1 00:00:00,000 --> 00:00:06,370 what we discussed last time in 11.6 2 00:00:06,370 --> 00:00:10,290 which was-- do you remember the topics we discussed? 3 00:00:10,290 --> 00:00:13,720 We discussed the valuation of a derivative. 4 00:00:13,720 --> 00:00:20,580 5 00:00:20,580 --> 00:00:23,408 What else have we discussed about them? 6 00:00:23,408 --> 00:00:27,836 I'm gonna split the fields, although they are related. 7 00:00:27,836 --> 00:00:37,184 Gradient and the steepest ascent and descent. 8 00:00:37,184 --> 00:00:41,120 9 00:00:41,120 --> 00:00:52,960 So in which direction will z equals 10 00:00:52,960 --> 00:00:57,647 f of x and y, differential cofunction, 11 00:00:57,647 --> 00:01:01,623 and with a derivative that is continuous? 12 00:01:01,623 --> 00:01:05,610 13 00:01:05,610 --> 00:01:12,114 So will this have the maximum rate of change? 14 00:01:12,114 --> 00:01:17,416 15 00:01:17,416 --> 00:01:18,690 This is just review. 16 00:01:18,690 --> 00:01:21,080 Why am I doing review? 17 00:01:21,080 --> 00:01:24,760 Well, after talking to you on a personal basis, 18 00:01:24,760 --> 00:01:29,500 like one-on-one basis by email and a little bit in person, 19 00:01:29,500 --> 00:01:33,695 I realized that you like very much when I review. 20 00:01:33,695 --> 00:01:37,350 When I briefly review some of the notions. 21 00:01:37,350 --> 00:01:42,140 I will give you the essentials for the section, that 22 00:01:42,140 --> 00:01:45,470 was 11.5 is embedded in this. 23 00:01:45,470 --> 00:01:46,680 Embedded. 24 00:01:46,680 --> 00:01:50,730 So 11.5 is embedded in 11.6, and in one shot 25 00:01:50,730 --> 00:01:53,450 we can talk of them. 26 00:01:53,450 --> 00:01:57,270 In 11.7 you're gonna see some extrema of functions 27 00:01:57,270 --> 00:02:00,455 of two variables, like max and min, 28 00:02:00,455 --> 00:02:05,500 and then we have 11.8, which is Lagrange multipliers. 29 00:02:05,500 --> 00:02:09,960 What have we done about this differentiable function 30 00:02:09,960 --> 00:02:13,124 with continuous partial derivatives? 31 00:02:13,124 --> 00:02:18,090 Let's assume that it's smooth, OK? 32 00:02:18,090 --> 00:02:25,600 And in that case, we define the partial-- 33 00:02:25,600 --> 00:02:30,914 the directional derivative in the direction of u at u, 34 00:02:30,914 --> 00:02:39,010 v with unit vector at the point x, y, but let's fix it 35 00:02:39,010 --> 00:02:44,110 x0, y0, using a limit of a difference quotient, 36 00:02:44,110 --> 00:02:48,020 just like any derivative should be introduced. 37 00:02:48,020 --> 00:02:51,660 But I'm not gonna repeat that definition. 38 00:02:51,660 --> 00:02:53,620 Why is that? 39 00:02:53,620 --> 00:02:58,620 Because I want you to give me an alternative definition, 40 00:02:58,620 --> 00:03:01,670 which is something that is simpler 41 00:03:01,670 --> 00:03:04,935 to use in applications, which is what? 42 00:03:04,935 --> 00:03:06,990 Who remembers? 43 00:03:06,990 --> 00:03:11,942 The partial derivative of f with respect to x 44 00:03:11,942 --> 00:03:13,930 measured at the point x0, y0. 45 00:03:13,930 --> 00:03:19,915 46 00:03:19,915 --> 00:03:22,880 I'm gonna use this color, and then I'll change the color. 47 00:03:22,880 --> 00:03:26,540 And I'll say times u1. 48 00:03:26,540 --> 00:03:35,960 Plus-- change the color again-- f sub y at x0, y0, times u2. 49 00:03:35,960 --> 00:03:40,720 50 00:03:40,720 --> 00:03:44,910 And this is just a times-- multiplication 51 00:03:44,910 --> 00:03:47,120 between real values. 52 00:03:47,120 --> 00:03:52,690 Because u1 and u2 will be nobody but the components-- 53 00:03:52,690 --> 00:03:57,570 the real value components of the unit vector direction 54 00:03:57,570 --> 00:03:58,570 that we have. 55 00:03:58,570 --> 00:04:03,280 So, guys, remember, direction in this books means unit vector. 56 00:04:03,280 --> 00:04:05,930 Every time I say direction, I should 57 00:04:05,930 --> 00:04:08,120 say that's a unit vector. 58 00:04:08,120 --> 00:04:14,171 Is there any way, any other way, to express this? 59 00:04:14,171 --> 00:04:18,853 Maybe with a vector multiplication of some sort-- 60 00:04:18,853 --> 00:04:20,290 multiplication between vectors. 61 00:04:20,290 --> 00:04:24,750 See, what if I had a pink vector with pink components, 62 00:04:24,750 --> 00:04:27,410 and a blue vector with blue components? 63 00:04:27,410 --> 00:04:28,900 I'm getting somewhere. 64 00:04:28,900 --> 00:04:29,700 And I'm sneaky. 65 00:04:29,700 --> 00:04:31,840 And you feel-- you know where I'm 66 00:04:31,840 --> 00:04:36,130 getting because you had chapter nine fresh in your mind 67 00:04:36,130 --> 00:04:40,600 and a certain product between vectors. 68 00:04:40,600 --> 00:04:42,818 So what is this? 69 00:04:42,818 --> 00:04:44,726 It's a dot product, excellent. 70 00:04:44,726 --> 00:04:46,400 Rachel, right? 71 00:04:46,400 --> 00:04:48,390 So it's a dot product. 72 00:04:48,390 --> 00:04:51,170 And I'm gonna run it, and it's going 73 00:04:51,170 --> 00:04:57,850 to be just the gradient of f at the point p0. 74 00:04:57,850 --> 00:05:00,825 But I'm going to write it again, x0, y0, 75 00:05:00,825 --> 00:05:03,002 although it drives me crazy to have 76 00:05:03,002 --> 00:05:05,882 to write that all the time. 77 00:05:05,882 --> 00:05:10,610 Dot product or scalar product with what vector u? 78 00:05:10,610 --> 00:05:13,540 I'm not gonna put a bar on that because the-- that would 79 00:05:13,540 --> 00:05:15,680 be like an oxymoron. 80 00:05:15,680 --> 00:05:18,200 The gradient is a bar thing. 81 00:05:18,200 --> 00:05:21,146 It's always a vector, so I'm not gonna write a bar. 82 00:05:21,146 --> 00:05:23,400 But I write a bar on u, reminding you 83 00:05:23,400 --> 00:05:27,340 that u is a free vector. 84 00:05:27,340 --> 00:05:28,950 All right, do you like this? 85 00:05:28,950 --> 00:05:29,450 Yes. 86 00:05:29,450 --> 00:05:33,580 It's a compactified form of the fluffy expression 87 00:05:33,580 --> 00:05:35,180 we had before. 88 00:05:35,180 --> 00:05:38,610 It's much easier to remember than the limit definition. 89 00:05:38,610 --> 00:05:41,960 Of course, it's equivalent to it. 90 00:05:41,960 --> 00:05:47,470 And in applications, I-- well, we ask about that all the time. 91 00:05:47,470 --> 00:05:48,420 What was the gradient? 92 00:05:48,420 --> 00:05:50,720 So see, in mathematics everything is related. 93 00:05:50,720 --> 00:05:53,545 And talking about-- speaking about the devil-- 94 00:05:53,545 --> 00:05:57,180 I mean, not you, but we weren't talking about you-- 95 00:05:57,180 --> 00:06:00,840 just this gradient. 96 00:06:00,840 --> 00:06:05,780 Gradient f at the point p will simply 97 00:06:05,780 --> 00:06:11,320 be the vector whose components in the direction of i and j 98 00:06:11,320 --> 00:06:13,934 are the partials. 99 00:06:13,934 --> 00:06:15,842 The partial derivative with respect 100 00:06:15,842 --> 00:06:20,870 to x, plus the partial derivative with respect to y. 101 00:06:20,870 --> 00:06:26,520 102 00:06:26,520 --> 00:06:30,900 OK, last time we dealt even with gradients of-- functions 103 00:06:30,900 --> 00:06:33,270 of more variables. 104 00:06:33,270 --> 00:06:35,670 If I have n variables, so x1, x2, 105 00:06:35,670 --> 00:06:41,240 x3, xn, then this vector will have n components-- f sub x1, 106 00:06:41,240 --> 00:06:43,830 f sub x2, f sub x3, f sub x4. 107 00:06:43,830 --> 00:06:44,930 Somebody stop me. 108 00:06:44,930 --> 00:06:46,400 F sub xn. 109 00:06:46,400 --> 00:06:50,120 So I have n of them, and it's an n [INAUDIBLE]. 110 00:06:50,120 --> 00:06:58,325 It's a sum ordered [INAUDIBLE]. 111 00:06:58,325 --> 00:06:58,825 OK. 112 00:06:58,825 --> 00:07:01,340 A set of n elements in this order. 113 00:07:01,340 --> 00:07:04,760 It's an order set, [INAUDIBLE] n. 114 00:07:04,760 --> 00:07:05,580 All right. 115 00:07:05,580 --> 00:07:06,910 So the order matters. 116 00:07:06,910 --> 00:07:11,350 Now, steepest ascent and descent. 117 00:07:11,350 --> 00:07:15,330 In which direction do I have the maximum rate of change? 118 00:07:15,330 --> 00:07:17,962 And the answer is-- this is Q and this 119 00:07:17,962 --> 00:07:24,960 is A-- the maximum rate of change 120 00:07:24,960 --> 00:07:42,350 happens in the direction of the gradient at every point-- 121 00:07:42,350 --> 00:07:45,210 every point of the domain where I 122 00:07:45,210 --> 00:07:48,810 have smoothness or [INAUDIBLE] or [INAUDIBLE] whatever. 123 00:07:48,810 --> 00:07:52,530 124 00:07:52,530 --> 00:07:53,290 All right. 125 00:07:53,290 --> 00:07:54,300 And what else? 126 00:07:54,300 --> 00:07:57,820 I claimed last time, but I didn't 127 00:07:57,820 --> 00:08:04,110 prove, that in that direction that I claimed c 128 00:08:04,110 --> 00:08:05,550 from [INAUDIBLE]. 129 00:08:05,550 --> 00:08:22,410 The directional derivative is maximized exactly 130 00:08:22,410 --> 00:08:26,460 in the direction of the gradient. 131 00:08:26,460 --> 00:08:31,940 132 00:08:31,940 --> 00:08:33,070 Can we prove that? 133 00:08:33,070 --> 00:08:34,610 Now we can prove it. 134 00:08:34,610 --> 00:08:37,500 Before I couldn't prove it because you couldn't see it, 135 00:08:37,500 --> 00:08:40,419 because I didn't look at it as a dot product. 136 00:08:40,419 --> 00:08:44,340 And we were all blind, like guiding each other in the dark. 137 00:08:44,340 --> 00:08:45,700 Me blind, you blind. 138 00:08:45,700 --> 00:08:47,130 We couldn't see. 139 00:08:47,130 --> 00:08:48,490 Now we can see. 140 00:08:48,490 --> 00:08:51,980 So now we can see how to prove my claim 141 00:08:51,980 --> 00:08:53,680 that the directional derivative, which 142 00:08:53,680 --> 00:08:57,670 is measuring the maximum-- measuring 143 00:08:57,670 --> 00:09:01,540 the instantaneous rate of change in a direction-- compass 144 00:09:01,540 --> 00:09:04,386 direction, like, what is that, east? 145 00:09:04,386 --> 00:09:06,930 East, northeast, southwest, whatever. 146 00:09:06,930 --> 00:09:09,680 Those are the compass directions like i, j, i 147 00:09:09,680 --> 00:09:11,910 plus j over square root 2 and so on, 148 00:09:11,910 --> 00:09:13,840 those are called compass directions 149 00:09:13,840 --> 00:09:17,900 because you hold the compass in your hand as horizontal 150 00:09:17,900 --> 00:09:21,870 as you can, and you refer to the floor. 151 00:09:21,870 --> 00:09:24,270 Even if you are on a slope. 152 00:09:24,270 --> 00:09:28,490 Maybe you imagine me on a slope, hiking or whatever. 153 00:09:28,490 --> 00:09:29,890 Going down, going up. 154 00:09:29,890 --> 00:09:33,250 But the compass should always be kept horizontal. 155 00:09:33,250 --> 00:09:34,077 Do you hike, Alex? 156 00:09:34,077 --> 00:09:34,660 STUDENT: Yeah. 157 00:09:34,660 --> 00:09:36,700 PROFESSOR: I'm sorry, I've put you on the spot. 158 00:09:36,700 --> 00:09:44,540 So whatever you do when you think of a path up or down 159 00:09:44,540 --> 00:09:47,900 is measured in the direction of the horizontal plane, 160 00:09:47,900 --> 00:09:49,238 the compass direction. 161 00:09:49,238 --> 00:09:51,920 And that is the gradient I was talking about. 162 00:09:51,920 --> 00:09:53,290 You see, it's a function. 163 00:09:53,290 --> 00:09:56,080 It's a vector in plane. 164 00:09:56,080 --> 00:10:00,222 That means a geographic compass direction. 165 00:10:00,222 --> 00:10:01,156 Prove it. 166 00:10:01,156 --> 00:10:02,150 Let's prove the claim. 167 00:10:02,150 --> 00:10:04,260 Let's prove the claim, because this is Tuesday, 168 00:10:04,260 --> 00:10:07,065 and it's almost weekend, and we have to prove something 169 00:10:07,065 --> 00:10:09,180 this week, right? 170 00:10:09,180 --> 00:10:14,720 Now, do you like what you see? 171 00:10:14,720 --> 00:10:18,790 Well, I have no idea. 172 00:10:18,790 --> 00:10:21,370 What if I want to measure this. 173 00:10:21,370 --> 00:10:25,520 This could be a negative number, but it doesn't matter. 174 00:10:25,520 --> 00:10:30,341 Assume that I take the dot product, 175 00:10:30,341 --> 00:10:34,574 and I think, well, it's a scalar product, it must be positive. 176 00:10:34,574 --> 00:10:36,502 What do I get? 177 00:10:36,502 --> 00:10:40,370 178 00:10:40,370 --> 00:10:44,010 This guy-- if I'm a physicist, I'm going to say this guy 179 00:10:44,010 --> 00:10:47,820 is the length of the first vector at p0. 180 00:10:47,820 --> 00:10:49,320 Gradient at p0. 181 00:10:49,320 --> 00:10:51,780 That's the length of u. 182 00:10:51,780 --> 00:10:54,350 Duh, that is 1. 183 00:10:54,350 --> 00:10:58,230 Last time I checked, u was unitary, so that's silly of me, 184 00:10:58,230 --> 00:11:00,610 but I'll write it anyway. 185 00:11:00,610 --> 00:11:05,360 Times the cosine of the angle between-- oh, cosine 186 00:11:05,360 --> 00:11:10,346 of the parenthesis angle between nabla f and u. 187 00:11:10,346 --> 00:11:15,310 188 00:11:15,310 --> 00:11:15,860 OK. 189 00:11:15,860 --> 00:11:17,460 When is this maximized? 190 00:11:17,460 --> 00:11:22,739 STUDENT: When the angle between nabla f and u is 0. 191 00:11:22,739 --> 00:11:23,530 PROFESSOR: Exactly. 192 00:11:23,530 --> 00:11:33,540 When this is pi, so this quantity is maximized-- gosh, 193 00:11:33,540 --> 00:11:35,496 I hate writing a lot. 194 00:11:35,496 --> 00:11:38,919 I had to submit homework in the past two days, 195 00:11:38,919 --> 00:11:43,800 and one was about biological research and the other one 196 00:11:43,800 --> 00:11:45,986 about stress management. 197 00:11:45,986 --> 00:11:49,318 The stress management class stresses me out the most. 198 00:11:49,318 --> 00:11:52,174 I shouldn't make it public. 199 00:11:52,174 --> 00:11:55,700 Really, because we have to write these essays of seven or eight 200 00:11:55,700 --> 00:11:58,320 pages every Tuesday, every end of the week. 201 00:11:58,320 --> 00:11:59,596 Twice a week. 202 00:11:59,596 --> 00:12:04,590 , So I realized how much I hate writing down a lot, 203 00:12:04,590 --> 00:12:07,020 and what a blessing it is to be a mathematician. 204 00:12:07,020 --> 00:12:08,660 You abbreviate everything. 205 00:12:08,660 --> 00:12:11,500 You compress everything. 206 00:12:11,500 --> 00:12:12,710 I love formulas. 207 00:12:12,710 --> 00:12:20,780 So what we have here is maximized when the cosine is 1. 208 00:12:20,780 --> 00:12:23,530 209 00:12:23,530 --> 00:12:26,220 And if you have become 0 to 2 pi open, 210 00:12:26,220 --> 00:12:33,300 then theta 0 is your only option. 211 00:12:33,300 --> 00:12:35,560 Well, if you take an absolute value, 212 00:12:35,560 --> 00:12:39,370 you could also have it in the other direction, 213 00:12:39,370 --> 00:12:43,060 cosine pi, but in that case you change the sign. 214 00:12:43,060 --> 00:12:46,720 So what you get-- you get a maximum. 215 00:12:46,720 --> 00:12:48,880 Let's say you hike, right? 216 00:12:48,880 --> 00:12:50,930 I'm just hiking in my brain. 217 00:12:50,930 --> 00:12:54,290 The maximum rate of change in this direction, 218 00:12:54,290 --> 00:12:56,560 climbing towards the peak. 219 00:12:56,560 --> 00:13:02,320 And then the steepest descent is the exactly minus gradient 220 00:13:02,320 --> 00:13:02,820 direction. 221 00:13:02,820 --> 00:13:07,150 So I could have 0 or pi for the angle. 222 00:13:07,150 --> 00:13:10,100 That's the philosophical meaning of that. 223 00:13:10,100 --> 00:13:10,690 All right. 224 00:13:10,690 --> 00:13:13,030 So the directional derivative, which is this guy, 225 00:13:13,030 --> 00:13:15,040 when does it become maximum? 226 00:13:15,040 --> 00:13:16,290 When the angle is 0. 227 00:13:16,290 --> 00:13:17,517 So I'm done. 228 00:13:17,517 --> 00:13:20,499 QED. 229 00:13:20,499 --> 00:13:21,990 What does it mean? 230 00:13:21,990 --> 00:13:28,810 That I know this happens when-- when direction u is 231 00:13:28,810 --> 00:13:30,366 the direction of the gradient. 232 00:13:30,366 --> 00:13:32,360 Can I write u equals gradient of f? 233 00:13:32,360 --> 00:13:34,180 Not quite. 234 00:13:34,180 --> 00:13:40,980 I should say divided by norm or magnitude. 235 00:13:40,980 --> 00:13:42,154 Why is that? 236 00:13:42,154 --> 00:13:44,320 And you say, Magdalena, didn't you say like 10 times 237 00:13:44,320 --> 00:13:46,840 that u is a unit vector? 238 00:13:46,840 --> 00:13:49,102 You want u to be a unit vector direction. 239 00:13:49,102 --> 00:13:52,009 So the direction should be the direction of the gradient 240 00:13:52,009 --> 00:13:54,050 in order to maximize this directional derivative. 241 00:13:54,050 --> 00:13:57,110 But then you have to take the gradient 242 00:13:57,110 --> 00:13:59,500 and divide it by its magnitude. 243 00:13:59,500 --> 00:14:00,292 Let's compute it. 244 00:14:00,292 --> 00:14:02,700 Let's see what we get. 245 00:14:02,700 --> 00:14:03,640 Let's see what we get. 246 00:14:03,640 --> 00:14:06,970 Now, I'm sorry about my beautiful handwriting, 247 00:14:06,970 --> 00:14:11,920 but I'm-- well, I'm gonna have to-- I have room here. 248 00:14:11,920 --> 00:14:14,340 And actually I can use this formula. 249 00:14:14,340 --> 00:14:16,900 So in the direction of the gradient, 250 00:14:16,900 --> 00:14:18,560 when u is the gradient. 251 00:14:18,560 --> 00:14:21,950 Let's take u to be-- what did I say over there? 252 00:14:21,950 --> 00:14:27,450 Gradient of f over the magnitude of gradient of f. 253 00:14:27,450 --> 00:14:29,720 I'll take this guy and drag him here. 254 00:14:29,720 --> 00:14:31,480 [INAUDIBLE] 255 00:14:31,480 --> 00:14:36,110 What will the value of the directional derivative be? 256 00:14:36,110 --> 00:14:39,740 We've done last time that-- the same thing on an example. 257 00:14:39,740 --> 00:14:43,680 We've done it on a function, beautiful f of x, y 258 00:14:43,680 --> 00:14:46,250 equals x squared plus y squared. 259 00:14:46,250 --> 00:14:48,750 This is the type of function I like, 260 00:14:48,750 --> 00:14:52,250 because they are the fastest to deal with. 261 00:14:52,250 --> 00:14:56,240 But anyway, we'll have all sorts of other functions. 262 00:14:56,240 --> 00:14:59,190 What am I going to write here? 263 00:14:59,190 --> 00:15:00,660 I have to write. 264 00:15:00,660 --> 00:15:03,400 Well, by definition, now, by my new way 265 00:15:03,400 --> 00:15:05,360 to look at the definition, I'm gonna 266 00:15:05,360 --> 00:15:09,570 have a gradient at the point, the vector, 267 00:15:09,570 --> 00:15:12,196 dot-- who is u again? 268 00:15:12,196 --> 00:15:16,590 The gradient this time, that special value, 269 00:15:16,590 --> 00:15:23,190 divided by absolute-- by magnitude of the gradient. 270 00:15:23,190 --> 00:15:25,890 But what in the world is that? 271 00:15:25,890 --> 00:15:29,410 272 00:15:29,410 --> 00:15:33,565 This animal is-- what if you take 273 00:15:33,565 --> 00:15:35,730 a vector multiplied by itself? 274 00:15:35,730 --> 00:15:36,336 Dot product. 275 00:15:36,336 --> 00:15:38,630 No, not dot product. 276 00:15:38,630 --> 00:15:42,870 What you get is the magnitude squared. 277 00:15:42,870 --> 00:15:43,430 All right. 278 00:15:43,430 --> 00:15:48,690 So although the pinkie guy and the pinkie guy 279 00:15:48,690 --> 00:15:54,100 are magnitude of f squared divided by magnitude of f. 280 00:15:54,100 --> 00:15:57,230 And Alex said, but wait, that's just-- I know. 281 00:15:57,230 --> 00:16:01,280 I didn't want to just jump ahead too fast. 282 00:16:01,280 --> 00:16:06,500 We get gradient of f in magnitude. 283 00:16:06,500 --> 00:16:07,980 So, beautiful. 284 00:16:07,980 --> 00:16:10,900 So we know who that maximum is. 285 00:16:10,900 --> 00:16:13,090 The maximum of the rate of change 286 00:16:13,090 --> 00:16:19,650 will be for-- this equals f of x, y. 287 00:16:19,650 --> 00:16:29,610 The max of the rate of change is-- what is that again? 288 00:16:29,610 --> 00:16:39,660 Magnitude of lambda f, in the direction nabla-- nabla f 289 00:16:39,660 --> 00:16:41,332 divided by its magnitude. 290 00:16:41,332 --> 00:16:43,980 291 00:16:43,980 --> 00:16:46,290 This is what we discovered. 292 00:16:46,290 --> 00:16:48,460 And now I'm going to ask you, what 293 00:16:48,460 --> 00:16:54,060 is the minimum rate of change at the same point? 294 00:16:54,060 --> 00:16:56,164 STUDENT: [INAUDIBLE]. 295 00:16:56,164 --> 00:16:57,580 PROFESSOR: Just parallel opposite. 296 00:16:57,580 --> 00:17:00,260 297 00:17:00,260 --> 00:17:06,160 So it's gonna-- I'm gonna have the so-called highest-- 298 00:17:06,160 --> 00:17:09,368 steepest, not highest, highest means maximum. 299 00:17:09,368 --> 00:17:10,460 The lowest value. 300 00:17:10,460 --> 00:17:12,770 So I'm going to have the lowest value, which 301 00:17:12,770 --> 00:17:14,680 indicates the steepest descent. 302 00:17:14,680 --> 00:17:20,481 Me going down on-- in the snow, I'm dreaming, on a sleigh, 303 00:17:20,481 --> 00:17:22,530 or on a plastic bag. 304 00:17:22,530 --> 00:17:26,670 That would give me the steepest descent, 305 00:17:26,670 --> 00:17:29,310 and the steepest descent will correspond to what? 306 00:17:29,310 --> 00:17:32,210 I'm going to make an NB. 307 00:17:32,210 --> 00:17:32,950 Nota bene. 308 00:17:32,950 --> 00:17:35,700 In Latin. 309 00:17:35,700 --> 00:17:45,900 Note the minimum will be minus magnitude 310 00:17:45,900 --> 00:17:51,545 of f, [INAUDIBLE] nabla f, in the opposite direction. 311 00:17:51,545 --> 00:17:53,410 Shall I write it in words? 312 00:17:53,410 --> 00:17:56,580 Let me write it in as O-P-P from opposite-- no, 313 00:17:56,580 --> 00:17:58,581 from opposite-- opposite direction. 314 00:17:58,581 --> 00:18:01,990 315 00:18:01,990 --> 00:18:03,630 What do I mean opposite direction? 316 00:18:03,630 --> 00:18:06,490 Opposite direction to the gradient. 317 00:18:06,490 --> 00:18:08,546 Which is the same direction, if you think about, 318 00:18:08,546 --> 00:18:09,670 because it's the same line. 319 00:18:09,670 --> 00:18:16,460 So it's going to be minus nabla f over magnitude of nabla f. 320 00:18:16,460 --> 00:18:19,370 It's like when we were in [INAUDIBLE], which 321 00:18:19,370 --> 00:18:24,210 was 1 minus x squared minus y squared, whatever it was-- 322 00:18:24,210 --> 00:18:31,410 we had i plus j for the descent, and minus i minus 323 00:18:31,410 --> 00:18:35,110 j after the ascent-- the steepest descent 324 00:18:35,110 --> 00:18:37,680 and the steepest ascent. 325 00:18:37,680 --> 00:18:40,210 We started with examples because it's easier 326 00:18:40,210 --> 00:18:42,660 to understand mathematics-- actually 327 00:18:42,660 --> 00:18:46,380 it's easier to understand anything on an example. 328 00:18:46,380 --> 00:18:48,630 And then-- if the example is good. 329 00:18:48,630 --> 00:18:50,440 If the example is bad, it's confusing. 330 00:18:50,440 --> 00:18:52,850 But if the example is good, you understand just 331 00:18:52,850 --> 00:18:56,437 about any concept, and then you move on to the theory, 332 00:18:56,437 --> 00:18:57,780 and this is the theory. 333 00:18:57,780 --> 00:18:59,030 And it looks very abstract. 334 00:18:59,030 --> 00:19:01,620 When somebody steps in this classroom 335 00:19:01,620 --> 00:19:05,880 and they haven't taken more than calc 2 they will get scared, 336 00:19:05,880 --> 00:19:09,092 and they will never want to take calc 3. 337 00:19:09,092 --> 00:19:13,070 Well, that's why-- I didn't want to scare you off yet. 338 00:19:13,070 --> 00:19:19,600 OK, so this is what you have to remember from section 11.6 339 00:19:19,600 --> 00:19:25,320 with 11.5 embedded in it. 340 00:19:25,320 --> 00:19:32,260 Now, one thing that I would like to see 341 00:19:32,260 --> 00:19:38,130 would be more examples and connections to other topics. 342 00:19:38,130 --> 00:19:42,190 So one example that I picked-- and I think it's a nice one. 343 00:19:42,190 --> 00:19:44,130 I just copied from the book. 344 00:19:44,130 --> 00:19:47,020 I usually don't bring cheat sheets. 345 00:19:47,020 --> 00:19:52,380 I don't like professors who bring books to the class 346 00:19:52,380 --> 00:19:55,167 and start reading out of the book. 347 00:19:55,167 --> 00:19:56,250 I think that's ridiculous. 348 00:19:56,250 --> 00:20:00,366 I mean, as if you guys couldn't read your own book at home. 349 00:20:00,366 --> 00:20:05,610 And I try to make up examples that are easier than the ones 350 00:20:05,610 --> 00:20:08,246 from the book to start with. 351 00:20:08,246 --> 00:20:13,066 But here's example 4, which is not so easy to deal with, 352 00:20:13,066 --> 00:20:15,000 but it's not hard either. 353 00:20:15,000 --> 00:20:18,090 And I picked it because I saw this browsing 354 00:20:18,090 --> 00:20:21,890 through the previous finals, I saw it 355 00:20:21,890 --> 00:20:24,866 as a pattern coming every now and then. 356 00:20:24,866 --> 00:20:30,800 Find the direction in which f increases or decreases 357 00:20:30,800 --> 00:20:32,283 most rapidly. 358 00:20:32,283 --> 00:20:34,300 I have to write down beautifully. 359 00:20:34,300 --> 00:20:52,510 Find the directions in which f increases or decreases most 360 00:20:52,510 --> 00:21:01,097 rapidly at p0 coordinates 2, 1. 361 00:21:01,097 --> 00:21:05,025 362 00:21:05,025 --> 00:21:11,240 And, what is the maximum-- that's 363 00:21:11,240 --> 00:21:20,835 a question-- what is the maximum rate of change or of increase? 364 00:21:20,835 --> 00:21:24,270 This type of problem is also covered in the Khan Academy 365 00:21:24,270 --> 00:21:28,220 videos and also the MIT library, but I don't 366 00:21:28,220 --> 00:21:30,087 feel they do a very good job. 367 00:21:30,087 --> 00:21:31,670 They cover it just lightly, as if they 368 00:21:31,670 --> 00:21:35,260 were afraid to speak too much about-- give too many examples 369 00:21:35,260 --> 00:21:37,730 and talk too much about the subject. 370 00:21:37,730 --> 00:21:42,560 So what do you guys want to do with this one? 371 00:21:42,560 --> 00:21:44,100 Help me solve the problem. 372 00:21:44,100 --> 00:21:45,570 That was kind of the idea. 373 00:21:45,570 --> 00:21:49,250 So we start with computing the what? 374 00:21:49,250 --> 00:21:52,732 The animal called-- what's the animal? 375 00:21:52,732 --> 00:21:53,720 STUDENT: Gradient. 376 00:21:53,720 --> 00:21:54,553 PROFESSOR: Gradient. 377 00:21:54,553 --> 00:21:55,720 Thank you very much. 378 00:21:55,720 --> 00:22:02,380 So we do that and we start differentiating. 379 00:22:02,380 --> 00:22:07,440 With respect to x, we get a product [INAUDIBLE]. 380 00:22:07,440 --> 00:22:13,230 So your product [INAUDIBLE] 1-- it's differentiated-- times e 381 00:22:13,230 --> 00:22:18,770 to the 2y minus x [INAUDIBLE] plus x undifferentiated. 382 00:22:18,770 --> 00:22:21,550 The second guy, prime. 383 00:22:21,550 --> 00:22:23,720 Copy and paste, Magdalena. 384 00:22:23,720 --> 00:22:25,510 Times-- don't forget the minus 1, 385 00:22:25,510 --> 00:22:28,900 because-- I'm talking to myself. 386 00:22:28,900 --> 00:22:33,620 Because if you do, you get a 0 on this in the final. 387 00:22:33,620 --> 00:22:34,957 I'm talking to myself. 388 00:22:34,957 --> 00:22:35,970 OK? 389 00:22:35,970 --> 00:22:36,470 All right. 390 00:22:36,470 --> 00:22:42,640 Plus, parenthesis-- the same procedure with respect to y. 391 00:22:42,640 --> 00:22:45,340 When I do it with respect to y, this 392 00:22:45,340 --> 00:22:47,340 is good review of the whole chapter. 393 00:22:47,340 --> 00:22:48,099 Yes, sir. 394 00:22:48,099 --> 00:22:48,974 STUDENT: [INAUDIBLE]. 395 00:22:48,974 --> 00:22:58,170 396 00:22:58,170 --> 00:23:00,740 PROFESSOR: This primed with respect to x. 397 00:23:00,740 --> 00:23:02,460 Am I doing something wrong? 398 00:23:02,460 --> 00:23:04,180 No. 399 00:23:04,180 --> 00:23:05,180 Are you with me? 400 00:23:05,180 --> 00:23:07,910 So this guy primed with respect to x, I'm 401 00:23:07,910 --> 00:23:13,350 going to write it es copied. 402 00:23:13,350 --> 00:23:17,810 And take that out, you maintain and differentiate with respect 403 00:23:17,810 --> 00:23:18,960 to x. 404 00:23:18,960 --> 00:23:22,488 So I did it right. 405 00:23:22,488 --> 00:23:29,170 OK, the second one, x is a constant for me right now. 406 00:23:29,170 --> 00:23:31,330 Who is the variable y? 407 00:23:31,330 --> 00:23:36,650 So I copy and paste e to the whole argument 408 00:23:36,650 --> 00:23:39,680 times-- I cover everything else with my hand, 409 00:23:39,680 --> 00:23:42,953 and I differentiate the argument with respect to y. 410 00:23:42,953 --> 00:23:47,150 And I get prime sub 2 and a j, and I say, thank 411 00:23:47,150 --> 00:23:48,770 god, this was a little bit long. 412 00:23:48,770 --> 00:23:53,000 You realize that if you make the slightest algebra mistake, 413 00:23:53,000 --> 00:23:55,010 it's all over for you. 414 00:23:55,010 --> 00:23:57,920 In that case, I ask my colleagues, 415 00:23:57,920 --> 00:24:02,610 what do you guys do when guy missed that or missed this? 416 00:24:02,610 --> 00:24:09,118 0, 0, no points-- OK, maybe a little bit, 417 00:24:09,118 --> 00:24:12,170 maybe a tiny bit of extra credit. 418 00:24:12,170 --> 00:24:14,300 But pay attention to your math. 419 00:24:14,300 --> 00:24:17,810 So you know what you need to do. 420 00:24:17,810 --> 00:24:24,100 Now I'm going to go on and say, but I am at the point 0. 421 00:24:24,100 --> 00:24:27,964 By the way, I really don't like what we did in the book. 422 00:24:27,964 --> 00:24:31,780 OK, I should not say that out loud, but it's too late. 423 00:24:31,780 --> 00:24:36,510 The book denotes that sometimes, well, we 424 00:24:36,510 --> 00:24:41,040 try not to do that too often, but not by F sub 0. 425 00:24:41,040 --> 00:24:44,040 Because some other books use that. 426 00:24:44,040 --> 00:24:45,190 I don't like that. 427 00:24:45,190 --> 00:24:48,960 So every time-- you should never do that. 428 00:24:48,960 --> 00:24:50,690 Because it gives the feeling that 429 00:24:50,690 --> 00:24:54,466 you're differentiating a constant or something. 430 00:24:54,466 --> 00:24:59,180 OK, so I always try to say, gradient [? up ?] 431 00:24:59,180 --> 00:25:02,825 is-- which means, I have a fixed value. 432 00:25:02,825 --> 00:25:05,550 But I don't fix the value before I took the gradient. 433 00:25:05,550 --> 00:25:07,750 This is too confusing as a notation. 434 00:25:07,750 --> 00:25:09,564 Don't do it. 435 00:25:09,564 --> 00:25:10,980 Close your eyes when you get to it 436 00:25:10,980 --> 00:25:12,188 when you're reading the book. 437 00:25:12,188 --> 00:25:20,763 OK, now I have to plug in instead of x sub 2-- 438 00:25:20,763 --> 00:25:25,150 I tried to remember that-- y equals 1. 439 00:25:25,150 --> 00:25:29,208 So I go 1 times e to the 2 times 1 minus 2 440 00:25:29,208 --> 00:25:32,660 plus 2 times e to the 2 times 1 minus-- 441 00:25:32,660 --> 00:25:38,750 I wish I had Data with me, I mean the guy from Star Trek. 442 00:25:38,750 --> 00:25:41,520 Because he could do this in just a fraction of a second 443 00:25:41,520 --> 00:25:46,202 without me having to bother with this whole thing. 444 00:25:46,202 --> 00:25:51,042 445 00:25:51,042 --> 00:25:52,512 STUDENT: But then if Data existed, 446 00:25:52,512 --> 00:25:53,678 why would we be math majors? 447 00:25:53,678 --> 00:25:57,170 PROFESSOR: Exactly, so we do this 448 00:25:57,170 --> 00:26:01,830 so that we can program people, I mean androids, 449 00:26:01,830 --> 00:26:06,290 and eventually learn how to clone ourselves. 450 00:26:06,290 --> 00:26:09,270 So let's see what we have. 451 00:26:09,270 --> 00:26:11,696 e to the 0 is 1. 452 00:26:11,696 --> 00:26:16,380 e to the 0 is 1, 1 minus 2-- are you guys with me? 453 00:26:16,380 --> 00:26:17,945 I'm going too fast? 454 00:26:17,945 --> 00:26:19,320 STUDENT: No, the [? board ?] says 455 00:26:19,320 --> 00:26:23,798 the y component is equal to 1, x component is equal to 2. 456 00:26:23,798 --> 00:26:27,700 PROFESSOR: x component equals to 2. 457 00:26:27,700 --> 00:26:31,034 x0 is 2, and y0 is 1. 458 00:26:31,034 --> 00:26:38,744 And I plug in x0 equals [INAUDIBLE]. 459 00:26:38,744 --> 00:26:44,816 So 2 times e to the 2 times 1 minus 2-- you think I like it? 460 00:26:44,816 --> 00:26:45,770 I don't like it. 461 00:26:45,770 --> 00:26:47,300 But anyway, it's my life. 462 00:26:47,300 --> 00:26:48,960 I have to go on. 463 00:26:48,960 --> 00:26:54,250 So we have 1 minus 2 plus minus 1, right? 464 00:26:54,250 --> 00:26:59,970 Minus 1i-- minus 1i sounds scary. 465 00:26:59,970 --> 00:27:09,250 OK, plus 4j-- minus i plus 4j. 466 00:27:09,250 --> 00:27:10,210 Is it lovely? 467 00:27:10,210 --> 00:27:11,740 No, I hate it. 468 00:27:11,740 --> 00:27:16,300 The magnitude is going to be square root of 17. 469 00:27:16,300 --> 00:27:17,160 But that's life. 470 00:27:17,160 --> 00:27:20,670 I mean, you as engineering majors 471 00:27:20,670 --> 00:27:23,276 see that all the time, and even worse than that. 472 00:27:23,276 --> 00:27:30,110 So I'm going to say the gradient of F at P0 in magnitude 473 00:27:30,110 --> 00:27:34,330 will be square root of 17. 474 00:27:34,330 --> 00:27:36,145 What is that? 475 00:27:36,145 --> 00:27:40,620 That is the maximum rate of change, right guys? 476 00:27:40,620 --> 00:27:44,320 But in which direction does that happen? 477 00:27:44,320 --> 00:27:47,660 478 00:27:47,660 --> 00:27:53,720 My beloved book says, in the direction of minus i plus 4j. 479 00:27:53,720 --> 00:27:58,130 That's our answer, so in the direction of-- 480 00:27:58,130 --> 00:27:59,620 STUDENT: Over root 17. 481 00:27:59,620 --> 00:28:01,610 PROFESSOR: No, let me tell you what. 482 00:28:01,610 --> 00:28:04,870 We fought about that as the authors when we wrote the book. 483 00:28:04,870 --> 00:28:09,150 So I said, if you're saying, in the direction of, 484 00:28:09,150 --> 00:28:11,395 then you have to say, over root 17. 485 00:28:11,395 --> 00:28:14,040 And my coauthor said, no, actually, Magdalena, 486 00:28:14,040 --> 00:28:16,515 as a matter of English-- since you're not a native, 487 00:28:16,515 --> 00:28:18,628 you don't understand. 488 00:28:18,628 --> 00:28:22,210 In the direction of a certain vector 489 00:28:22,210 --> 00:28:25,050 means the direction of a certain vector, 490 00:28:25,050 --> 00:28:27,520 the direction could be that. 491 00:28:27,520 --> 00:28:35,536 But I say equivalently, in the direction double dot 492 00:28:35,536 --> 00:28:39,440 minus i plus 4j over square root of 17, 493 00:28:39,440 --> 00:28:43,360 meaning that this is the direction. 494 00:28:43,360 --> 00:28:44,740 It's a matter of interpretation. 495 00:28:44,740 --> 00:28:46,800 I don't understand it, but it's your language. 496 00:28:46,800 --> 00:28:47,390 Yes, sir. 497 00:28:47,390 --> 00:28:48,265 STUDENT: [INAUDIBLE]. 498 00:28:48,265 --> 00:28:53,290 499 00:28:53,290 --> 00:28:54,969 PROFESSOR: I'm saying-- 500 00:28:54,969 --> 00:28:56,260 STUDENT: Which one do you like? 501 00:28:56,260 --> 00:28:57,110 PROFESSOR: Which one do I like? 502 00:28:57,110 --> 00:28:57,910 This one. 503 00:28:57,910 --> 00:28:59,310 STUDENT: [INAUDIBLE]. 504 00:28:59,310 --> 00:29:00,820 PROFESSOR: Thank you. 505 00:29:00,820 --> 00:29:04,116 So if we define direction to be a unit vector, 506 00:29:04,116 --> 00:29:08,580 let's be consistent and not say, of me, of you, of my cousin, 507 00:29:08,580 --> 00:29:11,060 of whatever. 508 00:29:11,060 --> 00:29:14,930 All right, was this hard? 509 00:29:14,930 --> 00:29:17,346 No. 510 00:29:17,346 --> 00:29:21,890 Do I have a caveat about this kind of problem? 511 00:29:21,890 --> 00:29:23,852 Yeah, that was kind of the idea. 512 00:29:23,852 --> 00:29:26,470 If I put that in the midterm, guys, 513 00:29:26,470 --> 00:29:28,930 please do it two or three times, make 514 00:29:28,930 --> 00:29:31,145 sure you didn't make any algebra mistakes. 515 00:29:31,145 --> 00:29:34,720 Because if you know the theory, I will still 516 00:29:34,720 --> 00:29:36,720 give you like 30% or something. 517 00:29:36,720 --> 00:29:38,940 But if you mess up with the algebra, 518 00:29:38,940 --> 00:29:45,080 I have no choice but giving up the 70%, whatever that is. 519 00:29:45,080 --> 00:29:49,065 I try to be fair and give you something for everything 520 00:29:49,065 --> 00:29:50,825 you do and know. 521 00:29:50,825 --> 00:29:52,886 But try not to mess it up too badly, 522 00:29:52,886 --> 00:29:54,540 because it's very easy to mess it up. 523 00:29:54,540 --> 00:29:55,860 Yes, sir. 524 00:29:55,860 --> 00:29:58,060 STUDENT: So which way is increasing? 525 00:29:58,060 --> 00:29:59,890 PROFESSOR: OK, find the direction 526 00:29:59,890 --> 00:30:03,750 in which r increases or decreases most rapidly. 527 00:30:03,750 --> 00:30:08,110 OK, the direction in which I could draw it, 528 00:30:08,110 --> 00:30:12,256 this is increasing in the direction of that. 529 00:30:12,256 --> 00:30:13,270 At what rate? 530 00:30:13,270 --> 00:30:15,273 At the rate of square root of 17. 531 00:30:15,273 --> 00:30:17,532 Good question. 532 00:30:17,532 --> 00:30:19,530 How about the other one? 533 00:30:19,530 --> 00:30:24,430 In the direction of plus i minus 4j over square root of 17, 534 00:30:24,430 --> 00:30:29,480 I get the rate of change minus square root of 17. 535 00:30:29,480 --> 00:30:30,220 And that's it. 536 00:30:30,220 --> 00:30:31,996 Do we have to say that? 537 00:30:31,996 --> 00:30:35,570 Ehh, I give you extra credit if you say that. 538 00:30:35,570 --> 00:30:38,724 But at this point, I'm saying I'm happy with what I just 539 00:30:38,724 --> 00:30:40,765 wrote on the board. 540 00:30:40,765 --> 00:30:41,640 STUDENT: [INAUDIBLE]. 541 00:30:41,640 --> 00:30:46,450 542 00:30:46,450 --> 00:30:49,110 PROFESSOR: Yeah, so it's like finding 543 00:30:49,110 --> 00:30:52,710 on which path are you going to get to the top 544 00:30:52,710 --> 00:30:56,300 the fastest when you climb a mountain. 545 00:30:56,300 --> 00:30:58,620 It's the same kind of question. 546 00:30:58,620 --> 00:31:00,810 Because it's all about z being an altitude. 547 00:31:00,810 --> 00:31:03,310 z equals F of x. 548 00:31:03,310 --> 00:31:07,254 I will go ahead and erase the whole thing. 549 00:31:07,254 --> 00:31:15,652 And I wish you good luck now. 550 00:31:15,652 --> 00:31:21,530 I wish you good luck because I was asked by [INAUDIBLE] 551 00:31:21,530 --> 00:31:28,540 to solve a problem like the one you gave me in the web work, 552 00:31:28,540 --> 00:31:30,340 but I don't remember it. 553 00:31:30,340 --> 00:31:31,660 But it's OK. 554 00:31:31,660 --> 00:31:35,780 it was a review of what we did last time. 555 00:31:35,780 --> 00:31:43,570 And we said, instead of that, where the tangent plane-- 556 00:31:43,570 --> 00:31:44,696 we know what that is. 557 00:31:44,696 --> 00:31:49,885 At P0 was-- guys, by the final I want this memorized. 558 00:31:49,885 --> 00:31:51,480 There is no question. 559 00:31:51,480 --> 00:31:55,770 So z minus z0 is like Taylor's formula 560 00:31:55,770 --> 00:31:59,010 in the linear approximation. 561 00:31:59,010 --> 00:31:59,830 You truncate. 562 00:31:59,830 --> 00:32:02,040 You throw away the second order, and so on. 563 00:32:02,040 --> 00:32:03,740 So what is that? 564 00:32:03,740 --> 00:32:06,460 f sub x at delta x. 565 00:32:06,460 --> 00:32:07,670 Oh, x equals 0. 566 00:32:07,670 --> 00:32:08,999 I'm lazy. 567 00:32:08,999 --> 00:32:12,671 Times x minus x0-- this is the delta x. 568 00:32:12,671 --> 00:32:16,810 This is the delta z. 569 00:32:16,810 --> 00:32:21,466 And this is not the round surface, curvy and everything. 570 00:32:21,466 --> 00:32:28,670 This is the plane approximation, the planar approximation-- 571 00:32:28,670 --> 00:32:39,110 A-P-P-R. I'm not done-- plus f sub y y minus y0. 572 00:32:39,110 --> 00:32:49,960 So this is the equation of the plane that's tangent pi at 0. 573 00:32:49,960 --> 00:32:53,380 And let me draw the surface pink. 574 00:32:53,380 --> 00:33:00,900 Because I'm a girl, and because I want to draw this in pink. 575 00:33:00,900 --> 00:33:06,540 Let's call it some S, for Surface. 576 00:33:06,540 --> 00:33:09,715 Can we paint an S here? 577 00:33:09,715 --> 00:33:18,160 OK, but if I'm giving the same picture 578 00:33:18,160 --> 00:33:24,020 for a different equation-- so I have F of x, y, z 579 00:33:24,020 --> 00:33:27,394 equals constant, that's the implicit form. 580 00:33:27,394 --> 00:33:30,780 581 00:33:30,780 --> 00:33:31,990 I make this face. 582 00:33:31,990 --> 00:33:33,400 Why do I make this face? 583 00:33:33,400 --> 00:33:36,380 Because I've got three confessions. 584 00:33:36,380 --> 00:33:40,330 I'm more like a priestess in mathematics. 585 00:33:40,330 --> 00:33:43,790 People don't like implicit differentiation. 586 00:33:43,790 --> 00:33:46,220 We will do a little bit of that today, 587 00:33:46,220 --> 00:33:50,275 because you told me your stories from implicit differentiation, 588 00:33:50,275 --> 00:33:51,665 and I got scared. 589 00:33:51,665 --> 00:33:53,910 Those were horror stories. 590 00:33:53,910 --> 00:33:56,500 And I don't want those to repeat in the final or the midterm. 591 00:33:56,500 --> 00:33:59,292 So I'm going to do something with implicit differentiation 592 00:33:59,292 --> 00:34:00,960 as well. 593 00:34:00,960 --> 00:34:03,590 What did I want to say? 594 00:34:03,590 --> 00:34:07,410 If we apply this, we'd be wrong. 595 00:34:07,410 --> 00:34:11,100 But we have to remember what the normal would be. 596 00:34:11,100 --> 00:34:16,900 And the normal was the gradient of big F. 597 00:34:16,900 --> 00:34:24,356 So those will be F sub xi plus F sub yj plus big F sub zk. 598 00:34:24,356 --> 00:34:29,379 Then force, such a surface, even implicitly, 599 00:34:29,379 --> 00:34:32,139 the tangent plane looks a little bit differently. 600 00:34:32,139 --> 00:34:33,790 But it's the same story. 601 00:34:33,790 --> 00:34:35,350 And I prove that. 602 00:34:35,350 --> 00:34:37,840 You may not believe it, or may not remember. 603 00:34:37,840 --> 00:34:42,710 But I proved that, that it's one and the same thing. 604 00:34:42,710 --> 00:34:44,170 So I get F sub x. 605 00:34:44,170 --> 00:34:46,620 I get 0i0. 606 00:34:46,620 --> 00:34:51,739 This was the A dot, times x minus x plus. 607 00:34:51,739 --> 00:34:53,270 What's next? 608 00:34:53,270 --> 00:34:57,280 From the coefficients coming from the gradient-- 609 00:34:57,280 --> 00:34:59,740 that was the gradient. 610 00:34:59,740 --> 00:35:02,380 611 00:35:02,380 --> 00:35:06,130 So this is F of x, y, z equals C. 612 00:35:06,130 --> 00:35:12,948 And the gradient was-- this is the normal, F sub x, F sub y, 613 00:35:12,948 --> 00:35:20,946 F sub z angular bracket equals gradient of F, which 614 00:35:20,946 --> 00:35:23,090 is more or less than normal. 615 00:35:23,090 --> 00:35:31,480 The unit normal will be just gradient of F over length of F. 616 00:35:31,480 --> 00:35:40,218 So let's continue-- F sub y at x0y0 times delta y. 617 00:35:40,218 --> 00:35:47,930 This is B. I had the problem with memorizing these, 618 00:35:47,930 --> 00:35:52,430 especially since when I was 18 I did not understand them 619 00:35:52,430 --> 00:35:55,570 whatsoever first year of college. 620 00:35:55,570 --> 00:36:00,520 I had to use markers, put them in markers 621 00:36:00,520 --> 00:36:02,620 and glue them to my closet. 622 00:36:02,620 --> 00:36:04,450 Because when I was 18, of course, 623 00:36:04,450 --> 00:36:06,570 I was looking in the mirror all the time. 624 00:36:06,570 --> 00:36:08,120 So whenever I got into the closet, 625 00:36:08,120 --> 00:36:11,860 opened the door to the mirror, next to the mirror 626 00:36:11,860 --> 00:36:14,260 there was this formula. 627 00:36:14,260 --> 00:36:19,990 So whether I liked it or not-- I didn't-- I memorized it just 628 00:36:19,990 --> 00:36:23,325 by seeing it every day when I opened the door. 629 00:36:23,325 --> 00:36:23,824 Yes, sir. 630 00:36:23,824 --> 00:36:25,656 STUDENT: That green one, should that be x minus 0 or z minus 0? 631 00:36:25,656 --> 00:36:26,530 PROFESSOR: z minus 0. 632 00:36:26,530 --> 00:36:28,000 That is my mistake. 633 00:36:28,000 --> 00:36:29,530 Thank you. 634 00:36:29,530 --> 00:36:35,366 So again, you have delta x, delta y, delta z. 635 00:36:35,366 --> 00:36:37,360 Thank you. 636 00:36:37,360 --> 00:36:38,960 Is it hard to memorize? 637 00:36:38,960 --> 00:36:41,470 No, not if you put it in markers. 638 00:36:41,470 --> 00:36:45,355 I think you will do just fine. 639 00:36:45,355 --> 00:36:47,580 What have we done as time? 640 00:36:47,580 --> 00:36:49,860 Let me review it really quickly. 641 00:36:49,860 --> 00:36:51,260 We said, wait a minute. 642 00:36:51,260 --> 00:36:53,505 How come they are one and the same? 643 00:36:53,505 --> 00:36:56,190 You said, oh I'm getting a headache. 644 00:36:56,190 --> 00:37:00,060 I don't understand why they are one and the same. 645 00:37:00,060 --> 00:37:06,670 And we said, yes, but you see, this guy is nothing but z 646 00:37:06,670 --> 00:37:09,590 minus F of x, y. 647 00:37:09,590 --> 00:37:14,870 So from this form, you can make it implicit it by pulling out F 648 00:37:14,870 --> 00:37:24,750 to the left and creating this big F of x, y, and z equals 0. 649 00:37:24,750 --> 00:37:26,340 And what does it mean? 650 00:37:26,340 --> 00:37:31,410 It means that F sub x will be oh my god, this has nothing 651 00:37:31,410 --> 00:37:35,641 to do with us minus F sub x. 652 00:37:35,641 --> 00:37:39,770 F sub y will be minus F sub y. 653 00:37:39,770 --> 00:37:41,150 Am I right? 654 00:37:41,150 --> 00:37:47,800 And F sub z, big F sub z, is simply-- there's no z here-- 1. 655 00:37:47,800 --> 00:37:56,243 So coming back to the guide's A, B, C, forget about A, 656 00:37:56,243 --> 00:38:00,240 B, C. I'll take the A, and I'll replace it with minus F 657 00:38:00,240 --> 00:38:03,810 sub x, which doesn't write. 658 00:38:03,810 --> 00:38:08,815 And it's time for him to go away-- minus F sub x. 659 00:38:08,815 --> 00:38:14,328 And this is F sub y with the minus. 660 00:38:14,328 --> 00:38:18,160 661 00:38:18,160 --> 00:38:21,530 And finally, Mr. C, who is happy, he is 1. 662 00:38:21,530 --> 00:38:23,980 So he says, I'm happy. 663 00:38:23,980 --> 00:38:26,536 You're going to separate me. 664 00:38:26,536 --> 00:38:29,000 We are going to separate him. 665 00:38:29,000 --> 00:38:34,400 So this equation is nothing but what? 666 00:38:34,400 --> 00:38:36,770 Let's write it from the left to the right. 667 00:38:36,770 --> 00:38:39,750 Let's keep the green guy in the left hand side. 668 00:38:39,750 --> 00:38:43,060 669 00:38:43,060 --> 00:38:46,042 And everybody else goes for a moving sale. 670 00:38:46,042 --> 00:38:48,614 The blue and the pink go away. 671 00:38:48,614 --> 00:38:51,180 And when they go to the other side, 672 00:38:51,180 --> 00:38:54,610 they have a minus, pick up a minus sign. 673 00:38:54,610 --> 00:38:58,259 But with the minus, the minus here is a plus sign. 674 00:38:58,259 --> 00:38:59,678 Are you guys with me? 675 00:38:59,678 --> 00:39:03,000 676 00:39:03,000 --> 00:39:07,410 So the blue guy has moved and changed sign. 677 00:39:07,410 --> 00:39:08,750 Where's the pink? 678 00:39:08,750 --> 00:39:12,010 The pink guy will also move. 679 00:39:12,010 --> 00:39:16,570 And he picked up this. 680 00:39:16,570 --> 00:39:21,320 So practically, this and that formula are one and the same. 681 00:39:21,320 --> 00:39:24,750 They're both used for the same tangent plane. 682 00:39:24,750 --> 00:39:28,210 It depends how you introduce the tangent plane. 683 00:39:28,210 --> 00:39:33,080 And [INAUDIBLE], before class started, 20 or 25 minutes 684 00:39:33,080 --> 00:39:38,080 before class, when I was in a hurry, I answered you briefly. 685 00:39:38,080 --> 00:39:42,380 You made some algebra mistake in the-- you got it? 686 00:39:42,380 --> 00:39:46,150 I would like to make up one like those. 687 00:39:46,150 --> 00:39:48,370 But I forgot what your surface was. 688 00:39:48,370 --> 00:39:49,280 Was it an ellipsoid? 689 00:39:49,280 --> 00:39:50,584 STUDENT: Ellipsoid. 690 00:39:50,584 --> 00:39:53,069 PROFESSOR: OK, I'll make up an ellipsoid. 691 00:39:53,069 --> 00:39:54,063 STUDENT: [INAUDIBLE]. 692 00:39:54,063 --> 00:39:58,039 PROFESSOR: Yeah, mhmm, if you have it with you. 693 00:39:58,039 --> 00:40:00,418 If you don't have it with you, that's fine. 694 00:40:00,418 --> 00:40:01,792 So I'm going to go ahead and keep 695 00:40:01,792 --> 00:40:03,880 just the implicit equation. 696 00:40:03,880 --> 00:40:07,462 Because the ellipsoid is given by the implicit equation. 697 00:40:07,462 --> 00:40:13,262 And everything else I will erase. 698 00:40:13,262 --> 00:40:15,657 And that was problem 24. 699 00:40:15,657 --> 00:40:16,520 How many problems? 700 00:40:16,520 --> 00:40:20,660 Well, you still have a lot, up to 49. 701 00:40:20,660 --> 00:40:21,680 STUDENT: 42. 702 00:40:21,680 --> 00:40:23,670 PROFESSOR: 42, OK, I reduced it. 703 00:40:23,670 --> 00:40:28,930 Now, when I sent you an email on Sunday, 704 00:40:28,930 --> 00:40:32,928 I said I was giving you an extension. 705 00:40:32,928 --> 00:40:35,338 STUDENT: Till March. 706 00:40:35,338 --> 00:40:37,085 PROFESSOR: Till a lot of March. 707 00:40:37,085 --> 00:40:41,410 Because I thought March the 2nd, and I gave a few more days. 708 00:40:41,410 --> 00:40:43,210 So you have one more week. 709 00:40:43,210 --> 00:40:44,440 STUDENT: When's spring break? 710 00:40:44,440 --> 00:40:46,220 PROFESSOR: Spring break is the 14th, 711 00:40:46,220 --> 00:40:48,735 but this is due on the 9th right? 712 00:40:48,735 --> 00:40:50,620 STUDENT: The 10th. 713 00:40:50,620 --> 00:40:53,940 PROFESSOR: The 10th-- maybe I don't remember. 714 00:40:53,940 --> 00:40:56,071 So what was your problem? 715 00:40:56,071 --> 00:40:57,514 What's your problem? 716 00:40:57,514 --> 00:40:58,957 What? 717 00:40:58,957 --> 00:41:00,400 Can I take it? 718 00:41:00,400 --> 00:41:05,700 719 00:41:05,700 --> 00:41:10,310 So you have an ellipsoid, which comes 720 00:41:10,310 --> 00:41:15,676 from the ellipse 3x squared [INAUDIBLE]. 721 00:41:15,676 --> 00:41:21,167 Then you have a 2z squared equals 9. 722 00:41:21,167 --> 00:41:24,660 Then it says, you look at the point P 723 00:41:24,660 --> 00:41:26,530 of coordinates minus 1-- I haven't even 724 00:41:26,530 --> 00:41:29,885 checked if it's correct, but it should be. 725 00:41:29,885 --> 00:41:33,600 So 3 plus-- I did not program the problem. 726 00:41:33,600 --> 00:41:38,590 3 plus 4, 7, plus 2, 9, so he did a good job. 727 00:41:38,590 --> 00:41:42,300 We want the tangent plane. 728 00:41:42,300 --> 00:41:45,150 I'll put it here. 729 00:41:45,150 --> 00:41:47,710 We want the tangent plane. 730 00:41:47,710 --> 00:41:49,630 How do we compute the tangent plane? 731 00:41:49,630 --> 00:41:54,490 You say, this is F of x, y, z, right? 732 00:41:54,490 --> 00:41:56,270 So F sub x equals 6x. 733 00:41:56,270 --> 00:41:58,790 F sub y equals 2y. 734 00:41:58,790 --> 00:42:01,960 F sub z equals 4z. 735 00:42:01,960 --> 00:42:06,800 Computing it at P0, what do we have? 736 00:42:06,800 --> 00:42:11,280 x is minus 1, y is minus 2, z is minus 1. 737 00:42:11,280 --> 00:42:18,210 I should get negative 6, negative 4, and minus 4. 738 00:42:18,210 --> 00:42:21,280 739 00:42:21,280 --> 00:42:24,990 And then I should plug in and get minus 6 times 740 00:42:24,990 --> 00:42:28,050 x minus minus 1. 741 00:42:28,050 --> 00:42:30,060 I have to pay attention myself. 742 00:42:30,060 --> 00:42:36,440 It's not easy to get the algebra right-- minus 4 times y 743 00:42:36,440 --> 00:42:45,126 plus 2 minus 4 times z plus 1 equals 0. 744 00:42:45,126 --> 00:42:50,740 And I hope I get what you got-- minus 6x minus 4y minus 4z, 745 00:42:50,740 --> 00:42:51,830 so many of those. 746 00:42:51,830 --> 00:42:53,320 You've got to divide by 2. 747 00:42:53,320 --> 00:42:55,290 I'm not getting that. 748 00:42:55,290 --> 00:42:59,465 And then minus 6-- I'm going to write it down. 749 00:42:59,465 --> 00:43:03,240 750 00:43:03,240 --> 00:43:05,330 So it's even, right? 751 00:43:05,330 --> 00:43:08,040 752 00:43:08,040 --> 00:43:15,862 The whole thing minus 18, divide by 2 should be-- divide by 2, 753 00:43:15,862 --> 00:43:18,680 and did you change the signs, [INAUDIBLE]? 754 00:43:18,680 --> 00:43:20,620 What's your password? 755 00:43:20,620 --> 00:43:21,576 No. 756 00:43:21,576 --> 00:43:27,230 [LAUGHING] Check if I'm getting the same thing you got. 757 00:43:27,230 --> 00:43:33,130 So I get 3x plus 2y plus 2z. 758 00:43:33,130 --> 00:43:37,188 I divide by minus 2, right, plus 9? 759 00:43:37,188 --> 00:43:39,428 Did you both get the same thing? 760 00:43:39,428 --> 00:43:40,780 STUDENT: [INAUDIBLE]. 761 00:43:40,780 --> 00:43:42,770 PROFESSOR: You didn't? 762 00:43:42,770 --> 00:43:47,255 Well, I'm trying to simplify all my answers 763 00:43:47,255 --> 00:43:49,990 with this simplification. 764 00:43:49,990 --> 00:43:52,527 So I guess of course if you enter it 765 00:43:52,527 --> 00:43:55,450 like that, it's going to work. 766 00:43:55,450 --> 00:44:03,660 Now, I need you guys to help me on this one. 767 00:44:03,660 --> 00:44:06,420 Find the parametric form-- it's so easy, 768 00:44:06,420 --> 00:44:10,640 but this is a problem session-- of the line passing 769 00:44:10,640 --> 00:44:12,790 through the same point that's perpendicular 770 00:44:12,790 --> 00:44:14,540 to the tangent plane. 771 00:44:14,540 --> 00:44:18,740 Express your answer in the parametric form of the type 772 00:44:18,740 --> 00:44:21,190 the one that you know that I don't like very much, 773 00:44:21,190 --> 00:44:30,500 but I will write it down-- a2t [INAUDIBLE] b2 a3t plus b3. 774 00:44:30,500 --> 00:44:34,550 This is how he wants me to write it, which I don't like. 775 00:44:34,550 --> 00:44:38,170 I would've even preferred it to be in symmetric form. 776 00:44:38,170 --> 00:44:39,220 It's the same. 777 00:44:39,220 --> 00:44:41,140 I'll put the t, and I'm fine. 778 00:44:41,140 --> 00:44:47,304 So x minus x0 over l, y minus y0 over m, 779 00:44:47,304 --> 00:44:51,780 z minus z0 over n, that was the symmetric form. 780 00:44:51,780 --> 00:44:55,242 I make it parametric by saying equal to t. 781 00:44:55,242 --> 00:44:58,546 So what were the parametric equations? 782 00:44:58,546 --> 00:45:01,850 x equals lt plus x0. 783 00:45:01,850 --> 00:45:03,280 That's the normal. 784 00:45:03,280 --> 00:45:06,240 y equals mt plus y0. 785 00:45:06,240 --> 00:45:09,170 z equals nt plus z0. 786 00:45:09,170 --> 00:45:14,263 So finally, my answer-- I'll check with [INAUDIBLE] answer 787 00:45:14,263 --> 00:45:20,110 in a second-- should be take the normal from the tangent plane, 788 00:45:20,110 --> 00:45:24,510 3, 2, 2, right? 789 00:45:24,510 --> 00:45:31,905 2t plus whatever, this is 2t plus z 790 00:45:31,905 --> 00:45:45,730 equals-- first it's 3, 3, 2, 2, 3, 2, 2, plus x0 y0 z0. 791 00:45:45,730 --> 00:45:47,915 So erase the pluses and minuses. 792 00:45:47,915 --> 00:45:50,595 793 00:45:50,595 --> 00:45:52,440 And those should be the equations. 794 00:45:52,440 --> 00:45:57,530 And I should write them down. 795 00:45:57,530 --> 00:45:59,700 It's OK, you have the same. 796 00:45:59,700 --> 00:46:05,426 Now you just have to take them, this, this, and that, 797 00:46:05,426 --> 00:46:08,884 and put it in this form. 798 00:46:08,884 --> 00:46:11,090 This is a combination problem. 799 00:46:11,090 --> 00:46:12,810 Why do I say combination problem? 800 00:46:12,810 --> 00:46:16,730 It's combining Chapter 11 with Chapter 9. 801 00:46:16,730 --> 00:46:18,810 This was the review from Chapter 9. 802 00:46:18,810 --> 00:46:21,860 803 00:46:21,860 --> 00:46:23,290 Did you have trouble understanding 804 00:46:23,290 --> 00:46:26,530 the radiant or the tangent planes 805 00:46:26,530 --> 00:46:29,660 or anything like that, implicit form, explicit form? 806 00:46:29,660 --> 00:46:37,200 Let me do an application, since I'm doing review anyway. 807 00:46:37,200 --> 00:46:39,515 I'm done with the Section 11.6. 808 00:46:39,515 --> 00:46:42,144 But before I want to go further, I 809 00:46:42,144 --> 00:46:44,454 want to do some review of Chapter 810 00:46:44,454 --> 00:47:00,640 11 sections 11.1 through 11.6. 811 00:47:00,640 --> 00:47:03,568 I said something about implicit differentiation 812 00:47:03,568 --> 00:47:06,920 being a headache for many of you. 813 00:47:06,920 --> 00:47:19,260 One person asked me, how do you compute z sub x and/or z 814 00:47:19,260 --> 00:47:32,300 sub y based on the equation x squared plus y squared plus z 815 00:47:32,300 --> 00:47:33,437 squared equals 5? 816 00:47:33,437 --> 00:47:36,240 817 00:47:36,240 --> 00:47:38,305 And of course this is implicit differentiation. 818 00:47:38,305 --> 00:47:46,480 819 00:47:46,480 --> 00:47:47,660 Why implicit? 820 00:47:47,660 --> 00:47:55,893 OK, because this is an implicit equation of the type F of x, y, 821 00:47:55,893 --> 00:48:00,340 z equals constant. 822 00:48:00,340 --> 00:48:02,390 When do we call it explicit? 823 00:48:02,390 --> 00:48:06,090 When one of the variables, x or y or z, 824 00:48:06,090 --> 00:48:10,390 is given explicitly in terms of the other two. 825 00:48:10,390 --> 00:48:14,545 So if this would be-- well, here it's hard to pull it out. 826 00:48:14,545 --> 00:48:17,670 But whether it be upper part then lower hemisphere, 827 00:48:17,670 --> 00:48:20,490 z would be plus or minus. 828 00:48:20,490 --> 00:48:23,590 So you have two caps, two hemispheres, 829 00:48:23,590 --> 00:48:27,940 plus/minus square root 5 minus x squared minus y squared. 830 00:48:27,940 --> 00:48:29,600 Well, that's two functions. 831 00:48:29,600 --> 00:48:31,590 We don't like that. 832 00:48:31,590 --> 00:48:34,440 We want to be able to do everything in one shot 833 00:48:34,440 --> 00:48:38,350 without splitting it into two different graphs. 834 00:48:38,350 --> 00:48:43,150 So how do we view z to be a function of x, 835 00:48:43,150 --> 00:48:44,710 you're going to ask yourself. 836 00:48:44,710 --> 00:48:49,400 You imagine inside this thing that x and y 837 00:48:49,400 --> 00:48:54,327 are independent variables-- independent variables. 838 00:48:54,327 --> 00:48:56,490 They can take whatever they want. 839 00:48:56,490 --> 00:48:57,520 One is temperature. 840 00:48:57,520 --> 00:48:58,290 One is time. 841 00:48:58,290 --> 00:49:00,510 They run like crazies. 842 00:49:00,510 --> 00:49:05,040 But z depends on both temperature and time, like us 843 00:49:05,040 --> 00:49:05,876 unfortunately. 844 00:49:05,876 --> 00:49:06,750 It's so cold outside. 845 00:49:06,750 --> 00:49:08,930 I hate it. 846 00:49:08,930 --> 00:49:12,500 OK, you promised me, and it came true. 847 00:49:12,500 --> 00:49:13,555 Who promised me? 848 00:49:13,555 --> 00:49:15,430 Matthew, I give you a brownie point for that. 849 00:49:15,430 --> 00:49:17,890 Because you said last week it's going to be 80 degrees. 850 00:49:17,890 --> 00:49:19,340 And it was. 851 00:49:19,340 --> 00:49:20,910 So the prophecy came true. 852 00:49:20,910 --> 00:49:24,890 On the other hand, it came back too bad. 853 00:49:24,890 --> 00:49:27,420 And of course it's not Matthew's fault. 854 00:49:27,420 --> 00:49:30,590 He didn't say what's going to happen this week. 855 00:49:30,590 --> 00:49:36,225 All right, in this case, implicit differentiation 856 00:49:36,225 --> 00:49:38,690 is just a philosophical thing. 857 00:49:38,690 --> 00:49:41,820 It's a very important philosophical step 858 00:49:41,820 --> 00:49:43,840 that you're taking-- think. 859 00:49:43,840 --> 00:49:48,394 860 00:49:48,394 --> 00:49:56,250 Think of z being a function of x and y. 861 00:49:56,250 --> 00:50:05,218 And two, differentiate z with respect to x. 862 00:50:05,218 --> 00:50:10,220 863 00:50:10,220 --> 00:50:13,700 So what do you mean, differentiate 864 00:50:13,700 --> 00:50:15,095 with respect to x? 865 00:50:15,095 --> 00:50:20,880 By differentiating the entire equation, 866 00:50:20,880 --> 00:50:30,540 both sides of an equation with respect to x. 867 00:50:30,540 --> 00:50:34,750 So for you, x is the wanted variable. 868 00:50:34,750 --> 00:50:36,510 y is like a constant. 869 00:50:36,510 --> 00:50:42,340 z is a function of x is not hard at all. 870 00:50:42,340 --> 00:50:47,850 So what is going to happen actually if you were to do it? 871 00:50:47,850 --> 00:50:49,764 Theoretically, you would go like that. 872 00:50:49,764 --> 00:50:53,990 If I'm going to differentiate this guy with respect to x, 873 00:50:53,990 --> 00:50:55,700 what is the philosophy? 874 00:50:55,700 --> 00:50:59,520 The chain rule tells me, differentiate F 875 00:50:59,520 --> 00:51:05,700 with respect to the first variable, and then times dx/dx. 876 00:51:05,700 --> 00:51:10,340 And you say, god, now that was silly, right? 877 00:51:10,340 --> 00:51:11,955 Differentiate with respect to x. 878 00:51:11,955 --> 00:51:13,960 That's the chain rule. 879 00:51:13,960 --> 00:51:16,560 Plus differentiate F with respect 880 00:51:16,560 --> 00:51:22,350 to the second place, second variable, and then 881 00:51:22,350 --> 00:51:27,544 say, dy with respect to x. 882 00:51:27,544 --> 00:51:30,195 But are dx and dy married? 883 00:51:30,195 --> 00:51:31,800 Do they depend on one another? 884 00:51:31,800 --> 00:51:35,260 Do they file an income tax return together? 885 00:51:35,260 --> 00:51:38,246 They don't want to have anything to do with one another. 886 00:51:38,246 --> 00:51:42,234 Thank god, so x and y are independent variables. 887 00:51:42,234 --> 00:51:45,820 If you're taking statistics or researching 888 00:51:45,820 --> 00:51:48,280 any other kind of physics, chemistry, 889 00:51:48,280 --> 00:51:50,520 you know that these are called independent variables, 890 00:51:50,520 --> 00:51:53,850 and this is called the dependent variable. 891 00:51:53,850 --> 00:51:57,070 And then you have what's called the constraint. 892 00:51:57,070 --> 00:52:01,770 In physics and engineering and mechanics, F of some variable 893 00:52:01,770 --> 00:52:05,460 equals C. It's called constraint usually. 894 00:52:05,460 --> 00:52:08,680 OK, so this guy is all silly. 895 00:52:08,680 --> 00:52:12,646 These guys don't want to have to do anything with one another. 896 00:52:12,646 --> 00:52:15,434 And then you get plus. 897 00:52:15,434 --> 00:52:20,990 Finally, dF with respect to the third place, 898 00:52:20,990 --> 00:52:25,010 and then that third place, z, is occupied by a function that's 899 00:52:25,010 --> 00:52:26,475 a function of x. 900 00:52:26,475 --> 00:52:28,360 So you go, dz/dx. 901 00:52:28,360 --> 00:52:30,880 Why del and not d? 902 00:52:30,880 --> 00:52:35,550 Because poor z is a function of two variables, x and y. 903 00:52:35,550 --> 00:52:37,700 So you cannot say, dz/dx. 904 00:52:37,700 --> 00:52:41,090 You have to say del z dx, equals 0. 905 00:52:41,090 --> 00:52:46,150 Thank god, I got to the end where I wanted to get. 906 00:52:46,150 --> 00:52:51,870 Now, if I want to see what's going on, it's a piece of cake. 907 00:52:51,870 --> 00:52:53,800 That's 1. 908 00:52:53,800 --> 00:52:59,270 And I get that Mr. z sub x, which other people write dz/dx, 909 00:52:59,270 --> 00:53:05,510 but I don't, because I don't like it-- I keep mixing x. 910 00:53:05,510 --> 00:53:11,560 Equals-- how do I pull this guy out? 911 00:53:11,560 --> 00:53:13,865 How do I substitute for that? 912 00:53:13,865 --> 00:53:21,685 I get Mr. First Fellow here to the other side. 913 00:53:21,685 --> 00:53:27,952 He's going to pick up a minus at whatever d 914 00:53:27,952 --> 00:53:34,590 I have divided by-- so this guy divided by this guy. 915 00:53:34,590 --> 00:53:45,750 916 00:53:45,750 --> 00:53:50,720 STUDENT: What happened to dF/dy, dy/dx? 917 00:53:50,720 --> 00:53:52,930 PROFESSOR: So again, that is a very good thing. 918 00:53:52,930 --> 00:53:54,620 So dF/dy was behaving. 919 00:53:54,620 --> 00:53:56,015 He was nice. 920 00:53:56,015 --> 00:54:00,160 But when we got to dy with respect to dx, y said, 921 00:54:00,160 --> 00:54:02,690 I'm not married to dx. 922 00:54:02,690 --> 00:54:04,900 I have nothing to do with dx. 923 00:54:04,900 --> 00:54:06,520 I'm independent from this. 924 00:54:06,520 --> 00:54:10,440 So dy/dx is 0. 925 00:54:10,440 --> 00:54:14,700 And so this guy disappears. dx/dx is 1. 926 00:54:14,700 --> 00:54:16,584 Duh, that's a piece of cake. 927 00:54:16,584 --> 00:54:17,526 So I'm done. 928 00:54:17,526 --> 00:54:22,900 This is actually a formula that looks sort of easy. 929 00:54:22,900 --> 00:54:25,640 But there is a lot hidden behind it. 930 00:54:25,640 --> 00:54:28,594 This is the implicit function theorem. 931 00:54:28,594 --> 00:54:31,480 932 00:54:31,480 --> 00:54:35,110 where you of course assume that these partial derivatives 933 00:54:35,110 --> 00:54:38,460 exist, are continuous, everything is nice. 934 00:54:38,460 --> 00:54:41,450 It's a beautiful result. People actually get 935 00:54:41,450 --> 00:54:45,750 to learn it only when they are big, I mean big mathematically, 936 00:54:45,750 --> 00:54:49,250 mature, in graduate school, first or second year 937 00:54:49,250 --> 00:54:50,060 of graduate school. 938 00:54:50,060 --> 00:54:52,025 We call that intermediate analysis 939 00:54:52,025 --> 00:54:55,170 or advanced-- very advanced-- calculus. 940 00:54:55,170 --> 00:54:57,138 Because this calculus is advanced enough, 941 00:54:57,138 --> 00:55:00,090 but I'm talking about graduate level calculus. 942 00:55:00,090 --> 00:55:03,010 And this is the so-called implicit function theorem. 943 00:55:03,010 --> 00:55:08,335 So if you will ever be even not necessarily a graduate student 944 00:55:08,335 --> 00:55:11,570 in mathematics, but a graduate student in physics or something 945 00:55:11,570 --> 00:55:16,095 related to pure science, remember this result. So let me 946 00:55:16,095 --> 00:55:18,420 see what's going to happen in practice. 947 00:55:18,420 --> 00:55:22,940 In practice, do we have to learn this? 948 00:55:22,940 --> 00:55:27,340 No, in practice we can build everything from scratch, again, 949 00:55:27,340 --> 00:55:32,100 just the way we did it with the formula. 950 00:55:32,100 --> 00:55:33,920 So for the example I gave you, it 951 00:55:33,920 --> 00:55:36,870 should be a piece of cake to do the differentiation. 952 00:55:36,870 --> 00:55:38,870 But I'm going to step by step. 953 00:55:38,870 --> 00:55:41,620 Step one, think. 954 00:55:41,620 --> 00:55:45,950 955 00:55:45,950 --> 00:55:47,250 You have to think. 956 00:55:47,250 --> 00:55:49,740 If you don't think, you cannot do math. 957 00:55:49,740 --> 00:55:53,110 So you have x squared plus y squared, 958 00:55:53,110 --> 00:55:56,980 the independent guys, and z, who is married to both of them. 959 00:55:56,980 --> 00:55:58,940 Or maybe z is the baby. 960 00:55:58,940 --> 00:56:02,473 These are two spouses that are independent from one another. 961 00:56:02,473 --> 00:56:04,180 And z is their baby. 962 00:56:04,180 --> 00:56:06,124 Because he depends on both of them. 963 00:56:06,124 --> 00:56:09,530 964 00:56:09,530 --> 00:56:12,295 So you thought you had a different approach 965 00:56:12,295 --> 00:56:16,710 to the problem, different vision of what's going on. 966 00:56:16,710 --> 00:56:23,090 Now finally, step two, differentiate with respect 967 00:56:23,090 --> 00:56:25,440 to one only, x only. 968 00:56:25,440 --> 00:56:28,892 969 00:56:28,892 --> 00:56:32,235 You could of course do the same process with respect to y. 970 00:56:32,235 --> 00:56:35,110 And in some of the final exam problems, 971 00:56:35,110 --> 00:56:38,176 we are asking, compute z sub x and z sub y. 972 00:56:38,176 --> 00:56:40,150 The secret is that-- maybe I shouldn't 973 00:56:40,150 --> 00:56:45,000 talk too much again-- when I grade those finals, if you 974 00:56:45,000 --> 00:56:47,610 do z sub x, I give you 100%. 975 00:56:47,610 --> 00:56:50,710 Because z sub y is the same. 976 00:56:50,710 --> 00:56:53,880 So I really don't care. 977 00:56:53,880 --> 00:56:56,800 Sometimes there are so many things 978 00:56:56,800 --> 00:57:00,795 to do that all I care is, did he or she cover 979 00:57:00,795 --> 00:57:03,310 the essential work? 980 00:57:03,310 --> 00:57:09,430 So with respect to x, x squared differentiated with respect 981 00:57:09,430 --> 00:57:11,235 to x-- 2x. 982 00:57:11,235 --> 00:57:15,620 Good first step, now, y squared differentiated with respect 983 00:57:15,620 --> 00:57:17,500 to x. 984 00:57:17,500 --> 00:57:19,290 0-- am I going to write 0? 985 00:57:19,290 --> 00:57:21,450 Yes, because I'm silly. 986 00:57:21,450 --> 00:57:24,564 But I don't have to. 987 00:57:24,564 --> 00:57:27,680 2 times z of x, y. 988 00:57:27,680 --> 00:57:32,310 989 00:57:32,310 --> 00:57:35,894 2 jumps down, z of x, y-- I'm not done with the chain rule. 990 00:57:35,894 --> 00:57:39,670 991 00:57:39,670 --> 00:57:41,086 STUDENT: z sub x. 992 00:57:41,086 --> 00:57:43,000 PROFESSOR: It's z sub x, very good. 993 00:57:43,000 --> 00:57:43,870 This is dz/dx. 994 00:57:43,870 --> 00:57:48,290 I'm not going to hide it completely like that. 995 00:57:48,290 --> 00:57:50,200 That is the same thing. 996 00:57:50,200 --> 00:57:52,250 y prime is 0, thank god. 997 00:57:52,250 --> 00:57:55,910 998 00:57:55,910 --> 00:57:59,545 So you say, if I were to keep in mind that that's 999 00:57:59,545 --> 00:58:06,080 the derivative of big F with respect to x, 1000 00:58:06,080 --> 00:58:08,920 I could plug in everything in here. 1001 00:58:08,920 --> 00:58:10,110 I could plug in the formula. 1002 00:58:10,110 --> 00:58:12,317 But why memorize the formula and plug it 1003 00:58:12,317 --> 00:58:16,380 in when you can do everything from scratch all over again? 1004 00:58:16,380 --> 00:58:18,510 Math is not about memorization. 1005 00:58:18,510 --> 00:58:22,570 If you are good, for example, some people here-- I'm 1006 00:58:22,570 --> 00:58:28,000 not going to name them-- are in sciences that involve 1007 00:58:28,000 --> 00:58:29,985 a lot of memorization. 1008 00:58:29,985 --> 00:58:31,670 More power to them. 1009 00:58:31,670 --> 00:58:34,373 I was not very good at that. 1010 00:58:34,373 --> 00:58:37,870 So I'm going to go ahead and write z sub x pulled down 1011 00:58:37,870 --> 00:58:42,950 minus 2x divided by 2z. 1012 00:58:42,950 --> 00:58:47,380 I'm too lazy to remind you that z is the baby, 1013 00:58:47,380 --> 00:58:50,246 and he depends on his parents x and y. 1014 00:58:50,246 --> 00:58:52,492 I'm not going to write that. 1015 00:58:52,492 --> 00:58:53,680 And that's the answer. 1016 00:58:53,680 --> 00:58:57,820 So you have minus x/z. 1017 00:58:57,820 --> 00:59:03,690 So for example, if somebody says, compute z sub x 1018 00:59:03,690 --> 00:59:17,400 at the point on the sphere, that is 0, root 5, and 0, 1019 00:59:17,400 --> 00:59:19,230 what do you have to do? 1020 00:59:19,230 --> 00:59:23,820 You have to say, z sub x equals-- 1021 00:59:23,820 --> 00:59:30,620 and now I'm asking you something that is minus 0/0. 1022 00:59:30,620 --> 00:59:35,980 1023 00:59:35,980 --> 00:59:49,400 Assuming that the expressions, the derivatives, are defined 1024 00:59:49,400 --> 00:59:56,590 and the denominator one is different from 0-- so 1025 00:59:56,590 --> 00:59:59,710 whenever you do the implicit function theorem, 1026 00:59:59,710 --> 01:00:04,790 you can apply with the condition that you are away 1027 01:00:04,790 --> 01:00:10,290 from points where derivative of F with respect to z are 0. 1028 01:00:10,290 --> 01:00:13,320 So this is a problem that's not well posed. 1029 01:00:13,320 --> 01:00:15,565 So to give you a well-posed problem, what 1030 01:00:15,565 --> 01:00:17,860 do I need to do on the final? 1031 01:00:17,860 --> 01:00:23,956 I have to say the same-- 2, 1, and 0. 1032 01:00:23,956 --> 01:00:26,890 1033 01:00:26,890 --> 01:00:27,868 STUDENT: z can't be 0. 1034 01:00:27,868 --> 01:00:30,112 PROFESSOR: No, I know. 1035 01:00:30,112 --> 01:00:35,022 So I go, z is 0 is too easy. 1036 01:00:35,022 --> 01:00:36,004 Let's have y to be 0. 1037 01:00:36,004 --> 01:00:36,712 STUDENT: 2, 0, 1. 1038 01:00:36,712 --> 01:00:39,932 PROFESSOR: Very good, x equals 2, z equals 1, excellent. 1039 01:00:39,932 --> 01:00:47,510 So z sub x at the point 2, 0, 1 will 1040 01:00:47,510 --> 01:00:54,780 be by the implicit function theorem minus 2/1 1041 01:00:54,780 --> 01:00:55,810 equals negative. 1042 01:00:55,810 --> 01:01:00,320 You see, that's a slope in a certain direction 1043 01:01:00,320 --> 01:01:06,128 if you were to look at z with respect to x in the plane x, z. 1044 01:01:06,128 --> 01:01:09,032 OK, what else? 1045 01:01:09,032 --> 01:01:13,410 Nothing-- that was review of chain rule and stuff. 1046 01:01:13,410 --> 01:01:15,390 And you have to review chain rule. 1047 01:01:15,390 --> 01:01:18,340 1048 01:01:18,340 --> 01:01:19,835 Make yourself a note. 1049 01:01:19,835 --> 01:01:22,735 Before the midterm, I have to memorize the chain rule. 1050 01:01:22,735 --> 01:01:23,676 Yes, sir. 1051 01:01:23,676 --> 01:01:24,551 STUDENT: [INAUDIBLE]. 1052 01:01:24,551 --> 01:01:28,210 1053 01:01:28,210 --> 01:01:32,120 PROFESSOR: I will do that either in the review session today 1054 01:01:32,120 --> 01:01:36,640 or in the review for the midterm, OK? 1055 01:01:36,640 --> 01:01:40,493 And I'm thinking about that. 1056 01:01:40,493 --> 01:01:43,774 In March, I want to dedicate at least 10 days 1057 01:01:43,774 --> 01:01:45,960 for the review for the midterm. 1058 01:01:45,960 --> 01:01:46,461 Yes, sir. 1059 01:01:46,461 --> 01:01:47,668 STUDENT: When is the midterm? 1060 01:01:47,668 --> 01:01:49,900 PROFESSOR: The midterm is on the 2nd of April. 1061 01:01:49,900 --> 01:01:55,103 1062 01:01:55,103 --> 01:01:57,905 Several people asked me-- OK, I forgot about that. 1063 01:01:57,905 --> 01:01:59,140 I have to tell you guys. 1064 01:01:59,140 --> 01:02:00,810 Several people asked me questions 1065 01:02:00,810 --> 01:02:03,740 by email about the midterm. 1066 01:02:03,740 --> 01:02:07,320 So the midterm-- write down for yourselves-- will 1067 01:02:07,320 --> 01:02:08,640 be over the following chapters. 1068 01:02:08,640 --> 01:02:11,380 1069 01:02:11,380 --> 01:02:20,864 Chapter 10, no Chapter 9. 1070 01:02:20,864 --> 01:02:23,274 Chapter 9 is [INAUDIBLE]. 1071 01:02:23,274 --> 01:02:30,900 Chapter 11 all, Chapter 10 only what 1072 01:02:30,900 --> 01:02:36,820 we have required-- 10.1, 10.2, and 10.3-- and Chapter 12, 1073 01:02:36,820 --> 01:02:43,760 all but Section 12.6. 1074 01:02:43,760 --> 01:02:46,830 Because I see that some of you study ahead of time. 1075 01:02:46,830 --> 01:02:48,280 More power to you. 1076 01:02:48,280 --> 01:02:49,800 You know what to read. 1077 01:02:49,800 --> 01:02:51,010 Skip Section 12.6. 1078 01:02:51,010 --> 01:02:57,480 And I'm planning to not give you anything after Chapter 13 1079 01:02:57,480 --> 01:02:58,180 on the midterm. 1080 01:02:58,180 --> 01:03:02,445 But of course, Chapter 13 will be on the final emphasized 1081 01:03:02,445 --> 01:03:06,555 in at least six problems out of the 15 problems 1082 01:03:06,555 --> 01:03:11,175 you'll have on the final, all right? 1083 01:03:11,175 --> 01:03:15,120 We still have plenty of time. 1084 01:03:15,120 --> 01:03:18,355 Chapter 9, guys, you were concerned about it. 1085 01:03:18,355 --> 01:03:21,560 It's some sort of embedded, you see? 1086 01:03:21,560 --> 01:03:23,170 Wherever you go, wherever you turn, 1087 01:03:23,170 --> 01:03:26,740 you bump into some parametric equations of a line 1088 01:03:26,740 --> 01:03:28,010 or bump into a tangent line. 1089 01:03:28,010 --> 01:03:33,650 That's the dot product that you dealt with, delta F dot N. 1090 01:03:33,650 --> 01:03:38,050 So it's like an obsession, repetitive review of Chapter 9 1091 01:03:38,050 --> 01:03:39,520 at ever step. 1092 01:03:39,520 --> 01:03:40,940 Vector spaces are very important. 1093 01:03:40,940 --> 01:03:44,860 Vectors in general are very important. 1094 01:03:44,860 --> 01:03:47,910 I'm going to move onto 11.7 right now. 1095 01:03:47,910 --> 01:03:49,494 We'll take a break. 1096 01:03:49,494 --> 01:03:53,270 Why don't we take a short break now, five minutes. 1097 01:03:53,270 --> 01:03:58,505 And then we have to go on until 2:50. 1098 01:03:58,505 --> 01:04:01,730 So practically we have one more hour. 1099 01:04:01,730 --> 01:04:03,530 Take a break, eat, drink something. 1100 01:04:03,530 --> 01:04:05,390 I don't want a big break. 1101 01:04:05,390 --> 01:04:07,650 Because then a big break we'll just fall asleep. 1102 01:04:07,650 --> 01:04:10,532 I'm tired as well. 1103 01:04:10,532 --> 01:04:13,022 So we have to keep going. 1104 01:04:13,022 --> 01:04:16,010 1105 01:04:16,010 --> 01:04:19,994 [BACKGROUND CHATTER] 1106 01:04:19,994 --> 01:08:40,145 1107 01:08:40,145 --> 01:08:41,142 PROFESSOR: All right. 1108 01:08:41,142 --> 01:08:43,790 I will start with a little bit of a review 1109 01:08:43,790 --> 01:08:46,096 of some friend of yours. 1110 01:08:46,096 --> 01:08:49,078 And since we are in Texas, of course 1111 01:08:49,078 --> 01:08:50,569 this counts as an obsession. 1112 01:08:50,569 --> 01:08:54,049 1113 01:08:54,049 --> 01:09:00,013 This is going to be extrema of functions of several variables. 1114 01:09:00,013 --> 01:09:15,420 1115 01:09:15,420 --> 01:09:16,810 Do I draw better lately? 1116 01:09:16,810 --> 01:09:20,310 I think I do. 1117 01:09:20,310 --> 01:09:22,029 That's why I stopped drinking coffee. 1118 01:09:22,029 --> 01:09:23,702 I'm drinking white tea. 1119 01:09:23,702 --> 01:09:26,354 It's good for you. 1120 01:09:26,354 --> 01:09:27,500 All right. 1121 01:09:27,500 --> 01:09:28,470 White tea. 1122 01:09:28,470 --> 01:09:30,675 For some reason, the black tea was giving me 1123 01:09:30,675 --> 01:09:33,370 the shaking and all that. 1124 01:09:33,370 --> 01:09:34,840 Too much black tea. 1125 01:09:34,840 --> 01:09:37,779 I don't know, maybe it has less caffeine. 1126 01:09:37,779 --> 01:09:39,701 Jasmine is good, green, or white. 1127 01:09:39,701 --> 01:09:41,956 STUDENT: I think green has less [INAUDIBLE]. 1128 01:09:41,956 --> 01:09:43,590 PROFESSOR: OK. 1129 01:09:43,590 --> 01:09:47,370 So above this saddle is a function 1130 01:09:47,370 --> 01:09:52,206 of two variables-- you know a lot already, 1131 01:09:52,206 --> 01:09:57,147 but I'm asking you to compute the partial derivatives 1132 01:09:57,147 --> 01:09:59,602 and the gradient. 1133 01:09:59,602 --> 01:10:01,490 And you're going to jump on it and say 1134 01:10:01,490 --> 01:10:03,730 I'm doing [INAUDIBLE] anyway. 1135 01:10:03,730 --> 01:10:08,890 So I've got 2x, and this is minus 2y. 1136 01:10:08,890 --> 01:10:11,140 If I want to ask you the differential 1137 01:10:11,140 --> 01:10:14,000 on the final or midterm, you will say 1138 01:10:14,000 --> 01:10:19,190 that f sub xdx plus x of ygy. 1139 01:10:19,190 --> 01:10:22,472 Everybody knows that. 1140 01:10:22,472 --> 01:10:23,750 Don't break my heart. 1141 01:10:23,750 --> 01:10:28,455 Don't say 2x minus y, because I'll never recover. 1142 01:10:28,455 --> 01:10:31,340 Every time I see that, I die 100 deaths. 1143 01:10:31,340 --> 01:10:34,755 So don't forget about the x and the y, 1144 01:10:34,755 --> 01:10:40,071 which are the important guys of infinitesimal elements. 1145 01:10:40,071 --> 01:10:42,406 This is a 1 form. 1146 01:10:42,406 --> 01:10:47,580 In mathematics, any combination of a dx and dy 1147 01:10:47,580 --> 01:10:50,505 in a linear combination in the 1 form. 1148 01:10:50,505 --> 01:10:54,540 It's a consecrated terminology. 1149 01:10:54,540 --> 01:10:56,500 But I'm not asking you about the differential. 1150 01:10:56,500 --> 01:10:59,560 I'm asking you about the gradient. 1151 01:10:59,560 --> 01:11:06,810 All righty, and that is a f sub xi plus f sub yk, which 1152 01:11:06,810 --> 01:11:11,000 is exactly 2xi minus 2yj. 1153 01:11:11,000 --> 01:11:14,690 1154 01:11:14,690 --> 01:11:18,270 And you say all right, but I want to take a look, 1155 01:11:18,270 --> 01:11:20,260 I always have started with examples. 1156 01:11:20,260 --> 01:11:23,095 Hopefully they are good. 1157 01:11:23,095 --> 01:11:27,415 Let's look at the tangent vectors to the surface. 1158 01:11:27,415 --> 01:11:30,163 We discussed about the notion of tangent vector 1159 01:11:30,163 --> 01:11:34,000 before, remember, when we had r sub u and r sub 1160 01:11:34,000 --> 01:11:35,560 v form the parametrization. 1161 01:11:35,560 --> 01:11:39,472 Now look at the tangent vectors for this graph 1162 01:11:39,472 --> 01:11:42,750 along the x direction going this way, 1163 01:11:42,750 --> 01:11:46,850 and along the y direction going this way. 1164 01:11:46,850 --> 01:11:51,715 We see that both of them are horizontal at the origin. 1165 01:11:51,715 --> 01:11:54,570 And that's a beautiful thing. 1166 01:11:54,570 --> 01:12:00,750 And so this origin is a so-called critical point. 1167 01:12:00,750 --> 01:12:05,222 Critical point for a differentiable function. 1168 01:12:05,222 --> 01:12:13,090 1169 01:12:13,090 --> 01:12:23,540 Z equals f of xy is a point in plane x0i0 1170 01:12:23,540 --> 01:12:30,918 where the partial derivatives vanish. 1171 01:12:30,918 --> 01:12:35,838 1172 01:12:35,838 --> 01:12:41,044 And according to the book, and many books, all don't exist. 1173 01:12:41,044 --> 01:12:45,410 Well I don't like that. 1174 01:12:45,410 --> 01:12:53,100 Even our book says if you have a function in calc 1-- 1175 01:12:53,100 --> 01:13:01,650 let's say b equal g of u, critical point. 1176 01:13:01,650 --> 01:13:05,440 Do you remember what a critical point was? 1177 01:13:05,440 --> 01:13:12,400 U0, in calc 1 we said either a point where g prime of u was 0, 1178 01:13:12,400 --> 01:13:16,570 or g prime of u 0 doesn't exist. 1179 01:13:16,570 --> 01:13:19,470 1180 01:13:19,470 --> 01:13:22,810 Although u is 0 is in the domain. 1181 01:13:22,810 --> 01:13:24,456 I don't like that. 1182 01:13:24,456 --> 01:13:27,100 You say wait a minute, why don't you like that? 1183 01:13:27,100 --> 01:13:30,020 I don't like that for many reasons practically. 1184 01:13:30,020 --> 01:13:37,630 If you have the absolute value function, 1185 01:13:37,630 --> 01:13:41,520 you'll say yeah, yeah, but look, I considered the corner 1186 01:13:41,520 --> 01:13:44,420 to be a point of non-differentiability, 1187 01:13:44,420 --> 01:13:49,882 but it's still an extreme value, a critical point. 1188 01:13:49,882 --> 01:13:53,790 According to our book in Calculus 1, yeah. 1189 01:13:53,790 --> 01:13:57,400 We extended this definition to ugly points, 1190 01:13:57,400 --> 01:14:00,050 points where you don't have a [? pick ?] or a value 1191 01:14:00,050 --> 01:14:04,660 or an inflection, but you have something ugly like 1192 01:14:04,660 --> 01:14:08,660 a [? cusp, ?] a corner, the ugliness. 1193 01:14:08,660 --> 01:14:10,380 I don't like that kind of ugliness, 1194 01:14:10,380 --> 01:14:14,660 because I want to have more information there. 1195 01:14:14,660 --> 01:14:19,841 I maybe even have a point with a bigger problem than that. 1196 01:14:19,841 --> 01:14:22,030 First of all, when I say critical point, 1197 01:14:22,030 --> 01:14:24,856 I have to assume the point is in the domain of the function. 1198 01:14:24,856 --> 01:14:28,153 But then what kind of ugliness I can have there? 1199 01:14:28,153 --> 01:14:30,200 I don't even want to think about it. 1200 01:14:30,200 --> 01:14:34,870 So in the context of my class-- in context 1201 01:14:34,870 --> 01:14:47,010 of my class-- calc 3 honors, I will denote a critical point. 1202 01:14:47,010 --> 01:14:51,150 1203 01:14:51,150 --> 01:15:01,312 Is the x0y0 such that f sub x at x0y0 is 0. 1204 01:15:01,312 --> 01:15:04,990 One slope is 0, the other slope is 0. 1205 01:15:04,990 --> 01:15:08,027 f sub y is x0y0, of course. 1206 01:15:08,027 --> 01:15:14,012 And no other are the points. 1207 01:15:14,012 --> 01:15:18,286 What am I going to call the [INAUDIBLE] points where 1208 01:15:18,286 --> 01:15:24,140 derivatives don't exist? 1209 01:15:24,140 --> 01:15:31,420 I simply say I have a singularity. 1210 01:15:31,420 --> 01:15:33,650 I have a singularity. 1211 01:15:33,650 --> 01:15:38,070 What type of singularity we can discuss in an advanced calculus 1212 01:15:38,070 --> 01:15:38,570 setting. 1213 01:15:38,570 --> 01:15:42,145 If you're math majors, you're going to have the chance 1214 01:15:42,145 --> 01:15:43,830 to discuss that later on. 1215 01:15:43,830 --> 01:15:49,065 So remember that I would prefer both in the context 1216 01:15:49,065 --> 01:15:54,600 of calculus 1 and calculus 3 to say critical value 1217 01:15:54,600 --> 01:15:58,550 is where the derivative becomes zero. 1218 01:15:58,550 --> 01:16:02,560 Not undefined, plus, minus, infinity, or something 1219 01:16:02,560 --> 01:16:05,870 really crazy, one on the left, one on the right. 1220 01:16:05,870 --> 01:16:09,625 So I don't want to have any kind of complications. 1221 01:16:09,625 --> 01:16:14,645 Now you may say, but I thought that since you 1222 01:16:14,645 --> 01:16:17,069 have those slopes both zero, that 1223 01:16:17,069 --> 01:16:21,420 means that the tangent plane at the point is horizontal. 1224 01:16:21,420 --> 01:16:23,072 And that's exactly what it is. 1225 01:16:23,072 --> 01:16:23,780 I agree with you. 1226 01:16:23,780 --> 01:16:26,520 If somebody would draw the tangent plane to the surface, 1227 01:16:26,520 --> 01:16:29,630 S-- S is for surface, but it's funny, 1228 01:16:29,630 --> 01:16:32,210 S is also coming from saddles. 1229 01:16:32,210 --> 01:16:36,920 So that's a saddle point, saddle surface. 1230 01:16:36,920 --> 01:16:39,030 Origin is so-called saddle point. 1231 01:16:39,030 --> 01:16:40,135 We don't know yet why. 1232 01:16:40,135 --> 01:16:42,698 1233 01:16:42,698 --> 01:16:49,280 The tangent plane at 0, at the origin, 1234 01:16:49,280 --> 01:16:52,950 will be 0.0 in this case. 1235 01:16:52,950 --> 01:16:54,070 Why? 1236 01:16:54,070 --> 01:16:55,680 Well, it's easy to see. 1237 01:16:55,680 --> 01:17:01,220 z minus 0 equals f sub x, x minus x0 plus f sub y, 1238 01:17:01,220 --> 01:17:02,550 y minus y0. 1239 01:17:02,550 --> 01:17:06,090 But this is 0 and that's 0, so z equals zero. 1240 01:17:06,090 --> 01:17:08,380 So thank you very much. 1241 01:17:08,380 --> 01:17:09,240 Poor horse. 1242 01:17:09,240 --> 01:17:14,540 I can take a horizontal plane, imaginary plane 1243 01:17:14,540 --> 01:17:22,190 and make it be tangent to the saddle in all directions 1244 01:17:22,190 --> 01:17:24,426 at the point in the middle. 1245 01:17:24,426 --> 01:17:29,850 1246 01:17:29,850 --> 01:17:30,531 All right. 1247 01:17:30,531 --> 01:17:32,614 STUDENT: So you're saying [? the critical ?] point 1248 01:17:32,614 --> 01:17:33,428 is where both-- 1249 01:17:33,428 --> 01:17:35,868 PROFESSOR: Where both partial derivatives vanish. 1250 01:17:35,868 --> 01:17:37,820 They have to both vanish. 1251 01:17:37,820 --> 01:17:40,138 In case of calculus 1, of course there 1252 01:17:40,138 --> 01:17:44,660 is only one derivative that vanishes at that point. 1253 01:17:44,660 --> 01:17:47,190 What if I were in-- now, you see, 1254 01:17:47,190 --> 01:17:49,926 the more you ask me questions, the more I think 1255 01:17:49,926 --> 01:17:51,515 And it's a dangerous thing. 1256 01:17:51,515 --> 01:17:56,150 What if I had z equals f of x1, x2, x3, xn? 1257 01:17:56,150 --> 01:17:59,310 Critical point would be where all the partial derivatives 1258 01:17:59,310 --> 01:18:00,275 will be zero. 1259 01:18:00,275 --> 01:18:02,690 And then the situation becomes more complicated, 1260 01:18:02,690 --> 01:18:05,020 but it's doable. 1261 01:18:05,020 --> 01:18:11,320 The other is the classification of special points. 1262 01:18:11,320 --> 01:18:25,280 Classification of critical points 1263 01:18:25,280 --> 01:18:30,501 based on second partial derivatives. 1264 01:18:30,501 --> 01:18:39,944 1265 01:18:39,944 --> 01:18:44,835 The objects you want to study in this case are several. 1266 01:18:44,835 --> 01:18:48,160 1267 01:18:48,160 --> 01:18:52,655 One of the most important ones is the so-called discriminant. 1268 01:18:52,655 --> 01:18:54,794 What is the discriminant? 1269 01:18:54,794 --> 01:18:58,540 You haven't talked about discriminants since a long time 1270 01:18:58,540 --> 01:18:59,350 ago. 1271 01:18:59,350 --> 01:19:02,040 And there is a relationship between discriminant 1272 01:19:02,040 --> 01:19:09,070 in high school algebra and discriminant in calculus 3. 1273 01:19:09,070 --> 01:19:10,860 The discriminant the way we define 1274 01:19:10,860 --> 01:19:13,970 it is D, or delta-- some people denote it 1275 01:19:13,970 --> 01:19:16,930 like this, some people by delta-- 1276 01:19:16,930 --> 01:19:19,360 and that is the following. 1277 01:19:19,360 --> 01:19:21,920 This is the determinant. 1278 01:19:21,920 --> 01:19:29,970 f sub xx, f sub xy, f sub yx, f sub yy, 1279 01:19:29,970 --> 01:19:32,300 computed at the point p0, which is critical. 1280 01:19:32,300 --> 01:19:35,900 1281 01:19:35,900 --> 01:19:40,440 So p0 first has to satisfy those two equations, 1282 01:19:40,440 --> 01:19:42,690 and then I'm going to have to compute the [INAUDIBLE] 1283 01:19:42,690 --> 01:19:44,560 at that point. 1284 01:19:44,560 --> 01:19:46,256 But you say wait a minute, Magdalena, 1285 01:19:46,256 --> 01:19:47,570 what the heck is this? 1286 01:19:47,570 --> 01:19:51,105 Well this is the second partial with respect 1287 01:19:51,105 --> 01:19:55,200 of x, one after the other, second partial with respect 1288 01:19:55,200 --> 01:19:56,880 to y, one after the other. 1289 01:19:56,880 --> 01:19:58,085 These guys are equal. 1290 01:19:58,085 --> 01:20:01,455 Remember that there was a German mathematician 1291 01:20:01,455 --> 01:20:06,600 whose name was Schwartz, the black cavalier, the black man. 1292 01:20:06,600 --> 01:20:08,510 Schwartz means black in German. 1293 01:20:08,510 --> 01:20:10,370 And he came up with this theorem that it 1294 01:20:10,370 --> 01:20:14,770 doesn't matter in which order you differentiate, f sub xy 1295 01:20:14,770 --> 01:20:19,250 or f sub yx is the same thing as long as the function is smooth. 1296 01:20:19,250 --> 01:20:22,320 So I'm very happy about that. 1297 01:20:22,320 --> 01:20:26,480 Now there are these other guys, A, B, 1298 01:20:26,480 --> 01:20:30,790 C. It's very easy to remember, it's from the song 1299 01:20:30,790 --> 01:20:33,070 that you all learned in kindergarten. 1300 01:20:33,070 --> 01:20:37,170 Once you know your ABC, you come back to the discriminant. 1301 01:20:37,170 --> 01:20:45,300 So f sub xx at the point p0, f sub xy at the point p0, 1302 01:20:45,300 --> 01:20:50,370 and f sub yy at the point p0. 1303 01:20:50,370 --> 01:20:53,140 Second partial with respect to x, second partial with respect 1304 01:20:53,140 --> 01:20:56,870 to x and y, mixed one, mixed derivative, and second 1305 01:20:56,870 --> 01:20:58,260 partial with respect to y. 1306 01:20:58,260 --> 01:21:01,170 1307 01:21:01,170 --> 01:21:06,960 You have to plug in the values for the p0 will be x0, y0. 1308 01:21:06,960 --> 01:21:09,890 The critical point you got from what? 1309 01:21:09,890 --> 01:21:12,690 From solving this system. 1310 01:21:12,690 --> 01:21:16,210 So you got x0y0 by solving that system. 1311 01:21:16,210 --> 01:21:21,930 Come back, plug in, compute those, get ABC as numbers. 1312 01:21:21,930 --> 01:21:26,093 And who is D going to be based on ABC? 1313 01:21:26,093 --> 01:21:30,020 According to the diagram that I drew, it's easy for you guys 1314 01:21:30,020 --> 01:21:35,910 to see that A and B and C are what? 1315 01:21:35,910 --> 01:21:46,390 Related to D. So D will simply be A, B, B, and C, 1316 01:21:46,390 --> 01:21:48,370 computed at the point p0. 1317 01:21:48,370 --> 01:21:51,340 1318 01:21:51,340 --> 01:21:54,340 So it's going to be now-- now that 1319 01:21:54,340 --> 01:21:56,556 will remind you of something. 1320 01:21:56,556 --> 01:21:58,448 AC minus B-squared. 1321 01:21:58,448 --> 01:22:02,710 1322 01:22:02,710 --> 01:22:03,525 OK? 1323 01:22:03,525 --> 01:22:07,310 When we had the quadratic formula in school-- 1324 01:22:07,310 --> 01:22:08,990 I'm not going to write it. 1325 01:22:08,990 --> 01:22:10,115 I'm going to write it here. 1326 01:22:10,115 --> 01:22:12,010 So what was the quadratic formula? 1327 01:22:12,010 --> 01:22:15,230 ax-squared plus bx plus c equals 0. 1328 01:22:15,230 --> 01:22:17,150 That was algebra. 1329 01:22:17,150 --> 01:22:18,060 Baby algebra. 1330 01:22:18,060 --> 01:22:19,000 What do we call that? 1331 01:22:19,000 --> 01:22:21,420 High school algebra? 1332 01:22:21,420 --> 01:22:28,500 x12 plus minus b plus minus square root of b-squared minus 1333 01:22:28,500 --> 01:22:33,520 4ac divided by 2a. 1334 01:22:33,520 --> 01:22:37,340 Now don't don;t know what kind of professors you had. 1335 01:22:37,340 --> 01:22:41,760 But I had a teacher when I was in high school. 1336 01:22:41,760 --> 01:22:44,550 Every time she taught me something and I did not 1337 01:22:44,550 --> 01:22:46,860 absorb it, she was all over me. 1338 01:22:46,860 --> 01:22:49,290 She was preparing me for some math competitions, 1339 01:22:49,290 --> 01:22:51,390 and she taught me a trick. 1340 01:22:51,390 --> 01:22:54,000 She said look, Magdalena, pay attention. 1341 01:22:54,000 --> 01:23:01,436 If b would be an even number-- take b to be 2b prime, 1342 01:23:01,436 --> 01:23:04,430 2-- give me another letter. 1343 01:23:04,430 --> 01:23:07,880 2 big B. Right? 1344 01:23:07,880 --> 01:23:11,864 Then, the quadratic formula would be easier to use. 1345 01:23:11,864 --> 01:23:17,660 Because in that case, you get x1 2 equals minus-- b is 2b. 1346 01:23:17,660 --> 01:23:20,856 So you have just 2b like that. 1347 01:23:20,856 --> 01:23:24,770 Plus minus square root 4b squared 1348 01:23:24,770 --> 01:23:29,080 minus 4ac divided by 2a. 1349 01:23:29,080 --> 01:23:31,460 She explained this to me once and then 1350 01:23:31,460 --> 01:23:34,860 she expected me to remember it for the rest of my life. 1351 01:23:34,860 --> 01:23:42,105 And then she said minus big B plus minus square root of bb 1352 01:23:42,105 --> 01:23:44,005 squared minus ac. 1353 01:23:44,005 --> 01:23:45,220 Do you see why? 1354 01:23:45,220 --> 01:23:50,135 It's because you pull out the factor of 4, square root of 4 1355 01:23:50,135 --> 01:23:51,298 is 2. 1356 01:23:51,298 --> 01:23:53,596 2, 2, and 2 simplify. 1357 01:23:53,596 --> 01:23:56,370 And then she gave me to solve problems. 1358 01:23:56,370 --> 01:23:57,479 STUDENT: What about the a? 1359 01:23:57,479 --> 01:23:58,520 STUDENT: How about the a? 1360 01:23:58,520 --> 01:24:00,436 STUDENT: Because you divide it by [INAUDIBLE]. 1361 01:24:00,436 --> 01:24:01,490 PROFESSOR: Divided by. 1362 01:24:01,490 --> 01:24:04,095 I forgot to write it down. 1363 01:24:04,095 --> 01:24:05,765 Because I didn't have space. 1364 01:24:05,765 --> 01:24:09,080 I said, I'm not going to bend and doodle. 1365 01:24:09,080 --> 01:24:17,964 So when you have x-squared plus 2x-- let's say minus 3. 1366 01:24:17,964 --> 01:24:20,376 And she gave me that. 1367 01:24:20,376 --> 01:24:21,675 And I said OK, let me do it. 1368 01:24:21,675 --> 01:24:22,250 Let me do it. 1369 01:24:22,250 --> 01:24:30,700 x1 2 minus 2 plus minus square root b-squared minus 4ac, 1370 01:24:30,700 --> 01:24:34,090 which is 12, divided by 2. 1371 01:24:34,090 --> 01:24:36,030 And she started screaming. 1372 01:24:36,030 --> 01:24:37,580 And she started screaming big time. 1373 01:24:37,580 --> 01:24:38,960 Do you know why? 1374 01:24:38,960 --> 01:24:44,340 She said, I just told you the half formula. 1375 01:24:44,340 --> 01:24:48,040 By half formula, I mean she meant this one. 1376 01:24:48,040 --> 01:24:50,850 So when-- and I said OK, OK, the half formula. 1377 01:24:50,850 --> 01:24:53,860 But then for maybe another seven years, 1378 01:24:53,860 --> 01:24:57,230 I did this with the formula-- with the formula 1379 01:24:57,230 --> 01:24:59,813 that everybody knows. 1380 01:24:59,813 --> 01:25:01,530 And at the end, I would remember I 1381 01:25:01,530 --> 01:25:03,000 could have done the half formula, 1382 01:25:03,000 --> 01:25:07,640 but I didn't do it because I'm in a routine. 1383 01:25:07,640 --> 01:25:11,510 So the way she wanted me to do this was what? 1384 01:25:11,510 --> 01:25:13,650 Who is the half of 2? 1385 01:25:13,650 --> 01:25:14,690 1. 1386 01:25:14,690 --> 01:25:21,265 So put minus 1 plus minus square root of big B 1387 01:25:21,265 --> 01:25:26,820 is 1-squared minus a times c, which 1388 01:25:26,820 --> 01:25:30,600 is plus 3, divided by 1 divided by nobody. 1389 01:25:30,600 --> 01:25:33,150 This way you don't have to simplify it further, 1390 01:25:33,150 --> 01:25:36,040 and you do it faster. 1391 01:25:36,040 --> 01:25:46,060 So you get minus 1 plus minus root 4, which is minus 5 and 3. 1392 01:25:46,060 --> 01:25:49,205 But of course, you could have done this by factoring. 1393 01:25:49,205 --> 01:25:51,510 So you could have said wait a minute. 1394 01:25:51,510 --> 01:25:55,020 Two numbers that multiply-- um-- 1395 01:25:55,020 --> 01:25:56,460 STUDENT: [INAUDIBLE] square root. 1396 01:25:56,460 --> 01:25:57,668 PROFESSOR: I didn't do right. 1397 01:25:57,668 --> 01:25:58,380 So it's 4-- 1398 01:25:58,380 --> 01:26:00,780 STUDENT: [INAUDIBLE]. 1399 01:26:00,780 --> 01:26:01,750 PROFESSOR: Yeah. 1400 01:26:01,750 --> 01:26:07,066 So you get x plus 5 times-- 1401 01:26:07,066 --> 01:26:09,400 STUDENT: It's x minus 1-- 1402 01:26:09,400 --> 01:26:10,999 [INTERPOSING VOICES] 1403 01:26:10,999 --> 01:26:12,082 PROFESSOR: Oh, I think I-- 1404 01:26:12,082 --> 01:26:12,955 STUDENT: Square root. 1405 01:26:12,955 --> 01:26:13,913 PROFESSOR: Square root. 1406 01:26:13,913 --> 01:26:15,550 I'm sorry, guys. 1407 01:26:15,550 --> 01:26:16,050 OK. 1408 01:26:16,050 --> 01:26:17,500 Thank you for that. 1409 01:26:17,500 --> 01:26:18,980 1 and minus 3. 1410 01:26:18,980 --> 01:26:25,350 So x plus 3 times x minus 1, which 1411 01:26:25,350 --> 01:26:28,905 is the same-- the exact same as x-squared plus 2x minus 3 1412 01:26:28,905 --> 01:26:30,210 equals. 1413 01:26:30,210 --> 01:26:31,160 All right? 1414 01:26:31,160 --> 01:26:37,380 So just the way she insisted that I learn the half formula. 1415 01:26:37,380 --> 01:26:42,020 I'm not insisting that you learn the half formula, god forbid. 1416 01:26:42,020 --> 01:26:45,540 But see here there is some more symmetry. 1417 01:26:45,540 --> 01:26:48,640 The four doesn't appear anymore. 1418 01:26:48,640 --> 01:26:52,480 b-squared minus 4ac appeared here, but here it doesn't. 1419 01:26:52,480 --> 01:26:55,365 Here you're going to have b-squared minus ac. 1420 01:26:55,365 --> 01:26:56,470 There is a reason. 1421 01:26:56,470 --> 01:27:00,090 This comes from a discriminant just like that. 1422 01:27:00,090 --> 01:27:02,480 And this is why I told you the whole secret 1423 01:27:02,480 --> 01:27:06,010 about the half quadratic formula. 1424 01:27:06,010 --> 01:27:08,206 Not because I wanted you to know about it, 1425 01:27:08,206 --> 01:27:13,512 but because I wanted you to see that there is a pattern here. 1426 01:27:13,512 --> 01:27:15,905 You have-- for the half formula, you 1427 01:27:15,905 --> 01:27:20,785 have plus minus square root of a new type of discriminant. 1428 01:27:20,785 --> 01:27:24,260 People even call this discriminant b-squared 1429 01:27:24,260 --> 01:27:25,282 minus 4ac. 1430 01:27:25,282 --> 01:27:26,710 b-squared minus ac. 1431 01:27:26,710 --> 01:27:30,510 So for us, it is ac minus b-squared. 1432 01:27:30,510 --> 01:27:35,490 It's just the opposite of that discriminant you have. 1433 01:27:35,490 --> 01:27:39,740 Now depending on the sign of this discriminant, 1434 01:27:39,740 --> 01:27:48,350 you can go ahead and classify the critical values you have. 1435 01:27:48,350 --> 01:27:51,600 So classification is the following. 1436 01:27:51,600 --> 01:27:56,970 Classification of special critical points. 1437 01:27:56,970 --> 01:27:59,850 1438 01:27:59,850 --> 01:28:07,230 If delta at p0 is negative, then p0 is a saddle point. 1439 01:28:07,230 --> 01:28:11,650 1440 01:28:11,650 --> 01:28:21,160 If delta at p0 is 0, nothing can be said yet 1441 01:28:21,160 --> 01:28:23,865 about the nature of the point. 1442 01:28:23,865 --> 01:28:26,870 So I make a face, a sad face. 1443 01:28:26,870 --> 01:28:41,200 If delta at p0 is greater than 0, then I have to ramify again. 1444 01:28:41,200 --> 01:28:47,356 And I get if a is positive, it's going to look like a smile. 1445 01:28:47,356 --> 01:28:48,930 Forget about this side. 1446 01:28:48,930 --> 01:28:51,330 It's going to look like a smile. 1447 01:28:51,330 --> 01:28:57,650 So it's going to be a valley point, what do we call that? 1448 01:28:57,650 --> 01:29:02,650 Relative minimum, or valley point. 1449 01:29:02,650 --> 01:29:05,137 Don't say valley point on the exam, OK? 1450 01:29:05,137 --> 01:29:05,931 Relative minimum. 1451 01:29:05,931 --> 01:29:10,040 If a is less than 0 at the point, 1452 01:29:10,040 --> 01:29:13,610 then locally the surface will look 1453 01:29:13,610 --> 01:29:21,030 like I have a peak-- a relative maximum Peaks and valleys. 1454 01:29:21,030 --> 01:29:24,207 Just the way you remember them in Calc 1. 1455 01:29:24,207 --> 01:29:25,790 Now it's a little bit more complicated 1456 01:29:25,790 --> 01:29:29,120 because the functions have two variables. 1457 01:29:29,120 --> 01:29:33,310 But some of the patterns can be recognized. 1458 01:29:33,310 --> 01:29:36,340 1459 01:29:36,340 --> 01:29:42,540 Let's go back to our original example 1460 01:29:42,540 --> 01:29:45,020 and say wait a minute, Magdalena. 1461 01:29:45,020 --> 01:29:47,800 You just gave us a saddle, but we didn't 1462 01:29:47,800 --> 01:29:49,760 do the whole classification. 1463 01:29:49,760 --> 01:29:54,822 Yes, we didn't, because I didn't go over the next steps. 1464 01:29:54,822 --> 01:29:57,430 z equals x-squared minus y-squared. 1465 01:29:57,430 --> 01:29:59,292 Again, we computed the gradient. 1466 01:29:59,292 --> 01:30:01,697 We computed the partial derivatives. 1467 01:30:01,697 --> 01:30:07,774 And then what was that one in finding the critical points? 1468 01:30:07,774 --> 01:30:11,790 So f sub x equals 0, f sub y equals 0. 1469 01:30:11,790 --> 01:30:14,340 Solve for x and y. 1470 01:30:14,340 --> 01:30:17,166 1471 01:30:17,166 --> 01:30:19,050 And that's good, because that's going 1472 01:30:19,050 --> 01:30:23,035 to give me a lot of information, a lot that 1473 01:30:23,035 --> 01:30:25,930 will give me exactly where the critical points may be. 1474 01:30:25,930 --> 01:30:32,210 So that is if and only if I need to solve 2x equals 0 minus 2y 1475 01:30:32,210 --> 01:30:33,890 equals 0. 1476 01:30:33,890 --> 01:30:36,150 Is this system hard to solve? 1477 01:30:36,150 --> 01:30:36,830 No. 1478 01:30:36,830 --> 01:30:39,200 That's exactly why I picked it. 1479 01:30:39,200 --> 01:30:41,490 Because it's easy to solve. 1480 01:30:41,490 --> 01:30:47,450 The only solution is x0 equals y0 equals 0. 1481 01:30:47,450 --> 01:30:52,400 So the origin-- that's exactly where 1482 01:30:52,400 --> 01:30:59,790 you put your butt on the saddle when you ride the horse. 1483 01:30:59,790 --> 01:31:04,190 That is the only critical point you have. 1484 01:31:04,190 --> 01:31:05,670 The only one. 1485 01:31:05,670 --> 01:31:11,520 Now if we want to classify that, what kind of-- is it a valley? 1486 01:31:11,520 --> 01:31:12,770 No. 1487 01:31:12,770 --> 01:31:15,405 It looks like a valley in the direction 1488 01:31:15,405 --> 01:31:20,940 of the axis of the horse, Because the saddle's 1489 01:31:20,940 --> 01:31:22,900 going to look like that. 1490 01:31:22,900 --> 01:31:23,650 This is the horse. 1491 01:31:23,650 --> 01:31:25,970 That's the head of the horse I'm petting. 1492 01:31:25,970 --> 01:31:29,150 And this is the tail of the horse. 1493 01:31:29,150 --> 01:31:35,040 So in this direction, the saddle will be shaped like a parabola, 1494 01:31:35,040 --> 01:31:36,190 like a valley. 1495 01:31:36,190 --> 01:31:40,560 But in the perpendicular direction, 1496 01:31:40,560 --> 01:31:43,250 it's going to be shaped going down, 1497 01:31:43,250 --> 01:31:45,280 like a parabola going down. 1498 01:31:45,280 --> 01:31:49,110 So it's neither a valley nor a peak. 1499 01:31:49,110 --> 01:31:51,250 It's a valley in one direction, and a peak 1500 01:31:51,250 --> 01:31:52,300 in another direction. 1501 01:31:52,300 --> 01:31:55,170 And that's the saddle point. 1502 01:31:55,170 --> 01:31:56,370 So say it again. 1503 01:31:56,370 --> 01:31:57,320 What is that? 1504 01:31:57,320 --> 01:32:00,730 It looks like a valley in one principle direction 1505 01:32:00,730 --> 01:32:05,450 and the peak in the other principle direction. 1506 01:32:05,450 --> 01:32:08,690 And then that's going to be a saddle point. 1507 01:32:08,690 --> 01:32:14,280 Indeed, how do we figure this out by the method I provided? 1508 01:32:14,280 --> 01:32:16,800 Well, who is A? 1509 01:32:16,800 --> 01:32:21,720 A is f sub xx at the point. 1510 01:32:21,720 --> 01:32:24,600 2x goes primed one time. 1511 01:32:24,600 --> 01:32:28,260 f sub x was 2x. 1512 01:32:28,260 --> 01:32:30,420 f sub y was 2y. 1513 01:32:30,420 --> 01:32:32,320 f sub xx is 2. 1514 01:32:32,320 --> 01:32:34,860 1515 01:32:34,860 --> 01:32:39,680 f sub xB is f sub xy. 1516 01:32:39,680 --> 01:32:41,680 What is that? 1517 01:32:41,680 --> 01:32:43,980 0. 1518 01:32:43,980 --> 01:32:46,480 Good, that makes my life easier. 1519 01:32:46,480 --> 01:32:49,920 C equals f sub yy. 1520 01:32:49,920 --> 01:32:52,270 What is that? 1521 01:32:52,270 --> 01:32:53,220 2. 1522 01:32:53,220 --> 01:32:55,880 OK, this is looking beautiful. 1523 01:32:55,880 --> 01:32:58,530 Because I don't have to plug in any values. 1524 01:32:58,530 --> 01:33:00,495 The D is there for me to see it. 1525 01:33:00,495 --> 01:33:04,540 And it's going to consist of the determinant having the elements 1526 01:33:04,540 --> 01:33:10,900 2, 0, 0, 2-- minus 2, minus 2. 1527 01:33:10,900 --> 01:33:14,350 I'm sorry, guys, I missed here the minus. 1528 01:33:14,350 --> 01:33:19,000 And it cost me my life-- 2x and minus 2y, and here minus 2. 1529 01:33:19,000 --> 01:33:20,800 STUDENT: It didn't cost you your life, 1530 01:33:20,800 --> 01:33:21,880 because you caught it before you were done with the problem. 1531 01:33:21,880 --> 01:33:23,065 PROFESSOR: I caught it up there. 1532 01:33:23,065 --> 01:33:24,280 I'm taking the final exam. 1533 01:33:24,280 --> 01:33:28,230 I still get 100%, because I caught it up 1534 01:33:28,230 --> 01:33:29,920 at the last minute. 1535 01:33:29,920 --> 01:33:34,480 So 2, 0, 0, minus 2-- I knew that I 1536 01:33:34,480 --> 01:33:35,750 had to get something negative. 1537 01:33:35,750 --> 01:33:39,620 So I said, for god's sake, I need to get a saddle point. 1538 01:33:39,620 --> 01:33:42,220 That's why it's the horse in the saddle. 1539 01:33:42,220 --> 01:33:46,832 So I knew I should get minus 4, negative. 1540 01:33:46,832 --> 01:33:50,733 All right, so the only thing I have 1541 01:33:50,733 --> 01:33:53,060 to say as a final answer is the only 1542 01:33:53,060 --> 01:33:57,440 critical point of this surface that I'm too lazy to write 1543 01:33:57,440 --> 01:33:59,260 about-- don't write that. 1544 01:33:59,260 --> 01:34:02,356 So the only critical point on the surface z 1545 01:34:02,356 --> 01:34:04,771 equals x squared minus y squared will 1546 01:34:04,771 --> 01:34:10,084 be at the origin O of corner 0, 0, 0 1547 01:34:10,084 --> 01:34:14,480 where the discriminant being negative 1548 01:34:14,480 --> 01:34:17,295 indicates it's going to be a saddle point. 1549 01:34:17,295 --> 01:34:20,480 And that's it-- nothing else. 1550 01:34:20,480 --> 01:34:22,154 You don't need more. 1551 01:34:22,154 --> 01:34:24,589 But there are more examples. 1552 01:34:24,589 --> 01:34:26,540 Because life is hard. 1553 01:34:26,540 --> 01:34:28,375 And I'm going to give you another example. 1554 01:34:28,375 --> 01:34:31,688 1555 01:34:31,688 --> 01:34:34,646 Well, OK, this one. 1556 01:34:34,646 --> 01:34:37,620 1557 01:34:37,620 --> 01:34:47,657 Suppose we have the surface-- that's 1558 01:34:47,657 --> 01:34:49,500 still going to be very easy. 1559 01:34:49,500 --> 01:34:51,880 But I want to make the first examples easy. 1560 01:34:51,880 --> 01:34:54,660 1561 01:34:54,660 --> 01:34:56,448 I have a reason why. 1562 01:34:56,448 --> 01:35:04,890 1563 01:35:04,890 --> 01:35:07,290 This is a function of two variables, right? 1564 01:35:07,290 --> 01:35:12,380 It's still a polynomial in two variables of order 2. 1565 01:35:12,380 --> 01:35:17,890 And how do I solve for the classification of the extrema? 1566 01:35:17,890 --> 01:35:23,590 I'm looking for local extrema, not absolute-- local extrema. 1567 01:35:23,590 --> 01:35:25,170 I'm not constrained. 1568 01:35:25,170 --> 01:35:27,620 I'm saying, what do you mean, no constraint? 1569 01:35:27,620 --> 01:35:30,790 Constrained would have been, let's say that x and y 1570 01:35:30,790 --> 01:35:33,590 are in the unit disc. 1571 01:35:33,590 --> 01:35:38,750 Or let's say x and y are on the circle x squared 1572 01:35:38,750 --> 01:35:40,340 plus y squared equals 1. 1573 01:35:40,340 --> 01:35:41,970 That would be a constraint. 1574 01:35:41,970 --> 01:35:44,520 But they're not constrained about anything. 1575 01:35:44,520 --> 01:35:47,490 x and y are real numbers. 1576 01:35:47,490 --> 01:35:51,310 They can take the whole plane as a domain. 1577 01:35:51,310 --> 01:35:59,220 So I get f sub x equals 0, f sub y equals 0, solve for x and y, 1578 01:35:59,220 --> 01:36:00,700 get the critical values. 1579 01:36:00,700 --> 01:36:05,070 I get very nice 2x. 1580 01:36:05,070 --> 01:36:06,180 I have to pay attention. 1581 01:36:06,180 --> 01:36:09,550 Because now this is not so easy anymore-- 1582 01:36:09,550 --> 01:36:16,690 plus prime with respect to x, 2y, prime with respect to x, 0, 1583 01:36:16,690 --> 01:36:25,670 prime with respect to x, plus 3, prime of this, OK, equals 0. 1584 01:36:25,670 --> 01:36:32,321 f sub y-- 0 plus prime with respect 1585 01:36:32,321 --> 01:36:41,030 to y, 2x, plus prime with respect to y, 2y, 1586 01:36:41,030 --> 01:36:46,822 plus nothing, prime with respect to y equals 0. 1587 01:36:46,822 --> 01:36:52,714 1588 01:36:52,714 --> 01:36:58,260 And now you have to be very smart. 1589 01:36:58,260 --> 01:37:03,022 Well, you have to be perceptive and tell me what I got. 1590 01:37:03,022 --> 01:37:04,990 What is this that we mean? 1591 01:37:04,990 --> 01:37:08,926 1592 01:37:08,926 --> 01:37:10,894 Look at this system. 1593 01:37:10,894 --> 01:37:12,862 It looks like crazy. 1594 01:37:12,862 --> 01:37:22,702 1595 01:37:22,702 --> 01:37:25,654 STUDENT: [INAUDIBLE] the origin or-- because can't you 1596 01:37:25,654 --> 01:37:28,114 just subtract it down? 1597 01:37:28,114 --> 01:37:30,082 PROFESSOR: Is this possible? 1598 01:37:30,082 --> 01:37:31,670 And what does this mean? 1599 01:37:31,670 --> 01:37:33,300 What do we call such a system? 1600 01:37:33,300 --> 01:37:38,120 1601 01:37:38,120 --> 01:37:42,153 Inconsistent system-- we call it inconsistent. 1602 01:37:42,153 --> 01:37:45,104 How can I make this problem to be possible, 1603 01:37:45,104 --> 01:37:47,760 to have some critical points? 1604 01:37:47,760 --> 01:37:50,225 STUDENT: If you add 3x. 1605 01:37:50,225 --> 01:37:52,197 PROFESSOR: How about that, just remove the 3x 1606 01:37:52,197 --> 01:37:55,660 and see what's going to happen. 1607 01:37:55,660 --> 01:37:59,460 Oh, in that case, I have something 1608 01:37:59,460 --> 01:38:06,010 that's over-determined, right? 1609 01:38:06,010 --> 01:38:11,740 I have something that tells me the same thing. 1610 01:38:11,740 --> 01:38:13,500 So I'm priming with respect to x. 1611 01:38:13,500 --> 01:38:14,835 I get that. 1612 01:38:14,835 --> 01:38:16,420 I'm priming with respect to y. 1613 01:38:16,420 --> 01:38:17,650 I get this. 1614 01:38:17,650 --> 01:38:18,740 I get 0. 1615 01:38:18,740 --> 01:38:21,860 So I don't even need the second equation. 1616 01:38:21,860 --> 01:38:27,992 And that means the critical point 1617 01:38:27,992 --> 01:38:40,536 is any point of the form-- shall I 1618 01:38:40,536 --> 01:38:44,488 put a Greek letter alpha minus alpha or lambda minus lambda? 1619 01:38:44,488 --> 01:38:45,950 What shall I do? 1620 01:38:45,950 --> 01:38:52,370 So any point that is situated on the second bisector, 1621 01:38:52,370 --> 01:38:54,100 I mean the x, y plane. 1622 01:38:54,100 --> 01:38:56,542 And this is the x, and this is the y. 1623 01:38:56,542 --> 01:39:00,770 And I say, what does it mean, x plus y equals 0? 1624 01:39:00,770 --> 01:39:03,120 Not this line-- don't draw it. 1625 01:39:03,120 --> 01:39:05,032 That is x equals y. 1626 01:39:05,032 --> 01:39:08,960 The other one, called the second bisector-- y 1627 01:39:08,960 --> 01:39:13,470 equals negative x, so not this one, the diagonal, 1628 01:39:13,470 --> 01:39:21,763 but the diagonal that's on the corridor, this one. 1629 01:39:21,763 --> 01:39:25,580 All right, so any point of the form alpha minus alpha, 1630 01:39:25,580 --> 01:39:27,624 here's the critical point. 1631 01:39:27,624 --> 01:39:33,076 The question is, how am I going to get to the classification 1632 01:39:33,076 --> 01:39:35,370 for such points? 1633 01:39:35,370 --> 01:39:36,730 Can anybody help me? 1634 01:39:36,730 --> 01:39:38,334 So step two-- 1635 01:39:38,334 --> 01:39:39,500 STUDENT: Solve the equation. 1636 01:39:39,500 --> 01:39:43,270 STUDENT: Solve alpha for one of the two variables first. 1637 01:39:43,270 --> 01:39:46,360 PROFESSOR: Take alpha minus alpha-- could be anything. 1638 01:39:46,360 --> 01:39:50,630 And then I'll say, f sub-- this is f sub x. 1639 01:39:50,630 --> 01:39:53,900 And this is f sub y. 1640 01:39:53,900 --> 01:39:55,160 What is f sub x? 1641 01:39:55,160 --> 01:39:56,700 f sub xx, I'm sorry. 1642 01:39:56,700 --> 01:39:59,586 1643 01:39:59,586 --> 01:40:01,847 STUDENT: [INAUDIBLE]. 1644 01:40:01,847 --> 01:40:02,472 PROFESSOR: Huh? 1645 01:40:02,472 --> 01:40:03,434 2. 1646 01:40:03,434 --> 01:40:05,410 OK, are you with me? 1647 01:40:05,410 --> 01:40:07,880 So you know what it is. 1648 01:40:07,880 --> 01:40:12,410 f sub xy equals? 1649 01:40:12,410 --> 01:40:13,250 STUDENT: 2. 1650 01:40:13,250 --> 01:40:15,390 PROFESSOR: 2. 1651 01:40:15,390 --> 01:40:17,810 f sub yy equals? 1652 01:40:17,810 --> 01:40:18,710 STUDENT: 2 1653 01:40:18,710 --> 01:40:21,660 PROFESSOR: 2-- that's the mystery man. 1654 01:40:21,660 --> 01:40:23,130 The book doesn't give this example, 1655 01:40:23,130 --> 01:40:24,680 and it drives me crazy. 1656 01:40:24,680 --> 01:40:28,540 And I wanted to give you some bad example where 1657 01:40:28,540 --> 01:40:30,940 the classification doesn't work. 1658 01:40:30,940 --> 01:40:35,470 Because we always cook up nice examples for you 1659 01:40:35,470 --> 01:40:38,925 and claim everything is beautiful. 1660 01:40:38,925 --> 01:40:41,410 Life is not always beautiful. 1661 01:40:41,410 --> 01:40:44,480 So you get 0. 1662 01:40:44,480 --> 01:40:48,270 In that case, nothing can be said with this classification. 1663 01:40:48,270 --> 01:40:50,490 I make a face, sad face. 1664 01:40:50,490 --> 01:40:52,220 So what do I hope? 1665 01:40:52,220 --> 01:40:59,416 To get to Maple or MATLAB and be able to draw that, or a TI-92 1666 01:40:59,416 --> 01:41:02,630 if my mother would give me $200 and some. 1667 01:41:02,630 --> 01:41:03,600 I told her. 1668 01:41:03,600 --> 01:41:08,020 She asked me what to buy for my birthday. 1669 01:41:08,020 --> 01:41:10,710 I have a TI-83 or something. 1670 01:41:10,710 --> 01:41:12,740 And it was cheap. 1671 01:41:12,740 --> 01:41:17,380 I bought it on eBay, and then I stopped using it. 1672 01:41:17,380 --> 01:41:20,920 And then I saw this TI-92 that can draw surfaces 1673 01:41:20,920 --> 01:41:22,510 in three dimensions. 1674 01:41:22,510 --> 01:41:24,080 And I said, this is like MATLAB. 1675 01:41:24,080 --> 01:41:25,850 You just carry it in your pocket. 1676 01:41:25,850 --> 01:41:29,390 It's only a little bit too expensive. 1677 01:41:29,390 --> 01:41:34,700 All right, how about another kind? 1678 01:41:34,700 --> 01:41:38,628 1679 01:41:38,628 --> 01:41:40,101 Look at this one. 1680 01:41:40,101 --> 01:41:55,350 1681 01:41:55,350 --> 01:41:58,590 You cannot tell with the naked eye. 1682 01:41:58,590 --> 01:42:02,140 But you can go ahead and do this step one 1683 01:42:02,140 --> 01:42:05,080 looking for critical values. 1684 01:42:05,080 --> 01:42:17,750 So the system, f sub x will be 6x plus 2y equals 0. 1685 01:42:17,750 --> 01:42:24,390 f sub y will be-- who's going to tell me? 1686 01:42:24,390 --> 01:42:29,760 2x plus 2y equals 0. 1687 01:42:29,760 --> 01:42:34,290 Now, by elimination or by substitution or by anything 1688 01:42:34,290 --> 01:42:36,800 I want, I subtract the second from the first. 1689 01:42:36,800 --> 01:42:38,360 What do I get? 1690 01:42:38,360 --> 01:42:41,340 I get 4x equals 0. 1691 01:42:41,340 --> 01:42:46,410 And that gives me the only possibility is x0 equals 0. 1692 01:42:46,410 --> 01:42:53,070 And then I say, OK, if my only one is 0, then y is 0. 1693 01:42:53,070 --> 01:42:54,010 0 is 0. 1694 01:42:54,010 --> 01:43:01,250 So I only have one critical point, which is the origin. 1695 01:43:01,250 --> 01:43:03,500 Now, do I know, what am I going to get? 1696 01:43:03,500 --> 01:43:05,710 Not unless I'm a genius and I can 1697 01:43:05,710 --> 01:43:07,910 see two steps ahead of time. 1698 01:43:07,910 --> 01:43:10,860 I would need to do ABC quickly in my head. 1699 01:43:10,860 --> 01:43:13,160 Some of you are able, thank god. 1700 01:43:13,160 --> 01:43:15,030 But some of you, like me, are not. 1701 01:43:15,030 --> 01:43:19,290 So I have to take a few seconds to see what's going on. 1702 01:43:19,290 --> 01:43:24,260 A-- f sub xx at the point is 0. 1703 01:43:24,260 --> 01:43:27,720 B-- f sub xy. 1704 01:43:27,720 --> 01:43:29,911 C-- f sub yy. 1705 01:43:29,911 --> 01:43:36,740 1706 01:43:36,740 --> 01:43:38,700 What do we do? 1707 01:43:38,700 --> 01:43:40,080 We get 6. 1708 01:43:40,080 --> 01:43:41,250 Are we happy about it? 1709 01:43:41,250 --> 01:43:44,910 We don't know yet, to be happy or not. 1710 01:43:44,910 --> 01:43:48,920 f sub xy or f sub yx, you see, Mr. Schwarz 1711 01:43:48,920 --> 01:43:52,050 is now happy that he proved to you 1712 01:43:52,050 --> 01:43:55,120 that it doesn't matter which order you're 1713 01:43:55,120 --> 01:43:58,790 taking for a polynomial that's a smooth function. 1714 01:43:58,790 --> 01:44:03,840 You always have the same. 1715 01:44:03,840 --> 01:44:08,160 And finally, C is 2. 1716 01:44:08,160 --> 01:44:12,725 And you are ready to do the D. And I could smell that D, 1717 01:44:12,725 --> 01:44:15,200 but I didn't want to say anything. 1718 01:44:15,200 --> 01:44:24,240 6, 2, 2, and 2-- is that a nice thing? 1719 01:44:24,240 --> 01:44:27,820 Yeah, we haven't encountered this example yet. 1720 01:44:27,820 --> 01:44:30,680 Because according to the classification, 1721 01:44:30,680 --> 01:44:32,410 this is greater than 0. 1722 01:44:32,410 --> 01:44:34,450 Does it really matter what value it is? 1723 01:44:34,450 --> 01:44:37,775 No, it only matters that it is positive. 1724 01:44:37,775 --> 01:44:42,350 And if it's positive, that means I can move on with my life 1725 01:44:42,350 --> 01:44:45,050 and look at the classification. 1726 01:44:45,050 --> 01:44:50,172 From this point where delta or v is positive, 1727 01:44:50,172 --> 01:44:57,100 I'm going to get a ramification into separate cases. 1728 01:44:57,100 --> 01:45:01,180 And who is going to tell me next what to do? 1729 01:45:01,180 --> 01:45:04,090 Look at A. Oh, by the way, talking 1730 01:45:04,090 --> 01:45:07,650 about the quadratic formula from school, 1731 01:45:07,650 --> 01:45:12,620 from kindergarten, when we computed 1732 01:45:12,620 --> 01:45:18,780 the-- I'll use the general one, minus b plus minus square root 1733 01:45:18,780 --> 01:45:21,470 of b squared minus 4ac over 2a. 1734 01:45:21,470 --> 01:45:27,110 1735 01:45:27,110 --> 01:45:33,630 We were afraid of some special cases 1736 01:45:33,630 --> 01:45:35,890 when we were looking at that. 1737 01:45:35,890 --> 01:45:37,696 Especially when delta was negative, 1738 01:45:37,696 --> 01:45:40,390 that was really imaginary and so on. 1739 01:45:40,390 --> 01:45:42,890 But one thing we remember from ninth grade-- 1740 01:45:42,890 --> 01:45:46,910 was this ninth grade or eighth grade? 1741 01:45:46,910 --> 01:45:51,540 The parabola opens up when a is positive. 1742 01:45:51,540 --> 01:45:57,565 Just the same way, something opens up when A, big A, 1743 01:45:57,565 --> 01:45:59,310 is positive here. 1744 01:45:59,310 --> 01:46:01,630 Then you have opening up. 1745 01:46:01,630 --> 01:46:05,590 When big A is negative, then you have opening down. 1746 01:46:05,590 --> 01:46:10,386 So remember-- I'm going to make smile here so you remember. 1747 01:46:10,386 --> 01:46:12,310 So I have it like that. 1748 01:46:12,310 --> 01:46:16,080 So I suspect that it's going to look 1749 01:46:16,080 --> 01:46:23,240 like a surface of some sort that maybe is not 1750 01:46:23,240 --> 01:46:25,740 surface of revolution. 1751 01:46:25,740 --> 01:46:27,290 You should tell me what it is. 1752 01:46:27,290 --> 01:46:31,530 You should think about this and do the cross sections with z 1753 01:46:31,530 --> 01:46:34,590 constant and tell me what surface that is. 1754 01:46:34,590 --> 01:46:36,965 But in any case, what do I care? 1755 01:46:36,965 --> 01:46:41,770 I care that I'm looking at the origin. 1756 01:46:41,770 --> 01:46:44,090 And this is where my special point is. 1757 01:46:44,090 --> 01:46:46,640 That's going to be the value point. 1758 01:46:46,640 --> 01:46:48,170 How do I know? 1759 01:46:48,170 --> 01:46:52,710 Because A, which is 6, is positive. 1760 01:46:52,710 --> 01:46:55,659 At this point, I know what I'm left with. 1761 01:46:55,659 --> 01:46:59,740 I know that my surface is going to look like a valley. 1762 01:46:59,740 --> 01:47:02,138 So how do I know again? 1763 01:47:02,138 --> 01:47:04,070 I'm not going to draw it. 1764 01:47:04,070 --> 01:47:07,890 But it's going to look something like that. 1765 01:47:07,890 --> 01:47:10,500 At the origin, this is going to be 7. 1766 01:47:10,500 --> 01:47:12,500 Are you guys with me? 1767 01:47:12,500 --> 01:47:15,390 And it's going to open up. 1768 01:47:15,390 --> 01:47:19,400 And so you should not attempt intersecting with z equals 5 1769 01:47:19,400 --> 01:47:20,160 or z equals 1. 1770 01:47:20,160 --> 01:47:21,992 Because you're not going to get anything. 1771 01:47:21,992 --> 01:47:24,540 But if you intersect, for example, at z 1772 01:47:24,540 --> 01:47:26,620 equals 9, what are you going to get? 1773 01:47:26,620 --> 01:47:31,300 If you intersect at z equals 9, you 1774 01:47:31,300 --> 01:47:37,110 get 3x plus 2xy plus y squared equals 2. 1775 01:47:37,110 --> 01:47:39,070 And what is that? 1776 01:47:39,070 --> 01:47:43,970 It's a rotated form of an ellipse. 1777 01:47:43,970 --> 01:47:47,500 It's hard to see, because it's missing [INAUDIBLE]. 1778 01:47:47,500 --> 01:47:51,080 But this is exactly what discriminant is saying. 1779 01:47:51,080 --> 01:47:55,760 So this is going to be an x. 1780 01:47:55,760 --> 01:47:58,660 Good, so I know what I'm going to get. 1781 01:47:58,660 --> 01:48:01,050 What do you have to say on the midterm 1782 01:48:01,050 --> 01:48:03,532 or on the final about this problem? 1783 01:48:03,532 --> 01:48:04,918 STUDENT: The point is-- 1784 01:48:04,918 --> 01:48:07,700 PROFESSOR: The point is the origin. 1785 01:48:07,700 --> 01:48:08,560 I classified it. 1786 01:48:08,560 --> 01:48:09,550 I got delta positive. 1787 01:48:09,550 --> 01:48:11,760 I got A positive. 1788 01:48:11,760 --> 01:48:13,010 So it's a valley. 1789 01:48:13,010 --> 01:48:14,530 It's a relative minimum. 1790 01:48:14,530 --> 01:48:15,590 And that's it. 1791 01:48:15,590 --> 01:48:21,850 I have a relative min at the point P of coordinates 0, 0, 1792 01:48:21,850 --> 01:48:23,100 and 7. 1793 01:48:23,100 --> 01:48:26,350 1794 01:48:26,350 --> 01:48:27,970 And that's the valley. 1795 01:48:27,970 --> 01:48:28,800 Yes, sir. 1796 01:48:28,800 --> 01:48:30,750 STUDENT: Why is it A that determines 1797 01:48:30,750 --> 01:48:33,792 whether it's a relative min or a relative max? 1798 01:48:33,792 --> 01:48:35,130 PROFESSOR: It's a whole story. 1799 01:48:35,130 --> 01:48:36,080 You can prove it. 1800 01:48:36,080 --> 01:48:40,340 I don't remember if we proved this in the book or not. 1801 01:48:40,340 --> 01:48:43,050 But it can be proved, so the fact 1802 01:48:43,050 --> 01:48:48,740 that it has to do with concavity and convexity. 1803 01:48:48,740 --> 01:48:54,000 When you had a second derivative, let's say, 1804 01:48:54,000 --> 01:48:56,590 what's the equivalent of the Calculus I 1805 01:48:56,590 --> 01:48:58,954 notion that you know about? 1806 01:48:58,954 --> 01:49:01,710 In Calculus I, you had functions of one variable, 1807 01:49:01,710 --> 01:49:04,160 and life was so easy like that. 1808 01:49:04,160 --> 01:49:08,280 And f prime positive meant that the function increased. 1809 01:49:08,280 --> 01:49:12,330 And f prime negative meant that the function decreased. 1810 01:49:12,330 --> 01:49:18,090 And f double prime was just like your-- you sense 1811 01:49:18,090 --> 01:49:22,070 that the second partials must have something 1812 01:49:22,070 --> 01:49:25,860 to do with it, especially the first one with respect to x. 1813 01:49:25,860 --> 01:49:28,620 If you were in plane, and you have 1814 01:49:28,620 --> 01:49:34,422 f double prime with respect to x, when was this a valley? 1815 01:49:34,422 --> 01:49:35,380 When you had the smile. 1816 01:49:35,380 --> 01:49:37,530 When did you have a smile? 1817 01:49:37,530 --> 01:49:42,908 When f double prime was positive, you have concave up. 1818 01:49:42,908 --> 01:49:45,253 When f double prime was negative, 1819 01:49:45,253 --> 01:49:47,984 you have concave down. 1820 01:49:47,984 --> 01:49:48,740 Remember, guys? 1821 01:49:48,740 --> 01:49:53,520 So you have a smile or a frown. 1822 01:49:53,520 --> 01:49:54,415 This is how we know. 1823 01:49:54,415 --> 01:49:58,500 For the same reason that would take about two pages 1824 01:49:58,500 --> 01:50:04,060 to write down the proof, you have a smile for A positive. 1825 01:50:04,060 --> 01:50:07,850 And the smile means actually in all directions 1826 01:50:07,850 --> 01:50:12,967 you have a smile locally around the origin. 1827 01:50:12,967 --> 01:50:17,760 OK, look in the book. 1828 01:50:17,760 --> 01:50:19,885 I'm not sure how much should we do. 1829 01:50:19,885 --> 01:50:23,525 Do we give a sketch of a proof, or we give the entire proof? 1830 01:50:23,525 --> 01:50:25,856 But more likely, a sketch. 1831 01:50:25,856 --> 01:50:28,748 1832 01:50:28,748 --> 01:50:31,220 Yes. 1833 01:50:31,220 --> 01:50:35,830 STUDENT: I asked the slightly wrong question, 1834 01:50:35,830 --> 01:50:37,325 but I answered it myself. 1835 01:50:37,325 --> 01:50:42,167 I wanted to ask, why is it dependent on A and not on C? 1836 01:50:42,167 --> 01:50:43,000 PROFESSOR: Not on C. 1837 01:50:43,000 --> 01:50:44,960 STUDENT: But then I realized that it is dependent on C 1838 01:50:44,960 --> 01:50:46,270 as well, because if A is positive, 1839 01:50:46,270 --> 01:50:47,344 then C must be positive. 1840 01:50:47,344 --> 01:50:50,732 PROFESSOR: Yes, yes, it is dependent on both. 1841 01:50:50,732 --> 01:50:52,184 STUDENT: OK, there we go. 1842 01:50:52,184 --> 01:50:55,100 That was my question. 1843 01:50:55,100 --> 01:50:56,760 PROFESSOR: So guys, remember. 1844 01:50:56,760 --> 01:51:03,070 Imagine what happens when you had no B, B was 0. 1845 01:51:03,070 --> 01:51:06,100 Then the matrix is diagonalizable. 1846 01:51:06,100 --> 01:51:11,935 And here you have A and C. And Alex 1847 01:51:11,935 --> 01:51:16,190 says, why would A be more important than C? 1848 01:51:16,190 --> 01:51:17,370 It's not. 1849 01:51:17,370 --> 01:51:20,680 But practically, if A is positive and C is negative, 1850 01:51:20,680 --> 01:51:24,170 that means these are the principal directions in which 1851 01:51:24,170 --> 01:51:28,705 one bends like a valley up and one bends like a peak down. 1852 01:51:28,705 --> 01:51:36,220 So this is what happens in the direction of x, f double prime 1853 01:51:36,220 --> 01:51:38,770 in the direction of x, kind of. 1854 01:51:38,770 --> 01:51:42,900 And this is in the direction of y. 1855 01:51:42,900 --> 01:51:45,180 So this is f double prime in the direction of y, 1856 01:51:45,180 --> 01:51:46,860 which we don't denote like that. 1857 01:51:46,860 --> 01:51:52,240 We call it f sub xx and f sub yy, which is A and C. 1858 01:51:52,240 --> 01:52:00,364 So A positive, A being 1 and C being negative 2, 1859 01:52:00,364 --> 01:52:05,274 means a valley here, means the valley meets the horse. 1860 01:52:05,274 --> 01:52:10,690 Look, I'm drawing the tail of the horse. 1861 01:52:10,690 --> 01:52:14,205 He's a little bit fat, this horse. 1862 01:52:14,205 --> 01:52:18,534 And that's his mane, his eye. 1863 01:52:18,534 --> 01:52:22,400 I'm just taking a break. 1864 01:52:22,400 --> 01:52:24,060 STUDENT: That's a pretty good drawing. 1865 01:52:24,060 --> 01:52:30,600 PROFESSOR: It looks more like a dog or a plush horse 1866 01:52:30,600 --> 01:52:31,770 or something. 1867 01:52:31,770 --> 01:52:36,680 So A equals 1, and C equals minus 2. 1868 01:52:36,680 --> 01:52:42,470 But if it were diagonalizable, and A would be 1 1869 01:52:42,470 --> 01:52:46,610 and C would be 7, both of them positive in any case, 1870 01:52:46,610 --> 01:52:51,790 then you'll have valley and valley, an x direction valley 1871 01:52:51,790 --> 01:52:53,640 and y direction valley. 1872 01:52:53,640 --> 01:52:56,506 So it has to be a valley everywhere. 1873 01:52:56,506 --> 01:53:01,436 These are the principal directions that I have 1 and 2. 1874 01:53:01,436 --> 01:53:06,200 But then the ultimate case, what happens 1875 01:53:06,200 --> 01:53:15,200 when A is negative and-- hmm, OK, 1876 01:53:15,200 --> 01:53:21,800 then either you have them both one positive, one negative, 1877 01:53:21,800 --> 01:53:27,356 or you have plus, plus and minus, minus. 1878 01:53:27,356 --> 01:53:32,510 And then you have this as your surface, right? 1879 01:53:32,510 --> 01:53:33,840 Which one is the x direction? 1880 01:53:33,840 --> 01:53:34,798 That's the y direction. 1881 01:53:34,798 --> 01:53:36,800 That's the second one. 1882 01:53:36,800 --> 01:53:39,940 The x direction is that. 1883 01:53:39,940 --> 01:53:43,240 In the x direction, you have a frown. 1884 01:53:43,240 --> 01:53:47,344 So f sub yy is negative. 1885 01:53:47,344 --> 01:53:51,224 In the y direction, you also have a frown. 1886 01:53:51,224 --> 01:53:53,140 So both of them are negative. 1887 01:53:53,140 --> 01:53:55,200 So you have a relative max. 1888 01:53:55,200 --> 01:53:58,910 Yes, sir, Matthew, tell me. 1889 01:53:58,910 --> 01:54:02,250 STUDENT: So isn't it possible to have both A and C positive, 1890 01:54:02,250 --> 01:54:06,700 but then yet still not be more positive than B squared? 1891 01:54:06,700 --> 01:54:10,975 PROFESSOR: No, because there's a theorem that-- 1892 01:54:10,975 --> 01:54:13,480 STUDENT: I was just wondering like numbers-wise. 1893 01:54:13,480 --> 01:54:16,870 PROFESSOR: You have this matrix. 1894 01:54:16,870 --> 01:54:19,790 And there is a theorem that shows you that you can actually 1895 01:54:19,790 --> 01:54:21,690 diagonalize this matrix. 1896 01:54:21,690 --> 01:54:25,074 You'll learn your linear algebra [INAUDIBLE]. 1897 01:54:25,074 --> 01:54:26,615 STUDENT: It makes sense, because when 1898 01:54:26,615 --> 01:54:29,810 you were saying A is this way, and that way there's no way 1899 01:54:29,810 --> 01:54:34,060 you could have 2 come up, and then yet, 1900 01:54:34,060 --> 01:54:36,525 not be a-- you know what I'm saying? 1901 01:54:36,525 --> 01:54:39,280 Because then they'd be less than 0. 1902 01:54:39,280 --> 01:54:43,210 PROFESSOR: You can if you don't have or the 2. 1903 01:54:43,210 --> 01:54:44,760 That's an excellent question. 1904 01:54:44,760 --> 01:54:47,740 If I would have x to the 4 y to the 4 1905 01:54:47,740 --> 01:54:52,080 added together, like Ax to the 4 plus By to the 4 plus 1906 01:54:52,080 --> 01:54:56,080 something, then I have the so-called monkey saddle. 1907 01:54:56,080 --> 01:54:57,500 That's so funny. 1908 01:54:57,500 --> 01:55:05,500 You can have something that looks like that. 1909 01:55:05,500 --> 01:55:07,910 So in your direction, you can have this. 1910 01:55:07,910 --> 01:55:10,272 Then I've reached two equal peaks 1911 01:55:10,272 --> 01:55:11,480 in the x and the y direction. 1912 01:55:11,480 --> 01:55:14,055 But in the between, I also went down. 1913 01:55:14,055 --> 01:55:18,440 So depending on a higher degree symmetric polynomial, 1914 01:55:18,440 --> 01:55:21,420 you can have a monkey saddle. 1915 01:55:21,420 --> 01:55:24,825 And then it's not just like you can predict what's 1916 01:55:24,825 --> 01:55:26,930 going to happen in between. 1917 01:55:26,930 --> 01:55:31,090 In between, if I go up, if I go valley 1918 01:55:31,090 --> 01:55:34,030 in the x direction and valley in the y direction, 1919 01:55:34,030 --> 01:55:36,665 I know that's going to be a valley everywhere-- no. 1920 01:55:36,665 --> 01:55:38,880 If a polynomial is high in order, 1921 01:55:38,880 --> 01:55:42,410 it can go down, valley, and up again, 1922 01:55:42,410 --> 01:55:45,960 and monkey saddle it looks like. 1923 01:55:45,960 --> 01:55:49,320 Guys, you have dealt with it when you 1924 01:55:49,320 --> 01:55:52,785 went to Luna Park or Joyland. 1925 01:55:52,785 --> 01:56:00,350 It's one of those things that look like-- I'm trying. 1926 01:56:00,350 --> 01:56:01,119 I cannot draw. 1927 01:56:01,119 --> 01:56:02,785 STUDENT: It sounds more like an octopus. 1928 01:56:02,785 --> 01:56:04,246 PROFESSOR: Like an octopus. 1929 01:56:04,246 --> 01:56:06,940 And one of those things-- exactly-- 1930 01:56:06,940 --> 01:56:11,340 that are shaped so that they are undulated, 1931 01:56:11,340 --> 01:56:13,936 in some directions are going up, in some directions 1932 01:56:13,936 --> 01:56:14,925 are going down. 1933 01:56:14,925 --> 01:56:16,730 STUDENT: Like an egg carton, almost? 1934 01:56:16,730 --> 01:56:19,910 PROFESSOR: Yeah, really undulated. 1935 01:56:19,910 --> 01:56:22,910 Imagine even a surface made of metal that's undulated 1936 01:56:22,910 --> 01:56:24,640 and rotating at the same time. 1937 01:56:24,640 --> 01:56:28,675 They have some of those in Disney World. 1938 01:56:28,675 --> 01:56:32,364 Have you been to Orlando? 1939 01:56:32,364 --> 01:56:35,360 STUDENT: I was there last semester. 1940 01:56:35,360 --> 01:56:38,155 PROFESSOR: But you didn't take me with you, which is bad. 1941 01:56:38,155 --> 01:56:42,442 Because that's one of my favorite places. 1942 01:56:42,442 --> 01:56:44,942 STUDENT: I was just trying to think of what you were talking 1943 01:56:44,942 --> 01:56:45,380 about so I could visualize it. 1944 01:56:45,380 --> 01:56:47,520 PROFESSOR: Maybe we could make a proposal 1945 01:56:47,520 --> 01:56:50,510 to teach Calculus III at Disney World 1946 01:56:50,510 --> 01:56:55,060 so that we could have examples of motion and surfaces 1947 01:56:55,060 --> 01:57:00,112 all around and study the motion of all sorts of gadgets, 1948 01:57:00,112 --> 01:57:01,180 velocity and trajectory. 1949 01:57:01,180 --> 01:57:06,310 1950 01:57:06,310 --> 01:57:09,520 Last night I couldn't sleep until 1:00, and I was thinking, 1951 01:57:09,520 --> 01:57:14,010 I gave examples of the winter sports 1952 01:57:14,010 --> 01:57:18,136 like bobsled and all sorts of skiing and so on. 1953 01:57:18,136 --> 01:57:22,020 But I never thought about a screw curve 1954 01:57:22,020 --> 01:57:28,000 with curvature and torsion that is based on the roller coaster. 1955 01:57:28,000 --> 01:57:29,530 And the roller coaster is actually 1956 01:57:29,530 --> 01:57:35,390 the best place to study the [INAUDIBLE], the velocity, 1957 01:57:35,390 --> 01:57:38,460 the tangent unit, the normal, the bi-normal. 1958 01:57:38,460 --> 01:57:42,910 And when you have in a plane the roller coaster 1959 01:57:42,910 --> 01:57:46,610 goes like that, like this and like that, like in a plane, 1960 01:57:46,610 --> 01:57:49,650 you have nothing but bending, which means curvature. 1961 01:57:49,650 --> 01:57:53,480 But then when the roller coaster goes away from the plane, 1962 01:57:53,480 --> 01:57:55,020 you have the torsion. 1963 01:57:55,020 --> 01:57:57,940 And that makes you sick really to the stomach. 1964 01:57:57,940 --> 01:58:00,910 So we would have to experience that to understand 1965 01:58:00,910 --> 01:58:02,660 Calculus III better. 1966 01:58:02,660 --> 01:58:05,785 So our next proposal is we ask the administration 1967 01:58:05,785 --> 01:58:10,920 instead of study abroad courses, the domestic study 1968 01:58:10,920 --> 01:58:15,570 at Disney World for Calc III. 1969 01:58:15,570 --> 01:58:16,790 It's Applied Calculus III. 1970 01:58:16,790 --> 01:58:20,490 1971 01:58:20,490 --> 01:58:24,420 OK, something else that I want you to do-- I 1972 01:58:24,420 --> 01:58:26,776 had prepared an example. 1973 01:58:26,776 --> 01:58:30,906 1974 01:58:30,906 --> 01:58:33,346 This is an absolute extrema. 1975 01:58:33,346 --> 01:58:40,670 1976 01:58:40,670 --> 01:58:43,670 And you say, what the heck are the absolute extrema? 1977 01:58:43,670 --> 01:58:47,590 Because she only talked to us about relative maximum 1978 01:58:47,590 --> 01:58:49,520 and relative minimum. 1979 01:58:49,520 --> 01:58:53,680 And she never said anything about absolute extrema. 1980 01:58:53,680 --> 01:58:56,495 1981 01:58:56,495 --> 01:58:58,900 And that will be the table. 1982 01:58:58,900 --> 01:59:02,280 And these will be the extrema. 1983 01:59:02,280 --> 01:59:05,730 I want to refresh your memory first just a little bit. 1984 01:59:05,730 --> 01:59:07,200 This will be the last example. 1985 01:59:07,200 --> 01:59:10,339 Because it's actually two examples in one. 1986 01:59:10,339 --> 01:59:13,237 1987 01:59:13,237 --> 01:59:26,210 And what if you have, let's say, f of x equals e to the minus 1988 01:59:26,210 --> 01:59:38,330 x squared over the interval minus 1, 1? 1989 01:59:38,330 --> 01:59:41,170 You are in Calc I. You will build a time 1990 01:59:41,170 --> 01:59:44,120 machine from Disney World. 1991 01:59:44,120 --> 01:59:50,020 And we went back in time when you actually took Calc I. 1992 01:59:50,020 --> 01:59:52,176 And you struggled with this at first. 1993 01:59:52,176 --> 01:59:53,928 But then you loved it so much that you 1994 01:59:53,928 --> 01:59:56,960 said, oh, that's my favorite problem on the final. 1995 01:59:56,960 --> 02:00:04,820 They asked us for two things-- relative extrema, min or max, 1996 02:00:04,820 --> 02:00:08,610 min/max theory, and they say absolute. 1997 02:00:08,610 --> 02:00:11,460 But for the absolute, your teacher said, attention, 1998 02:00:11,460 --> 02:00:15,140 you have to know how to get to the absolute. 1999 02:00:15,140 --> 02:00:20,387 You are constrained to be on the segment minus 1, 1. 2000 02:00:20,387 --> 02:00:21,970 You see, the fact that they introduced 2001 02:00:21,970 --> 02:00:24,530 this extra constraint and they don't 2002 02:00:24,530 --> 02:00:30,555 let you move with x on the whole real line is a big headache. 2003 02:00:30,555 --> 02:00:32,040 Why is that a big headache? 2004 02:00:32,040 --> 02:00:35,070 Your life would be much easier if it were just e 2005 02:00:35,070 --> 02:00:36,870 to the negative x squared. 2006 02:00:36,870 --> 02:00:40,160 Because in that case, you say, OK, 2007 02:00:40,160 --> 02:00:47,180 f prime of x equals minus 2xe to the minus x squared. 2008 02:00:47,180 --> 02:00:48,270 Piece of cake. 2009 02:00:48,270 --> 02:00:50,450 x0 is 0. 2010 02:00:50,450 --> 02:00:55,020 That's the only critical point. 2011 02:00:55,020 --> 02:00:59,670 And I want to study what kind of critical point that is. 2012 02:00:59,670 --> 02:01:03,770 So I have to do f double prime of x. 2013 02:01:03,770 --> 02:01:08,260 And if I don't know the product rule, I'm in trouble. 2014 02:01:08,260 --> 02:01:12,180 And I go, let's say, minus 2 times 2015 02:01:12,180 --> 02:01:15,650 e to the negative x squared from prime of this 2016 02:01:15,650 --> 02:01:20,425 and this non-prime, plus minus 2x un-prime times e 2017 02:01:20,425 --> 02:01:25,422 to the minus x squared times minus 2x again. 2018 02:01:25,422 --> 02:01:27,680 So it's a headache. 2019 02:01:27,680 --> 02:01:31,395 I pull out an e to the minus x squared. 2020 02:01:31,395 --> 02:01:37,609 And I have 4x squared-- 4x squared-- minus 2. 2021 02:01:37,609 --> 02:01:41,400 2022 02:01:41,400 --> 02:01:43,350 But you say, but wait a minute, Magdalena, 2023 02:01:43,350 --> 02:01:45,580 I'm not going to compute the inflection points. 2024 02:01:45,580 --> 02:01:48,940 The inflection points will be x equals 2025 02:01:48,940 --> 02:01:52,920 plus/minus 1 over root 2. 2026 02:01:52,920 --> 02:01:55,695 I only care about the critical point. 2027 02:01:55,695 --> 02:01:59,072 And the only critical point I have is at 0. 2028 02:01:59,072 --> 02:02:04,789 Compute f double prime of 0 to see if it's a smile or a frown. 2029 02:02:04,789 --> 02:02:07,570 2030 02:02:07,570 --> 02:02:09,690 And you do it. 2031 02:02:09,690 --> 02:02:14,630 And you plug in, and you say, e to the 0 is 1, 0 minus 2. 2032 02:02:14,630 --> 02:02:16,952 So you get a negative. 2033 02:02:16,952 --> 02:02:18,540 Do you care what it is? 2034 02:02:18,540 --> 02:02:20,210 No, but you care it's negative. 2035 02:02:20,210 --> 02:02:23,430 2036 02:02:23,430 --> 02:02:30,550 So at 0, so you draw, and you know 2037 02:02:30,550 --> 02:02:36,541 at 0 you're going to have some sort of a what? 2038 02:02:36,541 --> 02:02:37,894 Relative max. 2039 02:02:37,894 --> 02:02:40,491 2040 02:02:40,491 --> 02:02:40,990 Where? 2041 02:02:40,990 --> 02:02:44,480 At 0, and when you plug 0 again, 1. 2042 02:02:44,480 --> 02:02:45,660 So you draw a table. 2043 02:02:45,660 --> 02:02:52,050 And you say, relative max at 0, 1. 2044 02:02:52,050 --> 02:02:55,520 And then you're not done. 2045 02:02:55,520 --> 02:02:57,706 Because you say, wait a minute, I 2046 02:02:57,706 --> 02:03:03,650 am to study my function in Calc I at minus 1 and 1. 2047 02:03:03,650 --> 02:03:07,780 It's like you have a continuous picture, 2048 02:03:07,780 --> 02:03:13,110 and you chop, take scissors, and cut and cut at the extrema. 2049 02:03:13,110 --> 02:03:15,970 And there you can get additional points 2050 02:03:15,970 --> 02:03:19,720 where you can get a relative max or relative min. 2051 02:03:19,720 --> 02:03:25,170 Absolute max or min will be the lowest of all the values 2052 02:03:25,170 --> 02:03:28,360 and the highest of all the values. 2053 02:03:28,360 --> 02:03:35,270 OK, so I get at the point minus 1-- how shall I put here? 2054 02:03:35,270 --> 02:03:37,040 x equals minus 1. 2055 02:03:37,040 --> 02:03:38,680 What do I get for y? 2056 02:03:38,680 --> 02:03:41,080 And for plus 1, what do I get for y? 2057 02:03:41,080 --> 02:03:43,030 This is the question. 2058 02:03:43,030 --> 02:03:47,604 I plug it in, and I get minus minus 1/e. 2059 02:03:47,604 --> 02:03:52,200 2060 02:03:52,200 --> 02:03:56,180 And then when I have 1, what do I get? 2061 02:03:56,180 --> 02:03:59,650 1/e again. 2062 02:03:59,650 --> 02:04:01,760 So do I have a relative min here? 2063 02:04:01,760 --> 02:04:07,224 No, but I have an absolute something. 2064 02:04:07,224 --> 02:04:10,212 2065 02:04:10,212 --> 02:04:12,204 And what do I have here? 2066 02:04:12,204 --> 02:04:15,700 Here I have an absolute max. 2067 02:04:15,700 --> 02:04:22,230 So how do we check the absolute maxima and absolute minima? 2068 02:04:22,230 --> 02:04:23,960 We look for critical points. 2069 02:04:23,960 --> 02:04:26,560 We get many of them, finitely many of them. 2070 02:04:26,560 --> 02:04:29,520 We compute all the values of z for them, 2071 02:04:29,520 --> 02:04:32,740 all the function values. 2072 02:04:32,740 --> 02:04:36,050 And then we look at the end points, 2073 02:04:36,050 --> 02:04:38,490 and we compare all three of them, all the three 2074 02:04:38,490 --> 02:04:39,270 values in the end. 2075 02:04:39,270 --> 02:04:43,260 So in the end, you compare 1/e to 1/e to 1. 2076 02:04:43,260 --> 02:04:45,910 And that's all you can get. 2077 02:04:45,910 --> 02:04:49,456 So the lowest in one will be the highest in one. 2078 02:04:49,456 --> 02:04:50,920 Good. 2079 02:04:50,920 --> 02:04:55,160 In Calculus III, it's more complicated. 2080 02:04:55,160 --> 02:04:57,551 But it's not much more complicated. 2081 02:04:57,551 --> 02:05:02,230 Let's see what's going to happen. 2082 02:05:02,230 --> 02:05:04,040 You can have a critical point inside. 2083 02:05:04,040 --> 02:05:07,400 We are just praying we don't have too many. 2084 02:05:07,400 --> 02:05:10,290 So how do I get to step one? 2085 02:05:10,290 --> 02:05:15,160 Critical point means f sub x equals e to the x squared 2086 02:05:15,160 --> 02:05:18,890 minus y squared times 2x. 2087 02:05:18,890 --> 02:05:20,760 All righty, it looks good. 2088 02:05:20,760 --> 02:05:24,530 f sub y equals e to the x squared minus y squared times 2089 02:05:24,530 --> 02:05:25,610 minus y. 2090 02:05:25,610 --> 02:05:26,870 I'm full of hope. 2091 02:05:26,870 --> 02:05:30,420 Because I only have one critical point, thank god. 2092 02:05:30,420 --> 02:05:35,410 Origin is my only critical point. 2093 02:05:35,410 --> 02:05:37,590 I don't know what that is going to give me. 2094 02:05:37,590 --> 02:05:39,830 But it can give me a relative max 2095 02:05:39,830 --> 02:05:42,380 or relative min or a saddle. 2096 02:05:42,380 --> 02:05:45,760 I don't know what it's going to be. 2097 02:05:45,760 --> 02:05:49,890 Who tells me what that is going to be? 2098 02:05:49,890 --> 02:05:51,917 Well, did I do this further? 2099 02:05:51,917 --> 02:05:55,186 2100 02:05:55,186 --> 02:06:00,960 I did it further and a little bit lazy. 2101 02:06:00,960 --> 02:06:05,010 But I'm not asking the nature of the point. 2102 02:06:05,010 --> 02:06:08,870 So for the time being, I only want to see what happens at 0, 2103 02:06:08,870 --> 02:06:09,740 0. 2104 02:06:09,740 --> 02:06:12,210 So I have 1. 2105 02:06:12,210 --> 02:06:18,070 So in my table I will put, OK, this is x, y, and this is z. 2106 02:06:18,070 --> 02:06:20,920 For 0, 0, I'm interested. 2107 02:06:20,920 --> 02:06:23,733 Because that's the critical point inside the domain. 2108 02:06:23,733 --> 02:06:26,245 The domain will be the unities. 2109 02:06:26,245 --> 02:06:29,600 And inside the origin, something interesting happens. 2110 02:06:29,600 --> 02:06:31,340 I get a 1. 2111 02:06:31,340 --> 02:06:35,105 And I hope that's going to be my absolute something. 2112 02:06:35,105 --> 02:06:36,730 But I cannot be sure. 2113 02:06:36,730 --> 02:06:38,220 Why? 2114 02:06:38,220 --> 02:06:43,280 There may be other values coming from the boundary. 2115 02:06:43,280 --> 02:06:46,020 And just like in Calculus I, the only guys 2116 02:06:46,020 --> 02:06:48,640 that can give you other absolute max or min, 2117 02:06:48,640 --> 02:06:51,570 they can come from the boundary, nothing else. 2118 02:06:51,570 --> 02:06:55,320 Nowhere else in the interior of the disc am I going to look. 2119 02:06:55,320 --> 02:06:56,560 I'm not interested. 2120 02:06:56,560 --> 02:06:59,670 I'm only interested in x squared plus y squared equals 1. 2121 02:06:59,670 --> 02:07:02,610 This is where something can happen, 2122 02:07:02,610 --> 02:07:07,070 nothing else interesting in the inside, just like in Calc I. 2123 02:07:07,070 --> 02:07:12,280 So to take x squared plus y squared equals 1 into account, 2124 02:07:12,280 --> 02:07:16,255 I pull y squared, who is married to x-- the poor guy. 2125 02:07:16,255 --> 02:07:17,960 He's married to x. 2126 02:07:17,960 --> 02:07:21,910 He's dependent on x completely, y squared 2127 02:07:21,910 --> 02:07:23,710 equals 1 minus x squared. 2128 02:07:23,710 --> 02:07:27,810 And I have to push him back into the function. 2129 02:07:27,810 --> 02:07:36,960 So at the boundary, f becomes a function of one variable. 2130 02:07:36,960 --> 02:07:43,410 He becomes f of x only equals e to the x 2131 02:07:43,410 --> 02:07:47,465 squared minus 1 plus x squared. 2132 02:07:47,465 --> 02:07:50,380 Are you guys with me? 2133 02:07:50,380 --> 02:07:56,730 So f of x will become e to the 2x squared minus 1 2134 02:07:56,730 --> 02:08:02,570 along the boundary, along the circle, only here. 2135 02:08:02,570 --> 02:08:07,670 2136 02:08:07,670 --> 02:08:10,700 Now what else do I need to do? 2137 02:08:10,700 --> 02:08:14,290 I need to compute the critical values for this function of one 2138 02:08:14,290 --> 02:08:18,060 variable, just the way I did it in Calc I. 2139 02:08:18,060 --> 02:08:22,710 So f prime of x will give me e to the 2x squared 2140 02:08:22,710 --> 02:08:27,630 minus 1 times-- what comes down from the chain rule? 2141 02:08:27,630 --> 02:08:28,130 STUDENT: 4x. 2142 02:08:28,130 --> 02:08:31,440 PROFESSOR: 4x, so life is hard but not that hard. 2143 02:08:31,440 --> 02:08:35,490 Because I can get what? 2144 02:08:35,490 --> 02:08:39,560 I can get only x at 0 equals 0 here. 2145 02:08:39,560 --> 02:08:44,980 OK, so that's a critical point that comes from the boundary. 2146 02:08:44,980 --> 02:08:48,260 But guys, you have to pay attention. 2147 02:08:48,260 --> 02:08:53,280 When x is 0, how many y's can I have for that 0 2148 02:08:53,280 --> 02:08:54,520 on the boundary? 2149 02:08:54,520 --> 02:08:59,130 This is on the boundary-- on the boundary. 2150 02:08:59,130 --> 02:09:00,108 STUDENT: Two. 2151 02:09:00,108 --> 02:09:02,979 PROFESSOR: Two of them-- I can have 1, 2152 02:09:02,979 --> 02:09:04,020 or I can have negative 1. 2153 02:09:04,020 --> 02:09:07,450 2154 02:09:07,450 --> 02:09:10,160 There is one more tricky thing. 2155 02:09:10,160 --> 02:09:12,045 This is a function of one variable only. 2156 02:09:12,045 --> 02:09:15,860 But this stinking function is not 2157 02:09:15,860 --> 02:09:19,072 defined for arbitrary x real. 2158 02:09:19,072 --> 02:09:21,225 So I make a face again. 2159 02:09:21,225 --> 02:09:23,520 So I go, oh, headache. 2160 02:09:23,520 --> 02:09:24,490 Why? 2161 02:09:24,490 --> 02:09:26,140 x is constrained. 2162 02:09:26,140 --> 02:09:27,995 x is constrained, you see? 2163 02:09:27,995 --> 02:09:36,200 If you were inside the disc, x must be between minus 1 and 1. 2164 02:09:36,200 --> 02:09:40,160 So I have to take into account that x is not any real number, 2165 02:09:40,160 --> 02:09:43,870 but x is between minus 1 and 1. 2166 02:09:43,870 --> 02:09:46,490 Those are endpoints for this function. 2167 02:09:46,490 --> 02:09:48,435 And in Calc I, I learned, OK, I have 2168 02:09:48,435 --> 02:09:52,290 to also evaluate what happens at those endpoints. 2169 02:09:52,290 --> 02:09:55,990 But thank god that will exhaust my list, so I have a list. 2170 02:09:55,990 --> 02:10:00,210 Minus 1 for x and 1 for x-- thank god. 2171 02:10:00,210 --> 02:10:04,290 That will give you what y on the boundary? 2172 02:10:04,290 --> 02:10:07,280 When x is 1 and x is minus 1, you're 2173 02:10:07,280 --> 02:10:10,420 interested in what happens, maximization or minimization, 2174 02:10:10,420 --> 02:10:12,850 for this function at the endpoints. 2175 02:10:12,850 --> 02:10:20,990 But fortunately, since you are on the boundary, y must be 0. 2176 02:10:20,990 --> 02:10:25,480 Because that's how you got y out. 2177 02:10:25,480 --> 02:10:29,320 If x is plus/minus 1 on the boundary, y must be 0. 2178 02:10:29,320 --> 02:10:32,680 So my list contains how many interesting points? 2179 02:10:32,680 --> 02:10:36,140 One, two, three, four, five-- for all of them, 2180 02:10:36,140 --> 02:10:38,350 we need to compute, and we are done. 2181 02:10:38,350 --> 02:10:43,250 Of all of them, the lowest z is called absolute minimum. 2182 02:10:43,250 --> 02:10:45,510 And the highest z is the absolute maximum. 2183 02:10:45,510 --> 02:10:46,480 And we are done. 2184 02:10:46,480 --> 02:10:49,970 You guys need to help me, because I'm running out of gas. 2185 02:10:49,970 --> 02:10:52,570 So x is 0. 2186 02:10:52,570 --> 02:10:53,546 Y is 1. 2187 02:10:53,546 --> 02:10:55,220 What is z? 2188 02:10:55,220 --> 02:10:56,900 STUDENT: [INAUDIBLE]. 2189 02:10:56,900 --> 02:10:59,610 PROFESSOR: e to the minus 1, you were 2190 02:10:59,610 --> 02:11:02,280 fast, 1/e, thank you, guys. 2191 02:11:02,280 --> 02:11:06,072 So when x is 0 and y is minus 1? 2192 02:11:06,072 --> 02:11:07,320 STUDENT: [INAUDIBLE]. 2193 02:11:07,320 --> 02:11:08,190 PROFESSOR: Huh? 2194 02:11:08,190 --> 02:11:09,150 STUDENT: [INAUDIBLE]. 2195 02:11:09,150 --> 02:11:10,691 PROFESSOR: No, no, no, it's the same. 2196 02:11:10,691 --> 02:11:12,660 Because x is 0. y is minus 1. 2197 02:11:12,660 --> 02:11:17,130 I get e to the minus 1, which is 1/e. 2198 02:11:17,130 --> 02:11:19,260 So far, so good-- I'm circling all the guys 2199 02:11:19,260 --> 02:11:22,480 that I want to compare after. 2200 02:11:22,480 --> 02:11:28,630 So for the final four points there, what do I have? 2201 02:11:28,630 --> 02:11:34,320 My final candidates could be x equals plus/minus 1 2202 02:11:34,320 --> 02:11:39,436 and y equals 0-- e and e. 2203 02:11:39,436 --> 02:11:40,186 Who's the biggest? 2204 02:11:40,186 --> 02:11:41,190 Who's the smallest? 2205 02:11:41,190 --> 02:11:43,000 STUDENT: e is the biggest. 2206 02:11:43,000 --> 02:11:45,850 PROFESSOR: e is the biggest, and 1/e is the smallest. 2207 02:11:45,850 --> 02:11:48,688 So how do I write conclusion? 2208 02:11:48,688 --> 02:12:01,460 Conclusion-- we have two absolute maxima 2209 02:12:01,460 --> 02:12:08,280 at minus 1, 0 and 1, 0. 2210 02:12:08,280 --> 02:12:24,992 And we have two absolute minima at 0, minus 1 and 0, 1. 2211 02:12:24,992 --> 02:12:32,070 2212 02:12:32,070 --> 02:12:42,010 OK, now I have to-- now that's like a saddle. 2213 02:12:42,010 --> 02:12:44,195 Can you see it with the eyes of your imagination? 2214 02:12:44,195 --> 02:12:45,670 It's hard to see it. 2215 02:12:45,670 --> 02:12:48,510 2216 02:12:48,510 --> 02:12:49,930 This is the disc. 2217 02:12:49,930 --> 02:12:52,980 And the four points, the cardinal 2218 02:12:52,980 --> 02:12:58,330 points-- OK, this is the disc. 2219 02:12:58,330 --> 02:13:02,106 We are looking at this disc from perspective. 2220 02:13:02,106 --> 02:13:06,522 And the five points, one is in the middle. 2221 02:13:06,522 --> 02:13:09,180 2222 02:13:09,180 --> 02:13:11,960 One is here. 2223 02:13:11,960 --> 02:13:13,160 Minus 1 is 0. 2224 02:13:13,160 --> 02:13:14,610 One is here, 1, 0. 2225 02:13:14,610 --> 02:13:20,000 One is here, 0, minus 1, and one here, 0, 1. 2226 02:13:20,000 --> 02:13:25,770 At minus 1, 0 and 1, 0 I get the maximum. 2227 02:13:25,770 --> 02:13:29,420 So the way it's going to be shaped would be like that. 2228 02:13:29,420 --> 02:13:31,100 In this direction, it will be like that. 2229 02:13:31,100 --> 02:13:32,880 And cut the cake here. 2230 02:13:32,880 --> 02:13:34,770 You see it's like that. 2231 02:13:34,770 --> 02:13:36,930 It's going to be like this. 2232 02:13:36,930 --> 02:13:41,480 OK, passing through the origin, with my hands 2233 02:13:41,480 --> 02:13:45,570 I'm molding the surface made of Play-Doh or something for you. 2234 02:13:45,570 --> 02:13:49,090 So I'm starting here, and I'm going up. 2235 02:13:49,090 --> 02:13:51,980 And at this points, I'm here. 2236 02:13:51,980 --> 02:13:52,750 Are you with me? 2237 02:13:52,750 --> 02:13:55,070 The same height. 2238 02:13:55,070 --> 02:14:00,640 In the other direction, I'm going from 0. 2239 02:14:00,640 --> 02:14:04,050 But I'm not so high. 2240 02:14:04,050 --> 02:14:08,500 I'm going only up to-- what is 1/e? 2241 02:14:08,500 --> 02:14:12,240 About 1/3, meh, something like that. 2242 02:14:12,240 --> 02:14:18,320 So I'm going to get here. 2243 02:14:18,320 --> 02:14:23,050 So the problem is that one will be in between. 2244 02:14:23,050 --> 02:14:26,650 So if you really want to see what it looks like, 2245 02:14:26,650 --> 02:14:29,140 we are here at 1. 2246 02:14:29,140 --> 02:14:32,250 We grow from 1, altitude 1, you see? 2247 02:14:32,250 --> 02:14:39,840 We grow from 1 to about 2.71718283 for both of these. 2248 02:14:39,840 --> 02:14:47,560 And from 1, in this direction I have to go down to 1/e. 2249 02:14:47,560 --> 02:14:51,295 So it looks like that. 2250 02:14:51,295 --> 02:14:53,259 I'll try to draw, OK? 2251 02:14:53,259 --> 02:15:03,079 2252 02:15:03,079 --> 02:15:07,900 Do you see the patch around the origin? 2253 02:15:07,900 --> 02:15:11,130 So here's e. 2254 02:15:11,130 --> 02:15:16,170 And here's 1/e above the sea level. 2255 02:15:16,170 --> 02:15:19,400 And this is 1. 2256 02:15:19,400 --> 02:15:26,757 And you have one just like that in the back that is the-- it 2257 02:15:26,757 --> 02:15:27,840 still looks like a saddle. 2258 02:15:27,840 --> 02:15:28,560 It is a saddle. 2259 02:15:28,560 --> 02:15:29,720 It's symmetric. 2260 02:15:29,720 --> 02:15:33,742 But it's another kind of saddle. 2261 02:15:33,742 --> 02:15:35,200 There are all sorts of saddles made 2262 02:15:35,200 --> 02:15:38,960 in Texas, different ranches, different saddles. 2263 02:15:38,960 --> 02:15:44,550 So that was the harder one. 2264 02:15:44,550 --> 02:15:48,040 The ones that I actually saw on the finals, 2265 02:15:48,040 --> 02:15:49,740 some of the last three or four finals, 2266 02:15:49,740 --> 02:15:55,405 were much easier in the sense that the table you had to draw 2267 02:15:55,405 --> 02:16:00,860 was much shorter than this one-- in principle, 2268 02:16:00,860 --> 02:16:05,410 one critical value and one max and one min point. 2269 02:16:05,410 --> 02:16:09,710 But you have to be prepared more, rehearse more, 2270 02:16:09,710 --> 02:16:11,760 so when you see the problem in the midterm, 2271 02:16:11,760 --> 02:16:15,575 you say, oh, well that is easier than I'm used to. 2272 02:16:15,575 --> 02:16:17,900 That's the idea. 2273 02:16:17,900 --> 02:16:19,730 OK, go home. 2274 02:16:19,730 --> 02:16:21,530 Send me emails by WeBWorK. 2275 02:16:21,530 --> 02:16:25,130 We still have time to talk about the homework if you get stuck. 2276 02:16:25,130 --> 02:16:27,830 2277 02:16:27,830 --> 02:16:31,730 And I'll see you Thursday. 2278 02:16:31,730 --> 02:16:35,080 [BACKGROUND CHATTER] 2279 02:16:35,080 --> 02:16:59,326