,2:00:40.720 2:00:40.720,2:00:44.140 This fellow-- e to the t. 2:00:44.140,2:00:46.863 And how about that? 2:00:46.863,2:00:50.300 Negative e to the negative t. 2:00:50.300,2:00:53.246 STUDENT: I thought the arc[br]length was the square root of 1 2:00:53.246,2:00:56.192 plus f prime of t squared. 2:00:56.192,2:00:58.730 2:00:58.730,2:01:02.355 MAGDALENA TODA:[br]For a plane curve. 2:01:02.355,2:01:04.440 OK, let me remind you. 2:01:04.440,2:01:05.980 If you have a plane[br]curve y equals 2:01:05.980,2:01:12.467 f of x, then this thing[br]would become integral from A 2:01:12.467,2:01:17.740 to B square root of 1[br]plus f prime of x dx. 2:01:17.740,2:01:22.010 And that, did you do that[br]with your Calc II instructor? 2:01:22.010,2:01:25.740 How many of you[br]had Dr. Williams? 2:01:25.740,2:01:28.000 That was a wonderful[br]class, wasn't it? 2:01:28.000,2:01:29.380 And he taught that. 2:01:29.380,2:01:31.460 And of course he[br]was not supposed 2:01:31.460,2:01:36.120 to tell you that was the[br]speed of a parametric curve. 2:01:36.120,2:01:39.020 If you were to[br]parametrize here, x of t 2:01:39.020,2:01:44.000 was t and y of t[br]would be f of t. 2:01:44.000,2:01:45.450 He could have told you. 2:01:45.450,2:01:46.320 Maybe he told you. 2:01:46.320,2:01:47.470 Maybe you don't remember. 2:01:47.470,2:01:48.990 OK, let's forget about it. 2:01:48.990,2:01:50.340 That was Calc II. 2:01:50.340,2:01:54.120 Now, coming back here,[br]I have to list what? 2:01:54.120,2:01:57.916 Square root of 2 times[br]t prime is one k. 2:01:57.916,2:01:59.582 Who's going to help[br]me compute the speed 2:01:59.582,2:02:02.380 and put it in a nice formula? 2:02:02.380,2:02:04.163 Well, my god-- 2:02:04.163,2:02:04.996 STUDENT: [INAUDIBLE] 2:02:04.996,2:02:08.230 2:02:08.230,2:02:10.790 MAGDALENA TODA: Ahh,[br]you are too smart. 2:02:10.790,2:02:15.152 Today you had some what is[br]that called with caffeine 2:02:15.152,2:02:17.036 and vitamins and-- 2:02:17.036,2:02:18.920 STUDENT: You're[br]thinking of Red Bull. 2:02:18.920,2:02:20.340 MAGDALENA TODA: I know. 2:02:20.340,2:02:22.660 That was very nice. 2:02:22.660,2:02:23.740 I try to stay away. 2:02:23.740,2:02:28.223 What is that called[br]with the energy booster? 2:02:28.223,2:02:29.264 STUDENT: I wouldn't know. 2:02:29.264,2:02:30.491 STUDENT: 5-Hour Energy. 2:02:30.491,2:02:31.719 MAGDALENA TODA: 5-Hour, OK. 2:02:31.719,2:02:33.192 I used to have that. 2:02:33.192,2:02:36.670 When I had that, I could[br]anticipate two steps computing. 2:02:36.670,2:02:39.809 Just a joke, Alex,[br]don't take it up. 2:02:39.809,2:02:40.725 Very good observation. 2:02:40.725,2:02:43.460 So Alex saw. 2:02:43.460,2:02:45.650 He has a premonition. 2:02:45.650,2:02:48.820 He can see two steps in advance. 2:02:48.820,2:02:50.915 He said, OK, square that. 2:02:50.915,2:02:52.710 You have e to the 2t. 2:02:52.710,2:02:53.395 Square this. 2:02:53.395,2:02:55.606 The minus doesn't matter. 2:02:55.606,2:03:00.330 Plus e to the minus[br]2t, and square that. 2:03:00.330,2:03:02.560 Then he saw patterns. 2:03:02.560,2:03:06.130 Because he is the[br]wizard 101 today. 2:03:06.130,2:03:09.090 So what is the[br]witchcraft he performed? 2:03:09.090,2:03:10.470 Do you see? 2:03:10.470,2:03:13.350 Does anybody else[br]see the pattern? 2:03:13.350,2:03:15.360 [? Nateesh ?] sees the pattern. 2:03:15.360,2:03:16.719 Anybody illuminated? 2:03:16.719,2:03:18.010 I didn't see it from the start. 2:03:18.010,2:03:19.660 You guys saw it faster than me. 2:03:19.660,2:03:23.190 It took me about a[br]minute and a half 2:03:23.190,2:03:26.710 when I saw this[br]for the first time. 2:03:26.710,2:03:29.930 Is this a perfect square? 2:03:29.930,2:03:32.060 Of who? 2:03:32.060,2:03:36.350 e to the t plus e to[br]the minus 2 squared 2:03:36.350,2:03:40.390 is-- anybody else sees the[br]pattern I don't have candy. 2:03:40.390,2:03:44.210 Next time-- Alex,[br][INAUDIBLE], anybody else? 2:03:44.210,2:03:47.000 Do you now see the[br]pattern, e to the 2t plus 2:03:47.000,2:03:51.340 e to the minus 2t plus[br]twice the product? 2:03:51.340,2:03:54.470 And that's where the student[br]was having the problem. 2:03:54.470,2:03:56.550 Where do you see the product? 2:03:56.550,2:03:58.474 The product is 1. 2:03:58.474,2:03:59.920 The product is 1 doubled. 2:03:59.920,2:04:02.100 So you get 2. 2:04:02.100,2:04:06.690 So it's indeed exactly[br]the perfect square. 2:04:06.690,2:04:09.430 So once-- it was a she. 2:04:09.430,2:04:14.490 Once she saw the perfect[br]square, she was so happy. 2:04:14.490,2:04:16.850 Because you get square[br]root of the square. 2:04:16.850,2:04:19.560 You get e to the t[br]plus e to the minus t. 2:04:19.560,2:04:22.694 And that's a trivial thing[br]to integrate that you 2:04:22.694,2:04:23.860 have no problem integrating. 2:04:23.860,2:04:26.980 It's a positive[br]function, very beautiful. 2:04:26.980,2:04:31.880 The professor who gave this was[br]Dr. [INAUDIBLE] from Denmark. 2:04:31.880,2:04:34.730 He's one of the best[br]teachers we have. 2:04:34.730,2:04:40.690 But he makes up his[br]homework as far as I know. 2:04:40.690,2:04:43.200 I think in the sixth[br]edition, this edition, 2:04:43.200,2:04:48.770 we actually stole his idea,[br]and we made a problem like that 2:04:48.770,2:04:51.490 in the book somewhere. 2:04:51.490,2:04:55.190 We doubled the number of[br]problems more or less. 2:04:55.190,2:05:00.900 So if you are to compute[br]0 to 1 of the speed, 2:05:00.900,2:05:03.069 what is the speed? 2:05:03.069,2:05:05.534 The speed is this[br]beautiful thing. 2:05:05.534,2:05:09.971 Because you were able[br]to see the pattern. 2:05:09.971,2:05:12.764 If you're not able[br]to see that, do you 2:05:12.764,2:05:15.440 realize it's[br]impossible, practically, 2:05:15.440,2:05:17.940 for you to integrate by hand? 2:05:17.940,2:05:22.700 You have to go to a[br]calculator, Matlab, whatever. 2:05:22.700,2:05:23.830 So this is easy. 2:05:23.830,2:05:29.170 Why is that easy? e to the t[br]minus e to the minus t at 1 2:05:29.170,2:05:32.040 and at 0-- you compare them. 2:05:32.040,2:05:36.420 You get at 1 e minus[br]e to the minus 1 2:05:36.420,2:05:41.090 minus the fundamental theorem[br]of calc e to the 0 minus 2:05:41.090,2:05:42.500 e to the 0. 2:05:42.500,2:05:43.620 Well, that's silly. 2:05:43.620,2:05:45.440 Why is that silly? 2:05:45.440,2:05:49.170 Because I'm going to give it up. 2:05:49.170,2:05:52.110 So the answer was[br]e to the minus 1/e. 2:05:52.110,2:05:54.570 And she knew what[br]the answer would be. 2:05:54.570,2:05:57.030 But she didn't know why. 2:05:57.030,2:05:58.434 So she came back to me. 2:05:58.434,2:06:02.610 I don't know how the tutoring[br]center helped her figure 2:06:02.610,2:06:03.470 out the answer. 2:06:03.470,2:06:06.200 But she did not[br]understand the solution. 2:06:06.200,2:06:08.946 So I said, I'm not going to[br]take anymore people coming 2:06:08.946,2:06:11.020 from Professor [INAUDIBLE]. 2:06:11.020,2:06:12.830 I was also told it's not OK. 2:06:12.830,2:06:16.670 So don't go to another[br]professor with homework coming 2:06:16.670,2:06:18.410 for me or the other way around. 2:06:18.410,2:06:20.600 Because it's not OK. 2:06:20.600,2:06:25.310 But you can go to the tutoring[br]center asking them for hints. 2:06:25.310,2:06:30.216 They're open starting 9:00[br]AM and until around when? 2:06:30.216,2:06:31.560 Do you know? 2:06:31.560,2:06:32.990 They used to have until 4:00. 2:06:32.990,2:06:35.870 But now they're going to[br]work on an extended schedule 2:06:35.870,2:06:37.850 until 8:00 PM. 2:06:37.850,2:06:40.325 It's going to be[br]something crazy. 2:06:40.325,2:06:43.790 Now, the thing is, we want[br]the students to be better, 2:06:43.790,2:06:48.620 to do better, to not give[br]up, to be successful, 2:06:48.620,2:06:51.730 top one, two, three. 2:06:51.730,2:06:54.290 I'm a little bit[br]concerned, but maybe I 2:06:54.290,2:06:56.572 shouldn't be, about those hours. 2:06:56.572,2:06:59.939 So I don't know if they managed[br]to put a security camera 2:06:59.939,2:07:00.901 or not. 2:07:00.901,2:07:04.520 But having extended[br]hours may be a problem. 2:07:04.520,2:07:09.780 Take advantage of[br]those afternoon hours, 2:07:09.780,2:07:11.726 especially if you are busy. 2:07:11.726,2:07:18.698 Those late hours will[br]be a big help for you. 2:07:18.698,2:07:21.266 Do you know where it is? 2:07:21.266,2:07:23.656 Room 106 over there. 2:07:23.656,2:07:26.530 2:07:26.530,2:07:29.800 Any other questions related[br]to this type of problem 2:07:29.800,2:07:35.240 or related to anything[br]else in the material 2:07:35.240,2:07:38.870 that maybe I can[br]give you hints on, 2:07:38.870,2:07:40.970 at least the hint I'm[br]going to give you? 2:07:40.970,2:07:44.860 Sometimes I cannot stop, and[br]I just give the problem away. 2:07:44.860,2:07:46.330 I'm not supposed to do that. 2:07:46.330,2:07:50.750 2:07:50.750,2:07:54.303 Look at your WeBWorK, see what[br]kind of help I can give you. 2:07:54.303,2:07:56.428 You still have a[br]little bit of time. 2:07:56.428,2:07:57.261 STUDENT: [INAUDIBLE] 2:07:57.261,2:08:00.712 2:08:00.712,2:08:05.140 MAGDALENA TODA: That's[br]the maximum of what? 2:08:05.140,2:08:06.677 It was-- 2:08:06.677,2:08:07.510 STUDENT: [INAUDIBLE] 2:08:07.510,2:08:11.110 2:08:11.110,2:08:12.680 MAGDALENA TODA: Was[br]this the problem? 2:08:12.680,2:08:14.600 STUDENT: e to the 2x[br]or something like that. 2:08:14.600,2:08:15.560 MAGDALENA TODA:[br]Something like that? 2:08:15.560,2:08:16.060 I erased it. 2:08:16.060,2:08:19.400 STUDENT: You erased[br]that? [INAUDIBLE]. 2:08:19.400,2:08:21.330 I found an answer. 2:08:21.330,2:08:23.395 MAGDALENA TODA: It's[br]very computational I saw. 2:08:23.395,2:08:26.750 But before that, I[br]saw that seven of you 2:08:26.750,2:08:28.990 guys-- you two also did it. 2:08:28.990,2:08:33.714 So I wrote-- you have a[br]brownie waiting for that. 2:08:33.714,2:08:35.163 But then I erased it. 2:08:35.163,2:08:39.510 STUDENT: You erased the previous[br]one too in the homework one. 2:08:39.510,2:08:42.040 MAGDALENA TODA: Because[br]that had a bug in it. 2:08:42.040,2:08:45.400 That one, the one in the[br]homework one, had a bug in it. 2:08:45.400,2:08:46.965 It only worked for some data. 2:08:46.965,2:08:50.090 And for other data[br]it didn't work. 2:08:50.090,2:08:53.580 So every time you find[br]a bug, you tell me, 2:08:53.580,2:08:56.200 and I will tell the programmer[br]of those problems, who's 2:08:56.200,2:08:57.010 really careful. 2:08:57.010,2:09:02.423 But one in 1,000 you[br]are bound to find a bug. 2:09:02.423,2:09:06.207 And I'm going to[br]give you a chocolate 2:09:06.207,2:09:08.092 or something for every bug. 2:09:08.092,2:09:09.820 And any other questions? 2:09:09.820,2:09:14.695 2:09:14.695,2:09:17.665 STUDENT: So are you[br]saying this is too long? 2:09:17.665,2:09:20.140 MAGDALENA TODA: Actually,[br]it's very beautiful. 2:09:20.140,2:09:23.605 If you have a calculator,[br]it's easier to solve it. 2:09:23.605,2:09:25.585 You can do it by hand,[br]write it by hand, also. 2:09:25.585,2:09:27.257 But it's a long-- 2:09:27.257,2:09:28.090 STUDENT: [INAUDIBLE] 2:09:28.090,2:09:30.776 2:09:30.776,2:09:34.262 MAGDALENA TODA: Right,[br]so let's do it now 2:09:34.262,2:09:36.752 for anybody who wants to stay. 2:09:36.752,2:09:37.748 You don't have to stay. 2:09:37.748,2:09:39.740 So practicing what you do-- 2:09:39.740,2:09:44.974 [SIDE CONVERSATIONS] 2:09:44.974,2:11:55.449