1 00:00:00,000 --> 00:00:00,500 2 00:00:00,500 --> 00:00:03,087 People asked me if I'm going to go over homework. 3 00:00:03,087 --> 00:00:04,280 Of course I will. 4 00:00:04,280 --> 00:00:05,300 Let me explain. 5 00:00:05,300 --> 00:00:08,200 Out of the four hours you have, three 6 00:00:08,200 --> 00:00:11,090 should be more or less lecture time. 7 00:00:11,090 --> 00:00:14,470 And the fourth hour, which is the instructor's latitude, 8 00:00:14,470 --> 00:00:17,764 where they put it-- it's applications, problems, 9 00:00:17,764 --> 00:00:20,600 homework like problems, all sorts of practice for exams 10 00:00:20,600 --> 00:00:22,000 and so on. 11 00:00:22,000 --> 00:00:23,410 It's not a recitation. 12 00:00:23,410 --> 00:00:31,111 It's some sort of workshop that the instructor conducts himself 13 00:00:31,111 --> 00:00:33,083 personally. 14 00:00:33,083 --> 00:00:36,041 All right. 15 00:00:36,041 --> 00:00:38,834 If you don't have questions, I'm just 16 00:00:38,834 --> 00:00:42,030 going to go ahead and review a little bit of what 17 00:00:42,030 --> 00:00:44,580 we discussed last time. 18 00:00:44,580 --> 00:00:53,190 Something new and exciting was chapter 11, section 11.1. 19 00:00:53,190 --> 00:00:55,174 And we did 11.2. 20 00:00:55,174 --> 00:00:57,160 And what was that about? 21 00:00:57,160 --> 00:00:59,480 That was about functions of several variables. 22 00:00:59,480 --> 00:01:07,970 23 00:01:07,970 --> 00:01:10,430 And we discussed several examples, 24 00:01:10,430 --> 00:01:13,520 but then we focused our attention mainly 25 00:01:13,520 --> 00:01:19,280 to explicit functions, which means z equals f of x, y, 26 00:01:19,280 --> 00:01:21,383 of two variables. 27 00:01:21,383 --> 00:01:25,330 And we call this a graph because it is a graph. 28 00:01:25,330 --> 00:01:33,350 In 3D, it's a surface whose domain is on the floor. 29 00:01:33,350 --> 00:01:38,670 And the altitude is z, and that is the-- this is the-- OK. 30 00:01:38,670 --> 00:01:40,970 How many of you are non-math majors? 31 00:01:40,970 --> 00:01:43,390 Can you raise hands? 32 00:01:43,390 --> 00:01:44,500 Oh, OK. 33 00:01:44,500 --> 00:01:47,340 So you know a little bit about research 34 00:01:47,340 --> 00:01:49,910 from your own classes, science classes 35 00:01:49,910 --> 00:01:51,870 or from science fairs from school. 36 00:01:51,870 --> 00:01:55,990 These are the independent variables, x, y. 37 00:01:55,990 --> 00:01:58,430 And z is the dependent variable. 38 00:01:58,430 --> 00:02:01,400 We don't use this kind of terminology in this class. 39 00:02:01,400 --> 00:02:06,640 But so that you know-- we discussed domain last time. 40 00:02:06,640 --> 00:02:07,790 This was about what? 41 00:02:07,790 --> 00:02:10,550 Domain, range. 42 00:02:10,550 --> 00:02:12,435 After range, what did we do? 43 00:02:12,435 --> 00:02:14,790 We talked about level curves. 44 00:02:14,790 --> 00:02:17,616 What is the level curve? 45 00:02:17,616 --> 00:02:22,220 Level curves are curves x, y in the plane corresponding 46 00:02:22,220 --> 00:02:24,885 to f of x, y equals constant. 47 00:02:24,885 --> 00:02:27,555 48 00:02:27,555 --> 00:02:29,970 These are called level curves in plane, 49 00:02:29,970 --> 00:02:32,860 in the plane called x, y plane. 50 00:02:32,860 --> 00:02:36,180 51 00:02:36,180 --> 00:02:37,850 What else have we discussed? 52 00:02:37,850 --> 00:02:41,980 We went straight into 11.2. 53 00:02:41,980 --> 00:02:44,650 In 11.2, we were very happy to remember 54 00:02:44,650 --> 00:02:49,450 a little bit of Calculus 1, which was practically 55 00:02:49,450 --> 00:02:53,030 a review of limits from Calc 1. 56 00:02:53,030 --> 00:02:54,420 And what did we do? 57 00:02:54,420 --> 00:02:59,240 We did epsilon delta, which was not covered in Calculus 1. 58 00:02:59,240 --> 00:03:01,410 And where is Aaron? 59 00:03:01,410 --> 00:03:01,910 OK. 60 00:03:01,910 --> 00:03:04,680 Thank you, Aaron. 61 00:03:04,680 --> 00:03:07,460 And today, I was thinking, I want to show you actually 62 00:03:07,460 --> 00:03:12,640 an example that is quite easy of how you use epsilon 63 00:03:12,640 --> 00:03:20,328 delta for continuity, to show if the function is continuous, 64 00:03:20,328 --> 00:03:23,744 but for a function of true variables. 65 00:03:23,744 --> 00:03:25,208 And that's not hard. 66 00:03:25,208 --> 00:03:26,672 You may think, oh, my god. 67 00:03:26,672 --> 00:03:27,648 That must be hard. 68 00:03:27,648 --> 00:03:29,112 That's not hard at all. 69 00:03:29,112 --> 00:03:32,700 I'm going to move on to the second part of 11.2, which 70 00:03:32,700 --> 00:03:34,770 is continuity. 71 00:03:34,770 --> 00:03:38,240 11.2, second part. 72 00:03:38,240 --> 00:03:39,650 The first part was what? 73 00:03:39,650 --> 00:03:41,590 It was limits of functions, right, guys? 74 00:03:41,590 --> 00:03:45,160 We discussed properties of limits, 75 00:03:45,160 --> 00:03:49,810 algebraic properties of adding sums and taking a limit 76 00:03:49,810 --> 00:03:53,750 of a sum, taking a limit of a product of functions, 77 00:03:53,750 --> 00:03:58,500 taking the limit of a quotient of function, when it exists, 78 00:03:58,500 --> 00:04:00,290 when it doesn't. 79 00:04:00,290 --> 00:04:06,165 Now the second part of 11.2 is called continuity. 80 00:04:06,165 --> 00:04:08,095 Continuity of what? 81 00:04:08,095 --> 00:04:09,470 Well, I'm too lazy to right down, 82 00:04:09,470 --> 00:04:16,450 but it's continuity of functions of two variables, right? 83 00:04:16,450 --> 00:04:20,170 Now in Calc 1-- you reminded me last time. 84 00:04:20,170 --> 00:04:21,839 I tried to remind you. 85 00:04:21,839 --> 00:04:22,990 You tried to remind me. 86 00:04:22,990 --> 00:04:24,640 Let's remind each other. 87 00:04:24,640 --> 00:04:27,084 This is like a discussion. 88 00:04:27,084 --> 00:04:43,010 What was the meaning of f of x being a continuous function x0, 89 00:04:43,010 --> 00:04:46,640 which is part of the domain? 90 00:04:46,640 --> 00:04:48,305 x0 has to be in the domain. 91 00:04:48,305 --> 00:04:55,952 92 00:04:55,952 --> 00:04:58,045 This is if and only if what? 93 00:04:58,045 --> 00:04:59,650 Well, what kind of function is that? 94 00:04:59,650 --> 00:05:01,925 A one variable function, real value. 95 00:05:01,925 --> 00:05:05,770 It takes values on, let's say, an interval on the real line. 96 00:05:05,770 --> 00:05:11,058 What was the group of properties that 97 00:05:11,058 --> 00:05:14,080 have to be simultaneously satisfied, 98 00:05:14,080 --> 00:05:15,690 satisfied at the same time? 99 00:05:15,690 --> 00:05:18,350 100 00:05:18,350 --> 00:05:21,198 And you told me it has to be at the same time. 101 00:05:21,198 --> 00:05:24,530 And I was very happy because if one of the three conditions 102 00:05:24,530 --> 00:05:28,254 is missing, then goodbye, continuity. 103 00:05:28,254 --> 00:05:30,192 One? 104 00:05:30,192 --> 00:05:31,691 STUDENT: It's defined at that point. 105 00:05:31,691 --> 00:05:33,655 MAGDALENA TODA: Yes, sir. f of x0 is defined. 106 00:05:33,655 --> 00:05:36,620 107 00:05:36,620 --> 00:05:39,850 Actually, I said that here in the domain. 108 00:05:39,850 --> 00:05:43,070 I'll remove it because now I said it better. 109 00:05:43,070 --> 00:05:44,943 Two? 110 00:05:44,943 --> 00:05:46,215 STUDENT: The limit exists. 111 00:05:46,215 --> 00:05:47,298 MAGDALENA TODA: Very good. 112 00:05:47,298 --> 00:05:51,650 The limit, as I approach x0 with any kind of value 113 00:05:51,650 --> 00:05:59,855 closer and closer, exists and is finite. 114 00:05:59,855 --> 00:06:03,280 Let's give it a name. 115 00:06:03,280 --> 00:06:09,634 Let's call it L. 116 00:06:09,634 --> 00:06:11,092 STUDENT: [? The following value ?] 117 00:06:11,092 --> 00:06:12,100 equals the limit. 118 00:06:12,100 --> 00:06:13,141 MAGDALENA TODA: Yes, sir. 119 00:06:13,141 --> 00:06:14,085 That's the last thing. 120 00:06:14,085 --> 00:06:17,490 And I'm glad I didn't have to pull the truth out 121 00:06:17,490 --> 00:06:18,510 of your mouth. 122 00:06:18,510 --> 00:06:27,410 So the limit will-- the limit of f of x when x goes to x0 123 00:06:27,410 --> 00:06:28,890 equals f of x0. 124 00:06:28,890 --> 00:06:31,870 125 00:06:31,870 --> 00:06:33,977 No examples. 126 00:06:33,977 --> 00:06:38,510 You should know Calc 1, and you do. 127 00:06:38,510 --> 00:06:43,920 I'm just going to move on to Calc 3. 128 00:06:43,920 --> 00:06:48,500 And let's see what the definition of continuity 129 00:06:48,500 --> 00:06:52,075 would mean for us in Calc 3. 130 00:06:52,075 --> 00:07:00,470 Can anybody mimic the properties that-- well, f of x, y 131 00:07:00,470 --> 00:07:15,262 is said to be continuous at x0, y0 132 00:07:15,262 --> 00:07:34,320 if and only if the following conditions are-- my arm hurts. 133 00:07:34,320 --> 00:07:36,055 Are simultaneously satisfied. 134 00:07:36,055 --> 00:07:48,980 135 00:07:48,980 --> 00:07:53,240 I don't like professors who use PDF files or slides. 136 00:07:53,240 --> 00:07:53,740 Shh. 137 00:07:53,740 --> 00:07:54,240 OK. 138 00:07:54,240 --> 00:07:56,150 I don't want anything premade. 139 00:07:56,150 --> 00:08:00,810 The class is a construction, is working, 140 00:08:00,810 --> 00:08:04,630 is something like a work in progress. 141 00:08:04,630 --> 00:08:07,360 We are building things together. 142 00:08:07,360 --> 00:08:08,680 This is teamwork. 143 00:08:08,680 --> 00:08:11,110 If I come up with some slides that were 144 00:08:11,110 --> 00:08:13,080 made at home or a PDF file. 145 00:08:13,080 --> 00:08:14,420 First of all, it means I'm lazy. 146 00:08:14,420 --> 00:08:17,035 Second of all, it means that I'm not 147 00:08:17,035 --> 00:08:20,214 willing to take it one step at a time 148 00:08:20,214 --> 00:08:24,584 and show you how the idea's revealed. 149 00:08:24,584 --> 00:08:25,084 One. 150 00:08:25,084 --> 00:08:33,470 151 00:08:33,470 --> 00:08:34,230 Who is telling me? 152 00:08:34,230 --> 00:08:35,320 I'm not going to say it. 153 00:08:35,320 --> 00:08:36,332 It's a work in progress. 154 00:08:36,332 --> 00:08:39,171 155 00:08:39,171 --> 00:08:41,070 STUDENT: [INAUDIBLE] 156 00:08:41,070 --> 00:08:43,014 MAGDALENA TODA: f of-- 157 00:08:43,014 --> 00:08:44,450 STUDENT: [INAUDIBLE] 158 00:08:44,450 --> 00:08:48,180 MAGDALENA TODA: Of x0, y0 is defined. 159 00:08:48,180 --> 00:08:49,856 And why not? 160 00:08:49,856 --> 00:08:52,140 Well, just to have a silly [? pun ?]. 161 00:08:52,140 --> 00:08:52,640 Two. 162 00:08:52,640 --> 00:08:55,260 163 00:08:55,260 --> 00:09:02,922 Limit as the pair x, y approaches x0, x0-- and guys, 164 00:09:02,922 --> 00:09:05,840 when you close your eyes-- no you close your eyes-- 165 00:09:05,840 --> 00:09:09,410 and you imagine x, y going to x0, 166 00:09:09,410 --> 00:09:16,030 y0 by any possible paths in any possible way, 167 00:09:16,030 --> 00:09:21,570 it's not that you have a predetermined path to x0, y0, 168 00:09:21,570 --> 00:09:23,650 because you may be trapped. 169 00:09:23,650 --> 00:09:26,170 You may have-- as you've seen last time, you may have, 170 00:09:26,170 --> 00:09:28,540 coming from this direction, the limit will exist, 171 00:09:28,540 --> 00:09:29,990 will be this one. 172 00:09:29,990 --> 00:09:32,600 Coming from that direction, the limit will exist, 173 00:09:32,600 --> 00:09:34,600 would be another one. 174 00:09:34,600 --> 00:09:36,540 And then you don't have overall limits. 175 00:09:36,540 --> 00:09:41,450 So the limit-- when I call that, that means the overall limit 176 00:09:41,450 --> 00:09:51,250 exists, exists and equals L. It's finite. 177 00:09:51,250 --> 00:09:53,430 That's what I mean. 178 00:09:53,430 --> 00:09:57,880 And three, the value of the function at x0, 179 00:09:57,880 --> 00:10:04,510 y0 must be equal to the limit of the function that value 180 00:10:04,510 --> 00:10:08,382 as you approach it, x0, y0. 181 00:10:08,382 --> 00:10:11,550 And equals L, of course. 182 00:10:11,550 --> 00:10:12,650 So great. 183 00:10:12,650 --> 00:10:16,810 So it's so obvious that we are following 184 00:10:16,810 --> 00:10:19,358 exactly the same type of definition, 185 00:10:19,358 --> 00:10:22,214 the same type of pattern. 186 00:10:22,214 --> 00:10:28,660 I'm going to ask you to help me, to help 187 00:10:28,660 --> 00:10:34,720 me solve a harder problem that involves continuity. 188 00:10:34,720 --> 00:10:38,558 And I'm asking you, if I have the following function-- 189 00:10:38,558 --> 00:10:40,950 I'm going to erase the definition of continuity 190 00:10:40,950 --> 00:10:43,308 from Calc 1. 191 00:10:43,308 --> 00:10:45,720 I'm going to ask you, what if I have this funny function? 192 00:10:45,720 --> 00:10:49,082 You've seen it before, and I gave you a little bit 193 00:10:49,082 --> 00:10:50,384 of a warning about it. 194 00:10:50,384 --> 00:10:53,320 195 00:10:53,320 --> 00:11:00,860 Limit as x, y goes to 0, 0 of x squared 196 00:11:00,860 --> 00:11:07,170 plus y squared times sine of 1 over x squared plus y squared. 197 00:11:07,170 --> 00:11:10,188 198 00:11:10,188 --> 00:11:11,176 Does that exist? 199 00:11:11,176 --> 00:11:16,120 200 00:11:16,120 --> 00:11:16,920 And also-- 201 00:11:16,920 --> 00:11:20,320 STUDENT: It's actually-- so the limit is actually 202 00:11:20,320 --> 00:11:23,640 approaching a plane rather than a set of [INAUDIBLE]. 203 00:11:23,640 --> 00:11:26,850 MAGDALENA TODA: So well, actually, it's 204 00:11:26,850 --> 00:11:28,080 not approaching a plane. 205 00:11:28,080 --> 00:11:29,800 Let's see what's happening when-- 206 00:11:29,800 --> 00:11:30,740 STUDENT: Sorry, sorry. 207 00:11:30,740 --> 00:11:32,417 Not a plane, a [? line. ?] 208 00:11:32,417 --> 00:11:33,250 MAGDALENA TODA: Yes. 209 00:11:33,250 --> 00:11:36,037 STUDENT: And is the z-axis-- the entire z-axis is 0, 0? 210 00:11:36,037 --> 00:11:37,620 MAGDALENA TODA: So this is the z-axis. 211 00:11:37,620 --> 00:11:43,930 And that means exactly that x and y-- it will be 0. 212 00:11:43,930 --> 00:11:47,030 Now I am just looking at what happens 213 00:11:47,030 --> 00:11:50,530 in the plane, in the floor plane x, y. 214 00:11:50,530 --> 00:11:53,730 The pairs x, y are wiggly. 215 00:11:53,730 --> 00:11:56,220 They are like little wormy worms. 216 00:11:56,220 --> 00:12:00,680 And they float on the water on the floor. 217 00:12:00,680 --> 00:12:03,340 And these squiggly things approach 218 00:12:03,340 --> 00:12:05,870 x, y from any possible path. 219 00:12:05,870 --> 00:12:06,995 They go like this. 220 00:12:06,995 --> 00:12:09,040 They go like that. 221 00:12:09,040 --> 00:12:11,192 They go in every possible way. 222 00:12:11,192 --> 00:12:12,150 Let's see what happens. 223 00:12:12,150 --> 00:12:15,050 224 00:12:15,050 --> 00:12:17,550 Continuity-- is this continuous? 225 00:12:17,550 --> 00:12:19,840 Well, you say, Magdalena, come on. 226 00:12:19,840 --> 00:12:21,960 You cannot have this continuous at 0, 0, 227 00:12:21,960 --> 00:12:24,480 because it's undefined at 0, 0. 228 00:12:24,480 --> 00:12:24,980 Yes. 229 00:12:24,980 --> 00:12:27,410 But maybe I can extend it by continuity. 230 00:12:27,410 --> 00:12:31,670 So let me introduce-- this is my favorite, f of x, y. 231 00:12:31,670 --> 00:12:35,790 But I'll say, f of x, y is not defined at 0, 0. 232 00:12:35,790 --> 00:12:46,775 But how about g of x, y as being my f of x, y for any x, 233 00:12:46,775 --> 00:12:50,380 y different from 0, 0. 234 00:12:50,380 --> 00:12:55,540 And at the origin, at the very origin, I will say, 235 00:12:55,540 --> 00:12:59,105 I want to have-- when x, y equals 0, 236 00:12:59,105 --> 00:13:00,430 0, I want to have a value. 237 00:13:00,430 --> 00:13:05,460 Which value do you think might extend 238 00:13:05,460 --> 00:13:07,420 this function by continuity? 239 00:13:07,420 --> 00:13:08,890 STUDENT: The limit. 240 00:13:08,890 --> 00:13:10,650 MAGDALENA TODA: The limit if it exists 241 00:13:10,650 --> 00:13:15,960 and if-- well, you know already, I think, what the limit is 242 00:13:15,960 --> 00:13:18,780 because some of you thought about this at home 243 00:13:18,780 --> 00:13:20,160 for extra credit. 244 00:13:20,160 --> 00:13:21,550 So it's not fair, right? 245 00:13:21,550 --> 00:13:22,640 No, I'm just kidding. 246 00:13:22,640 --> 00:13:26,690 So I claim that maybe-- if I put a 0 here, 247 00:13:26,690 --> 00:13:28,894 will this be continuous? 248 00:13:28,894 --> 00:13:31,284 Will g be continuous? 249 00:13:31,284 --> 00:13:35,600 250 00:13:35,600 --> 00:13:42,060 So prove, prove either way, prove, justify your answer 251 00:13:42,060 --> 00:13:45,665 by a proof, a complete proof with epsilon delta. 252 00:13:45,665 --> 00:13:46,425 Proof. 253 00:13:46,425 --> 00:13:48,300 OK. 254 00:13:48,300 --> 00:13:48,800 OK. 255 00:13:48,800 --> 00:13:51,330 So now is a worried face. 256 00:13:51,330 --> 00:13:52,660 Like, oh, my god. 257 00:13:52,660 --> 00:13:54,965 This guy is worried because, oh, my god. 258 00:13:54,965 --> 00:13:55,786 Epsilon delta. 259 00:13:55,786 --> 00:13:57,530 Oh, my god. 260 00:13:57,530 --> 00:13:59,800 But the principle-- the intuition 261 00:13:59,800 --> 00:14:03,920 tells us that we should look first at some sort of a graph, 262 00:14:03,920 --> 00:14:05,290 just like Ryan pointed out. 263 00:14:05,290 --> 00:14:09,240 One should close their eyes and imagine a graph of a function 264 00:14:09,240 --> 00:14:16,800 with-- it's hard to visualize in 3D the graph of a function that 265 00:14:16,800 --> 00:14:18,740 is a surface. 266 00:14:18,740 --> 00:14:23,790 This is a surface. z equals the whole shebang. 267 00:14:23,790 --> 00:14:29,650 But when I'm going to look at the one dimensional case 268 00:14:29,650 --> 00:14:33,680 from last time, we remember the sine of 1/x 269 00:14:33,680 --> 00:14:35,210 was a crazy function. 270 00:14:35,210 --> 00:14:39,320 We called it the harmonica, well, 20-something years ago 271 00:14:39,320 --> 00:14:40,710 when I was in high school. 272 00:14:40,710 --> 00:14:42,736 I was in an advanced calculus class. 273 00:14:42,736 --> 00:14:46,170 And our teacher was not funny at all. 274 00:14:46,170 --> 00:14:49,260 He was also not teaching much, gave us a lot of homework, 275 00:14:49,260 --> 00:14:50,580 very challenging. 276 00:14:50,580 --> 00:14:54,432 So in order to make our life a little bit easier, 277 00:14:54,432 --> 00:14:57,228 we always worked in groups, which was allowed. 278 00:14:57,228 --> 00:15:00,960 So we called it a harmonica because it was oscillating 279 00:15:00,960 --> 00:15:02,800 like that to the point that-- you've seen 280 00:15:02,800 --> 00:15:06,186 the harmonica-- the accordion. 281 00:15:06,186 --> 00:15:12,940 When you bring it back to the-- harmonica came to my mind 282 00:15:12,940 --> 00:15:15,720 from the harmonic function. 283 00:15:15,720 --> 00:15:19,260 So the accordion is-- when you actually 284 00:15:19,260 --> 00:15:25,920 squeeze it, all that oscillation things, the cusps are 285 00:15:25,920 --> 00:15:28,490 closer and closer to a line. 286 00:15:28,490 --> 00:15:33,920 So what you have here is this kind of oscillation, 287 00:15:33,920 --> 00:15:37,630 very, very rapid oscillation for sine of 1/x. 288 00:15:37,630 --> 00:15:41,806 When we want to multiply by an x, what's going to happen? 289 00:15:41,806 --> 00:15:47,510 Well, this has not limit at 0 because it takes all the values 290 00:15:47,510 --> 00:15:49,600 infinitesimally close to 0. 291 00:15:49,600 --> 00:15:52,480 It keeps going through all the values between minus 1 and 1, 292 00:15:52,480 --> 00:15:53,230 closer and closer. 293 00:15:53,230 --> 00:15:55,560 So that was no good. 294 00:15:55,560 --> 00:16:03,887 But if we take this guy-- that's going to go to-- well, 295 00:16:03,887 --> 00:16:05,348 I cannot do better. 296 00:16:05,348 --> 00:16:07,296 MATLAB can do better than me. 297 00:16:07,296 --> 00:16:09,244 Mathematica can do better. 298 00:16:09,244 --> 00:16:10,090 You can do that. 299 00:16:10,090 --> 00:16:12,600 In most engineering classes, if you are-- 300 00:16:12,600 --> 00:16:15,690 who is an electrical engineering major? 301 00:16:15,690 --> 00:16:18,960 But even if you are not, you are going 302 00:16:18,960 --> 00:16:21,250 to see this type of function a lot. 303 00:16:21,250 --> 00:16:24,560 And you're going to see it again in differential equations. 304 00:16:24,560 --> 00:16:27,150 305 00:16:27,150 --> 00:16:30,745 How can I imagine-- this graph is hard to draw. 306 00:16:30,745 --> 00:16:34,060 Don't ask me to draw that. 307 00:16:34,060 --> 00:16:41,060 But ask me if I can use epsilon delta to prove continuity. 308 00:16:41,060 --> 00:16:44,965 So what would it mean, proving continuity? 309 00:16:44,965 --> 00:16:45,895 I have a feeling-- 310 00:16:45,895 --> 00:16:47,290 STUDENT: Well, actually, if this is-- going back to that graph, 311 00:16:47,290 --> 00:16:49,010 doesn't that graph look like-- 312 00:16:49,010 --> 00:16:50,301 MAGDALENA TODA: This goes to 0. 313 00:16:50,301 --> 00:16:53,840 The limit exists for x sine of 1/x, and it is 0. 314 00:16:53,840 --> 00:16:55,580 Why? 315 00:16:55,580 --> 00:16:57,170 Ryan? 316 00:16:57,170 --> 00:17:01,050 RYAN: Wouldn't the graph with the x squared plus 317 00:17:01,050 --> 00:17:02,900 y squared times that side-- wouldn't that 318 00:17:02,900 --> 00:17:06,040 just look like a ripple in a circle going out 319 00:17:06,040 --> 00:17:07,372 from the center? 320 00:17:07,372 --> 00:17:09,079 MAGDALENA TODA: Yeah, it will be ripples. 321 00:17:09,079 --> 00:17:10,680 STUDENT: Just like a [INAUDIBLE] from an epicenter 322 00:17:10,680 --> 00:17:11,960 going outwards [INAUDIBLE]. 323 00:17:11,960 --> 00:17:16,050 MAGDALENA TODA: And I think-- yes, we managed to-- you 324 00:17:16,050 --> 00:17:19,896 have a concentric image, right? 325 00:17:19,896 --> 00:17:20,480 STUDENT: Yeah. 326 00:17:20,480 --> 00:17:22,369 MAGDALENA TODA: Like those ripples, exactly like-- 327 00:17:22,369 --> 00:17:24,036 STUDENT: So that's what that looks like? 328 00:17:24,036 --> 00:17:26,550 MAGDALENA TODA: --when you throw a stone into the water, 329 00:17:26,550 --> 00:17:27,680 this kind of wave. 330 00:17:27,680 --> 00:17:30,730 But it's infinitesimally close. 331 00:17:30,730 --> 00:17:32,910 It's like acting weird. 332 00:17:32,910 --> 00:17:37,240 But then it sort of shrinks here. 333 00:17:37,240 --> 00:17:40,786 And that-- it imposes the limit 0. 334 00:17:40,786 --> 00:17:43,230 How come this goes to 0, you say? 335 00:17:43,230 --> 00:17:46,150 Well, Magdalena, this guy is crazy, right? 336 00:17:46,150 --> 00:17:49,155 Sine of 1/x goes between minus 1 and 1 337 00:17:49,155 --> 00:17:51,500 infinitely many times as I go close, close, 338 00:17:51,500 --> 00:17:57,930 closer and closer, more rapidly, more and more rapidly close 339 00:17:57,930 --> 00:17:58,890 to 0. 340 00:17:58,890 --> 00:18:00,535 This will oscillate more rapidly, 341 00:18:00,535 --> 00:18:03,190 more rapidly, and more rapidly. 342 00:18:03,190 --> 00:18:04,630 This is crazy, right? 343 00:18:04,630 --> 00:18:07,690 How does this guy, x-- how is this guy taming this guy? 344 00:18:07,690 --> 00:18:10,445 STUDENT: Because as 0 [INAUDIBLE]. 345 00:18:10,445 --> 00:18:12,570 Something really small times something [INAUDIBLE]. 346 00:18:12,570 --> 00:18:14,069 MAGDALENA TODA: Something very small 347 00:18:14,069 --> 00:18:17,900 that shrinks to 0 times something bounded. 348 00:18:17,900 --> 00:18:20,880 Ryan brought the main idea. 349 00:18:20,880 --> 00:18:25,160 If something goes strongly to 0, and that multiplies something 350 00:18:25,160 --> 00:18:28,490 that's bounded, bounded by a finite number, 351 00:18:28,490 --> 00:18:31,069 the whole problem will go to 0. 352 00:18:31,069 --> 00:18:32,961 Actually, you can prove that as a theorem. 353 00:18:32,961 --> 00:18:34,860 And some of you did. 354 00:18:34,860 --> 00:18:36,630 In most honors classes unfortunately, 355 00:18:36,630 --> 00:18:39,100 epsilon delta was not covered. 356 00:18:39,100 --> 00:18:43,350 So let's see how we prove this with epsilon delta. 357 00:18:43,350 --> 00:18:45,120 And, oh, my god. 358 00:18:45,120 --> 00:18:52,825 Many of you read from the book and may be able to help me. 359 00:18:52,825 --> 00:19:00,100 So what am I supposed to show with epsilon delta? 360 00:19:00,100 --> 00:19:09,860 The limit of x squared plus y squared sine of 1 over x 361 00:19:09,860 --> 00:19:14,560 squared plus y squared is 0 as I approach the origin 362 00:19:14,560 --> 00:19:19,735 with my pair, couple, x, y, which can go any one path that 363 00:19:19,735 --> 00:19:20,340 approaches 0. 364 00:19:20,340 --> 00:19:23,780 365 00:19:23,780 --> 00:19:27,575 So you say, oh, well, Magdalena, the Ryan principle-- this 366 00:19:27,575 --> 00:19:29,070 is the Ryan theorem. 367 00:19:29,070 --> 00:19:32,180 It's the same because this guy will be 368 00:19:32,180 --> 00:19:34,100 bounded between minus 1 and 1. 369 00:19:34,100 --> 00:19:37,630 I multiplied with a guy that very determinedly goes 370 00:19:37,630 --> 00:19:39,540 to 0 very strongly. 371 00:19:39,540 --> 00:19:41,160 And he knows where he's going. 372 00:19:41,160 --> 00:19:44,020 x squared plus y squared says, I know what I'm doing. 373 00:19:44,020 --> 00:19:45,880 I'm not going to change my mind. 374 00:19:45,880 --> 00:19:49,480 This is like the guy who changes his major too many times. 375 00:19:49,480 --> 00:19:52,210 And this guy knows what he's doing. 376 00:19:52,210 --> 00:19:54,920 He's going there, and he's a polynomial, goes to 0, 377 00:19:54,920 --> 00:19:56,290 0 very rapidly. 378 00:19:56,290 --> 00:20:00,520 Now it's clear what happens intuitively. 379 00:20:00,520 --> 00:20:02,910 But I'm a mathematician. 380 00:20:02,910 --> 00:20:07,040 And if I don't publish my proof, my article 381 00:20:07,040 --> 00:20:12,675 will be very nicely rejected by all the serious journals 382 00:20:12,675 --> 00:20:13,970 on the market. 383 00:20:13,970 --> 00:20:17,470 This is how it goes in mathematics. 384 00:20:17,470 --> 00:20:19,390 Even before journals existed, mathematicians 385 00:20:19,390 --> 00:20:23,110 had to show a rigorous proof of their work, 386 00:20:23,110 --> 00:20:25,720 of their conjecture. 387 00:20:25,720 --> 00:20:26,920 OK. 388 00:20:26,920 --> 00:20:35,080 So I go, for every epsilon positive, no matter how small, 389 00:20:35,080 --> 00:20:41,040 there must exist a delta positive, which 390 00:20:41,040 --> 00:20:51,990 depends on epsilon-- that depends on epsilon-- such that 391 00:20:51,990 --> 00:20:58,550 as soon as-- how did we write the distance? 392 00:20:58,550 --> 00:21:01,820 I'll write the distance again because I'm lazy. 393 00:21:01,820 --> 00:21:05,590 The distance between the point x, y and the origin 394 00:21:05,590 --> 00:21:07,931 is less than delta. 395 00:21:07,931 --> 00:21:16,720 It follows that the absolute value-- 396 00:21:16,720 --> 00:21:24,330 these are all real numbers-- of f of x, y or g of x, 397 00:21:24,330 --> 00:21:27,456 y-- g of x, y is the extension. 398 00:21:27,456 --> 00:21:32,150 399 00:21:32,150 --> 00:21:36,350 f of x, y minus 0, which I claim to be the limit, 400 00:21:36,350 --> 00:21:39,135 will be less than epsilon. 401 00:21:39,135 --> 00:21:40,500 So you go, oh, my god. 402 00:21:40,500 --> 00:21:42,830 What is this woman doing? 403 00:21:42,830 --> 00:21:43,820 It's not hard. 404 00:21:43,820 --> 00:21:45,980 I need your help though. 405 00:21:45,980 --> 00:21:48,540 I need your help to do that. 406 00:21:48,540 --> 00:21:53,100 So it's hard to see how you should-- you take any epsilon. 407 00:21:53,100 --> 00:21:58,300 You pick your favorite epsilon, infinitesimally small, 408 00:21:58,300 --> 00:22:01,120 any small number, but then you go, but then I 409 00:22:01,120 --> 00:22:03,400 have to show this delta exists. 410 00:22:03,400 --> 00:22:06,590 You have to grab that delta and say, you are my delta. 411 00:22:06,590 --> 00:22:08,990 You cannot escape me. 412 00:22:08,990 --> 00:22:10,926 I tell you who you are. 413 00:22:10,926 --> 00:22:13,840 And that's the hardest part in here, 414 00:22:13,840 --> 00:22:18,240 figuring out who that delta must be as a function of epsilon. 415 00:22:18,240 --> 00:22:19,060 Is that hard? 416 00:22:19,060 --> 00:22:21,320 How do you build such a construction? 417 00:22:21,320 --> 00:22:26,690 First of all, understand what proof. 418 00:22:26,690 --> 00:22:30,385 "Choose any positive epsilon." 419 00:22:30,385 --> 00:22:32,885 Then forget about him, because he's your friend, 420 00:22:32,885 --> 00:22:36,400 and he's going to do whatever you want to do with him. 421 00:22:36,400 --> 00:22:40,120 Delta, chasing after delta is going 422 00:22:40,120 --> 00:22:42,390 to be a little bit harder. 423 00:22:42,390 --> 00:22:56,010 "Chasing after delta with that property." 424 00:22:56,010 --> 00:22:58,430 Dot, dot, dot, dot, dot. 425 00:22:58,430 --> 00:22:59,700 What is this distance? 426 00:22:59,700 --> 00:23:01,950 You guys have helped me last time, 427 00:23:01,950 --> 00:23:04,670 you cannot let me down now. 428 00:23:04,670 --> 00:23:08,350 So as soon as this distance, your gradient distance 429 00:23:08,350 --> 00:23:10,540 is less than delta, you must have 430 00:23:10,540 --> 00:23:13,312 that f of x, y [INAUDIBLE]. 431 00:23:13,312 --> 00:23:15,170 Could you tell me what that would be? 432 00:23:15,170 --> 00:23:16,170 It was Euclidean, right? 433 00:23:16,170 --> 00:23:21,670 So I had squared root of-- did I? 434 00:23:21,670 --> 00:23:30,396 Square root of x minus 0 squared plus y minus 0 squared. 435 00:23:30,396 --> 00:23:33,170 You say, but that's silly, Magdalena. 436 00:23:33,170 --> 00:23:37,536 So you have to write it down like that? 437 00:23:37,536 --> 00:23:38,970 STUDENT: It's the [INAUDIBLE]. 438 00:23:38,970 --> 00:23:40,410 MAGDALENA TODA: Huh? 439 00:23:40,410 --> 00:23:42,290 Yeah. 440 00:23:42,290 --> 00:23:47,174 So square root of this plus square root of that 441 00:23:47,174 --> 00:23:53,030 plus then delta, that means what? 442 00:23:53,030 --> 00:24:00,351 If and only if x squared plus y squared is less than delta 443 00:24:00,351 --> 00:24:00,850 squared. 444 00:24:00,850 --> 00:24:08,160 445 00:24:08,160 --> 00:24:11,110 And what do I want to do, what do I want to build? 446 00:24:11,110 --> 00:24:15,090 447 00:24:15,090 --> 00:24:19,030 So we are thinking how to set up all this thing. 448 00:24:19,030 --> 00:24:21,209 How to choose the delta. 449 00:24:21,209 --> 00:24:23,061 How to choose the delta. 450 00:24:23,061 --> 00:24:25,830 451 00:24:25,830 --> 00:24:28,350 OK, so what do I-- what am I after? 452 00:24:28,350 --> 00:24:34,253 "I am after having" double dot. 453 00:24:34,253 --> 00:24:39,830 F of x, y must be Mr. Ugly. 454 00:24:39,830 --> 00:24:40,820 This one. 455 00:24:40,820 --> 00:24:46,480 So absolute value of x squared plus y squared, sine of 1 456 00:24:46,480 --> 00:24:51,241 over x squared plus y squared minus 0. 457 00:24:51,241 --> 00:24:51,740 Duh. 458 00:24:51,740 --> 00:24:55,170 I'm not going to write it. 459 00:24:55,170 --> 00:24:59,260 We all know what that means. 460 00:24:59,260 --> 00:25:00,090 Less than epsilon. 461 00:25:00,090 --> 00:25:05,680 This is what must follow as a conclusion. 462 00:25:05,680 --> 00:25:12,360 This is what must follow, must happen. 463 00:25:12,360 --> 00:25:13,334 Must happen. 464 00:25:13,334 --> 00:25:16,260 465 00:25:16,260 --> 00:25:17,630 Now I'm getting excited. 466 00:25:17,630 --> 00:25:18,130 Why? 467 00:25:18,130 --> 00:25:21,050 Because I am thinking. 468 00:25:21,050 --> 00:25:23,040 I started thinking. 469 00:25:23,040 --> 00:25:26,420 Once I started thinking, I'm dangerous, man. 470 00:25:26,420 --> 00:25:31,530 So here sine of 1 over x squared plus y squared is your friend. 471 00:25:31,530 --> 00:25:34,210 Why is that your friend? 472 00:25:34,210 --> 00:25:37,250 Sine of 1 over x squared plus y squared, this 473 00:25:37,250 --> 00:25:39,115 is always an absolute value. 474 00:25:39,115 --> 00:25:42,977 The absolute value of that is always less than 1. 475 00:25:42,977 --> 00:25:43,476 OK? 476 00:25:43,476 --> 00:25:45,290 STUDENT: Can't it be 4? 477 00:25:45,290 --> 00:25:50,280 MAGDALENA TODA: So-- so-- so what 478 00:25:50,280 --> 00:25:54,940 shall I take in terms of delta-- this is my question. 479 00:25:54,940 --> 00:25:57,400 What shall I take in terms of delta? 480 00:25:57,400 --> 00:26:03,590 "Delta equals 1 as a function of epsilon 481 00:26:03,590 --> 00:26:20,160 in order to have the conclusion satisfied." 482 00:26:20,160 --> 00:26:20,920 You say, OK. 483 00:26:20,920 --> 00:26:24,610 It's enough to choose delta like that function of epsilon, 484 00:26:24,610 --> 00:26:28,840 and I'm done, because then everything will be fine. 485 00:26:28,840 --> 00:26:33,510 So you chose your own epsilon, positive, small, or God 486 00:26:33,510 --> 00:26:34,480 gave you an epsilon. 487 00:26:34,480 --> 00:26:37,040 You don't care how you got the epsilon. 488 00:26:37,040 --> 00:26:38,423 The epsilon is arbitrary. 489 00:26:38,423 --> 00:26:40,910 You pick positive and small. 490 00:26:40,910 --> 00:26:44,610 Now, it's up to you to find delta. 491 00:26:44,610 --> 00:26:48,820 So what delta would satisfy everything? 492 00:26:48,820 --> 00:26:50,940 What delta would be good enough-- 493 00:26:50,940 --> 00:26:52,830 you don't care for all the good-- 494 00:26:52,830 --> 00:26:54,550 it's like when you get married. 495 00:26:54,550 --> 00:26:57,740 Do you care for all the people who'd match you? 496 00:26:57,740 --> 00:27:00,596 Hopefully not, because then you would probably 497 00:27:00,596 --> 00:27:05,416 have too large of a pool, and it's hard to choose. 498 00:27:05,416 --> 00:27:13,285 You only need one that satisfies that assumption, that satisfies 499 00:27:13,285 --> 00:27:14,974 all the conditions you have. 500 00:27:14,974 --> 00:27:18,780 So what is the delta that satisfies all the conditions 501 00:27:18,780 --> 00:27:20,047 that I have? 502 00:27:20,047 --> 00:27:20,880 [INTERPOSING VOICES] 503 00:27:20,880 --> 00:27:22,270 MAGDALENA TODA: [INAUDIBLE]. 504 00:27:22,270 --> 00:27:22,960 Who? 505 00:27:22,960 --> 00:27:25,350 [INTERPOSING VOICES] 506 00:27:25,350 --> 00:27:27,855 MAGDALENA TODA: For example, delta equals epsilon. 507 00:27:27,855 --> 00:27:28,785 Would that satisfy? 508 00:27:28,785 --> 00:27:31,984 509 00:27:31,984 --> 00:27:33,860 Well, let's see. 510 00:27:33,860 --> 00:27:37,410 If I take delta to be epsilon, then x 511 00:27:37,410 --> 00:27:40,410 squared plus y squared would be less than epsilon squared. 512 00:27:40,410 --> 00:27:47,331 Now the question is is epsilon squared less than epsilon? 513 00:27:47,331 --> 00:27:48,510 Not always. 514 00:27:48,510 --> 00:27:52,920 If epsilon is between 0 and 1, then epsilon squared 515 00:27:52,920 --> 00:27:54,200 is less then epsilon. 516 00:27:54,200 --> 00:27:59,200 But if I choose epsilon to be greater than 1, 517 00:27:59,200 --> 00:28:00,220 then oh, my God. 518 00:28:00,220 --> 00:28:02,740 Then if it's greater than 1, then epsilon squared 519 00:28:02,740 --> 00:28:06,790 is greater than 1-- greater than it. 520 00:28:06,790 --> 00:28:14,700 So what if I choose delta to be what? 521 00:28:14,700 --> 00:28:18,652 522 00:28:18,652 --> 00:28:19,595 STUDENT: 0? 523 00:28:19,595 --> 00:28:20,720 MAGDALENA TODA: No, no, no. 524 00:28:20,720 --> 00:28:22,090 Delta cannot be 0. 525 00:28:22,090 --> 00:28:26,330 So delta-- look, there exists delta strictly bigger than 0, 526 00:28:26,330 --> 00:28:28,690 that depends on epsilon. 527 00:28:28,690 --> 00:28:33,673 Maybe if epsilon is very small, in a way Alexander was right. 528 00:28:33,673 --> 00:28:37,350 But the delta [INAUDIBLE], we don't go with epsilon 529 00:28:37,350 --> 00:28:38,280 greater than 1. 530 00:28:38,280 --> 00:28:39,000 Come on. 531 00:28:39,000 --> 00:28:39,500 Be serious. 532 00:28:39,500 --> 00:28:42,300 Epsilon is always between 0 and 1. 533 00:28:42,300 --> 00:28:44,603 I mean, it's a lot smaller than that. 534 00:28:44,603 --> 00:28:46,640 It's infinitesimal small. 535 00:28:46,640 --> 00:28:49,480 So in the end, yes, in that case epsilon squared 536 00:28:49,480 --> 00:28:52,610 would be less than epsilon, which would be OK for us 537 00:28:52,610 --> 00:28:54,590 and that would be fine. 538 00:28:54,590 --> 00:28:56,150 OK? 539 00:28:56,150 --> 00:28:58,380 So that would be a possibility to say, hey, 540 00:28:58,380 --> 00:29:01,080 since epsilon-- Alexander, if you write that as a proof 541 00:29:01,080 --> 00:29:01,815 I'll be OK. 542 00:29:01,815 --> 00:29:04,900 You say, I took my epsilon to be a very small number, 543 00:29:04,900 --> 00:29:07,020 so anyway it's going to be less than 1. 544 00:29:07,020 --> 00:29:09,190 So epsilon squared is less than epsilon. 545 00:29:09,190 --> 00:29:14,090 So when I take delta to be epsilon, 546 00:29:14,090 --> 00:29:18,210 for sure this guy will be less than epsilon squared, which 547 00:29:18,210 --> 00:29:21,148 is less than epsilon, so I'm satisfied. 548 00:29:21,148 --> 00:29:22,615 I'll give you a 100%. 549 00:29:22,615 --> 00:29:24,082 I'm happy. 550 00:29:24,082 --> 00:29:25,152 Is that the only way? 551 00:29:25,152 --> 00:29:26,527 STUDENT: But what about the sine? 552 00:29:26,527 --> 00:29:27,505 What about [INAUDIBLE]. 553 00:29:27,505 --> 00:29:28,483 STUDENT: Yeah. 554 00:29:28,483 --> 00:29:30,108 MAGDALENA TODA: So this doesn't matter. 555 00:29:30,108 --> 00:29:32,395 Let me write it down. 556 00:29:32,395 --> 00:29:39,730 So note that x squared plus y squared sine of 1 557 00:29:39,730 --> 00:29:42,650 over x squared plus y square would always 558 00:29:42,650 --> 00:29:46,380 be less than absolute value of x squared 559 00:29:46,380 --> 00:29:49,970 plus y, which is positive. 560 00:29:49,970 --> 00:29:52,145 Why is that? 561 00:29:52,145 --> 00:29:53,200 Is this true? 562 00:29:53,200 --> 00:29:54,080 Yeah. 563 00:29:54,080 --> 00:29:55,441 Why is that? 564 00:29:55,441 --> 00:29:58,150 STUDENT: Because the sine can only be one of these negatives. 565 00:29:58,150 --> 00:30:00,445 MAGDALENA TODA: So in absolute value, 566 00:30:00,445 --> 00:30:05,780 sine of 1 over x squared plus y squared is always less than 1. 567 00:30:05,780 --> 00:30:08,525 STUDENT: Can't it equal 1? 568 00:30:08,525 --> 00:30:11,885 MAGDALENA TODA: Well, when does it equal 1? 569 00:30:11,885 --> 00:30:14,310 STUDENT: Wouldn't it be x squared plus y squared equals 1 570 00:30:14,310 --> 00:30:15,647 [INAUDIBLE]? 571 00:30:15,647 --> 00:30:17,230 MAGDALENA TODA: Less than or equal to. 572 00:30:17,230 --> 00:30:18,480 For some values it will. 573 00:30:18,480 --> 00:30:19,160 STUDENT: Yeah. 574 00:30:19,160 --> 00:30:19,659 OK. 575 00:30:19,659 --> 00:30:21,870 MAGDALENA TODA: Now, will that be a problem with us? 576 00:30:21,870 --> 00:30:22,120 No. 577 00:30:22,120 --> 00:30:23,060 Let's put it here. 578 00:30:23,060 --> 00:30:27,370 Less than or equal to x squared plus y squared, which 579 00:30:27,370 --> 00:30:35,465 has to be less than epsilon if and only if-- well, 580 00:30:35,465 --> 00:30:38,780 if delta is what? 581 00:30:38,780 --> 00:30:41,240 So, again, Alexander said, well, but if I take delta 582 00:30:41,240 --> 00:30:42,820 to be epsilon, I'm done. 583 00:30:42,820 --> 00:30:45,760 584 00:30:45,760 --> 00:30:46,740 STUDENT: [INAUDIBLE]. 585 00:30:46,740 --> 00:30:49,690 MAGDALENA TODA: How about square root? 586 00:30:49,690 --> 00:30:52,150 Can I take delta to be square root of epsilon. 587 00:30:52,150 --> 00:30:53,541 STUDENT: That's what I said. 588 00:30:53,541 --> 00:30:54,332 MAGDALENA TODA: No. 589 00:30:54,332 --> 00:30:55,694 You said epsilon. 590 00:30:55,694 --> 00:30:57,319 STUDENT: I said square root of epsilon. 591 00:30:57,319 --> 00:30:58,270 MAGDALENA TODA: OK. 592 00:30:58,270 --> 00:31:01,290 If delta is square root of epsilon, 593 00:31:01,290 --> 00:31:05,280 then everything will be perfect and it will be a perfect match. 594 00:31:05,280 --> 00:31:05,997 In what case? 595 00:31:05,997 --> 00:31:07,705 STUDENT: If epsilon is in between 0 and 1 596 00:31:07,705 --> 00:31:10,130 and if delta is equal to bigger than epsilon. 597 00:31:10,130 --> 00:31:13,060 598 00:31:13,060 --> 00:31:17,740 MAGDALENA TODA: So that's exactly the same assumption. 599 00:31:17,740 --> 00:31:22,480 Epsilon should be made in less than. 600 00:31:22,480 --> 00:31:24,327 STUDENT: But I thought delta was supposed 601 00:31:24,327 --> 00:31:25,910 to be less than epsilon in every case. 602 00:31:25,910 --> 00:31:29,340 So if epsilon is between 0 and 1, the square root of epsilon 603 00:31:29,340 --> 00:31:31,800 is going to be [INAUDIBLE]. 604 00:31:31,800 --> 00:31:38,330 MAGDALENA TODA: So when both of them are small, 605 00:31:38,330 --> 00:31:45,040 delta squared will be-- if I take delta-- so take delta 606 00:31:45,040 --> 00:31:47,698 to be square root of epsilon. 607 00:31:47,698 --> 00:31:50,118 STUDENT: Then anything less than 1 and greater than 0, 608 00:31:50,118 --> 00:31:51,784 epsilon would be great than [INAUDIBLE]. 609 00:31:51,784 --> 00:31:54,958 MAGDALENA TODA: "Delta to be square root of epsilon, 610 00:31:54,958 --> 00:32:01,570 then x squared plus y squared less than delta squared equals 611 00:32:01,570 --> 00:32:03,780 epsilon." 612 00:32:03,780 --> 00:32:11,805 Then x squared plus y squared sine of 1 613 00:32:11,805 --> 00:32:14,990 over x squared plus y squared less than 614 00:32:14,990 --> 00:32:17,440 or equal to x squared plus y squared. 615 00:32:17,440 --> 00:32:19,155 I dont' need the absolute value. 616 00:32:19,155 --> 00:32:20,380 I can [INAUDIBLE]. 617 00:32:20,380 --> 00:32:23,320 Less than epsilon [INAUDIBLE]. 618 00:32:23,320 --> 00:32:24,110 Qed. 619 00:32:24,110 --> 00:32:26,050 STUDENT: Well, but you told us delta 620 00:32:26,050 --> 00:32:27,505 has to be less than epsilon. 621 00:32:27,505 --> 00:32:28,475 Well, if-- 622 00:32:28,475 --> 00:32:31,390 MAGDALENA TODA: No, I didn't say that. 623 00:32:31,390 --> 00:32:35,325 I didn't say that delta has to be less than epsilon. 624 00:32:35,325 --> 00:32:35,825 Absolutely-- 625 00:32:35,825 --> 00:32:36,408 STUDENT: Yeah. 626 00:32:36,408 --> 00:32:38,759 You said for all the values of epsilon greater than 0, 627 00:32:38,759 --> 00:32:42,280 there's a value of delta that is greater than 0 that [INAUDIBLE] 628 00:32:42,280 --> 00:32:45,648 such that as soon as the distance between is less than 629 00:32:45,648 --> 00:32:46,959 delta-- I don't remember what-- 630 00:32:46,959 --> 00:32:48,250 MAGDALENA TODA: OK, so, again-- 631 00:32:48,250 --> 00:32:50,166 STUDENT: Such that the distance is less than-- 632 00:32:50,166 --> 00:32:52,255 MAGDALENA TODA: So again, for epsilon positive, 633 00:32:52,255 --> 00:32:56,600 there is a delta positive, very small. 634 00:32:56,600 --> 00:32:58,740 Very small means very small, OK? 635 00:32:58,740 --> 00:33:01,440 I'm not threatened by-- what? 636 00:33:01,440 --> 00:33:04,688 For epsilon greater than 0, very small, 637 00:33:04,688 --> 00:33:07,128 there is a delta greater than 0, very small, 638 00:33:07,128 --> 00:33:10,544 which depends on epsilon-- I didn't say it cannot be equal 639 00:33:10,544 --> 00:33:21,100 to epsilon-- that depends on epsilon such that whenever x, 640 00:33:21,100 --> 00:33:30,012 y is within delta distance from origin, 641 00:33:30,012 --> 00:33:45,170 [INAUDIBLE] that f of x, y is within epsilon of from l. 642 00:33:45,170 --> 00:33:47,840 643 00:33:47,840 --> 00:33:48,340 All right? 644 00:33:48,340 --> 00:33:52,720 And now I will actually give you another example where 645 00:33:52,720 --> 00:33:55,970 maybe delta will be epsilon. 646 00:33:55,970 --> 00:33:59,430 And let me challenge you with another problem that's 647 00:33:59,430 --> 00:34:00,530 not hard. 648 00:34:00,530 --> 00:34:01,295 OK? 649 00:34:01,295 --> 00:34:03,650 So let me give you the function g 650 00:34:03,650 --> 00:34:16,460 of x, y equals x sine of 1 over y as x, y. 651 00:34:16,460 --> 00:34:19,150 652 00:34:19,150 --> 00:34:29,782 y is equal [? to delta 0. ?] And let's say 0 for the rest. 653 00:34:29,782 --> 00:34:35,500 654 00:34:35,500 --> 00:34:48,510 Can you show-- can you check if g is continuous at 0, 0? 655 00:34:48,510 --> 00:34:55,510 656 00:34:55,510 --> 00:34:58,510 This is one of the problems in your book. 657 00:34:58,510 --> 00:35:02,300 So how do you check that with epsilon delta? 658 00:35:02,300 --> 00:35:04,130 Again, we recite the poetry. 659 00:35:04,130 --> 00:35:05,555 We have to say that. 660 00:35:05,555 --> 00:35:11,950 "For every epsilon positive, small, very small, 661 00:35:11,950 --> 00:35:16,000 there is a delta positive that depends 662 00:35:16,000 --> 00:35:33,830 on epsilon, such that as soon as--" how is the distance? 663 00:35:33,830 --> 00:35:42,424 Square root of x squared plus y squared is less than delta. 664 00:35:42,424 --> 00:35:46,861 This is the distance between point and origin. 665 00:35:46,861 --> 00:36:09,260 "It follows that absolute value of x sine of 1 over y minus--" 666 00:36:09,260 --> 00:36:12,247 so practically x, y no 0. 667 00:36:12,247 --> 00:36:16,223 x, y different from 0. 668 00:36:16,223 --> 00:36:17,720 OK? 669 00:36:17,720 --> 00:36:21,720 I"m careful here, because if y is 0, then I blow up. 670 00:36:21,720 --> 00:36:23,080 And I don't want to blow up. 671 00:36:23,080 --> 00:36:25,850 So x sine of 1 over y minus who? 672 00:36:25,850 --> 00:36:30,810 Minus 0 is less than epsilon. 673 00:36:30,810 --> 00:36:32,850 So now you're thinking, OK, you want me 674 00:36:32,850 --> 00:36:34,760 to prove there is such a delta? 675 00:36:34,760 --> 00:36:35,830 Yes. 676 00:36:35,830 --> 00:36:37,060 That depends on epsilon? 677 00:36:37,060 --> 00:36:38,850 Yes. 678 00:36:38,850 --> 00:36:40,430 And what would that delta be? 679 00:36:40,430 --> 00:36:43,710 The simplest choice you can have in this case. 680 00:36:43,710 --> 00:36:45,080 So you go, oh, my God. 681 00:36:45,080 --> 00:36:46,070 How do I do that? 682 00:36:46,070 --> 00:36:48,470 You have to always think backwards. 683 00:36:48,470 --> 00:36:58,790 So "we need to satisfy absolute value of x sine of 1 684 00:36:58,790 --> 00:37:02,430 over y less than epsilon." 685 00:37:02,430 --> 00:37:05,730 Is this hard? 686 00:37:05,730 --> 00:37:10,295 What is your advantage here? 687 00:37:10,295 --> 00:37:13,570 Do you have any advantage? 688 00:37:13,570 --> 00:37:19,860 Remark absolute value of x sine of 1 over y 689 00:37:19,860 --> 00:37:22,690 is smaller than who? 690 00:37:22,690 --> 00:37:26,670 Smaller than the product of absolute values. 691 00:37:26,670 --> 00:37:27,580 Say it again? 692 00:37:27,580 --> 00:37:28,430 Yes? 693 00:37:28,430 --> 00:37:32,005 STUDENT: But, like, for example, the only condition 694 00:37:32,005 --> 00:37:35,290 for that equation is that y must not be equal to 0. 695 00:37:35,290 --> 00:37:38,520 What if you used another point for x? 696 00:37:38,520 --> 00:37:43,170 Would the answer for delta be different? 697 00:37:43,170 --> 00:37:45,240 MAGDALENA TODA: Well, x is-- you can 698 00:37:45,240 --> 00:37:49,000 choose-- you were right here. 699 00:37:49,000 --> 00:37:52,990 You can say, OK, can you be more restrictive, Magdelena, 700 00:37:52,990 --> 00:37:58,710 and say, for every point of the type x equals 0 701 00:37:58,710 --> 00:38:01,470 and y not 0, it's still OK? 702 00:38:01,470 --> 00:38:03,490 Yes. 703 00:38:03,490 --> 00:38:07,000 So you could be a professional mathematician. 704 00:38:07,000 --> 00:38:14,322 So practically all I care about is x, y in the disk. 705 00:38:14,322 --> 00:38:15,410 What disk? 706 00:38:15,410 --> 00:38:16,810 What is this disk? 707 00:38:16,810 --> 00:38:24,150 Disk of radius 0 when-- what is the radius? 708 00:38:24,150 --> 00:38:31,930 Delta-- such that your y should not be 0. 709 00:38:31,930 --> 00:38:35,990 So a more rigorous point would be 710 00:38:35,990 --> 00:38:38,630 like take all the couples that are 711 00:38:38,630 --> 00:38:43,600 in this small disk of radius delta, 712 00:38:43,600 --> 00:38:46,020 except for those where y is 0. 713 00:38:46,020 --> 00:38:48,650 So what do you actually remove? 714 00:38:48,650 --> 00:38:54,860 You remove this stinking line. 715 00:38:54,860 --> 00:39:01,080 But everybody else in this disk, every couple in this disk 716 00:39:01,080 --> 00:39:03,865 should be happy, should be analyzed 717 00:39:03,865 --> 00:39:06,080 as part of this thread. 718 00:39:06,080 --> 00:39:08,100 Right? 719 00:39:08,100 --> 00:39:09,070 OK. 720 00:39:09,070 --> 00:39:13,180 x sine of 1 over y less than-- is that true? 721 00:39:13,180 --> 00:39:16,120 Is that less than the absolute value of x? 722 00:39:16,120 --> 00:39:16,930 STUDENT: Yeah. 723 00:39:16,930 --> 00:39:17,846 MAGDALENA TODA: Right. 724 00:39:17,846 --> 00:39:20,950 So it should be-- less than should be made 725 00:39:20,950 --> 00:39:23,570 should be less than epsilon. 726 00:39:23,570 --> 00:39:27,040 When is this happening on that occasion? 727 00:39:27,040 --> 00:39:28,320 If I take delta-- meh? 728 00:39:28,320 --> 00:39:29,570 STUDENT: When delta's epsilon. 729 00:39:29,570 --> 00:39:31,310 MAGDALENA TODA: So if I take-- very good. 730 00:39:31,310 --> 00:39:35,570 So Alex saw that, hey, Magdelena, your proof is over. 731 00:39:35,570 --> 00:39:37,700 And I mean it's over. 732 00:39:37,700 --> 00:39:42,990 Take delta, which is delta of epsilon, to be epsilon. 733 00:39:42,990 --> 00:39:44,350 You're done. 734 00:39:44,350 --> 00:39:45,520 Why? 735 00:39:45,520 --> 00:39:47,590 Let me explain what Alex wants, because he 736 00:39:47,590 --> 00:39:50,280 doesn't want to explain much, but it's not his job. 737 00:39:50,280 --> 00:39:51,260 He's not your teacher. 738 00:39:51,260 --> 00:39:51,920 Right? 739 00:39:51,920 --> 00:39:54,346 So why is this working? 740 00:39:54,346 --> 00:40:02,850 Because in this case, note that if I take delta 741 00:40:02,850 --> 00:40:05,650 to be exactly epsilon, what's going to happen? 742 00:40:05,650 --> 00:40:08,620 743 00:40:08,620 --> 00:40:13,760 x, Mr. x, could be positive or negative. 744 00:40:13,760 --> 00:40:15,950 See, x could be positive or negative. 745 00:40:15,950 --> 00:40:18,810 Let's take this guy and protect him in absolute value. 746 00:40:18,810 --> 00:40:23,350 He's always less than square root of x square plus y 747 00:40:23,350 --> 00:40:25,650 squared. 748 00:40:25,650 --> 00:40:27,200 Why is that, guys? 749 00:40:27,200 --> 00:40:30,730 STUDENT: Because y can't be 0. 750 00:40:30,730 --> 00:40:34,482 MAGDALENA TODA: So this is-- square it in your mind. 751 00:40:34,482 --> 00:40:36,690 You got x squared less than x squared plus y squared. 752 00:40:36,690 --> 00:40:39,000 So this is always true. 753 00:40:39,000 --> 00:40:40,640 Always satisfied. 754 00:40:40,640 --> 00:40:44,940 But we chose this to be less than delta, 755 00:40:44,940 --> 00:40:49,450 and if we choose delta to be epsilon, that's our choice. 756 00:40:49,450 --> 00:40:54,310 So God gave us the epsilon, but delta is our choice, 757 00:40:54,310 --> 00:40:57,090 because you have to prove you can do something 758 00:40:57,090 --> 00:40:57,910 with your life. 759 00:40:57,910 --> 00:40:58,410 Right? 760 00:40:58,410 --> 00:41:00,700 So delta equals epsilon. 761 00:41:00,700 --> 00:41:02,670 If you take delta equals epsilon, 762 00:41:02,670 --> 00:41:06,440 then you're done, because in that case absolute value 763 00:41:06,440 --> 00:41:11,980 of x is less than epsilon, and your conclusion, which is this, 764 00:41:11,980 --> 00:41:13,690 was satisfied. 765 00:41:13,690 --> 00:41:16,680 Now, if a student is really smart-- 766 00:41:16,680 --> 00:41:20,650 one time I had a student, I gave him this proof. 767 00:41:20,650 --> 00:41:22,380 That was several years ago in honors, 768 00:41:22,380 --> 00:41:24,620 because we don't do epsilon delta in non-honors. 769 00:41:24,620 --> 00:41:28,200 And we very rarely do it in honors as well. 770 00:41:28,200 --> 00:41:31,140 His proof consisted of this. 771 00:41:31,140 --> 00:41:34,060 Considering the fact that absolute value of sine 772 00:41:34,060 --> 00:41:38,300 is less than 1, if I take delta to be epsilon, 773 00:41:38,300 --> 00:41:39,870 that is sufficient. 774 00:41:39,870 --> 00:41:41,775 I'm done. 775 00:41:41,775 --> 00:41:44,470 And of course I gave him 100%, because this 776 00:41:44,470 --> 00:41:46,030 is the essence of the proof. 777 00:41:46,030 --> 00:41:48,090 He didn't show any details. 778 00:41:48,090 --> 00:41:52,180 And I thought, this is the kind of guy who is great. 779 00:41:52,180 --> 00:41:55,920 He's very smart, but he's not going to make a good teacher. 780 00:41:55,920 --> 00:41:59,380 So he's probably going to be the next researcher, 781 00:41:59,380 --> 00:42:04,810 the next astronaut, the next something else, but not-- 782 00:42:04,810 --> 00:42:11,040 And then, years later, he took advanced calculus. 783 00:42:11,040 --> 00:42:13,635 He graduated with a graduate degree 784 00:42:13,635 --> 00:42:17,600 in three years sponsored by the Air Force. 785 00:42:17,600 --> 00:42:20,750 And he works right now for the Air Force. 786 00:42:20,750 --> 00:42:24,420 He came out dressed as a captain. 787 00:42:24,420 --> 00:42:28,840 He came and gave a talk this year at Tech in a conference-- 788 00:42:28,840 --> 00:42:29,770 he was rushed. 789 00:42:29,770 --> 00:42:32,150 I mean, if I talk like that, my student 790 00:42:32,150 --> 00:42:33,810 wouldn't be able to follow me. 791 00:42:33,810 --> 00:42:38,320 But he was the same brilliant student that I remember. 792 00:42:38,320 --> 00:42:46,280 So he's working on some very important top secret projects. 793 00:42:46,280 --> 00:42:48,900 Very intelligent guy. 794 00:42:48,900 --> 00:42:52,520 And every now and than going to give talks at conferences. 795 00:42:52,520 --> 00:42:58,170 Like, research talks about what he's doing. 796 00:42:58,170 --> 00:43:01,958 In his class-- he took advanced calculus with me, 797 00:43:01,958 --> 00:43:04,090 which was actually graduate level [INAUDIBLE]-- 798 00:43:04,090 --> 00:43:09,180 I explained epsilon delta, and he had it very well understood. 799 00:43:09,180 --> 00:43:13,270 And after I left the classroom he explained it to his peers, 800 00:43:13,270 --> 00:43:15,050 to his classmates. 801 00:43:15,050 --> 00:43:16,885 And he explained it better than me. 802 00:43:16,885 --> 00:43:21,010 And I was there listening, and I remember being jealous, 803 00:43:21,010 --> 00:43:23,050 because although he was very rushed, 804 00:43:23,050 --> 00:43:27,430 he had a very clear understanding of how 805 00:43:27,430 --> 00:43:31,140 you take an epsilon, no matter how small, and then 806 00:43:31,140 --> 00:43:34,200 you take a little ball here, radius delta. 807 00:43:34,200 --> 00:43:38,680 So the image of that little ball will fit in that ball 808 00:43:38,680 --> 00:43:40,110 that you take here. 809 00:43:40,110 --> 00:43:43,510 So even if you shrink on the image, 810 00:43:43,510 --> 00:43:46,020 you can take this ball even smaller 811 00:43:46,020 --> 00:43:48,670 so the image will still fit inside. 812 00:43:48,670 --> 00:43:51,490 And I was going, gosh, this is the essence, 813 00:43:51,490 --> 00:43:54,660 but I wish I could convey it, because no book 814 00:43:54,660 --> 00:43:58,550 will say it just-- or show you how to do it with your hands. 815 00:43:58,550 --> 00:43:59,202 816 00:43:59,202 --> 00:44:00,035 STUDENT: [INAUDIBLE] 817 00:44:00,035 --> 00:44:00,951 MAGDALENA TODA: Right. 818 00:44:00,951 --> 00:44:04,985 So he was rushed, but he had a very clear picture 819 00:44:04,985 --> 00:44:06,965 of what is going on. 820 00:44:06,965 --> 00:44:07,955 OK. 821 00:44:07,955 --> 00:44:10,925 11.3 is a completely new start. 822 00:44:10,925 --> 00:44:13,895 And you are gonna read and be happy about that 823 00:44:13,895 --> 00:44:16,370 because that's partial derivatives. 824 00:44:16,370 --> 00:44:20,360 And you say, Magdalena, finally, this is piece of cake. 825 00:44:20,360 --> 00:44:22,700 You see, I know these things. 826 00:44:22,700 --> 00:44:25,760 I can do them in my-- in my sleep. 827 00:44:25,760 --> 00:44:29,960 So f of x and y is still a graph. 828 00:44:29,960 --> 00:44:33,115 And then you say, how do we introduce 829 00:44:33,115 --> 00:44:37,740 the partial derivative with respect to one variable only. 830 00:44:37,740 --> 00:44:39,768 You think, I draw the graph. 831 00:44:39,768 --> 00:44:41,640 OK. 832 00:44:41,640 --> 00:44:44,748 On this graph, I pick a point x0, y0. 833 00:44:44,748 --> 00:44:53,950 And if I were to take x to be 0, what is-- what is the z 834 00:44:53,950 --> 00:44:56,155 equals f of x0, y? 835 00:44:56,155 --> 00:45:02,940 836 00:45:02,940 --> 00:45:04,120 So I'll try to draw it. 837 00:45:04,120 --> 00:45:05,120 It's not easy. 838 00:45:05,120 --> 00:45:10,580 839 00:45:10,580 --> 00:45:16,410 This is x and y and z, and you want your x0 to be a constant. 840 00:45:16,410 --> 00:45:17,457 STUDENT: [INAUDIBLE] 841 00:45:17,457 --> 00:45:19,540 MAGDALENA TODA: It's a so-called coordinate curve. 842 00:45:19,540 --> 00:45:20,130 Very good. 843 00:45:20,130 --> 00:45:23,085 It's a curve, but I want to be good enough to draw it. 844 00:45:23,085 --> 00:45:25,140 So you guys have to wish me luck, 845 00:45:25,140 --> 00:45:28,296 because I don't-- didn't have enough coffee and I don't feel 846 00:45:28,296 --> 00:45:30,060 like I can draw very well. 847 00:45:30,060 --> 00:45:33,530 x0 is here. 848 00:45:33,530 --> 00:45:39,580 So x is there, so you cut with this board-- are 849 00:45:39,580 --> 00:45:40,880 you guys with me? 850 00:45:40,880 --> 00:45:44,115 You cut with this board at the level x0 over here. 851 00:45:44,115 --> 00:45:45,920 You cut. 852 00:45:45,920 --> 00:45:49,290 When you cut with this board-- you 853 00:45:49,290 --> 00:45:52,563 cut your surface with this board-- 854 00:45:52,563 --> 00:45:54,882 you get a curve like that. 855 00:45:54,882 --> 00:46:00,350 And we call that a curve f of x0, y. 856 00:46:00,350 --> 00:46:05,838 Some people who are a little bit in a hurry and smarter than me, 857 00:46:05,838 --> 00:46:07,782 they say x equals x0. 858 00:46:07,782 --> 00:46:09,726 That's called coordinate curve. 859 00:46:09,726 --> 00:46:16,550 860 00:46:16,550 --> 00:46:20,270 So, the thing is, this-- it's a curve in plane. 861 00:46:20,270 --> 00:46:21,324 This is the blue plane. 862 00:46:21,324 --> 00:46:22,490 I don't know how to call it. 863 00:46:22,490 --> 00:46:23,420 Pi. 864 00:46:23,420 --> 00:46:25,840 You know I love to call it pi. 865 00:46:25,840 --> 00:46:28,180 Since I'm in plane with a point in a curve-- 866 00:46:28,180 --> 00:46:33,570 a plane curve-- this curve has a slope at x0, y0. 867 00:46:33,570 --> 00:46:35,180 Can I draw that slope? 868 00:46:35,180 --> 00:46:36,730 I'll try. 869 00:46:36,730 --> 00:46:38,540 The slope of the blue line, though. 870 00:46:38,540 --> 00:46:39,976 Let me make it red. 871 00:46:39,976 --> 00:46:43,950 The slope of the red line-- now, if you don't have colors 872 00:46:43,950 --> 00:46:47,206 you can make it a dotted line. 873 00:46:47,206 --> 00:46:57,810 The slope of the dotted line is-- who the heck is that? 874 00:46:57,810 --> 00:47:07,140 The derivative of f with respect to y, because x0 is a constant. 875 00:47:07,140 --> 00:47:09,490 So how do we write that? 876 00:47:09,490 --> 00:47:12,800 Because x0 is sort of in our way, driving us crazy. 877 00:47:12,800 --> 00:47:15,000 Although he was fixed. 878 00:47:15,000 --> 00:47:18,410 We keep him fixed by keeping him in this plane. 879 00:47:18,410 --> 00:47:20,130 x0 is fixed. 880 00:47:20,130 --> 00:47:21,980 We have to write another notation. 881 00:47:21,980 --> 00:47:24,210 We cannot say f prime. 882 00:47:24,210 --> 00:47:27,490 Because f depends on two variables. 883 00:47:27,490 --> 00:47:31,600 f prime were for when we were babies in calculus 1. 884 00:47:31,600 --> 00:47:32,934 We cannot use f prime anymore. 885 00:47:32,934 --> 00:47:33,850 We have two variables. 886 00:47:33,850 --> 00:47:36,310 Life became too complicated. 887 00:47:36,310 --> 00:47:37,393 So we have to say-- 888 00:47:37,393 --> 00:47:38,184 STUDENT: Professor? 889 00:47:38,184 --> 00:47:40,701 MAGDALENA TODA: --instead of df dy-- yes, sir. 890 00:47:40,701 --> 00:47:41,700 May you use a subscript? 891 00:47:41,700 --> 00:47:45,340 MAGDALENA TODA: You use-- yeah, you can do that as well. 892 00:47:45,340 --> 00:47:47,300 That's what I do. 893 00:47:47,300 --> 00:47:49,070 Let me do both. 894 00:47:49,070 --> 00:47:55,751 f sub y at-- who was fixed? x0 and y. 895 00:47:55,751 --> 00:47:58,750 But this is my favorite notation. 896 00:47:58,750 --> 00:48:01,190 I'm going to make a face because I love it. 897 00:48:01,190 --> 00:48:02,760 This is what engineers love. 898 00:48:02,760 --> 00:48:04,820 This is what we physicists love. 899 00:48:04,820 --> 00:48:07,325 Mathematicians, though, are crazy people. 900 00:48:07,325 --> 00:48:08,240 They are. 901 00:48:08,240 --> 00:48:09,330 All of them. 902 00:48:09,330 --> 00:48:12,930 And they invented another notation. 903 00:48:12,930 --> 00:48:15,060 Do you remember that Mr. Leibniz, 904 00:48:15,060 --> 00:48:19,030 because he had nothing better to do, when he invented calculus, 905 00:48:19,030 --> 00:48:23,250 he did df dy, or df dx? 906 00:48:23,250 --> 00:48:24,250 What is that? 907 00:48:24,250 --> 00:48:27,140 That was the limit of delta f, delta y, right? 908 00:48:27,140 --> 00:48:28,550 That's what Leibniz did. 909 00:48:28,550 --> 00:48:30,840 He introduced this delta notation, 910 00:48:30,840 --> 00:48:34,510 and then he said if you have delta space over delta time, 911 00:48:34,510 --> 00:48:38,080 then shrink both, and you make a ratio in the limit, 912 00:48:38,080 --> 00:48:40,620 you should read-- you should write it df dy. 913 00:48:40,620 --> 00:48:44,106 And that's the so-called Leibniz notation, right? 914 00:48:44,106 --> 00:48:46,756 That was in calc 1. 915 00:48:46,756 --> 00:48:49,630 But I erased it because that was calc 1. 916 00:48:49,630 --> 00:48:54,180 Now, mathematicians, to imitate the Leibniz notation, 917 00:48:54,180 --> 00:48:57,870 they said, I cannot use df dy. 918 00:48:57,870 --> 00:49:00,610 So what the heck shall I use? 919 00:49:00,610 --> 00:49:02,650 After they thought for about a year, 920 00:49:02,650 --> 00:49:05,268 and I was reading through the history about how 921 00:49:05,268 --> 00:49:07,140 they invented this, they said, let's take 922 00:49:07,140 --> 00:49:09,860 the Greek-- the Greek d. 923 00:49:09,860 --> 00:49:12,140 Which is the del. 924 00:49:12,140 --> 00:49:13,990 That's partial. 925 00:49:13,990 --> 00:49:19,400 The del f, del y, at x0, y. 926 00:49:19,400 --> 00:49:22,400 When I was 20-- no, I was 18 when 927 00:49:22,400 --> 00:49:27,100 I saw this the first time-- I had the hardest time making 928 00:49:27,100 --> 00:49:27,925 this sign. 929 00:49:27,925 --> 00:49:29,590 It's all in the wrist. 930 00:49:29,590 --> 00:49:32,160 It's very-- OK. 931 00:49:32,160 --> 00:49:32,950 Now. 932 00:49:32,950 --> 00:49:33,590 df dy. 933 00:49:33,590 --> 00:49:35,900 If you don't like it, then what do you do? 934 00:49:35,900 --> 00:49:38,858 You can adopt this notation. 935 00:49:38,858 --> 00:49:41,710 And what is the meaning of this by definition? 936 00:49:41,710 --> 00:49:45,222 You say, you haven't even defined it, Magdalena. 937 00:49:45,222 --> 00:49:47,985 It has to be limit of a difference quotient, 938 00:49:47,985 --> 00:49:49,066 just like here. 939 00:49:49,066 --> 00:49:53,320 But we have to be happy and think of that. 940 00:49:53,320 --> 00:49:57,360 What is the delta f versus the delta y? 941 00:49:57,360 --> 00:49:59,320 It has to be like that. 942 00:49:59,320 --> 00:50:02,710 f of Mr. x0 is fixed. 943 00:50:02,710 --> 00:50:07,060 x0, comma, y. 944 00:50:07,060 --> 00:50:09,850 We have an increment in y. 945 00:50:09,850 --> 00:50:16,310 y plus delta y. y plus delta y minus-- that's 946 00:50:16,310 --> 00:50:18,220 the difference quotient. 947 00:50:18,220 --> 00:50:22,956 f of what-- the original point was, well-- 948 00:50:22,956 --> 00:50:24,450 STUDENT: x0, y0. 949 00:50:24,450 --> 00:50:26,800 MAGDALENA TODA: x0-- let me put y0 950 00:50:26,800 --> 00:50:29,850 because our original point was x0, y0. 951 00:50:29,850 --> 00:50:37,810 x0, y0 over-- over delta y. 952 00:50:37,810 --> 00:50:43,385 But if I am at x0, y0, I better put x0, y0 fixed point here. 953 00:50:43,385 --> 00:50:46,800 954 00:50:46,800 --> 00:50:51,580 And I would like you to photograph or put this thing-- 955 00:50:51,580 --> 00:50:54,525 STUDENT: So is that a delta that's in front of the f? 956 00:50:54,525 --> 00:50:56,400 MAGDALENA TODA: Let me review the whole thing 957 00:50:56,400 --> 00:50:58,830 because it's very important. 958 00:50:58,830 --> 00:51:00,810 Where shall I start, here, or here? 959 00:51:00,810 --> 00:51:01,860 It doesn't matter. 960 00:51:01,860 --> 00:51:02,960 So the limit-- 961 00:51:02,960 --> 00:51:05,330 STUDENT: [INAUDIBLE] start at m. 962 00:51:05,330 --> 00:51:06,205 MAGDALENA TODA: At m? 963 00:51:06,205 --> 00:51:06,704 At m. 964 00:51:06,704 --> 00:51:07,930 OK, I'll start at m. 965 00:51:07,930 --> 00:51:13,070 The slopes of this line at x0, y0, right at my point, 966 00:51:13,070 --> 00:51:18,640 will be, my favorite notation is f sub y at x0, 967 00:51:18,640 --> 00:51:22,030 y0, which means partial derivative of f with respect 968 00:51:22,030 --> 00:51:26,190 to y at the point-- fixed point x0, y0. 969 00:51:26,190 --> 00:51:30,670 Or, for most mathematicians, df-- of del-- del f, 970 00:51:30,670 --> 00:51:34,310 del y at x0, y0. 971 00:51:34,310 --> 00:51:38,880 Which is by definition the limit of this difference quotient. 972 00:51:38,880 --> 00:51:42,270 So x0 is held fixed in both cases. 973 00:51:42,270 --> 00:51:45,480 y0 is allowed to deviate a little bit. 974 00:51:45,480 --> 00:51:50,400 So y0 is fixed, but you displace it by a little delta, 975 00:51:50,400 --> 00:51:53,626 or by a little-- how did we denote that in calc 1, h? 976 00:51:53,626 --> 00:51:54,367 Little h? 977 00:51:54,367 --> 00:51:54,950 STUDENT: Yeah. 978 00:51:54,950 --> 00:51:56,616 MAGDALENA TODA: So delta y, sometimes it 979 00:51:56,616 --> 00:51:58,440 was called little h. 980 00:51:58,440 --> 00:52:00,950 And this is the same as little h. 981 00:52:00,950 --> 00:52:03,840 Over that h. 982 00:52:03,840 --> 00:52:07,370 Now you, without my help, because you 983 00:52:07,370 --> 00:52:10,870 have all the knowledge and you're smart, 984 00:52:10,870 --> 00:52:17,430 you should tell me how I define f sub x at x0, y0, 985 00:52:17,430 --> 00:52:22,192 and shut up, Magdalena, let people talk. 986 00:52:22,192 --> 00:52:23,947 This is hard. 987 00:52:23,947 --> 00:52:24,780 [INTERPOSING VOICES] 988 00:52:24,780 --> 00:52:26,090 MAGDALENA TODA: No. 989 00:52:26,090 --> 00:52:28,250 I hope not. 990 00:52:28,250 --> 00:52:31,515 As a limit of a difference quotient, 991 00:52:31,515 --> 00:52:34,570 so it's gonna be an instantaneous rate of change. 992 00:52:34,570 --> 00:52:37,062 That's the limit of a difference quotient. 993 00:52:37,062 --> 00:52:38,771 Limit of what? 994 00:52:38,771 --> 00:52:39,271 Shut up. 995 00:52:39,271 --> 00:52:40,534 I will zip my lips. 996 00:52:40,534 --> 00:52:41,380 STUDENT: Delta x 997 00:52:41,380 --> 00:52:42,860 MAGDALENA TODA: Delta x, excellent. 998 00:52:42,860 --> 00:52:44,492 Delta x going to 0. 999 00:52:44,492 --> 00:52:47,990 So you shrink-- you displace by a small displacement 1000 00:52:47,990 --> 00:52:50,254 only in the direction of x. 1001 00:52:50,254 --> 00:52:52,220 STUDENT: So f. 1002 00:52:52,220 --> 00:52:53,452 MAGDALENA TODA: f. 1003 00:52:53,452 --> 00:52:57,260 STUDENT: [INAUDIBLE] this time, x is changing, so-- 1004 00:52:57,260 --> 00:52:58,580 [INTERPOSING VOICES] 1005 00:52:58,580 --> 00:53:04,890 MAGDALENA TODA: X0 plus delta x, y0 is still fixed, 1006 00:53:04,890 --> 00:53:11,183 minus f of x0, y0. 1007 00:53:11,183 --> 00:53:13,115 Thank God this is always fixed. 1008 00:53:13,115 --> 00:53:14,081 I love this guy. 1009 00:53:14,081 --> 00:53:16,500 STUDENT: Delta-- 1010 00:53:16,500 --> 00:53:19,910 MAGDALENA TODA: Delta x, which is 1011 00:53:19,910 --> 00:53:23,200 like the h we were talking about. 1012 00:53:23,200 --> 00:53:24,680 Now in reality, you never do that. 1013 00:53:24,680 --> 00:53:28,770 You would die if for every exercise, derivation exercise, 1014 00:53:28,770 --> 00:53:31,500 you would have to compute a limit of a difference quotient. 1015 00:53:31,500 --> 00:53:33,050 You will go bananas. 1016 00:53:33,050 --> 00:53:34,510 What we do? 1017 00:53:34,510 --> 00:53:37,400 We do exactly the same thing. 1018 00:53:37,400 --> 00:53:38,710 How can I draw? 1019 00:53:38,710 --> 00:53:40,620 Can anybody help me draw? 1020 00:53:40,620 --> 00:53:46,280 For y0, I would need to take this other plane through y0. 1021 00:53:46,280 --> 00:53:47,332 Where is y0? 1022 00:53:47,332 --> 00:53:48,940 Here. 1023 00:53:48,940 --> 00:53:50,200 Is my drawing good enough? 1024 00:53:50,200 --> 00:53:51,598 I hope so. 1025 00:53:51,598 --> 00:53:56,470 So it's something like I have this plane with, 1026 00:53:56,470 --> 00:53:57,920 oh, do you see that, guys? 1027 00:53:57,920 --> 00:53:58,570 OK. 1028 00:53:58,570 --> 00:54:02,842 So what is that, the other curve, coordinate curve, look 1029 00:54:02,842 --> 00:54:03,342 like? 1030 00:54:03,342 --> 00:54:06,680 1031 00:54:06,680 --> 00:54:08,702 Oh my God. 1032 00:54:08,702 --> 00:54:10,260 Looks like that. 1033 00:54:10,260 --> 00:54:13,070 Through the same point, and then the slope 1034 00:54:13,070 --> 00:54:18,370 of the line will be a blue slope and the slope 1035 00:54:18,370 --> 00:54:23,710 will be f sub-- well OK. 1036 00:54:23,710 --> 00:54:28,130 So here I have in the red one, which was the blue one, 1037 00:54:28,130 --> 00:54:33,670 this is f sub y, and for this one, this is f sub x. 1038 00:54:33,670 --> 00:54:34,890 Right? 1039 00:54:34,890 --> 00:54:42,240 So guys, don't look at the picture. 1040 00:54:42,240 --> 00:54:43,240 The picture's confusing. 1041 00:54:43,240 --> 00:54:46,510 This is x coming towards me, right? 1042 00:54:46,510 --> 00:54:49,630 And y going there and z is going up. 1043 00:54:49,630 --> 00:54:52,150 This is the graph. 1044 00:54:52,150 --> 00:54:54,650 When I do the derivative with respect 1045 00:54:54,650 --> 00:55:00,830 to what is this, y, the derivative with respect to y, 1046 00:55:00,830 --> 00:55:04,070 with respect to y, y is my only variable, 1047 00:55:04,070 --> 00:55:06,640 so the curve will be like that. 1048 00:55:06,640 --> 00:55:11,480 And the slope will be for a curve that depends on y only. 1049 00:55:11,480 --> 00:55:14,355 When I do derivative with respect to x, 1050 00:55:14,355 --> 00:55:19,460 it's like I'm on top of a hill and I decide to go skiing. 1051 00:55:19,460 --> 00:55:22,310 And I'm-- and I point my skis like that, 1052 00:55:22,310 --> 00:55:27,100 and the slope is going down, and that's the x direction. 1053 00:55:27,100 --> 00:55:27,750 OK? 1054 00:55:27,750 --> 00:55:30,910 And what I'm going to describe as a skier 1055 00:55:30,910 --> 00:55:34,130 will be a plane curve going down in this direction. 1056 00:55:34,130 --> 00:55:35,854 Zzzzsssshh, like that. 1057 00:55:35,854 --> 00:55:40,900 And the slope at every point, the slope of the line, 1058 00:55:40,900 --> 00:55:44,720 of y trajectory, will be the derivative. 1059 00:55:44,720 --> 00:55:47,460 So I have a curve like that, and a curve like this. 1060 00:55:47,460 --> 00:55:50,140 And they're called coordinate curves. 1061 00:55:50,140 --> 00:55:51,480 Now this is hard. 1062 00:55:51,480 --> 00:55:53,260 You'll see how beautiful and easy 1063 00:55:53,260 --> 00:55:57,160 it is when you actually compute the partial derivatives 1064 00:55:57,160 --> 00:55:59,890 of functions by hand. 1065 00:55:59,890 --> 00:56:01,930 Examples? 1066 00:56:01,930 --> 00:56:08,100 Let's take f of x, y to be x squared plus y squared. 1067 00:56:08,100 --> 00:56:12,550 I'm asking you, who is f sub x at x, y? 1068 00:56:12,550 --> 00:56:17,560 Who is f sub x at 1 minus 1, 1, 0, OK. 1069 00:56:17,560 --> 00:56:20,630 Who is f sub y at x, y? 1070 00:56:20,630 --> 00:56:26,130 And who is f sub y at 3 and 2. 1071 00:56:26,130 --> 00:56:28,245 Since I make up my example-- I don't 1072 00:56:28,245 --> 00:56:30,460 want to copy the examples from the book, 1073 00:56:30,460 --> 00:56:35,020 because you are supposedly going to read the book. 1074 00:56:35,020 --> 00:56:40,230 This is-- should be another example, just for you. 1075 00:56:40,230 --> 00:56:44,370 1076 00:56:44,370 --> 00:56:49,080 So who's gonna help me-- I'm pausing a little bit-- who's 1077 00:56:49,080 --> 00:56:50,760 gonna help me here? 1078 00:56:50,760 --> 00:56:53,884 What's the answer here? 1079 00:56:53,884 --> 00:56:55,810 So how do I think? 1080 00:56:55,810 --> 00:57:00,060 I think I got-- when I prime with respect to x, y 1081 00:57:00,060 --> 00:57:01,465 is like a held constant. 1082 00:57:01,465 --> 00:57:02,950 He's held prisoner. 1083 00:57:02,950 --> 00:57:05,205 Poor guy cannot leave his cell. 1084 00:57:05,205 --> 00:57:06,240 That's awful. 1085 00:57:06,240 --> 00:57:09,450 So you prime with respect to x. 1086 00:57:09,450 --> 00:57:11,350 Because x is the only variable. 1087 00:57:11,350 --> 00:57:12,176 And he is-- 1088 00:57:12,176 --> 00:57:14,515 STUDENT: So then it's 2x plus y? 1089 00:57:14,515 --> 00:57:17,068 MAGDALENA TODA: 2x plus 0. 1090 00:57:17,068 --> 00:57:17,764 Plus 0. 1091 00:57:17,764 --> 00:57:20,316 Because y is a constant and when you prime a constant, 1092 00:57:20,316 --> 00:57:22,074 you get 0. 1093 00:57:22,074 --> 00:57:23,990 STUDENT: So when you take partial derivatives, 1094 00:57:23,990 --> 00:57:25,656 you-- when you're taking it with respect 1095 00:57:25,656 --> 00:57:28,610 to the first derivative, the first variable [INAUDIBLE] 1096 00:57:28,610 --> 00:57:30,110 MAGDALENA TODA: You don't completely 1097 00:57:30,110 --> 00:57:31,960 know because it might be multiplied. 1098 00:57:31,960 --> 00:57:33,440 But you view it as a constant. 1099 00:57:33,440 --> 00:57:35,320 So for you-- very good, Ryan. 1100 00:57:35,320 --> 00:57:38,090 So for you, it's like, as if y would be 7. 1101 00:57:38,090 --> 00:57:39,970 Imagine that y would be 7. 1102 00:57:39,970 --> 00:57:44,154 And then you have x squared plus 7 squared prime is u, right? 1103 00:57:44,154 --> 00:57:47,740 STUDENT: So then that means f of 1-- or f x of 1,0 1104 00:57:47,740 --> 00:57:48,479 is [INAUDIBLE] 1105 00:57:48,479 --> 00:57:49,562 MAGDALENA TODA: Very good. 1106 00:57:49,562 --> 00:57:50,870 STUDENT: OK. 1107 00:57:50,870 --> 00:57:54,668 And in this case, f sub y, what do you think it is? 1108 00:57:54,668 --> 00:57:56,439 STUDENT: 2y. 1109 00:57:56,439 --> 00:57:57,230 MAGDALENA TODA: 2y. 1110 00:57:57,230 --> 00:57:59,955 And what is f y of 3, 2? 1111 00:57:59,955 --> 00:58:01,092 STUDENT: 4. 1112 00:58:01,092 --> 00:58:02,050 MAGDALENA TODA: It's 4. 1113 00:58:02,050 --> 00:58:04,560 And you say, OK, that makes sense, that was easy. 1114 00:58:04,560 --> 00:58:06,580 Let's try something hard. 1115 00:58:06,580 --> 00:58:09,385 I'm going to build them on so many examples 1116 00:58:09,385 --> 00:58:12,670 that you say, stop, Magdalena, because I became 1117 00:58:12,670 --> 00:58:15,580 an expert in partial differentiation 1118 00:58:15,580 --> 00:58:19,290 and I-- now everything is so trivial that you have to stop. 1119 00:58:19,290 --> 00:58:38,380 So example A, example B. A was f of x, y [INAUDIBLE] x, y plus y 1120 00:58:38,380 --> 00:58:39,929 sine x. 1121 00:58:39,929 --> 00:58:41,470 And you say, wait, wait, wait, you're 1122 00:58:41,470 --> 00:58:44,240 giving me a little bit of trouble. 1123 00:58:44,240 --> 00:58:45,620 No, I don't mean to. 1124 00:58:45,620 --> 00:58:47,020 It's very easy. 1125 00:58:47,020 --> 00:58:50,260 Believe me guys, very, very easy. 1126 00:58:50,260 --> 00:58:55,354 We just have to think how we do this. 1127 00:58:55,354 --> 00:59:02,170 f sub x at 1 and 2, f sub y at x, y in general, 1128 00:59:02,170 --> 00:59:06,980 f sub y at 1 and 2, for God's sake. 1129 00:59:06,980 --> 00:59:08,270 OK. 1130 00:59:08,270 --> 00:59:09,885 All right. 1131 00:59:09,885 --> 00:59:19,380 And now, while you're staring at that, 1132 00:59:19,380 --> 00:59:23,595 I take out my beautiful colors that I paid $6 for. 1133 00:59:23,595 --> 00:59:26,145 1134 00:59:26,145 --> 00:59:31,620 The department told me that they don't buy different colors, 1135 00:59:31,620 --> 00:59:35,490 just two or three basic ones. 1136 00:59:35,490 --> 00:59:35,990 All right? 1137 00:59:35,990 --> 00:59:38,170 So what do we do? 1138 00:59:38,170 --> 00:59:40,620 STUDENT: First one will be the y. 1139 00:59:40,620 --> 00:59:43,320 MAGDALENA TODA: It's like y would be a constant 7, right, 1140 00:59:43,320 --> 00:59:46,710 but you have to keep in mind it's mister called y. 1141 00:59:46,710 --> 00:59:48,580 Which for you is a constant. 1142 00:59:48,580 --> 00:59:52,730 So you go, I'm priming this with respect to x only-- 1143 00:59:52,730 --> 00:59:54,514 STUDENT: Then you get y. 1144 00:59:54,514 --> 00:59:56,325 MAGDALENA TODA: Very good. 1145 00:59:56,325 --> 00:59:56,824 Plus-- 1146 00:59:56,824 --> 00:59:59,910 1147 00:59:59,910 --> 01:00:01,114 STUDENT: y cosine x. 1148 01:00:01,114 --> 01:00:01,596 MAGDALENA TODA: y cosine x. 1149 01:00:01,596 --> 01:00:02,096 Excellent. 1150 01:00:02,096 --> 01:00:03,524 And stop. 1151 01:00:03,524 --> 01:00:04,970 And stop. 1152 01:00:04,970 --> 01:00:06,185 Because that's all I have. 1153 01:00:06,185 --> 01:00:08,763 You see, it's not hard. 1154 01:00:08,763 --> 01:00:11,710 Let me put here a y. 1155 01:00:11,710 --> 01:00:13,443 OK. 1156 01:00:13,443 --> 01:00:19,040 And then, I plug a different color. 1157 01:00:19,040 --> 01:00:21,520 I'm a girl, of course I like different colors. 1158 01:00:21,520 --> 01:00:26,700 So 1, 2. x is 1, and y is 2. 1159 01:00:26,700 --> 01:00:30,270 2 plus 2 cosine 1. 1160 01:00:30,270 --> 01:00:33,314 And you say, oh, wait a minute, what is that cosine of 1? 1161 01:00:33,314 --> 01:00:33,814 Never mind. 1162 01:00:33,814 --> 01:00:34,790 Don't worry about it. 1163 01:00:34,790 --> 01:00:37,230 It's like cosine of 1, [INAUDIBLE] 1164 01:00:37,230 --> 01:00:41,134 plug it in the calculator, nobody cares. 1165 01:00:41,134 --> 01:00:44,550 Well, in the final, you don't have a calculator, 1166 01:00:44,550 --> 01:00:47,966 so you leave it like that. 1167 01:00:47,966 --> 01:00:49,430 Who cares? 1168 01:00:49,430 --> 01:00:52,750 It's just the perfect-- I would actually hate it 1169 01:00:52,750 --> 01:00:54,230 that you gave me-- because all you 1170 01:00:54,230 --> 01:00:56,480 could give me would be an approximation, a truncation, 1171 01:00:56,480 --> 01:00:58,220 with two decimals. 1172 01:00:58,220 --> 01:01:01,330 I prefer you give me the precise answer, which 1173 01:01:01,330 --> 01:01:03,960 is an exact answer like that. 1174 01:01:03,960 --> 01:01:04,750 f sub y. 1175 01:01:04,750 --> 01:01:07,500 Now, Mr. x is held prisoner. 1176 01:01:07,500 --> 01:01:09,006 He is a constant. 1177 01:01:09,006 --> 01:01:10,620 He cannot move. 1178 01:01:10,620 --> 01:01:11,800 Mr. y can move. 1179 01:01:11,800 --> 01:01:13,280 He has all the freedom. 1180 01:01:13,280 --> 01:01:16,603 So prime with respect to y, what do you have? 1181 01:01:16,603 --> 01:01:17,102 STUDENT: x-- 1182 01:01:17,102 --> 01:01:18,026 [INTERPOSING VOICES] 1183 01:01:18,026 --> 01:01:21,952 MAGDALENA TODA: x plus sine x is a constant. 1184 01:01:21,952 --> 01:01:25,200 So for God's sake, I'll write it. 1185 01:01:25,200 --> 01:01:30,680 So then I get 1, plug in x equals 1. y 1186 01:01:30,680 --> 01:01:31,960 doesn't appear in the picture. 1187 01:01:31,960 --> 01:01:33,110 I don't care. 1188 01:01:33,110 --> 01:01:35,132 1 plus sine 1. 1189 01:01:35,132 --> 01:01:38,340 1190 01:01:38,340 --> 01:01:39,820 And now comes-- don't erase. 1191 01:01:39,820 --> 01:01:42,240 Now comes the-- I mean, you cannot erase it. 1192 01:01:42,240 --> 01:01:44,750 I can erase it. 1193 01:01:44,750 --> 01:01:48,768 Comes this mean professor who says, wait a minute, 1194 01:01:48,768 --> 01:01:50,950 I want more. 1195 01:01:50,950 --> 01:01:53,601 Mathematicians always want more. 1196 01:01:53,601 --> 01:01:57,330 He goes, I want the second derivative. 1197 01:01:57,330 --> 01:02:01,040 f sub x x of x, y. 1198 01:02:01,040 --> 01:02:03,530 And you say, what in the world is that? 1199 01:02:03,530 --> 01:02:06,290 Even some mathematicians, they denote it 1200 01:02:06,290 --> 01:02:13,295 as del 2 f dx 2, which is d of-- d with respect 1201 01:02:13,295 --> 01:02:16,520 to x sub d u with respect to x. 1202 01:02:16,520 --> 01:02:17,811 What does it mean? 1203 01:02:17,811 --> 01:02:20,640 You take the first derivative and you derive it again. 1204 01:02:20,640 --> 01:02:23,181 And don't drink and derive because you'll be in trouble. 1205 01:02:23,181 --> 01:02:23,680 Right? 1206 01:02:23,680 --> 01:02:28,039 So you have d of dx primed again, with-- differentiated 1207 01:02:28,039 --> 01:02:30,410 again with respect to x. 1208 01:02:30,410 --> 01:02:31,370 Is that hard? 1209 01:02:31,370 --> 01:02:31,870 Uh-uh. 1210 01:02:31,870 --> 01:02:32,950 What you do? 1211 01:02:32,950 --> 01:02:36,065 In the-- don't do it here. 1212 01:02:36,065 --> 01:02:37,430 You do it in general, right? 1213 01:02:37,430 --> 01:02:43,000 With respect to x as a variable, y is again held as a prisoner, 1214 01:02:43,000 --> 01:02:44,570 constant. 1215 01:02:44,570 --> 01:02:47,680 So when you prime that y goes away. 1216 01:02:47,680 --> 01:02:50,710 You're gonna get 0. 1217 01:02:50,710 --> 01:02:54,580 I'll write 0 like a silly because we are just starters. 1218 01:02:54,580 --> 01:02:56,207 And what else? 1219 01:02:56,207 --> 01:02:57,700 STUDENT: Negative y sine of x. 1220 01:02:57,700 --> 01:02:59,900 MAGDALENA TODA: Minus y sine of x. 1221 01:02:59,900 --> 01:03:02,340 And I know you've gonna love this process. 1222 01:03:02,340 --> 01:03:04,940 You are becoming experts in that. 1223 01:03:04,940 --> 01:03:10,050 And in a way I'm a little bit sorry it's so easy, 1224 01:03:10,050 --> 01:03:13,180 but I guess not everybody gets it. 1225 01:03:13,180 --> 01:03:16,290 There are students who don't get it the first time. 1226 01:03:16,290 --> 01:03:17,980 So what do we get here? 1227 01:03:17,980 --> 01:03:18,907 Minus-- 1228 01:03:18,907 --> 01:03:21,770 STUDENT: 0. 1229 01:03:21,770 --> 01:03:25,821 MAGDALENA TODA: Please tell me-- sine 1, 0. 1230 01:03:25,821 --> 01:03:26,320 Good. 1231 01:03:26,320 --> 01:03:30,002 I could do the same thing for f y y. 1232 01:03:30,002 --> 01:03:34,800 I could do this thing-- what is f sub x y? 1233 01:03:34,800 --> 01:03:37,250 By definition f sub x y-- 1234 01:03:37,250 --> 01:03:39,975 STUDENT: Is that taking the derivative of the derivative 1235 01:03:39,975 --> 01:03:42,100 with respect-- is that taking the second derivative 1236 01:03:42,100 --> 01:03:44,058 with respect to y after you take the derivative 1237 01:03:44,058 --> 01:03:46,110 of the-- first derivative with respect to x? 1238 01:03:46,110 --> 01:03:47,026 MAGDALENA TODA: Right. 1239 01:03:47,026 --> 01:03:49,510 So when I write like that, because that's a little bit 1240 01:03:49,510 --> 01:03:54,202 confusing, when students ask me, which one is first? 1241 01:03:54,202 --> 01:03:57,830 First you do f sub x, and then you do y. 1242 01:03:57,830 --> 01:04:02,766 And then f sub y x would be the derivative with respect to y 1243 01:04:02,766 --> 01:04:04,650 primed again with respect to x. 1244 01:04:04,650 --> 01:04:07,480 Now, let me tell you the good news. 1245 01:04:07,480 --> 01:04:13,390 They-- the book doesn't call it any name, because we don't 1246 01:04:13,390 --> 01:04:14,886 like to call anybody names. 1247 01:04:14,886 --> 01:04:15,840 I'm just kidding. 1248 01:04:15,840 --> 01:04:23,710 It's called the Schwartz principle, 1249 01:04:23,710 --> 01:04:27,470 or the theorem of Schwartz. 1250 01:04:27,470 --> 01:04:30,974 When I told my co-authors, they said, who cares? 1251 01:04:30,974 --> 01:04:34,950 Well I care, because I was a student when my professors told 1252 01:04:34,950 --> 01:04:38,290 me that this German mathematician made 1253 01:04:38,290 --> 01:04:41,040 this discovery, which is so beautiful. 1254 01:04:41,040 --> 01:04:55,380 If f is twice differentiable with respect to x and y, 1255 01:04:55,380 --> 01:04:58,820 and the partial derivatives-- the second partial 1256 01:04:58,820 --> 01:05:14,750 derivatives-- are continuous, then, now in English 1257 01:05:14,750 --> 01:05:17,480 it would say it doesn't matter in which order 1258 01:05:17,480 --> 01:05:18,530 you differentiate. 1259 01:05:18,530 --> 01:05:20,920 The mixed ones are always the same. 1260 01:05:20,920 --> 01:05:22,190 Say what? 1261 01:05:22,190 --> 01:05:26,760 f sub x y equals f sub y x for every point. 1262 01:05:26,760 --> 01:05:31,590 For every-- do you remember what I taught you for every x, y 1263 01:05:31,590 --> 01:05:32,360 in the domain. 1264 01:05:32,360 --> 01:05:36,110 Or for every x, y where this happens. 1265 01:05:36,110 --> 01:05:38,410 So what does this mean? 1266 01:05:38,410 --> 01:05:41,219 That means that whether you differentiate 1267 01:05:41,219 --> 01:05:43,760 first with respect to x and then with respect to, y, or first 1268 01:05:43,760 --> 01:05:46,100 with respect to y and then with respect to x, 1269 01:05:46,100 --> 01:05:48,250 it doesn't matter. 1270 01:05:48,250 --> 01:05:50,900 The mixed partial derivatives are the same. 1271 01:05:50,900 --> 01:05:52,140 Which is wonderful. 1272 01:05:52,140 --> 01:05:55,110 I mean, this is one of the best things 1273 01:05:55,110 --> 01:05:58,100 that ever happened to us. 1274 01:05:58,100 --> 01:06:01,306 Let's see if this is true in our case. 1275 01:06:01,306 --> 01:06:03,864 I mean, of course it's true because it's a theorem, 1276 01:06:03,864 --> 01:06:06,154 if it weren't true I wouldn't teach it, 1277 01:06:06,154 --> 01:06:11,050 but let's verify it on a baby. 1278 01:06:11,050 --> 01:06:14,343 Not on a real baby, on a baby example. 1279 01:06:14,343 --> 01:06:15,230 Right? 1280 01:06:15,230 --> 01:06:21,040 So, f sub x is y plus y equals sine x primed again 1281 01:06:21,040 --> 01:06:22,782 with respect to y. 1282 01:06:22,782 --> 01:06:27,630 And what do we get out of it? 1283 01:06:27,630 --> 01:06:29,250 Cosine of x. 1284 01:06:29,250 --> 01:06:31,081 Are you guys with me? 1285 01:06:31,081 --> 01:06:35,130 So f sub x was y plus y equals sine x. 1286 01:06:35,130 --> 01:06:39,078 Take this guy again, put it here, 1287 01:06:39,078 --> 01:06:42,667 squeeze them up a little bit, divide by-- no. 1288 01:06:42,667 --> 01:06:47,780 Time with respect to y, x is a constant, what do you think? 1289 01:06:47,780 --> 01:06:48,794 Cosine of x, am I right? 1290 01:06:48,794 --> 01:06:49,960 STUDENT: 1 plus [INAUDIBLE]. 1291 01:06:49,960 --> 01:06:52,389 1292 01:06:52,389 --> 01:06:54,180 MAGDALENA TODA: That's what it starts with. 1293 01:06:54,180 --> 01:06:56,700 Plus [INAUDIBLE]. 1294 01:06:56,700 --> 01:07:02,030 So cosine of x, [INAUDIBLE] a constant, plus 1. 1295 01:07:02,030 --> 01:07:04,965 Another way to have done it is, like, wait a minute, 1296 01:07:04,965 --> 01:07:10,810 at this point I go, constant out-- are you with me?-- 1297 01:07:10,810 --> 01:07:14,630 constant out, prime with respect to y, equals sine x plus 1. 1298 01:07:14,630 --> 01:07:16,885 Thank you. 1299 01:07:16,885 --> 01:07:17,385 All right. 1300 01:07:17,385 --> 01:07:20,822 1301 01:07:20,822 --> 01:07:26,370 F sub yx is going to be f sub y. 1302 01:07:26,370 --> 01:07:32,336 x plus sine x, but I have to take it from here, 1303 01:07:32,336 --> 01:07:38,416 and I prime again with respect to x, and I get the same thing. 1304 01:07:38,416 --> 01:07:39,790 I don't know, maybe I'm dyslexic, 1305 01:07:39,790 --> 01:07:43,450 I go from the right to the left, what's the matter with me. 1306 01:07:43,450 --> 01:07:47,305 Instead of saying 1 plus, I go cosine of x plus 1. 1307 01:07:47,305 --> 01:07:53,024 1308 01:07:53,024 --> 01:07:54,476 So it's the same thing. 1309 01:07:54,476 --> 01:07:55,444 Yes, sir. 1310 01:07:55,444 --> 01:07:58,611 STUDENT:I'm looking at the f of xy from the-- 1311 01:07:58,611 --> 01:08:00,486 MAGDALENA TODA: Which one are you looking at? 1312 01:08:00,486 --> 01:08:01,460 Show me. 1313 01:08:01,460 --> 01:08:03,518 STUDENT: It's in the purple. 1314 01:08:03,518 --> 01:08:05,150 MAGDALENA TODA: It is in the purple. 1315 01:08:05,150 --> 01:08:05,680 STUDENT: It's that one right there. 1316 01:08:05,680 --> 01:08:06,140 So-- 1317 01:08:06,140 --> 01:08:06,725 MAGDALENA TODA: This one? 1318 01:08:06,725 --> 01:08:07,308 STUDENT: Mmhm. 1319 01:08:07,308 --> 01:08:10,550 So, I'm looking at the y plus y cosine x. 1320 01:08:10,550 --> 01:08:12,503 You got that from f of x. 1321 01:08:12,503 --> 01:08:14,196 MAGDALENA TODA: I got this from f of x, 1322 01:08:14,196 --> 01:08:16,640 and I prime it again, with respect to y. 1323 01:08:16,640 --> 01:08:18,920 The whole thing. 1324 01:08:18,920 --> 01:08:21,510 STUDENT: OK, so you're not writing that as a derivative? 1325 01:08:21,510 --> 01:08:25,274 You're just substituting that in for f of x? 1326 01:08:25,274 --> 01:08:27,475 MAGDALENA TODA: So, let me write it better, 1327 01:08:27,475 --> 01:08:30,783 because I was a little bit rushed, and I don't know, 1328 01:08:30,783 --> 01:08:32,167 silly or something. 1329 01:08:32,167 --> 01:08:35,135 When I prime this with respect to y-- 1330 01:08:35,135 --> 01:08:38,274 STUDENT: Then you get the cosine of x plus 1. 1331 01:08:38,274 --> 01:08:39,149 MAGDALENA TODA: Yeah. 1332 01:08:39,149 --> 01:08:42,426 I could say, I can take out all the constants. 1333 01:08:42,426 --> 01:08:43,160 STUDENT: OK. 1334 01:08:43,160 --> 01:08:46,210 MAGDALENA TODA: And that constant is this plus 1. 1335 01:08:46,210 --> 01:08:47,455 And that's all I'm left with. 1336 01:08:47,455 --> 01:08:47,955 Right? 1337 01:08:47,955 --> 01:08:51,609 It's the same thing as 1 plus cosine x, 1338 01:08:51,609 --> 01:08:53,879 which is a constant times y. 1339 01:08:53,879 --> 01:08:57,238 Prime this with respect to y, I get the constant. 1340 01:08:57,238 --> 01:09:04,020 It's the same principal as when you have bdy of 7y equals 7. 1341 01:09:04,020 --> 01:09:06,752 Right? 1342 01:09:06,752 --> 01:09:08,703 OK. 1343 01:09:08,703 --> 01:09:10,130 Is this too easy? 1344 01:09:10,130 --> 01:09:13,420 I'll give you a nicer function. 1345 01:09:13,420 --> 01:09:28,760 I'm imitating the one in WeBWorK [INAUDIBLE] 1346 01:09:28,760 --> 01:09:31,444 To make it harder for you. 1347 01:09:31,444 --> 01:09:34,354 Nothing I can make at this point is hard for you, 1348 01:09:34,354 --> 01:09:39,250 because you're becoming experts in partial differentiation, 1349 01:09:39,250 --> 01:09:41,720 and I cannot challenge you on that. 1350 01:09:41,720 --> 01:09:54,113 1351 01:09:54,113 --> 01:09:57,048 I'm just trying to make it harder for you. 1352 01:09:57,048 --> 01:09:59,004 And I'm trying to look up something. 1353 01:09:59,004 --> 01:10:02,930 1354 01:10:02,930 --> 01:10:03,970 OK, how about that? 1355 01:10:03,970 --> 01:10:06,670 1356 01:10:06,670 --> 01:10:09,110 This is harder than the ones you have in WeBWorK. 1357 01:10:09,110 --> 01:10:11,970 But that was kind of the idea-- that when 1358 01:10:11,970 --> 01:10:15,620 you go home, and open those WeBWorK problem sets, 1359 01:10:15,620 --> 01:10:17,430 that's a piece of cake. 1360 01:10:17,430 --> 01:10:20,600 What we did in class was harder. 1361 01:10:20,600 --> 01:10:23,960 When I was a graduate student, one professor said, 1362 01:10:23,960 --> 01:10:27,320 the easy examples are the ones that the professor's 1363 01:10:27,320 --> 01:10:29,740 supposed to write in class, on the board. 1364 01:10:29,740 --> 01:10:31,440 The hard examples are the ones that 1365 01:10:31,440 --> 01:10:34,290 are left for the students' homework. 1366 01:10:34,290 --> 01:10:35,750 I disagree. 1367 01:10:35,750 --> 01:10:37,760 I think it should be the other way around. 1368 01:10:37,760 --> 01:10:40,260 So f sub x. 1369 01:10:40,260 --> 01:10:43,380 1370 01:10:43,380 --> 01:10:50,552 That means bfdx for the pair xy, any xy. 1371 01:10:50,552 --> 01:10:53,560 I'm not specifying an x and a y. 1372 01:10:53,560 --> 01:10:56,180 I'm not making them a constant. 1373 01:10:56,180 --> 01:10:58,970 What am I going to have in this case? 1374 01:10:58,970 --> 01:11:03,850 Chain -- if I catch you not knowing the chain rule, 1375 01:11:03,850 --> 01:11:05,370 you fail the final. 1376 01:11:05,370 --> 01:11:12,590 Not really, but, OK, you get some penalty. 1377 01:11:12,590 --> 01:11:13,730 You know it. 1378 01:11:13,730 --> 01:11:16,110 Just pay attention to what you do. 1379 01:11:16,110 --> 01:11:18,340 I make my own mistakes sometimes. 1380 01:11:18,340 --> 01:11:21,160 So 1 over. 1381 01:11:21,160 --> 01:11:23,590 What do you do here when you differentiate 1382 01:11:23,590 --> 01:11:24,350 with respect to x? 1383 01:11:24,350 --> 01:11:31,600 You think, OK, from the outside to the inside, one at a time. 1384 01:11:31,600 --> 01:11:36,130 1 over the variable squared plus 1, right? 1385 01:11:36,130 --> 01:11:42,215 Whatever that variable, it's like you call variable 1386 01:11:42,215 --> 01:11:44,890 of the argument xy, right? 1387 01:11:44,890 --> 01:11:47,240 STUDENT: [INAUDIBLE] 1388 01:11:47,240 --> 01:11:49,558 MAGDALENA TODA: Square plus 1. 1389 01:11:49,558 --> 01:11:56,710 Times-- cover it with your hand-- prime with respect to x. 1390 01:11:56,710 --> 01:11:59,000 y, right? 1391 01:11:59,000 --> 01:12:00,395 Good! 1392 01:12:00,395 --> 01:12:01,325 And you're done. 1393 01:12:01,325 --> 01:12:02,720 You see how easy it was. 1394 01:12:02,720 --> 01:12:07,510 Just don't forget something because it can cost you points. 1395 01:12:07,510 --> 01:12:09,640 Are you guys with me? 1396 01:12:09,640 --> 01:12:13,380 So, once we are done with saying, 1 over argument 1397 01:12:13,380 --> 01:12:16,130 squared plus 1, I cover this with my hand, 1398 01:12:16,130 --> 01:12:20,110 xy prime with respect to 2x is y. 1399 01:12:20,110 --> 01:12:22,380 And I'm done. 1400 01:12:22,380 --> 01:12:23,250 And I'm done. 1401 01:12:23,250 --> 01:12:26,250 And here, pause. 1402 01:12:26,250 --> 01:12:29,680 What's the easiest way to do that? 1403 01:12:29,680 --> 01:12:32,010 You look at it like, she wants me to get 1404 01:12:32,010 --> 01:12:34,310 caught in the quotient rule. 1405 01:12:34,310 --> 01:12:37,310 She wants to catch me not knowing this rule, 1406 01:12:37,310 --> 01:12:40,250 while I can do better. 1407 01:12:40,250 --> 01:12:43,460 One way to do it would be numerator prime plus 1408 01:12:43,460 --> 01:12:47,740 denominator, minus numerator [INAUDIBLE] What's 1409 01:12:47,740 --> 01:12:50,410 the easier way to do it? 1410 01:12:50,410 --> 01:12:52,870 STUDENT: x squared plus y squared, all of it 1411 01:12:52,870 --> 01:12:53,855 to the negative one. 1412 01:12:53,855 --> 01:12:54,771 MAGDALENA TODA: Right. 1413 01:12:54,771 --> 01:12:56,890 So you say, hey, you cannot catch me, 1414 01:12:56,890 --> 01:13:00,680 I'm the gingerbread man. 1415 01:13:00,680 --> 01:13:01,280 Good! 1416 01:13:01,280 --> 01:13:03,100 That was a good idea. 1417 01:13:03,100 --> 01:13:10,330 Chain rule, and minus 1/2, times-- 1418 01:13:10,330 --> 01:13:11,760 who tells me what's next? 1419 01:13:11,760 --> 01:13:13,210 I'm not going to say a word. 1420 01:13:13,210 --> 01:13:15,258 STUDENT: 2x plus y squared. 1421 01:13:15,258 --> 01:13:19,170 No, it's 2x. 1422 01:13:19,170 --> 01:13:20,637 x squared plus y squared. 1423 01:13:20,637 --> 01:13:22,595 MAGDALENA TODA: From the outside to the inside. 1424 01:13:22,595 --> 01:13:25,340 From the outside-- to the what? 1425 01:13:25,340 --> 01:13:27,186 STUDENT: [INAUDIBLE] 1426 01:13:27,186 --> 01:13:28,061 MAGDALENA TODA: Good. 1427 01:13:28,061 --> 01:13:28,970 And now I'm done. 1428 01:13:28,970 --> 01:13:31,220 I don't see that anymore. 1429 01:13:31,220 --> 01:13:33,860 I focus to the core. 1430 01:13:33,860 --> 01:13:35,650 2x. 1431 01:13:35,650 --> 01:13:38,640 Times 2x. 1432 01:13:38,640 --> 01:13:42,390 And that is plenty. 1433 01:13:42,390 --> 01:13:45,250 OK, now, let me ask you a question. 1434 01:13:45,250 --> 01:13:51,340 What if you would ask a smart kid, 1435 01:13:51,340 --> 01:13:56,760 I don't know, somebody who knows that, 1436 01:13:56,760 --> 01:14:01,620 can you pose the f sub y of xy without doing the whole thing 1437 01:14:01,620 --> 01:14:03,370 all over again? 1438 01:14:03,370 --> 01:14:06,350 Can you sort of figure out what it would be? 1439 01:14:06,350 --> 01:14:08,700 The beautiful thing about x and y 1440 01:14:08,700 --> 01:14:11,042 is that these are symmetric polynomials. 1441 01:14:11,042 --> 01:14:12,750 What does it mean, symmetric polynomials? 1442 01:14:12,750 --> 01:14:19,260 That means, if you swap x and y, and you swap x and y, 1443 01:14:19,260 --> 01:14:20,810 it's the same thing. 1444 01:14:20,810 --> 01:14:23,300 Just think of that-- swapping x and y. 1445 01:14:23,300 --> 01:14:25,230 Swapping the roles of x and y. 1446 01:14:25,230 --> 01:14:28,250 So what do you think you're going to get? 1447 01:14:28,250 --> 01:14:31,130 OK, one student said, this is for smart people, 1448 01:14:31,130 --> 01:14:32,564 not for people like me. 1449 01:14:32,564 --> 01:14:34,890 And I said, well, OK, what's the matter with you? 1450 01:14:34,890 --> 01:14:36,240 I'm a hard worker. 1451 01:14:36,240 --> 01:14:39,910 I'm the kind of guy who takes the whole thing again, and does 1452 01:14:39,910 --> 01:14:42,180 the derivation from scratch. 1453 01:14:42,180 --> 01:14:45,325 And thinking back in high school, I think, even 1454 01:14:45,325 --> 01:14:47,810 for symmetric polynomials, 1455 01:14:47,810 --> 01:14:49,570 I'm sure that being smart and being 1456 01:14:49,570 --> 01:14:53,510 able to guess the whole thing-- but I 1457 01:14:53,510 --> 01:14:56,000 did the computation many times mechanically, 1458 01:14:56,000 --> 01:14:59,440 just in the same way, because I was a hard worker. 1459 01:14:59,440 --> 01:15:01,220 So what do you have in that case? 1460 01:15:01,220 --> 01:15:09,810 1/xy squared plus 1 times x plus-- the same kind of thing. 1461 01:15:09,810 --> 01:15:14,338 Attention, this is the symmetric polynomial, and I go to that. 1462 01:15:14,338 --> 01:15:17,280 And then times 2y. 1463 01:15:17,280 --> 01:15:20,610 So, see-- that kind of easy, fast thing. 1464 01:15:20,610 --> 01:15:24,340 Why is this a good observation when 1465 01:15:24,340 --> 01:15:26,015 you have symmetric polynomials? 1466 01:15:26,015 --> 01:15:28,710 If you are on the final and you don't have that much time, 1467 01:15:28,710 --> 01:15:33,540 or on any kind of exam when you are in a time-crunch. 1468 01:15:33,540 --> 01:15:36,120 Now, we want those exams so you are not 1469 01:15:36,120 --> 01:15:38,024 going to be in a time-crunch. 1470 01:15:38,024 --> 01:15:41,665 If there is something I hate, I hate a final of 2 hours 1471 01:15:41,665 --> 01:15:44,640 and a half with 25 serious problems, 1472 01:15:44,640 --> 01:15:48,260 and you know nobody can do that. 1473 01:15:48,260 --> 01:15:50,710 So, it happens a lot. 1474 01:15:50,710 --> 01:15:55,880 I see that-- one of my jobs is also to look at the finals 1475 01:15:55,880 --> 01:15:58,586 after people wrote them, and I still 1476 01:15:58,586 --> 01:16:05,256 do that every semester-- I see too many people making finals. 1477 01:16:05,256 --> 01:16:06,880 The finals are not supposed to be long. 1478 01:16:06,880 --> 01:16:10,990 The finals are supposed to be comprehensive, cover 1479 01:16:10,990 --> 01:16:16,300 everything, but not extensive. 1480 01:16:16,300 --> 01:16:21,060 So maybe you'll have 15 problems that cover practically 1481 01:16:21,060 --> 01:16:22,911 the material entirely. 1482 01:16:22,911 --> 01:16:23,410 Why? 1483 01:16:23,410 --> 01:16:29,080 Because every little problem can have two short questions. 1484 01:16:29,080 --> 01:16:30,530 You were done with a section, you 1485 01:16:30,530 --> 01:16:34,690 shot half of a chapter only one question. 1486 01:16:34,690 --> 01:16:39,660 This is one example just-- not involving [INAUDIBLE] 1487 01:16:39,660 --> 01:16:41,210 of an expression like that, no. 1488 01:16:41,210 --> 01:16:43,310 That's too time-consuming. 1489 01:16:43,310 --> 01:16:47,600 But maybe just tangent of x-squared plus y-squared, 1490 01:16:47,600 --> 01:16:49,964 find the partial derivatives. 1491 01:16:49,964 --> 01:16:53,380 That's a good exam question, and that's enough 1492 01:16:53,380 --> 01:16:55,895 when it comes to testing partials. 1493 01:16:55,895 --> 01:16:58,130 By the way, how much-- what is that? 1494 01:16:58,130 --> 01:17:00,832 And I'm going to let you go right now. 1495 01:17:00,832 --> 01:17:01,816 Use the bathroom. 1496 01:17:01,816 --> 01:17:05,240 And when you come back from the bathroom, we'll fill in this. 1497 01:17:05,240 --> 01:17:10,515 You know I am horrible in the sense that I want-- I'm greedy. 1498 01:17:10,515 --> 01:17:12,055 I need extra time. 1499 01:17:12,055 --> 01:17:15,340 I want to use more time. 1500 01:17:15,340 --> 01:17:17,850 I will do your problems from now on, 1501 01:17:17,850 --> 01:17:22,061 and you can use the bathroom, eat something, wash your hands. 1502 01:17:22,061 --> 01:17:26,498 1503 01:17:26,498 --> 01:17:28,470 I'll start in about five minutes. 1504 01:17:28,470 --> 01:17:29,456 Don't worry. 1505 01:17:29,456 --> 01:17:32,907 1506 01:17:32,907 --> 01:17:33,893 Alexander? 1507 01:17:33,893 --> 01:17:35,372 Are you here? 1508 01:17:35,372 --> 01:17:37,837 Come get this. 1509 01:17:37,837 --> 01:17:40,350 I apologize. 1510 01:17:40,350 --> 01:17:42,426 This is long due back to you. 1511 01:17:42,426 --> 01:17:43,420 STUDENT: Oh. 1512 01:17:43,420 --> 01:17:43,920 Thank you. 1513 01:17:43,920 --> 01:17:47,406 1514 01:17:47,406 --> 01:17:49,900 STUDENT: Is there an attendance sheet today? 1515 01:17:49,900 --> 01:17:53,110 MAGDALENA TODA: I will-- I'm making up one. 1516 01:17:53,110 --> 01:17:56,750 There is already on one side attendance. 1517 01:17:56,750 --> 01:17:58,650 Let's use the other side. 1518 01:17:58,650 --> 01:18:01,566 Put today's date. 1519 01:18:01,566 --> 01:18:02,066 [INAUDIBLE] 1520 01:18:02,066 --> 01:18:44,396 1521 01:18:44,396 --> 01:18:48,380 [SIDE CONVERSATIONS] 1522 01:18:48,380 --> 01:18:56,348 1523 01:18:56,348 --> 01:18:58,340 MAGDALENA TODA: They are spoiling me. 1524 01:18:58,340 --> 01:19:03,334 They give me new sprays every week. 1525 01:19:03,334 --> 01:19:05,298 [INAUDIBLE] take care of this. 1526 01:19:05,298 --> 01:19:09,226 [SIDE CONVERSATIONS] 1527 01:19:09,226 --> 01:19:14,136 1528 01:19:14,136 --> 01:19:17,082 MAGDALENA TODA: So I'm going to ask you something. 1529 01:19:17,082 --> 01:19:20,530 And you respond honestly. 1530 01:19:20,530 --> 01:19:24,810 Which chapter-- we already browsed through three chapters. 1531 01:19:24,810 --> 01:19:26,840 I mean, Chapter 9 was vector spaces, 1532 01:19:26,840 --> 01:19:29,540 and it was all review from-- from what? 1533 01:19:29,540 --> 01:19:30,540 From Calc 2. 1534 01:19:30,540 --> 01:19:34,610 Chapter 10 was curves in [INAUDIBLE] and curves 1535 01:19:34,610 --> 01:19:36,602 in space, practically. 1536 01:19:36,602 --> 01:19:40,540 1537 01:19:40,540 --> 01:19:46,626 And Chapter 11 is functions of several variables. 1538 01:19:46,626 --> 01:19:48,640 Now you have a flavor of all of them, 1539 01:19:48,640 --> 01:19:50,475 which one was hardest for you? 1540 01:19:50,475 --> 01:19:51,476 STUDENT: 9 and 10, both. 1541 01:19:51,476 --> 01:19:52,725 MAGDALENA TODA: 9 and 10 both. 1542 01:19:52,725 --> 01:19:53,710 STUDENT: [INAUDIBLE]. 1543 01:19:53,710 --> 01:19:56,420 MAGDALENA TODA: This is so much better than the other. 1544 01:19:56,420 --> 01:20:00,660 No, I think you guys actually-- it looks better, 1545 01:20:00,660 --> 01:20:06,696 because you've seen a lot more vectors and vector functions. 1546 01:20:06,696 --> 01:20:08,592 STUDENT: I didn't understand any of 9 or 10. 1547 01:20:08,592 --> 01:20:09,540 STUDENT: [INAUDIBLE]. 1548 01:20:09,540 --> 01:20:10,370 MAGDALENA TODA: Yes, ma'am. 1549 01:20:10,370 --> 01:20:12,370 STUDENT: Could you go over parametrization [INAUDIBLE]? 1550 01:20:12,370 --> 01:20:14,370 MAGDALENA TODA: I will go over that again. 1551 01:20:14,370 --> 01:20:18,370 And I will go over some other parametrizations today. 1552 01:20:18,370 --> 01:20:24,500 And I promised that at the end, in those 20 minutes, 1553 01:20:24,500 --> 01:20:28,380 I will do that problem that gave a few of you trouble. 1554 01:20:28,380 --> 01:20:29,181 Yes, sir? 1555 01:20:29,181 --> 01:20:30,805 STUDENT: Do we take the same final exam 1556 01:20:30,805 --> 01:20:33,230 as all the other [INAUDIBLE] classes? [INAUDIBLE]? 1557 01:20:33,230 --> 01:20:36,500 MAGDALENA TODA: Well, that's what I was asked yesterday. 1558 01:20:36,500 --> 01:20:43,150 So practically, it's at the latitude of the instructor who 1559 01:20:43,150 --> 01:20:45,320 teaches honors if they write their own final, 1560 01:20:45,320 --> 01:20:48,620 and in general make it harder, or they 1561 01:20:48,620 --> 01:20:51,060 take the general final like everybody else. 1562 01:20:51,060 --> 01:20:53,988 For your formative purposes, and as a study, 1563 01:20:53,988 --> 01:20:57,892 I would like you to take the general final, 1564 01:20:57,892 --> 01:21:01,350 because I want to see where you stand compared 1565 01:21:01,350 --> 01:21:02,780 to the rest of the population. 1566 01:21:02,780 --> 01:21:06,848 So you are my sample, and they are the entire student 1567 01:21:06,848 --> 01:21:08,394 population of Calc 3, I want to make 1568 01:21:08,394 --> 01:21:14,494 the statistical analysis of your performance compared to them. 1569 01:21:14,494 --> 01:21:16,446 STUDENT: So we'll take the regular one? 1570 01:21:16,446 --> 01:21:17,422 MAGDALENA TODA: Yeah. 1571 01:21:17,422 --> 01:21:19,005 For this one, I just have to make sure 1572 01:21:19,005 --> 01:21:22,302 that they also have that extra credit added in. 1573 01:21:22,302 --> 01:21:25,840 Because if I have too much extra credit in there, 1574 01:21:25,840 --> 01:21:27,310 well they also count that. 1575 01:21:27,310 --> 01:21:28,780 So that's what that means. 1576 01:21:28,780 --> 01:21:30,250 So we can [INAUDIBLE]. 1577 01:21:30,250 --> 01:21:34,180 1578 01:21:34,180 --> 01:21:35,690 All right. 1579 01:21:35,690 --> 01:21:37,015 Let me finish this exercise. 1580 01:21:37,015 --> 01:21:41,000 And then [? stop ?] [INAUDIBLE] and go 1581 01:21:41,000 --> 01:21:45,635 over some homework problems and some parametrization problems. 1582 01:21:45,635 --> 01:21:48,605 And I will see what else. 1583 01:21:48,605 --> 01:21:55,535 So tangent of [INAUDIBLE]. 1584 01:21:55,535 --> 01:21:59,495 1585 01:21:59,495 --> 01:22:00,485 Is this hard? 1586 01:22:00,485 --> 01:22:01,970 No, it's [INAUDIBLE]. 1587 01:22:01,970 --> 01:22:05,930 But you have to remind me, because I 1588 01:22:05,930 --> 01:22:09,420 pretend that I forgot-- let me pretend 1589 01:22:09,420 --> 01:22:14,250 that I forgot what the derivative [INAUDIBLE] notation 1590 01:22:14,250 --> 01:22:17,722 of tangent of t was. 1591 01:22:17,722 --> 01:22:19,786 STUDENT: Secant squared. 1592 01:22:19,786 --> 01:22:23,237 MAGDALENA TODA: You guys love that secant squared thingy. 1593 01:22:23,237 --> 01:22:26,195 1594 01:22:26,195 --> 01:22:30,632 Why do you like secant squared? 1595 01:22:30,632 --> 01:22:34,083 I, as a student, I didn't like expressing it like that. 1596 01:22:34,083 --> 01:22:35,562 I liked [INAUDIBLE]. 1597 01:22:35,562 --> 01:22:37,041 Of course, it's the same thing. 1598 01:22:37,041 --> 01:22:40,492 But I always like it like 1 over cosine [INAUDIBLE]. 1599 01:22:40,492 --> 01:22:45,422 1600 01:22:45,422 --> 01:22:47,720 And of course, I have to ask you something, 1601 01:22:47,720 --> 01:22:52,224 because I'm curious to see what you remember. 1602 01:22:52,224 --> 01:22:55,212 And you say yeah, curiosity killed the cat. 1603 01:22:55,212 --> 01:23:00,192 But where did the derivative exist? 1604 01:23:00,192 --> 01:23:06,720 Because maybe was that tangent of T-- 1605 01:23:06,720 --> 01:23:08,300 STUDENT: Wasn't it a quotient rule 1606 01:23:08,300 --> 01:23:10,260 of sine and [? cosine x? ?] 1607 01:23:10,260 --> 01:23:11,730 MAGDALENA TODA: Good. 1608 01:23:11,730 --> 01:23:15,170 I'm proud of you. 1609 01:23:15,170 --> 01:23:17,430 That is the answer. 1610 01:23:17,430 --> 01:23:23,170 So [? my ?] [? have ?] this blowing up, this blows up-- 1611 01:23:23,170 --> 01:23:29,520 blows up where cosine T was zero, right? 1612 01:23:29,520 --> 01:23:32,380 So where did that blow up? 1613 01:23:32,380 --> 01:23:36,780 [INAUDIBLE] blow up of cosine and zero [INAUDIBLE]. 1614 01:23:36,780 --> 01:23:40,740 The cosine was the shadow on the x-axis. 1615 01:23:40,740 --> 01:23:43,992 So here you blow up here, you blow up here, you blow up here, 1616 01:23:43,992 --> 01:23:44,700 you blow up here. 1617 01:23:44,700 --> 01:23:49,155 1618 01:23:49,155 --> 01:23:51,630 So [? what does ?] [INAUDIBLE]. 1619 01:23:51,630 --> 01:23:53,115 It should not be what? 1620 01:23:53,115 --> 01:23:55,496 STUDENT: Pi over 2. 1621 01:23:55,496 --> 01:23:56,370 MAGDALENA TODA: Yeah. 1622 01:23:56,370 --> 01:23:59,630 And can we express that OK, among 0pi, 1623 01:23:59,630 --> 01:24:03,320 let's say you go in between 0 and 2pi only. 1624 01:24:03,320 --> 01:24:08,300 I get rid of pi over 2 and 3pi over 2. 1625 01:24:08,300 --> 01:24:11,940 But if I express that in general for [INAUDIBLE] T 1626 01:24:11,940 --> 01:24:15,000 not restricted to 0 to T, what do I say? 1627 01:24:15,000 --> 01:24:15,960 STUDENT: It's k. 1628 01:24:15,960 --> 01:24:18,820 STUDENT: So it can [? never be ?] pi over 2 1629 01:24:18,820 --> 01:24:19,320 plus pi? 1630 01:24:19,320 --> 01:24:21,240 MAGDALENA TODA: 2k plus 1. 1631 01:24:21,240 --> 01:24:23,952 2k plus 1. 1632 01:24:23,952 --> 01:24:25,398 Odd number over-- 1633 01:24:25,398 --> 01:24:26,243 STUDENT: Pi over 2. 1634 01:24:26,243 --> 01:24:27,326 MAGDALENA TODA: Pi over 2. 1635 01:24:27,326 --> 01:24:28,290 Odd number, pi over 2. 1636 01:24:28,290 --> 01:24:30,200 And all the odd numbers are 2k plus 1. 1637 01:24:30,200 --> 01:24:30,700 Right? 1638 01:24:30,700 --> 01:24:32,146 All right. 1639 01:24:32,146 --> 01:24:38,990 So you have a not existence and-- OK. 1640 01:24:38,990 --> 01:24:39,640 Coming back. 1641 01:24:39,640 --> 01:24:42,410 I'm just playing, because we are still in the break. 1642 01:24:42,410 --> 01:24:44,290 Now we are ready. 1643 01:24:44,290 --> 01:24:50,195 What is dfdx, del f, del x, xy. 1644 01:24:50,195 --> 01:24:51,920 And what is del f, del y? 1645 01:24:51,920 --> 01:24:55,600 I'm not going to ask you for the second partial derivative. 1646 01:24:55,600 --> 01:24:57,380 We've had enough of that. 1647 01:24:57,380 --> 01:25:05,230 We also agreed that we have important results in that. 1648 01:25:05,230 --> 01:25:08,980 What is the final answer here? 1649 01:25:08,980 --> 01:25:13,940 STUDENT: [INAUDIBLE] plus x-squared [INAUDIBLE]. 1650 01:25:13,940 --> 01:25:15,490 MAGDALENA TODA: 1 over [INAUDIBLE]. 1651 01:25:15,490 --> 01:25:17,980 I love this one, OK? 1652 01:25:17,980 --> 01:25:20,480 Don't tell me what I want to [INAUDIBLE]. 1653 01:25:20,480 --> 01:25:22,376 I'm just kidding. 1654 01:25:22,376 --> 01:25:23,940 [INAUDIBLE] squared times-- 1655 01:25:23,940 --> 01:25:24,805 STUDENT: 2x. 1656 01:25:24,805 --> 01:25:26,450 MAGDALENA TODA: 2x, good. 1657 01:25:26,450 --> 01:25:27,700 How about the other one? 1658 01:25:27,700 --> 01:25:28,440 The same thing. 1659 01:25:28,440 --> 01:25:34,320 1660 01:25:34,320 --> 01:25:36,670 Times 2y. 1661 01:25:36,670 --> 01:25:41,840 1662 01:25:41,840 --> 01:25:43,300 OK. 1663 01:25:43,300 --> 01:25:46,280 I want to tell you something that I will repeat. 1664 01:25:46,280 --> 01:25:49,365 But you will see it all through the course. 1665 01:25:49,365 --> 01:25:52,180 There is a certain notion that Alexander, 1666 01:25:52,180 --> 01:25:54,435 who is not talking-- I'm just kidding, 1667 01:25:54,435 --> 01:25:58,400 you can talk-- he reminded me of gradient. 1668 01:25:58,400 --> 01:26:02,960 We don't talk about gradient until a few sections from now. 1669 01:26:02,960 --> 01:26:05,120 But I'd like to anticipate a little bit. 1670 01:26:05,120 --> 01:26:08,440 So the gradient of a function, wherever 1671 01:26:08,440 --> 01:26:15,080 the partial derivatives exist, with the partial derivative-- 1672 01:26:15,080 --> 01:26:21,215 that is, f sub x and f sub y exist-- 1673 01:26:21,215 --> 01:26:26,572 I'm going to have that delta f-- nabla f. 1674 01:26:26,572 --> 01:26:29,494 nabla is a [INAUDIBLE]. 1675 01:26:29,494 --> 01:26:34,340 Nable f at xy represents what? 1676 01:26:34,340 --> 01:26:34,850 The vector. 1677 01:26:34,850 --> 01:26:37,400 1678 01:26:37,400 --> 01:26:39,020 And I know you love vectors. 1679 01:26:39,020 --> 01:26:45,580 And that's why I'm going back to the vector notation f sub x 1680 01:26:45,580 --> 01:26:51,508 at xy times i, i being the standard vector i 1681 01:26:51,508 --> 01:26:59,130 unit along the x axis, f sub y at xy times j. 1682 01:26:59,130 --> 01:27:03,450 STUDENT: So it's just like the notation of [INAUDIBLE]? 1683 01:27:03,450 --> 01:27:05,322 MAGDALENA TODA: Just the vector notation. 1684 01:27:05,322 --> 01:27:08,010 How else could I write it? 1685 01:27:08,010 --> 01:27:13,280 Angular bracket, f sub x x at xy, comma, f sub y at xy. 1686 01:27:13,280 --> 01:27:16,935 And you know-- people who saw my videos, colleagues 1687 01:27:16,935 --> 01:27:19,770 who teach Calc 3 at the same time 1688 01:27:19,770 --> 01:27:25,110 said I have a tendency of not going by the book notations 1689 01:27:25,110 --> 01:27:28,260 all the time, and just give you the [? round ?] parentheses. 1690 01:27:28,260 --> 01:27:28,960 It's OK. 1691 01:27:28,960 --> 01:27:31,340 I mean, different books, different notations. 1692 01:27:31,340 --> 01:27:35,460 But what I mean is to represent the vector in the standard way 1693 01:27:35,460 --> 01:27:37,430 [INAUDIBLE]. 1694 01:27:37,430 --> 01:27:38,010 All right. 1695 01:27:38,010 --> 01:27:38,510 OK. 1696 01:27:38,510 --> 01:27:41,760 Can you have this notion for something 1697 01:27:41,760 --> 01:27:44,840 like a function of three variables? 1698 01:27:44,840 --> 01:27:45,870 Absolutely. 1699 01:27:45,870 --> 01:27:48,430 Now I'll give you an easy one. 1700 01:27:48,430 --> 01:27:50,920 Suppose that you have x-squared plus y-squared 1701 01:27:50,920 --> 01:27:54,402 plus z-squared equals 1. 1702 01:27:54,402 --> 01:28:00,130 And that is called-- let's call it names-- f of x, y, z. 1703 01:28:00,130 --> 01:28:16,800 Compute the gradient nabla f at any point x, y, z for f. 1704 01:28:16,800 --> 01:28:20,705 Find the meaning of that gradient-- of that-- find 1705 01:28:20,705 --> 01:28:29,460 the geometric meaning of it. 1706 01:28:29,460 --> 01:28:34,210 For this case, not in general, for this case. 1707 01:28:34,210 --> 01:28:35,860 So you say, wait, wait, Magdalena. 1708 01:28:35,860 --> 01:28:38,846 A-dah-dah, you're confusing me. 1709 01:28:38,846 --> 01:28:39,941 This is the gradient. 1710 01:28:39,941 --> 01:28:40,440 Hmm. 1711 01:28:40,440 --> 01:28:43,360 Depends on how many variables you have. 1712 01:28:43,360 --> 01:28:47,680 So you have to show a vector whose coordinates represent 1713 01:28:47,680 --> 01:28:50,800 the partial derivatives with respect to all the variables. 1714 01:28:50,800 --> 01:28:56,065 If I have n variables, I have f sub x1 comma f sub x2 comma 1715 01:28:56,065 --> 01:28:58,770 f sub x3 comma f sub xn, and stop. 1716 01:28:58,770 --> 01:28:59,570 Yes, sir. 1717 01:28:59,570 --> 01:29:03,610 STUDENT: If the formula was just f of xy, 1718 01:29:03,610 --> 01:29:05,160 wouldn't that be implicit? 1719 01:29:05,160 --> 01:29:06,640 MAGDALENA TODA: That is implicit. 1720 01:29:06,640 --> 01:29:08,510 That's exactly what I meant. 1721 01:29:08,510 --> 01:29:12,220 What's the geometric meaning of this animal? 1722 01:29:12,220 --> 01:29:13,920 Forget about the left hand side. 1723 01:29:13,920 --> 01:29:15,510 I'm going to clean it quickly. 1724 01:29:15,510 --> 01:29:16,740 What is that animal? 1725 01:29:16,740 --> 01:29:19,640 That is a hippopotamus. 1726 01:29:19,640 --> 01:29:20,500 What is that? 1727 01:29:20,500 --> 01:29:22,148 STUDENT: It's a sphere. 1728 01:29:22,148 --> 01:29:23,398 MAGDALENA TODA: It's a sphere. 1729 01:29:23,398 --> 01:29:24,847 But what kind of sphere? 1730 01:29:24,847 --> 01:29:27,950 Center 0, 0, 0 with radius 1. 1731 01:29:27,950 --> 01:29:30,150 What do we call that? 1732 01:29:30,150 --> 01:29:30,670 Unit sphere. 1733 01:29:30,670 --> 01:29:33,594 Do you know what notation that mathematicians 1734 01:29:33,594 --> 01:29:36,516 use for that object? 1735 01:29:36,516 --> 01:29:40,380 You don't know but I'll tell you. s1 is the sphere. 1736 01:29:40,380 --> 01:29:42,340 We have s2, I'm sorry, the sphere 1737 01:29:42,340 --> 01:29:45,190 of dimension 2, which means the surface. 1738 01:29:45,190 --> 01:29:47,127 s1 is the circle. 1739 01:29:47,127 --> 01:29:49,115 s1 is a circle. 1740 01:29:49,115 --> 01:29:51,580 s2 is a sphere. 1741 01:29:51,580 --> 01:29:54,860 So what is this number here for a mathematician? 1742 01:29:54,860 --> 01:29:59,050 That's the dimension of that kind of manifold. 1743 01:29:59,050 --> 01:30:02,310 So if I have just a circle, we call it s1 1744 01:30:02,310 --> 01:30:05,510 because there is only a one independent variable, which 1745 01:30:05,510 --> 01:30:08,000 is time, and we parameterize. 1746 01:30:08,000 --> 01:30:09,135 Why go clockwise? 1747 01:30:09,135 --> 01:30:09,745 Shame on me. 1748 01:30:09,745 --> 01:30:12,190 Go counterclockwise. 1749 01:30:12,190 --> 01:30:13,030 All right. 1750 01:30:13,030 --> 01:30:14,110 That's s1. 1751 01:30:14,110 --> 01:30:16,260 For s2, I have two degrees of freedom. 1752 01:30:16,260 --> 01:30:18,770 It's a surface. 1753 01:30:18,770 --> 01:30:23,005 On earth, what are those two degrees of freedom? 1754 01:30:23,005 --> 01:30:25,980 It's a riddle. 1755 01:30:25,980 --> 01:30:27,065 No extra credit. 1756 01:30:27,065 --> 01:30:30,320 STUDENT: The latitude and longitude? 1757 01:30:30,320 --> 01:30:31,882 MAGDALENA TODA: Who said it? 1758 01:30:31,882 --> 01:30:33,532 Who said it first? 1759 01:30:33,532 --> 01:30:35,300 STUDENT: [INAUDIBLE]. 1760 01:30:35,300 --> 01:30:40,185 MAGDALENA TODA: How many of you said it at the same time? 1761 01:30:40,185 --> 01:30:40,935 Alexander said it. 1762 01:30:40,935 --> 01:30:42,726 STUDENT: I know there was one other person. 1763 01:30:42,726 --> 01:30:43,910 I wasn't the only one. 1764 01:30:43,910 --> 01:30:44,660 STUDENT: I didn't. 1765 01:30:44,660 --> 01:30:47,490 1766 01:30:47,490 --> 01:30:48,882 STUDENT: [INAUDIBLE], sorry. 1767 01:30:48,882 --> 01:30:49,715 [INTERPOSING VOICES] 1768 01:30:49,715 --> 01:30:52,420 MAGDALENA TODA: I don't have enough. 1769 01:30:52,420 --> 01:30:54,340 STUDENT: I'll take the credit for it. 1770 01:30:54,340 --> 01:30:56,170 MAGDALENA TODA: [INAUDIBLE] extra credit. 1771 01:30:56,170 --> 01:30:58,620 OK, you choose. 1772 01:30:58,620 --> 01:30:59,560 These are good. 1773 01:30:59,560 --> 01:31:01,950 They are Valentine's hearts, chocolate [INAUDIBLE]. 1774 01:31:01,950 --> 01:31:04,791 1775 01:31:04,791 --> 01:31:05,290 Wilson. 1776 01:31:05,290 --> 01:31:09,385 1777 01:31:09,385 --> 01:31:12,040 I heard you saying Wilson. 1778 01:31:12,040 --> 01:31:13,008 I have more. 1779 01:31:13,008 --> 01:31:13,976 I have more. 1780 01:31:13,976 --> 01:31:17,364 These are cough drops, so I'm [INAUDIBLE]. 1781 01:31:17,364 --> 01:31:20,489 You set it right next time, Alexander. 1782 01:31:20,489 --> 01:31:21,968 STUDENT: [INAUDIBLE]. 1783 01:31:21,968 --> 01:31:22,954 MAGDALENA TODA: OK. 1784 01:31:22,954 --> 01:31:23,940 Anybody else? 1785 01:31:23,940 --> 01:31:26,398 Anybody needing cough drops? 1786 01:31:26,398 --> 01:31:26,898 OK. 1787 01:31:26,898 --> 01:31:27,884 I'll leave them here. 1788 01:31:27,884 --> 01:31:29,363 Just let me see. 1789 01:31:29,363 --> 01:31:31,830 Do I have more chocolate? 1790 01:31:31,830 --> 01:31:32,986 Eh, next time. 1791 01:31:32,986 --> 01:31:35,160 I'm going to get some before-- we have-- we 1792 01:31:35,160 --> 01:31:37,090 need before Valentine's, right? 1793 01:31:37,090 --> 01:31:37,970 So it's Thursday. 1794 01:31:37,970 --> 01:31:41,060 I'm going to bring you a lot more. 1795 01:31:41,060 --> 01:31:46,650 So in that case, what is the gradient of f? 1796 01:31:46,650 --> 01:31:47,970 An x, y, z. 1797 01:31:47,970 --> 01:31:48,470 Aha. 1798 01:31:48,470 --> 01:31:50,170 I have three variables. 1799 01:31:50,170 --> 01:31:52,520 What's the gradient? 1800 01:31:52,520 --> 01:31:56,050 I can write it as a bracket, angular notation. 1801 01:31:56,050 --> 01:31:58,070 Am I right? 1802 01:31:58,070 --> 01:32:02,790 Or I can write it 2xi plus 2ij plus 2zk. 1803 01:32:02,790 --> 01:32:06,590 Can anybody tell me why? 1804 01:32:06,590 --> 01:32:09,640 What in the world are these, 2x, 2y, 2z? 1805 01:32:09,640 --> 01:32:11,697 STUDENT: Those are the partial derivatives. 1806 01:32:11,697 --> 01:32:14,030 MAGDALENA TODA: They are exactly the partial derivatives 1807 01:32:14,030 --> 01:32:17,810 with respect to x, with respect to y, with respect to z. 1808 01:32:17,810 --> 01:32:19,410 Does this have a geometric meaning? 1809 01:32:19,410 --> 01:32:20,540 I don't know. 1810 01:32:20,540 --> 01:32:21,870 I have to draw. 1811 01:32:21,870 --> 01:32:24,210 And maybe when I draw, I get an idea. 1812 01:32:24,210 --> 01:32:29,105 1813 01:32:29,105 --> 01:32:31,526 Is this a unit vector? 1814 01:32:31,526 --> 01:32:32,482 Uh-uh. 1815 01:32:32,482 --> 01:32:33,980 It's not. 1816 01:32:33,980 --> 01:32:35,770 Nabla s, right. 1817 01:32:35,770 --> 01:32:36,490 In a way it is. 1818 01:32:36,490 --> 01:32:37,800 It's not a unit vector. 1819 01:32:37,800 --> 01:32:41,040 But if I were to [? uniterize ?] it-- 1820 01:32:41,040 --> 01:32:43,710 and you know very well what it means to [? uniterize it ?]. 1821 01:32:43,710 --> 01:32:44,732 It means to-- 1822 01:32:44,732 --> 01:32:45,690 STUDENT: Divide it by-- 1823 01:32:45,690 --> 01:32:47,440 MAGDALENA TODA: Divide it by its magnitude 1824 01:32:47,440 --> 01:32:51,075 and make it a unit vector that would have a meaning. 1825 01:32:51,075 --> 01:32:52,180 This is the sphere. 1826 01:32:52,180 --> 01:32:56,016 1827 01:32:56,016 --> 01:32:57,670 What if I make like this? 1828 01:32:57,670 --> 01:33:04,145 n equals nabla f over a magnitude of f. 1829 01:33:04,145 --> 01:33:10,050 And what is the meaning of that going to be? 1830 01:33:10,050 --> 01:33:12,066 Can you tell me what I'm going to get here? 1831 01:33:12,066 --> 01:33:18,830 1832 01:33:18,830 --> 01:33:24,870 In your head, compute the magnitude 1833 01:33:24,870 --> 01:33:29,430 and divide by the magnitude, and you have exactly 15 seconds 1834 01:33:29,430 --> 01:33:31,420 to tell me what it is. 1835 01:33:31,420 --> 01:33:32,845 STUDENT: [INAUDIBLE]. 1836 01:33:32,845 --> 01:33:34,470 MAGDALENA TODA: [? Ryan, ?] [? Ryan, ?] 1837 01:33:34,470 --> 01:33:36,170 you are in a Twilight Zone. 1838 01:33:36,170 --> 01:33:39,798 But I'm sure once I tell you, once I tell you, [INAUDIBLE]. 1839 01:33:39,798 --> 01:33:41,548 STUDENT: 1 divided by the square root of 2 1840 01:33:41,548 --> 01:33:42,922 for the [? i controller. ?] 1841 01:33:42,922 --> 01:33:43,797 STUDENT: [INAUDIBLE]. 1842 01:33:43,797 --> 01:33:47,610 1843 01:33:47,610 --> 01:33:49,730 MAGDALENA TODA: Well, OK. 1844 01:33:49,730 --> 01:33:51,125 Say it again, somebody. 1845 01:33:51,125 --> 01:33:52,990 STUDENT: x plus y plus z. 1846 01:33:52,990 --> 01:33:58,112 MAGDALENA TODA: xi plus yj plus zk, not x plus x, y, 1847 01:33:58,112 --> 01:33:59,820 z because that would be a mistake. 1848 01:33:59,820 --> 01:34:03,325 It would be a scalar function. [INAUDIBLE] has to be a vector. 1849 01:34:03,325 --> 01:34:07,190 If I am to draw this vector, how am I going to draw it? 1850 01:34:07,190 --> 01:34:10,030 Well, this is the position vector. 1851 01:34:10,030 --> 01:34:11,220 Say it again. 1852 01:34:11,220 --> 01:34:12,720 This is the position vector. 1853 01:34:12,720 --> 01:34:15,881 When I have a point on this stinking earth, whatever 1854 01:34:15,881 --> 01:34:21,100 it is, x, y, z, the position vector is x, y, z. 1855 01:34:21,100 --> 01:34:26,260 It's xi plus yj plus zk. 1856 01:34:26,260 --> 01:34:28,690 I have this identification between the point 1857 01:34:28,690 --> 01:34:29,639 and the vector. 1858 01:34:29,639 --> 01:34:30,430 This is our vector. 1859 01:34:30,430 --> 01:34:33,300 So I'm going to draw these needles, all these needles, 1860 01:34:33,300 --> 01:34:41,880 all these vectors whose tips are exactly on the sphere. 1861 01:34:41,880 --> 01:34:42,960 So why? 1862 01:34:42,960 --> 01:34:43,870 You say, OK. 1863 01:34:43,870 --> 01:34:46,470 I understand that is the position vector, 1864 01:34:46,470 --> 01:34:48,735 but why did you put an n here? 1865 01:34:48,735 --> 01:34:52,900 And anybody who answers that gets a cough drops. 1866 01:34:52,900 --> 01:34:54,737 STUDENT: [INAUDIBLE]. 1867 01:34:54,737 --> 01:34:56,070 MAGDALENA TODA: Because that is? 1868 01:34:56,070 --> 01:34:58,235 STUDENT: The normal to the surface. 1869 01:34:58,235 --> 01:34:59,360 MAGDALENA TODA: You get a-- 1870 01:34:59,360 --> 01:35:00,824 STUDENT: Yeah, cough drop. 1871 01:35:00,824 --> 01:35:02,288 MAGDALENA TODA: Two of them. 1872 01:35:02,288 --> 01:35:03,157 STUDENT: Aw, yeah. 1873 01:35:03,157 --> 01:35:04,240 MAGDALENA TODA: All right. 1874 01:35:04,240 --> 01:35:07,840 So that's the normal to the surface, which 1875 01:35:07,840 --> 01:35:11,087 would be a continuation of the position vector. 1876 01:35:11,087 --> 01:35:11,670 You see, guys? 1877 01:35:11,670 --> 01:35:14,450 So imagine you take your position vector. 1878 01:35:14,450 --> 01:35:15,840 This is the sphere. 1879 01:35:15,840 --> 01:35:17,580 It's like an egg. 1880 01:35:17,580 --> 01:35:20,550 And these tips are on the sphere. 1881 01:35:20,550 --> 01:35:24,950 If you continue from sitting on the sphere, 1882 01:35:24,950 --> 01:35:29,240 another radius vector colinear to that, 1883 01:35:29,240 --> 01:35:31,280 that would be the normal to the sphere. 1884 01:35:31,280 --> 01:35:36,480 So in topology, we have a name for that. 1885 01:35:36,480 --> 01:35:38,914 We call that the hairy ball. 1886 01:35:38,914 --> 01:35:41,970 The hairy ball in mathematics, I'm not kidding, 1887 01:35:41,970 --> 01:35:44,410 it's a concentrated notations. 1888 01:35:44,410 --> 01:35:47,620 You see it in graduate courses, if you're 1889 01:35:47,620 --> 01:35:50,157 going to become a graduate student in mathematics, 1890 01:35:50,157 --> 01:35:51,990 or you want to do a dual degree or whatever, 1891 01:35:51,990 --> 01:35:55,690 you're going to see the hairy ball, all those normal vectors 1892 01:35:55,690 --> 01:35:58,820 of length 1. 1893 01:35:58,820 --> 01:36:01,620 It's also called the normal field. 1894 01:36:01,620 --> 01:36:04,526 So if you ask Dr. Ibragimov, because he 1895 01:36:04,526 --> 01:36:08,900 is in this kind of field theory, [INAUDIBLE] normal field 1896 01:36:08,900 --> 01:36:10,080 to a surface. 1897 01:36:10,080 --> 01:36:13,010 But for the topologists or geometers, 1898 01:36:13,010 --> 01:36:15,300 they say, oh, that's the hairy ball. 1899 01:36:15,300 --> 01:36:18,860 So if you ask him what the hairy ball is, he will say, 1900 01:36:18,860 --> 01:36:21,800 why are you talking nonsense to me? 1901 01:36:21,800 --> 01:36:22,780 Right. 1902 01:36:22,780 --> 01:36:24,250 Exactly. 1903 01:36:24,250 --> 01:36:30,745 So here's where we stopped our intrusion in chapter 11. 1904 01:36:30,745 --> 01:36:33,129 It's going to be as fun as it was today 1905 01:36:33,129 --> 01:36:34,420 with these partial derivatives. 1906 01:36:34,420 --> 01:36:35,590 You're going to love them. 1907 01:36:35,590 --> 01:36:39,860 You have a lot of computations like the ones we did today. 1908 01:36:39,860 --> 01:36:42,590 Let's go back to something you hated, 1909 01:36:42,590 --> 01:36:45,630 which is the parameterizations. 1910 01:36:45,630 --> 01:36:48,590 So one of you-- no, three of you-- 1911 01:36:48,590 --> 01:36:51,606 asked me to redo one problem like the one 1912 01:36:51,606 --> 01:36:54,366 with the parameterization of a circle. 1913 01:36:54,366 --> 01:36:58,280 But now I have to pay attention to the data 1914 01:36:58,280 --> 01:36:59,920 that I come up with. 1915 01:36:59,920 --> 01:37:14,258 So write the parameterization of a circle of radius. 1916 01:37:14,258 --> 01:37:17,240 1917 01:37:17,240 --> 01:37:20,888 Do you want specific data or you want letters? 1918 01:37:20,888 --> 01:37:21,763 STUDENT: [INAUDIBLE]. 1919 01:37:21,763 --> 01:37:25,561 1920 01:37:25,561 --> 01:37:26,352 MAGDALENA TODA: OK. 1921 01:37:26,352 --> 01:37:30,480 Let's do it [INAUDIBLE] r, and then I'll give an example. 1922 01:37:30,480 --> 01:37:43,230 And center x0, y0 in plane where-- what is the point? 1923 01:37:43,230 --> 01:37:57,200 Where is the particle moving for time t equals 0? 1924 01:37:57,200 --> 01:37:59,260 Where is it located? 1925 01:37:59,260 --> 01:38:00,310 All right. 1926 01:38:00,310 --> 01:38:02,746 So review. 1927 01:38:02,746 --> 01:38:15,624 We had frame that we always picked at the origin. 1928 01:38:15,624 --> 01:38:23,490 That was bad because we could pick x0, y0 as a center, 1929 01:38:23,490 --> 01:38:25,115 and that has a separate radius. 1930 01:38:25,115 --> 01:38:31,905 1931 01:38:31,905 --> 01:38:39,050 And now, they want me to write a parameterization of a circle. 1932 01:38:39,050 --> 01:38:41,020 How do you achieve it? 1933 01:38:41,020 --> 01:38:49,400 You say the circle is x minus x0 squared plus y minus y0 squared 1934 01:38:49,400 --> 01:38:50,980 equals r squared. 1935 01:38:50,980 --> 01:38:53,740 And one of you asked me by email-- 1936 01:38:53,740 --> 01:38:56,650 and that was a good question-- you said, come on. 1937 01:38:56,650 --> 01:38:58,920 Look, it was [INAUDIBLE]. 1938 01:38:58,920 --> 01:39:02,760 So you said, I was quite good in math. 1939 01:39:02,760 --> 01:39:04,050 I was smart. 1940 01:39:04,050 --> 01:39:09,550 Why didn't I know the equations, the parametric equations, 1941 01:39:09,550 --> 01:39:11,490 or even this? 1942 01:39:11,490 --> 01:39:13,730 I'll tell you why. 1943 01:39:13,730 --> 01:39:15,850 This used to be covered in high school. 1944 01:39:15,850 --> 01:39:18,056 It's something called college algebra. 1945 01:39:18,056 --> 01:39:21,460 We had a chapter, either trigonometry 1946 01:39:21,460 --> 01:39:22,272 or college algebra. 1947 01:39:22,272 --> 01:39:24,520 We had a chapter called analytic geometry. 1948 01:39:24,520 --> 01:39:26,340 This is analytic geometry. 1949 01:39:26,340 --> 01:39:28,530 It's the same chapter in which you guys 1950 01:39:28,530 --> 01:39:33,510 covered conics, [INAUDIBLE], ellipse, [INAUDIBLE], parabola. 1951 01:39:33,510 --> 01:39:36,120 It's no longer covered in most high schools. 1952 01:39:36,120 --> 01:39:37,030 I asked around. 1953 01:39:37,030 --> 01:39:39,920 The teachers told me that we reduced 1954 01:39:39,920 --> 01:39:41,810 the geometric applications a lot, 1955 01:39:41,810 --> 01:39:47,920 according to the general standards that are imposed. 1956 01:39:47,920 --> 01:39:51,603 That's a pity, because you really need this in college. 1957 01:39:51,603 --> 01:39:52,590 All right. 1958 01:39:52,590 --> 01:39:55,520 So how do you come up with a parameterization? 1959 01:39:55,520 --> 01:40:01,060 You say, I would like to parameterize in such way 1960 01:40:01,060 --> 01:40:03,490 that this would be easy to understand 1961 01:40:03,490 --> 01:40:06,300 this for Pythagorean theorem. 1962 01:40:06,300 --> 01:40:07,450 Oh, OK. 1963 01:40:07,450 --> 01:40:10,395 So what is the Pythagorean theorem telling me? 1964 01:40:10,395 --> 01:40:14,240 It's telling you that if you are in a unit circle practically, 1965 01:40:14,240 --> 01:40:19,005 then this is cosine and theta and this is sine theta, 1966 01:40:19,005 --> 01:40:21,637 and the sum of cosine theta squared 1967 01:40:21,637 --> 01:40:24,050 plus sine theta squared is 1. 1968 01:40:24,050 --> 01:40:26,778 This is 1, so that is the Pythagorean theorem 1969 01:40:26,778 --> 01:40:28,722 [INAUDIBLE]. 1970 01:40:28,722 --> 01:40:38,230 So xy plus x0 should be cosine of theta times an R. Why an R? 1971 01:40:38,230 --> 01:40:41,920 Because I want, when I square, I want the R squared up. 1972 01:40:41,920 --> 01:40:46,230 And here, this guy inside will be our sine [? thing. ?] 1973 01:40:46,230 --> 01:40:47,550 Am I going to be in good shape? 1974 01:40:47,550 --> 01:40:51,450 Yes, because when I square this fellow squared 1975 01:40:51,450 --> 01:40:54,660 plus this fellow squared will give me exactly R squared. 1976 01:40:54,660 --> 01:40:58,300 And here is my [INAUDIBLE] smiley face. 1977 01:40:58,300 --> 01:41:01,260 So I want to understand what I'm doing. 1978 01:41:01,260 --> 01:41:05,440 x minus x0 must be R cosine theta. 1979 01:41:05,440 --> 01:41:09,106 y minus y0 is R sine theta. 1980 01:41:09,106 --> 01:41:13,860 Theta in general is an angular velocity, [INAUDIBLE]. 1981 01:41:13,860 --> 01:41:17,250 But it's also time, right? 1982 01:41:17,250 --> 01:41:19,290 It has the meaning of time parameter. 1983 01:41:19,290 --> 01:41:22,980 So when we wrote those-- and some of you are bored, 1984 01:41:22,980 --> 01:41:25,620 but I think it's not going to harm anybody 1985 01:41:25,620 --> 01:41:27,240 that I do this again. 1986 01:41:27,240 --> 01:41:36,406 R cosine of t plus x0 y is R sine t plus x0, or plus y0. 1987 01:41:36,406 --> 01:41:41,490 Now note, all those examples in web work, 1988 01:41:41,490 --> 01:41:43,840 they were not very imaginative. 1989 01:41:43,840 --> 01:41:47,580 They didn't mean for you to try other things. 1990 01:41:47,580 --> 01:41:53,630 Like if one would put here cosine of 5t or sine of 5t, 1991 01:41:53,630 --> 01:41:56,830 that person would move five times faster on the circle. 1992 01:41:56,830 --> 01:42:00,240 And instead of being back at 2 pi, in time 2 pi, 1993 01:42:00,240 --> 01:42:02,970 they would be there in time 2 pi over 5. 1994 01:42:02,970 --> 01:42:06,860 All the examples-- and each of you, it was randomized somehow. 1995 01:42:06,860 --> 01:42:09,730 Each of you has a different data set. 1996 01:42:09,730 --> 01:42:11,970 Different R, different x0 with 0, 1997 01:42:11,970 --> 01:42:15,570 and a different place where the particle is moving. 1998 01:42:15,570 --> 01:42:18,580 But no matter what they gave you, 1999 01:42:18,580 --> 01:42:21,910 it's a response to the same problem. 2000 01:42:21,910 --> 01:42:26,930 And at time t equals 0, you have M. Do 2001 01:42:26,930 --> 01:42:28,650 you want me to call it M0? 2002 01:42:28,650 --> 01:42:33,090 Yes, from my initial-- M0. 2003 01:42:33,090 --> 01:42:41,040 For t equals 0, you're going to have R plus x0. 2004 01:42:41,040 --> 01:42:44,670 And for t equals 0, you have y0. 2005 01:42:44,670 --> 01:42:50,425 So for example, Ryan had-- Ryan, I don't remember what you had. 2006 01:42:50,425 --> 01:42:54,232 You had some where theta R was-- 2007 01:42:54,232 --> 01:42:54,940 STUDENT: 4 and 8. 2008 01:42:54,940 --> 01:42:57,057 MAGDALENA TODA: 7. 2009 01:42:57,057 --> 01:42:58,015 You, what did you have? 2010 01:42:58,015 --> 01:43:00,400 STUDENT: No, R was 7 and x was 3, y was 1. 2011 01:43:00,400 --> 01:43:03,510 MAGDALENA TODA: R was 7 and x0 was-- 2012 01:43:03,510 --> 01:43:05,740 STUDENT: 3, 1. 2013 01:43:05,740 --> 01:43:11,570 MAGDALENA TODA: 3, 1 was x0, y0 so in that case, the point they 2014 01:43:11,570 --> 01:43:15,820 gave here was 7 plus 3. 2015 01:43:15,820 --> 01:43:16,825 Am I right, Ryan? 2016 01:43:16,825 --> 01:43:17,700 You can always check. 2017 01:43:17,700 --> 01:43:18,200 I remember. 2018 01:43:18,200 --> 01:43:22,010 It was 10 and God knows, and 10 and 1. 2019 01:43:22,010 --> 01:43:25,810 So all of the data that you had in that problem 2020 01:43:25,810 --> 01:43:30,480 was created so that you have these equations. 2021 01:43:30,480 --> 01:43:36,392 And at time 0, you were exactly at the time t equals 0 replaced 2022 01:43:36,392 --> 01:43:37,181 the t. 2023 01:43:37,181 --> 01:43:38,130 All right. 2024 01:43:38,130 --> 01:43:39,230 OK. 2025 01:43:39,230 --> 01:43:40,315 STUDENT: What's the M0? 2026 01:43:40,315 --> 01:43:42,280 What is-- 2027 01:43:42,280 --> 01:43:45,240 MAGDALENA TODA: M0 is Magdalena times 0. 2028 01:43:45,240 --> 01:43:46,820 I don't know. 2029 01:43:46,820 --> 01:43:51,045 I mean, it's the point where you are. 2030 01:43:51,045 --> 01:43:55,140 I couldn't come up with a better name. 2031 01:43:55,140 --> 01:44:01,569 So I'm going to erase here and I'll 2032 01:44:01,569 --> 01:44:08,280 get to another problem, which gave you guys a big headache. 2033 01:44:08,280 --> 01:44:16,570 And it's not so hard, but this is the computational problem, 2034 01:44:16,570 --> 01:44:18,046 very pretty in itself. 2035 01:44:18,046 --> 01:44:24,934 2036 01:44:24,934 --> 01:44:35,706 [INAUDIBLE] cosine t i plus e to the 3t sine t j plus e 2037 01:44:35,706 --> 01:44:36,540 to the 3tk. 2038 01:44:36,540 --> 01:44:40,040 2039 01:44:40,040 --> 01:44:43,850 And I think this was more or less in everybody's homework 2040 01:44:43,850 --> 01:44:45,480 the same. 2041 01:44:45,480 --> 01:44:51,680 There's a position vector given as parameterized form. 2042 01:44:51,680 --> 01:44:54,330 So since you love parameterization so much, 2043 01:44:54,330 --> 01:45:00,523 I'm going to remind you what that means for x and y and zr. 2044 01:45:00,523 --> 01:45:03,481 And what did they want from you? 2045 01:45:03,481 --> 01:45:07,920 I forget what number of the problem that was. 2046 01:45:07,920 --> 01:45:16,270 They wanted the length of the arc of a curve from t 2047 01:45:16,270 --> 01:45:18,604 equals-- I don't know. 2048 01:45:18,604 --> 01:45:19,580 STUDENT: 2 to 5. 2049 01:45:19,580 --> 01:45:21,044 MAGDALENA TODA: 2 to 5. 2050 01:45:21,044 --> 01:45:22,020 Thank you. 2051 01:45:22,020 --> 01:45:23,972 [INAUDIBLE] t equals 5. 2052 01:45:23,972 --> 01:45:30,316 So this is the beginning and the end of the curve, the beginning 2053 01:45:30,316 --> 01:45:32,268 and the end of a curve. 2054 01:45:32,268 --> 01:45:35,630 So what is that going to be [INAUDIBLE]? 2055 01:45:35,630 --> 01:45:40,500 How does [INAUDIBLE], which we have 2056 01:45:40,500 --> 01:45:46,828 to write down 2 to 5 magnitude of r prime at t, dt. 2057 01:45:46,828 --> 01:45:50,140 2058 01:45:50,140 --> 01:45:53,000 And I don't know. 2059 01:45:53,000 --> 01:45:56,980 But I want to review this because-- so what in the world? 2060 01:45:56,980 --> 01:45:59,540 Maybe I put this on the midterm or I 2061 01:45:59,540 --> 01:46:03,530 make it a little bit easier, but the same what I don't like, 2062 01:46:03,530 --> 01:46:05,070 it's time consuming. 2063 01:46:05,070 --> 01:46:07,780 But I can give you something a lot easier 2064 01:46:07,780 --> 01:46:10,570 that tests the concept, the idea, not 2065 01:46:10,570 --> 01:46:13,090 the computational power. 2066 01:46:13,090 --> 01:46:20,210 So r prime of t here with a little bit of attention, 2067 01:46:20,210 --> 01:46:25,100 of course, most of you computing this correctly. 2068 01:46:25,100 --> 01:46:28,010 You are just a little bit scared of what happened after that, 2069 01:46:28,010 --> 01:46:30,467 and you should not be scared because now I'll tell you 2070 01:46:30,467 --> 01:46:32,860 why you shouldn't be scared. 2071 01:46:32,860 --> 01:46:34,850 Chain rule, product rule. 2072 01:46:34,850 --> 01:46:37,890 So I have first prime-- 2073 01:46:37,890 --> 01:46:38,900 STUDENT: 3. 2074 01:46:38,900 --> 01:46:42,490 MAGDALENA TODA: 3 into the 3e second and [? time ?] 2075 01:46:42,490 --> 01:46:46,730 cosine t plus-- I'm going to do that later. 2076 01:46:46,730 --> 01:46:48,320 I know what you're thinking. 2077 01:46:48,320 --> 01:46:49,792 STUDENT: e 3t. 2078 01:46:49,792 --> 01:46:53,264 MAGDALENA TODA: e to the 3t minus sine. 2079 01:46:53,264 --> 01:46:55,600 I'm not worried about this minus now. 2080 01:46:55,600 --> 01:46:57,430 I'll take care of that later. 2081 01:46:57,430 --> 01:46:58,010 Times i. 2082 01:46:58,010 --> 01:47:00,710 2083 01:47:00,710 --> 01:47:03,390 Now with your permission-- when you 2084 01:47:03,390 --> 01:47:08,150 say, why is she not writing the whole thing in continuation? 2085 01:47:08,150 --> 01:47:09,470 Because I don't want to. 2086 01:47:09,470 --> 01:47:09,970 No. 2087 01:47:09,970 --> 01:47:13,170 Because I want to help you see what's going on. 2088 01:47:13,170 --> 01:47:16,310 You do the same kind of stuff for this individual one. 2089 01:47:16,310 --> 01:47:17,804 I want to put it right underneath. 2090 01:47:17,804 --> 01:47:21,290 If I put it right underneath, it's going to [? agree ?]. 2091 01:47:21,290 --> 01:47:23,966 Otherwise it's not going to [? agree ?]. 2092 01:47:23,966 --> 01:47:32,371 E to the 3t times sine t plus e to the 3t cosine t. 2093 01:47:32,371 --> 01:47:34,111 You didn't have a problem because you 2094 01:47:34,111 --> 01:47:36,000 know how to differentiate. 2095 01:47:36,000 --> 01:47:40,780 You started having the problem from this point on. 2096 01:47:40,780 --> 01:47:44,412 3 into the 3tk. 2097 01:47:44,412 --> 01:47:47,010 The problem came when you were supposed 2098 01:47:47,010 --> 01:47:55,510 to identify the coordinates and square them and squeeze them 2099 01:47:55,510 --> 01:47:57,280 under the same square root. 2100 01:47:57,280 --> 01:48:01,250 And that drove you crazy when you have enough. 2101 01:48:01,250 --> 01:48:04,210 Let me put the minus here to make it more obvious what's 2102 01:48:04,210 --> 01:48:06,360 going to happen. 2103 01:48:06,360 --> 01:48:08,120 When you're going to have problems 2104 01:48:08,120 --> 01:48:09,770 like that in differential equations, 2105 01:48:09,770 --> 01:48:14,670 you better have the eye for it, [INAUDIBLE]. 2106 01:48:14,670 --> 01:48:18,870 You should be able to recognize this is like a pattern. 2107 01:48:18,870 --> 01:48:26,585 Have you seen the movie A Beautiful Mind? 2108 01:48:26,585 --> 01:48:27,210 STUDENT: Yeah. 2109 01:48:27,210 --> 01:48:28,620 MAGDALENA TODA: OK, so Nash, when 2110 01:48:28,620 --> 01:48:34,180 he was writing with the finger on everything, on the walls 2111 01:48:34,180 --> 01:48:39,835 at Princeton, on the window, he was thinking of patterns. 2112 01:48:39,835 --> 01:48:42,270 He's actually trying to-- and it's 2113 01:48:42,270 --> 01:48:44,130 hard to visualize without drawing, 2114 01:48:44,130 --> 01:48:48,374 but this is what most of us recognize all the time when 2115 01:48:48,374 --> 01:48:51,212 a mathematician writes down some computations 2116 01:48:51,212 --> 01:48:52,905 in a different way. 2117 01:48:52,905 --> 01:48:58,420 All we hope for is to get a few steps behind that board 2118 01:48:58,420 --> 01:48:59,850 and see a pattern. 2119 01:48:59,850 --> 01:49:02,340 And when you do that, you see the pattern. 2120 01:49:02,340 --> 01:49:05,580 This is an a minus b and that's an a plus b. 2121 01:49:05,580 --> 01:49:08,685 And then you say, OK, if I'm going to square them, 2122 01:49:08,685 --> 01:49:10,630 what's going to happen? 2123 01:49:10,630 --> 01:49:15,330 When you square an a minus b and you square an a plus b 2124 01:49:15,330 --> 01:49:18,810 and you have this giggly guy there-- leave him there. 2125 01:49:18,810 --> 01:49:21,980 He's having too much fun. 2126 01:49:21,980 --> 01:49:28,194 You actually develop these guys and you put them one 2127 01:49:28,194 --> 01:49:31,530 under the other and say wow, what 2128 01:49:31,530 --> 01:49:34,280 a beautiful simplification. 2129 01:49:34,280 --> 01:49:36,760 When I'm going to add these guys, 2130 01:49:36,760 --> 01:49:40,340 this thing in the middle will simply will cancel out, 2131 01:49:40,340 --> 01:49:44,700 but the a squared will double and the b squared will double. 2132 01:49:44,700 --> 01:49:46,670 And that's the beauty of seeing pattern. 2133 01:49:46,670 --> 01:49:50,970 You see how there is something symmetric and magic 2134 01:49:50,970 --> 01:49:56,470 in mathematics that make the answer simplified. 2135 01:49:56,470 --> 01:50:01,360 And that allows you to compress your equations that originally 2136 01:50:01,360 --> 01:50:05,740 seemed to be a mess into something that's 2137 01:50:05,740 --> 01:50:08,595 more easily expressed. 2138 01:50:08,595 --> 01:50:11,450 So when you're going to compute this r 2139 01:50:11,450 --> 01:50:17,792 prime of t magic absolute value of the magnitude, that's 2140 01:50:17,792 --> 01:50:21,664 going to be square root of-- instead of writing all the 2141 01:50:21,664 --> 01:50:25,190 [INAUDIBLE], I hate writing and rewriting the whole thing 2142 01:50:25,190 --> 01:50:28,804 squared plus the whole thing squared plus this squared. 2143 01:50:28,804 --> 01:50:32,970 If I love to write so much, I'd be in humanities and not 2144 01:50:32,970 --> 01:50:34,780 in mathematics. 2145 01:50:34,780 --> 01:50:41,175 So as a mathematician, how am I going to write that? 2146 01:50:41,175 --> 01:50:44,379 As a mathematician, I'm going to use some sort of-- like the U 2147 01:50:44,379 --> 01:50:44,920 substitution. 2148 01:50:44,920 --> 01:50:48,910 So I say, I call this Mr. A, and I call this Mr. B. 2149 01:50:48,910 --> 01:50:50,966 And that's A minus B, and that's A plus B. 2150 01:50:50,966 --> 01:50:53,810 And that's somebody else. 2151 01:50:53,810 --> 01:50:57,470 So when I square the first guy, and I 2152 01:50:57,470 --> 01:51:00,668 square the second component, and I square the third component, 2153 01:51:00,668 --> 01:51:09,780 and I add them together, I'm going to get what? 2154 01:51:09,780 --> 01:51:15,810 Square root of 2A squared plus 2B squared. 2155 01:51:15,810 --> 01:51:19,010 Because I know that these are the first two. 2156 01:51:19,010 --> 01:51:21,290 This guy squared plus this guy squared 2157 01:51:21,290 --> 01:51:23,430 is going to be exactly 2A squared 2158 01:51:23,430 --> 01:51:25,974 plus 2B squared, nothing in the middle. 2159 01:51:25,974 --> 01:51:28,830 These guys cancel out. 2160 01:51:28,830 --> 01:51:30,636 STUDENT: A and B are not the same. 2161 01:51:30,636 --> 01:51:34,210 2162 01:51:34,210 --> 01:51:42,030 MAGDALENA TODA: Well, yeah, you're right. 2163 01:51:42,030 --> 01:51:45,869 Let me call-- you're right, this is the same, 2164 01:51:45,869 --> 01:51:47,630 but these are different. 2165 01:51:47,630 --> 01:51:51,995 So let me call them A prime plus B prime. 2166 01:51:51,995 --> 01:51:53,680 No, that's derivative. 2167 01:51:53,680 --> 01:51:56,244 Let me call them C and D-- very good, 2168 01:51:56,244 --> 01:52:03,810 thank you-- C squared plus 2CD plus D squared. 2169 01:52:03,810 --> 01:52:06,370 2170 01:52:06,370 --> 01:52:08,130 But the principle is the same. 2171 01:52:08,130 --> 01:52:11,492 So I'm going to have A squared plus C squared. 2172 01:52:11,492 --> 01:52:12,940 This goes away. 2173 01:52:12,940 --> 01:52:14,660 Why? 2174 01:52:14,660 --> 01:52:18,530 Because this times that is the same as this times that. 2175 01:52:18,530 --> 01:52:19,836 Say it again. 2176 01:52:19,836 --> 01:52:22,692 If we look in the middle, the middle term 2177 01:52:22,692 --> 01:52:28,298 will have 3e to the 3t cosine t times e to the 3t sine t. 2178 01:52:28,298 --> 01:52:33,030 Middle term here is 3e to the 3t e to the 3t sine and cosine. 2179 01:52:33,030 --> 01:52:36,440 So they will cancel out, this and that. 2180 01:52:36,440 --> 01:52:40,191 So here I have the sum of the square of A 2181 01:52:40,191 --> 01:52:45,907 plus the square of C. And here I'm 2182 01:52:45,907 --> 01:52:50,680 going to have the square of B plus the square of D. 2183 01:52:50,680 --> 01:52:54,388 OK, now when I square this and that, what do I get? 2184 01:52:54,388 --> 01:52:57,220 2185 01:52:57,220 --> 01:53:00,970 The beauty of that-- let me write it down then explicitly. 2186 01:53:00,970 --> 01:53:06,910 9e to the 3t cosine squared t remains from this guy. 2187 01:53:06,910 --> 01:53:08,790 Plus from the square of that, we'll 2188 01:53:08,790 --> 01:53:19,588 have 9e to the 3t-- no, just 3, 9 to the 6t, 9 to the 6t sine 2189 01:53:19,588 --> 01:53:21,540 squared. 2190 01:53:21,540 --> 01:53:22,870 So I take this guy. 2191 01:53:22,870 --> 01:53:23,545 I square it. 2192 01:53:23,545 --> 01:53:24,712 I take this guy. 2193 01:53:24,712 --> 01:53:26,920 I square it. 2194 01:53:26,920 --> 01:53:30,050 The middle terms will disappear, thank god. 2195 01:53:30,050 --> 01:53:33,490 Then I have this guy, I square it, that guy, I square it, 2196 01:53:33,490 --> 01:53:34,840 good. 2197 01:53:34,840 --> 01:53:41,290 Plus another parenthesis-- e to the 6t sine squared t plus e 2198 01:53:41,290 --> 01:53:44,266 to the 6t cosine squared t. 2199 01:53:44,266 --> 01:53:47,110 2200 01:53:47,110 --> 01:53:50,340 So even if they don't double because they're not 2201 01:53:50,340 --> 01:53:52,820 the same thing, what is the principle 2202 01:53:52,820 --> 01:53:54,390 that will make my life easier? 2203 01:53:54,390 --> 01:53:58,840 The same pattern of simplification. 2204 01:53:58,840 --> 01:54:00,730 What is that same pattern of simplification? 2205 01:54:00,730 --> 01:54:03,720 Look at the beauty of this guy and look 2206 01:54:03,720 --> 01:54:05,110 at the beauty of this guy. 2207 01:54:05,110 --> 01:54:06,790 And then there is something missing, 2208 01:54:06,790 --> 01:54:12,560 the happy guy that was quiet because I told him to be quiet. 2209 01:54:12,560 --> 01:54:17,162 That's 9e to the 6t. 2210 01:54:17,162 --> 01:54:18,488 He was there in the corner. 2211 01:54:18,488 --> 01:54:22,330 And you had to square this guy and square this guy 2212 01:54:22,330 --> 01:54:26,188 and square this guy and add them on top together. 2213 01:54:26,188 --> 01:54:27,649 Now what is the pattern? 2214 01:54:27,649 --> 01:54:35,441 The pattern is 9e to the 6t with 9e to the 6t, same guy. 2215 01:54:35,441 --> 01:54:38,190 The orange guys-- that's why I love the colors. 2216 01:54:38,190 --> 01:54:40,595 Cosine squared cosine squared will be 1. 2217 01:54:40,595 --> 01:54:47,400 Another pattern like that, I have e to the 6t, to the 6t, 2218 01:54:47,400 --> 01:54:52,260 and the same happy guys sine squared t, sine squared t, 2219 01:54:52,260 --> 01:54:54,660 add them together is 1. 2220 01:54:54,660 --> 01:55:00,505 So all in all, this mess is not a mess anymore. 2221 01:55:00,505 --> 01:55:11,290 So it becomes 9e to the 6t plus e to the 6t plus 9e to the 6t. 2222 01:55:11,290 --> 01:55:12,630 Are you guys with me? 2223 01:55:12,630 --> 01:55:17,960 All right, now how many e to the 6t's do we have? 2224 01:55:17,960 --> 01:55:25,850 9 plus 9 plus 1, 19, square root of 19 e to the 6t. 2225 01:55:25,850 --> 01:55:29,900 So when we integrate, we go integral 2226 01:55:29,900 --> 01:55:33,410 from 2 to 5 square root of 19. 2227 01:55:33,410 --> 01:55:34,850 Kick him out of your life. 2228 01:55:34,850 --> 01:55:36,990 He's just making your life harder. 2229 01:55:36,990 --> 01:55:40,065 And then you have square root of e to the 6t e to the 3t. 2230 01:55:40,065 --> 01:55:42,910 2231 01:55:42,910 --> 01:55:47,930 So after you kick the guy out, you 2232 01:55:47,930 --> 01:55:55,060 have e to the 3t divided by 3 between t equals 2 2233 01:55:55,060 --> 01:55:58,170 and t equals 5. 2234 01:55:58,170 --> 01:56:03,230 Actually, I took it right off the WeBWorK problem you had. 2235 01:56:03,230 --> 01:56:06,104 So if you type this in your WeBWorK-- 2236 01:56:06,104 --> 01:56:12,000 you probably already did-- you should get exactly the answer 2237 01:56:12,000 --> 01:56:13,202 as being correct. 2238 01:56:13,202 --> 01:56:17,800 2239 01:56:17,800 --> 01:56:24,160 On the exam, do not expect anything that long. 2240 01:56:24,160 --> 01:56:26,720 The idea of simplifying these patterns 2241 01:56:26,720 --> 01:56:31,780 by finding the sine cosine, sine squared plus cosine squared is 2242 01:56:31,780 --> 01:56:33,110 1, is still going to be there. 2243 01:56:33,110 --> 01:56:35,690 But don't expect anything that long. 2244 01:56:35,690 --> 01:56:43,369 Also, don't expect-- once you get to this state, 2245 01:56:43,369 --> 01:56:44,818 I don't want an answer. 2246 01:56:44,818 --> 01:56:46,267 This is the answer. 2247 01:56:46,267 --> 01:56:48,199 That's the precise answer. 2248 01:56:48,199 --> 01:56:52,560 I don't want any approximation or anything like that. 2249 01:56:52,560 --> 01:56:54,272 A few of you did this with a calculator. 2250 01:56:54,272 --> 01:56:57,655 Well, you will not have calculators in the final. 2251 01:56:57,655 --> 01:56:59,285 You are going to have easy problems. 2252 01:56:59,285 --> 01:57:03,170 If you did that with a calculator, 2253 01:57:03,170 --> 01:57:05,230 and you truncated your answer later, 2254 01:57:05,230 --> 01:57:11,270 and if you were within 0.01 of the correct answer, 2255 01:57:11,270 --> 01:57:12,340 you were fine. 2256 01:57:12,340 --> 01:57:14,861 But some people approximated too much. 2257 01:57:14,861 --> 01:57:16,785 And that's always a problem. 2258 01:57:16,785 --> 01:57:19,490 So it's always a good idea to enter something 2259 01:57:19,490 --> 01:57:23,860 like that in WeBWorK. 2260 01:57:23,860 --> 01:57:27,470 I said I wouldn't do it except in the last 20 minutes. 2261 01:57:27,470 --> 01:57:31,190 But I wanted to do something like that. 2262 01:57:31,190 --> 01:57:34,500 I want to give you another example, because you love 2263 01:57:34,500 --> 01:57:39,216 parametrization so much it just occurred to me that it would 2264 01:57:39,216 --> 01:57:41,940 be very, very helpful-- maybe, I don't 2265 01:57:41,940 --> 01:57:47,060 know-- to give you another problem similar to this one. 2266 01:57:47,060 --> 01:57:50,250 It's not in the book, but it was cooked up 2267 01:57:50,250 --> 01:57:53,698 by one of my colleagues for his homework. 2268 01:57:53,698 --> 01:58:02,554 So I'd like to show it to you. 2269 01:58:02,554 --> 01:58:06,490 2270 01:58:06,490 --> 01:58:09,584 e to the t i is a parametrization 2271 01:58:09,584 --> 01:58:13,240 of a [INAUDIBLE] space. 2272 01:58:13,240 --> 01:58:28,139 Plus e to the minus t j plus square root of 2 tk. 2273 01:58:28,139 --> 01:58:36,030 2274 01:58:36,030 --> 01:58:37,470 And how do I know? 2275 01:58:37,470 --> 01:58:41,102 Well, one of his students came to me 2276 01:58:41,102 --> 01:58:43,656 and asked for help with homework. 2277 01:58:43,656 --> 01:58:51,450 Well, we don't give help when it comes from another colleague. 2278 01:58:51,450 --> 01:58:55,790 So in the end, the student went to the tutoring center. 2279 01:58:55,790 --> 01:58:58,711 And the tutoring center helped only in parts. 2280 01:58:58,711 --> 01:59:00,520 She came back to me. 2281 01:59:00,520 --> 01:59:03,860 So what was the deal here? 2282 01:59:03,860 --> 01:59:13,662 Find f prime of t in the most simplified form 2283 01:59:13,662 --> 01:59:16,440 and find the absolute value r prime of t 2284 01:59:16,440 --> 01:59:17,830 in the most simplified form. 2285 01:59:17,830 --> 01:59:22,830 2286 01:59:22,830 --> 01:59:31,830 And find the length of the arc of this curve between t 2287 01:59:31,830 --> 01:59:33,824 equals 0 and t equals 1. 2288 01:59:33,824 --> 01:59:36,632 If this were given by a physicist, 2289 01:59:36,632 --> 01:59:39,760 how would that physicist reformulate the problem? 2290 01:59:39,760 --> 01:59:47,895 He would say-- he or she-- what is the distance travelled 2291 01:59:47,895 --> 01:59:54,450 by the particle between 0 seconds and 1 second? 2292 01:59:54,450 --> 01:59:56,125 So how do you write that? 2293 01:59:56,125 --> 02:00:03,550 Integral from 0 to 1 of r prime of t [INAUDIBLE]. 2294 02:00:03,550 --> 02:00:05,530 And you have to do the rest. 2295 02:00:05,530 --> 02:00:08,510 2296 02:00:08,510 --> 02:00:13,040 So arguably, this is the Chapter 10 review. 2297 02:00:13,040 --> 02:00:15,070 It's very useful for the midterm exam. 2298 02:00:15,070 --> 02:00:17,570 So although we are just doing this review, 2299 02:00:17,570 --> 02:00:20,690 you should not erase it from your memory. 2300 02:00:20,690 --> 02:00:24,380 Because I don't like to put surprise problems 2301 02:00:24,380 --> 02:00:25,250 on the midterm. 2302 02:00:25,250 --> 02:00:28,950 But if you worked a certain type of problem, 2303 02:00:28,950 --> 02:00:31,320 you may expect something like that. 2304 02:00:31,320 --> 02:00:33,720 Maybe it's different but in the same spirit. 2305 02:00:33,720 --> 02:00:37,690 r prime of t, who's going to help me with r prime of t? 2306 02:00:37,690 --> 02:00:40,720 2307 02:00:40,720 --> 02:00:44,140 This fellow-- e to the t. 2308 02:00:44,140 --> 02:00:46,863 And how about that? 2309 02:00:46,863 --> 02:00:50,300 Negative e to the negative t. 2310 02:00:50,300 --> 02:00:53,246 STUDENT: I thought the arc length was the square root of 1 2311 02:00:53,246 --> 02:00:56,192 plus f prime of t squared. 2312 02:00:56,192 --> 02:00:58,730 2313 02:00:58,730 --> 02:01:02,355 MAGDALENA TODA: For a plane curve. 2314 02:01:02,355 --> 02:01:04,440 OK, let me remind you. 2315 02:01:04,440 --> 02:01:05,980 If you have a plane curve y equals 2316 02:01:05,980 --> 02:01:12,467 f of x, then this thing would become integral from A 2317 02:01:12,467 --> 02:01:17,740 to B square root of 1 plus f prime of x dx. 2318 02:01:17,740 --> 02:01:22,010 And that, did you do that with your Calc II instructor? 2319 02:01:22,010 --> 02:01:25,740 How many of you had Dr. Williams? 2320 02:01:25,740 --> 02:01:28,000 That was a wonderful class, wasn't it? 2321 02:01:28,000 --> 02:01:29,380 And he taught that. 2322 02:01:29,380 --> 02:01:31,460 And of course he was not supposed 2323 02:01:31,460 --> 02:01:36,120 to tell you that was the speed of a parametric curve. 2324 02:01:36,120 --> 02:01:39,020 If you were to parametrize here, x of t 2325 02:01:39,020 --> 02:01:44,000 was t and y of t would be f of t. 2326 02:01:44,000 --> 02:01:45,450 He could have told you. 2327 02:01:45,450 --> 02:01:46,320 Maybe he told you. 2328 02:01:46,320 --> 02:01:47,470 Maybe you don't remember. 2329 02:01:47,470 --> 02:01:48,990 OK, let's forget about it. 2330 02:01:48,990 --> 02:01:50,340 That was Calc II. 2331 02:01:50,340 --> 02:01:54,120 Now, coming back here, I have to list what? 2332 02:01:54,120 --> 02:01:57,916 Square root of 2 times t prime is one k. 2333 02:01:57,916 --> 02:01:59,582 Who's going to help me compute the speed 2334 02:01:59,582 --> 02:02:02,380 and put it in a nice formula? 2335 02:02:02,380 --> 02:02:04,163 Well, my god-- 2336 02:02:04,163 --> 02:02:04,996 STUDENT: [INAUDIBLE] 2337 02:02:04,996 --> 02:02:08,230 2338 02:02:08,230 --> 02:02:10,790 MAGDALENA TODA: Ahh, you are too smart. 2339 02:02:10,790 --> 02:02:15,152 Today you had some what is that called with caffeine 2340 02:02:15,152 --> 02:02:17,036 and vitamins and-- 2341 02:02:17,036 --> 02:02:18,920 STUDENT: You're thinking of Red Bull. 2342 02:02:18,920 --> 02:02:20,340 MAGDALENA TODA: I know. 2343 02:02:20,340 --> 02:02:22,660 That was very nice. 2344 02:02:22,660 --> 02:02:23,740 I try to stay away. 2345 02:02:23,740 --> 02:02:28,223 What is that called with the energy booster? 2346 02:02:28,223 --> 02:02:29,264 STUDENT: I wouldn't know. 2347 02:02:29,264 --> 02:02:30,491 STUDENT: 5-Hour Energy. 2348 02:02:30,491 --> 02:02:31,719 MAGDALENA TODA: 5-Hour, OK. 2349 02:02:31,719 --> 02:02:33,192 I used to have that. 2350 02:02:33,192 --> 02:02:36,670 When I had that, I could anticipate two steps computing. 2351 02:02:36,670 --> 02:02:39,809 Just a joke, Alex, don't take it up. 2352 02:02:39,809 --> 02:02:40,725 Very good observation. 2353 02:02:40,725 --> 02:02:43,460 So Alex saw. 2354 02:02:43,460 --> 02:02:45,650 He has a premonition. 2355 02:02:45,650 --> 02:02:48,820 He can see two steps in advance. 2356 02:02:48,820 --> 02:02:50,915 He said, OK, square that. 2357 02:02:50,915 --> 02:02:52,710 You have e to the 2t. 2358 02:02:52,710 --> 02:02:53,395 Square this. 2359 02:02:53,395 --> 02:02:55,606 The minus doesn't matter. 2360 02:02:55,606 --> 02:03:00,330 Plus e to the minus 2t, and square that. 2361 02:03:00,330 --> 02:03:02,560 Then he saw patterns. 2362 02:03:02,560 --> 02:03:06,130 Because he is the wizard 101 today. 2363 02:03:06,130 --> 02:03:09,090 So what is the witchcraft he performed? 2364 02:03:09,090 --> 02:03:10,470 Do you see? 2365 02:03:10,470 --> 02:03:13,350 Does anybody else see the pattern? 2366 02:03:13,350 --> 02:03:15,360 [? Nateesh ?] sees the pattern. 2367 02:03:15,360 --> 02:03:16,719 Anybody illuminated? 2368 02:03:16,719 --> 02:03:18,010 I didn't see it from the start. 2369 02:03:18,010 --> 02:03:19,660 You guys saw it faster than me. 2370 02:03:19,660 --> 02:03:23,190 It took me about a minute and a half 2371 02:03:23,190 --> 02:03:26,710 when I saw this for the first time. 2372 02:03:26,710 --> 02:03:29,930 Is this a perfect square? 2373 02:03:29,930 --> 02:03:32,060 Of who? 2374 02:03:32,060 --> 02:03:36,350 e to the t plus e to the minus 2 squared 2375 02:03:36,350 --> 02:03:40,390 is-- anybody else sees the pattern I don't have candy. 2376 02:03:40,390 --> 02:03:44,210 Next time-- Alex, [INAUDIBLE], anybody else? 2377 02:03:44,210 --> 02:03:47,000 Do you now see the pattern, e to the 2t plus 2378 02:03:47,000 --> 02:03:51,340 e to the minus 2t plus twice the product? 2379 02:03:51,340 --> 02:03:54,470 And that's where the student was having the problem. 2380 02:03:54,470 --> 02:03:56,550 Where do you see the product? 2381 02:03:56,550 --> 02:03:58,474 The product is 1. 2382 02:03:58,474 --> 02:03:59,920 The product is 1 doubled. 2383 02:03:59,920 --> 02:04:02,100 So you get 2. 2384 02:04:02,100 --> 02:04:06,690 So it's indeed exactly the perfect square. 2385 02:04:06,690 --> 02:04:09,430 So once-- it was a she. 2386 02:04:09,430 --> 02:04:14,490 Once she saw the perfect square, she was so happy. 2387 02:04:14,490 --> 02:04:16,850 Because you get square root of the square. 2388 02:04:16,850 --> 02:04:19,560 You get e to the t plus e to the minus t. 2389 02:04:19,560 --> 02:04:22,694 And that's a trivial thing to integrate that you 2390 02:04:22,694 --> 02:04:23,860 have no problem integrating. 2391 02:04:23,860 --> 02:04:26,980 It's a positive function, very beautiful. 2392 02:04:26,980 --> 02:04:31,880 The professor who gave this was Dr. [INAUDIBLE] from Denmark. 2393 02:04:31,880 --> 02:04:34,730 He's one of the best teachers we have. 2394 02:04:34,730 --> 02:04:40,690 But he makes up his homework as far as I know. 2395 02:04:40,690 --> 02:04:43,200 I think in the sixth edition, this edition, 2396 02:04:43,200 --> 02:04:48,770 we actually stole his idea, and we made a problem like that 2397 02:04:48,770 --> 02:04:51,490 in the book somewhere. 2398 02:04:51,490 --> 02:04:55,190 We doubled the number of problems more or less. 2399 02:04:55,190 --> 02:05:00,900 So if you are to compute 0 to 1 of the speed, 2400 02:05:00,900 --> 02:05:03,069 what is the speed? 2401 02:05:03,069 --> 02:05:05,534 The speed is this beautiful thing. 2402 02:05:05,534 --> 02:05:09,971 Because you were able to see the pattern. 2403 02:05:09,971 --> 02:05:12,764 If you're not able to see that, do you 2404 02:05:12,764 --> 02:05:15,440 realize it's impossible, practically, 2405 02:05:15,440 --> 02:05:17,940 for you to integrate by hand? 2406 02:05:17,940 --> 02:05:22,700 You have to go to a calculator, Matlab, whatever. 2407 02:05:22,700 --> 02:05:23,830 So this is easy. 2408 02:05:23,830 --> 02:05:29,170 Why is that easy? e to the t minus e to the minus t at 1 2409 02:05:29,170 --> 02:05:32,040 and at 0-- you compare them. 2410 02:05:32,040 --> 02:05:36,420 You get at 1 e minus e to the minus 1 2411 02:05:36,420 --> 02:05:41,090 minus the fundamental theorem of calc e to the 0 minus 2412 02:05:41,090 --> 02:05:42,500 e to the 0. 2413 02:05:42,500 --> 02:05:43,620 Well, that's silly. 2414 02:05:43,620 --> 02:05:45,440 Why is that silly? 2415 02:05:45,440 --> 02:05:49,170 Because I'm going to give it up. 2416 02:05:49,170 --> 02:05:52,110 So the answer was e to the minus 1/e. 2417 02:05:52,110 --> 02:05:54,570 And she knew what the answer would be. 2418 02:05:54,570 --> 02:05:57,030 But she didn't know why. 2419 02:05:57,030 --> 02:05:58,434 So she came back to me. 2420 02:05:58,434 --> 02:06:02,610 I don't know how the tutoring center helped her figure 2421 02:06:02,610 --> 02:06:03,470 out the answer. 2422 02:06:03,470 --> 02:06:06,200 But she did not understand the solution. 2423 02:06:06,200 --> 02:06:08,946 So I said, I'm not going to take anymore people coming 2424 02:06:08,946 --> 02:06:11,020 from Professor [INAUDIBLE]. 2425 02:06:11,020 --> 02:06:12,830 I was also told it's not OK. 2426 02:06:12,830 --> 02:06:16,670 So don't go to another professor with homework coming 2427 02:06:16,670 --> 02:06:18,410 for me or the other way around. 2428 02:06:18,410 --> 02:06:20,600 Because it's not OK. 2429 02:06:20,600 --> 02:06:25,310 But you can go to the tutoring center asking them for hints. 2430 02:06:25,310 --> 02:06:30,216 They're open starting 9:00 AM and until around when? 2431 02:06:30,216 --> 02:06:31,560 Do you know? 2432 02:06:31,560 --> 02:06:32,990 They used to have until 4:00. 2433 02:06:32,990 --> 02:06:35,870 But now they're going to work on an extended schedule 2434 02:06:35,870 --> 02:06:37,850 until 8:00 PM. 2435 02:06:37,850 --> 02:06:40,325 It's going to be something crazy. 2436 02:06:40,325 --> 02:06:43,790 Now, the thing is, we want the students to be better, 2437 02:06:43,790 --> 02:06:48,620 to do better, to not give up, to be successful, 2438 02:06:48,620 --> 02:06:51,730 top one, two, three. 2439 02:06:51,730 --> 02:06:54,290 I'm a little bit concerned, but maybe I 2440 02:06:54,290 --> 02:06:56,572 shouldn't be, about those hours. 2441 02:06:56,572 --> 02:06:59,939 So I don't know if they managed to put a security camera 2442 02:06:59,939 --> 02:07:00,901 or not. 2443 02:07:00,901 --> 02:07:04,520 But having extended hours may be a problem. 2444 02:07:04,520 --> 02:07:09,780 Take advantage of those afternoon hours, 2445 02:07:09,780 --> 02:07:11,726 especially if you are busy. 2446 02:07:11,726 --> 02:07:18,698 Those late hours will be a big help for you. 2447 02:07:18,698 --> 02:07:21,266 Do you know where it is? 2448 02:07:21,266 --> 02:07:23,656 Room 106 over there. 2449 02:07:23,656 --> 02:07:26,530 2450 02:07:26,530 --> 02:07:29,800 Any other questions related to this type of problem 2451 02:07:29,800 --> 02:07:35,240 or related to anything else in the material 2452 02:07:35,240 --> 02:07:38,870 that maybe I can give you hints on, 2453 02:07:38,870 --> 02:07:40,970 at least the hint I'm going to give you? 2454 02:07:40,970 --> 02:07:44,860 Sometimes I cannot stop, and I just give the problem away. 2455 02:07:44,860 --> 02:07:46,330 I'm not supposed to do that. 2456 02:07:46,330 --> 02:07:50,750 2457 02:07:50,750 --> 02:07:54,303 Look at your WeBWorK, see what kind of help I can give you. 2458 02:07:54,303 --> 02:07:56,428 You still have a little bit of time. 2459 02:07:56,428 --> 02:07:57,261 STUDENT: [INAUDIBLE] 2460 02:07:57,261 --> 02:08:00,712 2461 02:08:00,712 --> 02:08:05,140 MAGDALENA TODA: That's the maximum of what? 2462 02:08:05,140 --> 02:08:06,677 It was-- 2463 02:08:06,677 --> 02:08:07,510 STUDENT: [INAUDIBLE] 2464 02:08:07,510 --> 02:08:11,110 2465 02:08:11,110 --> 02:08:12,680 MAGDALENA TODA: Was this the problem? 2466 02:08:12,680 --> 02:08:14,600 STUDENT: e to the 2x or something like that. 2467 02:08:14,600 --> 02:08:15,560 MAGDALENA TODA: Something like that? 2468 02:08:15,560 --> 02:08:16,060 I erased it. 2469 02:08:16,060 --> 02:08:19,400 STUDENT: You erased that? [INAUDIBLE]. 2470 02:08:19,400 --> 02:08:21,330 I found an answer. 2471 02:08:21,330 --> 02:08:23,395 MAGDALENA TODA: It's very computational I saw. 2472 02:08:23,395 --> 02:08:26,750 But before that, I saw that seven of you 2473 02:08:26,750 --> 02:08:28,990 guys-- you two also did it. 2474 02:08:28,990 --> 02:08:33,714 So I wrote-- you have a brownie waiting for that. 2475 02:08:33,714 --> 02:08:35,163 But then I erased it. 2476 02:08:35,163 --> 02:08:39,510 STUDENT: You erased the previous one too in the homework one. 2477 02:08:39,510 --> 02:08:42,040 MAGDALENA TODA: Because that had a bug in it. 2478 02:08:42,040 --> 02:08:45,400 That one, the one in the homework one, had a bug in it. 2479 02:08:45,400 --> 02:08:46,965 It only worked for some data. 2480 02:08:46,965 --> 02:08:50,090 And for other data it didn't work. 2481 02:08:50,090 --> 02:08:53,580 So every time you find a bug, you tell me, 2482 02:08:53,580 --> 02:08:56,200 and I will tell the programmer of those problems, who's 2483 02:08:56,200 --> 02:08:57,010 really careful. 2484 02:08:57,010 --> 02:09:02,423 But one in 1,000 you are bound to find a bug. 2485 02:09:02,423 --> 02:09:06,207 And I'm going to give you a chocolate 2486 02:09:06,207 --> 02:09:08,092 or something for every bug. 2487 02:09:08,092 --> 02:09:09,820 And any other questions? 2488 02:09:09,820 --> 02:09:14,695 2489 02:09:14,695 --> 02:09:17,665 STUDENT: So are you saying this is too long? 2490 02:09:17,665 --> 02:09:20,140 MAGDALENA TODA: Actually, it's very beautiful. 2491 02:09:20,140 --> 02:09:23,605 If you have a calculator, it's easier to solve it. 2492 02:09:23,605 --> 02:09:25,585 You can do it by hand, write it by hand, also. 2493 02:09:25,585 --> 02:09:27,257 But it's a long-- 2494 02:09:27,257 --> 02:09:28,090 STUDENT: [INAUDIBLE] 2495 02:09:28,090 --> 02:09:30,776 2496 02:09:30,776 --> 02:09:34,262 MAGDALENA TODA: Right, so let's do it now 2497 02:09:34,262 --> 02:09:36,752 for anybody who wants to stay. 2498 02:09:36,752 --> 02:09:37,748 You don't have to stay. 2499 02:09:37,748 --> 02:09:39,740 So practicing what you do-- 2500 02:09:39,740 --> 02:09:44,974 [SIDE CONVERSATIONS] 2501 02:09:44,974 --> 02:11:55,449