1 00:00:00,000 --> 00:00:01,467 MAGDALENA TODA: We have any people 2 00:00:01,467 --> 00:00:04,890 who finished the extra credit and are 3 00:00:04,890 --> 00:00:08,313 willing to give it to me today? 4 00:00:08,313 --> 00:00:10,758 I mean, you don't have to. 5 00:00:10,758 --> 00:00:12,720 That's why it's called extra credit. 6 00:00:12,720 --> 00:00:18,378 But I think it's good for extra practice 7 00:00:18,378 --> 00:00:21,354 and for the extra points. 8 00:00:21,354 --> 00:00:25,322 So hold on to it if you cannot give it to me right now. 9 00:00:25,322 --> 00:00:29,290 And I'll collect it at the end of the class. 10 00:00:29,290 --> 00:00:30,282 Today's a big day. 11 00:00:30,282 --> 00:00:33,258 We are starting a new chapter, Chapter 11. 12 00:00:33,258 --> 00:00:44,170 13 00:00:44,170 --> 00:00:47,394 So practically, we are going to discuss all 14 00:00:47,394 --> 00:00:50,618 through this chapter functions of several variables. 15 00:00:50,618 --> 00:01:02,026 16 00:01:02,026 --> 00:01:05,089 And you are going to ask me, wait a minute, 17 00:01:05,089 --> 00:01:12,330 why do we need functions in more than one variable? 18 00:01:12,330 --> 00:01:15,770 Well, we are all functions of many variables. 19 00:01:15,770 --> 00:01:18,765 I was freezing outside, and I was thinking, 20 00:01:18,765 --> 00:01:20,620 I'm a function of everything I eat. 21 00:01:20,620 --> 00:01:24,070 I'm a function of the temperature outside. 22 00:01:24,070 --> 00:01:26,500 Almost everything in our body is a function 23 00:01:26,500 --> 00:01:29,200 of hundreds of factors, actually, thousands. 24 00:01:29,200 --> 00:01:34,350 But we don't have the time and the precise information 25 00:01:34,350 --> 00:01:37,420 to analyze all the parameters that 26 00:01:37,420 --> 00:01:42,380 affect our physical condition every day. 27 00:01:42,380 --> 00:01:44,990 We are getting there. 28 00:01:44,990 --> 00:01:48,360 I'm going to give you just the simple case. 29 00:01:48,360 --> 00:01:53,312 So instead of y equals f of x type of function, one variable, 30 00:01:53,312 --> 00:01:57,420 we are going to look at functions of the types z 31 00:01:57,420 --> 00:01:58,583 equals f of xy. 32 00:01:58,583 --> 00:02:01,760 33 00:02:01,760 --> 00:02:03,750 Can I have many more? 34 00:02:03,750 --> 00:02:05,560 Absolutely I can. 35 00:02:05,560 --> 00:02:08,880 And that's kind of the idea, that I 36 00:02:08,880 --> 00:02:14,640 can have a function in an-- let me 37 00:02:14,640 --> 00:02:18,580 count-- n plus 1 dimensional space 38 00:02:18,580 --> 00:02:30,350 as being of the type xn plus 1 equals f of x1, x2, x3, x4. 39 00:02:30,350 --> 00:02:32,820 Somebody stop me. xn. 40 00:02:32,820 --> 00:02:33,320 Right. 41 00:02:33,320 --> 00:02:37,885 I have many variables. 42 00:02:37,885 --> 00:02:41,790 And that is a problem that affects everything. 43 00:02:41,790 --> 00:02:44,600 Our physical world is affected by many parameters. 44 00:02:44,600 --> 00:02:47,630 45 00:02:47,630 --> 00:02:49,850 In engineering problems, you've already 46 00:02:49,850 --> 00:02:51,400 seen some of these parameters. 47 00:02:51,400 --> 00:02:55,040 Can you give me some examples of parameters you've 48 00:02:55,040 --> 00:02:57,190 seen in engineering classes? 49 00:02:57,190 --> 00:03:00,740 x1, x2, x3 could be the Euclidean coordinates, right, 50 00:03:00,740 --> 00:03:03,630 for the three [? space. ?] But besides those, there was an x4. 51 00:03:03,630 --> 00:03:05,600 It could be? 52 00:03:05,600 --> 00:03:06,100 Time. 53 00:03:06,100 --> 00:03:07,133 Excellent, [INAUDIBLE]. 54 00:03:07,133 --> 00:03:08,402 More than that. 55 00:03:08,402 --> 00:03:09,690 I want more. 56 00:03:09,690 --> 00:03:11,234 I want x5. 57 00:03:11,234 --> 00:03:12,650 Who can think of another parameter 58 00:03:12,650 --> 00:03:17,083 that affects physical processes or chemical reactions? 59 00:03:17,083 --> 00:03:17,583 Yes, sir? 60 00:03:17,583 --> 00:03:17,936 STUDENT: Temperature. 61 00:03:17,936 --> 00:03:19,102 MAGDALENA TODA: Temperature. 62 00:03:19,102 --> 00:03:19,910 Excellent. 63 00:03:19,910 --> 00:03:21,310 Another very good idea. 64 00:03:21,310 --> 00:03:23,100 How about x6? 65 00:03:23,100 --> 00:03:24,590 I'm running out of imagination. 66 00:03:24,590 --> 00:03:28,910 But you have a lot more information than me. 67 00:03:28,910 --> 00:03:29,892 Pressure. 68 00:03:29,892 --> 00:03:35,620 Maybe I'm studying a process of somewhere up in the atmosphere. 69 00:03:35,620 --> 00:03:37,677 Maybe I'm in an airplane, and then it 70 00:03:37,677 --> 00:03:39,260 becomes a little bit more complicated, 71 00:03:39,260 --> 00:03:41,890 because I hate the way cabins are pressurized. 72 00:03:41,890 --> 00:03:45,060 I can feel very uneasy. 73 00:03:45,060 --> 00:03:46,990 My ears pop and so on. 74 00:03:46,990 --> 00:03:48,850 We can be in the bottom of the ocean. 75 00:03:48,850 --> 00:03:52,900 There are very many physical parameters 76 00:03:52,900 --> 00:03:56,252 that affect physical processes, chemical processes, 77 00:03:56,252 --> 00:03:57,960 biological processes. 78 00:03:57,960 --> 00:04:00,410 I don't know if this is fortunate or unfortunate, 79 00:04:00,410 --> 00:04:03,120 but I think that was the key to the existence 80 00:04:03,120 --> 00:04:07,400 of the universe in the first place-- all these parameters. 81 00:04:07,400 --> 00:04:08,030 OK. 82 00:04:08,030 --> 00:04:11,700 Let me give you a simple example of a function that 83 00:04:11,700 --> 00:04:13,395 looks like a graph. 84 00:04:13,395 --> 00:04:14,460 This is a graph. 85 00:04:14,460 --> 00:04:17,180 86 00:04:17,180 --> 00:04:19,470 And you say, wait a minute, wait a minute. 87 00:04:19,470 --> 00:04:22,470 Can I have functions of several variables that cannot be 88 00:04:22,470 --> 00:04:26,260 represented as graphs? 89 00:04:26,260 --> 00:04:28,100 Yeah. 90 00:04:28,100 --> 00:04:29,020 Absolutely. 91 00:04:29,020 --> 00:04:31,300 We will talk about that a little bit later. 92 00:04:31,300 --> 00:04:35,200 So if I were to give you an example that you've 93 00:04:35,200 --> 00:04:42,270 seen before, and I would say, give me a good approximation 94 00:04:42,270 --> 00:04:47,000 to a valley that is actually a quadric that we love and we 95 00:04:47,000 --> 00:04:51,750 studied before for the first time. 96 00:04:51,750 --> 00:04:59,210 That quadric is a beautiful object, a valley. 97 00:04:59,210 --> 00:05:02,840 Any imagination, recognition, recollection? 98 00:05:02,840 --> 00:05:04,930 I know I scared you enough for you 99 00:05:04,930 --> 00:05:08,130 to know the equations of those quadrics since some of you 100 00:05:08,130 --> 00:05:10,000 told me we watched all the videos, 101 00:05:10,000 --> 00:05:13,440 we read all the stinking book like never before. 102 00:05:13,440 --> 00:05:14,770 That was kind of the idea. 103 00:05:14,770 --> 00:05:16,970 I didn't want to scare you away. 104 00:05:16,970 --> 00:05:19,330 I wanted to scare you enough to read the book 105 00:05:19,330 --> 00:05:20,620 and watch the videos. 106 00:05:20,620 --> 00:05:26,240 And I'm talking about a valley that you've seen before. 107 00:05:26,240 --> 00:05:31,070 Many of you told me you like the University of Minnesota 108 00:05:31,070 --> 00:05:34,900 website that has the quadric gallery of quadrics. 109 00:05:34,900 --> 00:05:40,140 110 00:05:40,140 --> 00:05:43,550 So you've met this guy before. 111 00:05:43,550 --> 00:05:46,430 They show the general equation. 112 00:05:46,430 --> 00:05:50,010 But I said I like the circular paraboloid. 113 00:05:50,010 --> 00:05:53,810 So they talk about elliptic paraboloid. 114 00:05:53,810 --> 00:05:56,666 Which one do you think I prefer? 115 00:05:56,666 --> 00:05:59,056 The circular paraboloid. 116 00:05:59,056 --> 00:06:02,880 Give me an example of a circular paraboloid. 117 00:06:02,880 --> 00:06:05,270 STUDENT: A flashlight? 118 00:06:05,270 --> 00:06:06,720 Inside. 119 00:06:06,720 --> 00:06:09,295 MAGDALENA TODA: The expression, the mathematical equation. 120 00:06:09,295 --> 00:06:10,086 STUDENT: Oh, sorry. 121 00:06:10,086 --> 00:06:11,737 So it would be x squred plus y squared. 122 00:06:11,737 --> 00:06:12,820 MAGDALENA TODA: Very good. 123 00:06:12,820 --> 00:06:14,470 That's exactly what I had in mind. 124 00:06:14,470 --> 00:06:18,110 Of course, it could be over something, over r. 125 00:06:18,110 --> 00:06:18,610 All right. 126 00:06:18,610 --> 00:06:19,930 That's my favorite. 127 00:06:19,930 --> 00:06:23,590 Now, if I put the flashlight in here just like one of you 128 00:06:23,590 --> 00:06:29,585 said, or the sign on top of the z-axis. 129 00:06:29,585 --> 00:06:36,780 Then I'm going to look at the various-- we discussed 130 00:06:36,780 --> 00:06:38,110 that a little bit before. 131 00:06:38,110 --> 00:06:42,900 So various horizontal planes, they're going to cut. 132 00:06:42,900 --> 00:06:48,450 They're going to cut the surface in different circles, 133 00:06:48,450 --> 00:06:50,390 upon different circles. 134 00:06:50,390 --> 00:06:52,780 We love them, and we use them. 135 00:06:52,780 --> 00:06:54,940 And what did we do with them last time? 136 00:06:54,940 --> 00:06:59,830 We projected them on the floor. 137 00:06:59,830 --> 00:07:03,390 And by floor, I mean the what? 138 00:07:03,390 --> 00:07:10,080 By floor, I mean the xy plane. 139 00:07:10,080 --> 00:07:12,420 Plus this xy plane. 140 00:07:12,420 --> 00:07:14,450 I label it like you like it. 141 00:07:14,450 --> 00:07:17,290 You said you like it when I label it, 142 00:07:17,290 --> 00:07:19,990 so you have the imagination of a table. 143 00:07:19,990 --> 00:07:23,640 This is x and y and z. 144 00:07:23,640 --> 00:07:31,040 And so I gave you an example of a graph cut in with z equals 145 00:07:31,040 --> 00:07:33,400 constant positive or negative? 146 00:07:33,400 --> 00:07:36,080 Well, it better be positive, because for negative, I 147 00:07:36,080 --> 00:07:37,770 have no solutions. 148 00:07:37,770 --> 00:07:39,020 Positive or zero. 149 00:07:39,020 --> 00:07:42,360 Well, for zero I have a degenerate conic. 150 00:07:42,360 --> 00:07:46,180 A degenerate conic could be a point, 151 00:07:46,180 --> 00:07:48,820 or it could be a bunch of lines. 152 00:07:48,820 --> 00:07:52,432 In this case, all those circles-- doo-doo-doo-doo-doo-- 153 00:07:52,432 --> 00:07:55,750 a family of one parameter, family of circles. 154 00:07:55,750 --> 00:07:58,170 Like the ones that is-- a dolphin 155 00:07:58,170 --> 00:08:01,730 is now doing that in San Antonio, 156 00:08:01,730 --> 00:08:04,490 San Diego-- to take those old circles 157 00:08:04,490 --> 00:08:07,765 from the bottom of the sea, and bring them different sizes, 158 00:08:07,765 --> 00:08:09,030 and put them together. 159 00:08:09,030 --> 00:08:10,400 So they are very smart. 160 00:08:10,400 --> 00:08:11,820 I love dolphins. 161 00:08:11,820 --> 00:08:15,540 So we'll see 0 [INAUDIBLE] get a point. 162 00:08:15,540 --> 00:08:17,290 That's still a conic. 163 00:08:17,290 --> 00:08:18,962 It's a degenerate circle. 164 00:08:18,962 --> 00:08:21,582 Do you realize that's a limit case? 165 00:08:21,582 --> 00:08:22,536 It's really beautiful. 166 00:08:22,536 --> 00:08:23,410 You know what I mean? 167 00:08:23,410 --> 00:08:25,284 Circle on top of a circle on top of a circle, 168 00:08:25,284 --> 00:08:26,175 smaller and smaller. 169 00:08:26,175 --> 00:08:27,570 All right. 170 00:08:27,570 --> 00:08:30,470 So good. 171 00:08:30,470 --> 00:08:33,270 If I create shadows-- because that's 172 00:08:33,270 --> 00:08:35,350 why you guys wanted the source of light 173 00:08:35,350 --> 00:08:39,360 on top-- of the projections of these circles, 174 00:08:39,360 --> 00:08:42,630 I'm going to have them at the same color. 175 00:08:42,630 --> 00:08:47,991 But dotted lines because I think the book doesn't show them 176 00:08:47,991 --> 00:08:48,490 dotted. 177 00:08:48,490 --> 00:08:51,210 But on my way here, I was thinking, 178 00:08:51,210 --> 00:08:55,040 I think it's more beautiful if I draw them dotted. 179 00:08:55,040 --> 00:08:56,635 And how big is this circle? 180 00:08:56,635 --> 00:08:57,440 Well, god knows. 181 00:08:57,440 --> 00:09:01,990 I'm going to make a purple circle that is, of course, 182 00:09:01,990 --> 00:09:05,940 equal in size, equal in radius with the original purple 183 00:09:05,940 --> 00:09:07,080 circle. 184 00:09:07,080 --> 00:09:10,290 So the dotted purple circle, that's on the ground-- 185 00:09:10,290 --> 00:09:13,520 is just the projection of the continuous purple circle. 186 00:09:13,520 --> 00:09:16,450 It's identical in radius. 187 00:09:16,450 --> 00:09:26,810 So for the family of circles on the surface, 188 00:09:26,810 --> 00:09:37,830 I have a family of projections on the ground in the xy plane. 189 00:09:37,830 --> 00:09:42,420 And such a family of projections represents 190 00:09:42,420 --> 00:09:44,910 a bunch of level curves. 191 00:09:44,910 --> 00:09:47,112 We call this family of level curves. 192 00:09:47,112 --> 00:09:54,402 193 00:09:54,402 --> 00:09:55,860 OK? 194 00:09:55,860 --> 00:09:56,832 All right. 195 00:09:56,832 --> 00:09:59,070 So if you think about it, what are level curves? 196 00:09:59,070 --> 00:10:02,360 You view them as being in plane. 197 00:10:02,360 --> 00:10:02,910 Oh, my god. 198 00:10:02,910 --> 00:10:07,600 So I should view them as a bunch of points, a set of points. 199 00:10:07,600 --> 00:10:10,665 If I make it like that, that means 200 00:10:10,665 --> 00:10:14,100 I view this as an element of what? 201 00:10:14,100 --> 00:10:18,820 Element of the xy plane, right, with the property 202 00:10:18,820 --> 00:10:22,920 that f of x and y is a constant. 203 00:10:22,920 --> 00:10:25,848 204 00:10:25,848 --> 00:10:28,356 OK? 205 00:10:28,356 --> 00:10:30,690 In my case, I have a [INAUDIBLE] constant. 206 00:10:30,690 --> 00:10:34,270 In general, I have an arbitrary real constant. 207 00:10:34,270 --> 00:10:40,020 That's a level curve for level C, for the level 208 00:10:40,020 --> 00:10:46,400 C called the level, or altitude would be the same thing. 209 00:10:46,400 --> 00:10:50,110 So have you seen these guys in geography? 210 00:10:50,110 --> 00:10:54,120 What in the world are these level curves in geography? 211 00:10:54,120 --> 00:10:56,520 STUDENT: [INAUDIBLE] show the slope 212 00:10:56,520 --> 00:10:59,400 of a-- the steepness of a hill. 213 00:10:59,400 --> 00:11:02,200 MAGDALENA TODA: You've seen topographical maps. 214 00:11:02,200 --> 00:11:06,140 And I'm going to try and draw one of them. 215 00:11:06,140 --> 00:11:08,310 I don't know, guys, how-- excuse me. 216 00:11:08,310 --> 00:11:10,275 I'm not very good today at drawing. 217 00:11:10,275 --> 00:11:13,180 But I'll do my best. 218 00:11:13,180 --> 00:11:19,860 It could be a temperature map or pressure map. 219 00:11:19,860 --> 00:11:22,610 [INAUDIBLE] or whatever. 220 00:11:22,610 --> 00:11:29,090 Now I'll say, this is going to go-- well, 221 00:11:29,090 --> 00:11:31,610 I cannot draw the infinite family. 222 00:11:31,610 --> 00:11:34,416 I have a one-parameter family. 223 00:11:34,416 --> 00:11:42,840 And then I'll-- I'm dreaming of the sea, summer break already. 224 00:11:42,840 --> 00:11:44,596 You see what I'm doing. 225 00:11:44,596 --> 00:11:47,580 Do you know what I'm doing? 226 00:11:47,580 --> 00:11:51,610 That means I'm dreaming of the different depths of the sea. 227 00:11:51,610 --> 00:11:55,896 So for every such broad line, I have the same depth. 228 00:11:55,896 --> 00:11:59,380 The same altitude for every continuous rule. 229 00:11:59,380 --> 00:12:02,530 The same depth for every-- so OK. 230 00:12:02,530 --> 00:12:05,110 I'm not going to swim too far, because that's 231 00:12:05,110 --> 00:12:06,270 where the sharks are. 232 00:12:06,270 --> 00:12:10,096 And I cannot draw the sharks, but I ask you to imagine them. 233 00:12:10,096 --> 00:12:12,712 It's fundamental in a calculus class. 234 00:12:12,712 --> 00:12:18,490 So somewhere here I'm going to have-- 235 00:12:18,490 --> 00:12:21,326 what's the deepest-- guys, what's 236 00:12:21,326 --> 00:12:23,066 the deepest point in that? 237 00:12:23,066 --> 00:12:24,173 [? STUDENT: 11,300. ?] 238 00:12:24,173 --> 00:12:25,881 MAGDALENA TODA: And do you know the name? 239 00:12:25,881 --> 00:12:26,381 I know the-- 240 00:12:26,381 --> 00:12:27,381 STUDENT: Mariana Trench. 241 00:12:27,381 --> 00:12:28,695 MAGDALENA TODA: Mariana Trench. 242 00:12:28,695 --> 00:12:30,757 STUDENT: Trench. 243 00:12:30,757 --> 00:12:31,840 MAGDALENA TODA: All right. 244 00:12:31,840 --> 00:12:34,490 So these topographical are full of curves. 245 00:12:34,490 --> 00:12:37,300 These are level curves. 246 00:12:37,300 --> 00:12:39,580 So you didn't know, but there is a lot 247 00:12:39,580 --> 00:12:42,060 of mathematics in geography. 248 00:12:42,060 --> 00:12:43,860 And there is a lot of mathematics 249 00:12:43,860 --> 00:12:45,600 in-- oh, you knew it. 250 00:12:45,600 --> 00:12:47,480 When you watch the weather report, 251 00:12:47,480 --> 00:12:49,740 that's all mathematics, right? 252 00:12:49,740 --> 00:12:53,030 It shows you the distribution of temperatures everyday. 253 00:12:53,030 --> 00:12:55,490 That is what we can [INAUDIBLE] also 254 00:12:55,490 --> 00:13:00,550 care about other functions of several parameters, right? 255 00:13:00,550 --> 00:13:04,430 And those functions could be pressure, wind, whatever. 256 00:13:04,430 --> 00:13:05,650 OK. 257 00:13:05,650 --> 00:13:06,773 Speed of the wind. 258 00:13:06,773 --> 00:13:09,040 Something like that. 259 00:13:09,040 --> 00:13:12,350 I did not dare to look at the prediction 260 00:13:12,350 --> 00:13:14,670 of the weather for this place. 261 00:13:14,670 --> 00:13:16,670 This place used to be a beautiful place. 262 00:13:16,670 --> 00:13:22,530 300 days of the year of sunshine. 263 00:13:22,530 --> 00:13:23,755 Not anymore. 264 00:13:23,755 --> 00:13:26,550 So there is something fishy in Denmark 265 00:13:26,550 --> 00:13:29,010 and also something fishy in [INAUDIBLE]. 266 00:13:29,010 --> 00:13:30,250 The world is changing. 267 00:13:30,250 --> 00:13:34,720 So if you don't believe in global warming, think again, 268 00:13:34,720 --> 00:13:38,180 and global cooling, think again. 269 00:13:38,180 --> 00:13:39,130 All right. 270 00:13:39,130 --> 00:13:42,960 So unfortunately, I am afraid still 271 00:13:42,960 --> 00:13:45,622 to look at the temperatures for the next few days. 272 00:13:45,622 --> 00:13:46,122 But-- 273 00:13:46,122 --> 00:13:48,602 STUDENT: It's going to be 80 degrees on Tuesday. 274 00:13:48,602 --> 00:13:49,147 MAGDALENA TODA: Really? [? 275 00:13:49,147 --> 00:13:50,855 Well, see, I should have looked at it. ?] 276 00:13:50,855 --> 00:13:51,580 [LAUGHTER] 277 00:13:51,580 --> 00:13:54,710 I should gather the courage, because I 278 00:13:54,710 --> 00:13:57,140 knew-- when I was interviewed here 279 00:13:57,140 --> 00:14:00,480 for assistant professor, gosh, I was young. 280 00:14:00,480 --> 00:14:02,230 2001. 281 00:14:02,230 --> 00:14:03,830 And my interview was in mid-February. 282 00:14:03,830 --> 00:14:08,180 And birds were chirping, it was blue skies, beautiful flowers 283 00:14:08,180 --> 00:14:09,700 everywhere on campus. 284 00:14:09,700 --> 00:14:11,380 And I love the campus. 285 00:14:11,380 --> 00:14:12,910 OK. 286 00:14:12,910 --> 00:14:18,520 Give me an example of a surface that cannot be represented 287 00:14:18,520 --> 00:14:23,810 as a graph in its entirety as a whole graph. 288 00:14:23,810 --> 00:14:27,070 You gave me that before, and I was so proud of you. 289 00:14:27,070 --> 00:14:27,940 It was a-- 290 00:14:27,940 --> 00:14:30,840 291 00:14:30,840 --> 00:14:32,316 [LAUGHS] 292 00:14:32,316 --> 00:14:33,300 293 00:14:33,300 --> 00:14:35,730 What kind of surface am I trying to mimic? 294 00:14:35,730 --> 00:14:36,694 STUDENT: A saddle. 295 00:14:36,694 --> 00:14:39,590 296 00:14:39,590 --> 00:14:42,120 MAGDALENA TODA: That can be actually a graph. 297 00:14:42,120 --> 00:14:44,630 That's a good example of a graph. 298 00:14:44,630 --> 00:14:45,920 A saddle. 299 00:14:45,920 --> 00:14:49,030 But give me an example of a non-graph that 300 00:14:49,030 --> 00:14:51,720 is given as an implicit form. 301 00:14:51,720 --> 00:14:58,160 So graph or explicit is the same thing. 302 00:14:58,160 --> 00:15:00,880 z equals f of xy. 303 00:15:00,880 --> 00:15:02,590 Give me a non-graph. 304 00:15:02,590 --> 00:15:05,140 One of you said it. 305 00:15:05,140 --> 00:15:08,602 x squared plus y squared plus z squared equals 1. 306 00:15:08,602 --> 00:15:10,820 Why is this not a graph? 307 00:15:10,820 --> 00:15:12,940 Not a graph. 308 00:15:12,940 --> 00:15:14,163 Why is this not a graph? 309 00:15:14,163 --> 00:15:18,993 310 00:15:18,993 --> 00:15:22,374 STUDENT: [INAUDIBLE]. 311 00:15:22,374 --> 00:15:26,260 When you move it over to 1, you can't actually-- 312 00:15:26,260 --> 00:15:28,872 MAGDALENA TODA: You cannot but you can cut it. 313 00:15:28,872 --> 00:15:31,460 You can take a sword and-- I'm OK. 314 00:15:31,460 --> 00:15:33,720 I don't want to think about it. 315 00:15:33,720 --> 00:15:37,180 So z is going to be two graphs. 316 00:15:37,180 --> 00:15:41,970 So I can split this surface even in a parametric form 317 00:15:41,970 --> 00:15:44,860 as two different graphs. 318 00:15:44,860 --> 00:15:46,626 Different graphs. 319 00:15:46,626 --> 00:15:51,490 If I cut along-- I have this orange, or sphere, globe. 320 00:15:51,490 --> 00:15:54,070 And I cut it along a great circle. 321 00:15:54,070 --> 00:15:57,580 It doesn't have to be the equator. 322 00:15:57,580 --> 00:15:59,710 But you have to imagine something 323 00:15:59,710 --> 00:16:02,540 like the world and the equator. 324 00:16:02,540 --> 00:16:06,030 This is kind of in the unit sphere. 325 00:16:06,030 --> 00:16:09,110 Today I drank enough coffee to draw better. 326 00:16:09,110 --> 00:16:10,160 Why don't I draw better? 327 00:16:10,160 --> 00:16:12,081 I have no idea. 328 00:16:12,081 --> 00:16:15,910 So that's not bad, though. 329 00:16:15,910 --> 00:16:16,760 OK. 330 00:16:16,760 --> 00:16:18,090 So that's the unit sphere. 331 00:16:18,090 --> 00:16:18,950 What does it mean? 332 00:16:18,950 --> 00:16:21,770 It means it has radius how much? 333 00:16:21,770 --> 00:16:22,270 STUDENT: 1. 334 00:16:22,270 --> 00:16:23,020 MAGDALENA TODA: 1. 335 00:16:23,020 --> 00:16:25,670 Radius 1, and we are happy about it. 336 00:16:25,670 --> 00:16:29,270 And it has two graphs. 337 00:16:29,270 --> 00:16:31,769 It's not one graph, it's two graphs. 338 00:16:31,769 --> 00:16:34,184 So this is called implicit equation. 339 00:16:34,184 --> 00:16:36,715 This is your lab from-- I was chatting 340 00:16:36,715 --> 00:16:38,670 with-- instead of studying last night, 341 00:16:38,670 --> 00:16:40,960 I was chatting with you at midnight. 342 00:16:40,960 --> 00:16:45,080 And one of you said, if I had something I hated in calculus, 343 00:16:45,080 --> 00:16:48,160 it was the implicit differentiation. 344 00:16:48,160 --> 00:16:50,400 And I know this is your weak point. 345 00:16:50,400 --> 00:16:52,570 So we'll do a lot of implicit differentiation, 346 00:16:52,570 --> 00:16:54,770 so you become more comfortable. 347 00:16:54,770 --> 00:16:59,120 Usually we have one exercise in this differentiation at least 348 00:16:59,120 --> 00:17:01,500 on the final. 349 00:17:01,500 --> 00:17:04,579 So this is an implicit equation. 350 00:17:04,579 --> 00:17:09,690 And z is going to be two graphs-- 1 minus x 351 00:17:09,690 --> 00:17:11,125 squared minus y squared. 352 00:17:11,125 --> 00:17:13,858 So I have, like, two charts, two different charts. 353 00:17:13,858 --> 00:17:14,358 OK. 354 00:17:14,358 --> 00:17:17,150 355 00:17:17,150 --> 00:17:20,089 The upper hemisphere-- I'm talking geography, 356 00:17:20,089 --> 00:17:23,190 but that's how we talk in geometry as well. 357 00:17:23,190 --> 00:17:26,055 So geography right now is like geometry. 358 00:17:26,055 --> 00:17:28,150 I have a north pole. 359 00:17:28,150 --> 00:17:31,620 Somebody quickly give me the coordinates of the north pole. 360 00:17:31,620 --> 00:17:32,530 STUDENT: 0, 0, 1. 361 00:17:32,530 --> 00:17:33,530 MAGDALENA TODA: 0, 0, 1. 362 00:17:33,530 --> 00:17:34,780 Thank you, Brian. 363 00:17:34,780 --> 00:17:35,640 0, 0, 1. 364 00:17:35,640 --> 00:17:37,512 How about the south pole? 365 00:17:37,512 --> 00:17:38,720 STUDENT: 0, 0, minus 1. 366 00:17:38,720 --> 00:17:41,290 MAGDALENA TODA: 0, 0, minus 1. 367 00:17:41,290 --> 00:17:45,640 And write yourself a note, because as you know, 368 00:17:45,640 --> 00:17:48,570 I'm very absent-minded and I forget 369 00:17:48,570 --> 00:17:52,490 what I eat for lunch and so on. 370 00:17:52,490 --> 00:17:55,560 Remind me to talk to you sometime 371 00:17:55,560 --> 00:17:58,280 at the end of the chapter about stereographic projection. 372 00:17:58,280 --> 00:18:01,080 It's a very important mathematical notion 373 00:18:01,080 --> 00:18:03,785 that also has to do a little bit with geography. 374 00:18:03,785 --> 00:18:06,060 But it's a one-to-one correspondence 375 00:18:06,060 --> 00:18:08,960 between a certain part of a sphere 376 00:18:08,960 --> 00:18:11,876 and a certain huge part of a plane. 377 00:18:11,876 --> 00:18:14,050 Now, we're not going to talk about that now, 378 00:18:14,050 --> 00:18:16,030 because that's not [INAUDIBLE]. 379 00:18:16,030 --> 00:18:18,360 That's a little bit harder [INAUDIBLE]. 380 00:18:18,360 --> 00:18:20,690 You guys should now see this line, right? 381 00:18:20,690 --> 00:18:24,044 This should be beyond-- in the twilight zone, 382 00:18:24,044 --> 00:18:25,740 behind the sphere. 383 00:18:25,740 --> 00:18:27,280 OK? 384 00:18:27,280 --> 00:18:28,660 So you don't see it. 385 00:18:28,660 --> 00:18:31,410 And who is this? z equals 0. 386 00:18:31,410 --> 00:18:34,750 And so this green fellow should be 387 00:18:34,750 --> 00:18:39,150 the circle x squared plus y squared equals 1 388 00:18:39,150 --> 00:18:40,615 in the xy plane. 389 00:18:40,615 --> 00:18:43,285 390 00:18:43,285 --> 00:18:44,700 Good. 391 00:18:44,700 --> 00:18:47,030 So I have two graphs. 392 00:18:47,030 --> 00:18:54,870 Now, if I were to ask you, what is the domain 393 00:18:54,870 --> 00:18:59,250 and the range of the function? 394 00:18:59,250 --> 00:19:02,630 I'm going to erase the whole thing. 395 00:19:02,630 --> 00:19:10,040 What is the domain and the range of the related function, z, 396 00:19:10,040 --> 00:19:13,904 which gives the upper hemisphere? 397 00:19:13,904 --> 00:19:15,368 Upper hemisphere. 398 00:19:15,368 --> 00:19:17,320 It's a graph. 399 00:19:17,320 --> 00:19:20,810 And square root of 1 minus x squared minus y squared. 400 00:19:20,810 --> 00:19:23,220 You may stare at it until tomorrow. 401 00:19:23,220 --> 00:19:27,940 It's not hard to figure out what I mean by domain 402 00:19:27,940 --> 00:19:30,570 and range of such a function. 403 00:19:30,570 --> 00:19:33,300 You are familiar with domain and range 404 00:19:33,300 --> 00:19:37,330 for a function of one variable. 405 00:19:37,330 --> 00:19:39,890 For most of you, that's a piece of cake. 406 00:19:39,890 --> 00:19:41,800 That was even pre-calc wasn't it? 407 00:19:41,800 --> 00:19:44,341 It was in Calc 1. 408 00:19:44,341 --> 00:19:47,360 So most of you had algebra and pre-calc. 409 00:19:47,360 --> 00:19:51,980 Now, what is the domain of such a function? 410 00:19:51,980 --> 00:19:57,390 Domain of definition has to be a set of points, x and y in plane 411 00:19:57,390 --> 00:20:00,590 for which the function is defined. 412 00:20:00,590 --> 00:20:03,040 If the function is impossible to be defined 413 00:20:03,040 --> 00:20:05,995 for a certain pair, x, y, you kick that couple out 414 00:20:05,995 --> 00:20:07,917 and you say, never come back. 415 00:20:07,917 --> 00:20:09,140 Right? 416 00:20:09,140 --> 00:20:14,665 So what I mean by domain is those couples that we hate. 417 00:20:14,665 --> 00:20:16,430 Who we hate? 418 00:20:16,430 --> 00:20:20,970 The couples x, y for which x squared plus y squared is how? 419 00:20:20,970 --> 00:20:24,130 420 00:20:24,130 --> 00:20:25,422 What existence condition do I-- 421 00:20:25,422 --> 00:20:26,296 STUDENT: [INAUDIBLE]. 422 00:20:26,296 --> 00:20:27,440 MAGDALENA TODA: Yeah. 423 00:20:27,440 --> 00:20:30,230 You see this guy under the square root 424 00:20:30,230 --> 00:20:33,790 has to be positive or 0. 425 00:20:33,790 --> 00:20:35,280 Right? 426 00:20:35,280 --> 00:20:37,890 Otherwise, there is no square root in real numbers. 427 00:20:37,890 --> 00:20:39,920 That's going to be in imaginary numbers, 428 00:20:39,920 --> 00:20:41,630 and you can take a walk, because we 429 00:20:41,630 --> 00:20:45,220 are in real calculus in real time as well. 430 00:20:45,220 --> 00:20:48,670 So x squared plus y squared must be how? 431 00:20:48,670 --> 00:20:50,660 Less than or equal to 1. 432 00:20:50,660 --> 00:20:54,040 We call that a certain name. 433 00:20:54,040 --> 00:20:59,484 This is called a closed unit disk. 434 00:20:59,484 --> 00:21:03,160 Please remember, I'm teaching you a little bit more 435 00:21:03,160 --> 00:21:06,195 than a regular Calc 3 class. 436 00:21:06,195 --> 00:21:09,324 They will never make a distinction. 437 00:21:09,324 --> 00:21:10,365 What's closing with this? 438 00:21:10,365 --> 00:21:11,900 What's opening with this? 439 00:21:11,900 --> 00:21:14,670 Everything will come into place when you 440 00:21:14,670 --> 00:21:19,701 move on to advanced calculus. 441 00:21:19,701 --> 00:21:24,640 If I don't take the boundary-- so everything inside the disk 442 00:21:24,640 --> 00:21:28,460 except for the boundary, I have to put strictly less than 1. 443 00:21:28,460 --> 00:21:30,610 That's called open unit disk. 444 00:21:30,610 --> 00:21:35,080 For advanced calculus, this is [INAUDIBLE]. 445 00:21:35,080 --> 00:21:35,580 All right. 446 00:21:35,580 --> 00:21:37,320 This is just a parentheses. 447 00:21:37,320 --> 00:21:40,388 My domain is the closed one. 448 00:21:40,388 --> 00:21:43,145 What is the range? 449 00:21:43,145 --> 00:21:45,690 The range is going to be-- 450 00:21:45,690 --> 00:21:47,000 STUDENT: [INAUDIBLE]. 451 00:21:47,000 --> 00:21:49,900 MAGDALENA TODA: The altitude starts having values from-- 452 00:21:49,900 --> 00:21:51,092 STUDENT: Negative 1 to 1. 453 00:21:51,092 --> 00:21:51,758 STUDENT: 0 to 1. 454 00:21:51,758 --> 00:21:53,008 MAGDALENA TODA: So I'm 0 to 1. 455 00:21:53,008 --> 00:21:55,090 I'll only talk about the upper hemisphere. 456 00:21:55,090 --> 00:21:58,310 I should even erase, because I don't want it. 457 00:21:58,310 --> 00:21:59,333 So say it again, guys. 458 00:21:59,333 --> 00:22:00,350 STUDENT: 0 to 1. 459 00:22:00,350 --> 00:22:01,100 MAGDALENA TODA: 0. 460 00:22:01,100 --> 00:22:01,865 Open or closed? 461 00:22:01,865 --> 00:22:02,635 STUDENT: Open. 462 00:22:02,635 --> 00:22:03,301 STUDENT: Closed. 463 00:22:03,301 --> 00:22:05,740 STUDENT: Closed, closed. 464 00:22:05,740 --> 00:22:07,630 MAGDALENA TODA: Closed to? 465 00:22:07,630 --> 00:22:08,450 STUDENT: 1 closed. 466 00:22:08,450 --> 00:22:09,650 MAGDALENA TODA: 1 closed. 467 00:22:09,650 --> 00:22:10,150 Yes. 468 00:22:10,150 --> 00:22:13,874 Because that is the north pole. 469 00:22:13,874 --> 00:22:19,160 I've been meaning to give you this example. 470 00:22:19,160 --> 00:22:22,270 And give me the other example for the lower hemisphere. 471 00:22:22,270 --> 00:22:23,400 What's different? 472 00:22:23,400 --> 00:22:24,894 The same domain? 473 00:22:24,894 --> 00:22:25,935 STUDENT: It ranges from-- 474 00:22:25,935 --> 00:22:27,042 STUDENT: Negative 1. 475 00:22:27,042 --> 00:22:28,710 STUDENT: Negative 1 to 0. 476 00:22:28,710 --> 00:22:30,490 MAGDALENA TODA: Closed internal, right? 477 00:22:30,490 --> 00:22:33,860 When we include the endpoints, we call that closed interval. 478 00:22:33,860 --> 00:22:36,200 It has a certain topological sense. 479 00:22:36,200 --> 00:22:39,110 You haven't taken topology, but very soon, 480 00:22:39,110 --> 00:22:43,940 if you are a math major, or you are a double major, or some 481 00:22:43,940 --> 00:22:48,090 of you even-- they want to learn more about topology, 482 00:22:48,090 --> 00:22:51,735 you will learn what an open set is versus a closed set. 483 00:22:51,735 --> 00:22:53,630 Remember we called this closed. 484 00:22:53,630 --> 00:22:56,230 This is open. 485 00:22:56,230 --> 00:22:59,660 And if it's closed here and open there, it's neither. 486 00:22:59,660 --> 00:23:00,160 OK? 487 00:23:00,160 --> 00:23:02,940 Don't say anything about that. 488 00:23:02,940 --> 00:23:03,440 OK. 489 00:23:03,440 --> 00:23:08,110 To be closed, it has to be containing both endpoints. 490 00:23:08,110 --> 00:23:09,300 I'm going to erase this. 491 00:23:09,300 --> 00:23:12,260 492 00:23:12,260 --> 00:23:19,728 And this was, of course, 11.1. 493 00:23:19,728 --> 00:23:22,632 We are in the middle of it. 494 00:23:22,632 --> 00:23:28,440 In 11.1, one of you gave me a beautiful graph to think about. 495 00:23:28,440 --> 00:23:30,525 And I'm going to give you something to do, 496 00:23:30,525 --> 00:23:32,690 because I don't want you to get lazy. 497 00:23:32,690 --> 00:23:36,041 I'm very happy you came up with the saddle. 498 00:23:36,041 --> 00:23:38,908 499 00:23:38,908 --> 00:23:39,408 All right. 500 00:23:39,408 --> 00:23:41,480 We drew such a saddle. 501 00:23:41,480 --> 00:23:44,460 502 00:23:44,460 --> 00:23:46,881 And I did my best, but it's not hard. 503 00:23:46,881 --> 00:23:50,360 It's not easy to draw saddle. 504 00:23:50,360 --> 00:23:54,540 When I am looking at the coordinates, x, y, z, 505 00:23:54,540 --> 00:24:01,575 I have z equals minus y squared will look down. 506 00:24:01,575 --> 00:24:05,020 507 00:24:05,020 --> 00:24:06,810 Maybe I made it too fat. 508 00:24:06,810 --> 00:24:08,980 I'm really sorry. 509 00:24:08,980 --> 00:24:11,430 And down. 510 00:24:11,430 --> 00:24:12,410 This continues. 511 00:24:12,410 --> 00:24:21,230 512 00:24:21,230 --> 00:24:22,210 OK? 513 00:24:22,210 --> 00:24:26,920 And then what other thing did I want to point out? 514 00:24:26,920 --> 00:24:31,100 I want to point out-- do you see this? 515 00:24:31,100 --> 00:24:33,660 This should look a little bit more round. 516 00:24:33,660 --> 00:24:36,556 It doesn't look round enough here. 517 00:24:36,556 --> 00:24:38,290 STUDENT: Your'e drawing a saddle, right? 518 00:24:38,290 --> 00:24:40,289 MAGDALENA TODA: No, I'm drawing just the section 519 00:24:40,289 --> 00:24:41,970 z equals minus y squared. 520 00:24:41,970 --> 00:24:44,330 So I took x to be 0. 521 00:24:44,330 --> 00:24:47,880 And the purple line should be on this wall. 522 00:24:47,880 --> 00:24:50,160 I know you guys have enough imagination. 523 00:24:50,160 --> 00:24:54,045 So this is going to be z equals minus y 524 00:24:54,045 --> 00:24:58,700 squared drawn on yz wall. 525 00:24:58,700 --> 00:25:03,170 526 00:25:03,170 --> 00:25:05,590 I've done this before, but I'm just reviewing. 527 00:25:05,590 --> 00:25:08,360 What if it's y0? 528 00:25:08,360 --> 00:25:10,920 Then I have to draw on that wall. 529 00:25:10,920 --> 00:25:14,326 And I have to draw beautifully, which I am not-- don't always-- 530 00:25:14,326 --> 00:25:15,718 I can't always do. 531 00:25:15,718 --> 00:25:17,110 But I'll try. 532 00:25:17,110 --> 00:25:23,750 I have z equals x squared drawn on that wall. 533 00:25:23,750 --> 00:25:26,580 If I start drawing, I'll get fired. 534 00:25:26,580 --> 00:25:29,020 That I have this branch. 535 00:25:29,020 --> 00:25:32,852 I should go through that corner and go out of the room 536 00:25:32,852 --> 00:25:35,307 and continue with that branch. 537 00:25:35,307 --> 00:25:36,010 All right? 538 00:25:36,010 --> 00:25:39,480 539 00:25:39,480 --> 00:25:43,050 This is curved like that in this direction. 540 00:25:43,050 --> 00:25:45,460 And this other is curved like this. 541 00:25:45,460 --> 00:25:50,300 So if the guy is going to put his feet, 542 00:25:50,300 --> 00:25:54,590 where is the butt of the writer going to sit? 543 00:25:54,590 --> 00:25:57,450 He is here. 544 00:25:57,450 --> 00:25:59,605 And these are his legs. 545 00:25:59,605 --> 00:26:02,490 546 00:26:02,490 --> 00:26:06,144 And these are his cowboy boots. 547 00:26:06,144 --> 00:26:06,644 OK. 548 00:26:06,644 --> 00:26:08,117 Do they look like cowboy boots? 549 00:26:08,117 --> 00:26:10,572 No, I apologize. 550 00:26:10,572 --> 00:26:12,105 STUDENT: Looks like socks. 551 00:26:12,105 --> 00:26:12,980 MAGDALENA TODA: Yeah. 552 00:26:12,980 --> 00:26:15,770 They look more like Christmas socks. 553 00:26:15,770 --> 00:26:17,750 But anyway, it's a poor cowboy. 554 00:26:17,750 --> 00:26:22,880 555 00:26:22,880 --> 00:26:24,710 Let's lower the saddle a little bit. 556 00:26:24,710 --> 00:26:27,200 He cannot see the horse, OK? 557 00:26:27,200 --> 00:26:30,290 So the saddle. 558 00:26:30,290 --> 00:26:35,182 If I cross the saddle, this is the saddle. 559 00:26:35,182 --> 00:26:38,160 And these are his hands. 560 00:26:38,160 --> 00:26:41,115 And he is holding his hat. 561 00:26:41,115 --> 00:26:42,065 This is [INAUDIBLE]. 562 00:26:42,065 --> 00:26:45,865 And with one hand is on the horse. 563 00:26:45,865 --> 00:26:46,815 I don't know. 564 00:26:46,815 --> 00:26:48,320 It's very [INAUDIBLE]. 565 00:26:48,320 --> 00:26:56,760 So what I'm trying to draw looks something like this. 566 00:26:56,760 --> 00:26:57,660 Right? 567 00:26:57,660 --> 00:26:59,100 Eh. 568 00:26:59,100 --> 00:27:01,500 Sorry. 569 00:27:01,500 --> 00:27:02,240 More or less. 570 00:27:02,240 --> 00:27:03,943 It's an abstract picture. 571 00:27:03,943 --> 00:27:05,750 Very abstract picture. 572 00:27:05,750 --> 00:27:14,946 So with this in mind, if I were to look at the level curves, 573 00:27:14,946 --> 00:27:18,802 I'm going to ask you, what are the level curves? 574 00:27:18,802 --> 00:27:22,200 Oh, my god, what are the level curves? 575 00:27:22,200 --> 00:27:25,390 576 00:27:25,390 --> 00:27:27,810 You already have them in your WeBWorK homework. 577 00:27:27,810 --> 00:27:30,300 But for one point extra credit, I 578 00:27:30,300 --> 00:27:34,072 want you to draw them on the floor. 579 00:27:34,072 --> 00:27:37,690 Draw the level curves. 580 00:27:37,690 --> 00:27:39,250 Remember what those were? 581 00:27:39,250 --> 00:27:43,290 They were projections of the curves on the surface 582 00:27:43,290 --> 00:27:46,570 at the intersection with z equals c planes. 583 00:27:46,570 --> 00:27:48,840 You project them on the ground. 584 00:27:48,840 --> 00:27:50,280 What do you think they are? 585 00:27:50,280 --> 00:27:51,270 Think about it. 586 00:27:51,270 --> 00:27:53,280 What are these? 587 00:27:53,280 --> 00:27:58,460 If I take c, what if c is positive? 588 00:27:58,460 --> 00:28:01,540 589 00:28:01,540 --> 00:28:04,730 What if c is 0? 590 00:28:04,730 --> 00:28:12,255 What if c is less than 0? 591 00:28:12,255 --> 00:28:14,500 What am I going to have? 592 00:28:14,500 --> 00:28:18,460 Your imagination gives you c equals 1, Magdalena. 593 00:28:18,460 --> 00:28:19,870 Let's draw that. 594 00:28:19,870 --> 00:28:20,370 OK. 595 00:28:20,370 --> 00:28:21,940 Well, I'll try. 596 00:28:21,940 --> 00:28:23,650 a and b would be 1, right, guys? 597 00:28:23,650 --> 00:28:26,275 So a and b would be 1. 598 00:28:26,275 --> 00:28:27,100 This is a square. 599 00:28:27,100 --> 00:28:29,970 These would be the asymptotes. 600 00:28:29,970 --> 00:28:36,110 So very, very briefly, the hyperbola 601 00:28:36,110 --> 00:28:41,050 would be this one-- x squared minus y squared equals 1, 602 00:28:41,050 --> 00:28:42,222 right? 603 00:28:42,222 --> 00:28:45,130 If I have the last case for c equals 1, 604 00:28:45,130 --> 00:28:47,460 I'm going to have-- c equals negative 1-- I'm 605 00:28:47,460 --> 00:28:49,215 going to have the conjugate. 606 00:28:49,215 --> 00:28:50,660 Are you guys with me? 607 00:28:50,660 --> 00:28:57,790 So I'll have an a squared, asymptotes, conjugate. 608 00:28:57,790 --> 00:29:01,230 609 00:29:01,230 --> 00:29:05,060 What if I have different level c? c equals 1/2. c equals 2. 610 00:29:05,060 --> 00:29:08,000 c equals pi. c equals-- what are they? 611 00:29:08,000 --> 00:29:12,330 I'm going to get families of hyperbolas, 612 00:29:12,330 --> 00:29:14,720 trenches that look like that. 613 00:29:14,720 --> 00:29:16,470 Standard trenches and conjugate trenches. 614 00:29:16,470 --> 00:29:20,600 A multitude of them, an infinite family of such hyperbolas, 615 00:29:20,600 --> 00:29:22,480 an infinite family of such hyperbolas. 616 00:29:22,480 --> 00:29:24,542 I wanted to draw it. 617 00:29:24,542 --> 00:29:28,880 What do I get when c is 0? 618 00:29:28,880 --> 00:29:30,022 What are those? 619 00:29:30,022 --> 00:29:31,772 STUDENT: Don't you get, like, [INAUDIBLE]? 620 00:29:31,772 --> 00:29:34,985 621 00:29:34,985 --> 00:29:36,730 MAGDALENA TODA: They get-- very good. 622 00:29:36,730 --> 00:29:37,230 Why? 623 00:29:37,230 --> 00:29:40,650 x squared minus y squared equals 0 would lead 624 00:29:40,650 --> 00:29:44,920 me to y equals plus/minus 1. 625 00:29:44,920 --> 00:29:48,470 And who are those y equals plus/minus 1? 626 00:29:48,470 --> 00:29:49,590 Exactly. 627 00:29:49,590 --> 00:29:54,500 But exactly the first bisector, which is y equals x. 628 00:29:54,500 --> 00:29:56,410 They are [? then the ?] function. 629 00:29:56,410 --> 00:29:59,820 And the other one, y equals negative [? x. ?] So these 630 00:29:59,820 --> 00:30:01,160 are the asymptotes. 631 00:30:01,160 --> 00:30:05,620 So I'm going to get a-- you guys have to do this better than me. 632 00:30:05,620 --> 00:30:06,880 Sorry. 633 00:30:06,880 --> 00:30:08,790 These are all hyperbolic trenches. 634 00:30:08,790 --> 00:30:11,730 They are all going to infinity like that. 635 00:30:11,730 --> 00:30:15,330 And I'm sorry that I'm giving you 636 00:30:15,330 --> 00:30:17,090 a little bit too many hints. 637 00:30:17,090 --> 00:30:19,282 This is part of your homework, your WeBWorK. 638 00:30:19,282 --> 00:30:20,740 I shouldn't talk too much about it. 639 00:30:20,740 --> 00:30:23,820 640 00:30:23,820 --> 00:30:25,350 Any questions so far? 641 00:30:25,350 --> 00:30:26,495 Is this hard? 642 00:30:26,495 --> 00:30:28,000 Yes, sir? 643 00:30:28,000 --> 00:30:28,500 No. 644 00:30:28,500 --> 00:30:30,133 STUDENT: So [? spherically, ?] if you had z 645 00:30:30,133 --> 00:30:31,508 equals y squared minus x squared, 646 00:30:31,508 --> 00:30:33,890 it's that same picture, just flipped? 647 00:30:33,890 --> 00:30:39,962 648 00:30:39,962 --> 00:30:41,450 MAGDALENA TODA: What would it be? 649 00:30:41,450 --> 00:30:43,241 It would be the poor saddle-- or cowboy-- 650 00:30:43,241 --> 00:30:44,490 STUDENT: Would be upside down. 651 00:30:44,490 --> 00:30:46,340 MAGDALENA TODA: --would be upside down. 652 00:30:46,340 --> 00:30:49,650 Or projected in something like a mirror. 653 00:30:49,650 --> 00:30:51,100 I don't know how to say. 654 00:30:51,100 --> 00:30:52,850 It would be exactly upside down. 655 00:30:52,850 --> 00:30:55,530 So the reflection of that. 656 00:30:55,530 --> 00:30:59,234 So you take all the points. 657 00:30:59,234 --> 00:31:01,075 If you have-- I don't know. 658 00:31:01,075 --> 00:31:03,380 It's hard to draw a reflection in three dimensions. 659 00:31:03,380 --> 00:31:03,880 But-- 660 00:31:03,880 --> 00:31:04,963 STUDENT: No, I understand. 661 00:31:04,963 --> 00:31:09,430 MAGDALENA TODA: Practically every curve 662 00:31:09,430 --> 00:31:14,690 would be upside down with respect to the floor. 663 00:31:14,690 --> 00:31:15,650 OK. 664 00:31:15,650 --> 00:31:16,560 All right. 665 00:31:16,560 --> 00:31:20,600 I'm going to erase in one. 666 00:31:20,600 --> 00:31:24,060 And you say, well, you've taught us about these things, 667 00:31:24,060 --> 00:31:26,150 like the domain and range. 668 00:31:26,150 --> 00:31:30,530 But what about other notions, like continuity and stuff? 669 00:31:30,530 --> 00:31:33,300 670 00:31:33,300 --> 00:31:50,730 Let me move on to 11.2. 671 00:31:50,730 --> 00:32:00,192 Limits of functions of the type z equals f of xy. 672 00:32:00,192 --> 00:32:14,640 673 00:32:14,640 --> 00:32:20,030 So what do you remember about the limit 674 00:32:20,030 --> 00:32:23,130 of a function of one variable? 675 00:32:23,130 --> 00:32:23,630 Comparison. 676 00:32:23,630 --> 00:32:27,960 677 00:32:27,960 --> 00:32:36,255 What about the limit if you take [? z's, ?] I don't know. 678 00:32:36,255 --> 00:32:37,700 I should look stunned. 679 00:32:37,700 --> 00:32:38,700 And I should be stunned. 680 00:32:38,700 --> 00:32:49,430 Of a function of y equals f of x of one variable. 681 00:32:49,430 --> 00:32:56,730 682 00:32:56,730 --> 00:33:10,513 When do we say that f has a limit at a? 683 00:33:10,513 --> 00:33:12,473 684 00:33:12,473 --> 00:33:14,764 STUDENT: When the [INAUDIBLE] approaches from the right 685 00:33:14,764 --> 00:33:16,740 and the left to the same value. 686 00:33:16,740 --> 00:33:22,610 MAGDALENA TODA: Actually, that was the simpler definition. 687 00:33:22,610 --> 00:33:25,555 Let's think a little bit deeper. 688 00:33:25,555 --> 00:33:35,330 We say that f has a limit L at x equals a. 689 00:33:35,330 --> 00:33:40,550 That's kind of the idea, left and right limits. 690 00:33:40,550 --> 00:33:43,680 But not both of them have to exist, you see. 691 00:33:43,680 --> 00:33:45,532 Maybe only the limit from the left or limit 692 00:33:45,532 --> 00:33:46,978 from the right only exists. 693 00:33:46,978 --> 00:33:49,870 694 00:33:49,870 --> 00:34:04,370 If, for any choice of values of x, closer and closer, closer 695 00:34:04,370 --> 00:34:23,909 and closer to a, we get that F gets closer and closer to L. 696 00:34:23,909 --> 00:34:27,159 And this "any" I put in. 697 00:34:27,159 --> 00:34:33,500 My god, I put it in a red circle thing, 698 00:34:33,500 --> 00:34:40,030 because one could get subsequencies of a sequence. 699 00:34:40,030 --> 00:34:42,400 And for that subsequence thing, things 700 00:34:42,400 --> 00:34:44,994 look like I would have a limit. 701 00:34:44,994 --> 00:34:47,830 And then you say, well, but in the end, 702 00:34:47,830 --> 00:34:50,889 I don't have a limit, because I can get another subsequence 703 00:34:50,889 --> 00:34:52,350 of the sequence. 704 00:34:52,350 --> 00:34:59,030 And for that one, I'm not going to have a limit. 705 00:34:59,030 --> 00:35:04,150 Can you give me an example of some crazy function that 706 00:35:04,150 --> 00:35:08,670 does not have a limit at 0? 707 00:35:08,670 --> 00:35:12,131 Example of a crazy function. 708 00:35:12,131 --> 00:35:12,630 No. 709 00:35:12,630 --> 00:35:14,560 No, don't write "crazy." 710 00:35:14,560 --> 00:35:26,270 Of a function f of x that is not defined at 0 711 00:35:26,270 --> 00:35:43,855 and does not have limit at 0, although it 712 00:35:43,855 --> 00:35:53,414 is defined for values arbitrarily close to 0. 713 00:35:53,414 --> 00:35:59,706 714 00:35:59,706 --> 00:36:07,900 Moreover, I want that function to be drawn without-- I 715 00:36:07,900 --> 00:36:22,960 want the function to be drawn without leaving 716 00:36:22,960 --> 00:36:26,300 the paper when I draw. 717 00:36:26,300 --> 00:36:30,748 718 00:36:30,748 --> 00:36:31,248 [INAUDIBLE] 719 00:36:31,248 --> 00:36:34,566 720 00:36:34,566 --> 00:36:43,530 So something that would be defined on the whole 0 721 00:36:43,530 --> 00:36:58,660 infinity except for 0 that I can draw continuously 722 00:36:58,660 --> 00:37:03,400 except when I get to 0, I get some really bad behavior. 723 00:37:03,400 --> 00:37:07,560 I don't have a limit for that function. 724 00:37:07,560 --> 00:37:08,910 You are close to that. 725 00:37:08,910 --> 00:37:10,720 Sine of 1/x. 726 00:37:10,720 --> 00:37:13,110 STUDENT: I said y equals 1/x. 727 00:37:13,110 --> 00:37:15,324 MAGDALENA TODA: y equals 1/x. 728 00:37:15,324 --> 00:37:16,260 Very good. 729 00:37:16,260 --> 00:37:18,600 Let's see. 730 00:37:18,600 --> 00:37:20,004 STUDENT: Oh, yeah. [INAUDIBLE]. 731 00:37:20,004 --> 00:37:21,129 MAGDALENA TODA: Yeah, yeah. 732 00:37:21,129 --> 00:37:22,390 Both are excellent examples. 733 00:37:22,390 --> 00:37:24,530 So let's see. 734 00:37:24,530 --> 00:37:29,080 This guy is a very nice function. 735 00:37:29,080 --> 00:37:31,370 How do we draw him, or her? 736 00:37:31,370 --> 00:37:32,540 Well, it's a her, right? 737 00:37:32,540 --> 00:37:33,040 It's a she. 738 00:37:33,040 --> 00:37:33,990 It's a function. 739 00:37:33,990 --> 00:37:34,660 No, no. 740 00:37:34,660 --> 00:37:36,360 In English, it doesn't make any sense, 741 00:37:36,360 --> 00:37:40,732 but if I think French, Italian, Spanish, Romanian-- now 742 00:37:40,732 --> 00:37:44,143 I speak both Italian and Romanian-- 743 00:37:44,143 --> 00:37:47,220 we say it's a she, it's a feminine. 744 00:37:47,220 --> 00:37:52,345 So as I approach with values closer and closer and closer 745 00:37:52,345 --> 00:37:56,345 to 0, what happens to my poor function? 746 00:37:56,345 --> 00:37:58,911 It blows up. 747 00:37:58,911 --> 00:37:59,410 OK. 748 00:37:59,410 --> 00:38:05,040 So I have limit of 1/x from the right and from the left. 749 00:38:05,040 --> 00:38:08,310 If I take it from the left, I don't care. 750 00:38:08,310 --> 00:38:11,170 Let's take it only from the right. 751 00:38:11,170 --> 00:38:11,670 OK? 752 00:38:11,670 --> 00:38:17,910 753 00:38:17,910 --> 00:38:19,630 It's close to 0. 754 00:38:19,630 --> 00:38:21,210 That's going to blow up, right? 755 00:38:21,210 --> 00:38:24,560 756 00:38:24,560 --> 00:38:25,490 And I restrict it. 757 00:38:25,490 --> 00:38:30,455 So let's say, if I want the domain to be containing 758 00:38:30,455 --> 00:38:32,742 [? both, ?] that's also fine. 759 00:38:32,742 --> 00:38:35,730 So if you guys want, we can draw the other one. 760 00:38:35,730 --> 00:38:37,010 This goes to paradise. 761 00:38:37,010 --> 00:38:39,665 The other one, I'm not going to say where it goes. 762 00:38:39,665 --> 00:38:43,270 But it's the same idea, that as you approach 0 763 00:38:43,270 --> 00:38:45,710 with closer and closer and closer values, 764 00:38:45,710 --> 00:38:47,662 it's going to blow up. 765 00:38:47,662 --> 00:38:51,020 It's going to explode. 766 00:38:51,020 --> 00:38:54,010 This is a beautiful function. 767 00:38:54,010 --> 00:38:55,160 How beautiful [INAUDIBLE]. 768 00:38:55,160 --> 00:38:57,940 Beautiful with a bad behavior near 0. 769 00:38:57,940 --> 00:38:59,730 So I'm not going to have a limit. 770 00:38:59,730 --> 00:39:00,610 No limit. 771 00:39:00,610 --> 00:39:02,590 Some people say, limit exists and is infinity. 772 00:39:02,590 --> 00:39:05,265 But does infinity exist? 773 00:39:05,265 --> 00:39:07,382 Well, this is a really philosophical, 774 00:39:07,382 --> 00:39:10,570 religious notion, so I don't want to get into it. 775 00:39:10,570 --> 00:39:13,480 But in mathematics, we consider that unless the limit is 776 00:39:13,480 --> 00:39:16,730 finite, you cannot have a limit. 777 00:39:16,730 --> 00:39:20,850 So if the limit is plus/minus infinity, there is no limit. 778 00:39:20,850 --> 00:39:25,240 Could the limit be different or different subsequences? 779 00:39:25,240 --> 00:39:28,190 This is what I wanted to point out. 780 00:39:28,190 --> 00:39:34,270 If you try this guy, you are in real trouble on that guy. 781 00:39:34,270 --> 00:39:35,152 Why? 782 00:39:35,152 --> 00:39:36,568 You can have two. 783 00:39:36,568 --> 00:39:38,928 If you have a graphing calculator, which 784 00:39:38,928 --> 00:39:43,990 I'm going to be opposed to you being used in the classroom, 785 00:39:43,990 --> 00:39:46,160 you would probably see what happens. 786 00:39:46,160 --> 00:39:51,540 Sine is defined on all the real numbers. 787 00:39:51,540 --> 00:39:54,240 But you cannot have a value at 0, 788 00:39:54,240 --> 00:39:57,510 because the 1/x is not defined at 0. 789 00:39:57,510 --> 00:40:02,330 Imagine you get closer and closer to 0 from both sides. 790 00:40:02,330 --> 00:40:05,280 I cannot draw very beautifully. 791 00:40:05,280 --> 00:40:09,590 But as 1, this is plus 1 and this is minus 1. 792 00:40:09,590 --> 00:40:11,840 I'm going to have some behavior. 793 00:40:11,840 --> 00:40:15,160 And how many of you have seen that on a computer screen 794 00:40:15,160 --> 00:40:15,941 or calculator? 795 00:40:15,941 --> 00:40:16,440 You've seen. 796 00:40:16,440 --> 00:40:17,585 Yeah, you've seen. 797 00:40:17,585 --> 00:40:20,450 By the way, did you see the Lubbuck High? 798 00:40:20,450 --> 00:40:23,472 Was it in high school you saw it the first time in Calc 1 799 00:40:23,472 --> 00:40:25,140 or pre-calc? 800 00:40:25,140 --> 00:40:28,440 STUDENT: [INAUDIBLE] Algebra 1 with Mr. West. 801 00:40:28,440 --> 00:40:28,940 [INAUDIBLE] 802 00:40:28,940 --> 00:40:32,510 MAGDALENA TODA: So I'll try-- oh, guys, you 803 00:40:32,510 --> 00:40:34,240 have to be patient with me. 804 00:40:34,240 --> 00:40:38,150 I'm not leaving the poor board with the tip of my pencil. 805 00:40:38,150 --> 00:40:39,380 I'm not leaving him. 806 00:40:39,380 --> 00:40:42,358 I have continuity. 807 00:40:42,358 --> 00:40:45,810 As I got closer to this, I still have the [INAUDIBLE] property. 808 00:40:45,810 --> 00:40:47,150 Anyway, it's OK. 809 00:40:47,150 --> 00:40:48,410 I'm not leaving this. 810 00:40:48,410 --> 00:40:52,100 I am taking all the values possible between minus 1 and 1. 811 00:40:52,100 --> 00:40:54,820 So on intervals that are smaller, smaller, 812 00:40:54,820 --> 00:40:57,756 I'm really taking all the values between minus 1 and 1, 813 00:40:57,756 --> 00:41:01,390 and really rapidly-- [INAUDIBLE]. 814 00:41:01,390 --> 00:41:07,520 When I'm getting closer to 0, I'm not going to have a limit. 815 00:41:07,520 --> 00:41:10,065 But as somebody may say, but wait. 816 00:41:10,065 --> 00:41:12,270 When I have a sequence of values that 817 00:41:12,270 --> 00:41:14,360 is getting closer and closer to 0, 818 00:41:14,360 --> 00:41:18,610 is that no guarantee that I'm going to have a limit? 819 00:41:18,610 --> 00:41:20,120 Nope. 820 00:41:20,120 --> 00:41:20,965 It depends. 821 00:41:20,965 --> 00:41:25,475 If you say "any," it has to be for any choice of points, 822 00:41:25,475 --> 00:41:27,960 any choice of points that you go closer to 0. 823 00:41:27,960 --> 00:41:30,150 Not for one sequence of points that 824 00:41:30,150 --> 00:41:32,290 is getting closer and closer to 0. 825 00:41:32,290 --> 00:41:35,296 For example, if your choice of points is this, 826 00:41:35,296 --> 00:41:36,282 choice of points. 827 00:41:36,282 --> 00:41:39,740 828 00:41:39,740 --> 00:41:43,805 Getting closer to 0. 829 00:41:43,805 --> 00:41:49,540 [INAUDIBLE] xn equals 1 over 2 pi n. 830 00:41:49,540 --> 00:41:51,815 Isn't this going to 0? 831 00:41:51,815 --> 00:41:52,315 Yeah. 832 00:41:52,315 --> 00:41:54,090 It then goes to infinity. 833 00:41:54,090 --> 00:41:55,690 This sequence goes to 0. 834 00:41:55,690 --> 00:41:56,300 What is it? 835 00:41:56,300 --> 00:41:57,015 1 over 2 pi? 836 00:41:57,015 --> 00:41:58,070 1 over 4 pi? 837 00:41:58,070 --> 00:41:58,890 1 over 8 pi? 838 00:41:58,890 --> 00:41:59,980 1 over 16 pi? 839 00:41:59,980 --> 00:42:01,100 1 over 32 pi? 840 00:42:01,100 --> 00:42:02,550 1 over 64 pi? 841 00:42:02,550 --> 00:42:04,450 This is what my son is doing to me. 842 00:42:04,450 --> 00:42:06,182 And I say, please stop. 843 00:42:06,182 --> 00:42:07,130 OK? 844 00:42:07,130 --> 00:42:08,240 He's 10 years old. 845 00:42:08,240 --> 00:42:09,590 He's so funny. 846 00:42:09,590 --> 00:42:12,964 Now, another choice of points. 847 00:42:12,964 --> 00:42:20,490 848 00:42:20,490 --> 00:42:21,170 Ah. 849 00:42:21,170 --> 00:42:26,450 Somebody-- all of you are smart enough to do this. 850 00:42:26,450 --> 00:42:29,230 What do you think I'm going to pick? 851 00:42:29,230 --> 00:42:30,780 1 over what? 852 00:42:30,780 --> 00:42:33,604 And when [? other ?] something that goes to 0 853 00:42:33,604 --> 00:42:34,520 then goes to infinity. 854 00:42:34,520 --> 00:42:41,660 And I know that your professor showed you that. 855 00:42:41,660 --> 00:42:44,995 pi over 2 plus 2 pi n. 856 00:42:44,995 --> 00:42:45,870 Doesn't this go to 0? 857 00:42:45,870 --> 00:42:46,370 Yes. 858 00:42:46,370 --> 00:42:49,740 As n gets bigger and bigger, this is going to 0. 859 00:42:49,740 --> 00:42:51,040 However, there is no limit. 860 00:42:51,040 --> 00:42:51,810 Why? 861 00:42:51,810 --> 00:42:59,280 Well, for the first sequence, as xn goes to 0, f of xn 862 00:42:59,280 --> 00:43:04,060 goes to-- what is sine of-- OK, I 863 00:43:04,060 --> 00:43:05,870 am too lazy to write this down. 864 00:43:05,870 --> 00:43:11,015 Sine of 1 over 1 over-- of 1 over 1 over 2 pi? 865 00:43:11,015 --> 00:43:14,810 866 00:43:14,810 --> 00:43:16,780 STUDENT: It's the sine over 2 pi. 867 00:43:16,780 --> 00:43:20,650 MAGDALENA TODA: This is sine of 2 pi n. 868 00:43:20,650 --> 00:43:22,361 And how much is that? 869 00:43:22,361 --> 00:43:22,860 STUDENT: 0. 870 00:43:22,860 --> 00:43:24,210 MAGDALENA TODA: 0. 871 00:43:24,210 --> 00:43:25,590 So this is a 0. 872 00:43:25,590 --> 00:43:28,730 And this is a-- this converges to 0. 873 00:43:28,730 --> 00:43:31,400 So I say, oh, so maybe I have a limit, and that'll be 0. 874 00:43:31,400 --> 00:43:32,520 Wrong. 875 00:43:32,520 --> 00:43:35,730 That would be the rapid, stupid conclusion. 876 00:43:35,730 --> 00:43:38,690 If somebody jumps [? up, ?] I picked some points, 877 00:43:38,690 --> 00:43:41,820 I formed the sequence that gets closer and closer to 0. 878 00:43:41,820 --> 00:43:43,700 I'm sure that the limit exists. 879 00:43:43,700 --> 00:43:45,500 I've got a 0. 880 00:43:45,500 --> 00:43:48,850 Well, did you think of any possible choice? 881 00:43:48,850 --> 00:43:49,670 That's the problem. 882 00:43:49,670 --> 00:43:51,942 You have to have any possible choice. 883 00:43:51,942 --> 00:44:02,580 F of yn sine of 1 over pi over 2 plus 1 over 1 884 00:44:02,580 --> 00:44:09,030 over-- Magdalena-- pi over 2 plus 2 pi n. 885 00:44:09,030 --> 00:44:10,860 So we saw that this was 0. 886 00:44:10,860 --> 00:44:14,850 What happens to sine of 1 over 1 over sine of pi 887 00:44:14,850 --> 00:44:18,720 over 2 plus 2 pi n? 888 00:44:18,720 --> 00:44:19,890 And where does this go? 889 00:44:19,890 --> 00:44:21,132 It then goes to infinity. 890 00:44:21,132 --> 00:44:26,970 891 00:44:26,970 --> 00:44:29,250 This sequence goes to 0. 892 00:44:29,250 --> 00:44:32,630 What is f of the sequence going to? 893 00:44:32,630 --> 00:44:33,520 To another limit. 894 00:44:33,520 --> 00:44:36,030 So there is no limit. 895 00:44:36,030 --> 00:44:38,580 What's the limit of this subsequence? 896 00:44:38,580 --> 00:44:41,400 It's a constant one, right? 897 00:44:41,400 --> 00:44:45,930 Because look, what does it mean pi over 2 plus 2 pi n? 898 00:44:45,930 --> 00:44:49,240 Where am I on the unit trigonometric circle? 899 00:44:49,240 --> 00:44:50,690 [INTERPOSING VOICES] 900 00:44:50,690 --> 00:44:53,930 Always here, right? 901 00:44:53,930 --> 00:44:56,450 Always on the sort of like the north pole. 902 00:44:56,450 --> 00:44:59,220 So what is the sine of this north pole? 903 00:44:59,220 --> 00:44:59,900 STUDENT: 1. 904 00:44:59,900 --> 00:45:00,970 MAGDALENA TODA: Always 1. 905 00:45:00,970 --> 00:45:02,400 So I get the limit 1. 906 00:45:02,400 --> 00:45:06,482 So I'm done because there are two different limits. 907 00:45:06,482 --> 00:45:09,360 So pay attention to this type of problem. 908 00:45:09,360 --> 00:45:17,530 Somebody can get you in trouble with this kind of thing. 909 00:45:17,530 --> 00:45:20,220 On the other hand, I'm asking you, 910 00:45:20,220 --> 00:45:22,995 what if I want to make this a function of two variables? 911 00:45:22,995 --> 00:45:27,660 912 00:45:27,660 --> 00:45:30,960 So I'll say, one point extra credit. 913 00:45:30,960 --> 00:45:34,070 I'm giving you too much extra credit. 914 00:45:34,070 --> 00:45:36,210 Maybe I give you too much-- it's OK. 915 00:45:36,210 --> 00:45:39,930 One point extra credit-- put them together. 916 00:45:39,930 --> 00:45:43,340 917 00:45:43,340 --> 00:45:47,520 Does f-- do you like to do the f? 918 00:45:47,520 --> 00:45:51,400 I used big F, and then I changed it to little f. 919 00:45:51,400 --> 00:45:54,073 This time I have a function of two variables-- little 920 00:45:54,073 --> 00:46:01,178 f with xy-- to be sine of 1 over x squared plus y squared. 921 00:46:01,178 --> 00:46:09,442 Does this function have a limit at the point 0, 0? 922 00:46:09,442 --> 00:46:12,350 923 00:46:12,350 --> 00:46:15,540 So when I approach 0, 0, do I have a limit? 924 00:46:15,540 --> 00:46:16,680 OK. 925 00:46:16,680 --> 00:46:19,810 And you say, well, it depends how I approach that 0, 0. 926 00:46:19,810 --> 00:46:21,490 That's exactly the thing. 927 00:46:21,490 --> 00:46:23,210 Yes, sir. 928 00:46:23,210 --> 00:46:25,278 Oh, you didn't want to ask me. 929 00:46:25,278 --> 00:46:28,480 930 00:46:28,480 --> 00:46:37,115 And does f of xy equals-- let me give you 931 00:46:37,115 --> 00:46:41,080 another one, a really sexy one. x 932 00:46:41,080 --> 00:46:44,780 squared plus y squared times sine of 1 933 00:46:44,780 --> 00:46:48,460 over x squared plus y squared. 934 00:46:48,460 --> 00:46:55,053 Have a limit at 0, 0? 935 00:46:55,053 --> 00:47:00,030 936 00:47:00,030 --> 00:47:01,490 I don't know. 937 00:47:01,490 --> 00:47:04,220 Continuous it cannot be, because it's not defined there. 938 00:47:04,220 --> 00:47:04,720 Right? 939 00:47:04,720 --> 00:47:07,670 For a function to be continuous at a point, 940 00:47:07,670 --> 00:47:11,360 the function has to satisfy three conditions. 941 00:47:11,360 --> 00:47:14,590 The function has to be defined there at that point. 942 00:47:14,590 --> 00:47:16,560 The function has to have a limit there 943 00:47:16,560 --> 00:47:18,890 at that point of the domain. 944 00:47:18,890 --> 00:47:23,140 And the limit and the function value have to coincide. 945 00:47:23,140 --> 00:47:24,750 Three conditions. 946 00:47:24,750 --> 00:47:28,470 We will talk about continuity later. 947 00:47:28,470 --> 00:47:29,500 Hint. 948 00:47:29,500 --> 00:47:31,620 Magdalena, too many hints. 949 00:47:31,620 --> 00:47:33,680 This should remind you of somebody 950 00:47:33,680 --> 00:47:36,080 from the first variable calculus. 951 00:47:36,080 --> 00:47:37,970 It's a more challenging problem. 952 00:47:37,970 --> 00:47:40,490 That's why I gave it to extra credit. 953 00:47:40,490 --> 00:47:45,660 If I had x sine of 1/x, what would that look like? 954 00:47:45,660 --> 00:47:46,890 STUDENT: x times-- 955 00:47:46,890 --> 00:47:50,260 MAGDALENA TODA: x times sine of 1/x. 956 00:47:50,260 --> 00:47:55,458 When I approach 0 with-- so if I have-- I 957 00:47:55,458 --> 00:47:57,160 don't ask for an answer now. 958 00:47:57,160 --> 00:47:58,800 You go home, you think about it. 959 00:47:58,800 --> 00:48:00,310 You take the calculator. 960 00:48:00,310 --> 00:48:05,910 But keep in mind that your calculator can fool you. 961 00:48:05,910 --> 00:48:11,300 Sometimes it can show an image that misguides you. 962 00:48:11,300 --> 00:48:14,580 So you have to think how to do that. 963 00:48:14,580 --> 00:48:18,920 How about x times sine of 1/x when-- 964 00:48:18,920 --> 00:48:22,130 does it have a limit when x goes to 0? 965 00:48:22,130 --> 00:48:23,620 Is there such a limit? 966 00:48:23,620 --> 00:48:24,600 Does it exist? 967 00:48:24,600 --> 00:48:28,170 968 00:48:28,170 --> 00:48:31,420 So if such a limit would exist, maybe we 969 00:48:31,420 --> 00:48:35,600 can extend by continuity the function x times sine over x. 970 00:48:35,600 --> 00:48:36,610 What does it mean? 971 00:48:36,610 --> 00:48:38,670 Like, extend it, prolong it. 972 00:48:38,670 --> 00:48:44,415 And say, it's this 4x equals 0 and this if x is not 0. 973 00:48:44,415 --> 00:48:47,610 So this is obviously x is different from 0, right? 974 00:48:47,610 --> 00:48:49,710 Can we extend it by continuity? 975 00:48:49,710 --> 00:48:51,230 Think about the drawing. 976 00:48:51,230 --> 00:48:54,500 Think about the arguments. 977 00:48:54,500 --> 00:48:58,350 And I think it's time for me to keep the promise I made 978 00:48:58,350 --> 00:49:04,950 to [? Aaron, ?] because I see no way. 979 00:49:04,950 --> 00:49:07,860 Oh, my god, [? Aaron, ?] I see no way out. 980 00:49:07,860 --> 00:49:10,380 981 00:49:10,380 --> 00:49:14,490 The epsilon delta definition of limit. 982 00:49:14,490 --> 00:49:17,460 [? Right? ?] OK. 983 00:49:17,460 --> 00:49:21,070 So what does it mean for a real mathematician or somebody 984 00:49:21,070 --> 00:49:25,350 with a strong mathematical foundation and education 985 00:49:25,350 --> 00:49:27,430 that they know the true definition 986 00:49:27,430 --> 00:49:31,310 of a limit of a function of, let's say, one variable? 987 00:49:31,310 --> 00:49:34,910 The epsilon delta, the one your dad told you about. [INAUDIBLE] 988 00:49:34,910 --> 00:49:39,980 try to fool you when avoid it in undergraduate education. 989 00:49:39,980 --> 00:49:41,740 People try to avoid the epsilon delta, 990 00:49:41,740 --> 00:49:45,615 because they think the students will never, never understand 991 00:49:45,615 --> 00:49:50,470 it, because it's such an abstract one. 992 00:49:50,470 --> 00:49:51,844 I think I wasn't ready. 993 00:49:51,844 --> 00:49:52,760 I wasn't smart enough. 994 00:49:52,760 --> 00:49:58,015 I think I was 16 when I was getting ready for some math 995 00:49:58,015 --> 00:49:58,725 competitions. 996 00:49:58,725 --> 00:50:02,810 And one professor taught me the epsilon delta and said, 997 00:50:02,810 --> 00:50:04,700 do you understand it? 998 00:50:04,700 --> 00:50:07,310 My 16-year-old mind said, no. 999 00:50:07,310 --> 00:50:08,860 But guess what? 1000 00:50:08,860 --> 00:50:10,546 Some other people smarter than me, 1001 00:50:10,546 --> 00:50:12,410 they told me, when you first see it, 1002 00:50:12,410 --> 00:50:16,690 you don't understand it in any case. 1003 00:50:16,690 --> 00:50:19,950 So it takes a little bit more time to sink in. 1004 00:50:19,950 --> 00:50:21,876 So the same idea. 1005 00:50:21,876 --> 00:50:25,140 As I'm getting closer and closer and closer and closer 1006 00:50:25,140 --> 00:50:30,280 to an x0 with my x values from anywhere around-- left, 1007 00:50:30,280 --> 00:50:35,480 right-- I have to pick an arbitrary choice of points 1008 00:50:35,480 --> 00:50:39,940 going towards x0, I have to be sure that at the same time, 1009 00:50:39,940 --> 00:50:45,210 the corresponding sequence of values is going to L, 1010 00:50:45,210 --> 00:50:47,010 I can express that in epsilon delta. 1011 00:50:47,010 --> 00:50:50,860 1012 00:50:50,860 --> 00:50:51,935 So we say that. 1013 00:50:51,935 --> 00:50:59,705 1014 00:50:59,705 --> 00:51:13,178 f of x has limit L at x equals x0 if. 1015 00:51:13,178 --> 00:51:17,180 1016 00:51:17,180 --> 00:51:24,100 For every epsilon positive, any choice of an epsilon positive, 1017 00:51:24,100 --> 00:51:25,230 there is a delta. 1018 00:51:25,230 --> 00:51:27,110 There exists-- oh, OK, guys. 1019 00:51:27,110 --> 00:51:28,430 You don't know the symbols. 1020 00:51:28,430 --> 00:51:31,290 I'll write it in English. 1021 00:51:31,290 --> 00:51:36,450 For every epsilon positive, no matter 1022 00:51:36,450 --> 00:51:40,728 how small-- put parentheses, because you 1023 00:51:40,728 --> 00:51:47,150 are just [? tired-- ?] no matter how small, 1024 00:51:47,150 --> 00:51:55,510 there exists a delta number that depends on epsilon. 1025 00:51:55,510 --> 00:52:02,356 1026 00:52:02,356 --> 00:52:16,190 So that whenever x minus x0 is less than delta, 1027 00:52:16,190 --> 00:52:33,800 this would imply that f of x minus L, 1028 00:52:33,800 --> 00:52:37,098 that limit I taught you about in absolute value, 1029 00:52:37,098 --> 00:52:38,562 is less than epsilon. 1030 00:52:38,562 --> 00:52:48,340 1031 00:52:48,340 --> 00:52:50,370 What does this mean? 1032 00:52:50,370 --> 00:52:55,200 I'm going to try and draw something 1033 00:52:55,200 --> 00:52:58,441 that happens on a line. 1034 00:52:58,441 --> 00:53:00,370 So this is x0. 1035 00:53:00,370 --> 00:53:03,894 And these are my values of x. 1036 00:53:03,894 --> 00:53:05,060 They can come from anywhere. 1037 00:53:05,060 --> 00:53:08,760 1038 00:53:08,760 --> 00:53:12,032 And this is f of x. 1039 00:53:12,032 --> 00:53:16,700 And this is L. So it says, no matter-- this 1040 00:53:16,700 --> 00:53:19,150 says-- this is an abstract way of saying, 1041 00:53:19,150 --> 00:53:23,810 no matter how close, you see, for every epsilon positive, 1042 00:53:23,810 --> 00:53:27,280 no matter how close you get to the L. 1043 00:53:27,280 --> 00:53:30,600 I decide to be in this interval, very tiny epsilon. 1044 00:53:30,600 --> 00:53:32,260 L minus epsilon. 1045 00:53:32,260 --> 00:53:35,840 L plus epsilon L. You give me your favorite epsilon. 1046 00:53:35,840 --> 00:53:38,640 You say, Magdalena, pick something really small. 1047 00:53:38,640 --> 00:53:42,180 Big epsilon to be 0.00001. 1048 00:53:42,180 --> 00:53:43,950 How about that? 1049 00:53:43,950 --> 00:53:47,660 Well, if I really have a limit there, 1050 00:53:47,660 --> 00:53:54,290 an L at x0, that means that no matter how much you shrink 1051 00:53:54,290 --> 00:53:57,540 this interval for me, you can be mean and shrink it 1052 00:53:57,540 --> 00:53:59,320 as much as you want. 1053 00:53:59,320 --> 00:54:03,400 I will still find a small interval around x0. 1054 00:54:03,400 --> 00:54:06,660 1055 00:54:06,660 --> 00:54:08,620 [? But ?] I will still find the smaller 1056 00:54:08,620 --> 00:54:13,230 interval around x0, which is-- this would be x0 minus delta. 1057 00:54:13,230 --> 00:54:15,760 This would be x0 plus delta. 1058 00:54:15,760 --> 00:54:20,961 So that the image of this purple interval fits inside. 1059 00:54:20,961 --> 00:54:22,150 You say, what? 1060 00:54:22,150 --> 00:54:25,840 So that the image of this purple interval fits inside. 1061 00:54:25,840 --> 00:54:30,040 So f of x minus L, the distance is still that, less than xy. 1062 00:54:30,040 --> 00:54:30,550 Yes, sir? 1063 00:54:30,550 --> 00:54:32,710 STUDENT: Where'd you get epsilon [INAUDIBLE]? 1064 00:54:32,710 --> 00:54:34,440 MAGDALENA TODA: So epsilon has to be 1065 00:54:34,440 --> 00:54:38,252 chose no matter how small. 1066 00:54:38,252 --> 00:54:39,837 STUDENT: [INAUDIBLE]. 1067 00:54:39,837 --> 00:54:40,670 MAGDALENA TODA: Huh? 1068 00:54:40,670 --> 00:54:42,160 Real number. 1069 00:54:42,160 --> 00:54:46,470 So I'm saying, you should not set the epsilon to be 0.0001. 1070 00:54:46,470 --> 00:54:47,770 That would be a mistake. 1071 00:54:47,770 --> 00:54:50,890 You have to think of that number as being as small as you want, 1072 00:54:50,890 --> 00:54:54,590 infinitesimally small, smaller than any particle in physics 1073 00:54:54,590 --> 00:54:57,330 that you are aware about. 1074 00:54:57,330 --> 00:54:59,880 And this is what I had the problem understanding-- 1075 00:54:59,880 --> 00:55:03,530 that notion of-- not the notion of, hey, not 1076 00:55:03,530 --> 00:55:06,000 matter how close I am, I can still 1077 00:55:06,000 --> 00:55:12,430 get something even smaller around x0 that fits in this. 1078 00:55:12,430 --> 00:55:14,360 That's not what I had the problem with. 1079 00:55:14,360 --> 00:55:18,316 The notion is to perceive an infinitesimal. 1080 00:55:18,316 --> 00:55:21,515 Our mind is too limited to understand infinity. 1081 00:55:21,515 --> 00:55:24,408 It's like trying to understand God. 1082 00:55:24,408 --> 00:55:29,512 And the same limitation comes with microscopic problems. 1083 00:55:29,512 --> 00:55:31,470 Yeah, we can see some things on the microscope, 1084 00:55:31,470 --> 00:55:32,219 and we understand. 1085 00:55:32,219 --> 00:55:34,580 Ah, I understand I have this bacteria. 1086 00:55:34,580 --> 00:55:36,040 This is staph. 1087 00:55:36,040 --> 00:55:37,460 Oh, my god. 1088 00:55:37,460 --> 00:55:43,350 But then there are molecules, atoms, subatomic particles 1089 00:55:43,350 --> 00:55:46,875 that we don't understand, because our mind is really 1090 00:55:46,875 --> 00:55:49,060 [? small. ?] Imagine something smaller 1091 00:55:49,060 --> 00:55:50,900 than the subatomic particles. 1092 00:55:50,900 --> 00:55:54,520 That's the abstract notion of infinitesimally small. 1093 00:55:54,520 --> 00:55:58,870 So I'm saying, if I really have a limit L there, 1094 00:55:58,870 --> 00:56:03,470 that means no matter how small I have this ball around it, 1095 00:56:03,470 --> 00:56:06,630 I can still find a smaller ball that 1096 00:56:06,630 --> 00:56:10,001 fits-- whose image fits inside. 1097 00:56:10,001 --> 00:56:10,500 All right? 1098 00:56:10,500 --> 00:56:15,000 The same kind of definition-- I will try to generalize this. 1099 00:56:15,000 --> 00:56:19,660 Can you guys help me generalize this limit notion 1100 00:56:19,660 --> 00:56:24,790 to the notion of function of two variables? 1101 00:56:24,790 --> 00:56:29,040 1102 00:56:29,040 --> 00:56:41,400 So we say, that f of xy has the limit L at x0y0. 1103 00:56:41,400 --> 00:56:44,970 1104 00:56:44,970 --> 00:56:50,710 What was x0y0 when I talked about-- what 1105 00:56:50,710 --> 00:56:52,550 example did I give you guys? 1106 00:56:52,550 --> 00:56:55,050 Sine of 1 over x squared plus y squared, right? 1107 00:56:55,050 --> 00:56:56,220 Something like that. 1108 00:56:56,220 --> 00:56:56,850 I don't know. 1109 00:56:56,850 --> 00:56:59,730 I said, think of 0, 0. 1110 00:56:59,730 --> 00:57:01,810 That was the given point. 1111 00:57:01,810 --> 00:57:03,540 It has to be a fixed couple. 1112 00:57:03,540 --> 00:57:07,870 So you think of the origin, 0, 0, as being as a fixed couple. 1113 00:57:07,870 --> 00:57:12,390 Or you think of the point 1, 0 as being as a fixed couple 1114 00:57:12,390 --> 00:57:14,761 in that plane you look at. 1115 00:57:14,761 --> 00:57:18,420 That is the fixed couple. 1116 00:57:18,420 --> 00:57:21,460 If-- now somebody has to help me. 1117 00:57:21,460 --> 00:57:27,540 For every epsilon positive, no matter how small, 1118 00:57:27,540 --> 00:57:30,690 that's where I have a problem imagining infinitesimally 1119 00:57:30,690 --> 00:57:31,615 small. 1120 00:57:31,615 --> 00:57:34,650 There exists-- I no longer have this problem. 1121 00:57:34,650 --> 00:57:37,310 But I had it enough when I was in my 20s. 1122 00:57:37,310 --> 00:57:40,155 I don't want to go back to my 20s and have-- I mean, 1123 00:57:40,155 --> 00:57:41,065 I would love to. 1124 00:57:41,065 --> 00:57:42,890 [LAUGHTER] 1125 00:57:42,890 --> 00:57:46,420 To go having vacations with no worries and so on. 1126 00:57:46,420 --> 00:57:48,570 But I wouldn't like to go back to my 20s 1127 00:57:48,570 --> 00:57:50,470 and have to relearn all the mathematics. 1128 00:57:50,470 --> 00:57:50,970 Now way. 1129 00:57:50,970 --> 00:57:53,190 That was too much of a struggle. 1130 00:57:53,190 --> 00:58:00,310 There exists a delta positive that depends on epsilon. 1131 00:58:00,310 --> 00:58:02,820 What does it mean, depends on epsilon? 1132 00:58:02,820 --> 00:58:04,650 Because guys, imagine you make this epsilon 1133 00:58:04,650 --> 00:58:05,860 smaller and smaller. 1134 00:58:05,860 --> 00:58:08,100 You have to make delta smaller and smaller, 1135 00:58:08,100 --> 00:58:12,321 so that you can fit that little ball in the big ball. 1136 00:58:12,321 --> 00:58:12,820 OK? 1137 00:58:12,820 --> 00:58:19,768 That depends on epsilon, so that whenever-- now, 1138 00:58:19,768 --> 00:58:21,930 that is a big problem. 1139 00:58:21,930 --> 00:58:28,305 How do I say, distance between the point xy and the point 1140 00:58:28,305 --> 00:58:29,310 x0y0? 1141 00:58:29,310 --> 00:58:32,130 Oh, my god. 1142 00:58:32,130 --> 00:58:37,310 This is distance between xy and x0y0 is less than delta. 1143 00:58:37,310 --> 00:58:48,480 This would imply that-- well, this 1144 00:58:48,480 --> 00:58:54,090 is a function with values in R. This is in R. Real number. 1145 00:58:54,090 --> 00:58:55,470 So I don't have a problem. 1146 00:58:55,470 --> 00:58:57,460 I can use absolute value here. 1147 00:58:57,460 --> 00:59:11,330 Absolute value of f of the couple xy minus L 1148 00:59:11,330 --> 00:59:14,580 is less than epsilon. 1149 00:59:14,580 --> 00:59:19,080 The thing is, can you visualize that little ball, 1150 00:59:19,080 --> 00:59:20,990 that little disk? 1151 00:59:20,990 --> 00:59:22,310 What do I mean? 1152 00:59:22,310 --> 00:59:26,440 Being close, xy is me, right? 1153 00:59:26,440 --> 00:59:27,520 But I'm moving. 1154 00:59:27,520 --> 00:59:28,760 I'm the moving point. 1155 00:59:28,760 --> 00:59:30,350 I'm dancing around. 1156 00:59:30,350 --> 00:59:33,436 And [? Nateesh ?] is x0y0. 1157 00:59:33,436 --> 00:59:37,570 How do I say that I have to be close enough to him? 1158 00:59:37,570 --> 00:59:38,600 I cannot touch him. 1159 00:59:38,600 --> 00:59:39,710 That's against the rules. 1160 00:59:39,710 --> 00:59:42,270 That's considered [INAUDIBLE] harassment. 1161 00:59:42,270 --> 00:59:45,800 But I can come as close as I want. 1162 00:59:45,800 --> 00:59:49,210 So I say, the distance between me-- 1163 00:59:49,210 --> 00:59:52,100 I'm xy-- and [? Nateesh, ?] who is 1164 00:59:52,100 --> 00:59:58,264 fixed x0y0, has to be smaller than that small delta. 1165 00:59:58,264 --> 01:00:00,796 How do I represent that in plane mathematics? 1166 01:00:00,796 --> 01:00:02,004 STUDENT: Doesn't [INAUDIBLE]? 1167 01:00:02,004 --> 01:00:05,520 1168 01:00:05,520 --> 01:00:06,520 MAGDALENA TODA: Exactly. 1169 01:00:06,520 --> 01:00:09,480 So that delta has to be small enough so 1170 01:00:09,480 --> 01:00:16,970 that the image of f at me minus the limit is less than epsilon. 1171 01:00:16,970 --> 01:00:20,960 Now you understand why all the other teachers avoid 1172 01:00:20,960 --> 01:00:22,820 talking about this [? one. ?] So I 1173 01:00:22,820 --> 01:00:27,840 want to get small enough-- not too close-- but close enough 1174 01:00:27,840 --> 01:00:39,520 to him, so that my value-- I'm f of xy-- minus the limit, 1175 01:00:39,520 --> 01:00:42,170 the limit-- I have a preset limit. 1176 01:00:42,170 --> 01:00:45,140 All around [? Nateesh, ?] I can have different values, 1177 01:00:45,140 --> 01:00:47,490 no matter where I go. 1178 01:00:47,490 --> 01:00:51,060 My value at all these points around [? Nateesh ?] have 1179 01:00:51,060 --> 01:00:54,510 to be close enough to L. So I say, 1180 01:00:54,510 --> 01:00:57,550 well, you have to get close enough to L. 1181 01:00:57,550 --> 01:00:59,490 Somebody presents me an epsilon. 1182 01:00:59,490 --> 01:01:02,350 Then I have to reduce my distance to [? Nateesh ?] 1183 01:01:02,350 --> 01:01:03,990 depending to that epsilon. 1184 01:01:03,990 --> 01:01:07,959 Because otherwise, the image doesn't fit. 1185 01:01:07,959 --> 01:01:09,000 It's a little bit tricky. 1186 01:01:09,000 --> 01:01:10,960 STUDENT: So is this like the squeeze theorem kind of? 1187 01:01:10,960 --> 01:01:12,430 MAGDALENA TODA: It is the squeeze theorem. 1188 01:01:12,430 --> 01:01:12,930 STUDENT: Oh, all right. 1189 01:01:12,930 --> 01:01:13,721 MAGDALENA TODA: OK? 1190 01:01:13,721 --> 01:01:18,580 So the squeezing-- I ball into another [? ball ?] [? limit. ?] 1191 01:01:18,580 --> 01:01:21,030 This is why-- it's not a ball, but it's a-- 1192 01:01:21,030 --> 01:01:21,780 STUDENT: A circle. 1193 01:01:21,780 --> 01:01:22,655 MAGDALENA TODA: Disk. 1194 01:01:22,655 --> 01:01:23,720 A circle, right? 1195 01:01:23,720 --> 01:01:28,830 So how do we express that in Calc 3 in plain? 1196 01:01:28,830 --> 01:01:31,005 This is the [? ingredient, ?] distance d. 1197 01:01:31,005 --> 01:01:33,630 So Seth, can you tell me what is the distance between these two 1198 01:01:33,630 --> 01:01:34,790 points? 1199 01:01:34,790 --> 01:01:36,010 Square root of-- 1200 01:01:36,010 --> 01:01:37,360 STUDENT: [INAUDIBLE]. 1201 01:01:37,360 --> 01:01:41,750 MAGDALENA TODA: x minus x0 squared plus y minus y0 1202 01:01:41,750 --> 01:01:43,154 squared. 1203 01:01:43,154 --> 01:01:45,383 Now shut up. [? And I ?] am talking to myself. 1204 01:01:45,383 --> 01:01:45,856 STUDENT: Must be less than delta. 1205 01:01:45,856 --> 01:01:46,695 [LAUGHTER] 1206 01:01:46,695 --> 01:01:48,280 MAGDALENA TODA: Less than delta. 1207 01:01:48,280 --> 01:01:51,530 So instead of writing this, I need 1208 01:01:51,530 --> 01:01:53,726 to write that I can do that in my mind. 1209 01:01:53,726 --> 01:01:58,356 1210 01:01:58,356 --> 01:02:00,024 OK? 1211 01:02:00,024 --> 01:02:00,990 All right. 1212 01:02:00,990 --> 01:02:01,850 This is hard. 1213 01:02:01,850 --> 01:02:03,470 We need to sleep on that. 1214 01:02:03,470 --> 01:02:09,449 I have one or two more problems that are less hard-- nah, 1215 01:02:09,449 --> 01:02:11,490 they are still hard, but they are more intuitive, 1216 01:02:11,490 --> 01:02:14,570 that I would like to ask you about the limit. 1217 01:02:14,570 --> 01:02:16,940 I'm going to give you a function. 1218 01:02:16,940 --> 01:02:21,476 And we would have to visualize as I get closer to a point 1219 01:02:21,476 --> 01:02:24,570 where I am actually going. 1220 01:02:24,570 --> 01:02:30,170 So I have this nasty function, f of xy 1221 01:02:30,170 --> 01:02:34,960 equals xy over z squared plus y squared. 1222 01:02:34,960 --> 01:02:39,020 1223 01:02:39,020 --> 01:02:44,600 And I'm saying, [INAUDIBLE] the point is the origin. 1224 01:02:44,600 --> 01:02:47,030 I choose the origin. 1225 01:02:47,030 --> 01:02:47,590 Question. 1226 01:02:47,590 --> 01:02:53,020 Do I have a limit that's-- do I have a limit? 1227 01:02:53,020 --> 01:02:54,620 Not [? really ?] for me. 1228 01:02:54,620 --> 01:03:02,480 Does f have a limit at the origin? 1229 01:03:02,480 --> 01:03:06,400 1230 01:03:06,400 --> 01:03:09,820 You would have to imagine that you'd draw this function. 1231 01:03:09,820 --> 01:03:13,145 And except you cannot draw, and you really don't care to draw 1232 01:03:13,145 --> 01:03:13,780 it. 1233 01:03:13,780 --> 01:03:17,030 You only have to imagine that you have some abstract graph-- 1234 01:03:17,030 --> 01:03:19,050 z equals f of xy. 1235 01:03:19,050 --> 01:03:20,880 You don't care what it looks like. 1236 01:03:20,880 --> 01:03:24,100 But then you take points on the floor, 1237 01:03:24,100 --> 01:03:27,450 just like I did the exercise with [? Nateesh ?] before. 1238 01:03:27,450 --> 01:03:30,820 And you get closer and closer to the origin. 1239 01:03:30,820 --> 01:03:34,415 But no attention-- no matter what path I take, 1240 01:03:34,415 --> 01:03:36,850 I have to get the same limit. 1241 01:03:36,850 --> 01:03:37,548 What? 1242 01:03:37,548 --> 01:03:46,680 No matter what path I take towards [? Nateesh-- ?] 1243 01:03:46,680 --> 01:03:52,860 don't write that down-- towards [? z0y0, ?] I have to get 1244 01:03:52,860 --> 01:03:53,790 the same limit. 1245 01:03:53,790 --> 01:03:56,970 1246 01:03:56,970 --> 01:03:59,030 Do I? 1247 01:03:59,030 --> 01:04:04,030 Let's imagine with the eyes of your imaginations. 1248 01:04:04,030 --> 01:04:07,410 And [? Nateesh ?] is the point 0, 0. 1249 01:04:07,410 --> 01:04:10,652 And you are aspiring to get closer and closer to him 1250 01:04:10,652 --> 01:04:12,750 without touching him. 1251 01:04:12,750 --> 01:04:15,265 Because otherwise, he's going to sue you. 1252 01:04:15,265 --> 01:04:18,120 So what do we have here? 1253 01:04:18,120 --> 01:04:19,350 We have different paths? 1254 01:04:19,350 --> 01:04:21,390 How can I get closer? 1255 01:04:21,390 --> 01:04:25,550 Either on this path or maybe on this path. 1256 01:04:25,550 --> 01:04:28,210 Or maybe on this path. 1257 01:04:28,210 --> 01:04:31,590 Or maybe, if I had something to drink last night-- which 1258 01:04:31,590 --> 01:04:35,115 I did not, because after the age of 35, 1259 01:04:35,115 --> 01:04:37,032 I stopped drinking completely. 1260 01:04:37,032 --> 01:04:40,990 1261 01:04:40,990 --> 01:04:44,820 That's when I decided I want to be a mom, 1262 01:04:44,820 --> 01:04:47,270 and I didn't want to make a bad example. 1263 01:04:47,270 --> 01:04:50,450 So no matter what path you take, you can make it wiggly, 1264 01:04:50,450 --> 01:04:52,030 you can make it any way you want. 1265 01:04:52,030 --> 01:04:53,700 We are still approaching 0, 0. 1266 01:04:53,700 --> 01:04:55,925 You still have to get the same limit. 1267 01:04:55,925 --> 01:05:00,170 Oh, that's tricky, because it's also the same in life. 1268 01:05:00,170 --> 01:05:02,430 Depending on the path you take in life, 1269 01:05:02,430 --> 01:05:05,450 you have different results, different limits. 1270 01:05:05,450 --> 01:05:11,243 Now, what if I take the path number one, number two, number 1271 01:05:11,243 --> 01:05:12,720 three possibility. 1272 01:05:12,720 --> 01:05:16,900 And number [? blooie ?] is the drunken variant. 1273 01:05:16,900 --> 01:05:22,120 That is hard to implement in an exercise. 1274 01:05:22,120 --> 01:05:26,600 Imagine that I have limit along the path one. 1275 01:05:26,600 --> 01:05:28,290 Path one. 1276 01:05:28,290 --> 01:05:34,680 xy goes to 0, 0 of xy over x squared plus y squared. 1277 01:05:34,680 --> 01:05:36,970 Do you guys see what's going to happen? 1278 01:05:36,970 --> 01:05:40,660 So I'm along the-- OK, here it is. 1279 01:05:40,660 --> 01:05:46,590 This line, right, this is the x-axis, y-axis, z-axis. 1280 01:05:46,590 --> 01:05:48,530 What's special for the x-axis? 1281 01:05:48,530 --> 01:05:50,470 Who is 0? 1282 01:05:50,470 --> 01:05:52,700 STUDENT: x. 1283 01:05:52,700 --> 01:05:53,330 STUDENT: yz. 1284 01:05:53,330 --> 01:05:54,420 MAGDALENA TODA: y is 0. 1285 01:05:54,420 --> 01:05:57,400 So y is 0. 1286 01:05:57,400 --> 01:05:59,410 So y is 0. 1287 01:05:59,410 --> 01:06:00,360 Don't laugh at me. 1288 01:06:00,360 --> 01:06:03,470 I'm going to write like that because it's easier. 1289 01:06:03,470 --> 01:06:06,770 And it's going to be something like limit 1290 01:06:06,770 --> 01:06:13,738 when x approaches 0 of x over x squared. 1291 01:06:13,738 --> 01:06:15,690 STUDENT: It's 1/x. 1292 01:06:15,690 --> 01:06:17,950 MAGDALENA TODA: Times 0 up. 1293 01:06:17,950 --> 01:06:18,950 Oh, my god. 1294 01:06:18,950 --> 01:06:20,671 Is that-- how much is that? 1295 01:06:20,671 --> 01:06:21,170 STUDENT: 0. 1296 01:06:21,170 --> 01:06:21,290 STUDENT: 0. 1297 01:06:21,290 --> 01:06:22,039 MAGDALENA TODA: 0! 1298 01:06:22,039 --> 01:06:22,570 I'm happy. 1299 01:06:22,570 --> 01:06:23,980 I say, maybe I have the limit. 1300 01:06:23,980 --> 01:06:24,820 I have the limit 0. 1301 01:06:24,820 --> 01:06:27,120 No, never rush in life. 1302 01:06:27,120 --> 01:06:28,040 Check. 1303 01:06:28,040 --> 01:06:30,780 Experiment any other paths. 1304 01:06:30,780 --> 01:06:35,120 And it's actually very easy to see where I can go wrong. 1305 01:06:35,120 --> 01:06:39,630 If I take the path number two, I will get the same result. 1306 01:06:39,630 --> 01:06:41,470 You don't need a lot of imagination 1307 01:06:41,470 --> 01:06:44,355 to realize, hey, whether she does it for x 1308 01:06:44,355 --> 01:06:48,362 or does it for y, if she goes along the 2, what 1309 01:06:48,362 --> 01:06:49,790 the heck is going to happen? 1310 01:06:49,790 --> 01:06:51,030 y is going to shrink. 1311 01:06:51,030 --> 01:06:53,150 x will always be 0. 1312 01:06:53,150 --> 01:06:56,850 Because this means a point's like what? 1313 01:06:56,850 --> 01:06:58,850 0,1. 1314 01:06:58,850 --> 01:07:00,835 0, 1/2. 1315 01:07:00,835 --> 01:07:03,180 0, 1/n, and so on. 1316 01:07:03,180 --> 01:07:07,820 But plug them all in here, I get 0, 1/n times 0. 1317 01:07:07,820 --> 01:07:08,670 It's still 0. 1318 01:07:08,670 --> 01:07:10,460 So I still get 0. 1319 01:07:10,460 --> 01:07:12,310 Path two. 1320 01:07:12,310 --> 01:07:15,470 When I approach my-- xt goes to 0, 0. 1321 01:07:15,470 --> 01:07:18,790 The poor [? Nateesh ?] is waiting for an answer. 1322 01:07:18,790 --> 01:07:20,950 I still get 0. 1323 01:07:20,950 --> 01:07:23,645 Let's take not the drunken path, because I 1324 01:07:23,645 --> 01:07:25,520 don't know [? it unless ?] the sine function. 1325 01:07:25,520 --> 01:07:26,970 That is really crazy. 1326 01:07:26,970 --> 01:07:29,010 I'll take this one. 1327 01:07:29,010 --> 01:07:31,420 What is this one, in your opinion? 1328 01:07:31,420 --> 01:07:32,970 Is that going to help me? 1329 01:07:32,970 --> 01:07:35,720 I don't know, but I need some intuition. 1330 01:07:35,720 --> 01:07:39,550 Mathematicians need intuition and a lot of patience. 1331 01:07:39,550 --> 01:07:42,110 So what is your intuition? 1332 01:07:42,110 --> 01:07:45,120 The one in the middle, I'm going to start walking on that, OK, 1333 01:07:45,120 --> 01:07:46,542 until you tell me what it is. 1334 01:07:46,542 --> 01:07:47,500 STUDENT: y [INAUDIBLE]. 1335 01:07:47,500 --> 01:07:49,540 MAGDALENA TODA: y equals x is the first bisector 1336 01:07:49,540 --> 01:07:51,040 or the first quadrant. 1337 01:07:51,040 --> 01:07:54,580 And I'm very happy I can go both ways. 1338 01:07:54,580 --> 01:07:55,850 y equals x. 1339 01:07:55,850 --> 01:07:56,460 x [INAUDIBLE]. 1340 01:07:56,460 --> 01:08:07,130 So limit when x equals y, but the pair xy goes to 0,0. 1341 01:08:07,130 --> 01:08:07,980 I'm silly. 1342 01:08:07,980 --> 01:08:10,978 I can say that, well, Magdalena, this 1343 01:08:10,978 --> 01:08:15,540 is the pair xx, because x equals what? 1344 01:08:15,540 --> 01:08:16,899 Let me plug them in. 1345 01:08:16,899 --> 01:08:19,085 So it's like two people. 1346 01:08:19,085 --> 01:08:20,555 x and y are married. 1347 01:08:20,555 --> 01:08:22,180 They are a couple, a pair. 1348 01:08:22,180 --> 01:08:24,330 They look identical. 1349 01:08:24,330 --> 01:08:26,380 Sometimes it happens. 1350 01:08:26,380 --> 01:08:28,439 Like twins, they start looking alike, 1351 01:08:28,439 --> 01:08:30,819 dressing alike, and so on. 1352 01:08:30,819 --> 01:08:36,529 The x and the y have to receive the same letter. 1353 01:08:36,529 --> 01:08:41,029 And you have to tell me what in the world the limit will be. 1354 01:08:41,029 --> 01:08:43,818 1355 01:08:43,818 --> 01:08:44,359 STUDENT: 1/2. 1356 01:08:44,359 --> 01:08:45,520 MAGDALENA TODA: 1/2. 1357 01:08:45,520 --> 01:08:46,640 Oh, my god. 1358 01:08:46,640 --> 01:08:48,479 So now I'm deflated. 1359 01:08:48,479 --> 01:08:52,470 So now I realize that taking two different paths, 1360 01:08:52,470 --> 01:08:57,578 I show that I have-- on this path, I have 1/2. 1361 01:08:57,578 --> 01:09:00,069 On this path, I have 0. 1362 01:09:00,069 --> 01:09:01,149 I don't match. 1363 01:09:01,149 --> 01:09:02,710 I don't have an overall limit. 1364 01:09:02,710 --> 01:09:10,140 So the answer is, no overall limit. 1365 01:09:10,140 --> 01:09:10,960 Oh, my god. 1366 01:09:10,960 --> 01:09:14,640 So what you need to do, guys, is read 1367 01:09:14,640 --> 01:09:18,340 section 11.1 and section 11.2. 1368 01:09:18,340 --> 01:09:21,100 And I will ask you next time-- and you can lie, 1369 01:09:21,100 --> 01:09:22,892 you can do whatever. 1370 01:09:22,892 --> 01:09:26,368 Did the book explain better than me, 1371 01:09:26,368 --> 01:09:28,890 or I explain better than the book? 1372 01:09:28,890 --> 01:09:31,790 This type of example when the limit does not exist. 1373 01:09:31,790 --> 01:09:33,430 We are going to see more examples. 1374 01:09:33,430 --> 01:09:37,783 You are going to see examples where the limit does exist. 1375 01:09:37,783 --> 01:09:40,160 Now, one last thing. 1376 01:09:40,160 --> 01:09:46,640 When you have to compute limits of compositions of functions 1377 01:09:46,640 --> 01:09:48,529 whose limit exist-- for example, you 1378 01:09:48,529 --> 01:09:58,290 know that limit is xy goes to x0y0 of f 1379 01:09:58,290 --> 01:10:10,410 of xy [INAUDIBLE] limit of xy go to x0y0 of gxy 1380 01:10:10,410 --> 01:10:13,730 is L-- L-- L-- M-- M. 1381 01:10:13,730 --> 01:10:25,200 How are you going to compute the limit of alpha f plus beta g? 1382 01:10:25,200 --> 01:10:27,100 This is in the book. 1383 01:10:27,100 --> 01:10:32,650 But you don't need the book to understand that. 1384 01:10:32,650 --> 01:10:34,320 You will already give me the answer, 1385 01:10:34,320 --> 01:10:39,210 because this is the equivalent thing to the function of one 1386 01:10:39,210 --> 01:10:41,320 variable thing in Calc 1. 1387 01:10:41,320 --> 01:10:43,670 So if you would only have f of x or g of x, 1388 01:10:43,670 --> 01:10:45,410 it would be piece of cake. 1389 01:10:45,410 --> 01:10:46,501 What would you say? 1390 01:10:46,501 --> 01:10:47,375 STUDENT: [INAUDIBLE]. 1391 01:10:47,375 --> 01:10:48,291 MAGDALENA TODA: Right. 1392 01:10:48,291 --> 01:10:54,080 Alpha times L plus beta times M. Can you also 1393 01:10:54,080 --> 01:10:55,379 multiply functions. 1394 01:10:55,379 --> 01:10:55,920 Yes, you can. 1395 01:10:55,920 --> 01:11:07,160 Limit of fg as xy goes to x0 or y0-- will be LM. 1396 01:11:07,160 --> 01:11:09,750 How about-- now I'm going to jump to conclusion, hoping 1397 01:11:09,750 --> 01:11:13,170 that you are going to catch me. 1398 01:11:13,170 --> 01:11:15,652 You are going to catch me, and shout at me, 1399 01:11:15,652 --> 01:11:18,410 and say, ooh, pay attention, Magdalena, 1400 01:11:18,410 --> 01:11:21,560 you can make a mistake there. 1401 01:11:21,560 --> 01:11:26,430 I say it's L/M when I do the division rule, right? 1402 01:11:26,430 --> 01:11:28,222 Where should I pay attention? 1403 01:11:28,222 --> 01:11:29,761 STUDENT: M [INAUDIBLE]. 1404 01:11:29,761 --> 01:11:31,010 MAGDALENA TODA: Pay attention. 1405 01:11:31,010 --> 01:11:38,740 Sometimes you can have the-- right? 1406 01:11:38,740 --> 01:11:45,120 And this also has to exist as well. 1407 01:11:45,120 --> 01:11:46,560 STUDENT: [INAUDIBLE]. 1408 01:11:46,560 --> 01:11:51,070 MAGDALENA TODA: So one last-- how many minutes 1409 01:11:51,070 --> 01:11:53,810 have I spent with you? 1410 01:11:53,810 --> 01:11:57,926 I've spent with you a long number of hours of my life. 1411 01:11:57,926 --> 01:11:58,800 No, I'm just kidding. 1412 01:11:58,800 --> 01:12:04,234 So you have one hour and 15, a little bit more. 1413 01:12:04,234 --> 01:12:05,400 Do I have a little bit more? 1414 01:12:05,400 --> 01:12:05,900 Yes. 1415 01:12:05,900 --> 01:12:07,990 I have 15 minutes. 1416 01:12:07,990 --> 01:12:08,490 I have-- 1417 01:12:08,490 --> 01:12:09,615 STUDENT: So we get out at-- 1418 01:12:09,615 --> 01:12:10,450 [INTERPOSING VOICES] 1419 01:12:10,450 --> 01:12:11,241 MAGDALENA TODA: 50. 1420 01:12:11,241 --> 01:12:12,514 Five more minutes. 1421 01:12:12,514 --> 01:12:15,360 OK. 1422 01:12:15,360 --> 01:12:20,655 So I want to ask you what you remember about some 1423 01:12:20,655 --> 01:12:25,090 of your friends, the trig functions involved in limits. 1424 01:12:25,090 --> 01:12:28,030 1425 01:12:28,030 --> 01:12:32,000 Why did we study limits at the point 1426 01:12:32,000 --> 01:12:34,290 where the function's not defined? 1427 01:12:34,290 --> 01:12:35,430 Well, to heck with it. 1428 01:12:35,430 --> 01:12:36,030 We don't care. 1429 01:12:36,030 --> 01:12:37,950 The function is not defined at 0. 1430 01:12:37,950 --> 01:12:40,076 But the limit is. 1431 01:12:40,076 --> 01:12:42,916 And nobody showed you how to do the epsilon delta 1432 01:12:42,916 --> 01:12:44,255 to show anything like that. 1433 01:12:44,255 --> 01:12:48,526 1434 01:12:48,526 --> 01:12:49,987 OK. 1435 01:12:49,987 --> 01:12:52,422 Can you do that with epsilon delta? 1436 01:12:52,422 --> 01:12:57,790 1437 01:12:57,790 --> 01:13:00,352 Actually, you can do everything with epsilon delta. 1438 01:13:00,352 --> 01:13:02,310 But I'm not going to give you any extra credit. 1439 01:13:02,310 --> 01:13:07,538 So I trust you that you remember that. 1440 01:13:07,538 --> 01:13:08,900 1! 1441 01:13:08,900 --> 01:13:10,815 How about-- let me-- OK. 1442 01:13:10,815 --> 01:13:11,767 I am so proud of you. 1443 01:13:11,767 --> 01:13:12,850 Let me challenge you more. 1444 01:13:12,850 --> 01:13:14,690 Let me challenge you more. 1445 01:13:14,690 --> 01:13:17,950 Tangent of ax over bx. 1446 01:13:17,950 --> 01:13:19,500 x go to 0. 1447 01:13:19,500 --> 01:13:22,110 I asked this to a girl from Lubbock High. 1448 01:13:22,110 --> 01:13:23,450 She was in high school. 1449 01:13:23,450 --> 01:13:25,390 She knew the answer. 1450 01:13:25,390 --> 01:13:28,300 STUDENT: Oh, I can't disappoint everybody in getting this. 1451 01:13:28,300 --> 01:13:31,476 STUDENT: Is it 1/a? 1452 01:13:31,476 --> 01:13:32,350 Oh, I can't remember. 1453 01:13:32,350 --> 01:13:33,860 MAGDALENA TODA: Tell me what to do to be smart. 1454 01:13:33,860 --> 01:13:34,360 Right? 1455 01:13:34,360 --> 01:13:37,530 I have to be doing something smart. 1456 01:13:37,530 --> 01:13:40,050 She-- can you give me hint? 1457 01:13:40,050 --> 01:13:41,779 I'm your student and you say, well-- 1458 01:13:41,779 --> 01:13:42,320 STUDENT: ba-- 1459 01:13:42,320 --> 01:13:44,062 STUDENT: It's 0. 1460 01:13:44,062 --> 01:13:45,447 STUDENT: It's [INAUDIBLE]. 1461 01:13:45,447 --> 01:13:46,780 MAGDALENA TODA: Um, it's a what? 1462 01:13:46,780 --> 01:13:48,090 STUDENT: b/a? 1463 01:13:48,090 --> 01:13:49,590 MAGDALENA TODA: I'm not [INAUDIBLE]. 1464 01:13:49,590 --> 01:13:51,080 I don't think so. 1465 01:13:51,080 --> 01:13:52,730 So what should I do? 1466 01:13:52,730 --> 01:13:58,070 I should say, instead of bx-- that drives me nuts. 1467 01:13:58,070 --> 01:14:00,390 This goes on my nerves-- bx. 1468 01:14:00,390 --> 01:14:03,550 Like, maybe I go on your nerves. bx is ax, right? 1469 01:14:03,550 --> 01:14:06,745 If it were ax, I would be more constructive, 1470 01:14:06,745 --> 01:14:09,270 and I knew what to do. 1471 01:14:09,270 --> 01:14:13,190 I say replace bx with ax, compensate for it, 1472 01:14:13,190 --> 01:14:15,120 and divide by bx. 1473 01:14:15,120 --> 01:14:17,880 And I was trying to explain that to my son, 1474 01:14:17,880 --> 01:14:23,550 that if you have a fraction a/b, and then you write a/n 1475 01:14:23,550 --> 01:14:26,695 times n/b, it's the same thing. 1476 01:14:26,695 --> 01:14:28,555 Gosh, I had the problem with him. 1477 01:14:28,555 --> 01:14:33,310 And then I realized that he didn't do simplifications 1478 01:14:33,310 --> 01:14:34,870 in school. 1479 01:14:34,870 --> 01:14:41,180 So it took a little more hours to explain these things. 1480 01:14:41,180 --> 01:14:42,590 This is fourth grade. 1481 01:14:42,590 --> 01:14:45,150 I think I remember doing that in fourth grade. 1482 01:14:45,150 --> 01:14:47,430 Third grade, actually. 1483 01:14:47,430 --> 01:14:50,360 So these two guys disappear. 1484 01:14:50,360 --> 01:14:53,790 I haven't changed my problem at all. 1485 01:14:53,790 --> 01:14:57,920 But I've changed the status, the shape of my problem 1486 01:14:57,920 --> 01:15:01,340 to something I can mold, because this goes to somebody, 1487 01:15:01,340 --> 01:15:02,680 and this goes to somebody else. 1488 01:15:02,680 --> 01:15:05,170 Who is this fellow? 1489 01:15:05,170 --> 01:15:07,620 It's a limit that's a constant-- a/b. 1490 01:15:07,620 --> 01:15:09,260 Who is this fellow? 1491 01:15:09,260 --> 01:15:09,760 STUDENT: 1. 1492 01:15:09,760 --> 01:15:10,840 MAGDALENA TODA: 1. 1493 01:15:10,840 --> 01:15:15,660 Because tangent of x/x as x goes to 0 goes to 1 exactly 1494 01:15:15,660 --> 01:15:16,160 like that. 1495 01:15:16,160 --> 01:15:22,140 So limit of sine x over cosine x, that's tangent, right? 1496 01:15:22,140 --> 01:15:23,290 Over x. 1497 01:15:23,290 --> 01:15:25,020 You do it exactly the same. 1498 01:15:25,020 --> 01:15:32,410 It's limit of sine x/x times 1 over cosine x. 1499 01:15:32,410 --> 01:15:34,396 That's how we did it in high school. 1500 01:15:34,396 --> 01:15:35,025 This goes to 1. 1501 01:15:35,025 --> 01:15:36,640 This goes to 1. 1502 01:15:36,640 --> 01:15:37,390 So it's 1. 1503 01:15:37,390 --> 01:15:39,190 So thank you, this is 1. 1504 01:15:39,190 --> 01:15:43,286 I know I took a little more time to explain than I wanted to. 1505 01:15:43,286 --> 01:15:46,130 But now you are grown up. 1506 01:15:46,130 --> 01:15:49,260 In two minutes, you are going to be finishing 1507 01:15:49,260 --> 01:15:50,766 this section, more or less. 1508 01:15:50,766 --> 01:15:54,640 What if I put a function of two variables, 1509 01:15:54,640 --> 01:15:57,790 and I ask you what the limit will be, 1510 01:15:57,790 --> 01:16:01,290 if it's the same type of function. 1511 01:16:01,290 --> 01:16:03,490 So you say, oh, Magdalena, what you doing to us? 1512 01:16:03,490 --> 01:16:05,240 OK, we'll see it's fun. 1513 01:16:05,240 --> 01:16:06,010 This one's fun. 1514 01:16:06,010 --> 01:16:07,850 It's not like the one before. 1515 01:16:07,850 --> 01:16:11,080 This one is pretty beautiful. 1516 01:16:11,080 --> 01:16:12,770 It's nice to you. 1517 01:16:12,770 --> 01:16:14,950 It exists. 1518 01:16:14,950 --> 01:16:16,590 xy goes to 0, 0. 1519 01:16:16,590 --> 01:16:19,745 So you have to imagine some preferable function 1520 01:16:19,745 --> 01:16:22,070 in abstract thinking. 1521 01:16:22,070 --> 01:16:24,720 And you want it in a little disk here. 1522 01:16:24,720 --> 01:16:31,580 And xy, these are all points xy close enough to 0, 0, 1523 01:16:31,580 --> 01:16:34,030 in the neighborhood of 0, 0. 1524 01:16:34,030 --> 01:16:34,813 OK. 1525 01:16:34,813 --> 01:16:37,480 What's going to happen as you get closer and closer 1526 01:16:37,480 --> 01:16:40,380 and closer and closer with tinier and tinier and tinier 1527 01:16:40,380 --> 01:16:43,880 disks around 0, 0? 1528 01:16:43,880 --> 01:16:47,590 You're going to shrink so much. 1529 01:16:47,590 --> 01:16:49,290 What do you think this will going to be, 1530 01:16:49,290 --> 01:16:50,750 and how do I prove it? 1531 01:16:50,750 --> 01:16:52,130 STUDENT: [INAUDIBLE]. 1532 01:16:52,130 --> 01:16:53,966 MAGDALENA TODA: Who said it? 1533 01:16:53,966 --> 01:16:55,950 You, sir? [INAUDIBLE] going to go to 1. 1534 01:16:55,950 --> 01:16:57,770 And he's right. 1535 01:16:57,770 --> 01:17:00,940 He has the intuition. 1536 01:17:00,940 --> 01:17:03,204 A mathematician will tell you, prove it. 1537 01:17:03,204 --> 01:17:04,620 STUDENT: Um, well, let's see here. 1538 01:17:04,620 --> 01:17:06,432 MAGDALENA TODA: Can you prove? 1539 01:17:06,432 --> 01:17:09,897 STUDENT: You could use the right triangle proof, 1540 01:17:09,897 --> 01:17:11,980 but that would probably take way more [INAUDIBLE]. 1541 01:17:11,980 --> 01:17:12,940 MAGDALENA TODA: x and y are independent. 1542 01:17:12,940 --> 01:17:13,680 That's the problem. 1543 01:17:13,680 --> 01:17:15,721 They are married, but they are still independent. 1544 01:17:15,721 --> 01:17:17,220 It's a couple. 1545 01:17:17,220 --> 01:17:20,920 However, we can use polar coordinates. 1546 01:17:20,920 --> 01:17:22,775 Why is polar coordinates? 1547 01:17:22,775 --> 01:17:28,950 Well, in general, if we are in xy, it's a pair. 1548 01:17:28,950 --> 01:17:30,585 This is r, right? 1549 01:17:30,585 --> 01:17:33,590 So rx is r cosine theta. 1550 01:17:33,590 --> 01:17:35,443 y is r sine theta. 1551 01:17:35,443 --> 01:17:37,442 And I can get closer and closer to the original. 1552 01:17:37,442 --> 01:17:38,680 I don't care. 1553 01:17:38,680 --> 01:17:41,310 What happens about x squared plus y squared, 1554 01:17:41,310 --> 01:17:43,040 this is r squared. 1555 01:17:43,040 --> 01:17:44,275 And r is a real number. 1556 01:17:44,275 --> 01:17:47,440 And as you walk closer and closer to the original 1557 01:17:47,440 --> 01:17:52,800 without touching it, that r goes to 0. 1558 01:17:52,800 --> 01:17:53,850 It shrinks to 0. 1559 01:17:53,850 --> 01:17:58,260 So that r squared goes to 0 but never touches 0. 1560 01:17:58,260 --> 01:18:04,310 So this becomes limit as r goes to 0, the radius of that disk 1561 01:18:04,310 --> 01:18:06,291 goes to 0. 1562 01:18:06,291 --> 01:18:10,540 Sine of r squared over r squared. 1563 01:18:10,540 --> 01:18:13,870 But r squared could be replaced by the real function, t, 1564 01:18:13,870 --> 01:18:17,425 by the real parameter, lambda, by whatever you want. 1565 01:18:17,425 --> 01:18:19,440 So then it's 1. 1566 01:18:19,440 --> 01:18:23,390 And then Alexander was right. 1567 01:18:23,390 --> 01:18:26,207 He based it on, like, observation, intuition, 1568 01:18:26,207 --> 01:18:27,040 everything you want. 1569 01:18:27,040 --> 01:18:28,830 It was not a proof. 1570 01:18:28,830 --> 01:18:32,470 On a multiple-choice exam, he would be a lucky guy. 1571 01:18:32,470 --> 01:18:34,301 I don't want you to prove it. 1572 01:18:34,301 --> 01:18:36,950 But if I want you to prove it, you have to say, 1573 01:18:36,950 --> 01:18:39,530 Magdalena, I know polar coordinates, 1574 01:18:39,530 --> 01:18:41,930 and so I can do it. 1575 01:18:41,930 --> 01:18:45,460 And one last question for today. 1576 01:18:45,460 --> 01:18:49,570 Guys, I'm asking you, limit xy goes to 0, 0. 1577 01:18:49,570 --> 01:18:53,600 You will see some of these in your WeBWorK for Chapter 11 1578 01:18:53,600 --> 01:18:56,890 that's waiting for you, homework 3. 1579 01:18:56,890 --> 01:19:03,200 Tangent of 2 x squared plus y squared over 3 1580 01:19:03,200 --> 01:19:06,446 x squared plus y squared. 1581 01:19:06,446 --> 01:19:09,320 What is that? 1582 01:19:09,320 --> 01:19:10,278 2/3. 1583 01:19:10,278 --> 01:19:11,021 STUDENT: 2/3. 1584 01:19:11,021 --> 01:19:12,520 MAGDALENA TODA: Am I asking you why? 1585 01:19:12,520 --> 01:19:13,540 No, enough. 1586 01:19:13,540 --> 01:19:14,440 OK. 1587 01:19:14,440 --> 01:19:17,030 [INAUDIBLE] I gave you everything 1588 01:19:17,030 --> 01:19:20,920 you need to show that. 1589 01:19:20,920 --> 01:19:23,616 x squared plus y squared, again, is Mr. r squared. 1590 01:19:23,616 --> 01:19:24,592 It's OK. 1591 01:19:24,592 --> 01:19:29,472 I taught you that. a/b. a is 2, b is 3. 1592 01:19:29,472 --> 01:19:30,448 Is it hard? 1593 01:19:30,448 --> 01:19:31,912 It is not easy, for sure. 1594 01:19:31,912 --> 01:19:35,328 Calc 3 is really difficult compared to other topics 1595 01:19:35,328 --> 01:19:37,780 you are probably taking. 1596 01:19:37,780 --> 01:19:40,960 But I hope that I can convince you 1597 01:19:40,960 --> 01:19:45,440 that math, although difficult, [INAUDIBLE] Calc 3, 1598 01:19:45,440 --> 01:19:48,230 is also fun. 1599 01:19:48,230 --> 01:19:49,930 OK? 1600 01:19:49,930 --> 01:19:50,760 All right. 1601 01:19:50,760 --> 01:19:55,083 So I need attendance and I need the extra credit. 1602 01:19:55,083 --> 01:19:56,208 STUDENT: Yeah, [INAUDIBLE]. 1603 01:19:56,208 --> 01:19:59,080 1604 01:19:59,080 --> 01:20:01,920 MAGDALENA TODA: Before you go, you need to sign. 1605 01:20:01,920 --> 01:20:04,439