WEBVTT 00:00:00.000 --> 00:00:01.467 MAGDALENA TODA: We have any people 00:00:01.467 --> 00:00:04.890 who finished the extra credit and are 00:00:04.890 --> 00:00:08.313 willing to give it to me today? 00:00:08.313 --> 00:00:10.758 I mean, you don't have to. 00:00:10.758 --> 00:00:12.720 That's why it's called extra credit. 00:00:12.720 --> 00:00:18.378 But I think it's good for extra practice 00:00:18.378 --> 00:00:21.354 and for the extra points. 00:00:21.354 --> 00:00:25.322 So hold on to it if you cannot give it to me right now. 00:00:25.322 --> 00:00:29.290 And I'll collect it at the end of the class. 00:00:29.290 --> 00:00:30.282 Today's a big day. 00:00:30.282 --> 00:00:33.258 We are starting a new chapter, Chapter 11. 00:00:33.258 --> 00:00:44.170 00:00:44.170 --> 00:00:47.394 So practically, we are going to discuss all 00:00:47.394 --> 00:00:50.618 through this chapter functions of several variables. 00:00:50.618 --> 00:01:02.026 00:01:02.026 --> 00:01:05.089 And you are going to ask me, wait a minute, 00:01:05.089 --> 00:01:12.330 why do we need functions in more than one variable? 00:01:12.330 --> 00:01:15.770 Well, we are all functions of many variables. 00:01:15.770 --> 00:01:18.765 I was freezing outside, and I was thinking, 00:01:18.765 --> 00:01:20.620 I'm a function of everything I eat. 00:01:20.620 --> 00:01:24.070 I'm a function of the temperature outside. 00:01:24.070 --> 00:01:26.500 Almost everything in our body is a function 00:01:26.500 --> 00:01:29.200 of hundreds of factors, actually, thousands. 00:01:29.200 --> 00:01:34.350 But we don't have the time and the precise information 00:01:34.350 --> 00:01:37.420 to analyze all the parameters that 00:01:37.420 --> 00:01:42.380 affect our physical condition every day. 00:01:42.380 --> 00:01:44.990 We are getting there. 00:01:44.990 --> 00:01:48.360 I'm going to give you just the simple case. 00:01:48.360 --> 00:01:53.312 So instead of y equals f of x type of function, one variable, 00:01:53.312 --> 00:01:57.420 we are going to look at functions of the types z 00:01:57.420 --> 00:01:58.583 equals f of xy. 00:01:58.583 --> 00:02:01.760 00:02:01.760 --> 00:02:03.750 Can I have many more? 00:02:03.750 --> 00:02:05.560 Absolutely I can. 00:02:05.560 --> 00:02:08.880 And that's kind of the idea, that I 00:02:08.880 --> 00:02:14.640 can have a function in an-- let me 00:02:14.640 --> 00:02:18.580 count-- n plus 1 dimensional space 00:02:18.580 --> 00:02:30.350 as being of the type xn plus 1 equals f of x1, x2, x3, x4. 00:02:30.350 --> 00:02:32.820 Somebody stop me. xn. 00:02:32.820 --> 00:02:33.320 Right. 00:02:33.320 --> 00:02:37.885 I have many variables. 00:02:37.885 --> 00:02:41.790 And that is a problem that affects everything. 00:02:41.790 --> 00:02:44.600 Our physical world is affected by many parameters. 00:02:44.600 --> 00:02:47.630 00:02:47.630 --> 00:02:49.850 In engineering problems, you've already 00:02:49.850 --> 00:02:51.400 seen some of these parameters. 00:02:51.400 --> 00:02:55.040 Can you give me some examples of parameters you've 00:02:55.040 --> 00:02:57.190 seen in engineering classes? 00:02:57.190 --> 00:03:00.740 x1, x2, x3 could be the Euclidean coordinates, right, 00:03:00.740 --> 00:03:03.630 for the three [? space. ?] But besides those, there was an x4. 00:03:03.630 --> 00:03:05.600 It could be? 00:03:05.600 --> 00:03:06.100 Time. 00:03:06.100 --> 00:03:07.133 Excellent, [INAUDIBLE]. 00:03:07.133 --> 00:03:08.402 More than that. 00:03:08.402 --> 00:03:09.690 I want more. 00:03:09.690 --> 00:03:11.234 I want x5. 00:03:11.234 --> 00:03:12.650 Who can think of another parameter 00:03:12.650 --> 00:03:17.083 that affects physical processes or chemical reactions? 00:03:17.083 --> 00:03:17.583 Yes, sir? 00:03:17.583 --> 00:03:17.936 STUDENT: Temperature. 00:03:17.936 --> 00:03:19.102 MAGDALENA TODA: Temperature. 00:03:19.102 --> 00:03:19.910 Excellent. 00:03:19.910 --> 00:03:21.310 Another very good idea. 00:03:21.310 --> 00:03:23.100 How about x6? 00:03:23.100 --> 00:03:24.590 I'm running out of imagination. 00:03:24.590 --> 00:03:28.910 But you have a lot more information than me. 00:03:28.910 --> 00:03:29.892 Pressure. 00:03:29.892 --> 00:03:35.620 Maybe I'm studying a process of somewhere up in the atmosphere. 00:03:35.620 --> 00:03:37.677 Maybe I'm in an airplane, and then it 00:03:37.677 --> 00:03:39.260 becomes a little bit more complicated, 00:03:39.260 --> 00:03:41.890 because I hate the way cabins are pressurized. 00:03:41.890 --> 00:03:45.060 I can feel very uneasy. 00:03:45.060 --> 00:03:46.990 My ears pop and so on. 00:03:46.990 --> 00:03:48.850 We can be in the bottom of the ocean. 00:03:48.850 --> 00:03:52.900 There are very many physical parameters 00:03:52.900 --> 00:03:56.252 that affect physical processes, chemical processes, 00:03:56.252 --> 00:03:57.960 biological processes. 00:03:57.960 --> 00:04:00.410 I don't know if this is fortunate or unfortunate, 00:04:00.410 --> 00:04:03.120 but I think that was the key to the existence 00:04:03.120 --> 00:04:07.400 of the universe in the first place-- all these parameters. 00:04:07.400 --> 00:04:08.030 OK. 00:04:08.030 --> 00:04:11.700 Let me give you a simple example of a function that 00:04:11.700 --> 00:04:13.395 looks like a graph. 00:04:13.395 --> 00:04:14.460 This is a graph. 00:04:14.460 --> 00:04:17.180 00:04:17.180 --> 00:04:19.470 And you say, wait a minute, wait a minute. 00:04:19.470 --> 00:04:22.470 Can I have functions of several variables that cannot be 00:04:22.470 --> 00:04:26.260 represented as graphs? 00:04:26.260 --> 00:04:28.100 Yeah. 00:04:28.100 --> 00:04:29.020 Absolutely. 00:04:29.020 --> 00:04:31.300 We will talk about that a little bit later. 00:04:31.300 --> 00:04:35.200 So if I were to give you an example that you've 00:04:35.200 --> 00:04:42.270 seen before, and I would say, give me a good approximation 00:04:42.270 --> 00:04:47.000 to a valley that is actually a quadric that we love and we 00:04:47.000 --> 00:04:51.750 studied before for the first time. 00:04:51.750 --> 00:04:59.210 That quadric is a beautiful object, a valley. 00:04:59.210 --> 00:05:02.840 Any imagination, recognition, recollection? 00:05:02.840 --> 00:05:04.930 I know I scared you enough for you 00:05:04.930 --> 00:05:08.130 to know the equations of those quadrics since some of you 00:05:08.130 --> 00:05:10.000 told me we watched all the videos, 00:05:10.000 --> 00:05:13.440 we read all the stinking book like never before. 00:05:13.440 --> 00:05:14.770 That was kind of the idea. 00:05:14.770 --> 00:05:16.970 I didn't want to scare you away. 00:05:16.970 --> 00:05:19.330 I wanted to scare you enough to read the book 00:05:19.330 --> 00:05:20.620 and watch the videos. 00:05:20.620 --> 00:05:26.240 And I'm talking about a valley that you've seen before. 00:05:26.240 --> 00:05:31.070 Many of you told me you like the University of Minnesota 00:05:31.070 --> 00:05:34.900 website that has the quadric gallery of quadrics. 00:05:34.900 --> 00:05:40.140 00:05:40.140 --> 00:05:43.550 So you've met this guy before. 00:05:43.550 --> 00:05:46.430 They show the general equation. 00:05:46.430 --> 00:05:50.010 But I said I like the circular paraboloid. 00:05:50.010 --> 00:05:53.810 So they talk about elliptic paraboloid. 00:05:53.810 --> 00:05:56.666 Which one do you think I prefer? 00:05:56.666 --> 00:05:59.056 The circular paraboloid. 00:05:59.056 --> 00:06:02.880 Give me an example of a circular paraboloid. 00:06:02.880 --> 00:06:05.270 STUDENT: A flashlight? 00:06:05.270 --> 00:06:06.720 Inside. 00:06:06.720 --> 00:06:09.295 MAGDALENA TODA: The expression, the mathematical equation. 00:06:09.295 --> 00:06:10.086 STUDENT: Oh, sorry. 00:06:10.086 --> 00:06:11.737 So it would be x squred plus y squared. 00:06:11.737 --> 00:06:12.820 MAGDALENA TODA: Very good. 00:06:12.820 --> 00:06:14.470 That's exactly what I had in mind. 00:06:14.470 --> 00:06:18.110 Of course, it could be over something, over r. 00:06:18.110 --> 00:06:18.610 All right. 00:06:18.610 --> 00:06:19.930 That's my favorite. 00:06:19.930 --> 00:06:23.590 Now, if I put the flashlight in here just like one of you 00:06:23.590 --> 00:06:29.585 said, or the sign on top of the z-axis. 00:06:29.585 --> 00:06:36.780 Then I'm going to look at the various-- we discussed 00:06:36.780 --> 00:06:38.110 that a little bit before. 00:06:38.110 --> 00:06:42.900 So various horizontal planes, they're going to cut. 00:06:42.900 --> 00:06:48.450 They're going to cut the surface in different circles, 00:06:48.450 --> 00:06:50.390 upon different circles. 00:06:50.390 --> 00:06:52.780 We love them, and we use them. 00:06:52.780 --> 00:06:54.940 And what did we do with them last time? 00:06:54.940 --> 00:06:59.830 We projected them on the floor. 00:06:59.830 --> 00:07:03.390 And by floor, I mean the what? 00:07:03.390 --> 00:07:10.080 By floor, I mean the xy plane. 00:07:10.080 --> 00:07:12.420 Plus this xy plane. 00:07:12.420 --> 00:07:14.450 I label it like you like it. 00:07:14.450 --> 00:07:17.290 You said you like it when I label it, 00:07:17.290 --> 00:07:19.990 so you have the imagination of a table. 00:07:19.990 --> 00:07:23.640 This is x and y and z. 00:07:23.640 --> 00:07:31.040 And so I gave you an example of a graph cut in with z equals 00:07:31.040 --> 00:07:33.400 constant positive or negative? 00:07:33.400 --> 00:07:36.080 Well, it better be positive, because for negative, I 00:07:36.080 --> 00:07:37.770 have no solutions. 00:07:37.770 --> 00:07:39.020 Positive or zero. 00:07:39.020 --> 00:07:42.360 Well, for zero I have a degenerate conic. 00:07:42.360 --> 00:07:46.180 A degenerate conic could be a point, 00:07:46.180 --> 00:07:48.820 or it could be a bunch of lines. 00:07:48.820 --> 00:07:52.432 In this case, all those circles-- doo-doo-doo-doo-doo-- 00:07:52.432 --> 00:07:55.750 a family of one parameter, family of circles. 00:07:55.750 --> 00:07:58.170 Like the ones that is-- a dolphin 00:07:58.170 --> 00:08:01.730 is now doing that in San Antonio, 00:08:01.730 --> 00:08:04.490 San Diego-- to take those old circles 00:08:04.490 --> 00:08:07.765 from the bottom of the sea, and bring them different sizes, 00:08:07.765 --> 00:08:09.030 and put them together. 00:08:09.030 --> 00:08:10.400 So they are very smart. 00:08:10.400 --> 00:08:11.820 I love dolphins. 00:08:11.820 --> 00:08:15.540 So we'll see 0 [INAUDIBLE] get a point. 00:08:15.540 --> 00:08:17.290 That's still a conic. 00:08:17.290 --> 00:08:18.962 It's a degenerate circle. 00:08:18.962 --> 00:08:21.582 Do you realize that's a limit case? 00:08:21.582 --> 00:08:22.536 It's really beautiful. 00:08:22.536 --> 00:08:23.410 You know what I mean? 00:08:23.410 --> 00:08:25.284 Circle on top of a circle on top of a circle, 00:08:25.284 --> 00:08:26.175 smaller and smaller. 00:08:26.175 --> 00:08:27.570 All right. 00:08:27.570 --> 00:08:30.470 So good. 00:08:30.470 --> 00:08:33.270 If I create shadows-- because that's 00:08:33.270 --> 00:08:35.350 why you guys wanted the source of light 00:08:35.350 --> 00:08:39.360 on top-- of the projections of these circles, 00:08:39.360 --> 00:08:42.630 I'm going to have them at the same color. 00:08:42.630 --> 00:08:47.991 But dotted lines because I think the book doesn't show them 00:08:47.991 --> 00:08:48.490 dotted. 00:08:48.490 --> 00:08:51.210 But on my way here, I was thinking, 00:08:51.210 --> 00:08:55.040 I think it's more beautiful if I draw them dotted. 00:08:55.040 --> 00:08:56.635 And how big is this circle? 00:08:56.635 --> 00:08:57.440 Well, god knows. 00:08:57.440 --> 00:09:01.990 I'm going to make a purple circle that is, of course, 00:09:01.990 --> 00:09:05.940 equal in size, equal in radius with the original purple 00:09:05.940 --> 00:09:07.080 circle. 00:09:07.080 --> 00:09:10.290 So the dotted purple circle, that's on the ground-- 00:09:10.290 --> 00:09:13.520 is just the projection of the continuous purple circle. 00:09:13.520 --> 00:09:16.450 It's identical in radius. 00:09:16.450 --> 00:09:26.810 So for the family of circles on the surface, 00:09:26.810 --> 00:09:37.830 I have a family of projections on the ground in the xy plane. 00:09:37.830 --> 00:09:42.420 And such a family of projections represents 00:09:42.420 --> 00:09:44.910 a bunch of level curves. 00:09:44.910 --> 00:09:47.112 We call this family of level curves. 00:09:47.112 --> 00:09:54.402 00:09:54.402 --> 00:09:55.860 OK? 00:09:55.860 --> 00:09:56.832 All right. 00:09:56.832 --> 00:09:59.070 So if you think about it, what are level curves? 00:09:59.070 --> 00:10:02.360 You view them as being in plane. 00:10:02.360 --> 00:10:02.910 Oh, my god. 00:10:02.910 --> 00:10:07.600 So I should view them as a bunch of points, a set of points. 00:10:07.600 --> 00:10:10.665 If I make it like that, that means 00:10:10.665 --> 00:10:14.100 I view this as an element of what? 00:10:14.100 --> 00:10:18.820 Element of the xy plane, right, with the property 00:10:18.820 --> 00:10:22.920 that f of x and y is a constant. 00:10:22.920 --> 00:10:25.848 00:10:25.848 --> 00:10:28.356 OK? 00:10:28.356 --> 00:10:30.690 In my case, I have a [INAUDIBLE] constant. 00:10:30.690 --> 00:10:34.270 In general, I have an arbitrary real constant. 00:10:34.270 --> 00:10:40.020 That's a level curve for level C, for the level 00:10:40.020 --> 00:10:46.400 C called the level, or altitude would be the same thing. 00:10:46.400 --> 00:10:50.110 So have you seen these guys in geography? 00:10:50.110 --> 00:10:54.120 What in the world are these level curves in geography? 00:10:54.120 --> 00:10:56.520 STUDENT: [INAUDIBLE] show the slope 00:10:56.520 --> 00:10:59.400 of a-- the steepness of a hill. 00:10:59.400 --> 00:11:02.200 MAGDALENA TODA: You've seen topographical maps. 00:11:02.200 --> 00:11:06.140 And I'm going to try and draw one of them. 00:11:06.140 --> 00:11:08.310 I don't know, guys, how-- excuse me. 00:11:08.310 --> 00:11:10.275 I'm not very good today at drawing. 00:11:10.275 --> 00:11:13.180 But I'll do my best. 00:11:13.180 --> 00:11:19.860 It could be a temperature map or pressure map. 00:11:19.860 --> 00:11:22.610 [INAUDIBLE] or whatever. 00:11:22.610 --> 00:11:29.090 Now I'll say, this is going to go-- well, 00:11:29.090 --> 00:11:31.610 I cannot draw the infinite family. 00:11:31.610 --> 00:11:34.416 I have a one-parameter family. 00:11:34.416 --> 00:11:42.840 And then I'll-- I'm dreaming of the sea, summer break already. 00:11:42.840 --> 00:11:44.596 You see what I'm doing. 00:11:44.596 --> 00:11:47.580 Do you know what I'm doing? 00:11:47.580 --> 00:11:51.610 That means I'm dreaming of the different depths of the sea. 00:11:51.610 --> 00:11:55.896 So for every such broad line, I have the same depth. 00:11:55.896 --> 00:11:59.380 The same altitude for every continuous rule. 00:11:59.380 --> 00:12:02.530 The same depth for every-- so OK. 00:12:02.530 --> 00:12:05.110 I'm not going to swim too far, because that's 00:12:05.110 --> 00:12:06.270 where the sharks are. 00:12:06.270 --> 00:12:10.096 And I cannot draw the sharks, but I ask you to imagine them. 00:12:10.096 --> 00:12:12.712 It's fundamental in a calculus class. 00:12:12.712 --> 00:12:18.490 So somewhere here I'm going to have-- 00:12:18.490 --> 00:12:21.326 what's the deepest-- guys, what's 00:12:21.326 --> 00:12:23.066 the deepest point in that? 00:12:23.066 --> 00:12:24.173 [? STUDENT: 11,300. ?] 00:12:24.173 --> 00:12:25.881 MAGDALENA TODA: And do you know the name? 00:12:25.881 --> 00:12:26.381 I know the-- 00:12:26.381 --> 00:12:27.381 STUDENT: Mariana Trench. 00:12:27.381 --> 00:12:28.695 MAGDALENA TODA: Mariana Trench. 00:12:28.695 --> 00:12:30.757 STUDENT: Trench. 00:12:30.757 --> 00:12:31.840 MAGDALENA TODA: All right. 00:12:31.840 --> 00:12:34.490 So these topographical are full of curves. 00:12:34.490 --> 00:12:37.300 These are level curves. 00:12:37.300 --> 00:12:39.580 So you didn't know, but there is a lot 00:12:39.580 --> 00:12:42.060 of mathematics in geography. 00:12:42.060 --> 00:12:43.860 And there is a lot of mathematics 00:12:43.860 --> 00:12:45.600 in-- oh, you knew it. 00:12:45.600 --> 00:12:47.480 When you watch the weather report, 00:12:47.480 --> 00:12:49.740 that's all mathematics, right? 00:12:49.740 --> 00:12:53.030 It shows you the distribution of temperatures everyday. 00:12:53.030 --> 00:12:55.490 That is what we can [INAUDIBLE] also 00:12:55.490 --> 00:13:00.550 care about other functions of several parameters, right? 00:13:00.550 --> 00:13:04.430 And those functions could be pressure, wind, whatever. 00:13:04.430 --> 00:13:05.650 OK. 00:13:05.650 --> 00:13:06.773 Speed of the wind. 00:13:06.773 --> 00:13:09.040 Something like that. 00:13:09.040 --> 00:13:12.350 I did not dare to look at the prediction 00:13:12.350 --> 00:13:14.670 of the weather for this place. 00:13:14.670 --> 00:13:16.670 This place used to be a beautiful place. 00:13:16.670 --> 00:13:22.530 300 days of the year of sunshine. 00:13:22.530 --> 00:13:23.755 Not anymore. 00:13:23.755 --> 00:13:26.550 So there is something fishy in Denmark 00:13:26.550 --> 00:13:29.010 and also something fishy in [INAUDIBLE]. 00:13:29.010 --> 00:13:30.250 The world is changing. 00:13:30.250 --> 00:13:34.720 So if you don't believe in global warming, think again, 00:13:34.720 --> 00:13:38.180 and global cooling, think again. 00:13:38.180 --> 00:13:39.130 All right. 00:13:39.130 --> 00:13:42.960 So unfortunately, I am afraid still 00:13:42.960 --> 00:13:45.622 to look at the temperatures for the next few days. 00:13:45.622 --> 00:13:46.122 But-- 00:13:46.122 --> 00:13:48.602 STUDENT: It's going to be 80 degrees on Tuesday. 00:13:48.602 --> 00:13:49.147 MAGDALENA TODA: Really? [? 00:13:49.147 --> 00:13:50.855 Well, see, I should have looked at it. ?] 00:13:50.855 --> 00:13:51.580 [LAUGHTER] 00:13:51.580 --> 00:13:54.710 I should gather the courage, because I 00:13:54.710 --> 00:13:57.140 knew-- when I was interviewed here 00:13:57.140 --> 00:14:00.480 for assistant professor, gosh, I was young. 00:14:00.480 --> 00:14:02.230 2001. 00:14:02.230 --> 00:14:03.830 And my interview was in mid-February. 00:14:03.830 --> 00:14:08.180 And birds were chirping, it was blue skies, beautiful flowers 00:14:08.180 --> 00:14:09.700 everywhere on campus. 00:14:09.700 --> 00:14:11.380 And I love the campus. 00:14:11.380 --> 00:14:12.910 OK. 00:14:12.910 --> 00:14:18.520 Give me an example of a surface that cannot be represented 00:14:18.520 --> 00:14:23.810 as a graph in its entirety as a whole graph. 00:14:23.810 --> 00:14:27.070 You gave me that before, and I was so proud of you. 00:14:27.070 --> 00:14:27.940 It was a-- 00:14:27.940 --> 00:14:30.840 00:14:30.840 --> 00:14:32.316 [LAUGHS] 00:14:32.316 --> 00:14:33.300 00:14:33.300 --> 00:14:35.730 What kind of surface am I trying to mimic? 00:14:35.730 --> 00:14:36.694 STUDENT: A saddle. 00:14:36.694 --> 00:14:39.590 00:14:39.590 --> 00:14:42.120 MAGDALENA TODA: That can be actually a graph. 00:14:42.120 --> 00:14:44.630 That's a good example of a graph. 00:14:44.630 --> 00:14:45.920 A saddle. 00:14:45.920 --> 00:14:49.030 But give me an example of a non-graph that 00:14:49.030 --> 00:14:51.720 is given as an implicit form. 00:14:51.720 --> 00:14:58.160 So graph or explicit is the same thing. 00:14:58.160 --> 00:15:00.880 z equals f of xy. 00:15:00.880 --> 00:15:02.590 Give me a non-graph. 00:15:02.590 --> 00:15:05.140 One of you said it. 00:15:05.140 --> 00:15:08.602 x squared plus y squared plus z squared equals 1. 00:15:08.602 --> 00:15:10.820 Why is this not a graph? 00:15:10.820 --> 00:15:12.940 Not a graph. 00:15:12.940 --> 00:15:14.163 Why is this not a graph? 00:15:14.163 --> 00:15:18.993 00:15:18.993 --> 00:15:22.374 STUDENT: [INAUDIBLE]. 00:15:22.374 --> 00:15:26.260 When you move it over to 1, you can't actually-- 00:15:26.260 --> 00:15:28.872 MAGDALENA TODA: You cannot but you can cut it. 00:15:28.872 --> 00:15:31.460 You can take a sword and-- I'm OK. 00:15:31.460 --> 00:15:33.720 I don't want to think about it. 00:15:33.720 --> 00:15:37.180 So z is going to be two graphs. 00:15:37.180 --> 00:15:41.970 So I can split this surface even in a parametric form 00:15:41.970 --> 00:15:44.860 as two different graphs. 00:15:44.860 --> 00:15:46.626 Different graphs. 00:15:46.626 --> 00:15:51.490 If I cut along-- I have this orange, or sphere, globe. 00:15:51.490 --> 00:15:54.070 And I cut it along a great circle. 00:15:54.070 --> 00:15:57.580 It doesn't have to be the equator. 00:15:57.580 --> 00:15:59.710 But you have to imagine something 00:15:59.710 --> 00:16:02.540 like the world and the equator. 00:16:02.540 --> 00:16:06.030 This is kind of in the unit sphere. 00:16:06.030 --> 00:16:09.110 Today I drank enough coffee to draw better. 00:16:09.110 --> 00:16:10.160 Why don't I draw better? 00:16:10.160 --> 00:16:12.081 I have no idea. 00:16:12.081 --> 00:16:15.910 So that's not bad, though. 00:16:15.910 --> 00:16:16.760 OK. 00:16:16.760 --> 00:16:18.090 So that's the unit sphere. 00:16:18.090 --> 00:16:18.950 What does it mean? 00:16:18.950 --> 00:16:21.770 It means it has radius how much? 00:16:21.770 --> 00:16:22.270 STUDENT: 1. 00:16:22.270 --> 00:16:23.020 MAGDALENA TODA: 1. 00:16:23.020 --> 00:16:25.670 Radius 1, and we are happy about it. 00:16:25.670 --> 00:16:29.270 And it has two graphs. 00:16:29.270 --> 00:16:31.769 It's not one graph, it's two graphs. 00:16:31.769 --> 00:16:34.184 So this is called implicit equation. 00:16:34.184 --> 00:16:36.715 This is your lab from-- I was chatting 00:16:36.715 --> 00:16:38.670 with-- instead of studying last night, 00:16:38.670 --> 00:16:40.960 I was chatting with you at midnight. 00:16:40.960 --> 00:16:45.080 And one of you said, if I had something I hated in calculus, 00:16:45.080 --> 00:16:48.160 it was the implicit differentiation. 00:16:48.160 --> 00:16:50.400 And I know this is your weak point. 00:16:50.400 --> 00:16:52.570 So we'll do a lot of implicit differentiation, 00:16:52.570 --> 00:16:54.770 so you become more comfortable. 00:16:54.770 --> 00:16:59.120 Usually we have one exercise in this differentiation at least 00:16:59.120 --> 00:17:01.500 on the final. 00:17:01.500 --> 00:17:04.579 So this is an implicit equation. 00:17:04.579 --> 00:17:09.690 And z is going to be two graphs-- 1 minus x 00:17:09.690 --> 00:17:11.125 squared minus y squared. 00:17:11.125 --> 00:17:13.858 So I have, like, two charts, two different charts. 00:17:13.858 --> 00:17:14.358 OK. 00:17:14.358 --> 00:17:17.150 00:17:17.150 --> 00:17:20.089 The upper hemisphere-- I'm talking geography, 00:17:20.089 --> 00:17:23.190 but that's how we talk in geometry as well. 00:17:23.190 --> 00:17:26.055 So geography right now is like geometry. 00:17:26.055 --> 00:17:28.150 I have a north pole. 00:17:28.150 --> 00:17:31.620 Somebody quickly give me the coordinates of the north pole. 00:17:31.620 --> 00:17:32.530 STUDENT: 0, 0, 1. 00:17:32.530 --> 00:17:33.530 MAGDALENA TODA: 0, 0, 1. 00:17:33.530 --> 00:17:34.780 Thank you, Brian. 00:17:34.780 --> 00:17:35.640 0, 0, 1. 00:17:35.640 --> 00:17:37.512 How about the south pole? 00:17:37.512 --> 00:17:38.720 STUDENT: 0, 0, minus 1. 00:17:38.720 --> 00:17:41.290 MAGDALENA TODA: 0, 0, minus 1. 00:17:41.290 --> 00:17:45.640 And write yourself a note, because as you know, 00:17:45.640 --> 00:17:48.570 I'm very absent-minded and I forget 00:17:48.570 --> 00:17:52.490 what I eat for lunch and so on. 00:17:52.490 --> 00:17:55.560 Remind me to talk to you sometime 00:17:55.560 --> 00:17:58.280 at the end of the chapter about stereographic projection. 00:17:58.280 --> 00:18:01.080 It's a very important mathematical notion 00:18:01.080 --> 00:18:03.785 that also has to do a little bit with geography. 00:18:03.785 --> 00:18:06.060 But it's a one-to-one correspondence 00:18:06.060 --> 00:18:08.960 between a certain part of a sphere 00:18:08.960 --> 00:18:11.876 and a certain huge part of a plane. 00:18:11.876 --> 00:18:14.050 Now, we're not going to talk about that now, 00:18:14.050 --> 00:18:16.030 because that's not [INAUDIBLE]. 00:18:16.030 --> 00:18:18.360 That's a little bit harder [INAUDIBLE]. 00:18:18.360 --> 00:18:20.690 You guys should now see this line, right? 00:18:20.690 --> 00:18:24.044 This should be beyond-- in the twilight zone, 00:18:24.044 --> 00:18:25.740 behind the sphere. 00:18:25.740 --> 00:18:27.280 OK? 00:18:27.280 --> 00:18:28.660 So you don't see it. 00:18:28.660 --> 00:18:31.410 And who is this? z equals 0. 00:18:31.410 --> 00:18:34.750 And so this green fellow should be 00:18:34.750 --> 00:18:39.150 the circle x squared plus y squared equals 1 00:18:39.150 --> 00:18:40.615 in the xy plane. 00:18:40.615 --> 00:18:43.285 00:18:43.285 --> 00:18:44.700 Good. 00:18:44.700 --> 00:18:47.030 So I have two graphs. 00:18:47.030 --> 00:18:54.870 Now, if I were to ask you, what is the domain 00:18:54.870 --> 00:18:59.250 and the range of the function? 00:18:59.250 --> 00:19:02.630 I'm going to erase the whole thing. 00:19:02.630 --> 00:19:10.040 What is the domain and the range of the related function, z, 00:19:10.040 --> 00:19:13.904 which gives the upper hemisphere? 00:19:13.904 --> 00:19:15.368 Upper hemisphere. 00:19:15.368 --> 00:19:17.320 It's a graph. 00:19:17.320 --> 00:19:20.810 And square root of 1 minus x squared minus y squared. 00:19:20.810 --> 00:19:23.220 You may stare at it until tomorrow. 00:19:23.220 --> 00:19:27.940 It's not hard to figure out what I mean by domain 00:19:27.940 --> 00:19:30.570 and range of such a function. 00:19:30.570 --> 00:19:33.300 You are familiar with domain and range 00:19:33.300 --> 00:19:37.330 for a function of one variable. 00:19:37.330 --> 00:19:39.890 For most of you, that's a piece of cake. 00:19:39.890 --> 00:19:41.800 That was even pre-calc wasn't it? 00:19:41.800 --> 00:19:44.341 It was in Calc 1. 00:19:44.341 --> 00:19:47.360 So most of you had algebra and pre-calc. 00:19:47.360 --> 00:19:51.980 Now, what is the domain of such a function? 00:19:51.980 --> 00:19:57.390 Domain of definition has to be a set of points, x and y in plane 00:19:57.390 --> 00:20:00.590 for which the function is defined. 00:20:00.590 --> 00:20:03.040 If the function is impossible to be defined 00:20:03.040 --> 00:20:05.995 for a certain pair, x, y, you kick that couple out 00:20:05.995 --> 00:20:07.917 and you say, never come back. 00:20:07.917 --> 00:20:09.140 Right? 00:20:09.140 --> 00:20:14.665 So what I mean by domain is those couples that we hate. 00:20:14.665 --> 00:20:16.430 Who we hate? 00:20:16.430 --> 00:20:20.970 The couples x, y for which x squared plus y squared is how? 00:20:20.970 --> 00:20:24.130 00:20:24.130 --> 00:20:25.422 What existence condition do I-- 00:20:25.422 --> 00:20:26.296 STUDENT: [INAUDIBLE]. 00:20:26.296 --> 00:20:27.440 MAGDALENA TODA: Yeah. 00:20:27.440 --> 00:20:30.230 You see this guy under the square root 00:20:30.230 --> 00:20:33.790 has to be positive or 0. 00:20:33.790 --> 00:20:35.280 Right? 00:20:35.280 --> 00:20:37.890 Otherwise, there is no square root in real numbers. 00:20:37.890 --> 00:20:39.920 That's going to be in imaginary numbers, 00:20:39.920 --> 00:20:41.630 and you can take a walk, because we 00:20:41.630 --> 00:20:45.220 are in real calculus in real time as well. 00:20:45.220 --> 00:20:48.670 So x squared plus y squared must be how? 00:20:48.670 --> 00:20:50.660 Less than or equal to 1. 00:20:50.660 --> 00:20:54.040 We call that a certain name. 00:20:54.040 --> 00:20:59.484 This is called a closed unit disk. 00:20:59.484 --> 00:21:03.160 Please remember, I'm teaching you a little bit more 00:21:03.160 --> 00:21:06.195 than a regular Calc 3 class. 00:21:06.195 --> 00:21:09.324 They will never make a distinction. 00:21:09.324 --> 00:21:10.365 What's closing with this? 00:21:10.365 --> 00:21:11.900 What's opening with this? 00:21:11.900 --> 00:21:14.670 Everything will come into place when you 00:21:14.670 --> 00:21:19.701 move on to advanced calculus. 00:21:19.701 --> 00:21:24.640 If I don't take the boundary-- so everything inside the disk 00:21:24.640 --> 00:21:28.460 except for the boundary, I have to put strictly less than 1. 00:21:28.460 --> 00:21:30.610 That's called open unit disk. 00:21:30.610 --> 00:21:35.080 For advanced calculus, this is [INAUDIBLE]. 00:21:35.080 --> 00:21:35.580 All right. 00:21:35.580 --> 00:21:37.320 This is just a parentheses. 00:21:37.320 --> 00:21:40.388 My domain is the closed one. 00:21:40.388 --> 00:21:43.145 What is the range? 00:21:43.145 --> 00:21:45.690 The range is going to be-- 00:21:45.690 --> 00:21:47.000 STUDENT: [INAUDIBLE]. 00:21:47.000 --> 00:21:49.900 MAGDALENA TODA: The altitude starts having values from-- 00:21:49.900 --> 00:21:51.092 STUDENT: Negative 1 to 1. 00:21:51.092 --> 00:21:51.758 STUDENT: 0 to 1. 00:21:51.758 --> 00:21:53.008 MAGDALENA TODA: So I'm 0 to 1. 00:21:53.008 --> 00:21:55.090 I'll only talk about the upper hemisphere. 00:21:55.090 --> 00:21:58.310 I should even erase, because I don't want it. 00:21:58.310 --> 00:21:59.333 So say it again, guys. 00:21:59.333 --> 00:22:00.350 STUDENT: 0 to 1. 00:22:00.350 --> 00:22:01.100 MAGDALENA TODA: 0. 00:22:01.100 --> 00:22:01.865 Open or closed? 00:22:01.865 --> 00:22:02.635 STUDENT: Open. 00:22:02.635 --> 00:22:03.301 STUDENT: Closed. 00:22:03.301 --> 00:22:05.740 STUDENT: Closed, closed. 00:22:05.740 --> 00:22:07.630 MAGDALENA TODA: Closed to? 00:22:07.630 --> 00:22:08.450 STUDENT: 1 closed. 00:22:08.450 --> 00:22:09.650 MAGDALENA TODA: 1 closed. 00:22:09.650 --> 00:22:10.150 Yes. 00:22:10.150 --> 00:22:13.874 Because that is the north pole. 00:22:13.874 --> 00:22:19.160 I've been meaning to give you this example. 00:22:19.160 --> 00:22:22.270 And give me the other example for the lower hemisphere. 00:22:22.270 --> 00:22:23.400 What's different? 00:22:23.400 --> 00:22:24.894 The same domain? 00:22:24.894 --> 00:22:25.935 STUDENT: It ranges from-- 00:22:25.935 --> 00:22:27.042 STUDENT: Negative 1. 00:22:27.042 --> 00:22:28.710 STUDENT: Negative 1 to 0. 00:22:28.710 --> 00:22:30.490 MAGDALENA TODA: Closed internal, right? 00:22:30.490 --> 00:22:33.860 When we include the endpoints, we call that closed interval. 00:22:33.860 --> 00:22:36.200 It has a certain topological sense. 00:22:36.200 --> 00:22:39.110 You haven't taken topology, but very soon, 00:22:39.110 --> 00:22:43.940 if you are a math major, or you are a double major, or some 00:22:43.940 --> 00:22:48.090 of you even-- they want to learn more about topology, 00:22:48.090 --> 00:22:51.735 you will learn what an open set is versus a closed set. 00:22:51.735 --> 00:22:53.630 Remember we called this closed. 00:22:53.630 --> 00:22:56.230 This is open. 00:22:56.230 --> 00:22:59.660 And if it's closed here and open there, it's neither. 00:22:59.660 --> 00:23:00.160 OK? 00:23:00.160 --> 00:23:02.940 Don't say anything about that. 00:23:02.940 --> 00:23:03.440 OK. 00:23:03.440 --> 00:23:08.110 To be closed, it has to be containing both endpoints. 00:23:08.110 --> 00:23:09.300 I'm going to erase this. 00:23:09.300 --> 00:23:12.260 00:23:12.260 --> 00:23:19.728 And this was, of course, 11.1. 00:23:19.728 --> 00:23:22.632 We are in the middle of it. 00:23:22.632 --> 00:23:28.440 In 11.1, one of you gave me a beautiful graph to think about. 00:23:28.440 --> 00:23:30.525 And I'm going to give you something to do, 00:23:30.525 --> 00:23:32.690 because I don't want you to get lazy. 00:23:32.690 --> 00:23:36.041 I'm very happy you came up with the saddle. 00:23:36.041 --> 00:23:38.908 00:23:38.908 --> 00:23:39.408 All right. 00:23:39.408 --> 00:23:41.480 We drew such a saddle. 00:23:41.480 --> 00:23:44.460 00:23:44.460 --> 00:23:46.881 And I did my best, but it's not hard. 00:23:46.881 --> 00:23:50.360 It's not easy to draw saddle. 00:23:50.360 --> 00:23:54.540 When I am looking at the coordinates, x, y, z, 00:23:54.540 --> 00:24:01.575 I have z equals minus y squared will look down. 00:24:01.575 --> 00:24:05.020 00:24:05.020 --> 00:24:06.810 Maybe I made it too fat. 00:24:06.810 --> 00:24:08.980 I'm really sorry. 00:24:08.980 --> 00:24:11.430 And down. 00:24:11.430 --> 00:24:12.410 This continues. 00:24:12.410 --> 00:24:21.230 00:24:21.230 --> 00:24:22.210 OK? 00:24:22.210 --> 00:24:26.920 And then what other thing did I want to point out? 00:24:26.920 --> 00:24:31.100 I want to point out-- do you see this? 00:24:31.100 --> 00:24:33.660 This should look a little bit more round. 00:24:33.660 --> 00:24:36.556 It doesn't look round enough here. 00:24:36.556 --> 00:24:38.290 STUDENT: Your'e drawing a saddle, right? 00:24:38.290 --> 00:24:40.289 MAGDALENA TODA: No, I'm drawing just the section 00:24:40.289 --> 00:24:41.970 z equals minus y squared. 00:24:41.970 --> 00:24:44.330 So I took x to be 0. 00:24:44.330 --> 00:24:47.880 And the purple line should be on this wall. 00:24:47.880 --> 00:24:50.160 I know you guys have enough imagination. 00:24:50.160 --> 00:24:54.045 So this is going to be z equals minus y 00:24:54.045 --> 00:24:58.700 squared drawn on yz wall. 00:24:58.700 --> 00:25:03.170 00:25:03.170 --> 00:25:05.590 I've done this before, but I'm just reviewing. 00:25:05.590 --> 00:25:08.360 What if it's y0? 00:25:08.360 --> 00:25:10.920 Then I have to draw on that wall. 00:25:10.920 --> 00:25:14.326 And I have to draw beautifully, which I am not-- don't always-- 00:25:14.326 --> 00:25:15.718 I can't always do. 00:25:15.718 --> 00:25:17.110 But I'll try. 00:25:17.110 --> 00:25:23.750 I have z equals x squared drawn on that wall. 00:25:23.750 --> 00:25:26.580 If I start drawing, I'll get fired. 00:25:26.580 --> 00:25:29.020 That I have this branch. 00:25:29.020 --> 00:25:32.852 I should go through that corner and go out of the room 00:25:32.852 --> 00:25:35.307 and continue with that branch. 00:25:35.307 --> 00:25:36.010 All right? 00:25:36.010 --> 00:25:39.480 00:25:39.480 --> 00:25:43.050 This is curved like that in this direction. 00:25:43.050 --> 00:25:45.460 And this other is curved like this. 00:25:45.460 --> 00:25:50.300 So if the guy is going to put his feet, 00:25:50.300 --> 00:25:54.590 where is the butt of the writer going to sit? 00:25:54.590 --> 00:25:57.450 He is here. 00:25:57.450 --> 00:25:59.605 And these are his legs. 00:25:59.605 --> 00:26:02.490 00:26:02.490 --> 00:26:06.144 And these are his cowboy boots. 00:26:06.144 --> 00:26:06.644 OK. 00:26:06.644 --> 00:26:08.117 Do they look like cowboy boots? 00:26:08.117 --> 00:26:10.572 No, I apologize. 00:26:10.572 --> 00:26:12.105 STUDENT: Looks like socks. 00:26:12.105 --> 00:26:12.980 MAGDALENA TODA: Yeah. 00:26:12.980 --> 00:26:15.770 They look more like Christmas socks. 00:26:15.770 --> 00:26:17.750 But anyway, it's a poor cowboy. 00:26:17.750 --> 00:26:22.880 00:26:22.880 --> 00:26:24.710 Let's lower the saddle a little bit. 00:26:24.710 --> 00:26:27.200 He cannot see the horse, OK? 00:26:27.200 --> 00:26:30.290 So the saddle. 00:26:30.290 --> 00:26:35.182 If I cross the saddle, this is the saddle. 00:26:35.182 --> 00:26:38.160 And these are his hands. 00:26:38.160 --> 00:26:41.115 And he is holding his hat. 00:26:41.115 --> 00:26:42.065 This is [INAUDIBLE]. 00:26:42.065 --> 00:26:45.865 And with one hand is on the horse. 00:26:45.865 --> 00:26:46.815 I don't know. 00:26:46.815 --> 00:26:48.320 It's very [INAUDIBLE]. 00:26:48.320 --> 00:26:56.760 So what I'm trying to draw looks something like this. 00:26:56.760 --> 00:26:57.660 Right? 00:26:57.660 --> 00:26:59.100 Eh. 00:26:59.100 --> 00:27:01.500 Sorry. 00:27:01.500 --> 00:27:02.240 More or less. 00:27:02.240 --> 00:27:03.943 It's an abstract picture. 00:27:03.943 --> 00:27:05.750 Very abstract picture. 00:27:05.750 --> 00:27:14.946 So with this in mind, if I were to look at the level curves, 00:27:14.946 --> 00:27:18.802 I'm going to ask you, what are the level curves? 00:27:18.802 --> 00:27:22.200 Oh, my god, what are the level curves? 00:27:22.200 --> 00:27:25.390 00:27:25.390 --> 00:27:27.810 You already have them in your WeBWorK homework. 00:27:27.810 --> 00:27:30.300 But for one point extra credit, I 00:27:30.300 --> 00:27:34.072 want you to draw them on the floor. 00:27:34.072 --> 00:27:37.690 Draw the level curves. 00:27:37.690 --> 00:27:39.250 Remember what those were? 00:27:39.250 --> 00:27:43.290 They were projections of the curves on the surface 00:27:43.290 --> 00:27:46.570 at the intersection with z equals c planes. 00:27:46.570 --> 00:27:48.840 You project them on the ground. 00:27:48.840 --> 00:27:50.280 What do you think they are? 00:27:50.280 --> 00:27:51.270 Think about it. 00:27:51.270 --> 00:27:53.280 What are these? 00:27:53.280 --> 00:27:58.460 If I take c, what if c is positive? 00:27:58.460 --> 00:28:01.540 00:28:01.540 --> 00:28:04.730 What if c is 0? 00:28:04.730 --> 00:28:12.255 What if c is less than 0? 00:28:12.255 --> 00:28:14.500 What am I going to have? 00:28:14.500 --> 00:28:18.460 Your imagination gives you c equals 1, Magdalena. 00:28:18.460 --> 00:28:19.870 Let's draw that. 00:28:19.870 --> 00:28:20.370 OK. 00:28:20.370 --> 00:28:21.940 Well, I'll try. 00:28:21.940 --> 00:28:23.650 a and b would be 1, right, guys? 00:28:23.650 --> 00:28:26.275 So a and b would be 1. 00:28:26.275 --> 00:28:27.100 This is a square. 00:28:27.100 --> 00:28:29.970 These would be the asymptotes. 00:28:29.970 --> 00:28:36.110 So very, very briefly, the hyperbola 00:28:36.110 --> 00:28:41.050 would be this one-- x squared minus y squared equals 1, 00:28:41.050 --> 00:28:42.222 right? 00:28:42.222 --> 00:28:45.130 If I have the last case for c equals 1, 00:28:45.130 --> 00:28:47.460 I'm going to have-- c equals negative 1-- I'm 00:28:47.460 --> 00:28:49.215 going to have the conjugate. 00:28:49.215 --> 00:28:50.660 Are you guys with me? 00:28:50.660 --> 00:28:57.790 So I'll have an a squared, asymptotes, conjugate. 00:28:57.790 --> 00:29:01.230 00:29:01.230 --> 00:29:05.060 What if I have different level c? c equals 1/2. c equals 2. 00:29:05.060 --> 00:29:08.000 c equals pi. c equals-- what are they? 00:29:08.000 --> 00:29:12.330 I'm going to get families of hyperbolas, 00:29:12.330 --> 00:29:14.720 trenches that look like that. 00:29:14.720 --> 00:29:16.470 Standard trenches and conjugate trenches. 00:29:16.470 --> 00:29:20.600 A multitude of them, an infinite family of such hyperbolas, 00:29:20.600 --> 00:29:22.480 an infinite family of such hyperbolas. 00:29:22.480 --> 00:29:24.542 I wanted to draw it. 00:29:24.542 --> 00:29:28.880 What do I get when c is 0? 00:29:28.880 --> 00:29:30.022 What are those? 00:29:30.022 --> 00:29:31.772 STUDENT: Don't you get, like, [INAUDIBLE]? 00:29:31.772 --> 00:29:34.985 00:29:34.985 --> 00:29:36.730 MAGDALENA TODA: They get-- very good. 00:29:36.730 --> 00:29:37.230 Why? 00:29:37.230 --> 00:29:40.650 x squared minus y squared equals 0 would lead 00:29:40.650 --> 00:29:44.920 me to y equals plus/minus 1. 00:29:44.920 --> 00:29:48.470 And who are those y equals plus/minus 1? 00:29:48.470 --> 00:29:49.590 Exactly. 00:29:49.590 --> 00:29:54.500 But exactly the first bisector, which is y equals x. 00:29:54.500 --> 00:29:56.410 They are [? then the ?] function. 00:29:56.410 --> 00:29:59.820 And the other one, y equals negative [? x. ?] So these 00:29:59.820 --> 00:30:01.160 are the asymptotes. 00:30:01.160 --> 00:30:05.620 So I'm going to get a-- you guys have to do this better than me. 00:30:05.620 --> 00:30:06.880 Sorry. 00:30:06.880 --> 00:30:08.790 These are all hyperbolic trenches. 00:30:08.790 --> 00:30:11.730 They are all going to infinity like that. 00:30:11.730 --> 00:30:15.330 And I'm sorry that I'm giving you 00:30:15.330 --> 00:30:17.090 a little bit too many hints. 00:30:17.090 --> 00:30:19.282 This is part of your homework, your WeBWorK. 00:30:19.282 --> 00:30:20.740 I shouldn't talk too much about it. 00:30:20.740 --> 00:30:23.820 00:30:23.820 --> 00:30:25.350 Any questions so far? 00:30:25.350 --> 00:30:26.495 Is this hard? 00:30:26.495 --> 00:30:28.000 Yes, sir? 00:30:28.000 --> 00:30:28.500 No. 00:30:28.500 --> 00:30:30.133 STUDENT: So [? spherically, ?] if you had z 00:30:30.133 --> 00:30:31.508 equals y squared minus x squared, 00:30:31.508 --> 00:30:33.890 it's that same picture, just flipped? 00:30:33.890 --> 00:30:39.962 00:30:39.962 --> 00:30:41.450 MAGDALENA TODA: What would it be? 00:30:41.450 --> 00:30:43.241 It would be the poor saddle-- or cowboy-- 00:30:43.241 --> 00:30:44.490 STUDENT: Would be upside down. 00:30:44.490 --> 00:30:46.340 MAGDALENA TODA: --would be upside down. 00:30:46.340 --> 00:30:49.650 Or projected in something like a mirror. 00:30:49.650 --> 00:30:51.100 I don't know how to say. 00:30:51.100 --> 00:30:52.850 It would be exactly upside down. 00:30:52.850 --> 00:30:55.530 So the reflection of that. 00:30:55.530 --> 00:30:59.234 So you take all the points. 00:30:59.234 --> 00:31:01.075 If you have-- I don't know. 00:31:01.075 --> 00:31:03.380 It's hard to draw a reflection in three dimensions. 00:31:03.380 --> 00:31:03.880 But-- 00:31:03.880 --> 00:31:04.963 STUDENT: No, I understand. 00:31:04.963 --> 00:31:09.430 MAGDALENA TODA: Practically every curve 00:31:09.430 --> 00:31:14.690 would be upside down with respect to the floor. 00:31:14.690 --> 00:31:15.650 OK. 00:31:15.650 --> 00:31:16.560 All right. 00:31:16.560 --> 00:31:20.600 I'm going to erase in one. 00:31:20.600 --> 00:31:24.060 And you say, well, you've taught us about these things, 00:31:24.060 --> 00:31:26.150 like the domain and range. 00:31:26.150 --> 00:31:30.530 But what about other notions, like continuity and stuff? 00:31:30.530 --> 00:31:33.300 00:31:33.300 --> 00:31:50.730 Let me move on to 11.2. 00:31:50.730 --> 00:32:00.192 Limits of functions of the type z equals f of xy. 00:32:00.192 --> 00:32:14.640 00:32:14.640 --> 00:32:20.030 So what do you remember about the limit 00:32:20.030 --> 00:32:23.130 of a function of one variable? 00:32:23.130 --> 00:32:23.630 Comparison. 00:32:23.630 --> 00:32:27.960 00:32:27.960 --> 00:32:36.255 What about the limit if you take [? z's, ?] I don't know. 00:32:36.255 --> 00:32:37.700 I should look stunned. 00:32:37.700 --> 00:32:38.700 And I should be stunned. 00:32:38.700 --> 00:32:49.430 Of a function of y equals f of x of one variable. 00:32:49.430 --> 00:32:56.730 00:32:56.730 --> 00:33:10.513 When do we say that f has a limit at a? 00:33:10.513 --> 00:33:12.473 00:33:12.473 --> 00:33:14.764 STUDENT: When the [INAUDIBLE] approaches from the right 00:33:14.764 --> 00:33:16.740 and the left to the same value. 00:33:16.740 --> 00:33:22.610 MAGDALENA TODA: Actually, that was the simpler definition. 00:33:22.610 --> 00:33:25.555 Let's think a little bit deeper. 00:33:25.555 --> 00:33:35.330 We say that f has a limit L at x equals a. 00:33:35.330 --> 00:33:40.550 That's kind of the idea, left and right limits. 00:33:40.550 --> 00:33:43.680 But not both of them have to exist, you see. 00:33:43.680 --> 00:33:45.532 Maybe only the limit from the left or limit 00:33:45.532 --> 00:33:46.978 from the right only exists. 00:33:46.978 --> 00:33:49.870 00:33:49.870 --> 00:34:04.370 If, for any choice of values of x, closer and closer, closer 00:34:04.370 --> 00:34:23.909 and closer to a, we get that F gets closer and closer to L. 00:34:23.909 --> 00:34:27.159 And this "any" I put in. 00:34:27.159 --> 00:34:33.500 My god, I put it in a red circle thing, 00:34:33.500 --> 00:34:40.030 because one could get subsequencies of a sequence. 00:34:40.030 --> 00:34:42.400 And for that subsequence thing, things 00:34:42.400 --> 00:34:44.994 look like I would have a limit. 00:34:44.994 --> 00:34:47.830 And then you say, well, but in the end, 00:34:47.830 --> 00:34:50.889 I don't have a limit, because I can get another subsequence 00:34:50.889 --> 00:34:52.350 of the sequence. 00:34:52.350 --> 00:34:59.030 And for that one, I'm not going to have a limit. 00:34:59.030 --> 00:35:04.150 Can you give me an example of some crazy function that 00:35:04.150 --> 00:35:08.670 does not have a limit at 0? 00:35:08.670 --> 00:35:12.131 Example of a crazy function. 00:35:12.131 --> 00:35:12.630 No. 00:35:12.630 --> 00:35:14.560 No, don't write "crazy." 00:35:14.560 --> 00:35:26.270 Of a function f of x that is not defined at 0 00:35:26.270 --> 00:35:43.855 and does not have limit at 0, although it 00:35:43.855 --> 00:35:53.414 is defined for values arbitrarily close to 0. 00:35:53.414 --> 00:35:59.706 00:35:59.706 --> 00:36:07.900 Moreover, I want that function to be drawn without-- I 00:36:07.900 --> 00:36:22.960 want the function to be drawn without leaving 00:36:22.960 --> 00:36:26.300 the paper when I draw. 00:36:26.300 --> 00:36:30.748 00:36:30.748 --> 00:36:31.248 [INAUDIBLE] 00:36:31.248 --> 00:36:34.566 00:36:34.566 --> 00:36:43.530 So something that would be defined on the whole 0 00:36:43.530 --> 00:36:58.660 infinity except for 0 that I can draw continuously 00:36:58.660 --> 00:37:03.400 except when I get to 0, I get some really bad behavior. 00:37:03.400 --> 00:37:07.560 I don't have a limit for that function. 00:37:07.560 --> 00:37:08.910 You are close to that. 00:37:08.910 --> 00:37:10.720 Sine of 1/x. 00:37:10.720 --> 00:37:13.110 STUDENT: I said y equals 1/x. 00:37:13.110 --> 00:37:15.324 MAGDALENA TODA: y equals 1/x. 00:37:15.324 --> 00:37:16.260 Very good. 00:37:16.260 --> 00:37:18.600 Let's see. 00:37:18.600 --> 00:37:20.004 STUDENT: Oh, yeah. [INAUDIBLE]. 00:37:20.004 --> 00:37:21.129 MAGDALENA TODA: Yeah, yeah. 00:37:21.129 --> 00:37:22.390 Both are excellent examples. 00:37:22.390 --> 00:37:24.530 So let's see. 00:37:24.530 --> 00:37:29.080 This guy is a very nice function. 00:37:29.080 --> 00:37:31.370 How do we draw him, or her? 00:37:31.370 --> 00:37:32.540 Well, it's a her, right? 00:37:32.540 --> 00:37:33.040 It's a she. 00:37:33.040 --> 00:37:33.990 It's a function. 00:37:33.990 --> 00:37:34.660 No, no. 00:37:34.660 --> 00:37:36.360 In English, it doesn't make any sense, 00:37:36.360 --> 00:37:40.732 but if I think French, Italian, Spanish, Romanian-- now 00:37:40.732 --> 00:37:44.143 I speak both Italian and Romanian-- 00:37:44.143 --> 00:37:47.220 we say it's a she, it's a feminine. 00:37:47.220 --> 00:37:52.345 So as I approach with values closer and closer and closer 00:37:52.345 --> 00:37:56.345 to 0, what happens to my poor function? 00:37:56.345 --> 00:37:58.911 It blows up. 00:37:58.911 --> 00:37:59.410 OK. 00:37:59.410 --> 00:38:05.040 So I have limit of 1/x from the right and from the left. 00:38:05.040 --> 00:38:08.310 If I take it from the left, I don't care. 00:38:08.310 --> 00:38:11.170 Let's take it only from the right. 00:38:11.170 --> 00:38:11.670 OK? 00:38:11.670 --> 00:38:17.910 00:38:17.910 --> 00:38:19.630 It's close to 0. 00:38:19.630 --> 00:38:21.210 That's going to blow up, right? 00:38:21.210 --> 00:38:24.560 00:38:24.560 --> 00:38:25.490 And I restrict it. 00:38:25.490 --> 00:38:30.455 So let's say, if I want the domain to be containing 00:38:30.455 --> 00:38:32.742 [? both, ?] that's also fine. 00:38:32.742 --> 00:38:35.730 So if you guys want, we can draw the other one. 00:38:35.730 --> 00:38:37.010 This goes to paradise. 00:38:37.010 --> 00:38:39.665 The other one, I'm not going to say where it goes. 00:38:39.665 --> 00:38:43.270 But it's the same idea, that as you approach 0 00:38:43.270 --> 00:38:45.710 with closer and closer and closer values, 00:38:45.710 --> 00:38:47.662 it's going to blow up. 00:38:47.662 --> 00:38:51.020 It's going to explode. 00:38:51.020 --> 00:38:54.010 This is a beautiful function. 00:38:54.010 --> 00:38:55.160 How beautiful [INAUDIBLE]. 00:38:55.160 --> 00:38:57.940 Beautiful with a bad behavior near 0. 00:38:57.940 --> 00:38:59.730 So I'm not going to have a limit. 00:38:59.730 --> 00:39:00.610 No limit. 00:39:00.610 --> 00:39:02.590 Some people say, limit exists and is infinity. 00:39:02.590 --> 00:39:05.265 But does infinity exist? 00:39:05.265 --> 00:39:07.382 Well, this is a really philosophical, 00:39:07.382 --> 00:39:10.570 religious notion, so I don't want to get into it. 00:39:10.570 --> 00:39:13.480 But in mathematics, we consider that unless the limit is 00:39:13.480 --> 00:39:16.730 finite, you cannot have a limit. 00:39:16.730 --> 00:39:20.850 So if the limit is plus/minus infinity, there is no limit. 00:39:20.850 --> 00:39:25.240 Could the limit be different or different subsequences? 00:39:25.240 --> 00:39:28.190 This is what I wanted to point out. 00:39:28.190 --> 00:39:34.270 If you try this guy, you are in real trouble on that guy. 00:39:34.270 --> 00:39:35.152 Why? 00:39:35.152 --> 00:39:36.568 You can have two. 00:39:36.568 --> 00:39:38.928 If you have a graphing calculator, which 00:39:38.928 --> 00:39:43.990 I'm going to be opposed to you being used in the classroom, 00:39:43.990 --> 00:39:46.160 you would probably see what happens. 00:39:46.160 --> 00:39:51.540 Sine is defined on all the real numbers. 00:39:51.540 --> 00:39:54.240 But you cannot have a value at 0, 00:39:54.240 --> 00:39:57.510 because the 1/x is not defined at 0. 00:39:57.510 --> 00:40:02.330 Imagine you get closer and closer to 0 from both sides. 00:40:02.330 --> 00:40:05.280 I cannot draw very beautifully. 00:40:05.280 --> 00:40:09.590 But as 1, this is plus 1 and this is minus 1. 00:40:09.590 --> 00:40:11.840 I'm going to have some behavior. 00:40:11.840 --> 00:40:15.160 And how many of you have seen that on a computer screen 00:40:15.160 --> 00:40:15.941 or calculator? 00:40:15.941 --> 00:40:16.440 You've seen. 00:40:16.440 --> 00:40:17.585 Yeah, you've seen. 00:40:17.585 --> 00:40:20.450 By the way, did you see the Lubbuck High? 00:40:20.450 --> 00:40:23.472 Was it in high school you saw it the first time in Calc 1 00:40:23.472 --> 00:40:25.140 or pre-calc? 00:40:25.140 --> 00:40:28.440 STUDENT: [INAUDIBLE] Algebra 1 with Mr. West. 00:40:28.440 --> 00:40:28.940 [INAUDIBLE] 00:40:28.940 --> 00:40:32.510 MAGDALENA TODA: So I'll try-- oh, guys, you 00:40:32.510 --> 00:40:34.240 have to be patient with me. 00:40:34.240 --> 00:40:38.150 I'm not leaving the poor board with the tip of my pencil. 00:40:38.150 --> 00:40:39.380 I'm not leaving him. 00:40:39.380 --> 00:40:42.358 I have continuity. 00:40:42.358 --> 00:40:45.810 As I got closer to this, I still have the [INAUDIBLE] property. 00:40:45.810 --> 00:40:47.150 Anyway, it's OK. 00:40:47.150 --> 00:40:48.410 I'm not leaving this. 00:40:48.410 --> 00:40:52.100 I am taking all the values possible between minus 1 and 1. 00:40:52.100 --> 00:40:54.820 So on intervals that are smaller, smaller, 00:40:54.820 --> 00:40:57.756 I'm really taking all the values between minus 1 and 1, 00:40:57.756 --> 00:41:01.390 and really rapidly-- [INAUDIBLE]. 00:41:01.390 --> 00:41:07.520 When I'm getting closer to 0, I'm not going to have a limit. 00:41:07.520 --> 00:41:10.065 But as somebody may say, but wait. 00:41:10.065 --> 00:41:12.270 When I have a sequence of values that 00:41:12.270 --> 00:41:14.360 is getting closer and closer to 0, 00:41:14.360 --> 00:41:18.610 is that no guarantee that I'm going to have a limit? 00:41:18.610 --> 00:41:20.120 Nope. 00:41:20.120 --> 00:41:20.965 It depends. 00:41:20.965 --> 00:41:25.475 If you say "any," it has to be for any choice of points, 00:41:25.475 --> 00:41:27.960 any choice of points that you go closer to 0. 00:41:27.960 --> 00:41:30.150 Not for one sequence of points that 00:41:30.150 --> 00:41:32.290 is getting closer and closer to 0. 00:41:32.290 --> 00:41:35.296 For example, if your choice of points is this, 00:41:35.296 --> 00:41:36.282 choice of points. 00:41:36.282 --> 00:41:39.740 00:41:39.740 --> 00:41:43.805 Getting closer to 0. 00:41:43.805 --> 00:41:49.540 [INAUDIBLE] xn equals 1 over 2 pi n. 00:41:49.540 --> 00:41:51.815 Isn't this going to 0? 00:41:51.815 --> 00:41:52.315 Yeah. 00:41:52.315 --> 00:41:54.090 It then goes to infinity. 00:41:54.090 --> 00:41:55.690 This sequence goes to 0. 00:41:55.690 --> 00:41:56.300 What is it? 00:41:56.300 --> 00:41:57.015 1 over 2 pi? 00:41:57.015 --> 00:41:58.070 1 over 4 pi? 00:41:58.070 --> 00:41:58.890 1 over 8 pi? 00:41:58.890 --> 00:41:59.980 1 over 16 pi? 00:41:59.980 --> 00:42:01.100 1 over 32 pi? 00:42:01.100 --> 00:42:02.550 1 over 64 pi? 00:42:02.550 --> 00:42:04.450 This is what my son is doing to me. 00:42:04.450 --> 00:42:06.182 And I say, please stop. 00:42:06.182 --> 00:42:07.130 OK? 00:42:07.130 --> 00:42:08.240 He's 10 years old. 00:42:08.240 --> 00:42:09.590 He's so funny. 00:42:09.590 --> 00:42:12.964 Now, another choice of points. 00:42:12.964 --> 00:42:20.490 00:42:20.490 --> 00:42:21.170 Ah. 00:42:21.170 --> 00:42:26.450 Somebody-- all of you are smart enough to do this. 00:42:26.450 --> 00:42:29.230 What do you think I'm going to pick? 00:42:29.230 --> 00:42:30.780 1 over what? 00:42:30.780 --> 00:42:33.604 And when [? other ?] something that goes to 0 00:42:33.604 --> 00:42:34.520 then goes to infinity. 00:42:34.520 --> 00:42:41.660 And I know that your professor showed you that. 00:42:41.660 --> 00:42:44.995 pi over 2 plus 2 pi n. 00:42:44.995 --> 00:42:45.870 Doesn't this go to 0? 00:42:45.870 --> 00:42:46.370 Yes. 00:42:46.370 --> 00:42:49.740 As n gets bigger and bigger, this is going to 0. 00:42:49.740 --> 00:42:51.040 However, there is no limit. 00:42:51.040 --> 00:42:51.810 Why? 00:42:51.810 --> 00:42:59.280 Well, for the first sequence, as xn goes to 0, f of xn 00:42:59.280 --> 00:43:04.060 goes to-- what is sine of-- OK, I 00:43:04.060 --> 00:43:05.870 am too lazy to write this down. 00:43:05.870 --> 00:43:11.015 Sine of 1 over 1 over-- of 1 over 1 over 2 pi? 00:43:11.015 --> 00:43:14.810 00:43:14.810 --> 00:43:16.780 STUDENT: It's the sine over 2 pi. 00:43:16.780 --> 00:43:20.650 MAGDALENA TODA: This is sine of 2 pi n. 00:43:20.650 --> 00:43:22.361 And how much is that? 00:43:22.361 --> 00:43:22.860 STUDENT: 0. 00:43:22.860 --> 00:43:24.210 MAGDALENA TODA: 0. 00:43:24.210 --> 00:43:25.590 So this is a 0. 00:43:25.590 --> 00:43:28.730 And this is a-- this converges to 0. 00:43:28.730 --> 00:43:31.400 So I say, oh, so maybe I have a limit, and that'll be 0. 00:43:31.400 --> 00:43:32.520 Wrong. 00:43:32.520 --> 00:43:35.730 That would be the rapid, stupid conclusion. 00:43:35.730 --> 00:43:38.690 If somebody jumps [? up, ?] I picked some points, 00:43:38.690 --> 00:43:41.820 I formed the sequence that gets closer and closer to 0. 00:43:41.820 --> 00:43:43.700 I'm sure that the limit exists. 00:43:43.700 --> 00:43:45.500 I've got a 0. 00:43:45.500 --> 00:43:48.850 Well, did you think of any possible choice? 00:43:48.850 --> 00:43:49.670 That's the problem. 00:43:49.670 --> 00:43:51.942 You have to have any possible choice. 00:43:51.942 --> 00:44:02.580 F of yn sine of 1 over pi over 2 plus 1 over 1 00:44:02.580 --> 00:44:09.030 over-- Magdalena-- pi over 2 plus 2 pi n. 00:44:09.030 --> 00:44:10.860 So we saw that this was 0. 00:44:10.860 --> 00:44:14.850 What happens to sine of 1 over 1 over sine of pi 00:44:14.850 --> 00:44:18.720 over 2 plus 2 pi n? 00:44:18.720 --> 00:44:19.890 And where does this go? 00:44:19.890 --> 00:44:21.132 It then goes to infinity. 00:44:21.132 --> 00:44:26.970 00:44:26.970 --> 00:44:29.250 This sequence goes to 0. 00:44:29.250 --> 00:44:32.630 What is f of the sequence going to? 00:44:32.630 --> 00:44:33.520 To another limit. 00:44:33.520 --> 00:44:36.030 So there is no limit. 00:44:36.030 --> 00:44:38.580 What's the limit of this subsequence? 00:44:38.580 --> 00:44:41.400 It's a constant one, right? 00:44:41.400 --> 00:44:45.930 Because look, what does it mean pi over 2 plus 2 pi n? 00:44:45.930 --> 00:44:49.240 Where am I on the unit trigonometric circle? 00:44:49.240 --> 00:44:50.690 [INTERPOSING VOICES] 00:44:50.690 --> 00:44:53.930 Always here, right? 00:44:53.930 --> 00:44:56.450 Always on the sort of like the north pole. 00:44:56.450 --> 00:44:59.220 So what is the sine of this north pole? 00:44:59.220 --> 00:44:59.900 STUDENT: 1. 00:44:59.900 --> 00:45:00.970 MAGDALENA TODA: Always 1. 00:45:00.970 --> 00:45:02.400 So I get the limit 1. 00:45:02.400 --> 00:45:06.482 So I'm done because there are two different limits. 00:45:06.482 --> 00:45:09.360 So pay attention to this type of problem. 00:45:09.360 --> 00:45:17.530 Somebody can get you in trouble with this kind of thing. 00:45:17.530 --> 00:45:20.220 On the other hand, I'm asking you, 00:45:20.220 --> 00:45:22.995 what if I want to make this a function of two variables? 00:45:22.995 --> 00:45:27.660 00:45:27.660 --> 00:45:30.960 So I'll say, one point extra credit. 00:45:30.960 --> 00:45:34.070 I'm giving you too much extra credit. 00:45:34.070 --> 00:45:36.210 Maybe I give you too much-- it's OK. 00:45:36.210 --> 00:45:39.930 One point extra credit-- put them together. 00:45:39.930 --> 00:45:43.340 00:45:43.340 --> 00:45:47.520 Does f-- do you like to do the f? 00:45:47.520 --> 00:45:51.400 I used big F, and then I changed it to little f. 00:45:51.400 --> 00:45:54.073 This time I have a function of two variables-- little 00:45:54.073 --> 00:46:01.178 f with xy-- to be sine of 1 over x squared plus y squared. 00:46:01.178 --> 00:46:09.442 Does this function have a limit at the point 0, 0? 00:46:09.442 --> 00:46:12.350 00:46:12.350 --> 00:46:15.540 So when I approach 0, 0, do I have a limit? 00:46:15.540 --> 00:46:16.680 OK. 00:46:16.680 --> 00:46:19.810 And you say, well, it depends how I approach that 0, 0. 00:46:19.810 --> 00:46:21.490 That's exactly the thing. 00:46:21.490 --> 00:46:23.210 Yes, sir. 00:46:23.210 --> 00:46:25.278 Oh, you didn't want to ask me. 00:46:25.278 --> 00:46:28.480 00:46:28.480 --> 00:46:37.115 And does f of xy equals-- let me give you 00:46:37.115 --> 00:46:41.080 another one, a really sexy one. x 00:46:41.080 --> 00:46:44.780 squared plus y squared times sine of 1 00:46:44.780 --> 00:46:48.460 over x squared plus y squared. 00:46:48.460 --> 00:46:55.053 Have a limit at 0, 0? 00:46:55.053 --> 00:47:00.030 00:47:00.030 --> 00:47:01.490 I don't know. 00:47:01.490 --> 00:47:04.220 Continuous it cannot be, because it's not defined there. 00:47:04.220 --> 00:47:04.720 Right? 00:47:04.720 --> 00:47:07.670 For a function to be continuous at a point, 00:47:07.670 --> 00:47:11.360 the function has to satisfy three conditions. 00:47:11.360 --> 00:47:14.590 The function has to be defined there at that point. 00:47:14.590 --> 00:47:16.560 The function has to have a limit there 00:47:16.560 --> 00:47:18.890 at that point of the domain. 00:47:18.890 --> 00:47:23.140 And the limit and the function value have to coincide. 00:47:23.140 --> 00:47:24.750 Three conditions. 00:47:24.750 --> 00:47:28.470 We will talk about continuity later. 00:47:28.470 --> 00:47:29.500 Hint. 00:47:29.500 --> 00:47:31.620 Magdalena, too many hints. 00:47:31.620 --> 00:47:33.680 This should remind you of somebody 00:47:33.680 --> 00:47:36.080 from the first variable calculus. 00:47:36.080 --> 00:47:37.970 It's a more challenging problem. 00:47:37.970 --> 00:47:40.490 That's why I gave it to extra credit. 00:47:40.490 --> 00:47:45.660 If I had x sine of 1/x, what would that look like? 00:47:45.660 --> 00:47:46.890 STUDENT: x times-- 00:47:46.890 --> 00:47:50.260 MAGDALENA TODA: x times sine of 1/x. 00:47:50.260 --> 00:47:55.458 When I approach 0 with-- so if I have-- I 00:47:55.458 --> 00:47:57.160 don't ask for an answer now. 00:47:57.160 --> 00:47:58.800 You go home, you think about it. 00:47:58.800 --> 00:48:00.310 You take the calculator. 00:48:00.310 --> 00:48:05.910 But keep in mind that your calculator can fool you. 00:48:05.910 --> 00:48:11.300 Sometimes it can show an image that misguides you. 00:48:11.300 --> 00:48:14.580 So you have to think how to do that. 00:48:14.580 --> 00:48:18.920 How about x times sine of 1/x when-- 00:48:18.920 --> 00:48:22.130 does it have a limit when x goes to 0? 00:48:22.130 --> 00:48:23.620 Is there such a limit? 00:48:23.620 --> 00:48:24.600 Does it exist? 00:48:24.600 --> 00:48:28.170 00:48:28.170 --> 00:48:31.420 So if such a limit would exist, maybe we 00:48:31.420 --> 00:48:35.600 can extend by continuity the function x times sine over x. 00:48:35.600 --> 00:48:36.610 What does it mean? 00:48:36.610 --> 00:48:38.670 Like, extend it, prolong it. 00:48:38.670 --> 00:48:44.415 And say, it's this 4x equals 0 and this if x is not 0. 00:48:44.415 --> 00:48:47.610 So this is obviously x is different from 0, right? 00:48:47.610 --> 00:48:49.710 Can we extend it by continuity? 00:48:49.710 --> 00:48:51.230 Think about the drawing. 00:48:51.230 --> 00:48:54.500 Think about the arguments. 00:48:54.500 --> 00:48:58.350 And I think it's time for me to keep the promise I made 00:48:58.350 --> 00:49:04.950 to [? Aaron, ?] because I see no way. 00:49:04.950 --> 00:49:07.860 Oh, my god, [? Aaron, ?] I see no way out. 00:49:07.860 --> 00:49:10.380 00:49:10.380 --> 00:49:14.490 The epsilon delta definition of limit. 00:49:14.490 --> 00:49:17.460 [? Right? ?] OK. 00:49:17.460 --> 00:49:21.070 So what does it mean for a real mathematician or somebody 00:49:21.070 --> 00:49:25.350 with a strong mathematical foundation and education 00:49:25.350 --> 00:49:27.430 that they know the true definition 00:49:27.430 --> 00:49:31.310 of a limit of a function of, let's say, one variable? 00:49:31.310 --> 00:49:34.910 The epsilon delta, the one your dad told you about. [INAUDIBLE] 00:49:34.910 --> 00:49:39.980 try to fool you when avoid it in undergraduate education. 00:49:39.980 --> 00:49:41.740 People try to avoid the epsilon delta, 00:49:41.740 --> 00:49:45.615 because they think the students will never, never understand 00:49:45.615 --> 00:49:50.470 it, because it's such an abstract one. 00:49:50.470 --> 00:49:51.844 I think I wasn't ready. 00:49:51.844 --> 00:49:52.760 I wasn't smart enough. 00:49:52.760 --> 00:49:58.015 I think I was 16 when I was getting ready for some math 00:49:58.015 --> 00:49:58.725 competitions. 00:49:58.725 --> 00:50:02.810 And one professor taught me the epsilon delta and said, 00:50:02.810 --> 00:50:04.700 do you understand it? 00:50:04.700 --> 00:50:07.310 My 16-year-old mind said, no. 00:50:07.310 --> 00:50:08.860 But guess what? 00:50:08.860 --> 00:50:10.546 Some other people smarter than me, 00:50:10.546 --> 00:50:12.410 they told me, when you first see it, 00:50:12.410 --> 00:50:16.690 you don't understand it in any case. 00:50:16.690 --> 00:50:19.950 So it takes a little bit more time to sink in. 00:50:19.950 --> 00:50:21.876 So the same idea. 00:50:21.876 --> 00:50:25.140 As I'm getting closer and closer and closer and closer 00:50:25.140 --> 00:50:30.280 to an x0 with my x values from anywhere around-- left, 00:50:30.280 --> 00:50:35.480 right-- I have to pick an arbitrary choice of points 00:50:35.480 --> 00:50:39.940 going towards x0, I have to be sure that at the same time, 00:50:39.940 --> 00:50:45.210 the corresponding sequence of values is going to L, 00:50:45.210 --> 00:50:47.010 I can express that in epsilon delta. 00:50:47.010 --> 00:50:50.860 00:50:50.860 --> 00:50:51.935 So we say that. 00:50:51.935 --> 00:50:59.705 00:50:59.705 --> 00:51:13.178 f of x has limit L at x equals x0 if. 00:51:13.178 --> 00:51:17.180 00:51:17.180 --> 00:51:24.100 For every epsilon positive, any choice of an epsilon positive, 00:51:24.100 --> 00:51:25.230 there is a delta. 00:51:25.230 --> 00:51:27.110 There exists-- oh, OK, guys. 00:51:27.110 --> 00:51:28.430 You don't know the symbols. 00:51:28.430 --> 00:51:31.290 I'll write it in English. 00:51:31.290 --> 00:51:36.450 For every epsilon positive, no matter 00:51:36.450 --> 00:51:40.728 how small-- put parentheses, because you 00:51:40.728 --> 00:51:47.150 are just [? tired-- ?] no matter how small, 00:51:47.150 --> 00:51:55.510 there exists a delta number that depends on epsilon. 00:51:55.510 --> 00:52:02.356 00:52:02.356 --> 00:52:16.190 So that whenever x minus x0 is less than delta, 00:52:16.190 --> 00:52:33.800 this would imply that f of x minus L, 00:52:33.800 --> 00:52:37.098 that limit I taught you about in absolute value, 00:52:37.098 --> 00:52:38.562 is less than epsilon. 00:52:38.562 --> 00:52:48.340 00:52:48.340 --> 00:52:50.370 What does this mean? 00:52:50.370 --> 00:52:55.200 I'm going to try and draw something 00:52:55.200 --> 00:52:58.441 that happens on a line. 00:52:58.441 --> 00:53:00.370 So this is x0. 00:53:00.370 --> 00:53:03.894 And these are my values of x. 00:53:03.894 --> 00:53:05.060 They can come from anywhere. 00:53:05.060 --> 00:53:08.760 00:53:08.760 --> 00:53:12.032 And this is f of x. 00:53:12.032 --> 00:53:16.700 And this is L. So it says, no matter-- this 00:53:16.700 --> 00:53:19.150 says-- this is an abstract way of saying, 00:53:19.150 --> 00:53:23.810 no matter how close, you see, for every epsilon positive, 00:53:23.810 --> 00:53:27.280 no matter how close you get to the L. 00:53:27.280 --> 00:53:30.600 I decide to be in this interval, very tiny epsilon. 00:53:30.600 --> 00:53:32.260 L minus epsilon. 00:53:32.260 --> 00:53:35.840 L plus epsilon L. You give me your favorite epsilon. 00:53:35.840 --> 00:53:38.640 You say, Magdalena, pick something really small. 00:53:38.640 --> 00:53:42.180 Big epsilon to be 0.00001. 00:53:42.180 --> 00:53:43.950 How about that? 00:53:43.950 --> 00:53:47.660 Well, if I really have a limit there, 00:53:47.660 --> 00:53:54.290 an L at x0, that means that no matter how much you shrink 00:53:54.290 --> 00:53:57.540 this interval for me, you can be mean and shrink it 00:53:57.540 --> 00:53:59.320 as much as you want. 00:53:59.320 --> 00:54:03.400 I will still find a small interval around x0. 00:54:03.400 --> 00:54:06.660 00:54:06.660 --> 00:54:08.620 [? But ?] I will still find the smaller 00:54:08.620 --> 00:54:13.230 interval around x0, which is-- this would be x0 minus delta. 00:54:13.230 --> 00:54:15.760 This would be x0 plus delta. 00:54:15.760 --> 00:54:20.961 So that the image of this purple interval fits inside. 00:54:20.961 --> 00:54:22.150 You say, what? 00:54:22.150 --> 00:54:25.840 So that the image of this purple interval fits inside. 00:54:25.840 --> 00:54:30.040 So f of x minus L, the distance is still that, less than xy. 00:54:30.040 --> 00:54:30.550 Yes, sir? 00:54:30.550 --> 00:54:32.710 STUDENT: Where'd you get epsilon [INAUDIBLE]? 00:54:32.710 --> 00:54:34.440 MAGDALENA TODA: So epsilon has to be 00:54:34.440 --> 00:54:38.252 chose no matter how small. 00:54:38.252 --> 00:54:39.837 STUDENT: [INAUDIBLE]. 00:54:39.837 --> 00:54:40.670 MAGDALENA TODA: Huh? 00:54:40.670 --> 00:54:42.160 Real number. 00:54:42.160 --> 00:54:46.470 So I'm saying, you should not set the epsilon to be 0.0001. 00:54:46.470 --> 00:54:47.770 That would be a mistake. 00:54:47.770 --> 00:54:50.890 You have to think of that number as being as small as you want, 00:54:50.890 --> 00:54:54.590 infinitesimally small, smaller than any particle in physics 00:54:54.590 --> 00:54:57.330 that you are aware about. 00:54:57.330 --> 00:54:59.880 And this is what I had the problem understanding-- 00:54:59.880 --> 00:55:03.530 that notion of-- not the notion of, hey, not 00:55:03.530 --> 00:55:06.000 matter how close I am, I can still 00:55:06.000 --> 00:55:12.430 get something even smaller around x0 that fits in this. 00:55:12.430 --> 00:55:14.360 That's not what I had the problem with. 00:55:14.360 --> 00:55:18.316 The notion is to perceive an infinitesimal. 00:55:18.316 --> 00:55:21.515 Our mind is too limited to understand infinity. 00:55:21.515 --> 00:55:24.408 It's like trying to understand God. 00:55:24.408 --> 00:55:29.512 And the same limitation comes with microscopic problems. 00:55:29.512 --> 00:55:31.470 Yeah, we can see some things on the microscope, 00:55:31.470 --> 00:55:32.219 and we understand. 00:55:32.219 --> 00:55:34.580 Ah, I understand I have this bacteria. 00:55:34.580 --> 00:55:36.040 This is staph. 00:55:36.040 --> 00:55:37.460 Oh, my god. 00:55:37.460 --> 00:55:43.350 But then there are molecules, atoms, subatomic particles 00:55:43.350 --> 00:55:46.875 that we don't understand, because our mind is really 00:55:46.875 --> 00:55:49.060 [? small. ?] Imagine something smaller 00:55:49.060 --> 00:55:50.900 than the subatomic particles. 00:55:50.900 --> 00:55:54.520 That's the abstract notion of infinitesimally small. 00:55:54.520 --> 00:55:58.870 So I'm saying, if I really have a limit L there, 00:55:58.870 --> 00:56:03.470 that means no matter how small I have this ball around it, 00:56:03.470 --> 00:56:06.630 I can still find a smaller ball that 00:56:06.630 --> 00:56:10.001 fits-- whose image fits inside. 00:56:10.001 --> 00:56:10.500 All right? 00:56:10.500 --> 00:56:15.000 The same kind of definition-- I will try to generalize this. 00:56:15.000 --> 00:56:19.660 Can you guys help me generalize this limit notion 00:56:19.660 --> 00:56:24.790 to the notion of function of two variables? 00:56:24.790 --> 00:56:29.040 00:56:29.040 --> 00:56:41.400 So we say, that f of xy has the limit L at x0y0. 00:56:41.400 --> 00:56:44.970 00:56:44.970 --> 00:56:50.710 What was x0y0 when I talked about-- what 00:56:50.710 --> 00:56:52.550 example did I give you guys? 00:56:52.550 --> 00:56:55.050 Sine of 1 over x squared plus y squared, right? 00:56:55.050 --> 00:56:56.220 Something like that. 00:56:56.220 --> 00:56:56.850 I don't know. 00:56:56.850 --> 00:56:59.730 I said, think of 0, 0. 00:56:59.730 --> 00:57:01.810 That was the given point. 00:57:01.810 --> 00:57:03.540 It has to be a fixed couple. 00:57:03.540 --> 00:57:07.870 So you think of the origin, 0, 0, as being as a fixed couple. 00:57:07.870 --> 00:57:12.390 Or you think of the point 1, 0 as being as a fixed couple 00:57:12.390 --> 00:57:14.761 in that plane you look at. 00:57:14.761 --> 00:57:18.420 That is the fixed couple. 00:57:18.420 --> 00:57:21.460 If-- now somebody has to help me. 00:57:21.460 --> 00:57:27.540 For every epsilon positive, no matter how small, 00:57:27.540 --> 00:57:30.690 that's where I have a problem imagining infinitesimally 00:57:30.690 --> 00:57:31.615 small. 00:57:31.615 --> 00:57:34.650 There exists-- I no longer have this problem. 00:57:34.650 --> 00:57:37.310 But I had it enough when I was in my 20s. 00:57:37.310 --> 00:57:40.155 I don't want to go back to my 20s and have-- I mean, 00:57:40.155 --> 00:57:41.065 I would love to. 00:57:41.065 --> 00:57:42.890 [LAUGHTER] 00:57:42.890 --> 00:57:46.420 To go having vacations with no worries and so on. 00:57:46.420 --> 00:57:48.570 But I wouldn't like to go back to my 20s 00:57:48.570 --> 00:57:50.470 and have to relearn all the mathematics. 00:57:50.470 --> 00:57:50.970 Now way. 00:57:50.970 --> 00:57:53.190 That was too much of a struggle. 00:57:53.190 --> 00:58:00.310 There exists a delta positive that depends on epsilon. 00:58:00.310 --> 00:58:02.820 What does it mean, depends on epsilon? 00:58:02.820 --> 00:58:04.650 Because guys, imagine you make this epsilon 00:58:04.650 --> 00:58:05.860 smaller and smaller. 00:58:05.860 --> 00:58:08.100 You have to make delta smaller and smaller, 00:58:08.100 --> 00:58:12.321 so that you can fit that little ball in the big ball. 00:58:12.321 --> 00:58:12.820 OK? 00:58:12.820 --> 00:58:19.768 That depends on epsilon, so that whenever-- now, 00:58:19.768 --> 00:58:21.930 that is a big problem. 00:58:21.930 --> 00:58:28.305 How do I say, distance between the point xy and the point 00:58:28.305 --> 00:58:29.310 x0y0? 00:58:29.310 --> 00:58:32.130 Oh, my god. 00:58:32.130 --> 00:58:37.310 This is distance between xy and x0y0 is less than delta. 00:58:37.310 --> 00:58:48.480 This would imply that-- well, this 00:58:48.480 --> 00:58:54.090 is a function with values in R. This is in R. Real number. 00:58:54.090 --> 00:58:55.470 So I don't have a problem. 00:58:55.470 --> 00:58:57.460 I can use absolute value here. 00:58:57.460 --> 00:59:11.330 Absolute value of f of the couple xy minus L 00:59:11.330 --> 00:59:14.580 is less than epsilon. 00:59:14.580 --> 00:59:19.080 The thing is, can you visualize that little ball, 00:59:19.080 --> 00:59:20.990 that little disk? 00:59:20.990 --> 00:59:22.310 What do I mean? 00:59:22.310 --> 00:59:26.440 Being close, xy is me, right? 00:59:26.440 --> 00:59:27.520 But I'm moving. 00:59:27.520 --> 00:59:28.760 I'm the moving point. 00:59:28.760 --> 00:59:30.350 I'm dancing around. 00:59:30.350 --> 00:59:33.436 And [? Nateesh ?] is x0y0. 00:59:33.436 --> 00:59:37.570 How do I say that I have to be close enough to him? 00:59:37.570 --> 00:59:38.600 I cannot touch him. 00:59:38.600 --> 00:59:39.710 That's against the rules. 00:59:39.710 --> 00:59:42.270 That's considered [INAUDIBLE] harassment. 00:59:42.270 --> 00:59:45.800 But I can come as close as I want. 00:59:45.800 --> 00:59:49.210 So I say, the distance between me-- 00:59:49.210 --> 00:59:52.100 I'm xy-- and [? Nateesh, ?] who is 00:59:52.100 --> 00:59:58.264 fixed x0y0, has to be smaller than that small delta. 00:59:58.264 --> 01:00:00.796 How do I represent that in plane mathematics? 01:00:00.796 --> 01:00:02.004 STUDENT: Doesn't [INAUDIBLE]? 01:00:02.004 --> 01:00:05.520 01:00:05.520 --> 01:00:06.520 MAGDALENA TODA: Exactly. 01:00:06.520 --> 01:00:09.480 So that delta has to be small enough so 01:00:09.480 --> 01:00:16.970 that the image of f at me minus the limit is less than epsilon. 01:00:16.970 --> 01:00:20.960 Now you understand why all the other teachers avoid 01:00:20.960 --> 01:00:22.820 talking about this [? one. ?] So I 01:00:22.820 --> 01:00:27.840 want to get small enough-- not too close-- but close enough 01:00:27.840 --> 01:00:39.520 to him, so that my value-- I'm f of xy-- minus the limit, 01:00:39.520 --> 01:00:42.170 the limit-- I have a preset limit. 01:00:42.170 --> 01:00:45.140 All around [? Nateesh, ?] I can have different values, 01:00:45.140 --> 01:00:47.490 no matter where I go. 01:00:47.490 --> 01:00:51.060 My value at all these points around [? Nateesh ?] have 01:00:51.060 --> 01:00:54.510 to be close enough to L. So I say, 01:00:54.510 --> 01:00:57.550 well, you have to get close enough to L. 01:00:57.550 --> 01:00:59.490 Somebody presents me an epsilon. 01:00:59.490 --> 01:01:02.350 Then I have to reduce my distance to [? Nateesh ?] 01:01:02.350 --> 01:01:03.990 depending to that epsilon. 01:01:03.990 --> 01:01:07.959 Because otherwise, the image doesn't fit. 01:01:07.959 --> 01:01:09.000 It's a little bit tricky. 01:01:09.000 --> 01:01:10.960 STUDENT: So is this like the squeeze theorem kind of? 01:01:10.960 --> 01:01:12.430 MAGDALENA TODA: It is the squeeze theorem. 01:01:12.430 --> 01:01:12.930 STUDENT: Oh, all right. 01:01:12.930 --> 01:01:13.721 MAGDALENA TODA: OK? 01:01:13.721 --> 01:01:18.580 So the squeezing-- I ball into another [? ball ?] [? limit. ?] 01:01:18.580 --> 01:01:21.030 This is why-- it's not a ball, but it's a-- 01:01:21.030 --> 01:01:21.780 STUDENT: A circle. 01:01:21.780 --> 01:01:22.655 MAGDALENA TODA: Disk. 01:01:22.655 --> 01:01:23.720 A circle, right? 01:01:23.720 --> 01:01:28.830 So how do we express that in Calc 3 in plain? 01:01:28.830 --> 01:01:31.005 This is the [? ingredient, ?] distance d. 01:01:31.005 --> 01:01:33.630 So Seth, can you tell me what is the distance between these two 01:01:33.630 --> 01:01:34.790 points? 01:01:34.790 --> 01:01:36.010 Square root of-- 01:01:36.010 --> 01:01:37.360 STUDENT: [INAUDIBLE]. 01:01:37.360 --> 01:01:41.750 MAGDALENA TODA: x minus x0 squared plus y minus y0 01:01:41.750 --> 01:01:43.154 squared. 01:01:43.154 --> 01:01:45.383 Now shut up. [? And I ?] am talking to myself. 01:01:45.383 --> 01:01:45.856 STUDENT: Must be less than delta. 01:01:45.856 --> 01:01:46.695 [LAUGHTER] 01:01:46.695 --> 01:01:48.280 MAGDALENA TODA: Less than delta. 01:01:48.280 --> 01:01:51.530 So instead of writing this, I need 01:01:51.530 --> 01:01:53.726 to write that I can do that in my mind. 01:01:53.726 --> 01:01:58.356 01:01:58.356 --> 01:02:00.024 OK? 01:02:00.024 --> 01:02:00.990 All right. 01:02:00.990 --> 01:02:01.850 This is hard. 01:02:01.850 --> 01:02:03.470 We need to sleep on that. 01:02:03.470 --> 01:02:09.449 I have one or two more problems that are less hard-- nah, 01:02:09.449 --> 01:02:11.490 they are still hard, but they are more intuitive, 01:02:11.490 --> 01:02:14.570 that I would like to ask you about the limit. 01:02:14.570 --> 01:02:16.940 I'm going to give you a function. 01:02:16.940 --> 01:02:21.476 And we would have to visualize as I get closer to a point 01:02:21.476 --> 01:02:24.570 where I am actually going. 01:02:24.570 --> 01:02:30.170 So I have this nasty function, f of xy 01:02:30.170 --> 01:02:34.960 equals xy over z squared plus y squared. 01:02:34.960 --> 01:02:39.020 01:02:39.020 --> 01:02:44.600 And I'm saying, [INAUDIBLE] the point is the origin. 01:02:44.600 --> 01:02:47.030 I choose the origin. 01:02:47.030 --> 01:02:47.590 Question. 01:02:47.590 --> 01:02:53.020 Do I have a limit that's-- do I have a limit? 01:02:53.020 --> 01:02:54.620 Not [? really ?] for me. 01:02:54.620 --> 01:03:02.480 Does f have a limit at the origin? 01:03:02.480 --> 01:03:06.400 01:03:06.400 --> 01:03:09.820 You would have to imagine that you'd draw this function. 01:03:09.820 --> 01:03:13.145 And except you cannot draw, and you really don't care to draw 01:03:13.145 --> 01:03:13.780 it. 01:03:13.780 --> 01:03:17.030 You only have to imagine that you have some abstract graph-- 01:03:17.030 --> 01:03:19.050 z equals f of xy. 01:03:19.050 --> 01:03:20.880 You don't care what it looks like. 01:03:20.880 --> 01:03:24.100 But then you take points on the floor, 01:03:24.100 --> 01:03:27.450 just like I did the exercise with [? Nateesh ?] before. 01:03:27.450 --> 01:03:30.820 And you get closer and closer to the origin. 01:03:30.820 --> 01:03:34.415 But no attention-- no matter what path I take, 01:03:34.415 --> 01:03:36.850 I have to get the same limit. 01:03:36.850 --> 01:03:37.548 What? 01:03:37.548 --> 01:03:46.680 No matter what path I take towards [? Nateesh-- ?] 01:03:46.680 --> 01:03:52.860 don't write that down-- towards [? z0y0, ?] I have to get 01:03:52.860 --> 01:03:53.790 the same limit. 01:03:53.790 --> 01:03:56.970 01:03:56.970 --> 01:03:59.030 Do I? 01:03:59.030 --> 01:04:04.030 Let's imagine with the eyes of your imaginations. 01:04:04.030 --> 01:04:07.410 And [? Nateesh ?] is the point 0, 0. 01:04:07.410 --> 01:04:10.652 And you are aspiring to get closer and closer to him 01:04:10.652 --> 01:04:12.750 without touching him. 01:04:12.750 --> 01:04:15.265 Because otherwise, he's going to sue you. 01:04:15.265 --> 01:04:18.120 So what do we have here? 01:04:18.120 --> 01:04:19.350 We have different paths? 01:04:19.350 --> 01:04:21.390 How can I get closer? 01:04:21.390 --> 01:04:25.550 Either on this path or maybe on this path. 01:04:25.550 --> 01:04:28.210 Or maybe on this path. 01:04:28.210 --> 01:04:31.590 Or maybe, if I had something to drink last night-- which 01:04:31.590 --> 01:04:35.115 I did not, because after the age of 35, 01:04:35.115 --> 01:04:37.032 I stopped drinking completely. 01:04:37.032 --> 01:04:40.990 01:04:40.990 --> 01:04:44.820 That's when I decided I want to be a mom, 01:04:44.820 --> 01:04:47.270 and I didn't want to make a bad example. 01:04:47.270 --> 01:04:50.450 So no matter what path you take, you can make it wiggly, 01:04:50.450 --> 01:04:52.030 you can make it any way you want. 01:04:52.030 --> 01:04:53.700 We are still approaching 0, 0. 01:04:53.700 --> 01:04:55.925 You still have to get the same limit. 01:04:55.925 --> 01:05:00.170 Oh, that's tricky, because it's also the same in life. 01:05:00.170 --> 01:05:02.430 Depending on the path you take in life, 01:05:02.430 --> 01:05:05.450 you have different results, different limits. 01:05:05.450 --> 01:05:11.243 Now, what if I take the path number one, number two, number 01:05:11.243 --> 01:05:12.720 three possibility. 01:05:12.720 --> 01:05:16.900 And number [? blooie ?] is the drunken variant. 01:05:16.900 --> 01:05:22.120 That is hard to implement in an exercise. 01:05:22.120 --> 01:05:26.600 Imagine that I have limit along the path one. 01:05:26.600 --> 01:05:28.290 Path one. 01:05:28.290 --> 01:05:34.680 xy goes to 0, 0 of xy over x squared plus y squared. 01:05:34.680 --> 01:05:36.970 Do you guys see what's going to happen? 01:05:36.970 --> 01:05:40.660 So I'm along the-- OK, here it is. 01:05:40.660 --> 01:05:46.590 This line, right, this is the x-axis, y-axis, z-axis. 01:05:46.590 --> 01:05:48.530 What's special for the x-axis? 01:05:48.530 --> 01:05:50.470 Who is 0? 01:05:50.470 --> 01:05:52.700 STUDENT: x. 01:05:52.700 --> 01:05:53.330 STUDENT: yz. 01:05:53.330 --> 01:05:54.420 MAGDALENA TODA: y is 0. 01:05:54.420 --> 01:05:57.400 So y is 0. 01:05:57.400 --> 01:05:59.410 So y is 0. 01:05:59.410 --> 01:06:00.360 Don't laugh at me. 01:06:00.360 --> 01:06:03.470 I'm going to write like that because it's easier. 01:06:03.470 --> 01:06:06.770 And it's going to be something like limit 01:06:06.770 --> 01:06:13.738 when x approaches 0 of x over x squared. 01:06:13.738 --> 01:06:15.690 STUDENT: It's 1/x. 01:06:15.690 --> 01:06:17.950 MAGDALENA TODA: Times 0 up. 01:06:17.950 --> 01:06:18.950 Oh, my god. 01:06:18.950 --> 01:06:20.671 Is that-- how much is that? 01:06:20.671 --> 01:06:21.170 STUDENT: 0. 01:06:21.170 --> 01:06:21.290 STUDENT: 0. 01:06:21.290 --> 01:06:22.039 MAGDALENA TODA: 0! 01:06:22.039 --> 01:06:22.570 I'm happy. 01:06:22.570 --> 01:06:23.980 I say, maybe I have the limit. 01:06:23.980 --> 01:06:24.820 I have the limit 0. 01:06:24.820 --> 01:06:27.120 No, never rush in life. 01:06:27.120 --> 01:06:28.040 Check. 01:06:28.040 --> 01:06:30.780 Experiment any other paths. 01:06:30.780 --> 01:06:35.120 And it's actually very easy to see where I can go wrong. 01:06:35.120 --> 01:06:39.630 If I take the path number two, I will get the same result. 01:06:39.630 --> 01:06:41.470 You don't need a lot of imagination 01:06:41.470 --> 01:06:44.355 to realize, hey, whether she does it for x 01:06:44.355 --> 01:06:48.362 or does it for y, if she goes along the 2, what 01:06:48.362 --> 01:06:49.790 the heck is going to happen? 01:06:49.790 --> 01:06:51.030 y is going to shrink. 01:06:51.030 --> 01:06:53.150 x will always be 0. 01:06:53.150 --> 01:06:56.850 Because this means a point's like what? 01:06:56.850 --> 01:06:58.850 0,1. 01:06:58.850 --> 01:07:00.835 0, 1/2. 01:07:00.835 --> 01:07:03.180 0, 1/n, and so on. 01:07:03.180 --> 01:07:07.820 But plug them all in here, I get 0, 1/n times 0. 01:07:07.820 --> 01:07:08.670 It's still 0. 01:07:08.670 --> 01:07:10.460 So I still get 0. 01:07:10.460 --> 01:07:12.310 Path two. 01:07:12.310 --> 01:07:15.470 When I approach my-- xt goes to 0, 0. 01:07:15.470 --> 01:07:18.790 The poor [? Nateesh ?] is waiting for an answer. 01:07:18.790 --> 01:07:20.950 I still get 0. 01:07:20.950 --> 01:07:23.645 Let's take not the drunken path, because I 01:07:23.645 --> 01:07:25.520 don't know [? it unless ?] the sine function. 01:07:25.520 --> 01:07:26.970 That is really crazy. 01:07:26.970 --> 01:07:29.010 I'll take this one. 01:07:29.010 --> 01:07:31.420 What is this one, in your opinion? 01:07:31.420 --> 01:07:32.970 Is that going to help me? 01:07:32.970 --> 01:07:35.720 I don't know, but I need some intuition. 01:07:35.720 --> 01:07:39.550 Mathematicians need intuition and a lot of patience. 01:07:39.550 --> 01:07:42.110 So what is your intuition? 01:07:42.110 --> 01:07:45.120 The one in the middle, I'm going to start walking on that, OK, 01:07:45.120 --> 01:07:46.542 until you tell me what it is. 01:07:46.542 --> 01:07:47.500 STUDENT: y [INAUDIBLE]. 01:07:47.500 --> 01:07:49.540 MAGDALENA TODA: y equals x is the first bisector 01:07:49.540 --> 01:07:51.040 or the first quadrant. 01:07:51.040 --> 01:07:54.580 And I'm very happy I can go both ways. 01:07:54.580 --> 01:07:55.850 y equals x. 01:07:55.850 --> 01:07:56.460 x [INAUDIBLE]. 01:07:56.460 --> 01:08:07.130 So limit when x equals y, but the pair xy goes to 0,0. 01:08:07.130 --> 01:08:07.980 I'm silly. 01:08:07.980 --> 01:08:10.978 I can say that, well, Magdalena, this 01:08:10.978 --> 01:08:15.540 is the pair xx, because x equals what? 01:08:15.540 --> 01:08:16.899 Let me plug them in. 01:08:16.899 --> 01:08:19.085 So it's like two people. 01:08:19.085 --> 01:08:20.555 x and y are married. 01:08:20.555 --> 01:08:22.180 They are a couple, a pair. 01:08:22.180 --> 01:08:24.330 They look identical. 01:08:24.330 --> 01:08:26.380 Sometimes it happens. 01:08:26.380 --> 01:08:28.439 Like twins, they start looking alike, 01:08:28.439 --> 01:08:30.819 dressing alike, and so on. 01:08:30.819 --> 01:08:36.529 The x and the y have to receive the same letter. 01:08:36.529 --> 01:08:41.029 And you have to tell me what in the world the limit will be. 01:08:41.029 --> 01:08:43.818 01:08:43.818 --> 01:08:44.359 STUDENT: 1/2. 01:08:44.359 --> 01:08:45.520 MAGDALENA TODA: 1/2. 01:08:45.520 --> 01:08:46.640 Oh, my god. 01:08:46.640 --> 01:08:48.479 So now I'm deflated. 01:08:48.479 --> 01:08:52.470 So now I realize that taking two different paths, 01:08:52.470 --> 01:08:57.578 I show that I have-- on this path, I have 1/2. 01:08:57.578 --> 01:09:00.069 On this path, I have 0. 01:09:00.069 --> 01:09:01.149 I don't match. 01:09:01.149 --> 01:09:02.710 I don't have an overall limit. 01:09:02.710 --> 01:09:10.140 So the answer is, no overall limit. 01:09:10.140 --> 01:09:10.960 Oh, my god. 01:09:10.960 --> 01:09:14.640 So what you need to do, guys, is read 01:09:14.640 --> 01:09:18.340 section 11.1 and section 11.2. 01:09:18.340 --> 01:09:21.100 And I will ask you next time-- and you can lie, 01:09:21.100 --> 01:09:22.892 you can do whatever. 01:09:22.892 --> 01:09:26.368 Did the book explain better than me, 01:09:26.368 --> 01:09:28.890 or I explain better than the book? 01:09:28.890 --> 01:09:31.790 This type of example when the limit does not exist. 01:09:31.790 --> 01:09:33.430 We are going to see more examples. 01:09:33.430 --> 01:09:37.783 You are going to see examples where the limit does exist. 01:09:37.783 --> 01:09:40.160 Now, one last thing. 01:09:40.160 --> 01:09:46.640 When you have to compute limits of compositions of functions 01:09:46.640 --> 01:09:48.529 whose limit exist-- for example, you 01:09:48.529 --> 01:09:58.290 know that limit is xy goes to x0y0 of f 01:09:58.290 --> 01:10:10.410 of xy [INAUDIBLE] limit of xy go to x0y0 of gxy 01:10:10.410 --> 01:10:13.730 is L-- L-- L-- M-- M. 01:10:13.730 --> 01:10:25.200 How are you going to compute the limit of alpha f plus beta g? 01:10:25.200 --> 01:10:27.100 This is in the book. 01:10:27.100 --> 01:10:32.650 But you don't need the book to understand that. 01:10:32.650 --> 01:10:34.320 You will already give me the answer, 01:10:34.320 --> 01:10:39.210 because this is the equivalent thing to the function of one 01:10:39.210 --> 01:10:41.320 variable thing in Calc 1. 01:10:41.320 --> 01:10:43.670 So if you would only have f of x or g of x, 01:10:43.670 --> 01:10:45.410 it would be piece of cake. 01:10:45.410 --> 01:10:46.501 What would you say? 01:10:46.501 --> 01:10:47.375 STUDENT: [INAUDIBLE]. 01:10:47.375 --> 01:10:48.291 MAGDALENA TODA: Right. 01:10:48.291 --> 01:10:54.080 Alpha times L plus beta times M. Can you also 01:10:54.080 --> 01:10:55.379 multiply functions. 01:10:55.379 --> 01:10:55.920 Yes, you can. 01:10:55.920 --> 01:11:07.160 Limit of fg as xy goes to x0 or y0-- will be LM. 01:11:07.160 --> 01:11:09.750 How about-- now I'm going to jump to conclusion, hoping 01:11:09.750 --> 01:11:13.170 that you are going to catch me. 01:11:13.170 --> 01:11:15.652 You are going to catch me, and shout at me, 01:11:15.652 --> 01:11:18.410 and say, ooh, pay attention, Magdalena, 01:11:18.410 --> 01:11:21.560 you can make a mistake there. 01:11:21.560 --> 01:11:26.430 I say it's L/M when I do the division rule, right? 01:11:26.430 --> 01:11:28.222 Where should I pay attention? 01:11:28.222 --> 01:11:29.761 STUDENT: M [INAUDIBLE]. 01:11:29.761 --> 01:11:31.010 MAGDALENA TODA: Pay attention. 01:11:31.010 --> 01:11:38.740 Sometimes you can have the-- right? 01:11:38.740 --> 01:11:45.120 And this also has to exist as well. 01:11:45.120 --> 01:11:46.560 STUDENT: [INAUDIBLE]. 01:11:46.560 --> 01:11:51.070 MAGDALENA TODA: So one last-- how many minutes 01:11:51.070 --> 01:11:53.810 have I spent with you? 01:11:53.810 --> 01:11:57.926 I've spent with you a long number of hours of my life. 01:11:57.926 --> 01:11:58.800 No, I'm just kidding. 01:11:58.800 --> 01:12:04.234 So you have one hour and 15, a little bit more. 01:12:04.234 --> 01:12:05.400 Do I have a little bit more? 01:12:05.400 --> 01:12:05.900 Yes. 01:12:05.900 --> 01:12:07.990 I have 15 minutes. 01:12:07.990 --> 01:12:08.490 I have-- 01:12:08.490 --> 01:12:09.615 STUDENT: So we get out at-- 01:12:09.615 --> 01:12:10.450 [INTERPOSING VOICES] 01:12:10.450 --> 01:12:11.241 MAGDALENA TODA: 50. 01:12:11.241 --> 01:12:12.514 Five more minutes. 01:12:12.514 --> 01:12:15.360 OK. 01:12:15.360 --> 01:12:20.655 So I want to ask you what you remember about some 01:12:20.655 --> 01:12:25.090 of your friends, the trig functions involved in limits. 01:12:25.090 --> 01:12:28.030 01:12:28.030 --> 01:12:32.000 Why did we study limits at the point 01:12:32.000 --> 01:12:34.290 where the function's not defined? 01:12:34.290 --> 01:12:35.430 Well, to heck with it. 01:12:35.430 --> 01:12:36.030 We don't care. 01:12:36.030 --> 01:12:37.950 The function is not defined at 0. 01:12:37.950 --> 01:12:40.076 But the limit is. 01:12:40.076 --> 01:12:42.916 And nobody showed you how to do the epsilon delta 01:12:42.916 --> 01:12:44.255 to show anything like that. 01:12:44.255 --> 01:12:48.526 01:12:48.526 --> 01:12:49.987 OK. 01:12:49.987 --> 01:12:52.422 Can you do that with epsilon delta? 01:12:52.422 --> 01:12:57.790 01:12:57.790 --> 01:13:00.352 Actually, you can do everything with epsilon delta. 01:13:00.352 --> 01:13:02.310 But I'm not going to give you any extra credit. 01:13:02.310 --> 01:13:07.538 So I trust you that you remember that. 01:13:07.538 --> 01:13:08.900 1! 01:13:08.900 --> 01:13:10.815 How about-- let me-- OK. 01:13:10.815 --> 01:13:11.767 I am so proud of you. 01:13:11.767 --> 01:13:12.850 Let me challenge you more. 01:13:12.850 --> 01:13:14.690 Let me challenge you more. 01:13:14.690 --> 01:13:17.950 Tangent of ax over bx. 01:13:17.950 --> 01:13:19.500 x go to 0. 01:13:19.500 --> 01:13:22.110 I asked this to a girl from Lubbock High. 01:13:22.110 --> 01:13:23.450 She was in high school. 01:13:23.450 --> 01:13:25.390 She knew the answer. 01:13:25.390 --> 01:13:28.300 STUDENT: Oh, I can't disappoint everybody in getting this. 01:13:28.300 --> 01:13:31.476 STUDENT: Is it 1/a? 01:13:31.476 --> 01:13:32.350 Oh, I can't remember. 01:13:32.350 --> 01:13:33.860 MAGDALENA TODA: Tell me what to do to be smart. 01:13:33.860 --> 01:13:34.360 Right? 01:13:34.360 --> 01:13:37.530 I have to be doing something smart. 01:13:37.530 --> 01:13:40.050 She-- can you give me hint? 01:13:40.050 --> 01:13:41.779 I'm your student and you say, well-- 01:13:41.779 --> 01:13:42.320 STUDENT: ba-- 01:13:42.320 --> 01:13:44.062 STUDENT: It's 0. 01:13:44.062 --> 01:13:45.447 STUDENT: It's [INAUDIBLE]. 01:13:45.447 --> 01:13:46.780 MAGDALENA TODA: Um, it's a what? 01:13:46.780 --> 01:13:48.090 STUDENT: b/a? 01:13:48.090 --> 01:13:49.590 MAGDALENA TODA: I'm not [INAUDIBLE]. 01:13:49.590 --> 01:13:51.080 I don't think so. 01:13:51.080 --> 01:13:52.730 So what should I do? 01:13:52.730 --> 01:13:58.070 I should say, instead of bx-- that drives me nuts. 01:13:58.070 --> 01:14:00.390 This goes on my nerves-- bx. 01:14:00.390 --> 01:14:03.550 Like, maybe I go on your nerves. bx is ax, right? 01:14:03.550 --> 01:14:06.745 If it were ax, I would be more constructive, 01:14:06.745 --> 01:14:09.270 and I knew what to do. 01:14:09.270 --> 01:14:13.190 I say replace bx with ax, compensate for it, 01:14:13.190 --> 01:14:15.120 and divide by bx. 01:14:15.120 --> 01:14:17.880 And I was trying to explain that to my son, 01:14:17.880 --> 01:14:23.550 that if you have a fraction a/b, and then you write a/n 01:14:23.550 --> 01:14:26.695 times n/b, it's the same thing. 01:14:26.695 --> 01:14:28.555 Gosh, I had the problem with him. 01:14:28.555 --> 01:14:33.310 And then I realized that he didn't do simplifications 01:14:33.310 --> 01:14:34.870 in school. 01:14:34.870 --> 01:14:41.180 So it took a little more hours to explain these things. 01:14:41.180 --> 01:14:42.590 This is fourth grade. 01:14:42.590 --> 01:14:45.150 I think I remember doing that in fourth grade. 01:14:45.150 --> 01:14:47.430 Third grade, actually. 01:14:47.430 --> 01:14:50.360 So these two guys disappear. 01:14:50.360 --> 01:14:53.790 I haven't changed my problem at all. 01:14:53.790 --> 01:14:57.920 But I've changed the status, the shape of my problem 01:14:57.920 --> 01:15:01.340 to something I can mold, because this goes to somebody, 01:15:01.340 --> 01:15:02.680 and this goes to somebody else. 01:15:02.680 --> 01:15:05.170 Who is this fellow? 01:15:05.170 --> 01:15:07.620 It's a limit that's a constant-- a/b. 01:15:07.620 --> 01:15:09.260 Who is this fellow? 01:15:09.260 --> 01:15:09.760 STUDENT: 1. 01:15:09.760 --> 01:15:10.840 MAGDALENA TODA: 1. 01:15:10.840 --> 01:15:15.660 Because tangent of x/x as x goes to 0 goes to 1 exactly 01:15:15.660 --> 01:15:16.160 like that. 01:15:16.160 --> 01:15:22.140 So limit of sine x over cosine x, that's tangent, right? 01:15:22.140 --> 01:15:23.290 Over x. 01:15:23.290 --> 01:15:25.020 You do it exactly the same. 01:15:25.020 --> 01:15:32.410 It's limit of sine x/x times 1 over cosine x. 01:15:32.410 --> 01:15:34.396 That's how we did it in high school. 01:15:34.396 --> 01:15:35.025 This goes to 1. 01:15:35.025 --> 01:15:36.640 This goes to 1. 01:15:36.640 --> 01:15:37.390 So it's 1. 01:15:37.390 --> 01:15:39.190 So thank you, this is 1. 01:15:39.190 --> 01:15:43.286 I know I took a little more time to explain than I wanted to. 01:15:43.286 --> 01:15:46.130 But now you are grown up. 01:15:46.130 --> 01:15:49.260 In two minutes, you are going to be finishing 01:15:49.260 --> 01:15:50.766 this section, more or less. 01:15:50.766 --> 01:15:54.640 What if I put a function of two variables, 01:15:54.640 --> 01:15:57.790 and I ask you what the limit will be, 01:15:57.790 --> 01:16:01.290 if it's the same type of function. 01:16:01.290 --> 01:16:03.490 So you say, oh, Magdalena, what you doing to us? 01:16:03.490 --> 01:16:05.240 OK, we'll see it's fun. 01:16:05.240 --> 01:16:06.010 This one's fun. 01:16:06.010 --> 01:16:07.850 It's not like the one before. 01:16:07.850 --> 01:16:11.080 This one is pretty beautiful. 01:16:11.080 --> 01:16:12.770 It's nice to you. 01:16:12.770 --> 01:16:14.950 It exists. 01:16:14.950 --> 01:16:16.590 xy goes to 0, 0. 01:16:16.590 --> 01:16:19.745 So you have to imagine some preferable function 01:16:19.745 --> 01:16:22.070 in abstract thinking. 01:16:22.070 --> 01:16:24.720 And you want it in a little disk here. 01:16:24.720 --> 01:16:31.580 And xy, these are all points xy close enough to 0, 0, 01:16:31.580 --> 01:16:34.030 in the neighborhood of 0, 0. 01:16:34.030 --> 01:16:34.813 OK. 01:16:34.813 --> 01:16:37.480 What's going to happen as you get closer and closer 01:16:37.480 --> 01:16:40.380 and closer and closer with tinier and tinier and tinier 01:16:40.380 --> 01:16:43.880 disks around 0, 0? 01:16:43.880 --> 01:16:47.590 You're going to shrink so much. 01:16:47.590 --> 01:16:49.290 What do you think this will going to be, 01:16:49.290 --> 01:16:50.750 and how do I prove it? 01:16:50.750 --> 01:16:52.130 STUDENT: [INAUDIBLE]. 01:16:52.130 --> 01:16:53.966 MAGDALENA TODA: Who said it? 01:16:53.966 --> 01:16:55.950 You, sir? [INAUDIBLE] going to go to 1. 01:16:55.950 --> 01:16:57.770 And he's right. 01:16:57.770 --> 01:17:00.940 He has the intuition. 01:17:00.940 --> 01:17:03.204 A mathematician will tell you, prove it. 01:17:03.204 --> 01:17:04.620 STUDENT: Um, well, let's see here. 01:17:04.620 --> 01:17:06.432 MAGDALENA TODA: Can you prove? 01:17:06.432 --> 01:17:09.897 STUDENT: You could use the right triangle proof, 01:17:09.897 --> 01:17:11.980 but that would probably take way more [INAUDIBLE]. 01:17:11.980 --> 01:17:12.940 MAGDALENA TODA: x and y are independent. 01:17:12.940 --> 01:17:13.680 That's the problem. 01:17:13.680 --> 01:17:15.721 They are married, but they are still independent. 01:17:15.721 --> 01:17:17.220 It's a couple. 01:17:17.220 --> 01:17:20.920 However, we can use polar coordinates. 01:17:20.920 --> 01:17:22.775 Why is polar coordinates? 01:17:22.775 --> 01:17:28.950 Well, in general, if we are in xy, it's a pair. 01:17:28.950 --> 01:17:30.585 This is r, right? 01:17:30.585 --> 01:17:33.590 So rx is r cosine theta. 01:17:33.590 --> 01:17:35.443 y is r sine theta. 01:17:35.443 --> 01:17:37.442 And I can get closer and closer to the original. 01:17:37.442 --> 01:17:38.680 I don't care. 01:17:38.680 --> 01:17:41.310 What happens about x squared plus y squared, 01:17:41.310 --> 01:17:43.040 this is r squared. 01:17:43.040 --> 01:17:44.275 And r is a real number. 01:17:44.275 --> 01:17:47.440 And as you walk closer and closer to the original 01:17:47.440 --> 01:17:52.800 without touching it, that r goes to 0. 01:17:52.800 --> 01:17:53.850 It shrinks to 0. 01:17:53.850 --> 01:17:58.260 So that r squared goes to 0 but never touches 0. 01:17:58.260 --> 01:18:04.310 So this becomes limit as r goes to 0, the radius of that disk 01:18:04.310 --> 01:18:06.291 goes to 0. 01:18:06.291 --> 01:18:10.540 Sine of r squared over r squared. 01:18:10.540 --> 01:18:13.870 But r squared could be replaced by the real function, t, 01:18:13.870 --> 01:18:17.425 by the real parameter, lambda, by whatever you want. 01:18:17.425 --> 01:18:19.440 So then it's 1. 01:18:19.440 --> 01:18:23.390 And then Alexander was right. 01:18:23.390 --> 01:18:26.207 He based it on, like, observation, intuition, 01:18:26.207 --> 01:18:27.040 everything you want. 01:18:27.040 --> 01:18:28.830 It was not a proof. 01:18:28.830 --> 01:18:32.470 On a multiple-choice exam, he would be a lucky guy. 01:18:32.470 --> 01:18:34.301 I don't want you to prove it. 01:18:34.301 --> 01:18:36.950 But if I want you to prove it, you have to say, 01:18:36.950 --> 01:18:39.530 Magdalena, I know polar coordinates, 01:18:39.530 --> 01:18:41.930 and so I can do it. 01:18:41.930 --> 01:18:45.460 And one last question for today. 01:18:45.460 --> 01:18:49.570 Guys, I'm asking you, limit xy goes to 0, 0. 01:18:49.570 --> 01:18:53.600 You will see some of these in your WeBWorK for Chapter 11 01:18:53.600 --> 01:18:56.890 that's waiting for you, homework 3. 01:18:56.890 --> 01:19:03.200 Tangent of 2 x squared plus y squared over 3 01:19:03.200 --> 01:19:06.446 x squared plus y squared. 01:19:06.446 --> 01:19:09.320 What is that? 01:19:09.320 --> 01:19:10.278 2/3. 01:19:10.278 --> 01:19:11.021 STUDENT: 2/3. 01:19:11.021 --> 01:19:12.520 MAGDALENA TODA: Am I asking you why? 01:19:12.520 --> 01:19:13.540 No, enough. 01:19:13.540 --> 01:19:14.440 OK. 01:19:14.440 --> 01:19:17.030 [INAUDIBLE] I gave you everything 01:19:17.030 --> 01:19:20.920 you need to show that. 01:19:20.920 --> 01:19:23.616 x squared plus y squared, again, is Mr. r squared. 01:19:23.616 --> 01:19:24.592 It's OK. 01:19:24.592 --> 01:19:29.472 I taught you that. a/b. a is 2, b is 3. 01:19:29.472 --> 01:19:30.448 Is it hard? 01:19:30.448 --> 01:19:31.912 It is not easy, for sure. 01:19:31.912 --> 01:19:35.328 Calc 3 is really difficult compared to other topics 01:19:35.328 --> 01:19:37.780 you are probably taking. 01:19:37.780 --> 01:19:40.960 But I hope that I can convince you 01:19:40.960 --> 01:19:45.440 that math, although difficult, [INAUDIBLE] Calc 3, 01:19:45.440 --> 01:19:48.230 is also fun. 01:19:48.230 --> 01:19:49.930 OK? 01:19:49.930 --> 01:19:50.760 All right. 01:19:50.760 --> 01:19:55.083 So I need attendance and I need the extra credit. 01:19:55.083 --> 01:19:56.208 STUDENT: Yeah, [INAUDIBLE]. 01:19:56.208 --> 01:19:59.080 01:19:59.080 --> 01:20:01.920 MAGDALENA TODA: Before you go, you need to sign. 01:20:01.920 --> 01:20:04.439