[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:06.37,Default,,0000,0000,0000,,what we discussed\Nlast time in 11.6
Dialogue: 0,0:00:06.37,0:00:10.29,Default,,0000,0000,0000,,which was-- do you remember\Nthe topics we discussed?
Dialogue: 0,0:00:10.29,0:00:13.72,Default,,0000,0000,0000,,We discussed the\Nvaluation of a derivative.
Dialogue: 0,0:00:13.72,0:00:20.58,Default,,0000,0000,0000,,
Dialogue: 0,0:00:20.58,0:00:23.41,Default,,0000,0000,0000,,What else have we\Ndiscussed about them?
Dialogue: 0,0:00:23.41,0:00:27.84,Default,,0000,0000,0000,,I'm gonna split the fields,\Nalthough they are related.
Dialogue: 0,0:00:27.84,0:00:37.18,Default,,0000,0000,0000,,Gradient and the steepest\Nascent and descent.
Dialogue: 0,0:00:37.18,0:00:41.12,Default,,0000,0000,0000,,
Dialogue: 0,0:00:41.12,0:00:52.96,Default,,0000,0000,0000,,So in which direction\Nwill z equals
Dialogue: 0,0:00:52.96,0:00:57.65,Default,,0000,0000,0000,,f of x and y,\Ndifferential cofunction,
Dialogue: 0,0:00:57.65,0:01:01.62,Default,,0000,0000,0000,,and with a derivative\Nthat is continuous?
Dialogue: 0,0:01:01.62,0:01:05.61,Default,,0000,0000,0000,,
Dialogue: 0,0:01:05.61,0:01:12.11,Default,,0000,0000,0000,,So will this have the\Nmaximum rate of change?
Dialogue: 0,0:01:12.11,0:01:17.42,Default,,0000,0000,0000,,
Dialogue: 0,0:01:17.42,0:01:18.69,Default,,0000,0000,0000,,This is just review.
Dialogue: 0,0:01:18.69,0:01:21.08,Default,,0000,0000,0000,,Why am I doing review?
Dialogue: 0,0:01:21.08,0:01:24.76,Default,,0000,0000,0000,,Well, after talking to\Nyou on a personal basis,
Dialogue: 0,0:01:24.76,0:01:29.50,Default,,0000,0000,0000,,like one-on-one basis by email\Nand a little bit in person,
Dialogue: 0,0:01:29.50,0:01:33.70,Default,,0000,0000,0000,,I realized that you like\Nvery much when I review.
Dialogue: 0,0:01:33.70,0:01:37.35,Default,,0000,0000,0000,,When I briefly review\Nsome of the notions.
Dialogue: 0,0:01:37.35,0:01:42.14,Default,,0000,0000,0000,,I will give you the essentials\Nfor the section, that
Dialogue: 0,0:01:42.14,0:01:45.47,Default,,0000,0000,0000,,was 11.5 is embedded in this.
Dialogue: 0,0:01:45.47,0:01:46.68,Default,,0000,0000,0000,,Embedded.
Dialogue: 0,0:01:46.68,0:01:50.73,Default,,0000,0000,0000,,So 11.5 is embedded in\N11.6, and in one shot
Dialogue: 0,0:01:50.73,0:01:53.45,Default,,0000,0000,0000,,we can talk of them.
Dialogue: 0,0:01:53.45,0:01:57.27,Default,,0000,0000,0000,,In 11.7 you're gonna see\Nsome extrema of functions
Dialogue: 0,0:01:57.27,0:02:00.46,Default,,0000,0000,0000,,of two variables,\Nlike max and min,
Dialogue: 0,0:02:00.46,0:02:05.50,Default,,0000,0000,0000,,and then we have 11.8, which\Nis Lagrange multipliers.
Dialogue: 0,0:02:05.50,0:02:09.96,Default,,0000,0000,0000,,What have we done about\Nthis differentiable function
Dialogue: 0,0:02:09.96,0:02:13.12,Default,,0000,0000,0000,,with continuous\Npartial derivatives?
Dialogue: 0,0:02:13.12,0:02:18.09,Default,,0000,0000,0000,,Let's assume that\Nit's smooth, OK?
Dialogue: 0,0:02:18.09,0:02:25.60,Default,,0000,0000,0000,,And in that case, we\Ndefine the partial--
Dialogue: 0,0:02:25.60,0:02:30.91,Default,,0000,0000,0000,,the directional derivative\Nin the direction of u at u,
Dialogue: 0,0:02:30.91,0:02:39.01,Default,,0000,0000,0000,,v with unit vector at the\Npoint x, y, but let's fix it
Dialogue: 0,0:02:39.01,0:02:44.11,Default,,0000,0000,0000,,x0, y0, using a limit of\Na difference quotient,
Dialogue: 0,0:02:44.11,0:02:48.02,Default,,0000,0000,0000,,just like any derivative\Nshould be introduced.
Dialogue: 0,0:02:48.02,0:02:51.66,Default,,0000,0000,0000,,But I'm not gonna\Nrepeat that definition.
Dialogue: 0,0:02:51.66,0:02:53.62,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:02:53.62,0:02:58.62,Default,,0000,0000,0000,,Because I want you to give\Nme an alternative definition,
Dialogue: 0,0:02:58.62,0:03:01.67,Default,,0000,0000,0000,,which is something\Nthat is simpler
Dialogue: 0,0:03:01.67,0:03:04.94,Default,,0000,0000,0000,,to use in applications,\Nwhich is what?
Dialogue: 0,0:03:04.94,0:03:06.99,Default,,0000,0000,0000,,Who remembers?
Dialogue: 0,0:03:06.99,0:03:11.94,Default,,0000,0000,0000,,The partial derivative\Nof f with respect to x
Dialogue: 0,0:03:11.94,0:03:13.93,Default,,0000,0000,0000,,measured at the point x0, y0.
Dialogue: 0,0:03:13.93,0:03:19.92,Default,,0000,0000,0000,,
Dialogue: 0,0:03:19.92,0:03:22.88,Default,,0000,0000,0000,,I'm gonna use this color, and\Nthen I'll change the color.
Dialogue: 0,0:03:22.88,0:03:26.54,Default,,0000,0000,0000,,And I'll say times u1.
Dialogue: 0,0:03:26.54,0:03:35.96,Default,,0000,0000,0000,,Plus-- change the color again--\Nf sub y at x0, y0, times u2.
Dialogue: 0,0:03:35.96,0:03:40.72,Default,,0000,0000,0000,,
Dialogue: 0,0:03:40.72,0:03:44.91,Default,,0000,0000,0000,,And this is just a\Ntimes-- multiplication
Dialogue: 0,0:03:44.91,0:03:47.12,Default,,0000,0000,0000,,between real values.
Dialogue: 0,0:03:47.12,0:03:52.69,Default,,0000,0000,0000,,Because u1 and u2 will be\Nnobody but the components--
Dialogue: 0,0:03:52.69,0:03:57.57,Default,,0000,0000,0000,,the real value components\Nof the unit vector direction
Dialogue: 0,0:03:57.57,0:03:58.57,Default,,0000,0000,0000,,that we have.
Dialogue: 0,0:03:58.57,0:04:03.28,Default,,0000,0000,0000,,So, guys, remember, direction\Nin this books means unit vector.
Dialogue: 0,0:04:03.28,0:04:05.93,Default,,0000,0000,0000,,Every time I say\Ndirection, I should
Dialogue: 0,0:04:05.93,0:04:08.12,Default,,0000,0000,0000,,say that's a unit vector.
Dialogue: 0,0:04:08.12,0:04:14.17,Default,,0000,0000,0000,,Is there any way, any\Nother way, to express this?
Dialogue: 0,0:04:14.17,0:04:18.85,Default,,0000,0000,0000,,Maybe with a vector\Nmultiplication of some sort--
Dialogue: 0,0:04:18.85,0:04:20.29,Default,,0000,0000,0000,,multiplication between vectors.
Dialogue: 0,0:04:20.29,0:04:24.75,Default,,0000,0000,0000,,See, what if I had a pink\Nvector with pink components,
Dialogue: 0,0:04:24.75,0:04:27.41,Default,,0000,0000,0000,,and a blue vector\Nwith blue components?
Dialogue: 0,0:04:27.41,0:04:28.90,Default,,0000,0000,0000,,I'm getting somewhere.
Dialogue: 0,0:04:28.90,0:04:29.70,Default,,0000,0000,0000,,And I'm sneaky.
Dialogue: 0,0:04:29.70,0:04:31.84,Default,,0000,0000,0000,,And you feel--\Nyou know where I'm
Dialogue: 0,0:04:31.84,0:04:36.13,Default,,0000,0000,0000,,getting because you had\Nchapter nine fresh in your mind
Dialogue: 0,0:04:36.13,0:04:40.60,Default,,0000,0000,0000,,and a certain product\Nbetween vectors.
Dialogue: 0,0:04:40.60,0:04:42.82,Default,,0000,0000,0000,,So what is this?
Dialogue: 0,0:04:42.82,0:04:44.73,Default,,0000,0000,0000,,It's a dot product, excellent.
Dialogue: 0,0:04:44.73,0:04:46.40,Default,,0000,0000,0000,,Rachel, right?
Dialogue: 0,0:04:46.40,0:04:48.39,Default,,0000,0000,0000,,So it's a dot product.
Dialogue: 0,0:04:48.39,0:04:51.17,Default,,0000,0000,0000,,And I'm gonna run\Nit, and it's going
Dialogue: 0,0:04:51.17,0:04:57.85,Default,,0000,0000,0000,,to be just the gradient\Nof f at the point p0.
Dialogue: 0,0:04:57.85,0:05:00.82,Default,,0000,0000,0000,,But I'm going to write\Nit again, x0, y0,
Dialogue: 0,0:05:00.82,0:05:03.00,Default,,0000,0000,0000,,although it drives\Nme crazy to have
Dialogue: 0,0:05:03.00,0:05:05.88,Default,,0000,0000,0000,,to write that all the time.
Dialogue: 0,0:05:05.88,0:05:10.61,Default,,0000,0000,0000,,Dot product or scalar\Nproduct with what vector u?
Dialogue: 0,0:05:10.61,0:05:13.54,Default,,0000,0000,0000,,I'm not gonna put a bar on\Nthat because the-- that would
Dialogue: 0,0:05:13.54,0:05:15.68,Default,,0000,0000,0000,,be like an oxymoron.
Dialogue: 0,0:05:15.68,0:05:18.20,Default,,0000,0000,0000,,The gradient is a bar thing.
Dialogue: 0,0:05:18.20,0:05:21.15,Default,,0000,0000,0000,,It's always a vector, so\NI'm not gonna write a bar.
Dialogue: 0,0:05:21.15,0:05:23.40,Default,,0000,0000,0000,,But I write a bar\Non u, reminding you
Dialogue: 0,0:05:23.40,0:05:27.34,Default,,0000,0000,0000,,that u is a free vector.
Dialogue: 0,0:05:27.34,0:05:28.95,Default,,0000,0000,0000,,All right, do you like this?
Dialogue: 0,0:05:28.95,0:05:29.45,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:05:29.45,0:05:33.58,Default,,0000,0000,0000,,It's a compactified form\Nof the fluffy expression
Dialogue: 0,0:05:33.58,0:05:35.18,Default,,0000,0000,0000,,we had before.
Dialogue: 0,0:05:35.18,0:05:38.61,Default,,0000,0000,0000,,It's much easier to remember\Nthan the limit definition.
Dialogue: 0,0:05:38.61,0:05:41.96,Default,,0000,0000,0000,,Of course, it's\Nequivalent to it.
Dialogue: 0,0:05:41.96,0:05:47.47,Default,,0000,0000,0000,,And in applications, I-- well,\Nwe ask about that all the time.
Dialogue: 0,0:05:47.47,0:05:48.42,Default,,0000,0000,0000,,What was the gradient?
Dialogue: 0,0:05:48.42,0:05:50.72,Default,,0000,0000,0000,,So see, in mathematics\Neverything is related.
Dialogue: 0,0:05:50.72,0:05:53.54,Default,,0000,0000,0000,,And talking about--\Nspeaking about the devil--
Dialogue: 0,0:05:53.54,0:05:57.18,Default,,0000,0000,0000,,I mean, not you, but we\Nweren't talking about you--
Dialogue: 0,0:05:57.18,0:06:00.84,Default,,0000,0000,0000,,just this gradient.
Dialogue: 0,0:06:00.84,0:06:05.78,Default,,0000,0000,0000,,Gradient f at the\Npoint p will simply
Dialogue: 0,0:06:05.78,0:06:11.32,Default,,0000,0000,0000,,be the vector whose components\Nin the direction of i and j
Dialogue: 0,0:06:11.32,0:06:13.93,Default,,0000,0000,0000,,are the partials.
Dialogue: 0,0:06:13.93,0:06:15.84,Default,,0000,0000,0000,,The partial derivative\Nwith respect
Dialogue: 0,0:06:15.84,0:06:20.87,Default,,0000,0000,0000,,to x, plus the partial\Nderivative with respect to y.
Dialogue: 0,0:06:20.87,0:06:26.52,Default,,0000,0000,0000,,
Dialogue: 0,0:06:26.52,0:06:30.90,Default,,0000,0000,0000,,OK, last time we dealt even\Nwith gradients of-- functions
Dialogue: 0,0:06:30.90,0:06:33.27,Default,,0000,0000,0000,,of more variables.
Dialogue: 0,0:06:33.27,0:06:35.67,Default,,0000,0000,0000,,If I have n\Nvariables, so x1, x2,
Dialogue: 0,0:06:35.67,0:06:41.24,Default,,0000,0000,0000,,x3, xn, then this vector will\Nhave n components-- f sub x1,
Dialogue: 0,0:06:41.24,0:06:43.83,Default,,0000,0000,0000,,f sub x2, f sub x3, f sub x4.
Dialogue: 0,0:06:43.83,0:06:44.93,Default,,0000,0000,0000,,Somebody stop me.
Dialogue: 0,0:06:44.93,0:06:46.40,Default,,0000,0000,0000,,F sub xn.
Dialogue: 0,0:06:46.40,0:06:50.12,Default,,0000,0000,0000,,So I have n of them, and\Nit's an n [INAUDIBLE].
Dialogue: 0,0:06:50.12,0:06:58.32,Default,,0000,0000,0000,,It's a sum ordered [INAUDIBLE].
Dialogue: 0,0:06:58.32,0:06:58.82,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:06:58.82,0:07:01.34,Default,,0000,0000,0000,,A set of n elements\Nin this order.
Dialogue: 0,0:07:01.34,0:07:04.76,Default,,0000,0000,0000,,It's an order set,\N[INAUDIBLE] n.
Dialogue: 0,0:07:04.76,0:07:05.58,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:07:05.58,0:07:06.91,Default,,0000,0000,0000,,So the order matters.
Dialogue: 0,0:07:06.91,0:07:11.35,Default,,0000,0000,0000,,Now, steepest\Nascent and descent.
Dialogue: 0,0:07:11.35,0:07:15.33,Default,,0000,0000,0000,,In which direction do I have\Nthe maximum rate of change?
Dialogue: 0,0:07:15.33,0:07:17.96,Default,,0000,0000,0000,,And the answer is--\Nthis is Q and this
Dialogue: 0,0:07:17.96,0:07:24.96,Default,,0000,0000,0000,,is A-- the maximum\Nrate of change
Dialogue: 0,0:07:24.96,0:07:42.35,Default,,0000,0000,0000,,happens in the direction of\Nthe gradient at every point--
Dialogue: 0,0:07:42.35,0:07:45.21,Default,,0000,0000,0000,,every point of\Nthe domain where I
Dialogue: 0,0:07:45.21,0:07:48.81,Default,,0000,0000,0000,,have smoothness or [INAUDIBLE]\Nor [INAUDIBLE] whatever.
Dialogue: 0,0:07:48.81,0:07:52.53,Default,,0000,0000,0000,,
Dialogue: 0,0:07:52.53,0:07:53.29,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:07:53.29,0:07:54.30,Default,,0000,0000,0000,,And what else?
Dialogue: 0,0:07:54.30,0:07:57.82,Default,,0000,0000,0000,,I claimed last\Ntime, but I didn't
Dialogue: 0,0:07:57.82,0:08:04.11,Default,,0000,0000,0000,,prove, that in that\Ndirection that I claimed c
Dialogue: 0,0:08:04.11,0:08:05.55,Default,,0000,0000,0000,,from [INAUDIBLE].
Dialogue: 0,0:08:05.55,0:08:22.41,Default,,0000,0000,0000,,The directional derivative\Nis maximized exactly
Dialogue: 0,0:08:22.41,0:08:26.46,Default,,0000,0000,0000,,in the direction\Nof the gradient.
Dialogue: 0,0:08:26.46,0:08:31.94,Default,,0000,0000,0000,,
Dialogue: 0,0:08:31.94,0:08:33.07,Default,,0000,0000,0000,,Can we prove that?
Dialogue: 0,0:08:33.07,0:08:34.61,Default,,0000,0000,0000,,Now we can prove it.
Dialogue: 0,0:08:34.61,0:08:37.50,Default,,0000,0000,0000,,Before I couldn't prove it\Nbecause you couldn't see it,
Dialogue: 0,0:08:37.50,0:08:40.42,Default,,0000,0000,0000,,because I didn't look\Nat it as a dot product.
Dialogue: 0,0:08:40.42,0:08:44.34,Default,,0000,0000,0000,,And we were all blind, like\Nguiding each other in the dark.
Dialogue: 0,0:08:44.34,0:08:45.70,Default,,0000,0000,0000,,Me blind, you blind.
Dialogue: 0,0:08:45.70,0:08:47.13,Default,,0000,0000,0000,,We couldn't see.
Dialogue: 0,0:08:47.13,0:08:48.49,Default,,0000,0000,0000,,Now we can see.
Dialogue: 0,0:08:48.49,0:08:51.98,Default,,0000,0000,0000,,So now we can see\Nhow to prove my claim
Dialogue: 0,0:08:51.98,0:08:53.68,Default,,0000,0000,0000,,that the directional\Nderivative, which
Dialogue: 0,0:08:53.68,0:08:57.67,Default,,0000,0000,0000,,is measuring the\Nmaximum-- measuring
Dialogue: 0,0:08:57.67,0:09:01.54,Default,,0000,0000,0000,,the instantaneous rate of\Nchange in a direction-- compass
Dialogue: 0,0:09:01.54,0:09:04.39,Default,,0000,0000,0000,,direction, like,\Nwhat is that, east?
Dialogue: 0,0:09:04.39,0:09:06.93,Default,,0000,0000,0000,,East, northeast,\Nsouthwest, whatever.
Dialogue: 0,0:09:06.93,0:09:09.68,Default,,0000,0000,0000,,Those are the compass\Ndirections like i, j, i
Dialogue: 0,0:09:09.68,0:09:11.91,Default,,0000,0000,0000,,plus j over square\Nroot 2 and so on,
Dialogue: 0,0:09:11.91,0:09:13.84,Default,,0000,0000,0000,,those are called\Ncompass directions
Dialogue: 0,0:09:13.84,0:09:17.90,Default,,0000,0000,0000,,because you hold the compass\Nin your hand as horizontal
Dialogue: 0,0:09:17.90,0:09:21.87,Default,,0000,0000,0000,,as you can, and you\Nrefer to the floor.
Dialogue: 0,0:09:21.87,0:09:24.27,Default,,0000,0000,0000,,Even if you are on a slope.
Dialogue: 0,0:09:24.27,0:09:28.49,Default,,0000,0000,0000,,Maybe you imagine me on a\Nslope, hiking or whatever.
Dialogue: 0,0:09:28.49,0:09:29.89,Default,,0000,0000,0000,,Going down, going up.
Dialogue: 0,0:09:29.89,0:09:33.25,Default,,0000,0000,0000,,But the compass should\Nalways be kept horizontal.
Dialogue: 0,0:09:33.25,0:09:34.08,Default,,0000,0000,0000,,Do you hike, Alex?
Dialogue: 0,0:09:34.08,0:09:34.66,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:09:34.66,0:09:36.70,Default,,0000,0000,0000,,PROFESSOR: I'm sorry,\NI've put you on the spot.
Dialogue: 0,0:09:36.70,0:09:44.54,Default,,0000,0000,0000,,So whatever you do when you\Nthink of a path up or down
Dialogue: 0,0:09:44.54,0:09:47.90,Default,,0000,0000,0000,,is measured in the direction\Nof the horizontal plane,
Dialogue: 0,0:09:47.90,0:09:49.24,Default,,0000,0000,0000,,the compass direction.
Dialogue: 0,0:09:49.24,0:09:51.92,Default,,0000,0000,0000,,And that is the gradient\NI was talking about.
Dialogue: 0,0:09:51.92,0:09:53.29,Default,,0000,0000,0000,,You see, it's a function.
Dialogue: 0,0:09:53.29,0:09:56.08,Default,,0000,0000,0000,,It's a vector in plane.
Dialogue: 0,0:09:56.08,0:10:00.22,Default,,0000,0000,0000,,That means a geographic\Ncompass direction.
Dialogue: 0,0:10:00.22,0:10:01.16,Default,,0000,0000,0000,,Prove it.
Dialogue: 0,0:10:01.16,0:10:02.15,Default,,0000,0000,0000,,Let's prove the claim.
Dialogue: 0,0:10:02.15,0:10:04.26,Default,,0000,0000,0000,,Let's prove the claim,\Nbecause this is Tuesday,
Dialogue: 0,0:10:04.26,0:10:07.06,Default,,0000,0000,0000,,and it's almost weekend, and\Nwe have to prove something
Dialogue: 0,0:10:07.06,0:10:09.18,Default,,0000,0000,0000,,this week, right?
Dialogue: 0,0:10:09.18,0:10:14.72,Default,,0000,0000,0000,,Now, do you like what you see?
Dialogue: 0,0:10:14.72,0:10:18.79,Default,,0000,0000,0000,,Well, I have no idea.
Dialogue: 0,0:10:18.79,0:10:21.37,Default,,0000,0000,0000,,What if I want to measure this.
Dialogue: 0,0:10:21.37,0:10:25.52,Default,,0000,0000,0000,,This could be a negative\Nnumber, but it doesn't matter.
Dialogue: 0,0:10:25.52,0:10:30.34,Default,,0000,0000,0000,,Assume that I take\Nthe dot product,
Dialogue: 0,0:10:30.34,0:10:34.57,Default,,0000,0000,0000,,and I think, well, it's a scalar\Nproduct, it must be positive.
Dialogue: 0,0:10:34.57,0:10:36.50,Default,,0000,0000,0000,,What do I get?
Dialogue: 0,0:10:36.50,0:10:40.37,Default,,0000,0000,0000,,
Dialogue: 0,0:10:40.37,0:10:44.01,Default,,0000,0000,0000,,This guy-- if I'm a physicist,\NI'm going to say this guy
Dialogue: 0,0:10:44.01,0:10:47.82,Default,,0000,0000,0000,,is the length of the\Nfirst vector at p0.
Dialogue: 0,0:10:47.82,0:10:49.32,Default,,0000,0000,0000,,Gradient at p0.
Dialogue: 0,0:10:49.32,0:10:51.78,Default,,0000,0000,0000,,That's the length of u.
Dialogue: 0,0:10:51.78,0:10:54.35,Default,,0000,0000,0000,,Duh, that is 1.
Dialogue: 0,0:10:54.35,0:10:58.23,Default,,0000,0000,0000,,Last time I checked, u was\Nunitary, so that's silly of me,
Dialogue: 0,0:10:58.23,0:11:00.61,Default,,0000,0000,0000,,but I'll write it anyway.
Dialogue: 0,0:11:00.61,0:11:05.36,Default,,0000,0000,0000,,Times the cosine of the\Nangle between-- oh, cosine
Dialogue: 0,0:11:05.36,0:11:10.35,Default,,0000,0000,0000,,of the parenthesis angle\Nbetween nabla f and u.
Dialogue: 0,0:11:10.35,0:11:15.31,Default,,0000,0000,0000,,
Dialogue: 0,0:11:15.31,0:11:15.86,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:11:15.86,0:11:17.46,Default,,0000,0000,0000,,When is this maximized?
Dialogue: 0,0:11:17.46,0:11:22.74,Default,,0000,0000,0000,,STUDENT: When the angle\Nbetween nabla f and u is 0.
Dialogue: 0,0:11:22.74,0:11:23.53,Default,,0000,0000,0000,,PROFESSOR: Exactly.
Dialogue: 0,0:11:23.53,0:11:33.54,Default,,0000,0000,0000,,When this is pi, so this\Nquantity is maximized-- gosh,
Dialogue: 0,0:11:33.54,0:11:35.50,Default,,0000,0000,0000,,I hate writing a lot.
Dialogue: 0,0:11:35.50,0:11:38.92,Default,,0000,0000,0000,,I had to submit homework\Nin the past two days,
Dialogue: 0,0:11:38.92,0:11:43.80,Default,,0000,0000,0000,,and one was about biological\Nresearch and the other one
Dialogue: 0,0:11:43.80,0:11:45.99,Default,,0000,0000,0000,,about stress management.
Dialogue: 0,0:11:45.99,0:11:49.32,Default,,0000,0000,0000,,The stress management class\Nstresses me out the most.
Dialogue: 0,0:11:49.32,0:11:52.17,Default,,0000,0000,0000,,I shouldn't make it public.
Dialogue: 0,0:11:52.17,0:11:55.70,Default,,0000,0000,0000,,Really, because we have to write\Nthese essays of seven or eight
Dialogue: 0,0:11:55.70,0:11:58.32,Default,,0000,0000,0000,,pages every Tuesday,\Nevery end of the week.
Dialogue: 0,0:11:58.32,0:11:59.60,Default,,0000,0000,0000,,Twice a week.
Dialogue: 0,0:11:59.60,0:12:04.59,Default,,0000,0000,0000,,, So I realized how much\NI hate writing down a lot,
Dialogue: 0,0:12:04.59,0:12:07.02,Default,,0000,0000,0000,,and what a blessing it\Nis to be a mathematician.
Dialogue: 0,0:12:07.02,0:12:08.66,Default,,0000,0000,0000,,You abbreviate everything.
Dialogue: 0,0:12:08.66,0:12:11.50,Default,,0000,0000,0000,,You compress everything.
Dialogue: 0,0:12:11.50,0:12:12.71,Default,,0000,0000,0000,,I love formulas.
Dialogue: 0,0:12:12.71,0:12:20.78,Default,,0000,0000,0000,,So what we have here is\Nmaximized when the cosine is 1.
Dialogue: 0,0:12:20.78,0:12:23.53,Default,,0000,0000,0000,,
Dialogue: 0,0:12:23.53,0:12:26.22,Default,,0000,0000,0000,,And if you have\Nbecome 0 to 2 pi open,
Dialogue: 0,0:12:26.22,0:12:33.30,Default,,0000,0000,0000,,then theta 0 is\Nyour only option.
Dialogue: 0,0:12:33.30,0:12:35.56,Default,,0000,0000,0000,,Well, if you take\Nan absolute value,
Dialogue: 0,0:12:35.56,0:12:39.37,Default,,0000,0000,0000,,you could also have it\Nin the other direction,
Dialogue: 0,0:12:39.37,0:12:43.06,Default,,0000,0000,0000,,cosine pi, but in that\Ncase you change the sign.
Dialogue: 0,0:12:43.06,0:12:46.72,Default,,0000,0000,0000,,So what you get--\Nyou get a maximum.
Dialogue: 0,0:12:46.72,0:12:48.88,Default,,0000,0000,0000,,Let's say you hike, right?
Dialogue: 0,0:12:48.88,0:12:50.93,Default,,0000,0000,0000,,I'm just hiking in my brain.
Dialogue: 0,0:12:50.93,0:12:54.29,Default,,0000,0000,0000,,The maximum rate of\Nchange in this direction,
Dialogue: 0,0:12:54.29,0:12:56.56,Default,,0000,0000,0000,,climbing towards the peak.
Dialogue: 0,0:12:56.56,0:13:02.32,Default,,0000,0000,0000,,And then the steepest descent\Nis the exactly minus gradient
Dialogue: 0,0:13:02.32,0:13:02.82,Default,,0000,0000,0000,,direction.
Dialogue: 0,0:13:02.82,0:13:07.15,Default,,0000,0000,0000,,So I could have 0\Nor pi for the angle.
Dialogue: 0,0:13:07.15,0:13:10.10,Default,,0000,0000,0000,,That's the philosophical\Nmeaning of that.
Dialogue: 0,0:13:10.10,0:13:10.69,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:13:10.69,0:13:13.03,Default,,0000,0000,0000,,So the directional\Nderivative, which is this guy,
Dialogue: 0,0:13:13.03,0:13:15.04,Default,,0000,0000,0000,,when does it become maximum?
Dialogue: 0,0:13:15.04,0:13:16.29,Default,,0000,0000,0000,,When the angle is 0.
Dialogue: 0,0:13:16.29,0:13:17.52,Default,,0000,0000,0000,,So I'm done.
Dialogue: 0,0:13:17.52,0:13:20.50,Default,,0000,0000,0000,,QED.
Dialogue: 0,0:13:20.50,0:13:21.99,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:13:21.99,0:13:28.81,Default,,0000,0000,0000,,That I know this happens\Nwhen-- when direction u is
Dialogue: 0,0:13:28.81,0:13:30.37,Default,,0000,0000,0000,,the direction of the gradient.
Dialogue: 0,0:13:30.37,0:13:32.36,Default,,0000,0000,0000,,Can I write u equals\Ngradient of f?
Dialogue: 0,0:13:32.36,0:13:34.18,Default,,0000,0000,0000,,Not quite.
Dialogue: 0,0:13:34.18,0:13:40.98,Default,,0000,0000,0000,,I should say divided\Nby norm or magnitude.
Dialogue: 0,0:13:40.98,0:13:42.15,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:13:42.15,0:13:44.32,Default,,0000,0000,0000,,And you say, Magdalena,\Ndidn't you say like 10 times
Dialogue: 0,0:13:44.32,0:13:46.84,Default,,0000,0000,0000,,that u is a unit vector?
Dialogue: 0,0:13:46.84,0:13:49.10,Default,,0000,0000,0000,,You want u to be a\Nunit vector direction.
Dialogue: 0,0:13:49.10,0:13:52.01,Default,,0000,0000,0000,,So the direction should be\Nthe direction of the gradient
Dialogue: 0,0:13:52.01,0:13:54.05,Default,,0000,0000,0000,,in order to maximize this\Ndirectional derivative.
Dialogue: 0,0:13:54.05,0:13:57.11,Default,,0000,0000,0000,,But then you have\Nto take the gradient
Dialogue: 0,0:13:57.11,0:13:59.50,Default,,0000,0000,0000,,and divide it by its magnitude.
Dialogue: 0,0:13:59.50,0:14:00.29,Default,,0000,0000,0000,,Let's compute it.
Dialogue: 0,0:14:00.29,0:14:02.70,Default,,0000,0000,0000,,Let's see what we get.
Dialogue: 0,0:14:02.70,0:14:03.64,Default,,0000,0000,0000,,Let's see what we get.
Dialogue: 0,0:14:03.64,0:14:06.97,Default,,0000,0000,0000,,Now, I'm sorry about my\Nbeautiful handwriting,
Dialogue: 0,0:14:06.97,0:14:11.92,Default,,0000,0000,0000,,but I'm-- well, I'm gonna\Nhave to-- I have room here.
Dialogue: 0,0:14:11.92,0:14:14.34,Default,,0000,0000,0000,,And actually I can\Nuse this formula.
Dialogue: 0,0:14:14.34,0:14:16.90,Default,,0000,0000,0000,,So in the direction\Nof the gradient,
Dialogue: 0,0:14:16.90,0:14:18.56,Default,,0000,0000,0000,,when u is the gradient.
Dialogue: 0,0:14:18.56,0:14:21.95,Default,,0000,0000,0000,,Let's take u to be-- what\Ndid I say over there?
Dialogue: 0,0:14:21.95,0:14:27.45,Default,,0000,0000,0000,,Gradient of f over the\Nmagnitude of gradient of f.
Dialogue: 0,0:14:27.45,0:14:29.72,Default,,0000,0000,0000,,I'll take this guy\Nand drag him here.
Dialogue: 0,0:14:29.72,0:14:31.48,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:14:31.48,0:14:36.11,Default,,0000,0000,0000,,What will the value of the\Ndirectional derivative be?
Dialogue: 0,0:14:36.11,0:14:39.74,Default,,0000,0000,0000,,We've done last time that--\Nthe same thing on an example.
Dialogue: 0,0:14:39.74,0:14:43.68,Default,,0000,0000,0000,,We've done it on a function,\Nbeautiful f of x, y
Dialogue: 0,0:14:43.68,0:14:46.25,Default,,0000,0000,0000,,equals x squared plus y squared.
Dialogue: 0,0:14:46.25,0:14:48.75,Default,,0000,0000,0000,,This is the type\Nof function I like,
Dialogue: 0,0:14:48.75,0:14:52.25,Default,,0000,0000,0000,,because they are the\Nfastest to deal with.
Dialogue: 0,0:14:52.25,0:14:56.24,Default,,0000,0000,0000,,But anyway, we'll have all\Nsorts of other functions.
Dialogue: 0,0:14:56.24,0:14:59.19,Default,,0000,0000,0000,,What am I going to write here?
Dialogue: 0,0:14:59.19,0:15:00.66,Default,,0000,0000,0000,,I have to write.
Dialogue: 0,0:15:00.66,0:15:03.40,Default,,0000,0000,0000,,Well, by definition,\Nnow, by my new way
Dialogue: 0,0:15:03.40,0:15:05.36,Default,,0000,0000,0000,,to look at the\Ndefinition, I'm gonna
Dialogue: 0,0:15:05.36,0:15:09.57,Default,,0000,0000,0000,,have a gradient at\Nthe point, the vector,
Dialogue: 0,0:15:09.57,0:15:12.20,Default,,0000,0000,0000,,dot-- who is u again?
Dialogue: 0,0:15:12.20,0:15:16.59,Default,,0000,0000,0000,,The gradient this time,\Nthat special value,
Dialogue: 0,0:15:16.59,0:15:23.19,Default,,0000,0000,0000,,divided by absolute-- by\Nmagnitude of the gradient.
Dialogue: 0,0:15:23.19,0:15:25.89,Default,,0000,0000,0000,,But what in the world is that?
Dialogue: 0,0:15:25.89,0:15:29.41,Default,,0000,0000,0000,,
Dialogue: 0,0:15:29.41,0:15:33.56,Default,,0000,0000,0000,,This animal is--\Nwhat if you take
Dialogue: 0,0:15:33.56,0:15:35.73,Default,,0000,0000,0000,,a vector multiplied by itself?
Dialogue: 0,0:15:35.73,0:15:36.34,Default,,0000,0000,0000,,Dot product.
Dialogue: 0,0:15:36.34,0:15:38.63,Default,,0000,0000,0000,,No, not dot product.
Dialogue: 0,0:15:38.63,0:15:42.87,Default,,0000,0000,0000,,What you get is the\Nmagnitude squared.
Dialogue: 0,0:15:42.87,0:15:43.43,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:15:43.43,0:15:48.69,Default,,0000,0000,0000,,So although the pinkie\Nguy and the pinkie guy
Dialogue: 0,0:15:48.69,0:15:54.10,Default,,0000,0000,0000,,are magnitude of f squared\Ndivided by magnitude of f.
Dialogue: 0,0:15:54.10,0:15:57.23,Default,,0000,0000,0000,,And Alex said, but wait,\Nthat's just-- I know.
Dialogue: 0,0:15:57.23,0:16:01.28,Default,,0000,0000,0000,,I didn't want to just\Njump ahead too fast.
Dialogue: 0,0:16:01.28,0:16:06.50,Default,,0000,0000,0000,,We get gradient\Nof f in magnitude.
Dialogue: 0,0:16:06.50,0:16:07.98,Default,,0000,0000,0000,,So, beautiful.
Dialogue: 0,0:16:07.98,0:16:10.90,Default,,0000,0000,0000,,So we know who that maximum is.
Dialogue: 0,0:16:10.90,0:16:13.09,Default,,0000,0000,0000,,The maximum of\Nthe rate of change
Dialogue: 0,0:16:13.09,0:16:19.65,Default,,0000,0000,0000,,will be for-- this\Nequals f of x, y.
Dialogue: 0,0:16:19.65,0:16:29.61,Default,,0000,0000,0000,,The max of the rate of change\Nis-- what is that again?
Dialogue: 0,0:16:29.61,0:16:39.66,Default,,0000,0000,0000,,Magnitude of lambda f, in\Nthe direction nabla-- nabla f
Dialogue: 0,0:16:39.66,0:16:41.33,Default,,0000,0000,0000,,divided by its magnitude.
Dialogue: 0,0:16:41.33,0:16:43.98,Default,,0000,0000,0000,,
Dialogue: 0,0:16:43.98,0:16:46.29,Default,,0000,0000,0000,,This is what we discovered.
Dialogue: 0,0:16:46.29,0:16:48.46,Default,,0000,0000,0000,,And now I'm going\Nto ask you, what
Dialogue: 0,0:16:48.46,0:16:54.06,Default,,0000,0000,0000,,is the minimum rate of\Nchange at the same point?
Dialogue: 0,0:16:54.06,0:16:56.16,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:16:56.16,0:16:57.58,Default,,0000,0000,0000,,PROFESSOR: Just\Nparallel opposite.
Dialogue: 0,0:16:57.58,0:17:00.26,Default,,0000,0000,0000,,
Dialogue: 0,0:17:00.26,0:17:06.16,Default,,0000,0000,0000,,So it's gonna-- I'm gonna\Nhave the so-called highest--
Dialogue: 0,0:17:06.16,0:17:09.37,Default,,0000,0000,0000,,steepest, not highest,\Nhighest means maximum.
Dialogue: 0,0:17:09.37,0:17:10.46,Default,,0000,0000,0000,,The lowest value.
Dialogue: 0,0:17:10.46,0:17:12.77,Default,,0000,0000,0000,,So I'm going to have\Nthe lowest value, which
Dialogue: 0,0:17:12.77,0:17:14.68,Default,,0000,0000,0000,,indicates the steepest descent.
Dialogue: 0,0:17:14.68,0:17:20.48,Default,,0000,0000,0000,,Me going down on-- in the snow,\NI'm dreaming, on a sleigh,
Dialogue: 0,0:17:20.48,0:17:22.53,Default,,0000,0000,0000,,or on a plastic bag.
Dialogue: 0,0:17:22.53,0:17:26.67,Default,,0000,0000,0000,,That would give me\Nthe steepest descent,
Dialogue: 0,0:17:26.67,0:17:29.31,Default,,0000,0000,0000,,and the steepest descent\Nwill correspond to what?
Dialogue: 0,0:17:29.31,0:17:32.21,Default,,0000,0000,0000,,I'm going to make an NB.
Dialogue: 0,0:17:32.21,0:17:32.95,Default,,0000,0000,0000,,Nota bene.
Dialogue: 0,0:17:32.95,0:17:35.70,Default,,0000,0000,0000,,In Latin.
Dialogue: 0,0:17:35.70,0:17:45.90,Default,,0000,0000,0000,,Note the minimum will\Nbe minus magnitude
Dialogue: 0,0:17:45.90,0:17:51.54,Default,,0000,0000,0000,,of f, [INAUDIBLE] nabla f,\Nin the opposite direction.
Dialogue: 0,0:17:51.54,0:17:53.41,Default,,0000,0000,0000,,Shall I write it in words?
Dialogue: 0,0:17:53.41,0:17:56.58,Default,,0000,0000,0000,,Let me write it in as\NO-P-P from opposite-- no,
Dialogue: 0,0:17:56.58,0:17:58.58,Default,,0000,0000,0000,,from opposite--\Nopposite direction.
Dialogue: 0,0:17:58.58,0:18:01.99,Default,,0000,0000,0000,,
Dialogue: 0,0:18:01.99,0:18:03.63,Default,,0000,0000,0000,,What do I mean\Nopposite direction?
Dialogue: 0,0:18:03.63,0:18:06.49,Default,,0000,0000,0000,,Opposite direction\Nto the gradient.
Dialogue: 0,0:18:06.49,0:18:08.55,Default,,0000,0000,0000,,Which is the same direction,\Nif you think about,
Dialogue: 0,0:18:08.55,0:18:09.67,Default,,0000,0000,0000,,because it's the same line.
Dialogue: 0,0:18:09.67,0:18:16.46,Default,,0000,0000,0000,,So it's going to be minus nabla\Nf over magnitude of nabla f.
Dialogue: 0,0:18:16.46,0:18:19.37,Default,,0000,0000,0000,,It's like when we were\Nin [INAUDIBLE], which
Dialogue: 0,0:18:19.37,0:18:24.21,Default,,0000,0000,0000,,was 1 minus x squared minus\Ny squared, whatever it was--
Dialogue: 0,0:18:24.21,0:18:31.41,Default,,0000,0000,0000,,we had i plus j for the\Ndescent, and minus i minus
Dialogue: 0,0:18:31.41,0:18:35.11,Default,,0000,0000,0000,,j after the ascent--\Nthe steepest descent
Dialogue: 0,0:18:35.11,0:18:37.68,Default,,0000,0000,0000,,and the steepest ascent.
Dialogue: 0,0:18:37.68,0:18:40.21,Default,,0000,0000,0000,,We started with examples\Nbecause it's easier
Dialogue: 0,0:18:40.21,0:18:42.66,Default,,0000,0000,0000,,to understand\Nmathematics-- actually
Dialogue: 0,0:18:42.66,0:18:46.38,Default,,0000,0000,0000,,it's easier to understand\Nanything on an example.
Dialogue: 0,0:18:46.38,0:18:48.63,Default,,0000,0000,0000,,And then-- if the\Nexample is good.
Dialogue: 0,0:18:48.63,0:18:50.44,Default,,0000,0000,0000,,If the example is\Nbad, it's confusing.
Dialogue: 0,0:18:50.44,0:18:52.85,Default,,0000,0000,0000,,But if the example is\Ngood, you understand just
Dialogue: 0,0:18:52.85,0:18:56.44,Default,,0000,0000,0000,,about any concept, and then\Nyou move on to the theory,
Dialogue: 0,0:18:56.44,0:18:57.78,Default,,0000,0000,0000,,and this is the theory.
Dialogue: 0,0:18:57.78,0:18:59.03,Default,,0000,0000,0000,,And it looks very abstract.
Dialogue: 0,0:18:59.03,0:19:01.62,Default,,0000,0000,0000,,When somebody steps\Nin this classroom
Dialogue: 0,0:19:01.62,0:19:05.88,Default,,0000,0000,0000,,and they haven't taken more than\Ncalc 2 they will get scared,
Dialogue: 0,0:19:05.88,0:19:09.09,Default,,0000,0000,0000,,and they will never\Nwant to take calc 3.
Dialogue: 0,0:19:09.09,0:19:13.07,Default,,0000,0000,0000,,Well, that's why-- I didn't\Nwant to scare you off yet.
Dialogue: 0,0:19:13.07,0:19:19.60,Default,,0000,0000,0000,,OK, so this is what you have\Nto remember from section 11.6
Dialogue: 0,0:19:19.60,0:19:25.32,Default,,0000,0000,0000,,with 11.5 embedded in it.
Dialogue: 0,0:19:25.32,0:19:32.26,Default,,0000,0000,0000,,Now, one thing that\NI would like to see
Dialogue: 0,0:19:32.26,0:19:38.13,Default,,0000,0000,0000,,would be more examples and\Nconnections to other topics.
Dialogue: 0,0:19:38.13,0:19:42.19,Default,,0000,0000,0000,,So one example that I picked--\Nand I think it's a nice one.
Dialogue: 0,0:19:42.19,0:19:44.13,Default,,0000,0000,0000,,I just copied from the book.
Dialogue: 0,0:19:44.13,0:19:47.02,Default,,0000,0000,0000,,I usually don't\Nbring cheat sheets.
Dialogue: 0,0:19:47.02,0:19:52.38,Default,,0000,0000,0000,,I don't like professors who\Nbring books to the class
Dialogue: 0,0:19:52.38,0:19:55.17,Default,,0000,0000,0000,,and start reading\Nout of the book.
Dialogue: 0,0:19:55.17,0:19:56.25,Default,,0000,0000,0000,,I think that's ridiculous.
Dialogue: 0,0:19:56.25,0:20:00.37,Default,,0000,0000,0000,,I mean, as if you guys couldn't\Nread your own book at home.
Dialogue: 0,0:20:00.37,0:20:05.61,Default,,0000,0000,0000,,And I try to make up examples\Nthat are easier than the ones
Dialogue: 0,0:20:05.61,0:20:08.25,Default,,0000,0000,0000,,from the book to start with.
Dialogue: 0,0:20:08.25,0:20:13.07,Default,,0000,0000,0000,,But here's example 4, which\Nis not so easy to deal with,
Dialogue: 0,0:20:13.07,0:20:15.00,Default,,0000,0000,0000,,but it's not hard either.
Dialogue: 0,0:20:15.00,0:20:18.09,Default,,0000,0000,0000,,And I picked it because\NI saw this browsing
Dialogue: 0,0:20:18.09,0:20:21.89,Default,,0000,0000,0000,,through the previous\Nfinals, I saw it
Dialogue: 0,0:20:21.89,0:20:24.87,Default,,0000,0000,0000,,as a pattern coming\Nevery now and then.
Dialogue: 0,0:20:24.87,0:20:30.80,Default,,0000,0000,0000,,Find the direction in which\Nf increases or decreases
Dialogue: 0,0:20:30.80,0:20:32.28,Default,,0000,0000,0000,,most rapidly.
Dialogue: 0,0:20:32.28,0:20:34.30,Default,,0000,0000,0000,,I have to write\Ndown beautifully.
Dialogue: 0,0:20:34.30,0:20:52.51,Default,,0000,0000,0000,,Find the directions in which\Nf increases or decreases most
Dialogue: 0,0:20:52.51,0:21:01.10,Default,,0000,0000,0000,,rapidly at p0 coordinates 2, 1.
Dialogue: 0,0:21:01.10,0:21:05.02,Default,,0000,0000,0000,,
Dialogue: 0,0:21:05.02,0:21:11.24,Default,,0000,0000,0000,,And, what is the\Nmaximum-- that's
Dialogue: 0,0:21:11.24,0:21:20.84,Default,,0000,0000,0000,,a question-- what is the maximum\Nrate of change or of increase?
Dialogue: 0,0:21:20.84,0:21:24.27,Default,,0000,0000,0000,,This type of problem is also\Ncovered in the Khan Academy
Dialogue: 0,0:21:24.27,0:21:28.22,Default,,0000,0000,0000,,videos and also the MIT\Nlibrary, but I don't
Dialogue: 0,0:21:28.22,0:21:30.09,Default,,0000,0000,0000,,feel they do a very good job.
Dialogue: 0,0:21:30.09,0:21:31.67,Default,,0000,0000,0000,,They cover it just\Nlightly, as if they
Dialogue: 0,0:21:31.67,0:21:35.26,Default,,0000,0000,0000,,were afraid to speak too much\Nabout-- give too many examples
Dialogue: 0,0:21:35.26,0:21:37.73,Default,,0000,0000,0000,,and talk too much\Nabout the subject.
Dialogue: 0,0:21:37.73,0:21:42.56,Default,,0000,0000,0000,,So what do you guys want\Nto do with this one?
Dialogue: 0,0:21:42.56,0:21:44.10,Default,,0000,0000,0000,,Help me solve the problem.
Dialogue: 0,0:21:44.10,0:21:45.57,Default,,0000,0000,0000,,That was kind of the idea.
Dialogue: 0,0:21:45.57,0:21:49.25,Default,,0000,0000,0000,,So we start with\Ncomputing the what?
Dialogue: 0,0:21:49.25,0:21:52.73,Default,,0000,0000,0000,,The animal called--\Nwhat's the animal?
Dialogue: 0,0:21:52.73,0:21:53.72,Default,,0000,0000,0000,,STUDENT: Gradient.
Dialogue: 0,0:21:53.72,0:21:54.55,Default,,0000,0000,0000,,PROFESSOR: Gradient.
Dialogue: 0,0:21:54.55,0:21:55.72,Default,,0000,0000,0000,,Thank you very much.
Dialogue: 0,0:21:55.72,0:22:02.38,Default,,0000,0000,0000,,So we do that and we\Nstart differentiating.
Dialogue: 0,0:22:02.38,0:22:07.44,Default,,0000,0000,0000,,With respect to x, we get\Na product [INAUDIBLE].
Dialogue: 0,0:22:07.44,0:22:13.23,Default,,0000,0000,0000,,So your product [INAUDIBLE] 1--\Nit's differentiated-- times e
Dialogue: 0,0:22:13.23,0:22:18.77,Default,,0000,0000,0000,,to the 2y minus x [INAUDIBLE]\Nplus x undifferentiated.
Dialogue: 0,0:22:18.77,0:22:21.55,Default,,0000,0000,0000,,The second guy, prime.
Dialogue: 0,0:22:21.55,0:22:23.72,Default,,0000,0000,0000,,Copy and paste, Magdalena.
Dialogue: 0,0:22:23.72,0:22:25.51,Default,,0000,0000,0000,,Times-- don't\Nforget the minus 1,
Dialogue: 0,0:22:25.51,0:22:28.90,Default,,0000,0000,0000,,because-- I'm talking to myself.
Dialogue: 0,0:22:28.90,0:22:33.62,Default,,0000,0000,0000,,Because if you do, you get\Na 0 on this in the final.
Dialogue: 0,0:22:33.62,0:22:34.96,Default,,0000,0000,0000,,I'm talking to myself.
Dialogue: 0,0:22:34.96,0:22:35.97,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:22:35.97,0:22:36.47,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:22:36.47,0:22:42.64,Default,,0000,0000,0000,,Plus, parenthesis-- the same\Nprocedure with respect to y.
Dialogue: 0,0:22:42.64,0:22:45.34,Default,,0000,0000,0000,,When I do it with\Nrespect to y, this
Dialogue: 0,0:22:45.34,0:22:47.34,Default,,0000,0000,0000,,is good review of\Nthe whole chapter.
Dialogue: 0,0:22:47.34,0:22:48.10,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:22:48.10,0:22:48.97,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:22:48.97,0:22:58.17,Default,,0000,0000,0000,,
Dialogue: 0,0:22:58.17,0:23:00.74,Default,,0000,0000,0000,,PROFESSOR: This primed\Nwith respect to x.
Dialogue: 0,0:23:00.74,0:23:02.46,Default,,0000,0000,0000,,Am I doing something wrong?
Dialogue: 0,0:23:02.46,0:23:04.18,Default,,0000,0000,0000,,No.
Dialogue: 0,0:23:04.18,0:23:05.18,Default,,0000,0000,0000,,Are you with me?
Dialogue: 0,0:23:05.18,0:23:07.91,Default,,0000,0000,0000,,So this guy primed\Nwith respect to x, I'm
Dialogue: 0,0:23:07.91,0:23:13.35,Default,,0000,0000,0000,,going to write it es copied.
Dialogue: 0,0:23:13.35,0:23:17.81,Default,,0000,0000,0000,,And take that out, you maintain\Nand differentiate with respect
Dialogue: 0,0:23:17.81,0:23:18.96,Default,,0000,0000,0000,,to x.
Dialogue: 0,0:23:18.96,0:23:22.49,Default,,0000,0000,0000,,So I did it right.
Dialogue: 0,0:23:22.49,0:23:29.17,Default,,0000,0000,0000,,OK, the second one, x is a\Nconstant for me right now.
Dialogue: 0,0:23:29.17,0:23:31.33,Default,,0000,0000,0000,,Who is the variable y?
Dialogue: 0,0:23:31.33,0:23:36.65,Default,,0000,0000,0000,,So I copy and paste e\Nto the whole argument
Dialogue: 0,0:23:36.65,0:23:39.68,Default,,0000,0000,0000,,times-- I cover everything\Nelse with my hand,
Dialogue: 0,0:23:39.68,0:23:42.95,Default,,0000,0000,0000,,and I differentiate the\Nargument with respect to y.
Dialogue: 0,0:23:42.95,0:23:47.15,Default,,0000,0000,0000,,And I get prime sub 2\Nand a j, and I say, thank
Dialogue: 0,0:23:47.15,0:23:48.77,Default,,0000,0000,0000,,god, this was a little bit long.
Dialogue: 0,0:23:48.77,0:23:53.00,Default,,0000,0000,0000,,You realize that if you make\Nthe slightest algebra mistake,
Dialogue: 0,0:23:53.00,0:23:55.01,Default,,0000,0000,0000,,it's all over for you.
Dialogue: 0,0:23:55.01,0:23:57.92,Default,,0000,0000,0000,,In that case, I\Nask my colleagues,
Dialogue: 0,0:23:57.92,0:24:02.61,Default,,0000,0000,0000,,what do you guys do when guy\Nmissed that or missed this?
Dialogue: 0,0:24:02.61,0:24:09.12,Default,,0000,0000,0000,,0, 0, no points-- OK,\Nmaybe a little bit,
Dialogue: 0,0:24:09.12,0:24:12.17,Default,,0000,0000,0000,,maybe a tiny bit\Nof extra credit.
Dialogue: 0,0:24:12.17,0:24:14.30,Default,,0000,0000,0000,,But pay attention to your math.
Dialogue: 0,0:24:14.30,0:24:17.81,Default,,0000,0000,0000,,So you know what you need to do.
Dialogue: 0,0:24:17.81,0:24:24.10,Default,,0000,0000,0000,,Now I'm going to go on and\Nsay, but I am at the point 0.
Dialogue: 0,0:24:24.10,0:24:27.96,Default,,0000,0000,0000,,By the way, I really don't\Nlike what we did in the book.
Dialogue: 0,0:24:27.96,0:24:31.78,Default,,0000,0000,0000,,OK, I should not say that\Nout loud, but it's too late.
Dialogue: 0,0:24:31.78,0:24:36.51,Default,,0000,0000,0000,,The book denotes that\Nsometimes, well, we
Dialogue: 0,0:24:36.51,0:24:41.04,Default,,0000,0000,0000,,try not to do that too\Noften, but not by F sub 0.
Dialogue: 0,0:24:41.04,0:24:44.04,Default,,0000,0000,0000,,Because some other\Nbooks use that.
Dialogue: 0,0:24:44.04,0:24:45.19,Default,,0000,0000,0000,,I don't like that.
Dialogue: 0,0:24:45.19,0:24:48.96,Default,,0000,0000,0000,,So every time-- you\Nshould never do that.
Dialogue: 0,0:24:48.96,0:24:50.69,Default,,0000,0000,0000,,Because it gives\Nthe feeling that
Dialogue: 0,0:24:50.69,0:24:54.47,Default,,0000,0000,0000,,you're differentiating\Na constant or something.
Dialogue: 0,0:24:54.47,0:24:59.18,Default,,0000,0000,0000,,OK, so I always try to\Nsay, gradient [? up ?]
Dialogue: 0,0:24:59.18,0:25:02.82,Default,,0000,0000,0000,,is-- which means, I\Nhave a fixed value.
Dialogue: 0,0:25:02.82,0:25:05.55,Default,,0000,0000,0000,,But I don't fix the value\Nbefore I took the gradient.
Dialogue: 0,0:25:05.55,0:25:07.75,Default,,0000,0000,0000,,This is too confusing\Nas a notation.
Dialogue: 0,0:25:07.75,0:25:09.56,Default,,0000,0000,0000,,Don't do it.
Dialogue: 0,0:25:09.56,0:25:10.98,Default,,0000,0000,0000,,Close your eyes\Nwhen you get to it
Dialogue: 0,0:25:10.98,0:25:12.19,Default,,0000,0000,0000,,when you're reading the book.
Dialogue: 0,0:25:12.19,0:25:20.76,Default,,0000,0000,0000,,OK, now I have to plug\Nin instead of x sub 2--
Dialogue: 0,0:25:20.76,0:25:25.15,Default,,0000,0000,0000,,I tried to remember\Nthat-- y equals 1.
Dialogue: 0,0:25:25.15,0:25:29.21,Default,,0000,0000,0000,,So I go 1 times e to\Nthe 2 times 1 minus 2
Dialogue: 0,0:25:29.21,0:25:32.66,Default,,0000,0000,0000,,plus 2 times e to\Nthe 2 times 1 minus--
Dialogue: 0,0:25:32.66,0:25:38.75,Default,,0000,0000,0000,,I wish I had Data with me, I\Nmean the guy from Star Trek.
Dialogue: 0,0:25:38.75,0:25:41.52,Default,,0000,0000,0000,,Because he could do this in\Njust a fraction of a second
Dialogue: 0,0:25:41.52,0:25:46.20,Default,,0000,0000,0000,,without me having to bother\Nwith this whole thing.
Dialogue: 0,0:25:46.20,0:25:51.04,Default,,0000,0000,0000,,
Dialogue: 0,0:25:51.04,0:25:52.51,Default,,0000,0000,0000,,STUDENT: But then\Nif Data existed,
Dialogue: 0,0:25:52.51,0:25:53.68,Default,,0000,0000,0000,,why would we be math majors?
Dialogue: 0,0:25:53.68,0:25:57.17,Default,,0000,0000,0000,,PROFESSOR: Exactly,\Nso we do this
Dialogue: 0,0:25:57.17,0:26:01.83,Default,,0000,0000,0000,,so that we can program\Npeople, I mean androids,
Dialogue: 0,0:26:01.83,0:26:06.29,Default,,0000,0000,0000,,and eventually learn\Nhow to clone ourselves.
Dialogue: 0,0:26:06.29,0:26:09.27,Default,,0000,0000,0000,,So let's see what we have.
Dialogue: 0,0:26:09.27,0:26:11.70,Default,,0000,0000,0000,,e to the 0 is 1.
Dialogue: 0,0:26:11.70,0:26:16.38,Default,,0000,0000,0000,,e to the 0 is 1, 1 minus\N2-- are you guys with me?
Dialogue: 0,0:26:16.38,0:26:17.94,Default,,0000,0000,0000,,I'm going too fast?
Dialogue: 0,0:26:17.94,0:26:19.32,Default,,0000,0000,0000,,STUDENT: No, the\N[? board ?] says
Dialogue: 0,0:26:19.32,0:26:23.80,Default,,0000,0000,0000,,the y component is equal to\N1, x component is equal to 2.
Dialogue: 0,0:26:23.80,0:26:27.70,Default,,0000,0000,0000,,PROFESSOR: x\Ncomponent equals to 2.
Dialogue: 0,0:26:27.70,0:26:31.03,Default,,0000,0000,0000,,x0 is 2, and y0 is 1.
Dialogue: 0,0:26:31.03,0:26:38.74,Default,,0000,0000,0000,,And I plug in x0\Nequals [INAUDIBLE].
Dialogue: 0,0:26:38.74,0:26:44.82,Default,,0000,0000,0000,,So 2 times e to the 2 times 1\Nminus 2-- you think I like it?
Dialogue: 0,0:26:44.82,0:26:45.77,Default,,0000,0000,0000,,I don't like it.
Dialogue: 0,0:26:45.77,0:26:47.30,Default,,0000,0000,0000,,But anyway, it's my life.
Dialogue: 0,0:26:47.30,0:26:48.96,Default,,0000,0000,0000,,I have to go on.
Dialogue: 0,0:26:48.96,0:26:54.25,Default,,0000,0000,0000,,So we have 1 minus 2\Nplus minus 1, right?
Dialogue: 0,0:26:54.25,0:26:59.97,Default,,0000,0000,0000,,Minus 1i-- minus\N1i sounds scary.
Dialogue: 0,0:26:59.97,0:27:09.25,Default,,0000,0000,0000,,OK, plus 4j-- minus i plus 4j.
Dialogue: 0,0:27:09.25,0:27:10.21,Default,,0000,0000,0000,,Is it lovely?
Dialogue: 0,0:27:10.21,0:27:11.74,Default,,0000,0000,0000,,No, I hate it.
Dialogue: 0,0:27:11.74,0:27:16.30,Default,,0000,0000,0000,,The magnitude is going\Nto be square root of 17.
Dialogue: 0,0:27:16.30,0:27:17.16,Default,,0000,0000,0000,,But that's life.
Dialogue: 0,0:27:17.16,0:27:20.67,Default,,0000,0000,0000,,I mean, you as\Nengineering majors
Dialogue: 0,0:27:20.67,0:27:23.28,Default,,0000,0000,0000,,see that all the time,\Nand even worse than that.
Dialogue: 0,0:27:23.28,0:27:30.11,Default,,0000,0000,0000,,So I'm going to say the\Ngradient of F at P0 in magnitude
Dialogue: 0,0:27:30.11,0:27:34.33,Default,,0000,0000,0000,,will be square root of 17.
Dialogue: 0,0:27:34.33,0:27:36.14,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:27:36.14,0:27:40.62,Default,,0000,0000,0000,,That is the maximum rate\Nof change, right guys?
Dialogue: 0,0:27:40.62,0:27:44.32,Default,,0000,0000,0000,,But in which direction\Ndoes that happen?
Dialogue: 0,0:27:44.32,0:27:47.66,Default,,0000,0000,0000,,
Dialogue: 0,0:27:47.66,0:27:53.72,Default,,0000,0000,0000,,My beloved book says, in the\Ndirection of minus i plus 4j.
Dialogue: 0,0:27:53.72,0:27:58.13,Default,,0000,0000,0000,,That's our answer, so\Nin the direction of--
Dialogue: 0,0:27:58.13,0:27:59.62,Default,,0000,0000,0000,,STUDENT: Over root 17.
Dialogue: 0,0:27:59.62,0:28:01.61,Default,,0000,0000,0000,,PROFESSOR: No, let\Nme tell you what.
Dialogue: 0,0:28:01.61,0:28:04.87,Default,,0000,0000,0000,,We fought about that as the\Nauthors when we wrote the book.
Dialogue: 0,0:28:04.87,0:28:09.15,Default,,0000,0000,0000,,So I said, if you're\Nsaying, in the direction of,
Dialogue: 0,0:28:09.15,0:28:11.40,Default,,0000,0000,0000,,then you have to\Nsay, over root 17.
Dialogue: 0,0:28:11.40,0:28:14.04,Default,,0000,0000,0000,,And my coauthor said,\Nno, actually, Magdalena,
Dialogue: 0,0:28:14.04,0:28:16.52,Default,,0000,0000,0000,,as a matter of English--\Nsince you're not a native,
Dialogue: 0,0:28:16.52,0:28:18.63,Default,,0000,0000,0000,,you don't understand.
Dialogue: 0,0:28:18.63,0:28:22.21,Default,,0000,0000,0000,,In the direction\Nof a certain vector
Dialogue: 0,0:28:22.21,0:28:25.05,Default,,0000,0000,0000,,means the direction\Nof a certain vector,
Dialogue: 0,0:28:25.05,0:28:27.52,Default,,0000,0000,0000,,the direction could be that.
Dialogue: 0,0:28:27.52,0:28:35.54,Default,,0000,0000,0000,,But I say equivalently, in\Nthe direction double dot
Dialogue: 0,0:28:35.54,0:28:39.44,Default,,0000,0000,0000,,minus i plus 4j over\Nsquare root of 17,
Dialogue: 0,0:28:39.44,0:28:43.36,Default,,0000,0000,0000,,meaning that this\Nis the direction.
Dialogue: 0,0:28:43.36,0:28:44.74,Default,,0000,0000,0000,,It's a matter of interpretation.
Dialogue: 0,0:28:44.74,0:28:46.80,Default,,0000,0000,0000,,I don't understand it,\Nbut it's your language.
Dialogue: 0,0:28:46.80,0:28:47.39,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:28:47.39,0:28:48.26,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:28:48.26,0:28:53.29,Default,,0000,0000,0000,,
Dialogue: 0,0:28:53.29,0:28:54.97,Default,,0000,0000,0000,,PROFESSOR: I'm saying--
Dialogue: 0,0:28:54.97,0:28:56.26,Default,,0000,0000,0000,,STUDENT: Which one do you like?
Dialogue: 0,0:28:56.26,0:28:57.11,Default,,0000,0000,0000,,PROFESSOR: Which one do I like?
Dialogue: 0,0:28:57.11,0:28:57.91,Default,,0000,0000,0000,,This one.
Dialogue: 0,0:28:57.91,0:28:59.31,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:28:59.31,0:29:00.82,Default,,0000,0000,0000,,PROFESSOR: Thank you.
Dialogue: 0,0:29:00.82,0:29:04.12,Default,,0000,0000,0000,,So if we define direction\Nto be a unit vector,
Dialogue: 0,0:29:04.12,0:29:08.58,Default,,0000,0000,0000,,let's be consistent and not say,\Nof me, of you, of my cousin,
Dialogue: 0,0:29:08.58,0:29:11.06,Default,,0000,0000,0000,,of whatever.
Dialogue: 0,0:29:11.06,0:29:14.93,Default,,0000,0000,0000,,All right, was this hard?
Dialogue: 0,0:29:14.93,0:29:17.35,Default,,0000,0000,0000,,No.
Dialogue: 0,0:29:17.35,0:29:21.89,Default,,0000,0000,0000,,Do I have a caveat about\Nthis kind of problem?
Dialogue: 0,0:29:21.89,0:29:23.85,Default,,0000,0000,0000,,Yeah, that was kind of the idea.
Dialogue: 0,0:29:23.85,0:29:26.47,Default,,0000,0000,0000,,If I put that in\Nthe midterm, guys,
Dialogue: 0,0:29:26.47,0:29:28.93,Default,,0000,0000,0000,,please do it two or\Nthree times, make
Dialogue: 0,0:29:28.93,0:29:31.14,Default,,0000,0000,0000,,sure you didn't make\Nany algebra mistakes.
Dialogue: 0,0:29:31.14,0:29:34.72,Default,,0000,0000,0000,,Because if you know the\Ntheory, I will still
Dialogue: 0,0:29:34.72,0:29:36.72,Default,,0000,0000,0000,,give you like 30% or something.
Dialogue: 0,0:29:36.72,0:29:38.94,Default,,0000,0000,0000,,But if you mess up\Nwith the algebra,
Dialogue: 0,0:29:38.94,0:29:45.08,Default,,0000,0000,0000,,I have no choice but giving\Nup the 70%, whatever that is.
Dialogue: 0,0:29:45.08,0:29:49.06,Default,,0000,0000,0000,,I try to be fair and give\Nyou something for everything
Dialogue: 0,0:29:49.06,0:29:50.82,Default,,0000,0000,0000,,you do and know.
Dialogue: 0,0:29:50.82,0:29:52.89,Default,,0000,0000,0000,,But try not to mess\Nit up too badly,
Dialogue: 0,0:29:52.89,0:29:54.54,Default,,0000,0000,0000,,because it's very\Neasy to mess it up.
Dialogue: 0,0:29:54.54,0:29:55.86,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:29:55.86,0:29:58.06,Default,,0000,0000,0000,,STUDENT: So which\Nway is increasing?
Dialogue: 0,0:29:58.06,0:29:59.89,Default,,0000,0000,0000,,PROFESSOR: OK,\Nfind the direction
Dialogue: 0,0:29:59.89,0:30:03.75,Default,,0000,0000,0000,,in which r increases or\Ndecreases most rapidly.
Dialogue: 0,0:30:03.75,0:30:08.11,Default,,0000,0000,0000,,OK, the direction in\Nwhich I could draw it,
Dialogue: 0,0:30:08.11,0:30:12.26,Default,,0000,0000,0000,,this is increasing in\Nthe direction of that.
Dialogue: 0,0:30:12.26,0:30:13.27,Default,,0000,0000,0000,,At what rate?
Dialogue: 0,0:30:13.27,0:30:15.27,Default,,0000,0000,0000,,At the rate of\Nsquare root of 17.
Dialogue: 0,0:30:15.27,0:30:17.53,Default,,0000,0000,0000,,Good question.
Dialogue: 0,0:30:17.53,0:30:19.53,Default,,0000,0000,0000,,How about the other one?
Dialogue: 0,0:30:19.53,0:30:24.43,Default,,0000,0000,0000,,In the direction of plus i\Nminus 4j over square root of 17,
Dialogue: 0,0:30:24.43,0:30:29.48,Default,,0000,0000,0000,,I get the rate of change\Nminus square root of 17.
Dialogue: 0,0:30:29.48,0:30:30.22,Default,,0000,0000,0000,,And that's it.
Dialogue: 0,0:30:30.22,0:30:31.100,Default,,0000,0000,0000,,Do we have to say that?
Dialogue: 0,0:30:31.100,0:30:35.57,Default,,0000,0000,0000,,Ehh, I give you extra\Ncredit if you say that.
Dialogue: 0,0:30:35.57,0:30:38.72,Default,,0000,0000,0000,,But at this point, I'm saying\NI'm happy with what I just
Dialogue: 0,0:30:38.72,0:30:40.76,Default,,0000,0000,0000,,wrote on the board.
Dialogue: 0,0:30:40.76,0:30:41.64,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:30:41.64,0:30:46.45,Default,,0000,0000,0000,,
Dialogue: 0,0:30:46.45,0:30:49.11,Default,,0000,0000,0000,,PROFESSOR: Yeah, so\Nit's like finding
Dialogue: 0,0:30:49.11,0:30:52.71,Default,,0000,0000,0000,,on which path are you\Ngoing to get to the top
Dialogue: 0,0:30:52.71,0:30:56.30,Default,,0000,0000,0000,,the fastest when you\Nclimb a mountain.
Dialogue: 0,0:30:56.30,0:30:58.62,Default,,0000,0000,0000,,It's the same kind of question.
Dialogue: 0,0:30:58.62,0:31:00.81,Default,,0000,0000,0000,,Because it's all about\Nz being an altitude.
Dialogue: 0,0:31:00.81,0:31:03.31,Default,,0000,0000,0000,,z equals F of x.
Dialogue: 0,0:31:03.31,0:31:07.25,Default,,0000,0000,0000,,I will go ahead and\Nerase the whole thing.
Dialogue: 0,0:31:07.25,0:31:15.65,Default,,0000,0000,0000,,And I wish you good luck now.
Dialogue: 0,0:31:15.65,0:31:21.53,Default,,0000,0000,0000,,I wish you good luck because\NI was asked by [INAUDIBLE]
Dialogue: 0,0:31:21.53,0:31:28.54,Default,,0000,0000,0000,,to solve a problem like the one\Nyou gave me in the web work,
Dialogue: 0,0:31:28.54,0:31:30.34,Default,,0000,0000,0000,,but I don't remember it.
Dialogue: 0,0:31:30.34,0:31:31.66,Default,,0000,0000,0000,,But it's OK.
Dialogue: 0,0:31:31.66,0:31:35.78,Default,,0000,0000,0000,,it was a review of\Nwhat we did last time.
Dialogue: 0,0:31:35.78,0:31:43.57,Default,,0000,0000,0000,,And we said, instead of that,\Nwhere the tangent plane--
Dialogue: 0,0:31:43.57,0:31:44.70,Default,,0000,0000,0000,,we know what that is.
Dialogue: 0,0:31:44.70,0:31:49.88,Default,,0000,0000,0000,,At P0 was-- guys, by the\Nfinal I want this memorized.
Dialogue: 0,0:31:49.88,0:31:51.48,Default,,0000,0000,0000,,There is no question.
Dialogue: 0,0:31:51.48,0:31:55.77,Default,,0000,0000,0000,,So z minus z0 is\Nlike Taylor's formula
Dialogue: 0,0:31:55.77,0:31:59.01,Default,,0000,0000,0000,,in the linear approximation.
Dialogue: 0,0:31:59.01,0:31:59.83,Default,,0000,0000,0000,,You truncate.
Dialogue: 0,0:31:59.83,0:32:02.04,Default,,0000,0000,0000,,You throw away the\Nsecond order, and so on.
Dialogue: 0,0:32:02.04,0:32:03.74,Default,,0000,0000,0000,,So what is that?
Dialogue: 0,0:32:03.74,0:32:06.46,Default,,0000,0000,0000,,f sub x at delta x.
Dialogue: 0,0:32:06.46,0:32:07.67,Default,,0000,0000,0000,,Oh, x equals 0.
Dialogue: 0,0:32:07.67,0:32:08.100,Default,,0000,0000,0000,,I'm lazy.
Dialogue: 0,0:32:08.100,0:32:12.67,Default,,0000,0000,0000,,Times x minus x0--\Nthis is the delta x.
Dialogue: 0,0:32:12.67,0:32:16.81,Default,,0000,0000,0000,,This is the delta z.
Dialogue: 0,0:32:16.81,0:32:21.47,Default,,0000,0000,0000,,And this is not the round\Nsurface, curvy and everything.
Dialogue: 0,0:32:21.47,0:32:28.67,Default,,0000,0000,0000,,This is the plane approximation,\Nthe planar approximation--
Dialogue: 0,0:32:28.67,0:32:39.11,Default,,0000,0000,0000,,A-P-P-R. I'm not done--\Nplus f sub y y minus y0.
Dialogue: 0,0:32:39.11,0:32:49.96,Default,,0000,0000,0000,,So this is the equation of the\Nplane that's tangent pi at 0.
Dialogue: 0,0:32:49.96,0:32:53.38,Default,,0000,0000,0000,,And let me draw\Nthe surface pink.
Dialogue: 0,0:32:53.38,0:33:00.90,Default,,0000,0000,0000,,Because I'm a girl, and because\NI want to draw this in pink.
Dialogue: 0,0:33:00.90,0:33:06.54,Default,,0000,0000,0000,,Let's call it some\NS, for Surface.
Dialogue: 0,0:33:06.54,0:33:09.72,Default,,0000,0000,0000,,Can we paint an S here?
Dialogue: 0,0:33:09.72,0:33:18.16,Default,,0000,0000,0000,,OK, but if I'm giving\Nthe same picture
Dialogue: 0,0:33:18.16,0:33:24.02,Default,,0000,0000,0000,,for a different equation--\Nso I have F of x, y, z
Dialogue: 0,0:33:24.02,0:33:27.39,Default,,0000,0000,0000,,equals constant, that's\Nthe implicit form.
Dialogue: 0,0:33:27.39,0:33:30.78,Default,,0000,0000,0000,,
Dialogue: 0,0:33:30.78,0:33:31.99,Default,,0000,0000,0000,,I make this face.
Dialogue: 0,0:33:31.99,0:33:33.40,Default,,0000,0000,0000,,Why do I make this face?
Dialogue: 0,0:33:33.40,0:33:36.38,Default,,0000,0000,0000,,Because I've got\Nthree confessions.
Dialogue: 0,0:33:36.38,0:33:40.33,Default,,0000,0000,0000,,I'm more like a\Npriestess in mathematics.
Dialogue: 0,0:33:40.33,0:33:43.79,Default,,0000,0000,0000,,People don't like\Nimplicit differentiation.
Dialogue: 0,0:33:43.79,0:33:46.22,Default,,0000,0000,0000,,We will do a little\Nbit of that today,
Dialogue: 0,0:33:46.22,0:33:50.28,Default,,0000,0000,0000,,because you told me your stories\Nfrom implicit differentiation,
Dialogue: 0,0:33:50.28,0:33:51.66,Default,,0000,0000,0000,,and I got scared.
Dialogue: 0,0:33:51.66,0:33:53.91,Default,,0000,0000,0000,,Those were horror stories.
Dialogue: 0,0:33:53.91,0:33:56.50,Default,,0000,0000,0000,,And I don't want those to repeat\Nin the final or the midterm.
Dialogue: 0,0:33:56.50,0:33:59.29,Default,,0000,0000,0000,,So I'm going to do something\Nwith implicit differentiation
Dialogue: 0,0:33:59.29,0:34:00.96,Default,,0000,0000,0000,,as well.
Dialogue: 0,0:34:00.96,0:34:03.59,Default,,0000,0000,0000,,What did I want to say?
Dialogue: 0,0:34:03.59,0:34:07.41,Default,,0000,0000,0000,,If we apply this, we'd be wrong.
Dialogue: 0,0:34:07.41,0:34:11.10,Default,,0000,0000,0000,,But we have to remember\Nwhat the normal would be.
Dialogue: 0,0:34:11.10,0:34:16.90,Default,,0000,0000,0000,,And the normal was\Nthe gradient of big F.
Dialogue: 0,0:34:16.90,0:34:24.36,Default,,0000,0000,0000,,So those will be F sub xi plus\NF sub yj plus big F sub zk.
Dialogue: 0,0:34:24.36,0:34:29.38,Default,,0000,0000,0000,,Then force, such a\Nsurface, even implicitly,
Dialogue: 0,0:34:29.38,0:34:32.14,Default,,0000,0000,0000,,the tangent plane looks\Na little bit differently.
Dialogue: 0,0:34:32.14,0:34:33.79,Default,,0000,0000,0000,,But it's the same story.
Dialogue: 0,0:34:33.79,0:34:35.35,Default,,0000,0000,0000,,And I prove that.
Dialogue: 0,0:34:35.35,0:34:37.84,Default,,0000,0000,0000,,You may not believe it,\Nor may not remember.
Dialogue: 0,0:34:37.84,0:34:42.71,Default,,0000,0000,0000,,But I proved that, that\Nit's one and the same thing.
Dialogue: 0,0:34:42.71,0:34:44.17,Default,,0000,0000,0000,,So I get F sub x.
Dialogue: 0,0:34:44.17,0:34:46.62,Default,,0000,0000,0000,,I get 0i0.
Dialogue: 0,0:34:46.62,0:34:51.74,Default,,0000,0000,0000,,This was the A dot,\Ntimes x minus x plus.
Dialogue: 0,0:34:51.74,0:34:53.27,Default,,0000,0000,0000,,What's next?
Dialogue: 0,0:34:53.27,0:34:57.28,Default,,0000,0000,0000,,From the coefficients\Ncoming from the gradient--
Dialogue: 0,0:34:57.28,0:34:59.74,Default,,0000,0000,0000,,that was the gradient.
Dialogue: 0,0:34:59.74,0:35:02.38,Default,,0000,0000,0000,,
Dialogue: 0,0:35:02.38,0:35:06.13,Default,,0000,0000,0000,,So this is F of\Nx, y, z equals C.
Dialogue: 0,0:35:06.13,0:35:12.95,Default,,0000,0000,0000,,And the gradient was-- this is\Nthe normal, F sub x, F sub y,
Dialogue: 0,0:35:12.95,0:35:20.95,Default,,0000,0000,0000,,F sub z angular bracket\Nequals gradient of F, which
Dialogue: 0,0:35:20.95,0:35:23.09,Default,,0000,0000,0000,,is more or less than normal.
Dialogue: 0,0:35:23.09,0:35:31.48,Default,,0000,0000,0000,,The unit normal will be just\Ngradient of F over length of F.
Dialogue: 0,0:35:31.48,0:35:40.22,Default,,0000,0000,0000,,So let's continue-- F sub\Ny at x0y0 times delta y.
Dialogue: 0,0:35:40.22,0:35:47.93,Default,,0000,0000,0000,,This is B. I had the problem\Nwith memorizing these,
Dialogue: 0,0:35:47.93,0:35:52.43,Default,,0000,0000,0000,,especially since when I was\N18 I did not understand them
Dialogue: 0,0:35:52.43,0:35:55.57,Default,,0000,0000,0000,,whatsoever first\Nyear of college.
Dialogue: 0,0:35:55.57,0:36:00.52,Default,,0000,0000,0000,,I had to use markers,\Nput them in markers
Dialogue: 0,0:36:00.52,0:36:02.62,Default,,0000,0000,0000,,and glue them to my closet.
Dialogue: 0,0:36:02.62,0:36:04.45,Default,,0000,0000,0000,,Because when I\Nwas 18, of course,
Dialogue: 0,0:36:04.45,0:36:06.57,Default,,0000,0000,0000,,I was looking in the\Nmirror all the time.
Dialogue: 0,0:36:06.57,0:36:08.12,Default,,0000,0000,0000,,So whenever I got\Ninto the closet,
Dialogue: 0,0:36:08.12,0:36:11.86,Default,,0000,0000,0000,,opened the door to the\Nmirror, next to the mirror
Dialogue: 0,0:36:11.86,0:36:14.26,Default,,0000,0000,0000,,there was this formula.
Dialogue: 0,0:36:14.26,0:36:19.99,Default,,0000,0000,0000,,So whether I liked it or not--\NI didn't-- I memorized it just
Dialogue: 0,0:36:19.99,0:36:23.32,Default,,0000,0000,0000,,by seeing it every day\Nwhen I opened the door.
Dialogue: 0,0:36:23.32,0:36:23.82,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:36:23.82,0:36:25.66,Default,,0000,0000,0000,,STUDENT: That green one, should\Nthat be x minus 0 or z minus 0?
Dialogue: 0,0:36:25.66,0:36:26.53,Default,,0000,0000,0000,,PROFESSOR: z minus 0.
Dialogue: 0,0:36:26.53,0:36:28.00,Default,,0000,0000,0000,,That is my mistake.
Dialogue: 0,0:36:28.00,0:36:29.53,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:36:29.53,0:36:35.37,Default,,0000,0000,0000,,So again, you have delta\Nx, delta y, delta z.
Dialogue: 0,0:36:35.37,0:36:37.36,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:36:37.36,0:36:38.96,Default,,0000,0000,0000,,Is it hard to memorize?
Dialogue: 0,0:36:38.96,0:36:41.47,Default,,0000,0000,0000,,No, not if you\Nput it in markers.
Dialogue: 0,0:36:41.47,0:36:45.36,Default,,0000,0000,0000,,I think you will do just fine.
Dialogue: 0,0:36:45.36,0:36:47.58,Default,,0000,0000,0000,,What have we done as time?
Dialogue: 0,0:36:47.58,0:36:49.86,Default,,0000,0000,0000,,Let me review it really quickly.
Dialogue: 0,0:36:49.86,0:36:51.26,Default,,0000,0000,0000,,We said, wait a minute.
Dialogue: 0,0:36:51.26,0:36:53.50,Default,,0000,0000,0000,,How come they are\None and the same?
Dialogue: 0,0:36:53.50,0:36:56.19,Default,,0000,0000,0000,,You said, oh I'm\Ngetting a headache.
Dialogue: 0,0:36:56.19,0:37:00.06,Default,,0000,0000,0000,,I don't understand why\Nthey are one and the same.
Dialogue: 0,0:37:00.06,0:37:06.67,Default,,0000,0000,0000,,And we said, yes, but you\Nsee, this guy is nothing but z
Dialogue: 0,0:37:06.67,0:37:09.59,Default,,0000,0000,0000,,minus F of x, y.
Dialogue: 0,0:37:09.59,0:37:14.87,Default,,0000,0000,0000,,So from this form, you can make\Nit implicit it by pulling out F
Dialogue: 0,0:37:14.87,0:37:24.75,Default,,0000,0000,0000,,to the left and creating this\Nbig F of x, y, and z equals 0.
Dialogue: 0,0:37:24.75,0:37:26.34,Default,,0000,0000,0000,,And what does it mean?
Dialogue: 0,0:37:26.34,0:37:31.41,Default,,0000,0000,0000,,It means that F sub x will be\Noh my god, this has nothing
Dialogue: 0,0:37:31.41,0:37:35.64,Default,,0000,0000,0000,,to do with us minus F sub x.
Dialogue: 0,0:37:35.64,0:37:39.77,Default,,0000,0000,0000,,F sub y will be minus F sub y.
Dialogue: 0,0:37:39.77,0:37:41.15,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,0:37:41.15,0:37:47.80,Default,,0000,0000,0000,,And F sub z, big F sub z, is\Nsimply-- there's no z here-- 1.
Dialogue: 0,0:37:47.80,0:37:56.24,Default,,0000,0000,0000,,So coming back to the guide's\NA, B, C, forget about A,
Dialogue: 0,0:37:56.24,0:38:00.24,Default,,0000,0000,0000,,B, C. I'll take the A, and\NI'll replace it with minus F
Dialogue: 0,0:38:00.24,0:38:03.81,Default,,0000,0000,0000,,sub x, which doesn't write.
Dialogue: 0,0:38:03.81,0:38:08.82,Default,,0000,0000,0000,,And it's time for him to\Ngo away-- minus F sub x.
Dialogue: 0,0:38:08.82,0:38:14.33,Default,,0000,0000,0000,,And this is F sub\Ny with the minus.
Dialogue: 0,0:38:14.33,0:38:18.16,Default,,0000,0000,0000,,
Dialogue: 0,0:38:18.16,0:38:21.53,Default,,0000,0000,0000,,And finally, Mr. C,\Nwho is happy, he is 1.
Dialogue: 0,0:38:21.53,0:38:23.98,Default,,0000,0000,0000,,So he says, I'm happy.
Dialogue: 0,0:38:23.98,0:38:26.54,Default,,0000,0000,0000,,You're going to separate me.
Dialogue: 0,0:38:26.54,0:38:29.00,Default,,0000,0000,0000,,We are going to separate him.
Dialogue: 0,0:38:29.00,0:38:34.40,Default,,0000,0000,0000,,So this equation is\Nnothing but what?
Dialogue: 0,0:38:34.40,0:38:36.77,Default,,0000,0000,0000,,Let's write it from\Nthe left to the right.
Dialogue: 0,0:38:36.77,0:38:39.75,Default,,0000,0000,0000,,Let's keep the green guy\Nin the left hand side.
Dialogue: 0,0:38:39.75,0:38:43.06,Default,,0000,0000,0000,,
Dialogue: 0,0:38:43.06,0:38:46.04,Default,,0000,0000,0000,,And everybody else\Ngoes for a moving sale.
Dialogue: 0,0:38:46.04,0:38:48.61,Default,,0000,0000,0000,,The blue and the pink go away.
Dialogue: 0,0:38:48.61,0:38:51.18,Default,,0000,0000,0000,,And when they go\Nto the other side,
Dialogue: 0,0:38:51.18,0:38:54.61,Default,,0000,0000,0000,,they have a minus,\Npick up a minus sign.
Dialogue: 0,0:38:54.61,0:38:58.26,Default,,0000,0000,0000,,But with the minus, the\Nminus here is a plus sign.
Dialogue: 0,0:38:58.26,0:38:59.68,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:38:59.68,0:39:03.00,Default,,0000,0000,0000,,
Dialogue: 0,0:39:03.00,0:39:07.41,Default,,0000,0000,0000,,So the blue guy has\Nmoved and changed sign.
Dialogue: 0,0:39:07.41,0:39:08.75,Default,,0000,0000,0000,,Where's the pink?
Dialogue: 0,0:39:08.75,0:39:12.01,Default,,0000,0000,0000,,The pink guy will also move.
Dialogue: 0,0:39:12.01,0:39:16.57,Default,,0000,0000,0000,,And he picked up this.
Dialogue: 0,0:39:16.57,0:39:21.32,Default,,0000,0000,0000,,So practically, this and that\Nformula are one and the same.
Dialogue: 0,0:39:21.32,0:39:24.75,Default,,0000,0000,0000,,They're both used for\Nthe same tangent plane.
Dialogue: 0,0:39:24.75,0:39:28.21,Default,,0000,0000,0000,,It depends how you\Nintroduce the tangent plane.
Dialogue: 0,0:39:28.21,0:39:33.08,Default,,0000,0000,0000,,And [INAUDIBLE], before class\Nstarted, 20 or 25 minutes
Dialogue: 0,0:39:33.08,0:39:38.08,Default,,0000,0000,0000,,before class, when I was in a\Nhurry, I answered you briefly.
Dialogue: 0,0:39:38.08,0:39:42.38,Default,,0000,0000,0000,,You made some algebra\Nmistake in the-- you got it?
Dialogue: 0,0:39:42.38,0:39:46.15,Default,,0000,0000,0000,,I would like to make\Nup one like those.
Dialogue: 0,0:39:46.15,0:39:48.37,Default,,0000,0000,0000,,But I forgot what\Nyour surface was.
Dialogue: 0,0:39:48.37,0:39:49.28,Default,,0000,0000,0000,,Was it an ellipsoid?
Dialogue: 0,0:39:49.28,0:39:50.58,Default,,0000,0000,0000,,STUDENT: Ellipsoid.
Dialogue: 0,0:39:50.58,0:39:53.07,Default,,0000,0000,0000,,PROFESSOR: OK, I'll\Nmake up an ellipsoid.
Dialogue: 0,0:39:53.07,0:39:54.06,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:39:54.06,0:39:58.04,Default,,0000,0000,0000,,PROFESSOR: Yeah, mhmm,\Nif you have it with you.
Dialogue: 0,0:39:58.04,0:40:00.42,Default,,0000,0000,0000,,If you don't have it\Nwith you, that's fine.
Dialogue: 0,0:40:00.42,0:40:01.79,Default,,0000,0000,0000,,So I'm going to\Ngo ahead and keep
Dialogue: 0,0:40:01.79,0:40:03.88,Default,,0000,0000,0000,,just the implicit equation.
Dialogue: 0,0:40:03.88,0:40:07.46,Default,,0000,0000,0000,,Because the ellipsoid is given\Nby the implicit equation.
Dialogue: 0,0:40:07.46,0:40:13.26,Default,,0000,0000,0000,,And everything\Nelse I will erase.
Dialogue: 0,0:40:13.26,0:40:15.66,Default,,0000,0000,0000,,And that was problem 24.
Dialogue: 0,0:40:15.66,0:40:16.52,Default,,0000,0000,0000,,How many problems?
Dialogue: 0,0:40:16.52,0:40:20.66,Default,,0000,0000,0000,,Well, you still have\Na lot, up to 49.
Dialogue: 0,0:40:20.66,0:40:21.68,Default,,0000,0000,0000,,STUDENT: 42.
Dialogue: 0,0:40:21.68,0:40:23.67,Default,,0000,0000,0000,,PROFESSOR: 42, OK, I reduced it.
Dialogue: 0,0:40:23.67,0:40:28.93,Default,,0000,0000,0000,,Now, when I sent you\Nan email on Sunday,
Dialogue: 0,0:40:28.93,0:40:32.93,Default,,0000,0000,0000,,I said I was giving\Nyou an extension.
Dialogue: 0,0:40:32.93,0:40:35.34,Default,,0000,0000,0000,,STUDENT: Till March.
Dialogue: 0,0:40:35.34,0:40:37.08,Default,,0000,0000,0000,,PROFESSOR: Till a lot of March.
Dialogue: 0,0:40:37.08,0:40:41.41,Default,,0000,0000,0000,,Because I thought March the\N2nd, and I gave a few more days.
Dialogue: 0,0:40:41.41,0:40:43.21,Default,,0000,0000,0000,,So you have one more week.
Dialogue: 0,0:40:43.21,0:40:44.44,Default,,0000,0000,0000,,STUDENT: When's spring break?
Dialogue: 0,0:40:44.44,0:40:46.22,Default,,0000,0000,0000,,PROFESSOR: Spring\Nbreak is the 14th,
Dialogue: 0,0:40:46.22,0:40:48.74,Default,,0000,0000,0000,,but this is due\Non the 9th right?
Dialogue: 0,0:40:48.74,0:40:50.62,Default,,0000,0000,0000,,STUDENT: The 10th.
Dialogue: 0,0:40:50.62,0:40:53.94,Default,,0000,0000,0000,,PROFESSOR: The 10th--\Nmaybe I don't remember.
Dialogue: 0,0:40:53.94,0:40:56.07,Default,,0000,0000,0000,,So what was your problem?
Dialogue: 0,0:40:56.07,0:40:57.51,Default,,0000,0000,0000,,What's your problem?
Dialogue: 0,0:40:57.51,0:40:58.96,Default,,0000,0000,0000,,What?
Dialogue: 0,0:40:58.96,0:41:00.40,Default,,0000,0000,0000,,Can I take it?
Dialogue: 0,0:41:00.40,0:41:05.70,Default,,0000,0000,0000,,
Dialogue: 0,0:41:05.70,0:41:10.31,Default,,0000,0000,0000,,So you have an\Nellipsoid, which comes
Dialogue: 0,0:41:10.31,0:41:15.68,Default,,0000,0000,0000,,from the ellipse 3x\Nsquared [INAUDIBLE].
Dialogue: 0,0:41:15.68,0:41:21.17,Default,,0000,0000,0000,,Then you have a 2z\Nsquared equals 9.
Dialogue: 0,0:41:21.17,0:41:24.66,Default,,0000,0000,0000,,Then it says, you\Nlook at the point P
Dialogue: 0,0:41:24.66,0:41:26.53,Default,,0000,0000,0000,,of coordinates minus\N1-- I haven't even
Dialogue: 0,0:41:26.53,0:41:29.88,Default,,0000,0000,0000,,checked if it's correct,\Nbut it should be.
Dialogue: 0,0:41:29.88,0:41:33.60,Default,,0000,0000,0000,,So 3 plus-- I did not\Nprogram the problem.
Dialogue: 0,0:41:33.60,0:41:38.59,Default,,0000,0000,0000,,3 plus 4, 7, plus 2, 9,\Nso he did a good job.
Dialogue: 0,0:41:38.59,0:41:42.30,Default,,0000,0000,0000,,We want the tangent plane.
Dialogue: 0,0:41:42.30,0:41:45.15,Default,,0000,0000,0000,,I'll put it here.
Dialogue: 0,0:41:45.15,0:41:47.71,Default,,0000,0000,0000,,We want the tangent plane.
Dialogue: 0,0:41:47.71,0:41:49.63,Default,,0000,0000,0000,,How do we compute\Nthe tangent plane?
Dialogue: 0,0:41:49.63,0:41:54.49,Default,,0000,0000,0000,,You say, this is F\Nof x, y, z, right?
Dialogue: 0,0:41:54.49,0:41:56.27,Default,,0000,0000,0000,,So F sub x equals 6x.
Dialogue: 0,0:41:56.27,0:41:58.79,Default,,0000,0000,0000,,F sub y equals 2y.
Dialogue: 0,0:41:58.79,0:42:01.96,Default,,0000,0000,0000,,F sub z equals 4z.
Dialogue: 0,0:42:01.96,0:42:06.80,Default,,0000,0000,0000,,Computing it at P0,\Nwhat do we have?
Dialogue: 0,0:42:06.80,0:42:11.28,Default,,0000,0000,0000,,x is minus 1, y is\Nminus 2, z is minus 1.
Dialogue: 0,0:42:11.28,0:42:18.21,Default,,0000,0000,0000,,I should get negative 6,\Nnegative 4, and minus 4.
Dialogue: 0,0:42:18.21,0:42:21.28,Default,,0000,0000,0000,,
Dialogue: 0,0:42:21.28,0:42:24.99,Default,,0000,0000,0000,,And then I should plug\Nin and get minus 6 times
Dialogue: 0,0:42:24.99,0:42:28.05,Default,,0000,0000,0000,,x minus minus 1.
Dialogue: 0,0:42:28.05,0:42:30.06,Default,,0000,0000,0000,,I have to pay attention myself.
Dialogue: 0,0:42:30.06,0:42:36.44,Default,,0000,0000,0000,,It's not easy to get the\Nalgebra right-- minus 4 times y
Dialogue: 0,0:42:36.44,0:42:45.13,Default,,0000,0000,0000,,plus 2 minus 4 times\Nz plus 1 equals 0.
Dialogue: 0,0:42:45.13,0:42:50.74,Default,,0000,0000,0000,,And I hope I get what you got--\Nminus 6x minus 4y minus 4z,
Dialogue: 0,0:42:50.74,0:42:51.83,Default,,0000,0000,0000,,so many of those.
Dialogue: 0,0:42:51.83,0:42:53.32,Default,,0000,0000,0000,,You've got to divide by 2.
Dialogue: 0,0:42:53.32,0:42:55.29,Default,,0000,0000,0000,,I'm not getting that.
Dialogue: 0,0:42:55.29,0:42:59.46,Default,,0000,0000,0000,,And then minus 6-- I'm\Ngoing to write it down.
Dialogue: 0,0:42:59.46,0:43:03.24,Default,,0000,0000,0000,,
Dialogue: 0,0:43:03.24,0:43:05.33,Default,,0000,0000,0000,,So it's even, right?
Dialogue: 0,0:43:05.33,0:43:08.04,Default,,0000,0000,0000,,
Dialogue: 0,0:43:08.04,0:43:15.86,Default,,0000,0000,0000,,The whole thing minus 18, divide\Nby 2 should be-- divide by 2,
Dialogue: 0,0:43:15.86,0:43:18.68,Default,,0000,0000,0000,,and did you change the\Nsigns, [INAUDIBLE]?
Dialogue: 0,0:43:18.68,0:43:20.62,Default,,0000,0000,0000,,What's your password?
Dialogue: 0,0:43:20.62,0:43:21.58,Default,,0000,0000,0000,,No.
Dialogue: 0,0:43:21.58,0:43:27.23,Default,,0000,0000,0000,,[LAUGHING] Check if I'm\Ngetting the same thing you got.
Dialogue: 0,0:43:27.23,0:43:33.13,Default,,0000,0000,0000,,So I get 3x plus 2y plus 2z.
Dialogue: 0,0:43:33.13,0:43:37.19,Default,,0000,0000,0000,,I divide by minus\N2, right, plus 9?
Dialogue: 0,0:43:37.19,0:43:39.43,Default,,0000,0000,0000,,Did you both get the same thing?
Dialogue: 0,0:43:39.43,0:43:40.78,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:43:40.78,0:43:42.77,Default,,0000,0000,0000,,PROFESSOR: You didn't?
Dialogue: 0,0:43:42.77,0:43:47.26,Default,,0000,0000,0000,,Well, I'm trying to\Nsimplify all my answers
Dialogue: 0,0:43:47.26,0:43:49.99,Default,,0000,0000,0000,,with this simplification.
Dialogue: 0,0:43:49.99,0:43:52.53,Default,,0000,0000,0000,,So I guess of course\Nif you enter it
Dialogue: 0,0:43:52.53,0:43:55.45,Default,,0000,0000,0000,,like that, it's going to work.
Dialogue: 0,0:43:55.45,0:44:03.66,Default,,0000,0000,0000,,Now, I need you guys\Nto help me on this one.
Dialogue: 0,0:44:03.66,0:44:06.42,Default,,0000,0000,0000,,Find the parametric\Nform-- it's so easy,
Dialogue: 0,0:44:06.42,0:44:10.64,Default,,0000,0000,0000,,but this is a problem\Nsession-- of the line passing
Dialogue: 0,0:44:10.64,0:44:12.79,Default,,0000,0000,0000,,through the same point\Nthat's perpendicular
Dialogue: 0,0:44:12.79,0:44:14.54,Default,,0000,0000,0000,,to the tangent plane.
Dialogue: 0,0:44:14.54,0:44:18.74,Default,,0000,0000,0000,,Express your answer in the\Nparametric form of the type
Dialogue: 0,0:44:18.74,0:44:21.19,Default,,0000,0000,0000,,the one that you know that\NI don't like very much,
Dialogue: 0,0:44:21.19,0:44:30.50,Default,,0000,0000,0000,,but I will write it down-- a2t\N[INAUDIBLE] b2 a3t plus b3.
Dialogue: 0,0:44:30.50,0:44:34.55,Default,,0000,0000,0000,,This is how he wants me to\Nwrite it, which I don't like.
Dialogue: 0,0:44:34.55,0:44:38.17,Default,,0000,0000,0000,,I would've even preferred\Nit to be in symmetric form.
Dialogue: 0,0:44:38.17,0:44:39.22,Default,,0000,0000,0000,,It's the same.
Dialogue: 0,0:44:39.22,0:44:41.14,Default,,0000,0000,0000,,I'll put the t, and I'm fine.
Dialogue: 0,0:44:41.14,0:44:47.30,Default,,0000,0000,0000,,So x minus x0 over\Nl, y minus y0 over m,
Dialogue: 0,0:44:47.30,0:44:51.78,Default,,0000,0000,0000,,z minus z0 over n, that\Nwas the symmetric form.
Dialogue: 0,0:44:51.78,0:44:55.24,Default,,0000,0000,0000,,I make it parametric\Nby saying equal to t.
Dialogue: 0,0:44:55.24,0:44:58.55,Default,,0000,0000,0000,,So what were the\Nparametric equations?
Dialogue: 0,0:44:58.55,0:45:01.85,Default,,0000,0000,0000,,x equals lt plus x0.
Dialogue: 0,0:45:01.85,0:45:03.28,Default,,0000,0000,0000,,That's the normal.
Dialogue: 0,0:45:03.28,0:45:06.24,Default,,0000,0000,0000,,y equals mt plus y0.
Dialogue: 0,0:45:06.24,0:45:09.17,Default,,0000,0000,0000,,z equals nt plus z0.
Dialogue: 0,0:45:09.17,0:45:14.26,Default,,0000,0000,0000,,So finally, my answer-- I'll\Ncheck with [INAUDIBLE] answer
Dialogue: 0,0:45:14.26,0:45:20.11,Default,,0000,0000,0000,,in a second-- should be take the\Nnormal from the tangent plane,
Dialogue: 0,0:45:20.11,0:45:24.51,Default,,0000,0000,0000,,3, 2, 2, right?
Dialogue: 0,0:45:24.51,0:45:31.90,Default,,0000,0000,0000,,2t plus whatever,\Nthis is 2t plus z
Dialogue: 0,0:45:31.90,0:45:45.73,Default,,0000,0000,0000,,equals-- first it's 3, 3, 2,\N2, 3, 2, 2, plus x0 y0 z0.
Dialogue: 0,0:45:45.73,0:45:47.92,Default,,0000,0000,0000,,So erase the pluses and minuses.
Dialogue: 0,0:45:47.92,0:45:50.60,Default,,0000,0000,0000,,
Dialogue: 0,0:45:50.60,0:45:52.44,Default,,0000,0000,0000,,And those should\Nbe the equations.
Dialogue: 0,0:45:52.44,0:45:57.53,Default,,0000,0000,0000,,And I should write them down.
Dialogue: 0,0:45:57.53,0:45:59.70,Default,,0000,0000,0000,,It's OK, you have the same.
Dialogue: 0,0:45:59.70,0:46:05.43,Default,,0000,0000,0000,,Now you just have to take\Nthem, this, this, and that,
Dialogue: 0,0:46:05.43,0:46:08.88,Default,,0000,0000,0000,,and put it in this form.
Dialogue: 0,0:46:08.88,0:46:11.09,Default,,0000,0000,0000,,This is a combination problem.
Dialogue: 0,0:46:11.09,0:46:12.81,Default,,0000,0000,0000,,Why do I say\Ncombination problem?
Dialogue: 0,0:46:12.81,0:46:16.73,Default,,0000,0000,0000,,It's combining Chapter\N11 with Chapter 9.
Dialogue: 0,0:46:16.73,0:46:18.81,Default,,0000,0000,0000,,This was the review\Nfrom Chapter 9.
Dialogue: 0,0:46:18.81,0:46:21.86,Default,,0000,0000,0000,,
Dialogue: 0,0:46:21.86,0:46:23.29,Default,,0000,0000,0000,,Did you have trouble\Nunderstanding
Dialogue: 0,0:46:23.29,0:46:26.53,Default,,0000,0000,0000,,the radiant or\Nthe tangent planes
Dialogue: 0,0:46:26.53,0:46:29.66,Default,,0000,0000,0000,,or anything like that,\Nimplicit form, explicit form?
Dialogue: 0,0:46:29.66,0:46:37.20,Default,,0000,0000,0000,,Let me do an application,\Nsince I'm doing review anyway.
Dialogue: 0,0:46:37.20,0:46:39.52,Default,,0000,0000,0000,,I'm done with the Section 11.6.
Dialogue: 0,0:46:39.52,0:46:42.14,Default,,0000,0000,0000,,But before I want\Nto go further, I
Dialogue: 0,0:46:42.14,0:46:44.45,Default,,0000,0000,0000,,want to do some\Nreview of Chapter
Dialogue: 0,0:46:44.45,0:47:00.64,Default,,0000,0000,0000,,11 sections 11.1 through 11.6.
Dialogue: 0,0:47:00.64,0:47:03.57,Default,,0000,0000,0000,,I said something about\Nimplicit differentiation
Dialogue: 0,0:47:03.57,0:47:06.92,Default,,0000,0000,0000,,being a headache\Nfor many of you.
Dialogue: 0,0:47:06.92,0:47:19.26,Default,,0000,0000,0000,,One person asked me, how do\Nyou compute z sub x and/or z
Dialogue: 0,0:47:19.26,0:47:32.30,Default,,0000,0000,0000,,sub y based on the equation x\Nsquared plus y squared plus z
Dialogue: 0,0:47:32.30,0:47:33.44,Default,,0000,0000,0000,,squared equals 5?
Dialogue: 0,0:47:33.44,0:47:36.24,Default,,0000,0000,0000,,
Dialogue: 0,0:47:36.24,0:47:38.30,Default,,0000,0000,0000,,And of course this is\Nimplicit differentiation.
Dialogue: 0,0:47:38.30,0:47:46.48,Default,,0000,0000,0000,,
Dialogue: 0,0:47:46.48,0:47:47.66,Default,,0000,0000,0000,,Why implicit?
Dialogue: 0,0:47:47.66,0:47:55.89,Default,,0000,0000,0000,,OK, because this is an implicit\Nequation of the type F of x, y,
Dialogue: 0,0:47:55.89,0:48:00.34,Default,,0000,0000,0000,,z equals constant.
Dialogue: 0,0:48:00.34,0:48:02.39,Default,,0000,0000,0000,,When do we call it explicit?
Dialogue: 0,0:48:02.39,0:48:06.09,Default,,0000,0000,0000,,When one of the\Nvariables, x or y or z,
Dialogue: 0,0:48:06.09,0:48:10.39,Default,,0000,0000,0000,,is given explicitly in\Nterms of the other two.
Dialogue: 0,0:48:10.39,0:48:14.54,Default,,0000,0000,0000,,So if this would be-- well,\Nhere it's hard to pull it out.
Dialogue: 0,0:48:14.54,0:48:17.67,Default,,0000,0000,0000,,But whether it be upper\Npart then lower hemisphere,
Dialogue: 0,0:48:17.67,0:48:20.49,Default,,0000,0000,0000,,z would be plus or minus.
Dialogue: 0,0:48:20.49,0:48:23.59,Default,,0000,0000,0000,,So you have two caps,\Ntwo hemispheres,
Dialogue: 0,0:48:23.59,0:48:27.94,Default,,0000,0000,0000,,plus/minus square root 5 minus\Nx squared minus y squared.
Dialogue: 0,0:48:27.94,0:48:29.60,Default,,0000,0000,0000,,Well, that's two functions.
Dialogue: 0,0:48:29.60,0:48:31.59,Default,,0000,0000,0000,,We don't like that.
Dialogue: 0,0:48:31.59,0:48:34.44,Default,,0000,0000,0000,,We want to be able to do\Neverything in one shot
Dialogue: 0,0:48:34.44,0:48:38.35,Default,,0000,0000,0000,,without splitting it into\Ntwo different graphs.
Dialogue: 0,0:48:38.35,0:48:43.15,Default,,0000,0000,0000,,So how do we view z\Nto be a function of x,
Dialogue: 0,0:48:43.15,0:48:44.71,Default,,0000,0000,0000,,you're going to ask yourself.
Dialogue: 0,0:48:44.71,0:48:49.40,Default,,0000,0000,0000,,You imagine inside\Nthis thing that x and y
Dialogue: 0,0:48:49.40,0:48:54.33,Default,,0000,0000,0000,,are independent variables--\Nindependent variables.
Dialogue: 0,0:48:54.33,0:48:56.49,Default,,0000,0000,0000,,They can take\Nwhatever they want.
Dialogue: 0,0:48:56.49,0:48:57.52,Default,,0000,0000,0000,,One is temperature.
Dialogue: 0,0:48:57.52,0:48:58.29,Default,,0000,0000,0000,,One is time.
Dialogue: 0,0:48:58.29,0:49:00.51,Default,,0000,0000,0000,,They run like crazies.
Dialogue: 0,0:49:00.51,0:49:05.04,Default,,0000,0000,0000,,But z depends on both\Ntemperature and time, like us
Dialogue: 0,0:49:05.04,0:49:05.88,Default,,0000,0000,0000,,unfortunately.
Dialogue: 0,0:49:05.88,0:49:06.75,Default,,0000,0000,0000,,It's so cold outside.
Dialogue: 0,0:49:06.75,0:49:08.93,Default,,0000,0000,0000,,I hate it.
Dialogue: 0,0:49:08.93,0:49:12.50,Default,,0000,0000,0000,,OK, you promised me,\Nand it came true.
Dialogue: 0,0:49:12.50,0:49:13.56,Default,,0000,0000,0000,,Who promised me?
Dialogue: 0,0:49:13.56,0:49:15.43,Default,,0000,0000,0000,,Matthew, I give you a\Nbrownie point for that.
Dialogue: 0,0:49:15.43,0:49:17.89,Default,,0000,0000,0000,,Because you said last week\Nit's going to be 80 degrees.
Dialogue: 0,0:49:17.89,0:49:19.34,Default,,0000,0000,0000,,And it was.
Dialogue: 0,0:49:19.34,0:49:20.91,Default,,0000,0000,0000,,So the prophecy came true.
Dialogue: 0,0:49:20.91,0:49:24.89,Default,,0000,0000,0000,,On the other hand,\Nit came back too bad.
Dialogue: 0,0:49:24.89,0:49:27.42,Default,,0000,0000,0000,,And of course it's\Nnot Matthew's fault.
Dialogue: 0,0:49:27.42,0:49:30.59,Default,,0000,0000,0000,,He didn't say what's\Ngoing to happen this week.
Dialogue: 0,0:49:30.59,0:49:36.22,Default,,0000,0000,0000,,All right, in this case,\Nimplicit differentiation
Dialogue: 0,0:49:36.22,0:49:38.69,Default,,0000,0000,0000,,is just a philosophical thing.
Dialogue: 0,0:49:38.69,0:49:41.82,Default,,0000,0000,0000,,It's a very important\Nphilosophical step
Dialogue: 0,0:49:41.82,0:49:43.84,Default,,0000,0000,0000,,that you're taking-- think.
Dialogue: 0,0:49:43.84,0:49:48.39,Default,,0000,0000,0000,,
Dialogue: 0,0:49:48.39,0:49:56.25,Default,,0000,0000,0000,,Think of z being a\Nfunction of x and y.
Dialogue: 0,0:49:56.25,0:50:05.22,Default,,0000,0000,0000,,And two, differentiate\Nz with respect to x.
Dialogue: 0,0:50:05.22,0:50:10.22,Default,,0000,0000,0000,,
Dialogue: 0,0:50:10.22,0:50:13.70,Default,,0000,0000,0000,,So what do you\Nmean, differentiate
Dialogue: 0,0:50:13.70,0:50:15.10,Default,,0000,0000,0000,,with respect to x?
Dialogue: 0,0:50:15.10,0:50:20.88,Default,,0000,0000,0000,,By differentiating\Nthe entire equation,
Dialogue: 0,0:50:20.88,0:50:30.54,Default,,0000,0000,0000,,both sides of an equation\Nwith respect to x.
Dialogue: 0,0:50:30.54,0:50:34.75,Default,,0000,0000,0000,,So for you, x is\Nthe wanted variable.
Dialogue: 0,0:50:34.75,0:50:36.51,Default,,0000,0000,0000,,y is like a constant.
Dialogue: 0,0:50:36.51,0:50:42.34,Default,,0000,0000,0000,,z is a function of x\Nis not hard at all.
Dialogue: 0,0:50:42.34,0:50:47.85,Default,,0000,0000,0000,,So what is going to happen\Nactually if you were to do it?
Dialogue: 0,0:50:47.85,0:50:49.76,Default,,0000,0000,0000,,Theoretically, you\Nwould go like that.
Dialogue: 0,0:50:49.76,0:50:53.99,Default,,0000,0000,0000,,If I'm going to differentiate\Nthis guy with respect to x,
Dialogue: 0,0:50:53.99,0:50:55.70,Default,,0000,0000,0000,,what is the philosophy?
Dialogue: 0,0:50:55.70,0:50:59.52,Default,,0000,0000,0000,,The chain rule tells\Nme, differentiate F
Dialogue: 0,0:50:59.52,0:51:05.70,Default,,0000,0000,0000,,with respect to the first\Nvariable, and then times dx/dx.
Dialogue: 0,0:51:05.70,0:51:10.34,Default,,0000,0000,0000,,And you say, god, now\Nthat was silly, right?
Dialogue: 0,0:51:10.34,0:51:11.96,Default,,0000,0000,0000,,Differentiate with respect to x.
Dialogue: 0,0:51:11.96,0:51:13.96,Default,,0000,0000,0000,,That's the chain rule.
Dialogue: 0,0:51:13.96,0:51:16.56,Default,,0000,0000,0000,,Plus differentiate\NF with respect
Dialogue: 0,0:51:16.56,0:51:22.35,Default,,0000,0000,0000,,to the second place,\Nsecond variable, and then
Dialogue: 0,0:51:22.35,0:51:27.54,Default,,0000,0000,0000,,say, dy with respect to x.
Dialogue: 0,0:51:27.54,0:51:30.20,Default,,0000,0000,0000,,But are dx and dy married?
Dialogue: 0,0:51:30.20,0:51:31.80,Default,,0000,0000,0000,,Do they depend on one another?
Dialogue: 0,0:51:31.80,0:51:35.26,Default,,0000,0000,0000,,Do they file an income\Ntax return together?
Dialogue: 0,0:51:35.26,0:51:38.25,Default,,0000,0000,0000,,They don't want to have\Nanything to do with one another.
Dialogue: 0,0:51:38.25,0:51:42.23,Default,,0000,0000,0000,,Thank god, so x and y are\Nindependent variables.
Dialogue: 0,0:51:42.23,0:51:45.82,Default,,0000,0000,0000,,If you're taking\Nstatistics or researching
Dialogue: 0,0:51:45.82,0:51:48.28,Default,,0000,0000,0000,,any other kind of\Nphysics, chemistry,
Dialogue: 0,0:51:48.28,0:51:50.52,Default,,0000,0000,0000,,you know that these are\Ncalled independent variables,
Dialogue: 0,0:51:50.52,0:51:53.85,Default,,0000,0000,0000,,and this is called the\Ndependent variable.
Dialogue: 0,0:51:53.85,0:51:57.07,Default,,0000,0000,0000,,And then you have what's\Ncalled the constraint.
Dialogue: 0,0:51:57.07,0:52:01.77,Default,,0000,0000,0000,,In physics and engineering and\Nmechanics, F of some variable
Dialogue: 0,0:52:01.77,0:52:05.46,Default,,0000,0000,0000,,equals C. It's called\Nconstraint usually.
Dialogue: 0,0:52:05.46,0:52:08.68,Default,,0000,0000,0000,,OK, so this guy is all silly.
Dialogue: 0,0:52:08.68,0:52:12.65,Default,,0000,0000,0000,,These guys don't want to have\Nto do anything with one another.
Dialogue: 0,0:52:12.65,0:52:15.43,Default,,0000,0000,0000,,And then you get plus.
Dialogue: 0,0:52:15.43,0:52:20.99,Default,,0000,0000,0000,,Finally, dF with respect\Nto the third place,
Dialogue: 0,0:52:20.99,0:52:25.01,Default,,0000,0000,0000,,and then that third place, z,\Nis occupied by a function that's
Dialogue: 0,0:52:25.01,0:52:26.48,Default,,0000,0000,0000,,a function of x.
Dialogue: 0,0:52:26.48,0:52:28.36,Default,,0000,0000,0000,,So you go, dz/dx.
Dialogue: 0,0:52:28.36,0:52:30.88,Default,,0000,0000,0000,,Why del and not d?
Dialogue: 0,0:52:30.88,0:52:35.55,Default,,0000,0000,0000,,Because poor z is a function\Nof two variables, x and y.
Dialogue: 0,0:52:35.55,0:52:37.70,Default,,0000,0000,0000,,So you cannot say, dz/dx.
Dialogue: 0,0:52:37.70,0:52:41.09,Default,,0000,0000,0000,,You have to say\Ndel z dx, equals 0.
Dialogue: 0,0:52:41.09,0:52:46.15,Default,,0000,0000,0000,,Thank god, I got to the\Nend where I wanted to get.
Dialogue: 0,0:52:46.15,0:52:51.87,Default,,0000,0000,0000,,Now, if I want to see what's\Ngoing on, it's a piece of cake.
Dialogue: 0,0:52:51.87,0:52:53.80,Default,,0000,0000,0000,,That's 1.
Dialogue: 0,0:52:53.80,0:52:59.27,Default,,0000,0000,0000,,And I get that Mr. z sub x,\Nwhich other people write dz/dx,
Dialogue: 0,0:52:59.27,0:53:05.51,Default,,0000,0000,0000,,but I don't, because I don't\Nlike it-- I keep mixing x.
Dialogue: 0,0:53:05.51,0:53:11.56,Default,,0000,0000,0000,,Equals-- how do I\Npull this guy out?
Dialogue: 0,0:53:11.56,0:53:13.86,Default,,0000,0000,0000,,How do I substitute for that?
Dialogue: 0,0:53:13.86,0:53:21.68,Default,,0000,0000,0000,,I get Mr. First Fellow\Nhere to the other side.
Dialogue: 0,0:53:21.68,0:53:27.95,Default,,0000,0000,0000,,He's going to pick up\Na minus at whatever d
Dialogue: 0,0:53:27.95,0:53:34.59,Default,,0000,0000,0000,,I have divided by-- so this\Nguy divided by this guy.
Dialogue: 0,0:53:34.59,0:53:45.75,Default,,0000,0000,0000,,
Dialogue: 0,0:53:45.75,0:53:50.72,Default,,0000,0000,0000,,STUDENT: What happened\Nto dF/dy, dy/dx?
Dialogue: 0,0:53:50.72,0:53:52.93,Default,,0000,0000,0000,,PROFESSOR: So again, that\Nis a very good thing.
Dialogue: 0,0:53:52.93,0:53:54.62,Default,,0000,0000,0000,,So dF/dy was behaving.
Dialogue: 0,0:53:54.62,0:53:56.02,Default,,0000,0000,0000,,He was nice.
Dialogue: 0,0:53:56.02,0:54:00.16,Default,,0000,0000,0000,,But when we got to dy with\Nrespect to dx, y said,
Dialogue: 0,0:54:00.16,0:54:02.69,Default,,0000,0000,0000,,I'm not married to dx.
Dialogue: 0,0:54:02.69,0:54:04.90,Default,,0000,0000,0000,,I have nothing to do with dx.
Dialogue: 0,0:54:04.90,0:54:06.52,Default,,0000,0000,0000,,I'm independent from this.
Dialogue: 0,0:54:06.52,0:54:10.44,Default,,0000,0000,0000,,So dy/dx is 0.
Dialogue: 0,0:54:10.44,0:54:14.70,Default,,0000,0000,0000,,And so this guy\Ndisappears. dx/dx is 1.
Dialogue: 0,0:54:14.70,0:54:16.58,Default,,0000,0000,0000,,Duh, that's a piece of cake.
Dialogue: 0,0:54:16.58,0:54:17.53,Default,,0000,0000,0000,,So I'm done.
Dialogue: 0,0:54:17.53,0:54:22.90,Default,,0000,0000,0000,,This is actually a formula\Nthat looks sort of easy.
Dialogue: 0,0:54:22.90,0:54:25.64,Default,,0000,0000,0000,,But there is a lot\Nhidden behind it.
Dialogue: 0,0:54:25.64,0:54:28.59,Default,,0000,0000,0000,,This is the implicit\Nfunction theorem.
Dialogue: 0,0:54:28.59,0:54:31.48,Default,,0000,0000,0000,,
Dialogue: 0,0:54:31.48,0:54:35.11,Default,,0000,0000,0000,,where you of course assume\Nthat these partial derivatives
Dialogue: 0,0:54:35.11,0:54:38.46,Default,,0000,0000,0000,,exist, are continuous,\Neverything is nice.
Dialogue: 0,0:54:38.46,0:54:41.45,Default,,0000,0000,0000,,It's a beautiful result.\NPeople actually get
Dialogue: 0,0:54:41.45,0:54:45.75,Default,,0000,0000,0000,,to learn it only when they are\Nbig, I mean big mathematically,
Dialogue: 0,0:54:45.75,0:54:49.25,Default,,0000,0000,0000,,mature, in graduate school,\Nfirst or second year
Dialogue: 0,0:54:49.25,0:54:50.06,Default,,0000,0000,0000,,of graduate school.
Dialogue: 0,0:54:50.06,0:54:52.02,Default,,0000,0000,0000,,We call that\Nintermediate analysis
Dialogue: 0,0:54:52.02,0:54:55.17,Default,,0000,0000,0000,,or advanced-- very\Nadvanced-- calculus.
Dialogue: 0,0:54:55.17,0:54:57.14,Default,,0000,0000,0000,,Because this calculus\Nis advanced enough,
Dialogue: 0,0:54:57.14,0:55:00.09,Default,,0000,0000,0000,,but I'm talking about\Ngraduate level calculus.
Dialogue: 0,0:55:00.09,0:55:03.01,Default,,0000,0000,0000,,And this is the so-called\Nimplicit function theorem.
Dialogue: 0,0:55:03.01,0:55:08.34,Default,,0000,0000,0000,,So if you will ever be even not\Nnecessarily a graduate student
Dialogue: 0,0:55:08.34,0:55:11.57,Default,,0000,0000,0000,,in mathematics, but a graduate\Nstudent in physics or something
Dialogue: 0,0:55:11.57,0:55:16.10,Default,,0000,0000,0000,,related to pure science,\Nremember this result. So let me
Dialogue: 0,0:55:16.10,0:55:18.42,Default,,0000,0000,0000,,see what's going to\Nhappen in practice.
Dialogue: 0,0:55:18.42,0:55:22.94,Default,,0000,0000,0000,,In practice, do we\Nhave to learn this?
Dialogue: 0,0:55:22.94,0:55:27.34,Default,,0000,0000,0000,,No, in practice we can build\Neverything from scratch, again,
Dialogue: 0,0:55:27.34,0:55:32.10,Default,,0000,0000,0000,,just the way we did\Nit with the formula.
Dialogue: 0,0:55:32.10,0:55:33.92,Default,,0000,0000,0000,,So for the example\NI gave you, it
Dialogue: 0,0:55:33.92,0:55:36.87,Default,,0000,0000,0000,,should be a piece of cake\Nto do the differentiation.
Dialogue: 0,0:55:36.87,0:55:38.87,Default,,0000,0000,0000,,But I'm going to step by step.
Dialogue: 0,0:55:38.87,0:55:41.62,Default,,0000,0000,0000,,Step one, think.
Dialogue: 0,0:55:41.62,0:55:45.95,Default,,0000,0000,0000,,
Dialogue: 0,0:55:45.95,0:55:47.25,Default,,0000,0000,0000,,You have to think.
Dialogue: 0,0:55:47.25,0:55:49.74,Default,,0000,0000,0000,,If you don't think,\Nyou cannot do math.
Dialogue: 0,0:55:49.74,0:55:53.11,Default,,0000,0000,0000,,So you have x squared\Nplus y squared,
Dialogue: 0,0:55:53.11,0:55:56.98,Default,,0000,0000,0000,,the independent guys, and z,\Nwho is married to both of them.
Dialogue: 0,0:55:56.98,0:55:58.94,Default,,0000,0000,0000,,Or maybe z is the baby.
Dialogue: 0,0:55:58.94,0:56:02.47,Default,,0000,0000,0000,,These are two spouses that are\Nindependent from one another.
Dialogue: 0,0:56:02.47,0:56:04.18,Default,,0000,0000,0000,,And z is their baby.
Dialogue: 0,0:56:04.18,0:56:06.12,Default,,0000,0000,0000,,Because he depends\Non both of them.
Dialogue: 0,0:56:06.12,0:56:09.53,Default,,0000,0000,0000,,
Dialogue: 0,0:56:09.53,0:56:12.30,Default,,0000,0000,0000,,So you thought you had\Na different approach
Dialogue: 0,0:56:12.30,0:56:16.71,Default,,0000,0000,0000,,to the problem, different\Nvision of what's going on.
Dialogue: 0,0:56:16.71,0:56:23.09,Default,,0000,0000,0000,,Now finally, step two,\Ndifferentiate with respect
Dialogue: 0,0:56:23.09,0:56:25.44,Default,,0000,0000,0000,,to one only, x only.
Dialogue: 0,0:56:25.44,0:56:28.89,Default,,0000,0000,0000,,
Dialogue: 0,0:56:28.89,0:56:32.24,Default,,0000,0000,0000,,You could of course do the\Nsame process with respect to y.
Dialogue: 0,0:56:32.24,0:56:35.11,Default,,0000,0000,0000,,And in some of the\Nfinal exam problems,
Dialogue: 0,0:56:35.11,0:56:38.18,Default,,0000,0000,0000,,we are asking, compute\Nz sub x and z sub y.
Dialogue: 0,0:56:38.18,0:56:40.15,Default,,0000,0000,0000,,The secret is that--\Nmaybe I shouldn't
Dialogue: 0,0:56:40.15,0:56:45.00,Default,,0000,0000,0000,,talk too much again-- when\NI grade those finals, if you
Dialogue: 0,0:56:45.00,0:56:47.61,Default,,0000,0000,0000,,do z sub x, I give you 100%.
Dialogue: 0,0:56:47.61,0:56:50.71,Default,,0000,0000,0000,,Because z sub y is the same.
Dialogue: 0,0:56:50.71,0:56:53.88,Default,,0000,0000,0000,,So I really don't care.
Dialogue: 0,0:56:53.88,0:56:56.80,Default,,0000,0000,0000,,Sometimes there\Nare so many things
Dialogue: 0,0:56:56.80,0:57:00.80,Default,,0000,0000,0000,,to do that all I care\Nis, did he or she cover
Dialogue: 0,0:57:00.80,0:57:03.31,Default,,0000,0000,0000,,the essential work?
Dialogue: 0,0:57:03.31,0:57:09.43,Default,,0000,0000,0000,,So with respect to x, x squared\Ndifferentiated with respect
Dialogue: 0,0:57:09.43,0:57:11.24,Default,,0000,0000,0000,,to x-- 2x.
Dialogue: 0,0:57:11.24,0:57:15.62,Default,,0000,0000,0000,,Good first step, now, y squared\Ndifferentiated with respect
Dialogue: 0,0:57:15.62,0:57:17.50,Default,,0000,0000,0000,,to x.
Dialogue: 0,0:57:17.50,0:57:19.29,Default,,0000,0000,0000,,0-- am I going to write 0?
Dialogue: 0,0:57:19.29,0:57:21.45,Default,,0000,0000,0000,,Yes, because I'm silly.
Dialogue: 0,0:57:21.45,0:57:24.56,Default,,0000,0000,0000,,But I don't have to.
Dialogue: 0,0:57:24.56,0:57:27.68,Default,,0000,0000,0000,,2 times z of x, y.
Dialogue: 0,0:57:27.68,0:57:32.31,Default,,0000,0000,0000,,
Dialogue: 0,0:57:32.31,0:57:35.89,Default,,0000,0000,0000,,2 jumps down, z of x, y-- I'm\Nnot done with the chain rule.
Dialogue: 0,0:57:35.89,0:57:39.67,Default,,0000,0000,0000,,
Dialogue: 0,0:57:39.67,0:57:41.09,Default,,0000,0000,0000,,STUDENT: z sub x.
Dialogue: 0,0:57:41.09,0:57:43.00,Default,,0000,0000,0000,,PROFESSOR: It's z\Nsub x, very good.
Dialogue: 0,0:57:43.00,0:57:43.87,Default,,0000,0000,0000,,This is dz/dx.
Dialogue: 0,0:57:43.87,0:57:48.29,Default,,0000,0000,0000,,I'm not going to hide\Nit completely like that.
Dialogue: 0,0:57:48.29,0:57:50.20,Default,,0000,0000,0000,,That is the same thing.
Dialogue: 0,0:57:50.20,0:57:52.25,Default,,0000,0000,0000,,y prime is 0, thank god.
Dialogue: 0,0:57:52.25,0:57:55.91,Default,,0000,0000,0000,,
Dialogue: 0,0:57:55.91,0:57:59.54,Default,,0000,0000,0000,,So you say, if I were to\Nkeep in mind that that's
Dialogue: 0,0:57:59.54,0:58:06.08,Default,,0000,0000,0000,,the derivative of big\NF with respect to x,
Dialogue: 0,0:58:06.08,0:58:08.92,Default,,0000,0000,0000,,I could plug in\Neverything in here.
Dialogue: 0,0:58:08.92,0:58:10.11,Default,,0000,0000,0000,,I could plug in the formula.
Dialogue: 0,0:58:10.11,0:58:12.32,Default,,0000,0000,0000,,But why memorize the\Nformula and plug it
Dialogue: 0,0:58:12.32,0:58:16.38,Default,,0000,0000,0000,,in when you can do everything\Nfrom scratch all over again?
Dialogue: 0,0:58:16.38,0:58:18.51,Default,,0000,0000,0000,,Math is not about memorization.
Dialogue: 0,0:58:18.51,0:58:22.57,Default,,0000,0000,0000,,If you are good, for example,\Nsome people here-- I'm
Dialogue: 0,0:58:22.57,0:58:28.00,Default,,0000,0000,0000,,not going to name them--\Nare in sciences that involve
Dialogue: 0,0:58:28.00,0:58:29.98,Default,,0000,0000,0000,,a lot of memorization.
Dialogue: 0,0:58:29.98,0:58:31.67,Default,,0000,0000,0000,,More power to them.
Dialogue: 0,0:58:31.67,0:58:34.37,Default,,0000,0000,0000,,I was not very good at that.
Dialogue: 0,0:58:34.37,0:58:37.87,Default,,0000,0000,0000,,So I'm going to go ahead and\Nwrite z sub x pulled down
Dialogue: 0,0:58:37.87,0:58:42.95,Default,,0000,0000,0000,,minus 2x divided by 2z.
Dialogue: 0,0:58:42.95,0:58:47.38,Default,,0000,0000,0000,,I'm too lazy to remind\Nyou that z is the baby,
Dialogue: 0,0:58:47.38,0:58:50.25,Default,,0000,0000,0000,,and he depends on\Nhis parents x and y.
Dialogue: 0,0:58:50.25,0:58:52.49,Default,,0000,0000,0000,,I'm not going to write that.
Dialogue: 0,0:58:52.49,0:58:53.68,Default,,0000,0000,0000,,And that's the answer.
Dialogue: 0,0:58:53.68,0:58:57.82,Default,,0000,0000,0000,,So you have minus x/z.
Dialogue: 0,0:58:57.82,0:59:03.69,Default,,0000,0000,0000,,So for example, if somebody\Nsays, compute z sub x
Dialogue: 0,0:59:03.69,0:59:17.40,Default,,0000,0000,0000,,at the point on the sphere,\Nthat is 0, root 5, and 0,
Dialogue: 0,0:59:17.40,0:59:19.23,Default,,0000,0000,0000,,what do you have to do?
Dialogue: 0,0:59:19.23,0:59:23.82,Default,,0000,0000,0000,,You have to say,\Nz sub x equals--
Dialogue: 0,0:59:23.82,0:59:30.62,Default,,0000,0000,0000,,and now I'm asking you\Nsomething that is minus 0/0.
Dialogue: 0,0:59:30.62,0:59:35.98,Default,,0000,0000,0000,,
Dialogue: 0,0:59:35.98,0:59:49.40,Default,,0000,0000,0000,,Assuming that the expressions,\Nthe derivatives, are defined
Dialogue: 0,0:59:49.40,0:59:56.59,Default,,0000,0000,0000,,and the denominator one\Nis different from 0-- so
Dialogue: 0,0:59:56.59,0:59:59.71,Default,,0000,0000,0000,,whenever you do the\Nimplicit function theorem,
Dialogue: 0,0:59:59.71,1:00:04.79,Default,,0000,0000,0000,,you can apply with the\Ncondition that you are away
Dialogue: 0,1:00:04.79,1:00:10.29,Default,,0000,0000,0000,,from points where derivative\Nof F with respect to z are 0.
Dialogue: 0,1:00:10.29,1:00:13.32,Default,,0000,0000,0000,,So this is a problem\Nthat's not well posed.
Dialogue: 0,1:00:13.32,1:00:15.56,Default,,0000,0000,0000,,So to give you a\Nwell-posed problem, what
Dialogue: 0,1:00:15.56,1:00:17.86,Default,,0000,0000,0000,,do I need to do on the final?
Dialogue: 0,1:00:17.86,1:00:23.96,Default,,0000,0000,0000,,I have to say the\Nsame-- 2, 1, and 0.
Dialogue: 0,1:00:23.96,1:00:26.89,Default,,0000,0000,0000,,
Dialogue: 0,1:00:26.89,1:00:27.87,Default,,0000,0000,0000,,STUDENT: z can't be 0.
Dialogue: 0,1:00:27.87,1:00:30.11,Default,,0000,0000,0000,,PROFESSOR: No, I know.
Dialogue: 0,1:00:30.11,1:00:35.02,Default,,0000,0000,0000,,So I go, z is 0 is too easy.
Dialogue: 0,1:00:35.02,1:00:36.00,Default,,0000,0000,0000,,Let's have y to be 0.
Dialogue: 0,1:00:36.00,1:00:36.71,Default,,0000,0000,0000,,STUDENT: 2, 0, 1.
Dialogue: 0,1:00:36.71,1:00:39.93,Default,,0000,0000,0000,,PROFESSOR: Very good, x equals\N2, z equals 1, excellent.
Dialogue: 0,1:00:39.93,1:00:47.51,Default,,0000,0000,0000,,So z sub x at the\Npoint 2, 0, 1 will
Dialogue: 0,1:00:47.51,1:00:54.78,Default,,0000,0000,0000,,be by the implicit\Nfunction theorem minus 2/1
Dialogue: 0,1:00:54.78,1:00:55.81,Default,,0000,0000,0000,,equals negative.
Dialogue: 0,1:00:55.81,1:01:00.32,Default,,0000,0000,0000,,You see, that's a slope\Nin a certain direction
Dialogue: 0,1:01:00.32,1:01:06.13,Default,,0000,0000,0000,,if you were to look at z with\Nrespect to x in the plane x, z.
Dialogue: 0,1:01:06.13,1:01:09.03,Default,,0000,0000,0000,,OK, what else?
Dialogue: 0,1:01:09.03,1:01:13.41,Default,,0000,0000,0000,,Nothing-- that was review\Nof chain rule and stuff.
Dialogue: 0,1:01:13.41,1:01:15.39,Default,,0000,0000,0000,,And you have to\Nreview chain rule.
Dialogue: 0,1:01:15.39,1:01:18.34,Default,,0000,0000,0000,,
Dialogue: 0,1:01:18.34,1:01:19.84,Default,,0000,0000,0000,,Make yourself a note.
Dialogue: 0,1:01:19.84,1:01:22.74,Default,,0000,0000,0000,,Before the midterm, I have\Nto memorize the chain rule.
Dialogue: 0,1:01:22.74,1:01:23.68,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:01:23.68,1:01:24.55,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:01:24.55,1:01:28.21,Default,,0000,0000,0000,,
Dialogue: 0,1:01:28.21,1:01:32.12,Default,,0000,0000,0000,,PROFESSOR: I will do that either\Nin the review session today
Dialogue: 0,1:01:32.12,1:01:36.64,Default,,0000,0000,0000,,or in the review\Nfor the midterm, OK?
Dialogue: 0,1:01:36.64,1:01:40.49,Default,,0000,0000,0000,,And I'm thinking about that.
Dialogue: 0,1:01:40.49,1:01:43.77,Default,,0000,0000,0000,,In March, I want to\Ndedicate at least 10 days
Dialogue: 0,1:01:43.77,1:01:45.96,Default,,0000,0000,0000,,for the review for the midterm.
Dialogue: 0,1:01:45.96,1:01:46.46,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:01:46.46,1:01:47.67,Default,,0000,0000,0000,,STUDENT: When is the midterm?
Dialogue: 0,1:01:47.67,1:01:49.90,Default,,0000,0000,0000,,PROFESSOR: The midterm\Nis on the 2nd of April.
Dialogue: 0,1:01:49.90,1:01:55.10,Default,,0000,0000,0000,,
Dialogue: 0,1:01:55.10,1:01:57.90,Default,,0000,0000,0000,,Several people asked me--\NOK, I forgot about that.
Dialogue: 0,1:01:57.90,1:01:59.14,Default,,0000,0000,0000,,I have to tell you guys.
Dialogue: 0,1:01:59.14,1:02:00.81,Default,,0000,0000,0000,,Several people\Nasked me questions
Dialogue: 0,1:02:00.81,1:02:03.74,Default,,0000,0000,0000,,by email about the midterm.
Dialogue: 0,1:02:03.74,1:02:07.32,Default,,0000,0000,0000,,So the midterm-- write\Ndown for yourselves-- will
Dialogue: 0,1:02:07.32,1:02:08.64,Default,,0000,0000,0000,,be over the following chapters.
Dialogue: 0,1:02:08.64,1:02:11.38,Default,,0000,0000,0000,,
Dialogue: 0,1:02:11.38,1:02:20.86,Default,,0000,0000,0000,,Chapter 10, no Chapter 9.
Dialogue: 0,1:02:20.86,1:02:23.27,Default,,0000,0000,0000,,Chapter 9 is [INAUDIBLE].
Dialogue: 0,1:02:23.27,1:02:30.90,Default,,0000,0000,0000,,Chapter 11 all,\NChapter 10 only what
Dialogue: 0,1:02:30.90,1:02:36.82,Default,,0000,0000,0000,,we have required-- 10.1, 10.2,\Nand 10.3-- and Chapter 12,
Dialogue: 0,1:02:36.82,1:02:43.76,Default,,0000,0000,0000,,all but Section 12.6.
Dialogue: 0,1:02:43.76,1:02:46.83,Default,,0000,0000,0000,,Because I see that some of\Nyou study ahead of time.
Dialogue: 0,1:02:46.83,1:02:48.28,Default,,0000,0000,0000,,More power to you.
Dialogue: 0,1:02:48.28,1:02:49.80,Default,,0000,0000,0000,,You know what to read.
Dialogue: 0,1:02:49.80,1:02:51.01,Default,,0000,0000,0000,,Skip Section 12.6.
Dialogue: 0,1:02:51.01,1:02:57.48,Default,,0000,0000,0000,,And I'm planning to not give\Nyou anything after Chapter 13
Dialogue: 0,1:02:57.48,1:02:58.18,Default,,0000,0000,0000,,on the midterm.
Dialogue: 0,1:02:58.18,1:03:02.44,Default,,0000,0000,0000,,But of course, Chapter 13 will\Nbe on the final emphasized
Dialogue: 0,1:03:02.44,1:03:06.56,Default,,0000,0000,0000,,in at least six problems\Nout of the 15 problems
Dialogue: 0,1:03:06.56,1:03:11.18,Default,,0000,0000,0000,,you'll have on the\Nfinal, all right?
Dialogue: 0,1:03:11.18,1:03:15.12,Default,,0000,0000,0000,,We still have plenty of time.
Dialogue: 0,1:03:15.12,1:03:18.36,Default,,0000,0000,0000,,Chapter 9, guys, you\Nwere concerned about it.
Dialogue: 0,1:03:18.36,1:03:21.56,Default,,0000,0000,0000,,It's some sort of\Nembedded, you see?
Dialogue: 0,1:03:21.56,1:03:23.17,Default,,0000,0000,0000,,Wherever you go,\Nwherever you turn,
Dialogue: 0,1:03:23.17,1:03:26.74,Default,,0000,0000,0000,,you bump into some parametric\Nequations of a line
Dialogue: 0,1:03:26.74,1:03:28.01,Default,,0000,0000,0000,,or bump into a tangent line.
Dialogue: 0,1:03:28.01,1:03:33.65,Default,,0000,0000,0000,,That's the dot product that\Nyou dealt with, delta F dot N.
Dialogue: 0,1:03:33.65,1:03:38.05,Default,,0000,0000,0000,,So it's like an obsession,\Nrepetitive review of Chapter 9
Dialogue: 0,1:03:38.05,1:03:39.52,Default,,0000,0000,0000,,at ever step.
Dialogue: 0,1:03:39.52,1:03:40.94,Default,,0000,0000,0000,,Vector spaces are\Nvery important.
Dialogue: 0,1:03:40.94,1:03:44.86,Default,,0000,0000,0000,,Vectors in general\Nare very important.
Dialogue: 0,1:03:44.86,1:03:47.91,Default,,0000,0000,0000,,I'm going to move\Nonto 11.7 right now.
Dialogue: 0,1:03:47.91,1:03:49.49,Default,,0000,0000,0000,,We'll take a break.
Dialogue: 0,1:03:49.49,1:03:53.27,Default,,0000,0000,0000,,Why don't we take a short\Nbreak now, five minutes.
Dialogue: 0,1:03:53.27,1:03:58.50,Default,,0000,0000,0000,,And then we have to\Ngo on until 2:50.
Dialogue: 0,1:03:58.50,1:04:01.73,Default,,0000,0000,0000,,So practically we\Nhave one more hour.
Dialogue: 0,1:04:01.73,1:04:03.53,Default,,0000,0000,0000,,Take a break, eat,\Ndrink something.
Dialogue: 0,1:04:03.53,1:04:05.39,Default,,0000,0000,0000,,I don't want a big break.
Dialogue: 0,1:04:05.39,1:04:07.65,Default,,0000,0000,0000,,Because then a big break\Nwe'll just fall asleep.
Dialogue: 0,1:04:07.65,1:04:10.53,Default,,0000,0000,0000,,I'm tired as well.
Dialogue: 0,1:04:10.53,1:04:13.02,Default,,0000,0000,0000,,So we have to keep going.
Dialogue: 0,1:04:13.02,1:04:16.01,Default,,0000,0000,0000,,
Dialogue: 0,1:04:16.01,1:04:19.99,Default,,0000,0000,0000,,[BACKGROUND CHATTER]
Dialogue: 0,1:04:19.99,1:08:40.14,Default,,0000,0000,0000,,
Dialogue: 0,1:08:40.14,1:08:41.14,Default,,0000,0000,0000,,PROFESSOR: All right.
Dialogue: 0,1:08:41.14,1:08:43.79,Default,,0000,0000,0000,,I will start with a\Nlittle bit of a review
Dialogue: 0,1:08:43.79,1:08:46.10,Default,,0000,0000,0000,,of some friend of yours.
Dialogue: 0,1:08:46.10,1:08:49.08,Default,,0000,0000,0000,,And since we are\Nin Texas, of course
Dialogue: 0,1:08:49.08,1:08:50.57,Default,,0000,0000,0000,,this counts as an obsession.
Dialogue: 0,1:08:50.57,1:08:54.05,Default,,0000,0000,0000,,
Dialogue: 0,1:08:54.05,1:09:00.01,Default,,0000,0000,0000,,This is going to be extrema of\Nfunctions of several variables.
Dialogue: 0,1:09:00.01,1:09:15.42,Default,,0000,0000,0000,,
Dialogue: 0,1:09:15.42,1:09:16.81,Default,,0000,0000,0000,,Do I draw better lately?
Dialogue: 0,1:09:16.81,1:09:20.31,Default,,0000,0000,0000,,I think I do.
Dialogue: 0,1:09:20.31,1:09:22.03,Default,,0000,0000,0000,,That's why I stopped\Ndrinking coffee.
Dialogue: 0,1:09:22.03,1:09:23.70,Default,,0000,0000,0000,,I'm drinking white tea.
Dialogue: 0,1:09:23.70,1:09:26.35,Default,,0000,0000,0000,,It's good for you.
Dialogue: 0,1:09:26.35,1:09:27.50,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:09:27.50,1:09:28.47,Default,,0000,0000,0000,,White tea.
Dialogue: 0,1:09:28.47,1:09:30.68,Default,,0000,0000,0000,,For some reason, the\Nblack tea was giving me
Dialogue: 0,1:09:30.68,1:09:33.37,Default,,0000,0000,0000,,the shaking and all that.
Dialogue: 0,1:09:33.37,1:09:34.84,Default,,0000,0000,0000,,Too much black tea.
Dialogue: 0,1:09:34.84,1:09:37.78,Default,,0000,0000,0000,,I don't know, maybe\Nit has less caffeine.
Dialogue: 0,1:09:37.78,1:09:39.70,Default,,0000,0000,0000,,Jasmine is good,\Ngreen, or white.
Dialogue: 0,1:09:39.70,1:09:41.96,Default,,0000,0000,0000,,STUDENT: I think green\Nhas less [INAUDIBLE].
Dialogue: 0,1:09:41.96,1:09:43.59,Default,,0000,0000,0000,,PROFESSOR: OK.
Dialogue: 0,1:09:43.59,1:09:47.37,Default,,0000,0000,0000,,So above this\Nsaddle is a function
Dialogue: 0,1:09:47.37,1:09:52.21,Default,,0000,0000,0000,,of two variables-- you\Nknow a lot already,
Dialogue: 0,1:09:52.21,1:09:57.15,Default,,0000,0000,0000,,but I'm asking you to compute\Nthe partial derivatives
Dialogue: 0,1:09:57.15,1:09:59.60,Default,,0000,0000,0000,,and the gradient.
Dialogue: 0,1:09:59.60,1:10:01.49,Default,,0000,0000,0000,,And you're going to\Njump on it and say
Dialogue: 0,1:10:01.49,1:10:03.73,Default,,0000,0000,0000,,I'm doing [INAUDIBLE] anyway.
Dialogue: 0,1:10:03.73,1:10:08.89,Default,,0000,0000,0000,,So I've got 2x, and\Nthis is minus 2y.
Dialogue: 0,1:10:08.89,1:10:11.14,Default,,0000,0000,0000,,If I want to ask\Nyou the differential
Dialogue: 0,1:10:11.14,1:10:14.00,Default,,0000,0000,0000,,on the final or\Nmidterm, you will say
Dialogue: 0,1:10:14.00,1:10:19.19,Default,,0000,0000,0000,,that f sub xdx plus x of ygy.
Dialogue: 0,1:10:19.19,1:10:22.47,Default,,0000,0000,0000,,Everybody knows that.
Dialogue: 0,1:10:22.47,1:10:23.75,Default,,0000,0000,0000,,Don't break my heart.
Dialogue: 0,1:10:23.75,1:10:28.46,Default,,0000,0000,0000,,Don't say 2x minus y,\Nbecause I'll never recover.
Dialogue: 0,1:10:28.46,1:10:31.34,Default,,0000,0000,0000,,Every time I see that,\NI die 100 deaths.
Dialogue: 0,1:10:31.34,1:10:34.76,Default,,0000,0000,0000,,So don't forget about\Nthe x and the y,
Dialogue: 0,1:10:34.76,1:10:40.07,Default,,0000,0000,0000,,which are the important guys\Nof infinitesimal elements.
Dialogue: 0,1:10:40.07,1:10:42.41,Default,,0000,0000,0000,,This is a 1 form.
Dialogue: 0,1:10:42.41,1:10:47.58,Default,,0000,0000,0000,,In mathematics, any\Ncombination of a dx and dy
Dialogue: 0,1:10:47.58,1:10:50.50,Default,,0000,0000,0000,,in a linear combination\Nin the 1 form.
Dialogue: 0,1:10:50.50,1:10:54.54,Default,,0000,0000,0000,,It's a consecrated terminology.
Dialogue: 0,1:10:54.54,1:10:56.50,Default,,0000,0000,0000,,But I'm not asking you\Nabout the differential.
Dialogue: 0,1:10:56.50,1:10:59.56,Default,,0000,0000,0000,,I'm asking you\Nabout the gradient.
Dialogue: 0,1:10:59.56,1:11:06.81,Default,,0000,0000,0000,,All righty, and that is a f\Nsub xi plus f sub yk, which
Dialogue: 0,1:11:06.81,1:11:11.00,Default,,0000,0000,0000,,is exactly 2xi minus 2yj.
Dialogue: 0,1:11:11.00,1:11:14.69,Default,,0000,0000,0000,,
Dialogue: 0,1:11:14.69,1:11:18.27,Default,,0000,0000,0000,,And you say all right,\Nbut I want to take a look,
Dialogue: 0,1:11:18.27,1:11:20.26,Default,,0000,0000,0000,,I always have started\Nwith examples.
Dialogue: 0,1:11:20.26,1:11:23.10,Default,,0000,0000,0000,,Hopefully they are good.
Dialogue: 0,1:11:23.10,1:11:27.42,Default,,0000,0000,0000,,Let's look at the tangent\Nvectors to the surface.
Dialogue: 0,1:11:27.42,1:11:30.16,Default,,0000,0000,0000,,We discussed about the\Nnotion of tangent vector
Dialogue: 0,1:11:30.16,1:11:34.00,Default,,0000,0000,0000,,before, remember, when\Nwe had r sub u and r sub
Dialogue: 0,1:11:34.00,1:11:35.56,Default,,0000,0000,0000,,v form the parametrization.
Dialogue: 0,1:11:35.56,1:11:39.47,Default,,0000,0000,0000,,Now look at the tangent\Nvectors for this graph
Dialogue: 0,1:11:39.47,1:11:42.75,Default,,0000,0000,0000,,along the x direction\Ngoing this way,
Dialogue: 0,1:11:42.75,1:11:46.85,Default,,0000,0000,0000,,and along the y\Ndirection going this way.
Dialogue: 0,1:11:46.85,1:11:51.72,Default,,0000,0000,0000,,We see that both of them are\Nhorizontal at the origin.
Dialogue: 0,1:11:51.72,1:11:54.57,Default,,0000,0000,0000,,And that's a beautiful thing.
Dialogue: 0,1:11:54.57,1:12:00.75,Default,,0000,0000,0000,,And so this origin is a\Nso-called critical point.
Dialogue: 0,1:12:00.75,1:12:05.22,Default,,0000,0000,0000,,Critical point for a\Ndifferentiable function.
Dialogue: 0,1:12:05.22,1:12:13.09,Default,,0000,0000,0000,,
Dialogue: 0,1:12:13.09,1:12:23.54,Default,,0000,0000,0000,,Z equals f of xy is\Na point in plane x0i0
Dialogue: 0,1:12:23.54,1:12:30.92,Default,,0000,0000,0000,,where the partial\Nderivatives vanish.
Dialogue: 0,1:12:30.92,1:12:35.84,Default,,0000,0000,0000,,
Dialogue: 0,1:12:35.84,1:12:41.04,Default,,0000,0000,0000,,And according to the book, and\Nmany books, all don't exist.
Dialogue: 0,1:12:41.04,1:12:45.41,Default,,0000,0000,0000,,Well I don't like that.
Dialogue: 0,1:12:45.41,1:12:53.10,Default,,0000,0000,0000,,Even our book says if you\Nhave a function in calc 1--
Dialogue: 0,1:12:53.10,1:13:01.65,Default,,0000,0000,0000,,let's say b equal g\Nof u, critical point.
Dialogue: 0,1:13:01.65,1:13:05.44,Default,,0000,0000,0000,,Do you remember what\Na critical point was?
Dialogue: 0,1:13:05.44,1:13:12.40,Default,,0000,0000,0000,,U0, in calc 1 we said either a\Npoint where g prime of u was 0,
Dialogue: 0,1:13:12.40,1:13:16.57,Default,,0000,0000,0000,,or g prime of u 0 doesn't exist.
Dialogue: 0,1:13:16.57,1:13:19.47,Default,,0000,0000,0000,,
Dialogue: 0,1:13:19.47,1:13:22.81,Default,,0000,0000,0000,,Although u is 0\Nis in the domain.
Dialogue: 0,1:13:22.81,1:13:24.46,Default,,0000,0000,0000,,I don't like that.
Dialogue: 0,1:13:24.46,1:13:27.10,Default,,0000,0000,0000,,You say wait a minute,\Nwhy don't you like that?
Dialogue: 0,1:13:27.10,1:13:30.02,Default,,0000,0000,0000,,I don't like that for\Nmany reasons practically.
Dialogue: 0,1:13:30.02,1:13:37.63,Default,,0000,0000,0000,,If you have the\Nabsolute value function,
Dialogue: 0,1:13:37.63,1:13:41.52,Default,,0000,0000,0000,,you'll say yeah, yeah, but\Nlook, I considered the corner
Dialogue: 0,1:13:41.52,1:13:44.42,Default,,0000,0000,0000,,to be a point of\Nnon-differentiability,
Dialogue: 0,1:13:44.42,1:13:49.88,Default,,0000,0000,0000,,but it's still an extreme\Nvalue, a critical point.
Dialogue: 0,1:13:49.88,1:13:53.79,Default,,0000,0000,0000,,According to our book\Nin Calculus 1, yeah.
Dialogue: 0,1:13:53.79,1:13:57.40,Default,,0000,0000,0000,,We extended this\Ndefinition to ugly points,
Dialogue: 0,1:13:57.40,1:14:00.05,Default,,0000,0000,0000,,points where you don't have\Na [? pick ?] or a value
Dialogue: 0,1:14:00.05,1:14:04.66,Default,,0000,0000,0000,,or an inflection, but you\Nhave something ugly like
Dialogue: 0,1:14:04.66,1:14:08.66,Default,,0000,0000,0000,,a [? cusp, ?] a\Ncorner, the ugliness.
Dialogue: 0,1:14:08.66,1:14:10.38,Default,,0000,0000,0000,,I don't like that\Nkind of ugliness,
Dialogue: 0,1:14:10.38,1:14:14.66,Default,,0000,0000,0000,,because I want to have\Nmore information there.
Dialogue: 0,1:14:14.66,1:14:19.84,Default,,0000,0000,0000,,I maybe even have a point with\Na bigger problem than that.
Dialogue: 0,1:14:19.84,1:14:22.03,Default,,0000,0000,0000,,First of all, when I\Nsay critical point,
Dialogue: 0,1:14:22.03,1:14:24.86,Default,,0000,0000,0000,,I have to assume the point is\Nin the domain of the function.
Dialogue: 0,1:14:24.86,1:14:28.15,Default,,0000,0000,0000,,But then what kind of\Nugliness I can have there?
Dialogue: 0,1:14:28.15,1:14:30.20,Default,,0000,0000,0000,,I don't even want\Nto think about it.
Dialogue: 0,1:14:30.20,1:14:34.87,Default,,0000,0000,0000,,So in the context of\Nmy class-- in context
Dialogue: 0,1:14:34.87,1:14:47.01,Default,,0000,0000,0000,,of my class-- calc 3 honors, I\Nwill denote a critical point.
Dialogue: 0,1:14:47.01,1:14:51.15,Default,,0000,0000,0000,,
Dialogue: 0,1:14:51.15,1:15:01.31,Default,,0000,0000,0000,,Is the x0y0 such that\Nf sub x at x0y0 is 0.
Dialogue: 0,1:15:01.31,1:15:04.99,Default,,0000,0000,0000,,One slope is 0, the\Nother slope is 0.
Dialogue: 0,1:15:04.99,1:15:08.03,Default,,0000,0000,0000,,f sub y is x0y0, of course.
Dialogue: 0,1:15:08.03,1:15:14.01,Default,,0000,0000,0000,,And no other are the points.
Dialogue: 0,1:15:14.01,1:15:18.29,Default,,0000,0000,0000,,What am I going to call the\N[INAUDIBLE] points where
Dialogue: 0,1:15:18.29,1:15:24.14,Default,,0000,0000,0000,,derivatives don't exist?
Dialogue: 0,1:15:24.14,1:15:31.42,Default,,0000,0000,0000,,I simply say I\Nhave a singularity.
Dialogue: 0,1:15:31.42,1:15:33.65,Default,,0000,0000,0000,,I have a singularity.
Dialogue: 0,1:15:33.65,1:15:38.07,Default,,0000,0000,0000,,What type of singularity we can\Ndiscuss in an advanced calculus
Dialogue: 0,1:15:38.07,1:15:38.57,Default,,0000,0000,0000,,setting.
Dialogue: 0,1:15:38.57,1:15:42.14,Default,,0000,0000,0000,,If you're math majors, you're\Ngoing to have the chance
Dialogue: 0,1:15:42.14,1:15:43.83,Default,,0000,0000,0000,,to discuss that later on.
Dialogue: 0,1:15:43.83,1:15:49.06,Default,,0000,0000,0000,,So remember that I would\Nprefer both in the context
Dialogue: 0,1:15:49.06,1:15:54.60,Default,,0000,0000,0000,,of calculus 1 and calculus\N3 to say critical value
Dialogue: 0,1:15:54.60,1:15:58.55,Default,,0000,0000,0000,,is where the derivative\Nbecomes zero.
Dialogue: 0,1:15:58.55,1:16:02.56,Default,,0000,0000,0000,,Not undefined, plus, minus,\Ninfinity, or something
Dialogue: 0,1:16:02.56,1:16:05.87,Default,,0000,0000,0000,,really crazy, one on the\Nleft, one on the right.
Dialogue: 0,1:16:05.87,1:16:09.62,Default,,0000,0000,0000,,So I don't want to have\Nany kind of complications.
Dialogue: 0,1:16:09.62,1:16:14.64,Default,,0000,0000,0000,,Now you may say, but I\Nthought that since you
Dialogue: 0,1:16:14.64,1:16:17.07,Default,,0000,0000,0000,,have those slopes\Nboth zero, that
Dialogue: 0,1:16:17.07,1:16:21.42,Default,,0000,0000,0000,,means that the tangent plane\Nat the point is horizontal.
Dialogue: 0,1:16:21.42,1:16:23.07,Default,,0000,0000,0000,,And that's exactly what it is.
Dialogue: 0,1:16:23.07,1:16:23.78,Default,,0000,0000,0000,,I agree with you.
Dialogue: 0,1:16:23.78,1:16:26.52,Default,,0000,0000,0000,,If somebody would draw the\Ntangent plane to the surface,
Dialogue: 0,1:16:26.52,1:16:29.63,Default,,0000,0000,0000,,S-- S is for surface,\Nbut it's funny,
Dialogue: 0,1:16:29.63,1:16:32.21,Default,,0000,0000,0000,,S is also coming from saddles.
Dialogue: 0,1:16:32.21,1:16:36.92,Default,,0000,0000,0000,,So that's a saddle\Npoint, saddle surface.
Dialogue: 0,1:16:36.92,1:16:39.03,Default,,0000,0000,0000,,Origin is so-called\Nsaddle point.
Dialogue: 0,1:16:39.03,1:16:40.14,Default,,0000,0000,0000,,We don't know yet why.
Dialogue: 0,1:16:40.14,1:16:42.70,Default,,0000,0000,0000,,
Dialogue: 0,1:16:42.70,1:16:49.28,Default,,0000,0000,0000,,The tangent plane\Nat 0, at the origin,
Dialogue: 0,1:16:49.28,1:16:52.95,Default,,0000,0000,0000,,will be 0.0 in this case.
Dialogue: 0,1:16:52.95,1:16:54.07,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:16:54.07,1:16:55.68,Default,,0000,0000,0000,,Well, it's easy to see.
Dialogue: 0,1:16:55.68,1:17:01.22,Default,,0000,0000,0000,,z minus 0 equals f sub x,\Nx minus x0 plus f sub y,
Dialogue: 0,1:17:01.22,1:17:02.55,Default,,0000,0000,0000,,y minus y0.
Dialogue: 0,1:17:02.55,1:17:06.09,Default,,0000,0000,0000,,But this is 0 and that's\N0, so z equals zero.
Dialogue: 0,1:17:06.09,1:17:08.38,Default,,0000,0000,0000,,So thank you very much.
Dialogue: 0,1:17:08.38,1:17:09.24,Default,,0000,0000,0000,,Poor horse.
Dialogue: 0,1:17:09.24,1:17:14.54,Default,,0000,0000,0000,,I can take a horizontal\Nplane, imaginary plane
Dialogue: 0,1:17:14.54,1:17:22.19,Default,,0000,0000,0000,,and make it be tangent to\Nthe saddle in all directions
Dialogue: 0,1:17:22.19,1:17:24.43,Default,,0000,0000,0000,,at the point in the middle.
Dialogue: 0,1:17:24.43,1:17:29.85,Default,,0000,0000,0000,,
Dialogue: 0,1:17:29.85,1:17:30.53,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:17:30.53,1:17:32.61,Default,,0000,0000,0000,,STUDENT: So you're saying\N[? the critical ?] point
Dialogue: 0,1:17:32.61,1:17:33.43,Default,,0000,0000,0000,,is where both--
Dialogue: 0,1:17:33.43,1:17:35.87,Default,,0000,0000,0000,,PROFESSOR: Where both\Npartial derivatives vanish.
Dialogue: 0,1:17:35.87,1:17:37.82,Default,,0000,0000,0000,,They have to both vanish.
Dialogue: 0,1:17:37.82,1:17:40.14,Default,,0000,0000,0000,,In case of calculus\N1, of course there
Dialogue: 0,1:17:40.14,1:17:44.66,Default,,0000,0000,0000,,is only one derivative that\Nvanishes at that point.
Dialogue: 0,1:17:44.66,1:17:47.19,Default,,0000,0000,0000,,What if I were\Nin-- now, you see,
Dialogue: 0,1:17:47.19,1:17:49.93,Default,,0000,0000,0000,,the more you ask me\Nquestions, the more I think
Dialogue: 0,1:17:49.93,1:17:51.52,Default,,0000,0000,0000,,And it's a dangerous thing.
Dialogue: 0,1:17:51.52,1:17:56.15,Default,,0000,0000,0000,,What if I had z equals\Nf of x1, x2, x3, xn?
Dialogue: 0,1:17:56.15,1:17:59.31,Default,,0000,0000,0000,,Critical point would be where\Nall the partial derivatives
Dialogue: 0,1:17:59.31,1:18:00.28,Default,,0000,0000,0000,,will be zero.
Dialogue: 0,1:18:00.28,1:18:02.69,Default,,0000,0000,0000,,And then the situation\Nbecomes more complicated,
Dialogue: 0,1:18:02.69,1:18:05.02,Default,,0000,0000,0000,,but it's doable.
Dialogue: 0,1:18:05.02,1:18:11.32,Default,,0000,0000,0000,,The other is the classification\Nof special points.
Dialogue: 0,1:18:11.32,1:18:25.28,Default,,0000,0000,0000,,Classification of\Ncritical points
Dialogue: 0,1:18:25.28,1:18:30.50,Default,,0000,0000,0000,,based on second\Npartial derivatives.
Dialogue: 0,1:18:30.50,1:18:39.94,Default,,0000,0000,0000,,
Dialogue: 0,1:18:39.94,1:18:44.84,Default,,0000,0000,0000,,The objects you want to study\Nin this case are several.
Dialogue: 0,1:18:44.84,1:18:48.16,Default,,0000,0000,0000,,
Dialogue: 0,1:18:48.16,1:18:52.66,Default,,0000,0000,0000,,One of the most important ones\Nis the so-called discriminant.
Dialogue: 0,1:18:52.66,1:18:54.79,Default,,0000,0000,0000,,What is the discriminant?
Dialogue: 0,1:18:54.79,1:18:58.54,Default,,0000,0000,0000,,You haven't talked about\Ndiscriminants since a long time
Dialogue: 0,1:18:58.54,1:18:59.35,Default,,0000,0000,0000,,ago.
Dialogue: 0,1:18:59.35,1:19:02.04,Default,,0000,0000,0000,,And there is a relationship\Nbetween discriminant
Dialogue: 0,1:19:02.04,1:19:09.07,Default,,0000,0000,0000,,in high school algebra and\Ndiscriminant in calculus 3.
Dialogue: 0,1:19:09.07,1:19:10.86,Default,,0000,0000,0000,,The discriminant\Nthe way we define
Dialogue: 0,1:19:10.86,1:19:13.97,Default,,0000,0000,0000,,it is D, or delta--\Nsome people denote it
Dialogue: 0,1:19:13.97,1:19:16.93,Default,,0000,0000,0000,,like this, some\Npeople by delta--
Dialogue: 0,1:19:16.93,1:19:19.36,Default,,0000,0000,0000,,and that is the following.
Dialogue: 0,1:19:19.36,1:19:21.92,Default,,0000,0000,0000,,This is the determinant.
Dialogue: 0,1:19:21.92,1:19:29.97,Default,,0000,0000,0000,,f sub xx, f sub xy,\Nf sub yx, f sub yy,
Dialogue: 0,1:19:29.97,1:19:32.30,Default,,0000,0000,0000,,computed at the point\Np0, which is critical.
Dialogue: 0,1:19:32.30,1:19:35.90,Default,,0000,0000,0000,,
Dialogue: 0,1:19:35.90,1:19:40.44,Default,,0000,0000,0000,,So p0 first has to satisfy\Nthose two equations,
Dialogue: 0,1:19:40.44,1:19:42.69,Default,,0000,0000,0000,,and then I'm going to have\Nto compute the [INAUDIBLE]
Dialogue: 0,1:19:42.69,1:19:44.56,Default,,0000,0000,0000,,at that point.
Dialogue: 0,1:19:44.56,1:19:46.26,Default,,0000,0000,0000,,But you say wait a\Nminute, Magdalena,
Dialogue: 0,1:19:46.26,1:19:47.57,Default,,0000,0000,0000,,what the heck is this?
Dialogue: 0,1:19:47.57,1:19:51.10,Default,,0000,0000,0000,,Well this is the second\Npartial with respect
Dialogue: 0,1:19:51.10,1:19:55.20,Default,,0000,0000,0000,,of x, one after the other,\Nsecond partial with respect
Dialogue: 0,1:19:55.20,1:19:56.88,Default,,0000,0000,0000,,to y, one after the other.
Dialogue: 0,1:19:56.88,1:19:58.08,Default,,0000,0000,0000,,These guys are equal.
Dialogue: 0,1:19:58.08,1:20:01.46,Default,,0000,0000,0000,,Remember that there was\Na German mathematician
Dialogue: 0,1:20:01.46,1:20:06.60,Default,,0000,0000,0000,,whose name was Schwartz, the\Nblack cavalier, the black man.
Dialogue: 0,1:20:06.60,1:20:08.51,Default,,0000,0000,0000,,Schwartz means black in German.
Dialogue: 0,1:20:08.51,1:20:10.37,Default,,0000,0000,0000,,And he came up with\Nthis theorem that it
Dialogue: 0,1:20:10.37,1:20:14.77,Default,,0000,0000,0000,,doesn't matter in which order\Nyou differentiate, f sub xy
Dialogue: 0,1:20:14.77,1:20:19.25,Default,,0000,0000,0000,,or f sub yx is the same thing as\Nlong as the function is smooth.
Dialogue: 0,1:20:19.25,1:20:22.32,Default,,0000,0000,0000,,So I'm very happy about that.
Dialogue: 0,1:20:22.32,1:20:26.48,Default,,0000,0000,0000,,Now there are these\Nother guys, A, B,
Dialogue: 0,1:20:26.48,1:20:30.79,Default,,0000,0000,0000,,C. It's very easy to\Nremember, it's from the song
Dialogue: 0,1:20:30.79,1:20:33.07,Default,,0000,0000,0000,,that you all learned\Nin kindergarten.
Dialogue: 0,1:20:33.07,1:20:37.17,Default,,0000,0000,0000,,Once you know your ABC, you\Ncome back to the discriminant.
Dialogue: 0,1:20:37.17,1:20:45.30,Default,,0000,0000,0000,,So f sub xx at the point p0,\Nf sub xy at the point p0,
Dialogue: 0,1:20:45.30,1:20:50.37,Default,,0000,0000,0000,,and f sub yy at the point p0.
Dialogue: 0,1:20:50.37,1:20:53.14,Default,,0000,0000,0000,,Second partial with respect to\Nx, second partial with respect
Dialogue: 0,1:20:53.14,1:20:56.87,Default,,0000,0000,0000,,to x and y, mixed one,\Nmixed derivative, and second
Dialogue: 0,1:20:56.87,1:20:58.26,Default,,0000,0000,0000,,partial with respect to y.
Dialogue: 0,1:20:58.26,1:21:01.17,Default,,0000,0000,0000,,
Dialogue: 0,1:21:01.17,1:21:06.96,Default,,0000,0000,0000,,You have to plug in the values\Nfor the p0 will be x0, y0.
Dialogue: 0,1:21:06.96,1:21:09.89,Default,,0000,0000,0000,,The critical point\Nyou got from what?
Dialogue: 0,1:21:09.89,1:21:12.69,Default,,0000,0000,0000,,From solving this system.
Dialogue: 0,1:21:12.69,1:21:16.21,Default,,0000,0000,0000,,So you got x0y0 by\Nsolving that system.
Dialogue: 0,1:21:16.21,1:21:21.93,Default,,0000,0000,0000,,Come back, plug in, compute\Nthose, get ABC as numbers.
Dialogue: 0,1:21:21.93,1:21:26.09,Default,,0000,0000,0000,,And who is D going\Nto be based on ABC?
Dialogue: 0,1:21:26.09,1:21:30.02,Default,,0000,0000,0000,,According to the diagram that\NI drew, it's easy for you guys
Dialogue: 0,1:21:30.02,1:21:35.91,Default,,0000,0000,0000,,to see that A and\NB and C are what?
Dialogue: 0,1:21:35.91,1:21:46.39,Default,,0000,0000,0000,,Related to D. So D will\Nsimply be A, B, B, and C,
Dialogue: 0,1:21:46.39,1:21:48.37,Default,,0000,0000,0000,,computed at the point p0.
Dialogue: 0,1:21:48.37,1:21:51.34,Default,,0000,0000,0000,,
Dialogue: 0,1:21:51.34,1:21:54.34,Default,,0000,0000,0000,,So it's going to\Nbe now-- now that
Dialogue: 0,1:21:54.34,1:21:56.56,Default,,0000,0000,0000,,will remind you of something.
Dialogue: 0,1:21:56.56,1:21:58.45,Default,,0000,0000,0000,,AC minus B-squared.
Dialogue: 0,1:21:58.45,1:22:02.71,Default,,0000,0000,0000,,
Dialogue: 0,1:22:02.71,1:22:03.52,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:22:03.52,1:22:07.31,Default,,0000,0000,0000,,When we had the quadratic\Nformula in school--
Dialogue: 0,1:22:07.31,1:22:08.99,Default,,0000,0000,0000,,I'm not going to write it.
Dialogue: 0,1:22:08.99,1:22:10.12,Default,,0000,0000,0000,,I'm going to write it here.
Dialogue: 0,1:22:10.12,1:22:12.01,Default,,0000,0000,0000,,So what was the\Nquadratic formula?
Dialogue: 0,1:22:12.01,1:22:15.23,Default,,0000,0000,0000,,ax-squared plus bx\Nplus c equals 0.
Dialogue: 0,1:22:15.23,1:22:17.15,Default,,0000,0000,0000,,That was algebra.
Dialogue: 0,1:22:17.15,1:22:18.06,Default,,0000,0000,0000,,Baby algebra.
Dialogue: 0,1:22:18.06,1:22:19.00,Default,,0000,0000,0000,,What do we call that?
Dialogue: 0,1:22:19.00,1:22:21.42,Default,,0000,0000,0000,,High school algebra?
Dialogue: 0,1:22:21.42,1:22:28.50,Default,,0000,0000,0000,,x12 plus minus b plus minus\Nsquare root of b-squared minus
Dialogue: 0,1:22:28.50,1:22:33.52,Default,,0000,0000,0000,,4ac divided by 2a.
Dialogue: 0,1:22:33.52,1:22:37.34,Default,,0000,0000,0000,,Now don't don;t know what\Nkind of professors you had.
Dialogue: 0,1:22:37.34,1:22:41.76,Default,,0000,0000,0000,,But I had a teacher when\NI was in high school.
Dialogue: 0,1:22:41.76,1:22:44.55,Default,,0000,0000,0000,,Every time she taught me\Nsomething and I did not
Dialogue: 0,1:22:44.55,1:22:46.86,Default,,0000,0000,0000,,absorb it, she was all over me.
Dialogue: 0,1:22:46.86,1:22:49.29,Default,,0000,0000,0000,,She was preparing me for\Nsome math competitions,
Dialogue: 0,1:22:49.29,1:22:51.39,Default,,0000,0000,0000,,and she taught me a trick.
Dialogue: 0,1:22:51.39,1:22:54.00,Default,,0000,0000,0000,,She said look,\NMagdalena, pay attention.
Dialogue: 0,1:22:54.00,1:23:01.44,Default,,0000,0000,0000,,If b would be an even number--\Ntake b to be 2b prime,
Dialogue: 0,1:23:01.44,1:23:04.43,Default,,0000,0000,0000,,2-- give me another letter.
Dialogue: 0,1:23:04.43,1:23:07.88,Default,,0000,0000,0000,,2 big B. Right?
Dialogue: 0,1:23:07.88,1:23:11.86,Default,,0000,0000,0000,,Then, the quadratic formula\Nwould be easier to use.
Dialogue: 0,1:23:11.86,1:23:17.66,Default,,0000,0000,0000,,Because in that case, you get\Nx1 2 equals minus-- b is 2b.
Dialogue: 0,1:23:17.66,1:23:20.86,Default,,0000,0000,0000,,So you have just 2b like that.
Dialogue: 0,1:23:20.86,1:23:24.77,Default,,0000,0000,0000,,Plus minus square\Nroot 4b squared
Dialogue: 0,1:23:24.77,1:23:29.08,Default,,0000,0000,0000,,minus 4ac divided by 2a.
Dialogue: 0,1:23:29.08,1:23:31.46,Default,,0000,0000,0000,,She explained this\Nto me once and then
Dialogue: 0,1:23:31.46,1:23:34.86,Default,,0000,0000,0000,,she expected me to remember\Nit for the rest of my life.
Dialogue: 0,1:23:34.86,1:23:42.10,Default,,0000,0000,0000,,And then she said minus big B\Nplus minus square root of bb
Dialogue: 0,1:23:42.10,1:23:44.00,Default,,0000,0000,0000,,squared minus ac.
Dialogue: 0,1:23:44.00,1:23:45.22,Default,,0000,0000,0000,,Do you see why?
Dialogue: 0,1:23:45.22,1:23:50.14,Default,,0000,0000,0000,,It's because you pull out the\Nfactor of 4, square root of 4
Dialogue: 0,1:23:50.14,1:23:51.30,Default,,0000,0000,0000,,is 2.
Dialogue: 0,1:23:51.30,1:23:53.60,Default,,0000,0000,0000,,2, 2, and 2 simplify.
Dialogue: 0,1:23:53.60,1:23:56.37,Default,,0000,0000,0000,,And then she gave me\Nto solve problems.
Dialogue: 0,1:23:56.37,1:23:57.48,Default,,0000,0000,0000,,STUDENT: What about the a?
Dialogue: 0,1:23:57.48,1:23:58.52,Default,,0000,0000,0000,,STUDENT: How about the a?
Dialogue: 0,1:23:58.52,1:24:00.44,Default,,0000,0000,0000,,STUDENT: Because you\Ndivide it by [INAUDIBLE].
Dialogue: 0,1:24:00.44,1:24:01.49,Default,,0000,0000,0000,,PROFESSOR: Divided by.
Dialogue: 0,1:24:01.49,1:24:04.10,Default,,0000,0000,0000,,I forgot to write it down.
Dialogue: 0,1:24:04.10,1:24:05.76,Default,,0000,0000,0000,,Because I didn't have space.
Dialogue: 0,1:24:05.76,1:24:09.08,Default,,0000,0000,0000,,I said, I'm not going\Nto bend and doodle.
Dialogue: 0,1:24:09.08,1:24:17.96,Default,,0000,0000,0000,,So when you have x-squared\Nplus 2x-- let's say minus 3.
Dialogue: 0,1:24:17.96,1:24:20.38,Default,,0000,0000,0000,,And she gave me that.
Dialogue: 0,1:24:20.38,1:24:21.68,Default,,0000,0000,0000,,And I said OK, let me do it.
Dialogue: 0,1:24:21.68,1:24:22.25,Default,,0000,0000,0000,,Let me do it.
Dialogue: 0,1:24:22.25,1:24:30.70,Default,,0000,0000,0000,,x1 2 minus 2 plus minus square\Nroot b-squared minus 4ac,
Dialogue: 0,1:24:30.70,1:24:34.09,Default,,0000,0000,0000,,which is 12, divided by 2.
Dialogue: 0,1:24:34.09,1:24:36.03,Default,,0000,0000,0000,,And she started screaming.
Dialogue: 0,1:24:36.03,1:24:37.58,Default,,0000,0000,0000,,And she started\Nscreaming big time.
Dialogue: 0,1:24:37.58,1:24:38.96,Default,,0000,0000,0000,,Do you know why?
Dialogue: 0,1:24:38.96,1:24:44.34,Default,,0000,0000,0000,,She said, I just told\Nyou the half formula.
Dialogue: 0,1:24:44.34,1:24:48.04,Default,,0000,0000,0000,,By half formula, I mean\Nshe meant this one.
Dialogue: 0,1:24:48.04,1:24:50.85,Default,,0000,0000,0000,,So when-- and I said OK,\NOK, the half formula.
Dialogue: 0,1:24:50.85,1:24:53.86,Default,,0000,0000,0000,,But then for maybe\Nanother seven years,
Dialogue: 0,1:24:53.86,1:24:57.23,Default,,0000,0000,0000,,I did this with the\Nformula-- with the formula
Dialogue: 0,1:24:57.23,1:24:59.81,Default,,0000,0000,0000,,that everybody knows.
Dialogue: 0,1:24:59.81,1:25:01.53,Default,,0000,0000,0000,,And at the end, I\Nwould remember I
Dialogue: 0,1:25:01.53,1:25:03.00,Default,,0000,0000,0000,,could have done\Nthe half formula,
Dialogue: 0,1:25:03.00,1:25:07.64,Default,,0000,0000,0000,,but I didn't do it\Nbecause I'm in a routine.
Dialogue: 0,1:25:07.64,1:25:11.51,Default,,0000,0000,0000,,So the way she wanted\Nme to do this was what?
Dialogue: 0,1:25:11.51,1:25:13.65,Default,,0000,0000,0000,,Who is the half of 2?
Dialogue: 0,1:25:13.65,1:25:14.69,Default,,0000,0000,0000,,1.
Dialogue: 0,1:25:14.69,1:25:21.26,Default,,0000,0000,0000,,So put minus 1 plus minus\Nsquare root of big B
Dialogue: 0,1:25:21.26,1:25:26.82,Default,,0000,0000,0000,,is 1-squared minus\Na times c, which
Dialogue: 0,1:25:26.82,1:25:30.60,Default,,0000,0000,0000,,is plus 3, divided by\N1 divided by nobody.
Dialogue: 0,1:25:30.60,1:25:33.15,Default,,0000,0000,0000,,This way you don't have\Nto simplify it further,
Dialogue: 0,1:25:33.15,1:25:36.04,Default,,0000,0000,0000,,and you do it faster.
Dialogue: 0,1:25:36.04,1:25:46.06,Default,,0000,0000,0000,,So you get minus 1 plus minus\Nroot 4, which is minus 5 and 3.
Dialogue: 0,1:25:46.06,1:25:49.20,Default,,0000,0000,0000,,But of course, you could\Nhave done this by factoring.
Dialogue: 0,1:25:49.20,1:25:51.51,Default,,0000,0000,0000,,So you could have\Nsaid wait a minute.
Dialogue: 0,1:25:51.51,1:25:55.02,Default,,0000,0000,0000,,Two numbers that multiply-- um--
Dialogue: 0,1:25:55.02,1:25:56.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nsquare root.
Dialogue: 0,1:25:56.46,1:25:57.67,Default,,0000,0000,0000,,PROFESSOR: I didn't do right.
Dialogue: 0,1:25:57.67,1:25:58.38,Default,,0000,0000,0000,,So it's 4--
Dialogue: 0,1:25:58.38,1:26:00.78,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:26:00.78,1:26:01.75,Default,,0000,0000,0000,,PROFESSOR: Yeah.
Dialogue: 0,1:26:01.75,1:26:07.07,Default,,0000,0000,0000,,So you get x plus 5 times--
Dialogue: 0,1:26:07.07,1:26:09.40,Default,,0000,0000,0000,,STUDENT: It's x minus 1--
Dialogue: 0,1:26:09.40,1:26:10.100,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:26:10.100,1:26:12.08,Default,,0000,0000,0000,,PROFESSOR: Oh, I think I--
Dialogue: 0,1:26:12.08,1:26:12.96,Default,,0000,0000,0000,,STUDENT: Square root.
Dialogue: 0,1:26:12.96,1:26:13.91,Default,,0000,0000,0000,,PROFESSOR: Square root.
Dialogue: 0,1:26:13.91,1:26:15.55,Default,,0000,0000,0000,,I'm sorry, guys.
Dialogue: 0,1:26:15.55,1:26:16.05,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:26:16.05,1:26:17.50,Default,,0000,0000,0000,,Thank you for that.
Dialogue: 0,1:26:17.50,1:26:18.98,Default,,0000,0000,0000,,1 and minus 3.
Dialogue: 0,1:26:18.98,1:26:25.35,Default,,0000,0000,0000,,So x plus 3 times\Nx minus 1, which
Dialogue: 0,1:26:25.35,1:26:28.90,Default,,0000,0000,0000,,is the same-- the exact same\Nas x-squared plus 2x minus 3
Dialogue: 0,1:26:28.90,1:26:30.21,Default,,0000,0000,0000,,equals.
Dialogue: 0,1:26:30.21,1:26:31.16,Default,,0000,0000,0000,,All right?
Dialogue: 0,1:26:31.16,1:26:37.38,Default,,0000,0000,0000,,So just the way she insisted\Nthat I learn the half formula.
Dialogue: 0,1:26:37.38,1:26:42.02,Default,,0000,0000,0000,,I'm not insisting that you learn\Nthe half formula, god forbid.
Dialogue: 0,1:26:42.02,1:26:45.54,Default,,0000,0000,0000,,But see here there is\Nsome more symmetry.
Dialogue: 0,1:26:45.54,1:26:48.64,Default,,0000,0000,0000,,The four doesn't appear anymore.
Dialogue: 0,1:26:48.64,1:26:52.48,Default,,0000,0000,0000,,b-squared minus 4ac appeared\Nhere, but here it doesn't.
Dialogue: 0,1:26:52.48,1:26:55.36,Default,,0000,0000,0000,,Here you're going to\Nhave b-squared minus ac.
Dialogue: 0,1:26:55.36,1:26:56.47,Default,,0000,0000,0000,,There is a reason.
Dialogue: 0,1:26:56.47,1:27:00.09,Default,,0000,0000,0000,,This comes from a\Ndiscriminant just like that.
Dialogue: 0,1:27:00.09,1:27:02.48,Default,,0000,0000,0000,,And this is why I told\Nyou the whole secret
Dialogue: 0,1:27:02.48,1:27:06.01,Default,,0000,0000,0000,,about the half\Nquadratic formula.
Dialogue: 0,1:27:06.01,1:27:08.21,Default,,0000,0000,0000,,Not because I wanted\Nyou to know about it,
Dialogue: 0,1:27:08.21,1:27:13.51,Default,,0000,0000,0000,,but because I wanted you to see\Nthat there is a pattern here.
Dialogue: 0,1:27:13.51,1:27:15.90,Default,,0000,0000,0000,,You have-- for the\Nhalf formula, you
Dialogue: 0,1:27:15.90,1:27:20.78,Default,,0000,0000,0000,,have plus minus square root\Nof a new type of discriminant.
Dialogue: 0,1:27:20.78,1:27:24.26,Default,,0000,0000,0000,,People even call this\Ndiscriminant b-squared
Dialogue: 0,1:27:24.26,1:27:25.28,Default,,0000,0000,0000,,minus 4ac.
Dialogue: 0,1:27:25.28,1:27:26.71,Default,,0000,0000,0000,,b-squared minus ac.
Dialogue: 0,1:27:26.71,1:27:30.51,Default,,0000,0000,0000,,So for us, it is\Nac minus b-squared.
Dialogue: 0,1:27:30.51,1:27:35.49,Default,,0000,0000,0000,,It's just the opposite of\Nthat discriminant you have.
Dialogue: 0,1:27:35.49,1:27:39.74,Default,,0000,0000,0000,,Now depending on the sign\Nof this discriminant,
Dialogue: 0,1:27:39.74,1:27:48.35,Default,,0000,0000,0000,,you can go ahead and classify\Nthe critical values you have.
Dialogue: 0,1:27:48.35,1:27:51.60,Default,,0000,0000,0000,,So classification\Nis the following.
Dialogue: 0,1:27:51.60,1:27:56.97,Default,,0000,0000,0000,,Classification of\Nspecial critical points.
Dialogue: 0,1:27:56.97,1:27:59.85,Default,,0000,0000,0000,,
Dialogue: 0,1:27:59.85,1:28:07.23,Default,,0000,0000,0000,,If delta at p0 is negative,\Nthen p0 is a saddle point.
Dialogue: 0,1:28:07.23,1:28:11.65,Default,,0000,0000,0000,,
Dialogue: 0,1:28:11.65,1:28:21.16,Default,,0000,0000,0000,,If delta at p0 is 0,\Nnothing can be said yet
Dialogue: 0,1:28:21.16,1:28:23.86,Default,,0000,0000,0000,,about the nature of the point.
Dialogue: 0,1:28:23.86,1:28:26.87,Default,,0000,0000,0000,,So I make a face, a sad face.
Dialogue: 0,1:28:26.87,1:28:41.20,Default,,0000,0000,0000,,If delta at p0 is greater than\N0, then I have to ramify again.
Dialogue: 0,1:28:41.20,1:28:47.36,Default,,0000,0000,0000,,And I get if a is positive,\Nit's going to look like a smile.
Dialogue: 0,1:28:47.36,1:28:48.93,Default,,0000,0000,0000,,Forget about this side.
Dialogue: 0,1:28:48.93,1:28:51.33,Default,,0000,0000,0000,,It's going to look like a smile.
Dialogue: 0,1:28:51.33,1:28:57.65,Default,,0000,0000,0000,,So it's going to be a valley\Npoint, what do we call that?
Dialogue: 0,1:28:57.65,1:29:02.65,Default,,0000,0000,0000,,Relative minimum,\Nor valley point.
Dialogue: 0,1:29:02.65,1:29:05.14,Default,,0000,0000,0000,,Don't say valley\Npoint on the exam, OK?
Dialogue: 0,1:29:05.14,1:29:05.93,Default,,0000,0000,0000,,Relative minimum.
Dialogue: 0,1:29:05.93,1:29:10.04,Default,,0000,0000,0000,,If a is less than\N0 at the point,
Dialogue: 0,1:29:10.04,1:29:13.61,Default,,0000,0000,0000,,then locally the\Nsurface will look
Dialogue: 0,1:29:13.61,1:29:21.03,Default,,0000,0000,0000,,like I have a peak-- a relative\Nmaximum Peaks and valleys.
Dialogue: 0,1:29:21.03,1:29:24.21,Default,,0000,0000,0000,,Just the way you\Nremember them in Calc 1.
Dialogue: 0,1:29:24.21,1:29:25.79,Default,,0000,0000,0000,,Now it's a little\Nbit more complicated
Dialogue: 0,1:29:25.79,1:29:29.12,Default,,0000,0000,0000,,because the functions\Nhave two variables.
Dialogue: 0,1:29:29.12,1:29:33.31,Default,,0000,0000,0000,,But some of the patterns\Ncan be recognized.
Dialogue: 0,1:29:33.31,1:29:36.34,Default,,0000,0000,0000,,
Dialogue: 0,1:29:36.34,1:29:42.54,Default,,0000,0000,0000,,Let's go back to\Nour original example
Dialogue: 0,1:29:42.54,1:29:45.02,Default,,0000,0000,0000,,and say wait a\Nminute, Magdalena.
Dialogue: 0,1:29:45.02,1:29:47.80,Default,,0000,0000,0000,,You just gave us a\Nsaddle, but we didn't
Dialogue: 0,1:29:47.80,1:29:49.76,Default,,0000,0000,0000,,do the whole classification.
Dialogue: 0,1:29:49.76,1:29:54.82,Default,,0000,0000,0000,,Yes, we didn't, because I\Ndidn't go over the next steps.
Dialogue: 0,1:29:54.82,1:29:57.43,Default,,0000,0000,0000,,z equals x-squared\Nminus y-squared.
Dialogue: 0,1:29:57.43,1:29:59.29,Default,,0000,0000,0000,,Again, we computed the gradient.
Dialogue: 0,1:29:59.29,1:30:01.70,Default,,0000,0000,0000,,We computed the\Npartial derivatives.
Dialogue: 0,1:30:01.70,1:30:07.77,Default,,0000,0000,0000,,And then what was that one in\Nfinding the critical points?
Dialogue: 0,1:30:07.77,1:30:11.79,Default,,0000,0000,0000,,So f sub x equals\N0, f sub y equals 0.
Dialogue: 0,1:30:11.79,1:30:14.34,Default,,0000,0000,0000,,Solve for x and y.
Dialogue: 0,1:30:14.34,1:30:17.17,Default,,0000,0000,0000,,
Dialogue: 0,1:30:17.17,1:30:19.05,Default,,0000,0000,0000,,And that's good,\Nbecause that's going
Dialogue: 0,1:30:19.05,1:30:23.04,Default,,0000,0000,0000,,to give me a lot of\Ninformation, a lot that
Dialogue: 0,1:30:23.04,1:30:25.93,Default,,0000,0000,0000,,will give me exactly where\Nthe critical points may be.
Dialogue: 0,1:30:25.93,1:30:32.21,Default,,0000,0000,0000,,So that is if and only if I need\Nto solve 2x equals 0 minus 2y
Dialogue: 0,1:30:32.21,1:30:33.89,Default,,0000,0000,0000,,equals 0.
Dialogue: 0,1:30:33.89,1:30:36.15,Default,,0000,0000,0000,,Is this system hard to solve?
Dialogue: 0,1:30:36.15,1:30:36.83,Default,,0000,0000,0000,,No.
Dialogue: 0,1:30:36.83,1:30:39.20,Default,,0000,0000,0000,,That's exactly why I picked it.
Dialogue: 0,1:30:39.20,1:30:41.49,Default,,0000,0000,0000,,Because it's easy to solve.
Dialogue: 0,1:30:41.49,1:30:47.45,Default,,0000,0000,0000,,The only solution is\Nx0 equals y0 equals 0.
Dialogue: 0,1:30:47.45,1:30:52.40,Default,,0000,0000,0000,,So the origin--\Nthat's exactly where
Dialogue: 0,1:30:52.40,1:30:59.79,Default,,0000,0000,0000,,you put your butt on the\Nsaddle when you ride the horse.
Dialogue: 0,1:30:59.79,1:31:04.19,Default,,0000,0000,0000,,That is the only\Ncritical point you have.
Dialogue: 0,1:31:04.19,1:31:05.67,Default,,0000,0000,0000,,The only one.
Dialogue: 0,1:31:05.67,1:31:11.52,Default,,0000,0000,0000,,Now if we want to classify that,\Nwhat kind of-- is it a valley?
Dialogue: 0,1:31:11.52,1:31:12.77,Default,,0000,0000,0000,,No.
Dialogue: 0,1:31:12.77,1:31:15.40,Default,,0000,0000,0000,,It looks like a valley\Nin the direction
Dialogue: 0,1:31:15.40,1:31:20.94,Default,,0000,0000,0000,,of the axis of the horse,\NBecause the saddle's
Dialogue: 0,1:31:20.94,1:31:22.90,Default,,0000,0000,0000,,going to look like that.
Dialogue: 0,1:31:22.90,1:31:23.65,Default,,0000,0000,0000,,This is the horse.
Dialogue: 0,1:31:23.65,1:31:25.97,Default,,0000,0000,0000,,That's the head of\Nthe horse I'm petting.
Dialogue: 0,1:31:25.97,1:31:29.15,Default,,0000,0000,0000,,And this is the\Ntail of the horse.
Dialogue: 0,1:31:29.15,1:31:35.04,Default,,0000,0000,0000,,So in this direction, the saddle\Nwill be shaped like a parabola,
Dialogue: 0,1:31:35.04,1:31:36.19,Default,,0000,0000,0000,,like a valley.
Dialogue: 0,1:31:36.19,1:31:40.56,Default,,0000,0000,0000,,But in the\Nperpendicular direction,
Dialogue: 0,1:31:40.56,1:31:43.25,Default,,0000,0000,0000,,it's going to be\Nshaped going down,
Dialogue: 0,1:31:43.25,1:31:45.28,Default,,0000,0000,0000,,like a parabola going down.
Dialogue: 0,1:31:45.28,1:31:49.11,Default,,0000,0000,0000,,So it's neither a\Nvalley nor a peak.
Dialogue: 0,1:31:49.11,1:31:51.25,Default,,0000,0000,0000,,It's a valley in one\Ndirection, and a peak
Dialogue: 0,1:31:51.25,1:31:52.30,Default,,0000,0000,0000,,in another direction.
Dialogue: 0,1:31:52.30,1:31:55.17,Default,,0000,0000,0000,,And that's the saddle point.
Dialogue: 0,1:31:55.17,1:31:56.37,Default,,0000,0000,0000,,So say it again.
Dialogue: 0,1:31:56.37,1:31:57.32,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:31:57.32,1:32:00.73,Default,,0000,0000,0000,,It looks like a valley in\None principle direction
Dialogue: 0,1:32:00.73,1:32:05.45,Default,,0000,0000,0000,,and the peak in the other\Nprinciple direction.
Dialogue: 0,1:32:05.45,1:32:08.69,Default,,0000,0000,0000,,And then that's going\Nto be a saddle point.
Dialogue: 0,1:32:08.69,1:32:14.28,Default,,0000,0000,0000,,Indeed, how do we figure this\Nout by the method I provided?
Dialogue: 0,1:32:14.28,1:32:16.80,Default,,0000,0000,0000,,Well, who is A?
Dialogue: 0,1:32:16.80,1:32:21.72,Default,,0000,0000,0000,,A is f sub xx at the point.
Dialogue: 0,1:32:21.72,1:32:24.60,Default,,0000,0000,0000,,2x goes primed one time.
Dialogue: 0,1:32:24.60,1:32:28.26,Default,,0000,0000,0000,,f sub x was 2x.
Dialogue: 0,1:32:28.26,1:32:30.42,Default,,0000,0000,0000,,f sub y was 2y.
Dialogue: 0,1:32:30.42,1:32:32.32,Default,,0000,0000,0000,,f sub xx is 2.
Dialogue: 0,1:32:32.32,1:32:34.86,Default,,0000,0000,0000,,
Dialogue: 0,1:32:34.86,1:32:39.68,Default,,0000,0000,0000,,f sub xB is f sub xy.
Dialogue: 0,1:32:39.68,1:32:41.68,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:32:41.68,1:32:43.98,Default,,0000,0000,0000,,0.
Dialogue: 0,1:32:43.98,1:32:46.48,Default,,0000,0000,0000,,Good, that makes my life easier.
Dialogue: 0,1:32:46.48,1:32:49.92,Default,,0000,0000,0000,,C equals f sub yy.
Dialogue: 0,1:32:49.92,1:32:52.27,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:32:52.27,1:32:53.22,Default,,0000,0000,0000,,2.
Dialogue: 0,1:32:53.22,1:32:55.88,Default,,0000,0000,0000,,OK, this is looking beautiful.
Dialogue: 0,1:32:55.88,1:32:58.53,Default,,0000,0000,0000,,Because I don't have\Nto plug in any values.
Dialogue: 0,1:32:58.53,1:33:00.50,Default,,0000,0000,0000,,The D is there for me to see it.
Dialogue: 0,1:33:00.50,1:33:04.54,Default,,0000,0000,0000,,And it's going to consist of the\Ndeterminant having the elements
Dialogue: 0,1:33:04.54,1:33:10.90,Default,,0000,0000,0000,,2, 0, 0, 2-- minus 2, minus 2.
Dialogue: 0,1:33:10.90,1:33:14.35,Default,,0000,0000,0000,,I'm sorry, guys, I\Nmissed here the minus.
Dialogue: 0,1:33:14.35,1:33:19.00,Default,,0000,0000,0000,,And it cost me my life-- 2x\Nand minus 2y, and here minus 2.
Dialogue: 0,1:33:19.00,1:33:20.80,Default,,0000,0000,0000,,STUDENT: It didn't\Ncost you your life,
Dialogue: 0,1:33:20.80,1:33:21.88,Default,,0000,0000,0000,,because you caught it before\Nyou were done with the problem.
Dialogue: 0,1:33:21.88,1:33:23.06,Default,,0000,0000,0000,,PROFESSOR: I caught it up there.
Dialogue: 0,1:33:23.06,1:33:24.28,Default,,0000,0000,0000,,I'm taking the final exam.
Dialogue: 0,1:33:24.28,1:33:28.23,Default,,0000,0000,0000,,I still get 100%,\Nbecause I caught it up
Dialogue: 0,1:33:28.23,1:33:29.92,Default,,0000,0000,0000,,at the last minute.
Dialogue: 0,1:33:29.92,1:33:34.48,Default,,0000,0000,0000,,So 2, 0, 0, minus\N2-- I knew that I
Dialogue: 0,1:33:34.48,1:33:35.75,Default,,0000,0000,0000,,had to get something negative.
Dialogue: 0,1:33:35.75,1:33:39.62,Default,,0000,0000,0000,,So I said, for god's sake, I\Nneed to get a saddle point.
Dialogue: 0,1:33:39.62,1:33:42.22,Default,,0000,0000,0000,,That's why it's the\Nhorse in the saddle.
Dialogue: 0,1:33:42.22,1:33:46.83,Default,,0000,0000,0000,,So I knew I should\Nget minus 4, negative.
Dialogue: 0,1:33:46.83,1:33:50.73,Default,,0000,0000,0000,,All right, so the\Nonly thing I have
Dialogue: 0,1:33:50.73,1:33:53.06,Default,,0000,0000,0000,,to say as a final\Nanswer is the only
Dialogue: 0,1:33:53.06,1:33:57.44,Default,,0000,0000,0000,,critical point of this surface\Nthat I'm too lazy to write
Dialogue: 0,1:33:57.44,1:33:59.26,Default,,0000,0000,0000,,about-- don't write that.
Dialogue: 0,1:33:59.26,1:34:02.36,Default,,0000,0000,0000,,So the only critical\Npoint on the surface z
Dialogue: 0,1:34:02.36,1:34:04.77,Default,,0000,0000,0000,,equals x squared\Nminus y squared will
Dialogue: 0,1:34:04.77,1:34:10.08,Default,,0000,0000,0000,,be at the origin O\Nof corner 0, 0, 0
Dialogue: 0,1:34:10.08,1:34:14.48,Default,,0000,0000,0000,,where the discriminant\Nbeing negative
Dialogue: 0,1:34:14.48,1:34:17.30,Default,,0000,0000,0000,,indicates it's going\Nto be a saddle point.
Dialogue: 0,1:34:17.30,1:34:20.48,Default,,0000,0000,0000,,And that's it-- nothing else.
Dialogue: 0,1:34:20.48,1:34:22.15,Default,,0000,0000,0000,,You don't need more.
Dialogue: 0,1:34:22.15,1:34:24.59,Default,,0000,0000,0000,,But there are more examples.
Dialogue: 0,1:34:24.59,1:34:26.54,Default,,0000,0000,0000,,Because life is hard.
Dialogue: 0,1:34:26.54,1:34:28.38,Default,,0000,0000,0000,,And I'm going to give\Nyou another example.
Dialogue: 0,1:34:28.38,1:34:31.69,Default,,0000,0000,0000,,
Dialogue: 0,1:34:31.69,1:34:34.65,Default,,0000,0000,0000,,Well, OK, this one.
Dialogue: 0,1:34:34.65,1:34:37.62,Default,,0000,0000,0000,,
Dialogue: 0,1:34:37.62,1:34:47.66,Default,,0000,0000,0000,,Suppose we have the\Nsurface-- that's
Dialogue: 0,1:34:47.66,1:34:49.50,Default,,0000,0000,0000,,still going to be very easy.
Dialogue: 0,1:34:49.50,1:34:51.88,Default,,0000,0000,0000,,But I want to make the\Nfirst examples easy.
Dialogue: 0,1:34:51.88,1:34:54.66,Default,,0000,0000,0000,,
Dialogue: 0,1:34:54.66,1:34:56.45,Default,,0000,0000,0000,,I have a reason why.
Dialogue: 0,1:34:56.45,1:35:04.89,Default,,0000,0000,0000,,
Dialogue: 0,1:35:04.89,1:35:07.29,Default,,0000,0000,0000,,This is a function of\Ntwo variables, right?
Dialogue: 0,1:35:07.29,1:35:12.38,Default,,0000,0000,0000,,It's still a polynomial in\Ntwo variables of order 2.
Dialogue: 0,1:35:12.38,1:35:17.89,Default,,0000,0000,0000,,And how do I solve for the\Nclassification of the extrema?
Dialogue: 0,1:35:17.89,1:35:23.59,Default,,0000,0000,0000,,I'm looking for local extrema,\Nnot absolute-- local extrema.
Dialogue: 0,1:35:23.59,1:35:25.17,Default,,0000,0000,0000,,I'm not constrained.
Dialogue: 0,1:35:25.17,1:35:27.62,Default,,0000,0000,0000,,I'm saying, what do you\Nmean, no constraint?
Dialogue: 0,1:35:27.62,1:35:30.79,Default,,0000,0000,0000,,Constrained would have\Nbeen, let's say that x and y
Dialogue: 0,1:35:30.79,1:35:33.59,Default,,0000,0000,0000,,are in the unit disc.
Dialogue: 0,1:35:33.59,1:35:38.75,Default,,0000,0000,0000,,Or let's say x and y are\Non the circle x squared
Dialogue: 0,1:35:38.75,1:35:40.34,Default,,0000,0000,0000,,plus y squared equals 1.
Dialogue: 0,1:35:40.34,1:35:41.97,Default,,0000,0000,0000,,That would be a constraint.
Dialogue: 0,1:35:41.97,1:35:44.52,Default,,0000,0000,0000,,But they're not\Nconstrained about anything.
Dialogue: 0,1:35:44.52,1:35:47.49,Default,,0000,0000,0000,,x and y are real numbers.
Dialogue: 0,1:35:47.49,1:35:51.31,Default,,0000,0000,0000,,They can take the whole\Nplane as a domain.
Dialogue: 0,1:35:51.31,1:35:59.22,Default,,0000,0000,0000,,So I get f sub x equals 0, f sub\Ny equals 0, solve for x and y,
Dialogue: 0,1:35:59.22,1:36:00.70,Default,,0000,0000,0000,,get the critical values.
Dialogue: 0,1:36:00.70,1:36:05.07,Default,,0000,0000,0000,,I get very nice 2x.
Dialogue: 0,1:36:05.07,1:36:06.18,Default,,0000,0000,0000,,I have to pay attention.
Dialogue: 0,1:36:06.18,1:36:09.55,Default,,0000,0000,0000,,Because now this is\Nnot so easy anymore--
Dialogue: 0,1:36:09.55,1:36:16.69,Default,,0000,0000,0000,,plus prime with respect to x,\N2y, prime with respect to x, 0,
Dialogue: 0,1:36:16.69,1:36:25.67,Default,,0000,0000,0000,,prime with respect to x, plus\N3, prime of this, OK, equals 0.
Dialogue: 0,1:36:25.67,1:36:32.32,Default,,0000,0000,0000,,f sub y-- 0 plus\Nprime with respect
Dialogue: 0,1:36:32.32,1:36:41.03,Default,,0000,0000,0000,,to y, 2x, plus prime\Nwith respect to y, 2y,
Dialogue: 0,1:36:41.03,1:36:46.82,Default,,0000,0000,0000,,plus nothing, prime with\Nrespect to y equals 0.
Dialogue: 0,1:36:46.82,1:36:52.71,Default,,0000,0000,0000,,
Dialogue: 0,1:36:52.71,1:36:58.26,Default,,0000,0000,0000,,And now you have\Nto be very smart.
Dialogue: 0,1:36:58.26,1:37:03.02,Default,,0000,0000,0000,,Well, you have to be perceptive\Nand tell me what I got.
Dialogue: 0,1:37:03.02,1:37:04.99,Default,,0000,0000,0000,,What is this that we mean?
Dialogue: 0,1:37:04.99,1:37:08.93,Default,,0000,0000,0000,,
Dialogue: 0,1:37:08.93,1:37:10.89,Default,,0000,0000,0000,,Look at this system.
Dialogue: 0,1:37:10.89,1:37:12.86,Default,,0000,0000,0000,,It looks like crazy.
Dialogue: 0,1:37:12.86,1:37:22.70,Default,,0000,0000,0000,,
Dialogue: 0,1:37:22.70,1:37:25.65,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] the\Norigin or-- because can't you
Dialogue: 0,1:37:25.65,1:37:28.11,Default,,0000,0000,0000,,just subtract it down?
Dialogue: 0,1:37:28.11,1:37:30.08,Default,,0000,0000,0000,,PROFESSOR: Is this possible?
Dialogue: 0,1:37:30.08,1:37:31.67,Default,,0000,0000,0000,,And what does this mean?
Dialogue: 0,1:37:31.67,1:37:33.30,Default,,0000,0000,0000,,What do we call such a system?
Dialogue: 0,1:37:33.30,1:37:38.12,Default,,0000,0000,0000,,
Dialogue: 0,1:37:38.12,1:37:42.15,Default,,0000,0000,0000,,Inconsistent system--\Nwe call it inconsistent.
Dialogue: 0,1:37:42.15,1:37:45.10,Default,,0000,0000,0000,,How can I make this\Nproblem to be possible,
Dialogue: 0,1:37:45.10,1:37:47.76,Default,,0000,0000,0000,,to have some critical points?
Dialogue: 0,1:37:47.76,1:37:50.22,Default,,0000,0000,0000,,STUDENT: If you add 3x.
Dialogue: 0,1:37:50.22,1:37:52.20,Default,,0000,0000,0000,,PROFESSOR: How about\Nthat, just remove the 3x
Dialogue: 0,1:37:52.20,1:37:55.66,Default,,0000,0000,0000,,and see what's going to happen.
Dialogue: 0,1:37:55.66,1:37:59.46,Default,,0000,0000,0000,,Oh, in that case,\NI have something
Dialogue: 0,1:37:59.46,1:38:06.01,Default,,0000,0000,0000,,that's over-determined, right?
Dialogue: 0,1:38:06.01,1:38:11.74,Default,,0000,0000,0000,,I have something that\Ntells me the same thing.
Dialogue: 0,1:38:11.74,1:38:13.50,Default,,0000,0000,0000,,So I'm priming\Nwith respect to x.
Dialogue: 0,1:38:13.50,1:38:14.84,Default,,0000,0000,0000,,I get that.
Dialogue: 0,1:38:14.84,1:38:16.42,Default,,0000,0000,0000,,I'm priming with respect to y.
Dialogue: 0,1:38:16.42,1:38:17.65,Default,,0000,0000,0000,,I get this.
Dialogue: 0,1:38:17.65,1:38:18.74,Default,,0000,0000,0000,,I get 0.
Dialogue: 0,1:38:18.74,1:38:21.86,Default,,0000,0000,0000,,So I don't even need\Nthe second equation.
Dialogue: 0,1:38:21.86,1:38:27.99,Default,,0000,0000,0000,,And that means\Nthe critical point
Dialogue: 0,1:38:27.99,1:38:40.54,Default,,0000,0000,0000,,is any point of\Nthe form-- shall I
Dialogue: 0,1:38:40.54,1:38:44.49,Default,,0000,0000,0000,,put a Greek letter alpha minus\Nalpha or lambda minus lambda?
Dialogue: 0,1:38:44.49,1:38:45.95,Default,,0000,0000,0000,,What shall I do?
Dialogue: 0,1:38:45.95,1:38:52.37,Default,,0000,0000,0000,,So any point that is situated\Non the second bisector,
Dialogue: 0,1:38:52.37,1:38:54.10,Default,,0000,0000,0000,,I mean the x, y plane.
Dialogue: 0,1:38:54.10,1:38:56.54,Default,,0000,0000,0000,,And this is the x,\Nand this is the y.
Dialogue: 0,1:38:56.54,1:39:00.77,Default,,0000,0000,0000,,And I say, what does it\Nmean, x plus y equals 0?
Dialogue: 0,1:39:00.77,1:39:03.12,Default,,0000,0000,0000,,Not this line-- don't draw it.
Dialogue: 0,1:39:03.12,1:39:05.03,Default,,0000,0000,0000,,That is x equals y.
Dialogue: 0,1:39:05.03,1:39:08.96,Default,,0000,0000,0000,,The other one, called\Nthe second bisector-- y
Dialogue: 0,1:39:08.96,1:39:13.47,Default,,0000,0000,0000,,equals negative x, so not\Nthis one, the diagonal,
Dialogue: 0,1:39:13.47,1:39:21.76,Default,,0000,0000,0000,,but the diagonal that's\Non the corridor, this one.
Dialogue: 0,1:39:21.76,1:39:25.58,Default,,0000,0000,0000,,All right, so any point of\Nthe form alpha minus alpha,
Dialogue: 0,1:39:25.58,1:39:27.62,Default,,0000,0000,0000,,here's the critical point.
Dialogue: 0,1:39:27.62,1:39:33.08,Default,,0000,0000,0000,,The question is, how am I going\Nto get to the classification
Dialogue: 0,1:39:33.08,1:39:35.37,Default,,0000,0000,0000,,for such points?
Dialogue: 0,1:39:35.37,1:39:36.73,Default,,0000,0000,0000,,Can anybody help me?
Dialogue: 0,1:39:36.73,1:39:38.33,Default,,0000,0000,0000,,So step two--
Dialogue: 0,1:39:38.33,1:39:39.50,Default,,0000,0000,0000,,STUDENT: Solve the equation.
Dialogue: 0,1:39:39.50,1:39:43.27,Default,,0000,0000,0000,,STUDENT: Solve alpha for one\Nof the two variables first.
Dialogue: 0,1:39:43.27,1:39:46.36,Default,,0000,0000,0000,,PROFESSOR: Take alpha minus\Nalpha-- could be anything.
Dialogue: 0,1:39:46.36,1:39:50.63,Default,,0000,0000,0000,,And then I'll say, f\Nsub-- this is f sub x.
Dialogue: 0,1:39:50.63,1:39:53.90,Default,,0000,0000,0000,,And this is f sub y.
Dialogue: 0,1:39:53.90,1:39:55.16,Default,,0000,0000,0000,,What is f sub x?
Dialogue: 0,1:39:55.16,1:39:56.70,Default,,0000,0000,0000,,f sub xx, I'm sorry.
Dialogue: 0,1:39:56.70,1:39:59.59,Default,,0000,0000,0000,,
Dialogue: 0,1:39:59.59,1:40:01.85,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:40:01.85,1:40:02.47,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,1:40:02.47,1:40:03.43,Default,,0000,0000,0000,,2.
Dialogue: 0,1:40:03.43,1:40:05.41,Default,,0000,0000,0000,,OK, are you with me?
Dialogue: 0,1:40:05.41,1:40:07.88,Default,,0000,0000,0000,,So you know what it is.
Dialogue: 0,1:40:07.88,1:40:12.41,Default,,0000,0000,0000,,f sub xy equals?
Dialogue: 0,1:40:12.41,1:40:13.25,Default,,0000,0000,0000,,STUDENT: 2.
Dialogue: 0,1:40:13.25,1:40:15.39,Default,,0000,0000,0000,,PROFESSOR: 2.
Dialogue: 0,1:40:15.39,1:40:17.81,Default,,0000,0000,0000,,f sub yy equals?
Dialogue: 0,1:40:17.81,1:40:18.71,Default,,0000,0000,0000,,STUDENT: 2
Dialogue: 0,1:40:18.71,1:40:21.66,Default,,0000,0000,0000,,PROFESSOR: 2-- that's\Nthe mystery man.
Dialogue: 0,1:40:21.66,1:40:23.13,Default,,0000,0000,0000,,The book doesn't\Ngive this example,
Dialogue: 0,1:40:23.13,1:40:24.68,Default,,0000,0000,0000,,and it drives me crazy.
Dialogue: 0,1:40:24.68,1:40:28.54,Default,,0000,0000,0000,,And I wanted to give you\Nsome bad example where
Dialogue: 0,1:40:28.54,1:40:30.94,Default,,0000,0000,0000,,the classification doesn't work.
Dialogue: 0,1:40:30.94,1:40:35.47,Default,,0000,0000,0000,,Because we always cook\Nup nice examples for you
Dialogue: 0,1:40:35.47,1:40:38.92,Default,,0000,0000,0000,,and claim everything\Nis beautiful.
Dialogue: 0,1:40:38.92,1:40:41.41,Default,,0000,0000,0000,,Life is not always beautiful.
Dialogue: 0,1:40:41.41,1:40:44.48,Default,,0000,0000,0000,,So you get 0.
Dialogue: 0,1:40:44.48,1:40:48.27,Default,,0000,0000,0000,,In that case, nothing can be\Nsaid with this classification.
Dialogue: 0,1:40:48.27,1:40:50.49,Default,,0000,0000,0000,,I make a face, sad face.
Dialogue: 0,1:40:50.49,1:40:52.22,Default,,0000,0000,0000,,So what do I hope?
Dialogue: 0,1:40:52.22,1:40:59.42,Default,,0000,0000,0000,,To get to Maple or MATLAB and\Nbe able to draw that, or a TI-92
Dialogue: 0,1:40:59.42,1:41:02.63,Default,,0000,0000,0000,,if my mother would\Ngive me $200 and some.
Dialogue: 0,1:41:02.63,1:41:03.60,Default,,0000,0000,0000,,I told her.
Dialogue: 0,1:41:03.60,1:41:08.02,Default,,0000,0000,0000,,She asked me what to\Nbuy for my birthday.
Dialogue: 0,1:41:08.02,1:41:10.71,Default,,0000,0000,0000,,I have a TI-83 or something.
Dialogue: 0,1:41:10.71,1:41:12.74,Default,,0000,0000,0000,,And it was cheap.
Dialogue: 0,1:41:12.74,1:41:17.38,Default,,0000,0000,0000,,I bought it on eBay, and\Nthen I stopped using it.
Dialogue: 0,1:41:17.38,1:41:20.92,Default,,0000,0000,0000,,And then I saw this TI-92\Nthat can draw surfaces
Dialogue: 0,1:41:20.92,1:41:22.51,Default,,0000,0000,0000,,in three dimensions.
Dialogue: 0,1:41:22.51,1:41:24.08,Default,,0000,0000,0000,,And I said, this is like MATLAB.
Dialogue: 0,1:41:24.08,1:41:25.85,Default,,0000,0000,0000,,You just carry it\Nin your pocket.
Dialogue: 0,1:41:25.85,1:41:29.39,Default,,0000,0000,0000,,It's only a little\Nbit too expensive.
Dialogue: 0,1:41:29.39,1:41:34.70,Default,,0000,0000,0000,,All right, how\Nabout another kind?
Dialogue: 0,1:41:34.70,1:41:38.63,Default,,0000,0000,0000,,
Dialogue: 0,1:41:38.63,1:41:40.10,Default,,0000,0000,0000,,Look at this one.
Dialogue: 0,1:41:40.10,1:41:55.35,Default,,0000,0000,0000,,
Dialogue: 0,1:41:55.35,1:41:58.59,Default,,0000,0000,0000,,You cannot tell\Nwith the naked eye.
Dialogue: 0,1:41:58.59,1:42:02.14,Default,,0000,0000,0000,,But you can go ahead\Nand do this step one
Dialogue: 0,1:42:02.14,1:42:05.08,Default,,0000,0000,0000,,looking for critical values.
Dialogue: 0,1:42:05.08,1:42:17.75,Default,,0000,0000,0000,,So the system, f sub x will\Nbe 6x plus 2y equals 0.
Dialogue: 0,1:42:17.75,1:42:24.39,Default,,0000,0000,0000,,f sub y will be--\Nwho's going to tell me?
Dialogue: 0,1:42:24.39,1:42:29.76,Default,,0000,0000,0000,,2x plus 2y equals 0.
Dialogue: 0,1:42:29.76,1:42:34.29,Default,,0000,0000,0000,,Now, by elimination or by\Nsubstitution or by anything
Dialogue: 0,1:42:34.29,1:42:36.80,Default,,0000,0000,0000,,I want, I subtract the\Nsecond from the first.
Dialogue: 0,1:42:36.80,1:42:38.36,Default,,0000,0000,0000,,What do I get?
Dialogue: 0,1:42:38.36,1:42:41.34,Default,,0000,0000,0000,,I get 4x equals 0.
Dialogue: 0,1:42:41.34,1:42:46.41,Default,,0000,0000,0000,,And that gives me the only\Npossibility is x0 equals 0.
Dialogue: 0,1:42:46.41,1:42:53.07,Default,,0000,0000,0000,,And then I say, OK, if my\Nonly one is 0, then y is 0.
Dialogue: 0,1:42:53.07,1:42:54.01,Default,,0000,0000,0000,,0 is 0.
Dialogue: 0,1:42:54.01,1:43:01.25,Default,,0000,0000,0000,,So I only have one critical\Npoint, which is the origin.
Dialogue: 0,1:43:01.25,1:43:03.50,Default,,0000,0000,0000,,Now, do I know, what\Nam I going to get?
Dialogue: 0,1:43:03.50,1:43:05.71,Default,,0000,0000,0000,,Not unless I'm a\Ngenius and I can
Dialogue: 0,1:43:05.71,1:43:07.91,Default,,0000,0000,0000,,see two steps ahead of time.
Dialogue: 0,1:43:07.91,1:43:10.86,Default,,0000,0000,0000,,I would need to do ABC\Nquickly in my head.
Dialogue: 0,1:43:10.86,1:43:13.16,Default,,0000,0000,0000,,Some of you are able, thank god.
Dialogue: 0,1:43:13.16,1:43:15.03,Default,,0000,0000,0000,,But some of you,\Nlike me, are not.
Dialogue: 0,1:43:15.03,1:43:19.29,Default,,0000,0000,0000,,So I have to take a few\Nseconds to see what's going on.
Dialogue: 0,1:43:19.29,1:43:24.26,Default,,0000,0000,0000,,A-- f sub xx at the point is 0.
Dialogue: 0,1:43:24.26,1:43:27.72,Default,,0000,0000,0000,,B-- f sub xy.
Dialogue: 0,1:43:27.72,1:43:29.91,Default,,0000,0000,0000,,C-- f sub yy.
Dialogue: 0,1:43:29.91,1:43:36.74,Default,,0000,0000,0000,,
Dialogue: 0,1:43:36.74,1:43:38.70,Default,,0000,0000,0000,,What do we do?
Dialogue: 0,1:43:38.70,1:43:40.08,Default,,0000,0000,0000,,We get 6.
Dialogue: 0,1:43:40.08,1:43:41.25,Default,,0000,0000,0000,,Are we happy about it?
Dialogue: 0,1:43:41.25,1:43:44.91,Default,,0000,0000,0000,,We don't know yet,\Nto be happy or not.
Dialogue: 0,1:43:44.91,1:43:48.92,Default,,0000,0000,0000,,f sub xy or f sub yx,\Nyou see, Mr. Schwarz
Dialogue: 0,1:43:48.92,1:43:52.05,Default,,0000,0000,0000,,is now happy that\Nhe proved to you
Dialogue: 0,1:43:52.05,1:43:55.12,Default,,0000,0000,0000,,that it doesn't matter\Nwhich order you're
Dialogue: 0,1:43:55.12,1:43:58.79,Default,,0000,0000,0000,,taking for a polynomial\Nthat's a smooth function.
Dialogue: 0,1:43:58.79,1:44:03.84,Default,,0000,0000,0000,,You always have the same.
Dialogue: 0,1:44:03.84,1:44:08.16,Default,,0000,0000,0000,,And finally, C is 2.
Dialogue: 0,1:44:08.16,1:44:12.72,Default,,0000,0000,0000,,And you are ready to do the\ND. And I could smell that D,
Dialogue: 0,1:44:12.72,1:44:15.20,Default,,0000,0000,0000,,but I didn't want\Nto say anything.
Dialogue: 0,1:44:15.20,1:44:24.24,Default,,0000,0000,0000,,6, 2, 2, and 2-- is\Nthat a nice thing?
Dialogue: 0,1:44:24.24,1:44:27.82,Default,,0000,0000,0000,,Yeah, we haven't encountered\Nthis example yet.
Dialogue: 0,1:44:27.82,1:44:30.68,Default,,0000,0000,0000,,Because according to\Nthe classification,
Dialogue: 0,1:44:30.68,1:44:32.41,Default,,0000,0000,0000,,this is greater than 0.
Dialogue: 0,1:44:32.41,1:44:34.45,Default,,0000,0000,0000,,Does it really matter\Nwhat value it is?
Dialogue: 0,1:44:34.45,1:44:37.78,Default,,0000,0000,0000,,No, it only matters\Nthat it is positive.
Dialogue: 0,1:44:37.78,1:44:42.35,Default,,0000,0000,0000,,And if it's positive, that\Nmeans I can move on with my life
Dialogue: 0,1:44:42.35,1:44:45.05,Default,,0000,0000,0000,,and look at the classification.
Dialogue: 0,1:44:45.05,1:44:50.17,Default,,0000,0000,0000,,From this point where\Ndelta or v is positive,
Dialogue: 0,1:44:50.17,1:44:57.10,Default,,0000,0000,0000,,I'm going to get a ramification\Ninto separate cases.
Dialogue: 0,1:44:57.10,1:45:01.18,Default,,0000,0000,0000,,And who is going to\Ntell me next what to do?
Dialogue: 0,1:45:01.18,1:45:04.09,Default,,0000,0000,0000,,Look at A. Oh, by\Nthe way, talking
Dialogue: 0,1:45:04.09,1:45:07.65,Default,,0000,0000,0000,,about the quadratic\Nformula from school,
Dialogue: 0,1:45:07.65,1:45:12.62,Default,,0000,0000,0000,,from kindergarten,\Nwhen we computed
Dialogue: 0,1:45:12.62,1:45:18.78,Default,,0000,0000,0000,,the-- I'll use the general one,\Nminus b plus minus square root
Dialogue: 0,1:45:18.78,1:45:21.47,Default,,0000,0000,0000,,of b squared minus 4ac over 2a.
Dialogue: 0,1:45:21.47,1:45:27.11,Default,,0000,0000,0000,,
Dialogue: 0,1:45:27.11,1:45:33.63,Default,,0000,0000,0000,,We were afraid of\Nsome special cases
Dialogue: 0,1:45:33.63,1:45:35.89,Default,,0000,0000,0000,,when we were looking at that.
Dialogue: 0,1:45:35.89,1:45:37.70,Default,,0000,0000,0000,,Especially when\Ndelta was negative,
Dialogue: 0,1:45:37.70,1:45:40.39,Default,,0000,0000,0000,,that was really\Nimaginary and so on.
Dialogue: 0,1:45:40.39,1:45:42.89,Default,,0000,0000,0000,,But one thing we remember\Nfrom ninth grade--
Dialogue: 0,1:45:42.89,1:45:46.91,Default,,0000,0000,0000,,was this ninth grade\Nor eighth grade?
Dialogue: 0,1:45:46.91,1:45:51.54,Default,,0000,0000,0000,,The parabola opens up\Nwhen a is positive.
Dialogue: 0,1:45:51.54,1:45:57.56,Default,,0000,0000,0000,,Just the same way, something\Nopens up when A, big A,
Dialogue: 0,1:45:57.56,1:45:59.31,Default,,0000,0000,0000,,is positive here.
Dialogue: 0,1:45:59.31,1:46:01.63,Default,,0000,0000,0000,,Then you have opening up.
Dialogue: 0,1:46:01.63,1:46:05.59,Default,,0000,0000,0000,,When big A is negative,\Nthen you have opening down.
Dialogue: 0,1:46:05.59,1:46:10.39,Default,,0000,0000,0000,,So remember-- I'm going to make\Nsmile here so you remember.
Dialogue: 0,1:46:10.39,1:46:12.31,Default,,0000,0000,0000,,So I have it like that.
Dialogue: 0,1:46:12.31,1:46:16.08,Default,,0000,0000,0000,,So I suspect that\Nit's going to look
Dialogue: 0,1:46:16.08,1:46:23.24,Default,,0000,0000,0000,,like a surface of some\Nsort that maybe is not
Dialogue: 0,1:46:23.24,1:46:25.74,Default,,0000,0000,0000,,surface of revolution.
Dialogue: 0,1:46:25.74,1:46:27.29,Default,,0000,0000,0000,,You should tell me what it is.
Dialogue: 0,1:46:27.29,1:46:31.53,Default,,0000,0000,0000,,You should think about this and\Ndo the cross sections with z
Dialogue: 0,1:46:31.53,1:46:34.59,Default,,0000,0000,0000,,constant and tell me\Nwhat surface that is.
Dialogue: 0,1:46:34.59,1:46:36.96,Default,,0000,0000,0000,,But in any case, what do I care?
Dialogue: 0,1:46:36.96,1:46:41.77,Default,,0000,0000,0000,,I care that I'm\Nlooking at the origin.
Dialogue: 0,1:46:41.77,1:46:44.09,Default,,0000,0000,0000,,And this is where\Nmy special point is.
Dialogue: 0,1:46:44.09,1:46:46.64,Default,,0000,0000,0000,,That's going to be\Nthe value point.
Dialogue: 0,1:46:46.64,1:46:48.17,Default,,0000,0000,0000,,How do I know?
Dialogue: 0,1:46:48.17,1:46:52.71,Default,,0000,0000,0000,,Because A, which\Nis 6, is positive.
Dialogue: 0,1:46:52.71,1:46:55.66,Default,,0000,0000,0000,,At this point, I know\Nwhat I'm left with.
Dialogue: 0,1:46:55.66,1:46:59.74,Default,,0000,0000,0000,,I know that my surface is\Ngoing to look like a valley.
Dialogue: 0,1:46:59.74,1:47:02.14,Default,,0000,0000,0000,,So how do I know again?
Dialogue: 0,1:47:02.14,1:47:04.07,Default,,0000,0000,0000,,I'm not going to draw it.
Dialogue: 0,1:47:04.07,1:47:07.89,Default,,0000,0000,0000,,But it's going to look\Nsomething like that.
Dialogue: 0,1:47:07.89,1:47:10.50,Default,,0000,0000,0000,,At the origin, this\Nis going to be 7.
Dialogue: 0,1:47:10.50,1:47:12.50,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:47:12.50,1:47:15.39,Default,,0000,0000,0000,,And it's going to open up.
Dialogue: 0,1:47:15.39,1:47:19.40,Default,,0000,0000,0000,,And so you should not attempt\Nintersecting with z equals 5
Dialogue: 0,1:47:19.40,1:47:20.16,Default,,0000,0000,0000,,or z equals 1.
Dialogue: 0,1:47:20.16,1:47:21.99,Default,,0000,0000,0000,,Because you're not\Ngoing to get anything.
Dialogue: 0,1:47:21.99,1:47:24.54,Default,,0000,0000,0000,,But if you intersect,\Nfor example, at z
Dialogue: 0,1:47:24.54,1:47:26.62,Default,,0000,0000,0000,,equals 9, what are\Nyou going to get?
Dialogue: 0,1:47:26.62,1:47:31.30,Default,,0000,0000,0000,,If you intersect\Nat z equals 9, you
Dialogue: 0,1:47:31.30,1:47:37.11,Default,,0000,0000,0000,,get 3x plus 2xy plus\Ny squared equals 2.
Dialogue: 0,1:47:37.11,1:47:39.07,Default,,0000,0000,0000,,And what is that?
Dialogue: 0,1:47:39.07,1:47:43.97,Default,,0000,0000,0000,,It's a rotated\Nform of an ellipse.
Dialogue: 0,1:47:43.97,1:47:47.50,Default,,0000,0000,0000,,It's hard to see, because\Nit's missing [INAUDIBLE].
Dialogue: 0,1:47:47.50,1:47:51.08,Default,,0000,0000,0000,,But this is exactly what\Ndiscriminant is saying.
Dialogue: 0,1:47:51.08,1:47:55.76,Default,,0000,0000,0000,,So this is going to be an x.
Dialogue: 0,1:47:55.76,1:47:58.66,Default,,0000,0000,0000,,Good, so I know what\NI'm going to get.
Dialogue: 0,1:47:58.66,1:48:01.05,Default,,0000,0000,0000,,What do you have to\Nsay on the midterm
Dialogue: 0,1:48:01.05,1:48:03.53,Default,,0000,0000,0000,,or on the final\Nabout this problem?
Dialogue: 0,1:48:03.53,1:48:04.92,Default,,0000,0000,0000,,STUDENT: The point is--
Dialogue: 0,1:48:04.92,1:48:07.70,Default,,0000,0000,0000,,PROFESSOR: The\Npoint is the origin.
Dialogue: 0,1:48:07.70,1:48:08.56,Default,,0000,0000,0000,,I classified it.
Dialogue: 0,1:48:08.56,1:48:09.55,Default,,0000,0000,0000,,I got delta positive.
Dialogue: 0,1:48:09.55,1:48:11.76,Default,,0000,0000,0000,,I got A positive.
Dialogue: 0,1:48:11.76,1:48:13.01,Default,,0000,0000,0000,,So it's a valley.
Dialogue: 0,1:48:13.01,1:48:14.53,Default,,0000,0000,0000,,It's a relative minimum.
Dialogue: 0,1:48:14.53,1:48:15.59,Default,,0000,0000,0000,,And that's it.
Dialogue: 0,1:48:15.59,1:48:21.85,Default,,0000,0000,0000,,I have a relative min at the\Npoint P of coordinates 0, 0,
Dialogue: 0,1:48:21.85,1:48:23.10,Default,,0000,0000,0000,,and 7.
Dialogue: 0,1:48:23.10,1:48:26.35,Default,,0000,0000,0000,,
Dialogue: 0,1:48:26.35,1:48:27.97,Default,,0000,0000,0000,,And that's the valley.
Dialogue: 0,1:48:27.97,1:48:28.80,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:48:28.80,1:48:30.75,Default,,0000,0000,0000,,STUDENT: Why is it\NA that determines
Dialogue: 0,1:48:30.75,1:48:33.79,Default,,0000,0000,0000,,whether it's a relative\Nmin or a relative max?
Dialogue: 0,1:48:33.79,1:48:35.13,Default,,0000,0000,0000,,PROFESSOR: It's a whole story.
Dialogue: 0,1:48:35.13,1:48:36.08,Default,,0000,0000,0000,,You can prove it.
Dialogue: 0,1:48:36.08,1:48:40.34,Default,,0000,0000,0000,,I don't remember if we proved\Nthis in the book or not.
Dialogue: 0,1:48:40.34,1:48:43.05,Default,,0000,0000,0000,,But it can be\Nproved, so the fact
Dialogue: 0,1:48:43.05,1:48:48.74,Default,,0000,0000,0000,,that it has to do with\Nconcavity and convexity.
Dialogue: 0,1:48:48.74,1:48:54.00,Default,,0000,0000,0000,,When you had a second\Nderivative, let's say,
Dialogue: 0,1:48:54.00,1:48:56.59,Default,,0000,0000,0000,,what's the equivalent\Nof the Calculus I
Dialogue: 0,1:48:56.59,1:48:58.95,Default,,0000,0000,0000,,notion that you know about?
Dialogue: 0,1:48:58.95,1:49:01.71,Default,,0000,0000,0000,,In Calculus I, you had\Nfunctions of one variable,
Dialogue: 0,1:49:01.71,1:49:04.16,Default,,0000,0000,0000,,and life was so easy like that.
Dialogue: 0,1:49:04.16,1:49:08.28,Default,,0000,0000,0000,,And f prime positive meant\Nthat the function increased.
Dialogue: 0,1:49:08.28,1:49:12.33,Default,,0000,0000,0000,,And f prime negative meant\Nthat the function decreased.
Dialogue: 0,1:49:12.33,1:49:18.09,Default,,0000,0000,0000,,And f double prime was\Njust like your-- you sense
Dialogue: 0,1:49:18.09,1:49:22.07,Default,,0000,0000,0000,,that the second partials\Nmust have something
Dialogue: 0,1:49:22.07,1:49:25.86,Default,,0000,0000,0000,,to do with it, especially the\Nfirst one with respect to x.
Dialogue: 0,1:49:25.86,1:49:28.62,Default,,0000,0000,0000,,If you were in\Nplane, and you have
Dialogue: 0,1:49:28.62,1:49:34.42,Default,,0000,0000,0000,,f double prime with respect\Nto x, when was this a valley?
Dialogue: 0,1:49:34.42,1:49:35.38,Default,,0000,0000,0000,,When you had the smile.
Dialogue: 0,1:49:35.38,1:49:37.53,Default,,0000,0000,0000,,When did you have a smile?
Dialogue: 0,1:49:37.53,1:49:42.91,Default,,0000,0000,0000,,When f double prime was\Npositive, you have concave up.
Dialogue: 0,1:49:42.91,1:49:45.25,Default,,0000,0000,0000,,When f double\Nprime was negative,
Dialogue: 0,1:49:45.25,1:49:47.98,Default,,0000,0000,0000,,you have concave down.
Dialogue: 0,1:49:47.98,1:49:48.74,Default,,0000,0000,0000,,Remember, guys?
Dialogue: 0,1:49:48.74,1:49:53.52,Default,,0000,0000,0000,,So you have a smile or a frown.
Dialogue: 0,1:49:53.52,1:49:54.42,Default,,0000,0000,0000,,This is how we know.
Dialogue: 0,1:49:54.42,1:49:58.50,Default,,0000,0000,0000,,For the same reason that\Nwould take about two pages
Dialogue: 0,1:49:58.50,1:50:04.06,Default,,0000,0000,0000,,to write down the proof, you\Nhave a smile for A positive.
Dialogue: 0,1:50:04.06,1:50:07.85,Default,,0000,0000,0000,,And the smile means\Nactually in all directions
Dialogue: 0,1:50:07.85,1:50:12.97,Default,,0000,0000,0000,,you have a smile locally\Naround the origin.
Dialogue: 0,1:50:12.97,1:50:17.76,Default,,0000,0000,0000,,OK, look in the book.
Dialogue: 0,1:50:17.76,1:50:19.88,Default,,0000,0000,0000,,I'm not sure how\Nmuch should we do.
Dialogue: 0,1:50:19.88,1:50:23.52,Default,,0000,0000,0000,,Do we give a sketch of a proof,\Nor we give the entire proof?
Dialogue: 0,1:50:23.52,1:50:25.86,Default,,0000,0000,0000,,But more likely, a sketch.
Dialogue: 0,1:50:25.86,1:50:28.75,Default,,0000,0000,0000,,
Dialogue: 0,1:50:28.75,1:50:31.22,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:50:31.22,1:50:35.83,Default,,0000,0000,0000,,STUDENT: I asked the\Nslightly wrong question,
Dialogue: 0,1:50:35.83,1:50:37.32,Default,,0000,0000,0000,,but I answered it myself.
Dialogue: 0,1:50:37.32,1:50:42.17,Default,,0000,0000,0000,,I wanted to ask, why is it\Ndependent on A and not on C?
Dialogue: 0,1:50:42.17,1:50:43.00,Default,,0000,0000,0000,,PROFESSOR: Not on C.
Dialogue: 0,1:50:43.00,1:50:44.96,Default,,0000,0000,0000,,STUDENT: But then I realized\Nthat it is dependent on C
Dialogue: 0,1:50:44.96,1:50:46.27,Default,,0000,0000,0000,,as well, because\Nif A is positive,
Dialogue: 0,1:50:46.27,1:50:47.34,Default,,0000,0000,0000,,then C must be positive.
Dialogue: 0,1:50:47.34,1:50:50.73,Default,,0000,0000,0000,,PROFESSOR: Yes, yes, it\Nis dependent on both.
Dialogue: 0,1:50:50.73,1:50:52.18,Default,,0000,0000,0000,,STUDENT: OK, there we go.
Dialogue: 0,1:50:52.18,1:50:55.10,Default,,0000,0000,0000,,That was my question.
Dialogue: 0,1:50:55.10,1:50:56.76,Default,,0000,0000,0000,,PROFESSOR: So guys, remember.
Dialogue: 0,1:50:56.76,1:51:03.07,Default,,0000,0000,0000,,Imagine what happens when\Nyou had no B, B was 0.
Dialogue: 0,1:51:03.07,1:51:06.10,Default,,0000,0000,0000,,Then the matrix\Nis diagonalizable.
Dialogue: 0,1:51:06.10,1:51:11.94,Default,,0000,0000,0000,,And here you have\NA and C. And Alex
Dialogue: 0,1:51:11.94,1:51:16.19,Default,,0000,0000,0000,,says, why would A be\Nmore important than C?
Dialogue: 0,1:51:16.19,1:51:17.37,Default,,0000,0000,0000,,It's not.
Dialogue: 0,1:51:17.37,1:51:20.68,Default,,0000,0000,0000,,But practically, if A is\Npositive and C is negative,
Dialogue: 0,1:51:20.68,1:51:24.17,Default,,0000,0000,0000,,that means these are the\Nprincipal directions in which
Dialogue: 0,1:51:24.17,1:51:28.70,Default,,0000,0000,0000,,one bends like a valley up and\None bends like a peak down.
Dialogue: 0,1:51:28.70,1:51:36.22,Default,,0000,0000,0000,,So this is what happens in the\Ndirection of x, f double prime
Dialogue: 0,1:51:36.22,1:51:38.77,Default,,0000,0000,0000,,in the direction of x, kind of.
Dialogue: 0,1:51:38.77,1:51:42.90,Default,,0000,0000,0000,,And this is in the\Ndirection of y.
Dialogue: 0,1:51:42.90,1:51:45.18,Default,,0000,0000,0000,,So this is f double prime\Nin the direction of y,
Dialogue: 0,1:51:45.18,1:51:46.86,Default,,0000,0000,0000,,which we don't denote like that.
Dialogue: 0,1:51:46.86,1:51:52.24,Default,,0000,0000,0000,,We call it f sub xx and f\Nsub yy, which is A and C.
Dialogue: 0,1:51:52.24,1:52:00.36,Default,,0000,0000,0000,,So A positive, A being 1\Nand C being negative 2,
Dialogue: 0,1:52:00.36,1:52:05.27,Default,,0000,0000,0000,,means a valley here, means\Nthe valley meets the horse.
Dialogue: 0,1:52:05.27,1:52:10.69,Default,,0000,0000,0000,,Look, I'm drawing the\Ntail of the horse.
Dialogue: 0,1:52:10.69,1:52:14.20,Default,,0000,0000,0000,,He's a little bit\Nfat, this horse.
Dialogue: 0,1:52:14.20,1:52:18.53,Default,,0000,0000,0000,,And that's his mane, his eye.
Dialogue: 0,1:52:18.53,1:52:22.40,Default,,0000,0000,0000,,I'm just taking a break.
Dialogue: 0,1:52:22.40,1:52:24.06,Default,,0000,0000,0000,,STUDENT: That's a\Npretty good drawing.
Dialogue: 0,1:52:24.06,1:52:30.60,Default,,0000,0000,0000,,PROFESSOR: It looks more\Nlike a dog or a plush horse
Dialogue: 0,1:52:30.60,1:52:31.77,Default,,0000,0000,0000,,or something.
Dialogue: 0,1:52:31.77,1:52:36.68,Default,,0000,0000,0000,,So A equals 1, and\NC equals minus 2.
Dialogue: 0,1:52:36.68,1:52:42.47,Default,,0000,0000,0000,,But if it were diagonalizable,\Nand A would be 1
Dialogue: 0,1:52:42.47,1:52:46.61,Default,,0000,0000,0000,,and C would be 7, both of\Nthem positive in any case,
Dialogue: 0,1:52:46.61,1:52:51.79,Default,,0000,0000,0000,,then you'll have valley and\Nvalley, an x direction valley
Dialogue: 0,1:52:51.79,1:52:53.64,Default,,0000,0000,0000,,and y direction valley.
Dialogue: 0,1:52:53.64,1:52:56.51,Default,,0000,0000,0000,,So it has to be a\Nvalley everywhere.
Dialogue: 0,1:52:56.51,1:53:01.44,Default,,0000,0000,0000,,These are the principal\Ndirections that I have 1 and 2.
Dialogue: 0,1:53:01.44,1:53:06.20,Default,,0000,0000,0000,,But then the ultimate\Ncase, what happens
Dialogue: 0,1:53:06.20,1:53:15.20,Default,,0000,0000,0000,,when A is negative\Nand-- hmm, OK,
Dialogue: 0,1:53:15.20,1:53:21.80,Default,,0000,0000,0000,,then either you have them both\None positive, one negative,
Dialogue: 0,1:53:21.80,1:53:27.36,Default,,0000,0000,0000,,or you have plus,\Nplus and minus, minus.
Dialogue: 0,1:53:27.36,1:53:32.51,Default,,0000,0000,0000,,And then you have this\Nas your surface, right?
Dialogue: 0,1:53:32.51,1:53:33.84,Default,,0000,0000,0000,,Which one is the x direction?
Dialogue: 0,1:53:33.84,1:53:34.80,Default,,0000,0000,0000,,That's the y direction.
Dialogue: 0,1:53:34.80,1:53:36.80,Default,,0000,0000,0000,,That's the second one.
Dialogue: 0,1:53:36.80,1:53:39.94,Default,,0000,0000,0000,,The x direction is that.
Dialogue: 0,1:53:39.94,1:53:43.24,Default,,0000,0000,0000,,In the x direction,\Nyou have a frown.
Dialogue: 0,1:53:43.24,1:53:47.34,Default,,0000,0000,0000,,So f sub yy is negative.
Dialogue: 0,1:53:47.34,1:53:51.22,Default,,0000,0000,0000,,In the y direction,\Nyou also have a frown.
Dialogue: 0,1:53:51.22,1:53:53.14,Default,,0000,0000,0000,,So both of them are negative.
Dialogue: 0,1:53:53.14,1:53:55.20,Default,,0000,0000,0000,,So you have a relative max.
Dialogue: 0,1:53:55.20,1:53:58.91,Default,,0000,0000,0000,,Yes, sir, Matthew, tell me.
Dialogue: 0,1:53:58.91,1:54:02.25,Default,,0000,0000,0000,,STUDENT: So isn't it possible\Nto have both A and C positive,
Dialogue: 0,1:54:02.25,1:54:06.70,Default,,0000,0000,0000,,but then yet still not be\Nmore positive than B squared?
Dialogue: 0,1:54:06.70,1:54:10.98,Default,,0000,0000,0000,,PROFESSOR: No, because\Nthere's a theorem that--
Dialogue: 0,1:54:10.98,1:54:13.48,Default,,0000,0000,0000,,STUDENT: I was just\Nwondering like numbers-wise.
Dialogue: 0,1:54:13.48,1:54:16.87,Default,,0000,0000,0000,,PROFESSOR: You have this matrix.
Dialogue: 0,1:54:16.87,1:54:19.79,Default,,0000,0000,0000,,And there is a theorem that\Nshows you that you can actually
Dialogue: 0,1:54:19.79,1:54:21.69,Default,,0000,0000,0000,,diagonalize this matrix.
Dialogue: 0,1:54:21.69,1:54:25.07,Default,,0000,0000,0000,,You'll learn your linear\Nalgebra [INAUDIBLE].
Dialogue: 0,1:54:25.07,1:54:26.62,Default,,0000,0000,0000,,STUDENT: It makes\Nsense, because when
Dialogue: 0,1:54:26.62,1:54:29.81,Default,,0000,0000,0000,,you were saying A is this way,\Nand that way there's no way
Dialogue: 0,1:54:29.81,1:54:34.06,Default,,0000,0000,0000,,you could have 2 come\Nup, and then yet,
Dialogue: 0,1:54:34.06,1:54:36.52,Default,,0000,0000,0000,,not be a-- you know\Nwhat I'm saying?
Dialogue: 0,1:54:36.52,1:54:39.28,Default,,0000,0000,0000,,Because then they'd\Nbe less than 0.
Dialogue: 0,1:54:39.28,1:54:43.21,Default,,0000,0000,0000,,PROFESSOR: You can if\Nyou don't have or the 2.
Dialogue: 0,1:54:43.21,1:54:44.76,Default,,0000,0000,0000,,That's an excellent question.
Dialogue: 0,1:54:44.76,1:54:47.74,Default,,0000,0000,0000,,If I would have x\Nto the 4 y to the 4
Dialogue: 0,1:54:47.74,1:54:52.08,Default,,0000,0000,0000,,added together, like Ax to\Nthe 4 plus By to the 4 plus
Dialogue: 0,1:54:52.08,1:54:56.08,Default,,0000,0000,0000,,something, then I have the\Nso-called monkey saddle.
Dialogue: 0,1:54:56.08,1:54:57.50,Default,,0000,0000,0000,,That's so funny.
Dialogue: 0,1:54:57.50,1:55:05.50,Default,,0000,0000,0000,,You can have something\Nthat looks like that.
Dialogue: 0,1:55:05.50,1:55:07.91,Default,,0000,0000,0000,,So in your direction,\Nyou can have this.
Dialogue: 0,1:55:07.91,1:55:10.27,Default,,0000,0000,0000,,Then I've reached\Ntwo equal peaks
Dialogue: 0,1:55:10.27,1:55:11.48,Default,,0000,0000,0000,,in the x and the y direction.
Dialogue: 0,1:55:11.48,1:55:14.06,Default,,0000,0000,0000,,But in the between,\NI also went down.
Dialogue: 0,1:55:14.06,1:55:18.44,Default,,0000,0000,0000,,So depending on a higher\Ndegree symmetric polynomial,
Dialogue: 0,1:55:18.44,1:55:21.42,Default,,0000,0000,0000,,you can have a monkey saddle.
Dialogue: 0,1:55:21.42,1:55:24.82,Default,,0000,0000,0000,,And then it's not just\Nlike you can predict what's
Dialogue: 0,1:55:24.82,1:55:26.93,Default,,0000,0000,0000,,going to happen in between.
Dialogue: 0,1:55:26.93,1:55:31.09,Default,,0000,0000,0000,,In between, if I go\Nup, if I go valley
Dialogue: 0,1:55:31.09,1:55:34.03,Default,,0000,0000,0000,,in the x direction and\Nvalley in the y direction,
Dialogue: 0,1:55:34.03,1:55:36.66,Default,,0000,0000,0000,,I know that's going to be\Na valley everywhere-- no.
Dialogue: 0,1:55:36.66,1:55:38.88,Default,,0000,0000,0000,,If a polynomial\Nis high in order,
Dialogue: 0,1:55:38.88,1:55:42.41,Default,,0000,0000,0000,,it can go down,\Nvalley, and up again,
Dialogue: 0,1:55:42.41,1:55:45.96,Default,,0000,0000,0000,,and monkey saddle it looks like.
Dialogue: 0,1:55:45.96,1:55:49.32,Default,,0000,0000,0000,,Guys, you have dealt\Nwith it when you
Dialogue: 0,1:55:49.32,1:55:52.78,Default,,0000,0000,0000,,went to Luna Park or Joyland.
Dialogue: 0,1:55:52.78,1:56:00.35,Default,,0000,0000,0000,,It's one of those things\Nthat look like-- I'm trying.
Dialogue: 0,1:56:00.35,1:56:01.12,Default,,0000,0000,0000,,I cannot draw.
Dialogue: 0,1:56:01.12,1:56:02.78,Default,,0000,0000,0000,,STUDENT: It sounds\Nmore like an octopus.
Dialogue: 0,1:56:02.78,1:56:04.25,Default,,0000,0000,0000,,PROFESSOR: Like an octopus.
Dialogue: 0,1:56:04.25,1:56:06.94,Default,,0000,0000,0000,,And one of those\Nthings-- exactly--
Dialogue: 0,1:56:06.94,1:56:11.34,Default,,0000,0000,0000,,that are shaped so that\Nthey are undulated,
Dialogue: 0,1:56:11.34,1:56:13.94,Default,,0000,0000,0000,,in some directions are\Ngoing up, in some directions
Dialogue: 0,1:56:13.94,1:56:14.92,Default,,0000,0000,0000,,are going down.
Dialogue: 0,1:56:14.92,1:56:16.73,Default,,0000,0000,0000,,STUDENT: Like an\Negg carton, almost?
Dialogue: 0,1:56:16.73,1:56:19.91,Default,,0000,0000,0000,,PROFESSOR: Yeah,\Nreally undulated.
Dialogue: 0,1:56:19.91,1:56:22.91,Default,,0000,0000,0000,,Imagine even a surface made\Nof metal that's undulated
Dialogue: 0,1:56:22.91,1:56:24.64,Default,,0000,0000,0000,,and rotating at the same time.
Dialogue: 0,1:56:24.64,1:56:28.68,Default,,0000,0000,0000,,They have some of\Nthose in Disney World.
Dialogue: 0,1:56:28.68,1:56:32.36,Default,,0000,0000,0000,,Have you been to Orlando?
Dialogue: 0,1:56:32.36,1:56:35.36,Default,,0000,0000,0000,,STUDENT: I was\Nthere last semester.
Dialogue: 0,1:56:35.36,1:56:38.16,Default,,0000,0000,0000,,PROFESSOR: But you didn't take\Nme with you, which is bad.
Dialogue: 0,1:56:38.16,1:56:42.44,Default,,0000,0000,0000,,Because that's one of\Nmy favorite places.
Dialogue: 0,1:56:42.44,1:56:44.94,Default,,0000,0000,0000,,STUDENT: I was just trying to\Nthink of what you were talking
Dialogue: 0,1:56:44.94,1:56:45.38,Default,,0000,0000,0000,,about so I could visualize it.
Dialogue: 0,1:56:45.38,1:56:47.52,Default,,0000,0000,0000,,PROFESSOR: Maybe we\Ncould make a proposal
Dialogue: 0,1:56:47.52,1:56:50.51,Default,,0000,0000,0000,,to teach Calculus\NIII at Disney World
Dialogue: 0,1:56:50.51,1:56:55.06,Default,,0000,0000,0000,,so that we could have examples\Nof motion and surfaces
Dialogue: 0,1:56:55.06,1:57:00.11,Default,,0000,0000,0000,,all around and study the\Nmotion of all sorts of gadgets,
Dialogue: 0,1:57:00.11,1:57:01.18,Default,,0000,0000,0000,,velocity and trajectory.
Dialogue: 0,1:57:01.18,1:57:06.31,Default,,0000,0000,0000,,
Dialogue: 0,1:57:06.31,1:57:09.52,Default,,0000,0000,0000,,Last night I couldn't sleep\Nuntil 1:00, and I was thinking,
Dialogue: 0,1:57:09.52,1:57:14.01,Default,,0000,0000,0000,,I gave examples of\Nthe winter sports
Dialogue: 0,1:57:14.01,1:57:18.14,Default,,0000,0000,0000,,like bobsled and all\Nsorts of skiing and so on.
Dialogue: 0,1:57:18.14,1:57:22.02,Default,,0000,0000,0000,,But I never thought\Nabout a screw curve
Dialogue: 0,1:57:22.02,1:57:28.00,Default,,0000,0000,0000,,with curvature and torsion that\Nis based on the roller coaster.
Dialogue: 0,1:57:28.00,1:57:29.53,Default,,0000,0000,0000,,And the roller\Ncoaster is actually
Dialogue: 0,1:57:29.53,1:57:35.39,Default,,0000,0000,0000,,the best place to study the\N[INAUDIBLE], the velocity,
Dialogue: 0,1:57:35.39,1:57:38.46,Default,,0000,0000,0000,,the tangent unit, the\Nnormal, the bi-normal.
Dialogue: 0,1:57:38.46,1:57:42.91,Default,,0000,0000,0000,,And when you have in a\Nplane the roller coaster
Dialogue: 0,1:57:42.91,1:57:46.61,Default,,0000,0000,0000,,goes like that, like this and\Nlike that, like in a plane,
Dialogue: 0,1:57:46.61,1:57:49.65,Default,,0000,0000,0000,,you have nothing but bending,\Nwhich means curvature.
Dialogue: 0,1:57:49.65,1:57:53.48,Default,,0000,0000,0000,,But then when the roller coaster\Ngoes away from the plane,
Dialogue: 0,1:57:53.48,1:57:55.02,Default,,0000,0000,0000,,you have the torsion.
Dialogue: 0,1:57:55.02,1:57:57.94,Default,,0000,0000,0000,,And that makes you sick\Nreally to the stomach.
Dialogue: 0,1:57:57.94,1:58:00.91,Default,,0000,0000,0000,,So we would have to\Nexperience that to understand
Dialogue: 0,1:58:00.91,1:58:02.66,Default,,0000,0000,0000,,Calculus III better.
Dialogue: 0,1:58:02.66,1:58:05.78,Default,,0000,0000,0000,,So our next proposal is\Nwe ask the administration
Dialogue: 0,1:58:05.78,1:58:10.92,Default,,0000,0000,0000,,instead of study abroad\Ncourses, the domestic study
Dialogue: 0,1:58:10.92,1:58:15.57,Default,,0000,0000,0000,,at Disney World for Calc III.
Dialogue: 0,1:58:15.57,1:58:16.79,Default,,0000,0000,0000,,It's Applied Calculus III.
Dialogue: 0,1:58:16.79,1:58:20.49,Default,,0000,0000,0000,,
Dialogue: 0,1:58:20.49,1:58:24.42,Default,,0000,0000,0000,,OK, something else that\NI want you to do-- I
Dialogue: 0,1:58:24.42,1:58:26.78,Default,,0000,0000,0000,,had prepared an example.
Dialogue: 0,1:58:26.78,1:58:30.91,Default,,0000,0000,0000,,
Dialogue: 0,1:58:30.91,1:58:33.35,Default,,0000,0000,0000,,This is an absolute extrema.
Dialogue: 0,1:58:33.35,1:58:40.67,Default,,0000,0000,0000,,
Dialogue: 0,1:58:40.67,1:58:43.67,Default,,0000,0000,0000,,And you say, what the heck\Nare the absolute extrema?
Dialogue: 0,1:58:43.67,1:58:47.59,Default,,0000,0000,0000,,Because she only talked to\Nus about relative maximum
Dialogue: 0,1:58:47.59,1:58:49.52,Default,,0000,0000,0000,,and relative minimum.
Dialogue: 0,1:58:49.52,1:58:53.68,Default,,0000,0000,0000,,And she never said anything\Nabout absolute extrema.
Dialogue: 0,1:58:53.68,1:58:56.50,Default,,0000,0000,0000,,
Dialogue: 0,1:58:56.50,1:58:58.90,Default,,0000,0000,0000,,And that will be the table.
Dialogue: 0,1:58:58.90,1:59:02.28,Default,,0000,0000,0000,,And these will be the extrema.
Dialogue: 0,1:59:02.28,1:59:05.73,Default,,0000,0000,0000,,I want to refresh your memory\Nfirst just a little bit.
Dialogue: 0,1:59:05.73,1:59:07.20,Default,,0000,0000,0000,,This will be the last example.
Dialogue: 0,1:59:07.20,1:59:10.34,Default,,0000,0000,0000,,Because it's actually\Ntwo examples in one.
Dialogue: 0,1:59:10.34,1:59:13.24,Default,,0000,0000,0000,,
Dialogue: 0,1:59:13.24,1:59:26.21,Default,,0000,0000,0000,,And what if you have, let's say,\Nf of x equals e to the minus
Dialogue: 0,1:59:26.21,1:59:38.33,Default,,0000,0000,0000,,x squared over the\Ninterval minus 1, 1?
Dialogue: 0,1:59:38.33,1:59:41.17,Default,,0000,0000,0000,,You are in Calc I.\NYou will build a time
Dialogue: 0,1:59:41.17,1:59:44.12,Default,,0000,0000,0000,,machine from Disney World.
Dialogue: 0,1:59:44.12,1:59:50.02,Default,,0000,0000,0000,,And we went back in time when\Nyou actually took Calc I.
Dialogue: 0,1:59:50.02,1:59:52.18,Default,,0000,0000,0000,,And you struggled\Nwith this at first.
Dialogue: 0,1:59:52.18,1:59:53.93,Default,,0000,0000,0000,,But then you loved\Nit so much that you
Dialogue: 0,1:59:53.93,1:59:56.96,Default,,0000,0000,0000,,said, oh, that's my favorite\Nproblem on the final.
Dialogue: 0,1:59:56.96,2:00:04.82,Default,,0000,0000,0000,,They asked us for two things--\Nrelative extrema, min or max,
Dialogue: 0,2:00:04.82,2:00:08.61,Default,,0000,0000,0000,,min/max theory, and\Nthey say absolute.
Dialogue: 0,2:00:08.61,2:00:11.46,Default,,0000,0000,0000,,But for the absolute, your\Nteacher said, attention,
Dialogue: 0,2:00:11.46,2:00:15.14,Default,,0000,0000,0000,,you have to know how\Nto get to the absolute.
Dialogue: 0,2:00:15.14,2:00:20.39,Default,,0000,0000,0000,,You are constrained to be\Non the segment minus 1, 1.
Dialogue: 0,2:00:20.39,2:00:21.97,Default,,0000,0000,0000,,You see, the fact\Nthat they introduced
Dialogue: 0,2:00:21.97,2:00:24.53,Default,,0000,0000,0000,,this extra constraint\Nand they don't
Dialogue: 0,2:00:24.53,2:00:30.56,Default,,0000,0000,0000,,let you move with x on the whole\Nreal line is a big headache.
Dialogue: 0,2:00:30.56,2:00:32.04,Default,,0000,0000,0000,,Why is that a big headache?
Dialogue: 0,2:00:32.04,2:00:35.07,Default,,0000,0000,0000,,Your life would be much\Neasier if it were just e
Dialogue: 0,2:00:35.07,2:00:36.87,Default,,0000,0000,0000,,to the negative x squared.
Dialogue: 0,2:00:36.87,2:00:40.16,Default,,0000,0000,0000,,Because in that\Ncase, you say, OK,
Dialogue: 0,2:00:40.16,2:00:47.18,Default,,0000,0000,0000,,f prime of x equals minus\N2xe to the minus x squared.
Dialogue: 0,2:00:47.18,2:00:48.27,Default,,0000,0000,0000,,Piece of cake.
Dialogue: 0,2:00:48.27,2:00:50.45,Default,,0000,0000,0000,,x0 is 0.
Dialogue: 0,2:00:50.45,2:00:55.02,Default,,0000,0000,0000,,That's the only critical point.
Dialogue: 0,2:00:55.02,2:00:59.67,Default,,0000,0000,0000,,And I want to study what kind\Nof critical point that is.
Dialogue: 0,2:00:59.67,2:01:03.77,Default,,0000,0000,0000,,So I have to do f\Ndouble prime of x.
Dialogue: 0,2:01:03.77,2:01:08.26,Default,,0000,0000,0000,,And if I don't know the\Nproduct rule, I'm in trouble.
Dialogue: 0,2:01:08.26,2:01:12.18,Default,,0000,0000,0000,,And I go, let's\Nsay, minus 2 times
Dialogue: 0,2:01:12.18,2:01:15.65,Default,,0000,0000,0000,,e to the negative x\Nsquared from prime of this
Dialogue: 0,2:01:15.65,2:01:20.42,Default,,0000,0000,0000,,and this non-prime, plus\Nminus 2x un-prime times e
Dialogue: 0,2:01:20.42,2:01:25.42,Default,,0000,0000,0000,,to the minus x squared\Ntimes minus 2x again.
Dialogue: 0,2:01:25.42,2:01:27.68,Default,,0000,0000,0000,,So it's a headache.
Dialogue: 0,2:01:27.68,2:01:31.40,Default,,0000,0000,0000,,I pull out an e to\Nthe minus x squared.
Dialogue: 0,2:01:31.40,2:01:37.61,Default,,0000,0000,0000,,And I have 4x squared--\N4x squared-- minus 2.
Dialogue: 0,2:01:37.61,2:01:41.40,Default,,0000,0000,0000,,
Dialogue: 0,2:01:41.40,2:01:43.35,Default,,0000,0000,0000,,But you say, but wait\Na minute, Magdalena,
Dialogue: 0,2:01:43.35,2:01:45.58,Default,,0000,0000,0000,,I'm not going to compute\Nthe inflection points.
Dialogue: 0,2:01:45.58,2:01:48.94,Default,,0000,0000,0000,,The inflection points\Nwill be x equals
Dialogue: 0,2:01:48.94,2:01:52.92,Default,,0000,0000,0000,,plus/minus 1 over root 2.
Dialogue: 0,2:01:52.92,2:01:55.70,Default,,0000,0000,0000,,I only care about\Nthe critical point.
Dialogue: 0,2:01:55.70,2:01:59.07,Default,,0000,0000,0000,,And the only critical\Npoint I have is at 0.
Dialogue: 0,2:01:59.07,2:02:04.79,Default,,0000,0000,0000,,Compute f double prime of 0 to\Nsee if it's a smile or a frown.
Dialogue: 0,2:02:04.79,2:02:07.57,Default,,0000,0000,0000,,
Dialogue: 0,2:02:07.57,2:02:09.69,Default,,0000,0000,0000,,And you do it.
Dialogue: 0,2:02:09.69,2:02:14.63,Default,,0000,0000,0000,,And you plug in, and you say,\Ne to the 0 is 1, 0 minus 2.
Dialogue: 0,2:02:14.63,2:02:16.95,Default,,0000,0000,0000,,So you get a negative.
Dialogue: 0,2:02:16.95,2:02:18.54,Default,,0000,0000,0000,,Do you care what it is?
Dialogue: 0,2:02:18.54,2:02:20.21,Default,,0000,0000,0000,,No, but you care it's negative.
Dialogue: 0,2:02:20.21,2:02:23.43,Default,,0000,0000,0000,,
Dialogue: 0,2:02:23.43,2:02:30.55,Default,,0000,0000,0000,,So at 0, so you\Ndraw, and you know
Dialogue: 0,2:02:30.55,2:02:36.54,Default,,0000,0000,0000,,at 0 you're going to\Nhave some sort of a what?
Dialogue: 0,2:02:36.54,2:02:37.89,Default,,0000,0000,0000,,Relative max.
Dialogue: 0,2:02:37.89,2:02:40.49,Default,,0000,0000,0000,,
Dialogue: 0,2:02:40.49,2:02:40.99,Default,,0000,0000,0000,,Where?
Dialogue: 0,2:02:40.99,2:02:44.48,Default,,0000,0000,0000,,At 0, and when you\Nplug 0 again, 1.
Dialogue: 0,2:02:44.48,2:02:45.66,Default,,0000,0000,0000,,So you draw a table.
Dialogue: 0,2:02:45.66,2:02:52.05,Default,,0000,0000,0000,,And you say,\Nrelative max at 0, 1.
Dialogue: 0,2:02:52.05,2:02:55.52,Default,,0000,0000,0000,,And then you're not done.
Dialogue: 0,2:02:55.52,2:02:57.71,Default,,0000,0000,0000,,Because you say,\Nwait a minute, I
Dialogue: 0,2:02:57.71,2:03:03.65,Default,,0000,0000,0000,,am to study my function in\NCalc I at minus 1 and 1.
Dialogue: 0,2:03:03.65,2:03:07.78,Default,,0000,0000,0000,,It's like you have a\Ncontinuous picture,
Dialogue: 0,2:03:07.78,2:03:13.11,Default,,0000,0000,0000,,and you chop, take scissors,\Nand cut and cut at the extrema.
Dialogue: 0,2:03:13.11,2:03:15.97,Default,,0000,0000,0000,,And there you can\Nget additional points
Dialogue: 0,2:03:15.97,2:03:19.72,Default,,0000,0000,0000,,where you can get a relative\Nmax or relative min.
Dialogue: 0,2:03:19.72,2:03:25.17,Default,,0000,0000,0000,,Absolute max or min will be\Nthe lowest of all the values
Dialogue: 0,2:03:25.17,2:03:28.36,Default,,0000,0000,0000,,and the highest\Nof all the values.
Dialogue: 0,2:03:28.36,2:03:35.27,Default,,0000,0000,0000,,OK, so I get at the point\Nminus 1-- how shall I put here?
Dialogue: 0,2:03:35.27,2:03:37.04,Default,,0000,0000,0000,,x equals minus 1.
Dialogue: 0,2:03:37.04,2:03:38.68,Default,,0000,0000,0000,,What do I get for y?
Dialogue: 0,2:03:38.68,2:03:41.08,Default,,0000,0000,0000,,And for plus 1,\Nwhat do I get for y?
Dialogue: 0,2:03:41.08,2:03:43.03,Default,,0000,0000,0000,,This is the question.
Dialogue: 0,2:03:43.03,2:03:47.60,Default,,0000,0000,0000,,I plug it in, and I\Nget minus minus 1/e.
Dialogue: 0,2:03:47.60,2:03:52.20,Default,,0000,0000,0000,,
Dialogue: 0,2:03:52.20,2:03:56.18,Default,,0000,0000,0000,,And then when I have\N1, what do I get?
Dialogue: 0,2:03:56.18,2:03:59.65,Default,,0000,0000,0000,,1/e again.
Dialogue: 0,2:03:59.65,2:04:01.76,Default,,0000,0000,0000,,So do I have a\Nrelative min here?
Dialogue: 0,2:04:01.76,2:04:07.22,Default,,0000,0000,0000,,No, but I have an\Nabsolute something.
Dialogue: 0,2:04:07.22,2:04:10.21,Default,,0000,0000,0000,,
Dialogue: 0,2:04:10.21,2:04:12.20,Default,,0000,0000,0000,,And what do I have here?
Dialogue: 0,2:04:12.20,2:04:15.70,Default,,0000,0000,0000,,Here I have an absolute max.
Dialogue: 0,2:04:15.70,2:04:22.23,Default,,0000,0000,0000,,So how do we check the absolute\Nmaxima and absolute minima?
Dialogue: 0,2:04:22.23,2:04:23.96,Default,,0000,0000,0000,,We look for critical points.
Dialogue: 0,2:04:23.96,2:04:26.56,Default,,0000,0000,0000,,We get many of them,\Nfinitely many of them.
Dialogue: 0,2:04:26.56,2:04:29.52,Default,,0000,0000,0000,,We compute all the\Nvalues of z for them,
Dialogue: 0,2:04:29.52,2:04:32.74,Default,,0000,0000,0000,,all the function values.
Dialogue: 0,2:04:32.74,2:04:36.05,Default,,0000,0000,0000,,And then we look\Nat the end points,
Dialogue: 0,2:04:36.05,2:04:38.49,Default,,0000,0000,0000,,and we compare all three\Nof them, all the three
Dialogue: 0,2:04:38.49,2:04:39.27,Default,,0000,0000,0000,,values in the end.
Dialogue: 0,2:04:39.27,2:04:43.26,Default,,0000,0000,0000,,So in the end, you\Ncompare 1/e to 1/e to 1.
Dialogue: 0,2:04:43.26,2:04:45.91,Default,,0000,0000,0000,,And that's all you can get.
Dialogue: 0,2:04:45.91,2:04:49.46,Default,,0000,0000,0000,,So the lowest in one will\Nbe the highest in one.
Dialogue: 0,2:04:49.46,2:04:50.92,Default,,0000,0000,0000,,Good.
Dialogue: 0,2:04:50.92,2:04:55.16,Default,,0000,0000,0000,,In Calculus III, it's\Nmore complicated.
Dialogue: 0,2:04:55.16,2:04:57.55,Default,,0000,0000,0000,,But it's not much\Nmore complicated.
Dialogue: 0,2:04:57.55,2:05:02.23,Default,,0000,0000,0000,,Let's see what's\Ngoing to happen.
Dialogue: 0,2:05:02.23,2:05:04.04,Default,,0000,0000,0000,,You can have a\Ncritical point inside.
Dialogue: 0,2:05:04.04,2:05:07.40,Default,,0000,0000,0000,,We are just praying we\Ndon't have too many.
Dialogue: 0,2:05:07.40,2:05:10.29,Default,,0000,0000,0000,,So how do I get to step one?
Dialogue: 0,2:05:10.29,2:05:15.16,Default,,0000,0000,0000,,Critical point means f sub\Nx equals e to the x squared
Dialogue: 0,2:05:15.16,2:05:18.89,Default,,0000,0000,0000,,minus y squared times 2x.
Dialogue: 0,2:05:18.89,2:05:20.76,Default,,0000,0000,0000,,All righty, it looks good.
Dialogue: 0,2:05:20.76,2:05:24.53,Default,,0000,0000,0000,,f sub y equals e to the x\Nsquared minus y squared times
Dialogue: 0,2:05:24.53,2:05:25.61,Default,,0000,0000,0000,,minus y.
Dialogue: 0,2:05:25.61,2:05:26.87,Default,,0000,0000,0000,,I'm full of hope.
Dialogue: 0,2:05:26.87,2:05:30.42,Default,,0000,0000,0000,,Because I only have one\Ncritical point, thank god.
Dialogue: 0,2:05:30.42,2:05:35.41,Default,,0000,0000,0000,,Origin is my only\Ncritical point.
Dialogue: 0,2:05:35.41,2:05:37.59,Default,,0000,0000,0000,,I don't know what that\Nis going to give me.
Dialogue: 0,2:05:37.59,2:05:39.83,Default,,0000,0000,0000,,But it can give\Nme a relative max
Dialogue: 0,2:05:39.83,2:05:42.38,Default,,0000,0000,0000,,or relative min or a saddle.
Dialogue: 0,2:05:42.38,2:05:45.76,Default,,0000,0000,0000,,I don't know what\Nit's going to be.
Dialogue: 0,2:05:45.76,2:05:49.89,Default,,0000,0000,0000,,Who tells me what\Nthat is going to be?
Dialogue: 0,2:05:49.89,2:05:51.92,Default,,0000,0000,0000,,Well, did I do this further?
Dialogue: 0,2:05:51.92,2:05:55.19,Default,,0000,0000,0000,,
Dialogue: 0,2:05:55.19,2:06:00.96,Default,,0000,0000,0000,,I did it further and\Na little bit lazy.
Dialogue: 0,2:06:00.96,2:06:05.01,Default,,0000,0000,0000,,But I'm not asking the\Nnature of the point.
Dialogue: 0,2:06:05.01,2:06:08.87,Default,,0000,0000,0000,,So for the time being, I only\Nwant to see what happens at 0,
Dialogue: 0,2:06:08.87,2:06:09.74,Default,,0000,0000,0000,,0.
Dialogue: 0,2:06:09.74,2:06:12.21,Default,,0000,0000,0000,,So I have 1.
Dialogue: 0,2:06:12.21,2:06:18.07,Default,,0000,0000,0000,,So in my table I will put, OK,\Nthis is x, y, and this is z.
Dialogue: 0,2:06:18.07,2:06:20.92,Default,,0000,0000,0000,,For 0, 0, I'm interested.
Dialogue: 0,2:06:20.92,2:06:23.73,Default,,0000,0000,0000,,Because that's the critical\Npoint inside the domain.
Dialogue: 0,2:06:23.73,2:06:26.24,Default,,0000,0000,0000,,The domain will be the unities.
Dialogue: 0,2:06:26.24,2:06:29.60,Default,,0000,0000,0000,,And inside the origin,\Nsomething interesting happens.
Dialogue: 0,2:06:29.60,2:06:31.34,Default,,0000,0000,0000,,I get a 1.
Dialogue: 0,2:06:31.34,2:06:35.10,Default,,0000,0000,0000,,And I hope that's going to\Nbe my absolute something.
Dialogue: 0,2:06:35.10,2:06:36.73,Default,,0000,0000,0000,,But I cannot be sure.
Dialogue: 0,2:06:36.73,2:06:38.22,Default,,0000,0000,0000,,Why?
Dialogue: 0,2:06:38.22,2:06:43.28,Default,,0000,0000,0000,,There may be other values\Ncoming from the boundary.
Dialogue: 0,2:06:43.28,2:06:46.02,Default,,0000,0000,0000,,And just like in\NCalculus I, the only guys
Dialogue: 0,2:06:46.02,2:06:48.64,Default,,0000,0000,0000,,that can give you other\Nabsolute max or min,
Dialogue: 0,2:06:48.64,2:06:51.57,Default,,0000,0000,0000,,they can come from the\Nboundary, nothing else.
Dialogue: 0,2:06:51.57,2:06:55.32,Default,,0000,0000,0000,,Nowhere else in the interior\Nof the disc am I going to look.
Dialogue: 0,2:06:55.32,2:06:56.56,Default,,0000,0000,0000,,I'm not interested.
Dialogue: 0,2:06:56.56,2:06:59.67,Default,,0000,0000,0000,,I'm only interested in x\Nsquared plus y squared equals 1.
Dialogue: 0,2:06:59.67,2:07:02.61,Default,,0000,0000,0000,,This is where\Nsomething can happen,
Dialogue: 0,2:07:02.61,2:07:07.07,Default,,0000,0000,0000,,nothing else interesting in the\Ninside, just like in Calc I.
Dialogue: 0,2:07:07.07,2:07:12.28,Default,,0000,0000,0000,,So to take x squared plus y\Nsquared equals 1 into account,
Dialogue: 0,2:07:12.28,2:07:16.26,Default,,0000,0000,0000,,I pull y squared, who is\Nmarried to x-- the poor guy.
Dialogue: 0,2:07:16.26,2:07:17.96,Default,,0000,0000,0000,,He's married to x.
Dialogue: 0,2:07:17.96,2:07:21.91,Default,,0000,0000,0000,,He's dependent on x\Ncompletely, y squared
Dialogue: 0,2:07:21.91,2:07:23.71,Default,,0000,0000,0000,,equals 1 minus x squared.
Dialogue: 0,2:07:23.71,2:07:27.81,Default,,0000,0000,0000,,And I have to push him\Nback into the function.
Dialogue: 0,2:07:27.81,2:07:36.96,Default,,0000,0000,0000,,So at the boundary, f becomes\Na function of one variable.
Dialogue: 0,2:07:36.96,2:07:43.41,Default,,0000,0000,0000,,He becomes f of x\Nonly equals e to the x
Dialogue: 0,2:07:43.41,2:07:47.46,Default,,0000,0000,0000,,squared minus 1 plus x squared.
Dialogue: 0,2:07:47.46,2:07:50.38,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,2:07:50.38,2:07:56.73,Default,,0000,0000,0000,,So f of x will become e\Nto the 2x squared minus 1
Dialogue: 0,2:07:56.73,2:08:02.57,Default,,0000,0000,0000,,along the boundary, along\Nthe circle, only here.
Dialogue: 0,2:08:02.57,2:08:07.67,Default,,0000,0000,0000,,
Dialogue: 0,2:08:07.67,2:08:10.70,Default,,0000,0000,0000,,Now what else do I need to do?
Dialogue: 0,2:08:10.70,2:08:14.29,Default,,0000,0000,0000,,I need to compute the critical\Nvalues for this function of one
Dialogue: 0,2:08:14.29,2:08:18.06,Default,,0000,0000,0000,,variable, just the way\NI did it in Calc I.
Dialogue: 0,2:08:18.06,2:08:22.71,Default,,0000,0000,0000,,So f prime of x will give\Nme e to the 2x squared
Dialogue: 0,2:08:22.71,2:08:27.63,Default,,0000,0000,0000,,minus 1 times-- what comes\Ndown from the chain rule?
Dialogue: 0,2:08:27.63,2:08:28.13,Default,,0000,0000,0000,,STUDENT: 4x.
Dialogue: 0,2:08:28.13,2:08:31.44,Default,,0000,0000,0000,,PROFESSOR: 4x, so life is\Nhard but not that hard.
Dialogue: 0,2:08:31.44,2:08:35.49,Default,,0000,0000,0000,,Because I can get what?
Dialogue: 0,2:08:35.49,2:08:39.56,Default,,0000,0000,0000,,I can get only x\Nat 0 equals 0 here.
Dialogue: 0,2:08:39.56,2:08:44.98,Default,,0000,0000,0000,,OK, so that's a critical point\Nthat comes from the boundary.
Dialogue: 0,2:08:44.98,2:08:48.26,Default,,0000,0000,0000,,But guys, you have\Nto pay attention.
Dialogue: 0,2:08:48.26,2:08:53.28,Default,,0000,0000,0000,,When x is 0, how many\Ny's can I have for that 0
Dialogue: 0,2:08:53.28,2:08:54.52,Default,,0000,0000,0000,,on the boundary?
Dialogue: 0,2:08:54.52,2:08:59.13,Default,,0000,0000,0000,,This is on the boundary--\Non the boundary.
Dialogue: 0,2:08:59.13,2:09:00.11,Default,,0000,0000,0000,,STUDENT: Two.
Dialogue: 0,2:09:00.11,2:09:02.98,Default,,0000,0000,0000,,PROFESSOR: Two of\Nthem-- I can have 1,
Dialogue: 0,2:09:02.98,2:09:04.02,Default,,0000,0000,0000,,or I can have negative 1.
Dialogue: 0,2:09:04.02,2:09:07.45,Default,,0000,0000,0000,,
Dialogue: 0,2:09:07.45,2:09:10.16,Default,,0000,0000,0000,,There is one more tricky thing.
Dialogue: 0,2:09:10.16,2:09:12.04,Default,,0000,0000,0000,,This is a function\Nof one variable only.
Dialogue: 0,2:09:12.04,2:09:15.86,Default,,0000,0000,0000,,But this stinking\Nfunction is not
Dialogue: 0,2:09:15.86,2:09:19.07,Default,,0000,0000,0000,,defined for arbitrary x real.
Dialogue: 0,2:09:19.07,2:09:21.22,Default,,0000,0000,0000,,So I make a face again.
Dialogue: 0,2:09:21.22,2:09:23.52,Default,,0000,0000,0000,,So I go, oh, headache.
Dialogue: 0,2:09:23.52,2:09:24.49,Default,,0000,0000,0000,,Why?
Dialogue: 0,2:09:24.49,2:09:26.14,Default,,0000,0000,0000,,x is constrained.
Dialogue: 0,2:09:26.14,2:09:27.100,Default,,0000,0000,0000,,x is constrained, you see?
Dialogue: 0,2:09:27.100,2:09:36.20,Default,,0000,0000,0000,,If you were inside the disc, x\Nmust be between minus 1 and 1.
Dialogue: 0,2:09:36.20,2:09:40.16,Default,,0000,0000,0000,,So I have to take into account\Nthat x is not any real number,
Dialogue: 0,2:09:40.16,2:09:43.87,Default,,0000,0000,0000,,but x is between minus 1 and 1.
Dialogue: 0,2:09:43.87,2:09:46.49,Default,,0000,0000,0000,,Those are endpoints\Nfor this function.
Dialogue: 0,2:09:46.49,2:09:48.44,Default,,0000,0000,0000,,And in Calc I, I\Nlearned, OK, I have
Dialogue: 0,2:09:48.44,2:09:52.29,Default,,0000,0000,0000,,to also evaluate what\Nhappens at those endpoints.
Dialogue: 0,2:09:52.29,2:09:55.99,Default,,0000,0000,0000,,But thank god that will exhaust\Nmy list, so I have a list.
Dialogue: 0,2:09:55.99,2:10:00.21,Default,,0000,0000,0000,,Minus 1 for x and 1\Nfor x-- thank god.
Dialogue: 0,2:10:00.21,2:10:04.29,Default,,0000,0000,0000,,That will give you\Nwhat y on the boundary?
Dialogue: 0,2:10:04.29,2:10:07.28,Default,,0000,0000,0000,,When x is 1 and x\Nis minus 1, you're
Dialogue: 0,2:10:07.28,2:10:10.42,Default,,0000,0000,0000,,interested in what happens,\Nmaximization or minimization,
Dialogue: 0,2:10:10.42,2:10:12.85,Default,,0000,0000,0000,,for this function\Nat the endpoints.
Dialogue: 0,2:10:12.85,2:10:20.99,Default,,0000,0000,0000,,But fortunately, since you are\Non the boundary, y must be 0.
Dialogue: 0,2:10:20.99,2:10:25.48,Default,,0000,0000,0000,,Because that's\Nhow you got y out.
Dialogue: 0,2:10:25.48,2:10:29.32,Default,,0000,0000,0000,,If x is plus/minus 1 on\Nthe boundary, y must be 0.
Dialogue: 0,2:10:29.32,2:10:32.68,Default,,0000,0000,0000,,So my list contains how\Nmany interesting points?
Dialogue: 0,2:10:32.68,2:10:36.14,Default,,0000,0000,0000,,One, two, three, four,\Nfive-- for all of them,
Dialogue: 0,2:10:36.14,2:10:38.35,Default,,0000,0000,0000,,we need to compute,\Nand we are done.
Dialogue: 0,2:10:38.35,2:10:43.25,Default,,0000,0000,0000,,Of all of them, the lowest z\Nis called absolute minimum.
Dialogue: 0,2:10:43.25,2:10:45.51,Default,,0000,0000,0000,,And the highest z is\Nthe absolute maximum.
Dialogue: 0,2:10:45.51,2:10:46.48,Default,,0000,0000,0000,,And we are done.
Dialogue: 0,2:10:46.48,2:10:49.97,Default,,0000,0000,0000,,You guys need to help me,\Nbecause I'm running out of gas.
Dialogue: 0,2:10:49.97,2:10:52.57,Default,,0000,0000,0000,,So x is 0.
Dialogue: 0,2:10:52.57,2:10:53.55,Default,,0000,0000,0000,,Y is 1.
Dialogue: 0,2:10:53.55,2:10:55.22,Default,,0000,0000,0000,,What is z?
Dialogue: 0,2:10:55.22,2:10:56.90,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:10:56.90,2:10:59.61,Default,,0000,0000,0000,,PROFESSOR: e to the\Nminus 1, you were
Dialogue: 0,2:10:59.61,2:11:02.28,Default,,0000,0000,0000,,fast, 1/e, thank you, guys.
Dialogue: 0,2:11:02.28,2:11:06.07,Default,,0000,0000,0000,,So when x is 0 and y is minus 1?
Dialogue: 0,2:11:06.07,2:11:07.32,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:11:07.32,2:11:08.19,Default,,0000,0000,0000,,PROFESSOR: Huh?
Dialogue: 0,2:11:08.19,2:11:09.15,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,2:11:09.15,2:11:10.69,Default,,0000,0000,0000,,PROFESSOR: No, no,\Nno, it's the same.
Dialogue: 0,2:11:10.69,2:11:12.66,Default,,0000,0000,0000,,Because x is 0. y is minus 1.
Dialogue: 0,2:11:12.66,2:11:17.13,Default,,0000,0000,0000,,I get e to the minus\N1, which is 1/e.
Dialogue: 0,2:11:17.13,2:11:19.26,Default,,0000,0000,0000,,So far, so good-- I'm\Ncircling all the guys
Dialogue: 0,2:11:19.26,2:11:22.48,Default,,0000,0000,0000,,that I want to compare after.
Dialogue: 0,2:11:22.48,2:11:28.63,Default,,0000,0000,0000,,So for the final four points\Nthere, what do I have?
Dialogue: 0,2:11:28.63,2:11:34.32,Default,,0000,0000,0000,,My final candidates could\Nbe x equals plus/minus 1
Dialogue: 0,2:11:34.32,2:11:39.44,Default,,0000,0000,0000,,and y equals 0-- e and e.
Dialogue: 0,2:11:39.44,2:11:40.19,Default,,0000,0000,0000,,Who's the biggest?
Dialogue: 0,2:11:40.19,2:11:41.19,Default,,0000,0000,0000,,Who's the smallest?
Dialogue: 0,2:11:41.19,2:11:43.00,Default,,0000,0000,0000,,STUDENT: e is the biggest.
Dialogue: 0,2:11:43.00,2:11:45.85,Default,,0000,0000,0000,,PROFESSOR: e is the biggest,\Nand 1/e is the smallest.
Dialogue: 0,2:11:45.85,2:11:48.69,Default,,0000,0000,0000,,So how do I write conclusion?
Dialogue: 0,2:11:48.69,2:12:01.46,Default,,0000,0000,0000,,Conclusion-- we have\Ntwo absolute maxima
Dialogue: 0,2:12:01.46,2:12:08.28,Default,,0000,0000,0000,,at minus 1, 0 and 1, 0.
Dialogue: 0,2:12:08.28,2:12:24.99,Default,,0000,0000,0000,,And we have two absolute\Nminima at 0, minus 1 and 0, 1.
Dialogue: 0,2:12:24.99,2:12:32.07,Default,,0000,0000,0000,,
Dialogue: 0,2:12:32.07,2:12:42.01,Default,,0000,0000,0000,,OK, now I have to-- now\Nthat's like a saddle.
Dialogue: 0,2:12:42.01,2:12:44.20,Default,,0000,0000,0000,,Can you see it with the\Neyes of your imagination?
Dialogue: 0,2:12:44.20,2:12:45.67,Default,,0000,0000,0000,,It's hard to see it.
Dialogue: 0,2:12:45.67,2:12:48.51,Default,,0000,0000,0000,,
Dialogue: 0,2:12:48.51,2:12:49.93,Default,,0000,0000,0000,,This is the disc.
Dialogue: 0,2:12:49.93,2:12:52.98,Default,,0000,0000,0000,,And the four\Npoints, the cardinal
Dialogue: 0,2:12:52.98,2:12:58.33,Default,,0000,0000,0000,,points-- OK, this is the disc.
Dialogue: 0,2:12:58.33,2:13:02.11,Default,,0000,0000,0000,,We are looking at this\Ndisc from perspective.
Dialogue: 0,2:13:02.11,2:13:06.52,Default,,0000,0000,0000,,And the five points,\None is in the middle.
Dialogue: 0,2:13:06.52,2:13:09.18,Default,,0000,0000,0000,,
Dialogue: 0,2:13:09.18,2:13:11.96,Default,,0000,0000,0000,,One is here.
Dialogue: 0,2:13:11.96,2:13:13.16,Default,,0000,0000,0000,,Minus 1 is 0.
Dialogue: 0,2:13:13.16,2:13:14.61,Default,,0000,0000,0000,,One is here, 1, 0.
Dialogue: 0,2:13:14.61,2:13:20.00,Default,,0000,0000,0000,,One is here, 0, minus\N1, and one here, 0, 1.
Dialogue: 0,2:13:20.00,2:13:25.77,Default,,0000,0000,0000,,At minus 1, 0 and 1,\N0 I get the maximum.
Dialogue: 0,2:13:25.77,2:13:29.42,Default,,0000,0000,0000,,So the way it's going to be\Nshaped would be like that.
Dialogue: 0,2:13:29.42,2:13:31.10,Default,,0000,0000,0000,,In this direction,\Nit will be like that.
Dialogue: 0,2:13:31.10,2:13:32.88,Default,,0000,0000,0000,,And cut the cake here.
Dialogue: 0,2:13:32.88,2:13:34.77,Default,,0000,0000,0000,,You see it's like that.
Dialogue: 0,2:13:34.77,2:13:36.93,Default,,0000,0000,0000,,It's going to be like this.
Dialogue: 0,2:13:36.93,2:13:41.48,Default,,0000,0000,0000,,OK, passing through the\Norigin, with my hands
Dialogue: 0,2:13:41.48,2:13:45.57,Default,,0000,0000,0000,,I'm molding the surface made of\NPlay-Doh or something for you.
Dialogue: 0,2:13:45.57,2:13:49.09,Default,,0000,0000,0000,,So I'm starting here,\Nand I'm going up.
Dialogue: 0,2:13:49.09,2:13:51.98,Default,,0000,0000,0000,,And at this points, I'm here.
Dialogue: 0,2:13:51.98,2:13:52.75,Default,,0000,0000,0000,,Are you with me?
Dialogue: 0,2:13:52.75,2:13:55.07,Default,,0000,0000,0000,,The same height.
Dialogue: 0,2:13:55.07,2:14:00.64,Default,,0000,0000,0000,,In the other direction,\NI'm going from 0.
Dialogue: 0,2:14:00.64,2:14:04.05,Default,,0000,0000,0000,,But I'm not so high.
Dialogue: 0,2:14:04.05,2:14:08.50,Default,,0000,0000,0000,,I'm going only up\Nto-- what is 1/e?
Dialogue: 0,2:14:08.50,2:14:12.24,Default,,0000,0000,0000,,About 1/3, meh,\Nsomething like that.
Dialogue: 0,2:14:12.24,2:14:18.32,Default,,0000,0000,0000,,So I'm going to get here.
Dialogue: 0,2:14:18.32,2:14:23.05,Default,,0000,0000,0000,,So the problem is that\None will be in between.
Dialogue: 0,2:14:23.05,2:14:26.65,Default,,0000,0000,0000,,So if you really want to\Nsee what it looks like,
Dialogue: 0,2:14:26.65,2:14:29.14,Default,,0000,0000,0000,,we are here at 1.
Dialogue: 0,2:14:29.14,2:14:32.25,Default,,0000,0000,0000,,We grow from 1,\Naltitude 1, you see?
Dialogue: 0,2:14:32.25,2:14:39.84,Default,,0000,0000,0000,,We grow from 1 to about\N2.71718283 for both of these.
Dialogue: 0,2:14:39.84,2:14:47.56,Default,,0000,0000,0000,,And from 1, in this direction\NI have to go down to 1/e.
Dialogue: 0,2:14:47.56,2:14:51.30,Default,,0000,0000,0000,,So it looks like that.
Dialogue: 0,2:14:51.30,2:14:53.26,Default,,0000,0000,0000,,I'll try to draw, OK?
Dialogue: 0,2:14:53.26,2:15:03.08,Default,,0000,0000,0000,,
Dialogue: 0,2:15:03.08,2:15:07.90,Default,,0000,0000,0000,,Do you see the patch\Naround the origin?
Dialogue: 0,2:15:07.90,2:15:11.13,Default,,0000,0000,0000,,So here's e.
Dialogue: 0,2:15:11.13,2:15:16.17,Default,,0000,0000,0000,,And here's 1/e\Nabove the sea level.
Dialogue: 0,2:15:16.17,2:15:19.40,Default,,0000,0000,0000,,And this is 1.
Dialogue: 0,2:15:19.40,2:15:26.76,Default,,0000,0000,0000,,And you have one just like that\Nin the back that is the-- it
Dialogue: 0,2:15:26.76,2:15:27.84,Default,,0000,0000,0000,,still looks like a saddle.
Dialogue: 0,2:15:27.84,2:15:28.56,Default,,0000,0000,0000,,It is a saddle.
Dialogue: 0,2:15:28.56,2:15:29.72,Default,,0000,0000,0000,,It's symmetric.
Dialogue: 0,2:15:29.72,2:15:33.74,Default,,0000,0000,0000,,But it's another kind of saddle.
Dialogue: 0,2:15:33.74,2:15:35.20,Default,,0000,0000,0000,,There are all sorts\Nof saddles made
Dialogue: 0,2:15:35.20,2:15:38.96,Default,,0000,0000,0000,,in Texas, different\Nranches, different saddles.
Dialogue: 0,2:15:38.96,2:15:44.55,Default,,0000,0000,0000,,So that was the harder one.
Dialogue: 0,2:15:44.55,2:15:48.04,Default,,0000,0000,0000,,The ones that I actually\Nsaw on the finals,
Dialogue: 0,2:15:48.04,2:15:49.74,Default,,0000,0000,0000,,some of the last\Nthree or four finals,
Dialogue: 0,2:15:49.74,2:15:55.40,Default,,0000,0000,0000,,were much easier in the sense\Nthat the table you had to draw
Dialogue: 0,2:15:55.40,2:16:00.86,Default,,0000,0000,0000,,was much shorter than\Nthis one-- in principle,
Dialogue: 0,2:16:00.86,2:16:05.41,Default,,0000,0000,0000,,one critical value and\None max and one min point.
Dialogue: 0,2:16:05.41,2:16:09.71,Default,,0000,0000,0000,,But you have to be prepared\Nmore, rehearse more,
Dialogue: 0,2:16:09.71,2:16:11.76,Default,,0000,0000,0000,,so when you see the\Nproblem in the midterm,
Dialogue: 0,2:16:11.76,2:16:15.58,Default,,0000,0000,0000,,you say, oh, well that is\Neasier than I'm used to.
Dialogue: 0,2:16:15.58,2:16:17.90,Default,,0000,0000,0000,,That's the idea.
Dialogue: 0,2:16:17.90,2:16:19.73,Default,,0000,0000,0000,,OK, go home.
Dialogue: 0,2:16:19.73,2:16:21.53,Default,,0000,0000,0000,,Send me emails by WeBWorK.
Dialogue: 0,2:16:21.53,2:16:25.13,Default,,0000,0000,0000,,We still have time to talk about\Nthe homework if you get stuck.
Dialogue: 0,2:16:25.13,2:16:27.83,Default,,0000,0000,0000,,
Dialogue: 0,2:16:27.83,2:16:31.73,Default,,0000,0000,0000,,And I'll see you Thursday.
Dialogue: 0,2:16:31.73,2:16:35.08,Default,,0000,0000,0000,,[BACKGROUND CHATTER]
Dialogue: 0,2:16:35.08,2:16:59.33,Default,,0000,0000,0000,,