[Script Info]
Title:
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.50,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.50,0:00:03.09,Default,,0000,0000,0000,,People asked me if I'm\Ngoing to go over homework.
Dialogue: 0,0:00:03.09,0:00:04.28,Default,,0000,0000,0000,,Of course I will.
Dialogue: 0,0:00:04.28,0:00:05.30,Default,,0000,0000,0000,,Let me explain.
Dialogue: 0,0:00:05.30,0:00:08.20,Default,,0000,0000,0000,,Out of the four\Nhours you have, three
Dialogue: 0,0:00:08.20,0:00:11.09,Default,,0000,0000,0000,,should be more or\Nless lecture time.
Dialogue: 0,0:00:11.09,0:00:14.47,Default,,0000,0000,0000,,And the fourth hour, which\Nis the instructor's latitude,
Dialogue: 0,0:00:14.47,0:00:17.76,Default,,0000,0000,0000,,where they put it-- it's\Napplications, problems,
Dialogue: 0,0:00:17.76,0:00:20.60,Default,,0000,0000,0000,,homework like problems, all\Nsorts of practice for exams
Dialogue: 0,0:00:20.60,0:00:22.00,Default,,0000,0000,0000,,and so on.
Dialogue: 0,0:00:22.00,0:00:23.41,Default,,0000,0000,0000,,It's not a recitation.
Dialogue: 0,0:00:23.41,0:00:31.11,Default,,0000,0000,0000,,It's some sort of workshop that\Nthe instructor conducts himself
Dialogue: 0,0:00:31.11,0:00:33.08,Default,,0000,0000,0000,,personally.
Dialogue: 0,0:00:33.08,0:00:36.04,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:00:36.04,0:00:38.83,Default,,0000,0000,0000,,If you don't have\Nquestions, I'm just
Dialogue: 0,0:00:38.83,0:00:42.03,Default,,0000,0000,0000,,going to go ahead and\Nreview a little bit of what
Dialogue: 0,0:00:42.03,0:00:44.58,Default,,0000,0000,0000,,we discussed last time.
Dialogue: 0,0:00:44.58,0:00:53.19,Default,,0000,0000,0000,,Something new and exciting\Nwas chapter 11, section 11.1.
Dialogue: 0,0:00:53.19,0:00:55.17,Default,,0000,0000,0000,,And we did 11.2.
Dialogue: 0,0:00:55.17,0:00:57.16,Default,,0000,0000,0000,,And what was that about?
Dialogue: 0,0:00:57.16,0:00:59.48,Default,,0000,0000,0000,,That was about functions\Nof several variables.
Dialogue: 0,0:00:59.48,0:01:07.97,Default,,0000,0000,0000,,
Dialogue: 0,0:01:07.97,0:01:10.43,Default,,0000,0000,0000,,And we discussed\Nseveral examples,
Dialogue: 0,0:01:10.43,0:01:13.52,Default,,0000,0000,0000,,but then we focused\Nour attention mainly
Dialogue: 0,0:01:13.52,0:01:19.28,Default,,0000,0000,0000,,to explicit functions, which\Nmeans z equals f of x, y,
Dialogue: 0,0:01:19.28,0:01:21.38,Default,,0000,0000,0000,,of two variables.
Dialogue: 0,0:01:21.38,0:01:25.33,Default,,0000,0000,0000,,And we call this a graph\Nbecause it is a graph.
Dialogue: 0,0:01:25.33,0:01:33.35,Default,,0000,0000,0000,,In 3D, it's a surface whose\Ndomain is on the floor.
Dialogue: 0,0:01:33.35,0:01:38.67,Default,,0000,0000,0000,,And the altitude is z, and\Nthat is the-- this is the-- OK.
Dialogue: 0,0:01:38.67,0:01:40.97,Default,,0000,0000,0000,,How many of you are\Nnon-math majors?
Dialogue: 0,0:01:40.97,0:01:43.39,Default,,0000,0000,0000,,Can you raise hands?
Dialogue: 0,0:01:43.39,0:01:44.50,Default,,0000,0000,0000,,Oh, OK.
Dialogue: 0,0:01:44.50,0:01:47.34,Default,,0000,0000,0000,,So you know a little\Nbit about research
Dialogue: 0,0:01:47.34,0:01:49.91,Default,,0000,0000,0000,,from your own classes,\Nscience classes
Dialogue: 0,0:01:49.91,0:01:51.87,Default,,0000,0000,0000,,or from science\Nfairs from school.
Dialogue: 0,0:01:51.87,0:01:55.99,Default,,0000,0000,0000,,These are the independent\Nvariables, x, y.
Dialogue: 0,0:01:55.99,0:01:58.43,Default,,0000,0000,0000,,And z is the dependent variable.
Dialogue: 0,0:01:58.43,0:02:01.40,Default,,0000,0000,0000,,We don't use this kind of\Nterminology in this class.
Dialogue: 0,0:02:01.40,0:02:06.64,Default,,0000,0000,0000,,But so that you know-- we\Ndiscussed domain last time.
Dialogue: 0,0:02:06.64,0:02:07.79,Default,,0000,0000,0000,,This was about what?
Dialogue: 0,0:02:07.79,0:02:10.55,Default,,0000,0000,0000,,Domain, range.
Dialogue: 0,0:02:10.55,0:02:12.44,Default,,0000,0000,0000,,After range, what did we do?
Dialogue: 0,0:02:12.44,0:02:14.79,Default,,0000,0000,0000,,We talked about level curves.
Dialogue: 0,0:02:14.79,0:02:17.62,Default,,0000,0000,0000,,What is the level curve?
Dialogue: 0,0:02:17.62,0:02:22.22,Default,,0000,0000,0000,,Level curves are curves x,\Ny in the plane corresponding
Dialogue: 0,0:02:22.22,0:02:24.88,Default,,0000,0000,0000,,to f of x, y equals constant.
Dialogue: 0,0:02:24.88,0:02:27.56,Default,,0000,0000,0000,,
Dialogue: 0,0:02:27.56,0:02:29.97,Default,,0000,0000,0000,,These are called\Nlevel curves in plane,
Dialogue: 0,0:02:29.97,0:02:32.86,Default,,0000,0000,0000,,in the plane called x, y plane.
Dialogue: 0,0:02:32.86,0:02:36.18,Default,,0000,0000,0000,,
Dialogue: 0,0:02:36.18,0:02:37.85,Default,,0000,0000,0000,,What else have we discussed?
Dialogue: 0,0:02:37.85,0:02:41.98,Default,,0000,0000,0000,,We went straight into 11.2.
Dialogue: 0,0:02:41.98,0:02:44.65,Default,,0000,0000,0000,,In 11.2, we were very\Nhappy to remember
Dialogue: 0,0:02:44.65,0:02:49.45,Default,,0000,0000,0000,,a little bit of Calculus\N1, which was practically
Dialogue: 0,0:02:49.45,0:02:53.03,Default,,0000,0000,0000,,a review of limits from Calc 1.
Dialogue: 0,0:02:53.03,0:02:54.42,Default,,0000,0000,0000,,And what did we do?
Dialogue: 0,0:02:54.42,0:02:59.24,Default,,0000,0000,0000,,We did epsilon delta, which\Nwas not covered in Calculus 1.
Dialogue: 0,0:02:59.24,0:03:01.41,Default,,0000,0000,0000,,And where is Aaron?
Dialogue: 0,0:03:01.41,0:03:01.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:03:01.91,0:03:04.68,Default,,0000,0000,0000,,Thank you, Aaron.
Dialogue: 0,0:03:04.68,0:03:07.46,Default,,0000,0000,0000,,And today, I was thinking,\NI want to show you actually
Dialogue: 0,0:03:07.46,0:03:12.64,Default,,0000,0000,0000,,an example that is quite\Neasy of how you use epsilon
Dialogue: 0,0:03:12.64,0:03:20.33,Default,,0000,0000,0000,,delta for continuity, to show\Nif the function is continuous,
Dialogue: 0,0:03:20.33,0:03:23.74,Default,,0000,0000,0000,,but for a function\Nof true variables.
Dialogue: 0,0:03:23.74,0:03:25.21,Default,,0000,0000,0000,,And that's not hard.
Dialogue: 0,0:03:25.21,0:03:26.67,Default,,0000,0000,0000,,You may think, oh, my god.
Dialogue: 0,0:03:26.67,0:03:27.65,Default,,0000,0000,0000,,That must be hard.
Dialogue: 0,0:03:27.65,0:03:29.11,Default,,0000,0000,0000,,That's not hard at all.
Dialogue: 0,0:03:29.11,0:03:32.70,Default,,0000,0000,0000,,I'm going to move on to the\Nsecond part of 11.2, which
Dialogue: 0,0:03:32.70,0:03:34.77,Default,,0000,0000,0000,,is continuity.
Dialogue: 0,0:03:34.77,0:03:38.24,Default,,0000,0000,0000,,11.2, second part.
Dialogue: 0,0:03:38.24,0:03:39.65,Default,,0000,0000,0000,,The first part was what?
Dialogue: 0,0:03:39.65,0:03:41.59,Default,,0000,0000,0000,,It was limits of\Nfunctions, right, guys?
Dialogue: 0,0:03:41.59,0:03:45.16,Default,,0000,0000,0000,,We discussed\Nproperties of limits,
Dialogue: 0,0:03:45.16,0:03:49.81,Default,,0000,0000,0000,,algebraic properties of\Nadding sums and taking a limit
Dialogue: 0,0:03:49.81,0:03:53.75,Default,,0000,0000,0000,,of a sum, taking a limit\Nof a product of functions,
Dialogue: 0,0:03:53.75,0:03:58.50,Default,,0000,0000,0000,,taking the limit of a quotient\Nof function, when it exists,
Dialogue: 0,0:03:58.50,0:04:00.29,Default,,0000,0000,0000,,when it doesn't.
Dialogue: 0,0:04:00.29,0:04:06.16,Default,,0000,0000,0000,,Now the second part of\N11.2 is called continuity.
Dialogue: 0,0:04:06.16,0:04:08.10,Default,,0000,0000,0000,,Continuity of what?
Dialogue: 0,0:04:08.10,0:04:09.47,Default,,0000,0000,0000,,Well, I'm too lazy\Nto right down,
Dialogue: 0,0:04:09.47,0:04:16.45,Default,,0000,0000,0000,,but it's continuity of functions\Nof two variables, right?
Dialogue: 0,0:04:16.45,0:04:20.17,Default,,0000,0000,0000,,Now in Calc 1-- you\Nreminded me last time.
Dialogue: 0,0:04:20.17,0:04:21.84,Default,,0000,0000,0000,,I tried to remind you.
Dialogue: 0,0:04:21.84,0:04:22.99,Default,,0000,0000,0000,,You tried to remind me.
Dialogue: 0,0:04:22.99,0:04:24.64,Default,,0000,0000,0000,,Let's remind each other.
Dialogue: 0,0:04:24.64,0:04:27.08,Default,,0000,0000,0000,,This is like a discussion.
Dialogue: 0,0:04:27.08,0:04:43.01,Default,,0000,0000,0000,,What was the meaning of f of x\Nbeing a continuous function x0,
Dialogue: 0,0:04:43.01,0:04:46.64,Default,,0000,0000,0000,,which is part of the domain?
Dialogue: 0,0:04:46.64,0:04:48.30,Default,,0000,0000,0000,,x0 has to be in the domain.
Dialogue: 0,0:04:48.30,0:04:55.95,Default,,0000,0000,0000,,
Dialogue: 0,0:04:55.95,0:04:58.04,Default,,0000,0000,0000,,This is if and only if what?
Dialogue: 0,0:04:58.04,0:04:59.65,Default,,0000,0000,0000,,Well, what kind of\Nfunction is that?
Dialogue: 0,0:04:59.65,0:05:01.92,Default,,0000,0000,0000,,A one variable\Nfunction, real value.
Dialogue: 0,0:05:01.92,0:05:05.77,Default,,0000,0000,0000,,It takes values on, let's say,\Nan interval on the real line.
Dialogue: 0,0:05:05.77,0:05:11.06,Default,,0000,0000,0000,,What was the group\Nof properties that
Dialogue: 0,0:05:11.06,0:05:14.08,Default,,0000,0000,0000,,have to be\Nsimultaneously satisfied,
Dialogue: 0,0:05:14.08,0:05:15.69,Default,,0000,0000,0000,,satisfied at the same time?
Dialogue: 0,0:05:15.69,0:05:18.35,Default,,0000,0000,0000,,
Dialogue: 0,0:05:18.35,0:05:21.20,Default,,0000,0000,0000,,And you told me it has\Nto be at the same time.
Dialogue: 0,0:05:21.20,0:05:24.53,Default,,0000,0000,0000,,And I was very happy because\Nif one of the three conditions
Dialogue: 0,0:05:24.53,0:05:28.25,Default,,0000,0000,0000,,is missing, then\Ngoodbye, continuity.
Dialogue: 0,0:05:28.25,0:05:30.19,Default,,0000,0000,0000,,One?
Dialogue: 0,0:05:30.19,0:05:31.69,Default,,0000,0000,0000,,STUDENT: It's defined\Nat that point.
Dialogue: 0,0:05:31.69,0:05:33.66,Default,,0000,0000,0000,,MAGDALENA TODA: Yes,\Nsir. f of x0 is defined.
Dialogue: 0,0:05:33.66,0:05:36.62,Default,,0000,0000,0000,,
Dialogue: 0,0:05:36.62,0:05:39.85,Default,,0000,0000,0000,,Actually, I said that\Nhere in the domain.
Dialogue: 0,0:05:39.85,0:05:43.07,Default,,0000,0000,0000,,I'll remove it because\Nnow I said it better.
Dialogue: 0,0:05:43.07,0:05:44.94,Default,,0000,0000,0000,,Two?
Dialogue: 0,0:05:44.94,0:05:46.22,Default,,0000,0000,0000,,STUDENT: The limit exists.
Dialogue: 0,0:05:46.22,0:05:47.30,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:05:47.30,0:05:51.65,Default,,0000,0000,0000,,The limit, as I approach\Nx0 with any kind of value
Dialogue: 0,0:05:51.65,0:05:59.86,Default,,0000,0000,0000,,closer and closer,\Nexists and is finite.
Dialogue: 0,0:05:59.86,0:06:03.28,Default,,0000,0000,0000,,Let's give it a name.
Dialogue: 0,0:06:03.28,0:06:09.63,Default,,0000,0000,0000,,Let's call it L.
Dialogue: 0,0:06:09.63,0:06:11.09,Default,,0000,0000,0000,,STUDENT:\N[? The following value ?]
Dialogue: 0,0:06:11.09,0:06:12.10,Default,,0000,0000,0000,,equals the limit.
Dialogue: 0,0:06:12.10,0:06:13.14,Default,,0000,0000,0000,,MAGDALENA TODA: Yes, sir.
Dialogue: 0,0:06:13.14,0:06:14.08,Default,,0000,0000,0000,,That's the last thing.
Dialogue: 0,0:06:14.08,0:06:17.49,Default,,0000,0000,0000,,And I'm glad I didn't\Nhave to pull the truth out
Dialogue: 0,0:06:17.49,0:06:18.51,Default,,0000,0000,0000,,of your mouth.
Dialogue: 0,0:06:18.51,0:06:27.41,Default,,0000,0000,0000,,So the limit will-- the limit\Nof f of x when x goes to x0
Dialogue: 0,0:06:27.41,0:06:28.89,Default,,0000,0000,0000,,equals f of x0.
Dialogue: 0,0:06:28.89,0:06:31.87,Default,,0000,0000,0000,,
Dialogue: 0,0:06:31.87,0:06:33.98,Default,,0000,0000,0000,,No examples.
Dialogue: 0,0:06:33.98,0:06:38.51,Default,,0000,0000,0000,,You should know\NCalc 1, and you do.
Dialogue: 0,0:06:38.51,0:06:43.92,Default,,0000,0000,0000,,I'm just going to\Nmove on to Calc 3.
Dialogue: 0,0:06:43.92,0:06:48.50,Default,,0000,0000,0000,,And let's see what the\Ndefinition of continuity
Dialogue: 0,0:06:48.50,0:06:52.08,Default,,0000,0000,0000,,would mean for us in Calc 3.
Dialogue: 0,0:06:52.08,0:07:00.47,Default,,0000,0000,0000,,Can anybody mimic the properties\Nthat-- well, f of x, y
Dialogue: 0,0:07:00.47,0:07:15.26,Default,,0000,0000,0000,,is said to be\Ncontinuous at x0, y0
Dialogue: 0,0:07:15.26,0:07:34.32,Default,,0000,0000,0000,,if and only if the following\Nconditions are-- my arm hurts.
Dialogue: 0,0:07:34.32,0:07:36.06,Default,,0000,0000,0000,,Are simultaneously satisfied.
Dialogue: 0,0:07:36.06,0:07:48.98,Default,,0000,0000,0000,,
Dialogue: 0,0:07:48.98,0:07:53.24,Default,,0000,0000,0000,,I don't like professors who\Nuse PDF files or slides.
Dialogue: 0,0:07:53.24,0:07:53.74,Default,,0000,0000,0000,,Shh.
Dialogue: 0,0:07:53.74,0:07:54.24,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:07:54.24,0:07:56.15,Default,,0000,0000,0000,,I don't want anything premade.
Dialogue: 0,0:07:56.15,0:08:00.81,Default,,0000,0000,0000,,The class is a\Nconstruction, is working,
Dialogue: 0,0:08:00.81,0:08:04.63,Default,,0000,0000,0000,,is something like\Na work in progress.
Dialogue: 0,0:08:04.63,0:08:07.36,Default,,0000,0000,0000,,We are building things together.
Dialogue: 0,0:08:07.36,0:08:08.68,Default,,0000,0000,0000,,This is teamwork.
Dialogue: 0,0:08:08.68,0:08:11.11,Default,,0000,0000,0000,,If I come up with\Nsome slides that were
Dialogue: 0,0:08:11.11,0:08:13.08,Default,,0000,0000,0000,,made at home or a PDF file.
Dialogue: 0,0:08:13.08,0:08:14.42,Default,,0000,0000,0000,,First of all, it means I'm lazy.
Dialogue: 0,0:08:14.42,0:08:17.04,Default,,0000,0000,0000,,Second of all, it\Nmeans that I'm not
Dialogue: 0,0:08:17.04,0:08:20.21,Default,,0000,0000,0000,,willing to take it\None step at a time
Dialogue: 0,0:08:20.21,0:08:24.58,Default,,0000,0000,0000,,and show you how\Nthe idea's revealed.
Dialogue: 0,0:08:24.58,0:08:25.08,Default,,0000,0000,0000,,One.
Dialogue: 0,0:08:25.08,0:08:33.47,Default,,0000,0000,0000,,
Dialogue: 0,0:08:33.47,0:08:34.23,Default,,0000,0000,0000,,Who is telling me?
Dialogue: 0,0:08:34.23,0:08:35.32,Default,,0000,0000,0000,,I'm not going to say it.
Dialogue: 0,0:08:35.32,0:08:36.33,Default,,0000,0000,0000,,It's a work in progress.
Dialogue: 0,0:08:36.33,0:08:39.17,Default,,0000,0000,0000,,
Dialogue: 0,0:08:39.17,0:08:41.07,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:08:41.07,0:08:43.01,Default,,0000,0000,0000,,MAGDALENA TODA: f of--
Dialogue: 0,0:08:43.01,0:08:44.45,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:08:44.45,0:08:48.18,Default,,0000,0000,0000,,MAGDALENA TODA: Of\Nx0, y0 is defined.
Dialogue: 0,0:08:48.18,0:08:49.86,Default,,0000,0000,0000,,And why not?
Dialogue: 0,0:08:49.86,0:08:52.14,Default,,0000,0000,0000,,Well, just to have\Na silly [? pun ?].
Dialogue: 0,0:08:52.14,0:08:52.64,Default,,0000,0000,0000,,Two.
Dialogue: 0,0:08:52.64,0:08:55.26,Default,,0000,0000,0000,,
Dialogue: 0,0:08:55.26,0:09:02.92,Default,,0000,0000,0000,,Limit as the pair x, y\Napproaches x0, x0-- and guys,
Dialogue: 0,0:09:02.92,0:09:05.84,Default,,0000,0000,0000,,when you close your eyes--\Nno you close your eyes--
Dialogue: 0,0:09:05.84,0:09:09.41,Default,,0000,0000,0000,,and you imagine\Nx, y going to x0,
Dialogue: 0,0:09:09.41,0:09:16.03,Default,,0000,0000,0000,,y0 by any possible paths\Nin any possible way,
Dialogue: 0,0:09:16.03,0:09:21.57,Default,,0000,0000,0000,,it's not that you have a\Npredetermined path to x0, y0,
Dialogue: 0,0:09:21.57,0:09:23.65,Default,,0000,0000,0000,,because you may be trapped.
Dialogue: 0,0:09:23.65,0:09:26.17,Default,,0000,0000,0000,,You may have-- as you've\Nseen last time, you may have,
Dialogue: 0,0:09:26.17,0:09:28.54,Default,,0000,0000,0000,,coming from this direction,\Nthe limit will exist,
Dialogue: 0,0:09:28.54,0:09:29.99,Default,,0000,0000,0000,,will be this one.
Dialogue: 0,0:09:29.99,0:09:32.60,Default,,0000,0000,0000,,Coming from that direction,\Nthe limit will exist,
Dialogue: 0,0:09:32.60,0:09:34.60,Default,,0000,0000,0000,,would be another one.
Dialogue: 0,0:09:34.60,0:09:36.54,Default,,0000,0000,0000,,And then you don't\Nhave overall limits.
Dialogue: 0,0:09:36.54,0:09:41.45,Default,,0000,0000,0000,,So the limit-- when I call that,\Nthat means the overall limit
Dialogue: 0,0:09:41.45,0:09:51.25,Default,,0000,0000,0000,,exists, exists and\Nequals L. It's finite.
Dialogue: 0,0:09:51.25,0:09:53.43,Default,,0000,0000,0000,,That's what I mean.
Dialogue: 0,0:09:53.43,0:09:57.88,Default,,0000,0000,0000,,And three, the value\Nof the function at x0,
Dialogue: 0,0:09:57.88,0:10:04.51,Default,,0000,0000,0000,,y0 must be equal to the limit\Nof the function that value
Dialogue: 0,0:10:04.51,0:10:08.38,Default,,0000,0000,0000,,as you approach it, x0, y0.
Dialogue: 0,0:10:08.38,0:10:11.55,Default,,0000,0000,0000,,And equals L, of course.
Dialogue: 0,0:10:11.55,0:10:12.65,Default,,0000,0000,0000,,So great.
Dialogue: 0,0:10:12.65,0:10:16.81,Default,,0000,0000,0000,,So it's so obvious\Nthat we are following
Dialogue: 0,0:10:16.81,0:10:19.36,Default,,0000,0000,0000,,exactly the same\Ntype of definition,
Dialogue: 0,0:10:19.36,0:10:22.21,Default,,0000,0000,0000,,the same type of pattern.
Dialogue: 0,0:10:22.21,0:10:28.66,Default,,0000,0000,0000,,I'm going to ask you\Nto help me, to help
Dialogue: 0,0:10:28.66,0:10:34.72,Default,,0000,0000,0000,,me solve a harder problem\Nthat involves continuity.
Dialogue: 0,0:10:34.72,0:10:38.56,Default,,0000,0000,0000,,And I'm asking you, if I\Nhave the following function--
Dialogue: 0,0:10:38.56,0:10:40.95,Default,,0000,0000,0000,,I'm going to erase the\Ndefinition of continuity
Dialogue: 0,0:10:40.95,0:10:43.31,Default,,0000,0000,0000,,from Calc 1.
Dialogue: 0,0:10:43.31,0:10:45.72,Default,,0000,0000,0000,,I'm going to ask you, what if\NI have this funny function?
Dialogue: 0,0:10:45.72,0:10:49.08,Default,,0000,0000,0000,,You've seen it before, and\NI gave you a little bit
Dialogue: 0,0:10:49.08,0:10:50.38,Default,,0000,0000,0000,,of a warning about it.
Dialogue: 0,0:10:50.38,0:10:53.32,Default,,0000,0000,0000,,
Dialogue: 0,0:10:53.32,0:11:00.86,Default,,0000,0000,0000,,Limit as x, y goes\Nto 0, 0 of x squared
Dialogue: 0,0:11:00.86,0:11:07.17,Default,,0000,0000,0000,,plus y squared times sine of 1\Nover x squared plus y squared.
Dialogue: 0,0:11:07.17,0:11:10.19,Default,,0000,0000,0000,,
Dialogue: 0,0:11:10.19,0:11:11.18,Default,,0000,0000,0000,,Does that exist?
Dialogue: 0,0:11:11.18,0:11:16.12,Default,,0000,0000,0000,,
Dialogue: 0,0:11:16.12,0:11:16.92,Default,,0000,0000,0000,,And also--
Dialogue: 0,0:11:16.92,0:11:20.32,Default,,0000,0000,0000,,STUDENT: It's actually--\Nso the limit is actually
Dialogue: 0,0:11:20.32,0:11:23.64,Default,,0000,0000,0000,,approaching a plane rather\Nthan a set of [INAUDIBLE].
Dialogue: 0,0:11:23.64,0:11:26.85,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nwell, actually, it's
Dialogue: 0,0:11:26.85,0:11:28.08,Default,,0000,0000,0000,,not approaching a plane.
Dialogue: 0,0:11:28.08,0:11:29.80,Default,,0000,0000,0000,,Let's see what's\Nhappening when--
Dialogue: 0,0:11:29.80,0:11:30.74,Default,,0000,0000,0000,,STUDENT: Sorry, sorry.
Dialogue: 0,0:11:30.74,0:11:32.42,Default,,0000,0000,0000,,Not a plane, a [? line. ?]
Dialogue: 0,0:11:32.42,0:11:33.25,Default,,0000,0000,0000,,MAGDALENA TODA: Yes.
Dialogue: 0,0:11:33.25,0:11:36.04,Default,,0000,0000,0000,,STUDENT: And is the z-axis--\Nthe entire z-axis is 0, 0?
Dialogue: 0,0:11:36.04,0:11:37.62,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nthis is the z-axis.
Dialogue: 0,0:11:37.62,0:11:43.93,Default,,0000,0000,0000,,And that means exactly that\Nx and y-- it will be 0.
Dialogue: 0,0:11:43.93,0:11:47.03,Default,,0000,0000,0000,,Now I am just looking\Nat what happens
Dialogue: 0,0:11:47.03,0:11:50.53,Default,,0000,0000,0000,,in the plane, in the\Nfloor plane x, y.
Dialogue: 0,0:11:50.53,0:11:53.73,Default,,0000,0000,0000,,The pairs x, y are wiggly.
Dialogue: 0,0:11:53.73,0:11:56.22,Default,,0000,0000,0000,,They are like\Nlittle wormy worms.
Dialogue: 0,0:11:56.22,0:12:00.68,Default,,0000,0000,0000,,And they float on the\Nwater on the floor.
Dialogue: 0,0:12:00.68,0:12:03.34,Default,,0000,0000,0000,,And these squiggly\Nthings approach
Dialogue: 0,0:12:03.34,0:12:05.87,Default,,0000,0000,0000,,x, y from any possible path.
Dialogue: 0,0:12:05.87,0:12:06.100,Default,,0000,0000,0000,,They go like this.
Dialogue: 0,0:12:06.100,0:12:09.04,Default,,0000,0000,0000,,They go like that.
Dialogue: 0,0:12:09.04,0:12:11.19,Default,,0000,0000,0000,,They go in every possible way.
Dialogue: 0,0:12:11.19,0:12:12.15,Default,,0000,0000,0000,,Let's see what happens.
Dialogue: 0,0:12:12.15,0:12:15.05,Default,,0000,0000,0000,,
Dialogue: 0,0:12:15.05,0:12:17.55,Default,,0000,0000,0000,,Continuity-- is this continuous?
Dialogue: 0,0:12:17.55,0:12:19.84,Default,,0000,0000,0000,,Well, you say,\NMagdalena, come on.
Dialogue: 0,0:12:19.84,0:12:21.96,Default,,0000,0000,0000,,You cannot have this\Ncontinuous at 0, 0,
Dialogue: 0,0:12:21.96,0:12:24.48,Default,,0000,0000,0000,,because it's undefined at 0, 0.
Dialogue: 0,0:12:24.48,0:12:24.98,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:12:24.98,0:12:27.41,Default,,0000,0000,0000,,But maybe I can extend\Nit by continuity.
Dialogue: 0,0:12:27.41,0:12:31.67,Default,,0000,0000,0000,,So let me introduce-- this\Nis my favorite, f of x, y.
Dialogue: 0,0:12:31.67,0:12:35.79,Default,,0000,0000,0000,,But I'll say, f of x, y\Nis not defined at 0, 0.
Dialogue: 0,0:12:35.79,0:12:46.78,Default,,0000,0000,0000,,But how about g of x, y as\Nbeing my f of x, y for any x,
Dialogue: 0,0:12:46.78,0:12:50.38,Default,,0000,0000,0000,,y different from 0, 0.
Dialogue: 0,0:12:50.38,0:12:55.54,Default,,0000,0000,0000,,And at the origin, at the\Nvery origin, I will say,
Dialogue: 0,0:12:55.54,0:12:59.10,Default,,0000,0000,0000,,I want to have--\Nwhen x, y equals 0,
Dialogue: 0,0:12:59.10,0:13:00.43,Default,,0000,0000,0000,,0, I want to have a value.
Dialogue: 0,0:13:00.43,0:13:05.46,Default,,0000,0000,0000,,Which value do you\Nthink might extend
Dialogue: 0,0:13:05.46,0:13:07.42,Default,,0000,0000,0000,,this function by continuity?
Dialogue: 0,0:13:07.42,0:13:08.89,Default,,0000,0000,0000,,STUDENT: The limit.
Dialogue: 0,0:13:08.89,0:13:10.65,Default,,0000,0000,0000,,MAGDALENA TODA: The\Nlimit if it exists
Dialogue: 0,0:13:10.65,0:13:15.96,Default,,0000,0000,0000,,and if-- well, you know already,\NI think, what the limit is
Dialogue: 0,0:13:15.96,0:13:18.78,Default,,0000,0000,0000,,because some of you\Nthought about this at home
Dialogue: 0,0:13:18.78,0:13:20.16,Default,,0000,0000,0000,,for extra credit.
Dialogue: 0,0:13:20.16,0:13:21.55,Default,,0000,0000,0000,,So it's not fair, right?
Dialogue: 0,0:13:21.55,0:13:22.64,Default,,0000,0000,0000,,No, I'm just kidding.
Dialogue: 0,0:13:22.64,0:13:26.69,Default,,0000,0000,0000,,So I claim that maybe--\Nif I put a 0 here,
Dialogue: 0,0:13:26.69,0:13:28.89,Default,,0000,0000,0000,,will this be continuous?
Dialogue: 0,0:13:28.89,0:13:31.28,Default,,0000,0000,0000,,Will g be continuous?
Dialogue: 0,0:13:31.28,0:13:35.60,Default,,0000,0000,0000,,
Dialogue: 0,0:13:35.60,0:13:42.06,Default,,0000,0000,0000,,So prove, prove either way,\Nprove, justify your answer
Dialogue: 0,0:13:42.06,0:13:45.66,Default,,0000,0000,0000,,by a proof, a complete\Nproof with epsilon delta.
Dialogue: 0,0:13:45.66,0:13:46.42,Default,,0000,0000,0000,,Proof.
Dialogue: 0,0:13:46.42,0:13:48.30,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:13:48.30,0:13:48.80,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:13:48.80,0:13:51.33,Default,,0000,0000,0000,,So now is a worried face.
Dialogue: 0,0:13:51.33,0:13:52.66,Default,,0000,0000,0000,,Like, oh, my god.
Dialogue: 0,0:13:52.66,0:13:54.96,Default,,0000,0000,0000,,This guy is worried\Nbecause, oh, my god.
Dialogue: 0,0:13:54.96,0:13:55.79,Default,,0000,0000,0000,,Epsilon delta.
Dialogue: 0,0:13:55.79,0:13:57.53,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,0:13:57.53,0:13:59.80,Default,,0000,0000,0000,,But the principle--\Nthe intuition
Dialogue: 0,0:13:59.80,0:14:03.92,Default,,0000,0000,0000,,tells us that we should look\Nfirst at some sort of a graph,
Dialogue: 0,0:14:03.92,0:14:05.29,Default,,0000,0000,0000,,just like Ryan pointed out.
Dialogue: 0,0:14:05.29,0:14:09.24,Default,,0000,0000,0000,,One should close their eyes and\Nimagine a graph of a function
Dialogue: 0,0:14:09.24,0:14:16.80,Default,,0000,0000,0000,,with-- it's hard to visualize in\N3D the graph of a function that
Dialogue: 0,0:14:16.80,0:14:18.74,Default,,0000,0000,0000,,is a surface.
Dialogue: 0,0:14:18.74,0:14:23.79,Default,,0000,0000,0000,,This is a surface. z\Nequals the whole shebang.
Dialogue: 0,0:14:23.79,0:14:29.65,Default,,0000,0000,0000,,But when I'm going to look\Nat the one dimensional case
Dialogue: 0,0:14:29.65,0:14:33.68,Default,,0000,0000,0000,,from last time, we\Nremember the sine of 1/x
Dialogue: 0,0:14:33.68,0:14:35.21,Default,,0000,0000,0000,,was a crazy function.
Dialogue: 0,0:14:35.21,0:14:39.32,Default,,0000,0000,0000,,We called it the harmonica,\Nwell, 20-something years ago
Dialogue: 0,0:14:39.32,0:14:40.71,Default,,0000,0000,0000,,when I was in high school.
Dialogue: 0,0:14:40.71,0:14:42.74,Default,,0000,0000,0000,,I was in an advanced\Ncalculus class.
Dialogue: 0,0:14:42.74,0:14:46.17,Default,,0000,0000,0000,,And our teacher was\Nnot funny at all.
Dialogue: 0,0:14:46.17,0:14:49.26,Default,,0000,0000,0000,,He was also not teaching much,\Ngave us a lot of homework,
Dialogue: 0,0:14:49.26,0:14:50.58,Default,,0000,0000,0000,,very challenging.
Dialogue: 0,0:14:50.58,0:14:54.43,Default,,0000,0000,0000,,So in order to make our\Nlife a little bit easier,
Dialogue: 0,0:14:54.43,0:14:57.23,Default,,0000,0000,0000,,we always worked in\Ngroups, which was allowed.
Dialogue: 0,0:14:57.23,0:15:00.96,Default,,0000,0000,0000,,So we called it a harmonica\Nbecause it was oscillating
Dialogue: 0,0:15:00.96,0:15:02.80,Default,,0000,0000,0000,,like that to the point\Nthat-- you've seen
Dialogue: 0,0:15:02.80,0:15:06.19,Default,,0000,0000,0000,,the harmonica-- the accordion.
Dialogue: 0,0:15:06.19,0:15:12.94,Default,,0000,0000,0000,,When you bring it back to\Nthe-- harmonica came to my mind
Dialogue: 0,0:15:12.94,0:15:15.72,Default,,0000,0000,0000,,from the harmonic function.
Dialogue: 0,0:15:15.72,0:15:19.26,Default,,0000,0000,0000,,So the accordion is--\Nwhen you actually
Dialogue: 0,0:15:19.26,0:15:25.92,Default,,0000,0000,0000,,squeeze it, all that oscillation\Nthings, the cusps are
Dialogue: 0,0:15:25.92,0:15:28.49,Default,,0000,0000,0000,,closer and closer to a line.
Dialogue: 0,0:15:28.49,0:15:33.92,Default,,0000,0000,0000,,So what you have here is\Nthis kind of oscillation,
Dialogue: 0,0:15:33.92,0:15:37.63,Default,,0000,0000,0000,,very, very rapid\Noscillation for sine of 1/x.
Dialogue: 0,0:15:37.63,0:15:41.81,Default,,0000,0000,0000,,When we want to multiply by\Nan x, what's going to happen?
Dialogue: 0,0:15:41.81,0:15:47.51,Default,,0000,0000,0000,,Well, this has not limit at 0\Nbecause it takes all the values
Dialogue: 0,0:15:47.51,0:15:49.60,Default,,0000,0000,0000,,infinitesimally close to 0.
Dialogue: 0,0:15:49.60,0:15:52.48,Default,,0000,0000,0000,,It keeps going through all the\Nvalues between minus 1 and 1,
Dialogue: 0,0:15:52.48,0:15:53.23,Default,,0000,0000,0000,,closer and closer.
Dialogue: 0,0:15:53.23,0:15:55.56,Default,,0000,0000,0000,,So that was no good.
Dialogue: 0,0:15:55.56,0:16:03.89,Default,,0000,0000,0000,,But if we take this guy--\Nthat's going to go to-- well,
Dialogue: 0,0:16:03.89,0:16:05.35,Default,,0000,0000,0000,,I cannot do better.
Dialogue: 0,0:16:05.35,0:16:07.30,Default,,0000,0000,0000,,MATLAB can do better than me.
Dialogue: 0,0:16:07.30,0:16:09.24,Default,,0000,0000,0000,,Mathematica can do better.
Dialogue: 0,0:16:09.24,0:16:10.09,Default,,0000,0000,0000,,You can do that.
Dialogue: 0,0:16:10.09,0:16:12.60,Default,,0000,0000,0000,,In most engineering\Nclasses, if you are--
Dialogue: 0,0:16:12.60,0:16:15.69,Default,,0000,0000,0000,,who is an electrical\Nengineering major?
Dialogue: 0,0:16:15.69,0:16:18.96,Default,,0000,0000,0000,,But even if you are\Nnot, you are going
Dialogue: 0,0:16:18.96,0:16:21.25,Default,,0000,0000,0000,,to see this type\Nof function a lot.
Dialogue: 0,0:16:21.25,0:16:24.56,Default,,0000,0000,0000,,And you're going to see it\Nagain in differential equations.
Dialogue: 0,0:16:24.56,0:16:27.15,Default,,0000,0000,0000,,
Dialogue: 0,0:16:27.15,0:16:30.74,Default,,0000,0000,0000,,How can I imagine-- this\Ngraph is hard to draw.
Dialogue: 0,0:16:30.74,0:16:34.06,Default,,0000,0000,0000,,Don't ask me to draw that.
Dialogue: 0,0:16:34.06,0:16:41.06,Default,,0000,0000,0000,,But ask me if I can use epsilon\Ndelta to prove continuity.
Dialogue: 0,0:16:41.06,0:16:44.96,Default,,0000,0000,0000,,So what would it mean,\Nproving continuity?
Dialogue: 0,0:16:44.96,0:16:45.90,Default,,0000,0000,0000,,I have a feeling--
Dialogue: 0,0:16:45.90,0:16:47.29,Default,,0000,0000,0000,,STUDENT: Well, actually, if this\Nis-- going back to that graph,
Dialogue: 0,0:16:47.29,0:16:49.01,Default,,0000,0000,0000,,doesn't that graph look like--
Dialogue: 0,0:16:49.01,0:16:50.30,Default,,0000,0000,0000,,MAGDALENA TODA: This goes to 0.
Dialogue: 0,0:16:50.30,0:16:53.84,Default,,0000,0000,0000,,The limit exists for x\Nsine of 1/x, and it is 0.
Dialogue: 0,0:16:53.84,0:16:55.58,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:16:55.58,0:16:57.17,Default,,0000,0000,0000,,Ryan?
Dialogue: 0,0:16:57.17,0:17:01.05,Default,,0000,0000,0000,,RYAN: Wouldn't the graph\Nwith the x squared plus
Dialogue: 0,0:17:01.05,0:17:02.90,Default,,0000,0000,0000,,y squared times that\Nside-- wouldn't that
Dialogue: 0,0:17:02.90,0:17:06.04,Default,,0000,0000,0000,,just look like a ripple\Nin a circle going out
Dialogue: 0,0:17:06.04,0:17:07.37,Default,,0000,0000,0000,,from the center?
Dialogue: 0,0:17:07.37,0:17:09.08,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah,\Nit will be ripples.
Dialogue: 0,0:17:09.08,0:17:10.68,Default,,0000,0000,0000,,STUDENT: Just like a\N[INAUDIBLE] from an epicenter
Dialogue: 0,0:17:10.68,0:17:11.96,Default,,0000,0000,0000,,going outwards [INAUDIBLE].
Dialogue: 0,0:17:11.96,0:17:16.05,Default,,0000,0000,0000,,MAGDALENA TODA: And I think--\Nyes, we managed to-- you
Dialogue: 0,0:17:16.05,0:17:19.90,Default,,0000,0000,0000,,have a concentric image, right?
Dialogue: 0,0:17:19.90,0:17:20.48,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:17:20.48,0:17:22.37,Default,,0000,0000,0000,,MAGDALENA TODA: Like those\Nripples, exactly like--
Dialogue: 0,0:17:22.37,0:17:24.04,Default,,0000,0000,0000,,STUDENT: So that's\Nwhat that looks like?
Dialogue: 0,0:17:24.04,0:17:26.55,Default,,0000,0000,0000,,MAGDALENA TODA: --when you\Nthrow a stone into the water,
Dialogue: 0,0:17:26.55,0:17:27.68,Default,,0000,0000,0000,,this kind of wave.
Dialogue: 0,0:17:27.68,0:17:30.73,Default,,0000,0000,0000,,But it's infinitesimally close.
Dialogue: 0,0:17:30.73,0:17:32.91,Default,,0000,0000,0000,,It's like acting weird.
Dialogue: 0,0:17:32.91,0:17:37.24,Default,,0000,0000,0000,,But then it sort\Nof shrinks here.
Dialogue: 0,0:17:37.24,0:17:40.79,Default,,0000,0000,0000,,And that-- it\Nimposes the limit 0.
Dialogue: 0,0:17:40.79,0:17:43.23,Default,,0000,0000,0000,,How come this goes\Nto 0, you say?
Dialogue: 0,0:17:43.23,0:17:46.15,Default,,0000,0000,0000,,Well, Magdalena, this\Nguy is crazy, right?
Dialogue: 0,0:17:46.15,0:17:49.16,Default,,0000,0000,0000,,Sine of 1/x goes\Nbetween minus 1 and 1
Dialogue: 0,0:17:49.16,0:17:51.50,Default,,0000,0000,0000,,infinitely many times\Nas I go close, close,
Dialogue: 0,0:17:51.50,0:17:57.93,Default,,0000,0000,0000,,closer and closer, more rapidly,\Nmore and more rapidly close
Dialogue: 0,0:17:57.93,0:17:58.89,Default,,0000,0000,0000,,to 0.
Dialogue: 0,0:17:58.89,0:18:00.54,Default,,0000,0000,0000,,This will oscillate\Nmore rapidly,
Dialogue: 0,0:18:00.54,0:18:03.19,Default,,0000,0000,0000,,more rapidly, and more rapidly.
Dialogue: 0,0:18:03.19,0:18:04.63,Default,,0000,0000,0000,,This is crazy, right?
Dialogue: 0,0:18:04.63,0:18:07.69,Default,,0000,0000,0000,,How does this guy, x-- how\Nis this guy taming this guy?
Dialogue: 0,0:18:07.69,0:18:10.44,Default,,0000,0000,0000,,STUDENT: Because\Nas 0 [INAUDIBLE].
Dialogue: 0,0:18:10.44,0:18:12.57,Default,,0000,0000,0000,,Something really small\Ntimes something [INAUDIBLE].
Dialogue: 0,0:18:12.57,0:18:14.07,Default,,0000,0000,0000,,MAGDALENA TODA:\NSomething very small
Dialogue: 0,0:18:14.07,0:18:17.90,Default,,0000,0000,0000,,that shrinks to 0 times\Nsomething bounded.
Dialogue: 0,0:18:17.90,0:18:20.88,Default,,0000,0000,0000,,Ryan brought the main idea.
Dialogue: 0,0:18:20.88,0:18:25.16,Default,,0000,0000,0000,,If something goes strongly to\N0, and that multiplies something
Dialogue: 0,0:18:25.16,0:18:28.49,Default,,0000,0000,0000,,that's bounded, bounded\Nby a finite number,
Dialogue: 0,0:18:28.49,0:18:31.07,Default,,0000,0000,0000,,the whole problem will go to 0.
Dialogue: 0,0:18:31.07,0:18:32.96,Default,,0000,0000,0000,,Actually, you can prove\Nthat as a theorem.
Dialogue: 0,0:18:32.96,0:18:34.86,Default,,0000,0000,0000,,And some of you did.
Dialogue: 0,0:18:34.86,0:18:36.63,Default,,0000,0000,0000,,In most honors\Nclasses unfortunately,
Dialogue: 0,0:18:36.63,0:18:39.10,Default,,0000,0000,0000,,epsilon delta was not covered.
Dialogue: 0,0:18:39.10,0:18:43.35,Default,,0000,0000,0000,,So let's see how we prove\Nthis with epsilon delta.
Dialogue: 0,0:18:43.35,0:18:45.12,Default,,0000,0000,0000,,And, oh, my god.
Dialogue: 0,0:18:45.12,0:18:52.82,Default,,0000,0000,0000,,Many of you read from the book\Nand may be able to help me.
Dialogue: 0,0:18:52.82,0:19:00.10,Default,,0000,0000,0000,,So what am I supposed to\Nshow with epsilon delta?
Dialogue: 0,0:19:00.10,0:19:09.86,Default,,0000,0000,0000,,The limit of x squared plus\Ny squared sine of 1 over x
Dialogue: 0,0:19:09.86,0:19:14.56,Default,,0000,0000,0000,,squared plus y squared is\N0 as I approach the origin
Dialogue: 0,0:19:14.56,0:19:19.74,Default,,0000,0000,0000,,with my pair, couple, x, y,\Nwhich can go any one path that
Dialogue: 0,0:19:19.74,0:19:20.34,Default,,0000,0000,0000,,approaches 0.
Dialogue: 0,0:19:20.34,0:19:23.78,Default,,0000,0000,0000,,
Dialogue: 0,0:19:23.78,0:19:27.58,Default,,0000,0000,0000,,So you say, oh, well, Magdalena,\Nthe Ryan principle-- this
Dialogue: 0,0:19:27.58,0:19:29.07,Default,,0000,0000,0000,,is the Ryan theorem.
Dialogue: 0,0:19:29.07,0:19:32.18,Default,,0000,0000,0000,,It's the same because\Nthis guy will be
Dialogue: 0,0:19:32.18,0:19:34.10,Default,,0000,0000,0000,,bounded between minus 1 and 1.
Dialogue: 0,0:19:34.10,0:19:37.63,Default,,0000,0000,0000,,I multiplied with a guy\Nthat very determinedly goes
Dialogue: 0,0:19:37.63,0:19:39.54,Default,,0000,0000,0000,,to 0 very strongly.
Dialogue: 0,0:19:39.54,0:19:41.16,Default,,0000,0000,0000,,And he knows where he's going.
Dialogue: 0,0:19:41.16,0:19:44.02,Default,,0000,0000,0000,,x squared plus y squared\Nsays, I know what I'm doing.
Dialogue: 0,0:19:44.02,0:19:45.88,Default,,0000,0000,0000,,I'm not going to change my mind.
Dialogue: 0,0:19:45.88,0:19:49.48,Default,,0000,0000,0000,,This is like the guy who changes\Nhis major too many times.
Dialogue: 0,0:19:49.48,0:19:52.21,Default,,0000,0000,0000,,And this guy knows\Nwhat he's doing.
Dialogue: 0,0:19:52.21,0:19:54.92,Default,,0000,0000,0000,,He's going there, and he's\Na polynomial, goes to 0,
Dialogue: 0,0:19:54.92,0:19:56.29,Default,,0000,0000,0000,,0 very rapidly.
Dialogue: 0,0:19:56.29,0:20:00.52,Default,,0000,0000,0000,,Now it's clear what\Nhappens intuitively.
Dialogue: 0,0:20:00.52,0:20:02.91,Default,,0000,0000,0000,,But I'm a mathematician.
Dialogue: 0,0:20:02.91,0:20:07.04,Default,,0000,0000,0000,,And if I don't publish\Nmy proof, my article
Dialogue: 0,0:20:07.04,0:20:12.68,Default,,0000,0000,0000,,will be very nicely rejected\Nby all the serious journals
Dialogue: 0,0:20:12.68,0:20:13.97,Default,,0000,0000,0000,,on the market.
Dialogue: 0,0:20:13.97,0:20:17.47,Default,,0000,0000,0000,,This is how it goes\Nin mathematics.
Dialogue: 0,0:20:17.47,0:20:19.39,Default,,0000,0000,0000,,Even before journals\Nexisted, mathematicians
Dialogue: 0,0:20:19.39,0:20:23.11,Default,,0000,0000,0000,,had to show a rigorous\Nproof of their work,
Dialogue: 0,0:20:23.11,0:20:25.72,Default,,0000,0000,0000,,of their conjecture.
Dialogue: 0,0:20:25.72,0:20:26.92,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:20:26.92,0:20:35.08,Default,,0000,0000,0000,,So I go, for every epsilon\Npositive, no matter how small,
Dialogue: 0,0:20:35.08,0:20:41.04,Default,,0000,0000,0000,,there must exist a\Ndelta positive, which
Dialogue: 0,0:20:41.04,0:20:51.99,Default,,0000,0000,0000,,depends on epsilon-- that\Ndepends on epsilon-- such that
Dialogue: 0,0:20:51.99,0:20:58.55,Default,,0000,0000,0000,,as soon as-- how did\Nwe write the distance?
Dialogue: 0,0:20:58.55,0:21:01.82,Default,,0000,0000,0000,,I'll write the distance\Nagain because I'm lazy.
Dialogue: 0,0:21:01.82,0:21:05.59,Default,,0000,0000,0000,,The distance between the\Npoint x, y and the origin
Dialogue: 0,0:21:05.59,0:21:07.93,Default,,0000,0000,0000,,is less than delta.
Dialogue: 0,0:21:07.93,0:21:16.72,Default,,0000,0000,0000,,It follows that the\Nabsolute value--
Dialogue: 0,0:21:16.72,0:21:24.33,Default,,0000,0000,0000,,these are all real numbers--\Nof f of x, y or g of x,
Dialogue: 0,0:21:24.33,0:21:27.46,Default,,0000,0000,0000,,y-- g of x, y is the extension.
Dialogue: 0,0:21:27.46,0:21:32.15,Default,,0000,0000,0000,,
Dialogue: 0,0:21:32.15,0:21:36.35,Default,,0000,0000,0000,,f of x, y minus 0, which\NI claim to be the limit,
Dialogue: 0,0:21:36.35,0:21:39.14,Default,,0000,0000,0000,,will be less than epsilon.
Dialogue: 0,0:21:39.14,0:21:40.50,Default,,0000,0000,0000,,So you go, oh, my god.
Dialogue: 0,0:21:40.50,0:21:42.83,Default,,0000,0000,0000,,What is this woman doing?
Dialogue: 0,0:21:42.83,0:21:43.82,Default,,0000,0000,0000,,It's not hard.
Dialogue: 0,0:21:43.82,0:21:45.98,Default,,0000,0000,0000,,I need your help though.
Dialogue: 0,0:21:45.98,0:21:48.54,Default,,0000,0000,0000,,I need your help to do that.
Dialogue: 0,0:21:48.54,0:21:53.10,Default,,0000,0000,0000,,So it's hard to see how you\Nshould-- you take any epsilon.
Dialogue: 0,0:21:53.10,0:21:58.30,Default,,0000,0000,0000,,You pick your favorite\Nepsilon, infinitesimally small,
Dialogue: 0,0:21:58.30,0:22:01.12,Default,,0000,0000,0000,,any small number, but\Nthen you go, but then I
Dialogue: 0,0:22:01.12,0:22:03.40,Default,,0000,0000,0000,,have to show this delta exists.
Dialogue: 0,0:22:03.40,0:22:06.59,Default,,0000,0000,0000,,You have to grab that delta\Nand say, you are my delta.
Dialogue: 0,0:22:06.59,0:22:08.99,Default,,0000,0000,0000,,You cannot escape me.
Dialogue: 0,0:22:08.99,0:22:10.93,Default,,0000,0000,0000,,I tell you who you are.
Dialogue: 0,0:22:10.93,0:22:13.84,Default,,0000,0000,0000,,And that's the\Nhardest part in here,
Dialogue: 0,0:22:13.84,0:22:18.24,Default,,0000,0000,0000,,figuring out who that delta must\Nbe as a function of epsilon.
Dialogue: 0,0:22:18.24,0:22:19.06,Default,,0000,0000,0000,,Is that hard?
Dialogue: 0,0:22:19.06,0:22:21.32,Default,,0000,0000,0000,,How do you build\Nsuch a construction?
Dialogue: 0,0:22:21.32,0:22:26.69,Default,,0000,0000,0000,,First of all,\Nunderstand what proof.
Dialogue: 0,0:22:26.69,0:22:30.38,Default,,0000,0000,0000,,"Choose any positive epsilon."
Dialogue: 0,0:22:30.38,0:22:32.88,Default,,0000,0000,0000,,Then forget about him,\Nbecause he's your friend,
Dialogue: 0,0:22:32.88,0:22:36.40,Default,,0000,0000,0000,,and he's going to do whatever\Nyou want to do with him.
Dialogue: 0,0:22:36.40,0:22:40.12,Default,,0000,0000,0000,,Delta, chasing\Nafter delta is going
Dialogue: 0,0:22:40.12,0:22:42.39,Default,,0000,0000,0000,,to be a little bit harder.
Dialogue: 0,0:22:42.39,0:22:56.01,Default,,0000,0000,0000,,"Chasing after delta\Nwith that property."
Dialogue: 0,0:22:56.01,0:22:58.43,Default,,0000,0000,0000,,Dot, dot, dot, dot, dot.
Dialogue: 0,0:22:58.43,0:22:59.70,Default,,0000,0000,0000,,What is this distance?
Dialogue: 0,0:22:59.70,0:23:01.95,Default,,0000,0000,0000,,You guys have\Nhelped me last time,
Dialogue: 0,0:23:01.95,0:23:04.67,Default,,0000,0000,0000,,you cannot let me down now.
Dialogue: 0,0:23:04.67,0:23:08.35,Default,,0000,0000,0000,,So as soon as this distance,\Nyour gradient distance
Dialogue: 0,0:23:08.35,0:23:10.54,Default,,0000,0000,0000,,is less than delta,\Nyou must have
Dialogue: 0,0:23:10.54,0:23:13.31,Default,,0000,0000,0000,,that f of x, y [INAUDIBLE].
Dialogue: 0,0:23:13.31,0:23:15.17,Default,,0000,0000,0000,,Could you tell me\Nwhat that would be?
Dialogue: 0,0:23:15.17,0:23:16.17,Default,,0000,0000,0000,,It was Euclidean, right?
Dialogue: 0,0:23:16.17,0:23:21.67,Default,,0000,0000,0000,,So I had squared\Nroot of-- did I?
Dialogue: 0,0:23:21.67,0:23:30.40,Default,,0000,0000,0000,,Square root of x minus 0\Nsquared plus y minus 0 squared.
Dialogue: 0,0:23:30.40,0:23:33.17,Default,,0000,0000,0000,,You say, but that's\Nsilly, Magdalena.
Dialogue: 0,0:23:33.17,0:23:37.54,Default,,0000,0000,0000,,So you have to write\Nit down like that?
Dialogue: 0,0:23:37.54,0:23:38.97,Default,,0000,0000,0000,,STUDENT: It's the [INAUDIBLE].
Dialogue: 0,0:23:38.97,0:23:40.41,Default,,0000,0000,0000,,MAGDALENA TODA: Huh?
Dialogue: 0,0:23:40.41,0:23:42.29,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:23:42.29,0:23:47.17,Default,,0000,0000,0000,,So square root of this\Nplus square root of that
Dialogue: 0,0:23:47.17,0:23:53.03,Default,,0000,0000,0000,,plus then delta,\Nthat means what?
Dialogue: 0,0:23:53.03,0:24:00.35,Default,,0000,0000,0000,,If and only if x squared plus\Ny squared is less than delta
Dialogue: 0,0:24:00.35,0:24:00.85,Default,,0000,0000,0000,,squared.
Dialogue: 0,0:24:00.85,0:24:08.16,Default,,0000,0000,0000,,
Dialogue: 0,0:24:08.16,0:24:11.11,Default,,0000,0000,0000,,And what do I want to do,\Nwhat do I want to build?
Dialogue: 0,0:24:11.11,0:24:15.09,Default,,0000,0000,0000,,
Dialogue: 0,0:24:15.09,0:24:19.03,Default,,0000,0000,0000,,So we are thinking how\Nto set up all this thing.
Dialogue: 0,0:24:19.03,0:24:21.21,Default,,0000,0000,0000,,How to choose the delta.
Dialogue: 0,0:24:21.21,0:24:23.06,Default,,0000,0000,0000,,How to choose the delta.
Dialogue: 0,0:24:23.06,0:24:25.83,Default,,0000,0000,0000,,
Dialogue: 0,0:24:25.83,0:24:28.35,Default,,0000,0000,0000,,OK, so what do I--\Nwhat am I after?
Dialogue: 0,0:24:28.35,0:24:34.25,Default,,0000,0000,0000,,"I am after having" double dot.
Dialogue: 0,0:24:34.25,0:24:39.83,Default,,0000,0000,0000,,F of x, y must be Mr. Ugly.
Dialogue: 0,0:24:39.83,0:24:40.82,Default,,0000,0000,0000,,This one.
Dialogue: 0,0:24:40.82,0:24:46.48,Default,,0000,0000,0000,,So absolute value of x squared\Nplus y squared, sine of 1
Dialogue: 0,0:24:46.48,0:24:51.24,Default,,0000,0000,0000,,over x squared plus\Ny squared minus 0.
Dialogue: 0,0:24:51.24,0:24:51.74,Default,,0000,0000,0000,,Duh.
Dialogue: 0,0:24:51.74,0:24:55.17,Default,,0000,0000,0000,,I'm not going to write it.
Dialogue: 0,0:24:55.17,0:24:59.26,Default,,0000,0000,0000,,We all know what that means.
Dialogue: 0,0:24:59.26,0:25:00.09,Default,,0000,0000,0000,,Less than epsilon.
Dialogue: 0,0:25:00.09,0:25:05.68,Default,,0000,0000,0000,,This is what must\Nfollow as a conclusion.
Dialogue: 0,0:25:05.68,0:25:12.36,Default,,0000,0000,0000,,This is what must\Nfollow, must happen.
Dialogue: 0,0:25:12.36,0:25:13.33,Default,,0000,0000,0000,,Must happen.
Dialogue: 0,0:25:13.33,0:25:16.26,Default,,0000,0000,0000,,
Dialogue: 0,0:25:16.26,0:25:17.63,Default,,0000,0000,0000,,Now I'm getting excited.
Dialogue: 0,0:25:17.63,0:25:18.13,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:25:18.13,0:25:21.05,Default,,0000,0000,0000,,Because I am thinking.
Dialogue: 0,0:25:21.05,0:25:23.04,Default,,0000,0000,0000,,I started thinking.
Dialogue: 0,0:25:23.04,0:25:26.42,Default,,0000,0000,0000,,Once I started thinking,\NI'm dangerous, man.
Dialogue: 0,0:25:26.42,0:25:31.53,Default,,0000,0000,0000,,So here sine of 1 over x squared\Nplus y squared is your friend.
Dialogue: 0,0:25:31.53,0:25:34.21,Default,,0000,0000,0000,,Why is that your friend?
Dialogue: 0,0:25:34.21,0:25:37.25,Default,,0000,0000,0000,,Sine of 1 over x squared\Nplus y squared, this
Dialogue: 0,0:25:37.25,0:25:39.12,Default,,0000,0000,0000,,is always an absolute value.
Dialogue: 0,0:25:39.12,0:25:42.98,Default,,0000,0000,0000,,The absolute value of that\Nis always less than 1.
Dialogue: 0,0:25:42.98,0:25:43.48,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:25:43.48,0:25:45.29,Default,,0000,0000,0000,,STUDENT: Can't it be 4?
Dialogue: 0,0:25:45.29,0:25:50.28,Default,,0000,0000,0000,,MAGDALENA TODA:\NSo-- so-- so what
Dialogue: 0,0:25:50.28,0:25:54.94,Default,,0000,0000,0000,,shall I take in terms of\Ndelta-- this is my question.
Dialogue: 0,0:25:54.94,0:25:57.40,Default,,0000,0000,0000,,What shall I take\Nin terms of delta?
Dialogue: 0,0:25:57.40,0:26:03.59,Default,,0000,0000,0000,,"Delta equals 1 as a\Nfunction of epsilon
Dialogue: 0,0:26:03.59,0:26:20.16,Default,,0000,0000,0000,,in order to have the\Nconclusion satisfied."
Dialogue: 0,0:26:20.16,0:26:20.92,Default,,0000,0000,0000,,You say, OK.
Dialogue: 0,0:26:20.92,0:26:24.61,Default,,0000,0000,0000,,It's enough to choose delta\Nlike that function of epsilon,
Dialogue: 0,0:26:24.61,0:26:28.84,Default,,0000,0000,0000,,and I'm done, because then\Neverything will be fine.
Dialogue: 0,0:26:28.84,0:26:33.51,Default,,0000,0000,0000,,So you chose your own epsilon,\Npositive, small, or God
Dialogue: 0,0:26:33.51,0:26:34.48,Default,,0000,0000,0000,,gave you an epsilon.
Dialogue: 0,0:26:34.48,0:26:37.04,Default,,0000,0000,0000,,You don't care how\Nyou got the epsilon.
Dialogue: 0,0:26:37.04,0:26:38.42,Default,,0000,0000,0000,,The epsilon is arbitrary.
Dialogue: 0,0:26:38.42,0:26:40.91,Default,,0000,0000,0000,,You pick positive and small.
Dialogue: 0,0:26:40.91,0:26:44.61,Default,,0000,0000,0000,,Now, it's up to\Nyou to find delta.
Dialogue: 0,0:26:44.61,0:26:48.82,Default,,0000,0000,0000,,So what delta would\Nsatisfy everything?
Dialogue: 0,0:26:48.82,0:26:50.94,Default,,0000,0000,0000,,What delta would\Nbe good enough--
Dialogue: 0,0:26:50.94,0:26:52.83,Default,,0000,0000,0000,,you don't care\Nfor all the good--
Dialogue: 0,0:26:52.83,0:26:54.55,Default,,0000,0000,0000,,it's like when you get married.
Dialogue: 0,0:26:54.55,0:26:57.74,Default,,0000,0000,0000,,Do you care for all the\Npeople who'd match you?
Dialogue: 0,0:26:57.74,0:27:00.60,Default,,0000,0000,0000,,Hopefully not, because\Nthen you would probably
Dialogue: 0,0:27:00.60,0:27:05.42,Default,,0000,0000,0000,,have too large of a pool,\Nand it's hard to choose.
Dialogue: 0,0:27:05.42,0:27:13.28,Default,,0000,0000,0000,,You only need one that satisfies\Nthat assumption, that satisfies
Dialogue: 0,0:27:13.28,0:27:14.97,Default,,0000,0000,0000,,all the conditions you have.
Dialogue: 0,0:27:14.97,0:27:18.78,Default,,0000,0000,0000,,So what is the delta that\Nsatisfies all the conditions
Dialogue: 0,0:27:18.78,0:27:20.05,Default,,0000,0000,0000,,that I have?
Dialogue: 0,0:27:20.05,0:27:20.88,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:27:20.88,0:27:22.27,Default,,0000,0000,0000,,MAGDALENA TODA: [INAUDIBLE].
Dialogue: 0,0:27:22.27,0:27:22.96,Default,,0000,0000,0000,,Who?
Dialogue: 0,0:27:22.96,0:27:25.35,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:27:25.35,0:27:27.86,Default,,0000,0000,0000,,MAGDALENA TODA: For example,\Ndelta equals epsilon.
Dialogue: 0,0:27:27.86,0:27:28.78,Default,,0000,0000,0000,,Would that satisfy?
Dialogue: 0,0:27:28.78,0:27:31.98,Default,,0000,0000,0000,,
Dialogue: 0,0:27:31.98,0:27:33.86,Default,,0000,0000,0000,,Well, let's see.
Dialogue: 0,0:27:33.86,0:27:37.41,Default,,0000,0000,0000,,If I take delta to\Nbe epsilon, then x
Dialogue: 0,0:27:37.41,0:27:40.41,Default,,0000,0000,0000,,squared plus y squared would\Nbe less than epsilon squared.
Dialogue: 0,0:27:40.41,0:27:47.33,Default,,0000,0000,0000,,Now the question is is epsilon\Nsquared less than epsilon?
Dialogue: 0,0:27:47.33,0:27:48.51,Default,,0000,0000,0000,,Not always.
Dialogue: 0,0:27:48.51,0:27:52.92,Default,,0000,0000,0000,,If epsilon is between 0\Nand 1, then epsilon squared
Dialogue: 0,0:27:52.92,0:27:54.20,Default,,0000,0000,0000,,is less then epsilon.
Dialogue: 0,0:27:54.20,0:27:59.20,Default,,0000,0000,0000,,But if I choose epsilon\Nto be greater than 1,
Dialogue: 0,0:27:59.20,0:28:00.22,Default,,0000,0000,0000,,then oh, my God.
Dialogue: 0,0:28:00.22,0:28:02.74,Default,,0000,0000,0000,,Then if it's greater than\N1, then epsilon squared
Dialogue: 0,0:28:02.74,0:28:06.79,Default,,0000,0000,0000,,is greater than 1--\Ngreater than it.
Dialogue: 0,0:28:06.79,0:28:14.70,Default,,0000,0000,0000,,So what if I choose\Ndelta to be what?
Dialogue: 0,0:28:14.70,0:28:18.65,Default,,0000,0000,0000,,
Dialogue: 0,0:28:18.65,0:28:19.60,Default,,0000,0000,0000,,STUDENT: 0?
Dialogue: 0,0:28:19.60,0:28:20.72,Default,,0000,0000,0000,,MAGDALENA TODA: No, no, no.
Dialogue: 0,0:28:20.72,0:28:22.09,Default,,0000,0000,0000,,Delta cannot be 0.
Dialogue: 0,0:28:22.09,0:28:26.33,Default,,0000,0000,0000,,So delta-- look, there exists\Ndelta strictly bigger than 0,
Dialogue: 0,0:28:26.33,0:28:28.69,Default,,0000,0000,0000,,that depends on epsilon.
Dialogue: 0,0:28:28.69,0:28:33.67,Default,,0000,0000,0000,,Maybe if epsilon is very small,\Nin a way Alexander was right.
Dialogue: 0,0:28:33.67,0:28:37.35,Default,,0000,0000,0000,,But the delta [INAUDIBLE],\Nwe don't go with epsilon
Dialogue: 0,0:28:37.35,0:28:38.28,Default,,0000,0000,0000,,greater than 1.
Dialogue: 0,0:28:38.28,0:28:39.00,Default,,0000,0000,0000,,Come on.
Dialogue: 0,0:28:39.00,0:28:39.50,Default,,0000,0000,0000,,Be serious.
Dialogue: 0,0:28:39.50,0:28:42.30,Default,,0000,0000,0000,,Epsilon is always\Nbetween 0 and 1.
Dialogue: 0,0:28:42.30,0:28:44.60,Default,,0000,0000,0000,,I mean, it's a lot\Nsmaller than that.
Dialogue: 0,0:28:44.60,0:28:46.64,Default,,0000,0000,0000,,It's infinitesimal small.
Dialogue: 0,0:28:46.64,0:28:49.48,Default,,0000,0000,0000,,So in the end, yes, in\Nthat case epsilon squared
Dialogue: 0,0:28:49.48,0:28:52.61,Default,,0000,0000,0000,,would be less than epsilon,\Nwhich would be OK for us
Dialogue: 0,0:28:52.61,0:28:54.59,Default,,0000,0000,0000,,and that would be fine.
Dialogue: 0,0:28:54.59,0:28:56.15,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:28:56.15,0:28:58.38,Default,,0000,0000,0000,,So that would be a\Npossibility to say, hey,
Dialogue: 0,0:28:58.38,0:29:01.08,Default,,0000,0000,0000,,since epsilon-- Alexander,\Nif you write that as a proof
Dialogue: 0,0:29:01.08,0:29:01.82,Default,,0000,0000,0000,,I'll be OK.
Dialogue: 0,0:29:01.82,0:29:04.90,Default,,0000,0000,0000,,You say, I took my epsilon\Nto be a very small number,
Dialogue: 0,0:29:04.90,0:29:07.02,Default,,0000,0000,0000,,so anyway it's going\Nto be less than 1.
Dialogue: 0,0:29:07.02,0:29:09.19,Default,,0000,0000,0000,,So epsilon squared\Nis less than epsilon.
Dialogue: 0,0:29:09.19,0:29:14.09,Default,,0000,0000,0000,,So when I take\Ndelta to be epsilon,
Dialogue: 0,0:29:14.09,0:29:18.21,Default,,0000,0000,0000,,for sure this guy will be less\Nthan epsilon squared, which
Dialogue: 0,0:29:18.21,0:29:21.15,Default,,0000,0000,0000,,is less than epsilon,\Nso I'm satisfied.
Dialogue: 0,0:29:21.15,0:29:22.62,Default,,0000,0000,0000,,I'll give you a 100%.
Dialogue: 0,0:29:22.62,0:29:24.08,Default,,0000,0000,0000,,I'm happy.
Dialogue: 0,0:29:24.08,0:29:25.15,Default,,0000,0000,0000,,Is that the only way?
Dialogue: 0,0:29:25.15,0:29:26.53,Default,,0000,0000,0000,,STUDENT: But what\Nabout the sine?
Dialogue: 0,0:29:26.53,0:29:27.50,Default,,0000,0000,0000,,What about [INAUDIBLE].
Dialogue: 0,0:29:27.50,0:29:28.48,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:29:28.48,0:29:30.11,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nthis doesn't matter.
Dialogue: 0,0:29:30.11,0:29:32.40,Default,,0000,0000,0000,,Let me write it down.
Dialogue: 0,0:29:32.40,0:29:39.73,Default,,0000,0000,0000,,So note that x squared\Nplus y squared sine of 1
Dialogue: 0,0:29:39.73,0:29:42.65,Default,,0000,0000,0000,,over x squared plus\Ny square would always
Dialogue: 0,0:29:42.65,0:29:46.38,Default,,0000,0000,0000,,be less than absolute\Nvalue of x squared
Dialogue: 0,0:29:46.38,0:29:49.97,Default,,0000,0000,0000,,plus y, which is positive.
Dialogue: 0,0:29:49.97,0:29:52.14,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:29:52.14,0:29:53.20,Default,,0000,0000,0000,,Is this true?
Dialogue: 0,0:29:53.20,0:29:54.08,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:29:54.08,0:29:55.44,Default,,0000,0000,0000,,Why is that?
Dialogue: 0,0:29:55.44,0:29:58.15,Default,,0000,0000,0000,,STUDENT: Because the sine can\Nonly be one of these negatives.
Dialogue: 0,0:29:58.15,0:30:00.44,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nin absolute value,
Dialogue: 0,0:30:00.44,0:30:05.78,Default,,0000,0000,0000,,sine of 1 over x squared plus y\Nsquared is always less than 1.
Dialogue: 0,0:30:05.78,0:30:08.52,Default,,0000,0000,0000,,STUDENT: Can't it equal 1?
Dialogue: 0,0:30:08.52,0:30:11.88,Default,,0000,0000,0000,,MAGDALENA TODA: Well,\Nwhen does it equal 1?
Dialogue: 0,0:30:11.88,0:30:14.31,Default,,0000,0000,0000,,STUDENT: Wouldn't it be x\Nsquared plus y squared equals 1
Dialogue: 0,0:30:14.31,0:30:15.65,Default,,0000,0000,0000,,[INAUDIBLE]?
Dialogue: 0,0:30:15.65,0:30:17.23,Default,,0000,0000,0000,,MAGDALENA TODA: Less\Nthan or equal to.
Dialogue: 0,0:30:17.23,0:30:18.48,Default,,0000,0000,0000,,For some values it will.
Dialogue: 0,0:30:18.48,0:30:19.16,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:30:19.16,0:30:19.66,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:30:19.66,0:30:21.87,Default,,0000,0000,0000,,MAGDALENA TODA: Now, will\Nthat be a problem with us?
Dialogue: 0,0:30:21.87,0:30:22.12,Default,,0000,0000,0000,,No.
Dialogue: 0,0:30:22.12,0:30:23.06,Default,,0000,0000,0000,,Let's put it here.
Dialogue: 0,0:30:23.06,0:30:27.37,Default,,0000,0000,0000,,Less than or equal to x\Nsquared plus y squared, which
Dialogue: 0,0:30:27.37,0:30:35.46,Default,,0000,0000,0000,,has to be less than epsilon\Nif and only if-- well,
Dialogue: 0,0:30:35.46,0:30:38.78,Default,,0000,0000,0000,,if delta is what?
Dialogue: 0,0:30:38.78,0:30:41.24,Default,,0000,0000,0000,,So, again, Alexander said,\Nwell, but if I take delta
Dialogue: 0,0:30:41.24,0:30:42.82,Default,,0000,0000,0000,,to be epsilon, I'm done.
Dialogue: 0,0:30:42.82,0:30:45.76,Default,,0000,0000,0000,,
Dialogue: 0,0:30:45.76,0:30:46.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:30:46.74,0:30:49.69,Default,,0000,0000,0000,,MAGDALENA TODA: How\Nabout square root?
Dialogue: 0,0:30:49.69,0:30:52.15,Default,,0000,0000,0000,,Can I take delta to be\Nsquare root of epsilon.
Dialogue: 0,0:30:52.15,0:30:53.54,Default,,0000,0000,0000,,STUDENT: That's what I said.
Dialogue: 0,0:30:53.54,0:30:54.33,Default,,0000,0000,0000,,MAGDALENA TODA: No.
Dialogue: 0,0:30:54.33,0:30:55.69,Default,,0000,0000,0000,,You said epsilon.
Dialogue: 0,0:30:55.69,0:30:57.32,Default,,0000,0000,0000,,STUDENT: I said square\Nroot of epsilon.
Dialogue: 0,0:30:57.32,0:30:58.27,Default,,0000,0000,0000,,MAGDALENA TODA: OK.
Dialogue: 0,0:30:58.27,0:31:01.29,Default,,0000,0000,0000,,If delta is square\Nroot of epsilon,
Dialogue: 0,0:31:01.29,0:31:05.28,Default,,0000,0000,0000,,then everything will be perfect\Nand it will be a perfect match.
Dialogue: 0,0:31:05.28,0:31:05.100,Default,,0000,0000,0000,,In what case?
Dialogue: 0,0:31:05.100,0:31:07.70,Default,,0000,0000,0000,,STUDENT: If epsilon\Nis in between 0 and 1
Dialogue: 0,0:31:07.70,0:31:10.13,Default,,0000,0000,0000,,and if delta is equal\Nto bigger than epsilon.
Dialogue: 0,0:31:10.13,0:31:13.06,Default,,0000,0000,0000,,
Dialogue: 0,0:31:13.06,0:31:17.74,Default,,0000,0000,0000,,MAGDALENA TODA: So that's\Nexactly the same assumption.
Dialogue: 0,0:31:17.74,0:31:22.48,Default,,0000,0000,0000,,Epsilon should be\Nmade in less than.
Dialogue: 0,0:31:22.48,0:31:24.33,Default,,0000,0000,0000,,STUDENT: But I thought\Ndelta was supposed
Dialogue: 0,0:31:24.33,0:31:25.91,Default,,0000,0000,0000,,to be less than\Nepsilon in every case.
Dialogue: 0,0:31:25.91,0:31:29.34,Default,,0000,0000,0000,,So if epsilon is between 0 and\N1, the square root of epsilon
Dialogue: 0,0:31:29.34,0:31:31.80,Default,,0000,0000,0000,,is going to be [INAUDIBLE].
Dialogue: 0,0:31:31.80,0:31:38.33,Default,,0000,0000,0000,,MAGDALENA TODA: So when\Nboth of them are small,
Dialogue: 0,0:31:38.33,0:31:45.04,Default,,0000,0000,0000,,delta squared will be-- if\NI take delta-- so take delta
Dialogue: 0,0:31:45.04,0:31:47.70,Default,,0000,0000,0000,,to be square root of epsilon.
Dialogue: 0,0:31:47.70,0:31:50.12,Default,,0000,0000,0000,,STUDENT: Then anything less\Nthan 1 and greater than 0,
Dialogue: 0,0:31:50.12,0:31:51.78,Default,,0000,0000,0000,,epsilon would be great\Nthan [INAUDIBLE].
Dialogue: 0,0:31:51.78,0:31:54.96,Default,,0000,0000,0000,,MAGDALENA TODA: "Delta to\Nbe square root of epsilon,
Dialogue: 0,0:31:54.96,0:32:01.57,Default,,0000,0000,0000,,then x squared plus y squared\Nless than delta squared equals
Dialogue: 0,0:32:01.57,0:32:03.78,Default,,0000,0000,0000,,epsilon."
Dialogue: 0,0:32:03.78,0:32:11.80,Default,,0000,0000,0000,,Then x squared plus\Ny squared sine of 1
Dialogue: 0,0:32:11.80,0:32:14.99,Default,,0000,0000,0000,,over x squared plus\Ny squared less than
Dialogue: 0,0:32:14.99,0:32:17.44,Default,,0000,0000,0000,,or equal to x squared\Nplus y squared.
Dialogue: 0,0:32:17.44,0:32:19.16,Default,,0000,0000,0000,,I dont' need the absolute value.
Dialogue: 0,0:32:19.16,0:32:20.38,Default,,0000,0000,0000,,I can [INAUDIBLE].
Dialogue: 0,0:32:20.38,0:32:23.32,Default,,0000,0000,0000,,Less than epsilon [INAUDIBLE].
Dialogue: 0,0:32:23.32,0:32:24.11,Default,,0000,0000,0000,,Qed.
Dialogue: 0,0:32:24.11,0:32:26.05,Default,,0000,0000,0000,,STUDENT: Well, but\Nyou told us delta
Dialogue: 0,0:32:26.05,0:32:27.50,Default,,0000,0000,0000,,has to be less than epsilon.
Dialogue: 0,0:32:27.50,0:32:28.48,Default,,0000,0000,0000,,Well, if--
Dialogue: 0,0:32:28.48,0:32:31.39,Default,,0000,0000,0000,,MAGDALENA TODA: No,\NI didn't say that.
Dialogue: 0,0:32:31.39,0:32:35.32,Default,,0000,0000,0000,,I didn't say that delta has\Nto be less than epsilon.
Dialogue: 0,0:32:35.32,0:32:35.82,Default,,0000,0000,0000,,Absolutely--
Dialogue: 0,0:32:35.82,0:32:36.41,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:32:36.41,0:32:38.76,Default,,0000,0000,0000,,You said for all the values\Nof epsilon greater than 0,
Dialogue: 0,0:32:38.76,0:32:42.28,Default,,0000,0000,0000,,there's a value of delta that is\Ngreater than 0 that [INAUDIBLE]
Dialogue: 0,0:32:42.28,0:32:45.65,Default,,0000,0000,0000,,such that as soon as the\Ndistance between is less than
Dialogue: 0,0:32:45.65,0:32:46.96,Default,,0000,0000,0000,,delta-- I don't remember what--
Dialogue: 0,0:32:46.96,0:32:48.25,Default,,0000,0000,0000,,MAGDALENA TODA: OK, so, again--
Dialogue: 0,0:32:48.25,0:32:50.17,Default,,0000,0000,0000,,STUDENT: Such that the\Ndistance is less than--
Dialogue: 0,0:32:50.17,0:32:52.26,Default,,0000,0000,0000,,MAGDALENA TODA: So again,\Nfor epsilon positive,
Dialogue: 0,0:32:52.26,0:32:56.60,Default,,0000,0000,0000,,there is a delta\Npositive, very small.
Dialogue: 0,0:32:56.60,0:32:58.74,Default,,0000,0000,0000,,Very small means very small, OK?
Dialogue: 0,0:32:58.74,0:33:01.44,Default,,0000,0000,0000,,I'm not threatened by-- what?
Dialogue: 0,0:33:01.44,0:33:04.69,Default,,0000,0000,0000,,For epsilon greater\Nthan 0, very small,
Dialogue: 0,0:33:04.69,0:33:07.13,Default,,0000,0000,0000,,there is a delta greater\Nthan 0, very small,
Dialogue: 0,0:33:07.13,0:33:10.54,Default,,0000,0000,0000,,which depends on epsilon-- I\Ndidn't say it cannot be equal
Dialogue: 0,0:33:10.54,0:33:21.10,Default,,0000,0000,0000,,to epsilon-- that depends on\Nepsilon such that whenever x,
Dialogue: 0,0:33:21.10,0:33:30.01,Default,,0000,0000,0000,,y is within delta\Ndistance from origin,
Dialogue: 0,0:33:30.01,0:33:45.17,Default,,0000,0000,0000,,[INAUDIBLE] that f of x, y\Nis within epsilon of from l.
Dialogue: 0,0:33:45.17,0:33:47.84,Default,,0000,0000,0000,,
Dialogue: 0,0:33:47.84,0:33:48.34,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:33:48.34,0:33:52.72,Default,,0000,0000,0000,,And now I will actually give\Nyou another example where
Dialogue: 0,0:33:52.72,0:33:55.97,Default,,0000,0000,0000,,maybe delta will be epsilon.
Dialogue: 0,0:33:55.97,0:33:59.43,Default,,0000,0000,0000,,And let me challenge you\Nwith another problem that's
Dialogue: 0,0:33:59.43,0:34:00.53,Default,,0000,0000,0000,,not hard.
Dialogue: 0,0:34:00.53,0:34:01.30,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:34:01.30,0:34:03.65,Default,,0000,0000,0000,,So let me give\Nyou the function g
Dialogue: 0,0:34:03.65,0:34:16.46,Default,,0000,0000,0000,,of x, y equals x sine\Nof 1 over y as x, y.
Dialogue: 0,0:34:16.46,0:34:19.15,Default,,0000,0000,0000,,
Dialogue: 0,0:34:19.15,0:34:29.78,Default,,0000,0000,0000,,y is equal [? to delta 0. ?]\NAnd let's say 0 for the rest.
Dialogue: 0,0:34:29.78,0:34:35.50,Default,,0000,0000,0000,,
Dialogue: 0,0:34:35.50,0:34:48.51,Default,,0000,0000,0000,,Can you show-- can you check\Nif g is continuous at 0, 0?
Dialogue: 0,0:34:48.51,0:34:55.51,Default,,0000,0000,0000,,
Dialogue: 0,0:34:55.51,0:34:58.51,Default,,0000,0000,0000,,This is one of the\Nproblems in your book.
Dialogue: 0,0:34:58.51,0:35:02.30,Default,,0000,0000,0000,,So how do you check\Nthat with epsilon delta?
Dialogue: 0,0:35:02.30,0:35:04.13,Default,,0000,0000,0000,,Again, we recite the poetry.
Dialogue: 0,0:35:04.13,0:35:05.56,Default,,0000,0000,0000,,We have to say that.
Dialogue: 0,0:35:05.56,0:35:11.95,Default,,0000,0000,0000,,"For every epsilon\Npositive, small, very small,
Dialogue: 0,0:35:11.95,0:35:16.00,Default,,0000,0000,0000,,there is a delta\Npositive that depends
Dialogue: 0,0:35:16.00,0:35:33.83,Default,,0000,0000,0000,,on epsilon, such that as soon\Nas--" how is the distance?
Dialogue: 0,0:35:33.83,0:35:42.42,Default,,0000,0000,0000,,Square root of x squared plus\Ny squared is less than delta.
Dialogue: 0,0:35:42.42,0:35:46.86,Default,,0000,0000,0000,,This is the distance\Nbetween point and origin.
Dialogue: 0,0:35:46.86,0:36:09.26,Default,,0000,0000,0000,,"It follows that absolute value\Nof x sine of 1 over y minus--"
Dialogue: 0,0:36:09.26,0:36:12.25,Default,,0000,0000,0000,,so practically x, y no 0.
Dialogue: 0,0:36:12.25,0:36:16.22,Default,,0000,0000,0000,,x, y different from 0.
Dialogue: 0,0:36:16.22,0:36:17.72,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:36:17.72,0:36:21.72,Default,,0000,0000,0000,,I"m careful here, because\Nif y is 0, then I blow up.
Dialogue: 0,0:36:21.72,0:36:23.08,Default,,0000,0000,0000,,And I don't want to blow up.
Dialogue: 0,0:36:23.08,0:36:25.85,Default,,0000,0000,0000,,So x sine of 1 over y minus who?
Dialogue: 0,0:36:25.85,0:36:30.81,Default,,0000,0000,0000,,Minus 0 is less than epsilon.
Dialogue: 0,0:36:30.81,0:36:32.85,Default,,0000,0000,0000,,So now you're thinking,\NOK, you want me
Dialogue: 0,0:36:32.85,0:36:34.76,Default,,0000,0000,0000,,to prove there is such a delta?
Dialogue: 0,0:36:34.76,0:36:35.83,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:36:35.83,0:36:37.06,Default,,0000,0000,0000,,That depends on epsilon?
Dialogue: 0,0:36:37.06,0:36:38.85,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:36:38.85,0:36:40.43,Default,,0000,0000,0000,,And what would that delta be?
Dialogue: 0,0:36:40.43,0:36:43.71,Default,,0000,0000,0000,,The simplest choice you\Ncan have in this case.
Dialogue: 0,0:36:43.71,0:36:45.08,Default,,0000,0000,0000,,So you go, oh, my God.
Dialogue: 0,0:36:45.08,0:36:46.07,Default,,0000,0000,0000,,How do I do that?
Dialogue: 0,0:36:46.07,0:36:48.47,Default,,0000,0000,0000,,You have to always\Nthink backwards.
Dialogue: 0,0:36:48.47,0:36:58.79,Default,,0000,0000,0000,,So "we need to satisfy\Nabsolute value of x sine of 1
Dialogue: 0,0:36:58.79,0:37:02.43,Default,,0000,0000,0000,,over y less than epsilon."
Dialogue: 0,0:37:02.43,0:37:05.73,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:37:05.73,0:37:10.30,Default,,0000,0000,0000,,What is your advantage here?
Dialogue: 0,0:37:10.30,0:37:13.57,Default,,0000,0000,0000,,Do you have any advantage?
Dialogue: 0,0:37:13.57,0:37:19.86,Default,,0000,0000,0000,,Remark absolute value\Nof x sine of 1 over y
Dialogue: 0,0:37:19.86,0:37:22.69,Default,,0000,0000,0000,,is smaller than who?
Dialogue: 0,0:37:22.69,0:37:26.67,Default,,0000,0000,0000,,Smaller than the product\Nof absolute values.
Dialogue: 0,0:37:26.67,0:37:27.58,Default,,0000,0000,0000,,Say it again?
Dialogue: 0,0:37:27.58,0:37:28.43,Default,,0000,0000,0000,,Yes?
Dialogue: 0,0:37:28.43,0:37:32.00,Default,,0000,0000,0000,,STUDENT: But, like, for\Nexample, the only condition
Dialogue: 0,0:37:32.00,0:37:35.29,Default,,0000,0000,0000,,for that equation is that\Ny must not be equal to 0.
Dialogue: 0,0:37:35.29,0:37:38.52,Default,,0000,0000,0000,,What if you used\Nanother point for x?
Dialogue: 0,0:37:38.52,0:37:43.17,Default,,0000,0000,0000,,Would the answer for\Ndelta be different?
Dialogue: 0,0:37:43.17,0:37:45.24,Default,,0000,0000,0000,,MAGDALENA TODA:\NWell, x is-- you can
Dialogue: 0,0:37:45.24,0:37:49.00,Default,,0000,0000,0000,,choose-- you were right here.
Dialogue: 0,0:37:49.00,0:37:52.99,Default,,0000,0000,0000,,You can say, OK, can you be\Nmore restrictive, Magdelena,
Dialogue: 0,0:37:52.99,0:37:58.71,Default,,0000,0000,0000,,and say, for every point\Nof the type x equals 0
Dialogue: 0,0:37:58.71,0:38:01.47,Default,,0000,0000,0000,,and y not 0, it's still OK?
Dialogue: 0,0:38:01.47,0:38:03.49,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:38:03.49,0:38:07.00,Default,,0000,0000,0000,,So you could be a\Nprofessional mathematician.
Dialogue: 0,0:38:07.00,0:38:14.32,Default,,0000,0000,0000,,So practically all I care\Nabout is x, y in the disk.
Dialogue: 0,0:38:14.32,0:38:15.41,Default,,0000,0000,0000,,What disk?
Dialogue: 0,0:38:15.41,0:38:16.81,Default,,0000,0000,0000,,What is this disk?
Dialogue: 0,0:38:16.81,0:38:24.15,Default,,0000,0000,0000,,Disk of radius 0 when--\Nwhat is the radius?
Dialogue: 0,0:38:24.15,0:38:31.93,Default,,0000,0000,0000,,Delta-- such that your\Ny should not be 0.
Dialogue: 0,0:38:31.93,0:38:35.99,Default,,0000,0000,0000,,So a more rigorous\Npoint would be
Dialogue: 0,0:38:35.99,0:38:38.63,Default,,0000,0000,0000,,like take all the\Ncouples that are
Dialogue: 0,0:38:38.63,0:38:43.60,Default,,0000,0000,0000,,in this small disk\Nof radius delta,
Dialogue: 0,0:38:43.60,0:38:46.02,Default,,0000,0000,0000,,except for those where y is 0.
Dialogue: 0,0:38:46.02,0:38:48.65,Default,,0000,0000,0000,,So what do you actually remove?
Dialogue: 0,0:38:48.65,0:38:54.86,Default,,0000,0000,0000,,You remove this stinking line.
Dialogue: 0,0:38:54.86,0:39:01.08,Default,,0000,0000,0000,,But everybody else in this\Ndisk, every couple in this disk
Dialogue: 0,0:39:01.08,0:39:03.86,Default,,0000,0000,0000,,should be happy,\Nshould be analyzed
Dialogue: 0,0:39:03.86,0:39:06.08,Default,,0000,0000,0000,,as part of this thread.
Dialogue: 0,0:39:06.08,0:39:08.10,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:39:08.10,0:39:09.07,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:39:09.07,0:39:13.18,Default,,0000,0000,0000,,x sine of 1 over y less\Nthan-- is that true?
Dialogue: 0,0:39:13.18,0:39:16.12,Default,,0000,0000,0000,,Is that less than the\Nabsolute value of x?
Dialogue: 0,0:39:16.12,0:39:16.93,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:39:16.93,0:39:17.85,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,0:39:17.85,0:39:20.95,Default,,0000,0000,0000,,So it should be-- less\Nthan should be made
Dialogue: 0,0:39:20.95,0:39:23.57,Default,,0000,0000,0000,,should be less than epsilon.
Dialogue: 0,0:39:23.57,0:39:27.04,Default,,0000,0000,0000,,When is this happening\Non that occasion?
Dialogue: 0,0:39:27.04,0:39:28.32,Default,,0000,0000,0000,,If I take delta-- meh?
Dialogue: 0,0:39:28.32,0:39:29.57,Default,,0000,0000,0000,,STUDENT: When delta's epsilon.
Dialogue: 0,0:39:29.57,0:39:31.31,Default,,0000,0000,0000,,MAGDALENA TODA: So if\NI take-- very good.
Dialogue: 0,0:39:31.31,0:39:35.57,Default,,0000,0000,0000,,So Alex saw that, hey,\NMagdelena, your proof is over.
Dialogue: 0,0:39:35.57,0:39:37.70,Default,,0000,0000,0000,,And I mean it's over.
Dialogue: 0,0:39:37.70,0:39:42.99,Default,,0000,0000,0000,,Take delta, which is delta\Nof epsilon, to be epsilon.
Dialogue: 0,0:39:42.99,0:39:44.35,Default,,0000,0000,0000,,You're done.
Dialogue: 0,0:39:44.35,0:39:45.52,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:39:45.52,0:39:47.59,Default,,0000,0000,0000,,Let me explain what\NAlex wants, because he
Dialogue: 0,0:39:47.59,0:39:50.28,Default,,0000,0000,0000,,doesn't want to explain\Nmuch, but it's not his job.
Dialogue: 0,0:39:50.28,0:39:51.26,Default,,0000,0000,0000,,He's not your teacher.
Dialogue: 0,0:39:51.26,0:39:51.92,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:39:51.92,0:39:54.35,Default,,0000,0000,0000,,So why is this working?
Dialogue: 0,0:39:54.35,0:40:02.85,Default,,0000,0000,0000,,Because in this case,\Nnote that if I take delta
Dialogue: 0,0:40:02.85,0:40:05.65,Default,,0000,0000,0000,,to be exactly epsilon,\Nwhat's going to happen?
Dialogue: 0,0:40:05.65,0:40:08.62,Default,,0000,0000,0000,,
Dialogue: 0,0:40:08.62,0:40:13.76,Default,,0000,0000,0000,,x, Mr. x, could be\Npositive or negative.
Dialogue: 0,0:40:13.76,0:40:15.95,Default,,0000,0000,0000,,See, x could be\Npositive or negative.
Dialogue: 0,0:40:15.95,0:40:18.81,Default,,0000,0000,0000,,Let's take this guy and\Nprotect him in absolute value.
Dialogue: 0,0:40:18.81,0:40:23.35,Default,,0000,0000,0000,,He's always less than square\Nroot of x square plus y
Dialogue: 0,0:40:23.35,0:40:25.65,Default,,0000,0000,0000,,squared.
Dialogue: 0,0:40:25.65,0:40:27.20,Default,,0000,0000,0000,,Why is that, guys?
Dialogue: 0,0:40:27.20,0:40:30.73,Default,,0000,0000,0000,,STUDENT: Because y can't be 0.
Dialogue: 0,0:40:30.73,0:40:34.48,Default,,0000,0000,0000,,MAGDALENA TODA: So this\Nis-- square it in your mind.
Dialogue: 0,0:40:34.48,0:40:36.69,Default,,0000,0000,0000,,You got x squared less than\Nx squared plus y squared.
Dialogue: 0,0:40:36.69,0:40:39.00,Default,,0000,0000,0000,,So this is always true.
Dialogue: 0,0:40:39.00,0:40:40.64,Default,,0000,0000,0000,,Always satisfied.
Dialogue: 0,0:40:40.64,0:40:44.94,Default,,0000,0000,0000,,But we chose this to\Nbe less than delta,
Dialogue: 0,0:40:44.94,0:40:49.45,Default,,0000,0000,0000,,and if we choose delta to be\Nepsilon, that's our choice.
Dialogue: 0,0:40:49.45,0:40:54.31,Default,,0000,0000,0000,,So God gave us the epsilon,\Nbut delta is our choice,
Dialogue: 0,0:40:54.31,0:40:57.09,Default,,0000,0000,0000,,because you have to prove\Nyou can do something
Dialogue: 0,0:40:57.09,0:40:57.91,Default,,0000,0000,0000,,with your life.
Dialogue: 0,0:40:57.91,0:40:58.41,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:40:58.41,0:41:00.70,Default,,0000,0000,0000,,So delta equals epsilon.
Dialogue: 0,0:41:00.70,0:41:02.67,Default,,0000,0000,0000,,If you take delta\Nequals epsilon,
Dialogue: 0,0:41:02.67,0:41:06.44,Default,,0000,0000,0000,,then you're done, because\Nin that case absolute value
Dialogue: 0,0:41:06.44,0:41:11.98,Default,,0000,0000,0000,,of x is less than epsilon, and\Nyour conclusion, which is this,
Dialogue: 0,0:41:11.98,0:41:13.69,Default,,0000,0000,0000,,was satisfied.
Dialogue: 0,0:41:13.69,0:41:16.68,Default,,0000,0000,0000,,Now, if a student\Nis really smart--
Dialogue: 0,0:41:16.68,0:41:20.65,Default,,0000,0000,0000,,one time I had a student,\NI gave him this proof.
Dialogue: 0,0:41:20.65,0:41:22.38,Default,,0000,0000,0000,,That was several\Nyears ago in honors,
Dialogue: 0,0:41:22.38,0:41:24.62,Default,,0000,0000,0000,,because we don't do epsilon\Ndelta in non-honors.
Dialogue: 0,0:41:24.62,0:41:28.20,Default,,0000,0000,0000,,And we very rarely do\Nit in honors as well.
Dialogue: 0,0:41:28.20,0:41:31.14,Default,,0000,0000,0000,,His proof consisted of this.
Dialogue: 0,0:41:31.14,0:41:34.06,Default,,0000,0000,0000,,Considering the fact that\Nabsolute value of sine
Dialogue: 0,0:41:34.06,0:41:38.30,Default,,0000,0000,0000,,is less than 1, if I\Ntake delta to be epsilon,
Dialogue: 0,0:41:38.30,0:41:39.87,Default,,0000,0000,0000,,that is sufficient.
Dialogue: 0,0:41:39.87,0:41:41.78,Default,,0000,0000,0000,,I'm done.
Dialogue: 0,0:41:41.78,0:41:44.47,Default,,0000,0000,0000,,And of course I gave\Nhim 100%, because this
Dialogue: 0,0:41:44.47,0:41:46.03,Default,,0000,0000,0000,,is the essence of the proof.
Dialogue: 0,0:41:46.03,0:41:48.09,Default,,0000,0000,0000,,He didn't show any details.
Dialogue: 0,0:41:48.09,0:41:52.18,Default,,0000,0000,0000,,And I thought, this is the\Nkind of guy who is great.
Dialogue: 0,0:41:52.18,0:41:55.92,Default,,0000,0000,0000,,He's very smart, but he's not\Ngoing to make a good teacher.
Dialogue: 0,0:41:55.92,0:41:59.38,Default,,0000,0000,0000,,So he's probably going to\Nbe the next researcher,
Dialogue: 0,0:41:59.38,0:42:04.81,Default,,0000,0000,0000,,the next astronaut, the next\Nsomething else, but not--
Dialogue: 0,0:42:04.81,0:42:11.04,Default,,0000,0000,0000,,And then, years later, he\Ntook advanced calculus.
Dialogue: 0,0:42:11.04,0:42:13.64,Default,,0000,0000,0000,,He graduated with\Na graduate degree
Dialogue: 0,0:42:13.64,0:42:17.60,Default,,0000,0000,0000,,in three years sponsored\Nby the Air Force.
Dialogue: 0,0:42:17.60,0:42:20.75,Default,,0000,0000,0000,,And he works right\Nnow for the Air Force.
Dialogue: 0,0:42:20.75,0:42:24.42,Default,,0000,0000,0000,,He came out dressed\Nas a captain.
Dialogue: 0,0:42:24.42,0:42:28.84,Default,,0000,0000,0000,,He came and gave a talk this\Nyear at Tech in a conference--
Dialogue: 0,0:42:28.84,0:42:29.77,Default,,0000,0000,0000,,he was rushed.
Dialogue: 0,0:42:29.77,0:42:32.15,Default,,0000,0000,0000,,I mean, if I talk\Nlike that, my student
Dialogue: 0,0:42:32.15,0:42:33.81,Default,,0000,0000,0000,,wouldn't be able to follow me.
Dialogue: 0,0:42:33.81,0:42:38.32,Default,,0000,0000,0000,,But he was the same brilliant\Nstudent that I remember.
Dialogue: 0,0:42:38.32,0:42:46.28,Default,,0000,0000,0000,,So he's working on some very\Nimportant top secret projects.
Dialogue: 0,0:42:46.28,0:42:48.90,Default,,0000,0000,0000,,Very intelligent guy.
Dialogue: 0,0:42:48.90,0:42:52.52,Default,,0000,0000,0000,,And every now and than going\Nto give talks at conferences.
Dialogue: 0,0:42:52.52,0:42:58.17,Default,,0000,0000,0000,,Like, research talks\Nabout what he's doing.
Dialogue: 0,0:42:58.17,0:43:01.96,Default,,0000,0000,0000,,In his class-- he took\Nadvanced calculus with me,
Dialogue: 0,0:43:01.96,0:43:04.09,Default,,0000,0000,0000,,which was actually graduate\Nlevel [INAUDIBLE]--
Dialogue: 0,0:43:04.09,0:43:09.18,Default,,0000,0000,0000,,I explained epsilon delta, and\Nhe had it very well understood.
Dialogue: 0,0:43:09.18,0:43:13.27,Default,,0000,0000,0000,,And after I left the classroom\Nhe explained it to his peers,
Dialogue: 0,0:43:13.27,0:43:15.05,Default,,0000,0000,0000,,to his classmates.
Dialogue: 0,0:43:15.05,0:43:16.88,Default,,0000,0000,0000,,And he explained\Nit better than me.
Dialogue: 0,0:43:16.88,0:43:21.01,Default,,0000,0000,0000,,And I was there listening,\Nand I remember being jealous,
Dialogue: 0,0:43:21.01,0:43:23.05,Default,,0000,0000,0000,,because although\Nhe was very rushed,
Dialogue: 0,0:43:23.05,0:43:27.43,Default,,0000,0000,0000,,he had a very clear\Nunderstanding of how
Dialogue: 0,0:43:27.43,0:43:31.14,Default,,0000,0000,0000,,you take an epsilon, no\Nmatter how small, and then
Dialogue: 0,0:43:31.14,0:43:34.20,Default,,0000,0000,0000,,you take a little ball\Nhere, radius delta.
Dialogue: 0,0:43:34.20,0:43:38.68,Default,,0000,0000,0000,,So the image of that little\Nball will fit in that ball
Dialogue: 0,0:43:38.68,0:43:40.11,Default,,0000,0000,0000,,that you take here.
Dialogue: 0,0:43:40.11,0:43:43.51,Default,,0000,0000,0000,,So even if you\Nshrink on the image,
Dialogue: 0,0:43:43.51,0:43:46.02,Default,,0000,0000,0000,,you can take this\Nball even smaller
Dialogue: 0,0:43:46.02,0:43:48.67,Default,,0000,0000,0000,,so the image will\Nstill fit inside.
Dialogue: 0,0:43:48.67,0:43:51.49,Default,,0000,0000,0000,,And I was going, gosh,\Nthis is the essence,
Dialogue: 0,0:43:51.49,0:43:54.66,Default,,0000,0000,0000,,but I wish I could convey\Nit, because no book
Dialogue: 0,0:43:54.66,0:43:58.55,Default,,0000,0000,0000,,will say it just-- or show you\Nhow to do it with your hands.
Dialogue: 0,0:43:58.55,0:43:59.20,Default,,0000,0000,0000,,
Dialogue: 0,0:43:59.20,0:44:00.04,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:44:00.04,0:44:00.95,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,0:44:00.95,0:44:04.98,Default,,0000,0000,0000,,So he was rushed, but he\Nhad a very clear picture
Dialogue: 0,0:44:04.98,0:44:06.96,Default,,0000,0000,0000,,of what is going on.
Dialogue: 0,0:44:06.96,0:44:07.96,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:44:07.96,0:44:10.92,Default,,0000,0000,0000,,11.3 is a completely new start.
Dialogue: 0,0:44:10.92,0:44:13.90,Default,,0000,0000,0000,,And you are gonna read\Nand be happy about that
Dialogue: 0,0:44:13.90,0:44:16.37,Default,,0000,0000,0000,,because that's\Npartial derivatives.
Dialogue: 0,0:44:16.37,0:44:20.36,Default,,0000,0000,0000,,And you say, Magdalena,\Nfinally, this is piece of cake.
Dialogue: 0,0:44:20.36,0:44:22.70,Default,,0000,0000,0000,,You see, I know these things.
Dialogue: 0,0:44:22.70,0:44:25.76,Default,,0000,0000,0000,,I can do them in\Nmy-- in my sleep.
Dialogue: 0,0:44:25.76,0:44:29.96,Default,,0000,0000,0000,,So f of x and y\Nis still a graph.
Dialogue: 0,0:44:29.96,0:44:33.12,Default,,0000,0000,0000,,And then you say,\Nhow do we introduce
Dialogue: 0,0:44:33.12,0:44:37.74,Default,,0000,0000,0000,,the partial derivative with\Nrespect to one variable only.
Dialogue: 0,0:44:37.74,0:44:39.77,Default,,0000,0000,0000,,You think, I draw the graph.
Dialogue: 0,0:44:39.77,0:44:41.64,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:44:41.64,0:44:44.75,Default,,0000,0000,0000,,On this graph, I\Npick a point x0, y0.
Dialogue: 0,0:44:44.75,0:44:53.95,Default,,0000,0000,0000,,And if I were to take x to\Nbe 0, what is-- what is the z
Dialogue: 0,0:44:53.95,0:44:56.16,Default,,0000,0000,0000,,equals f of x0, y?
Dialogue: 0,0:44:56.16,0:45:02.94,Default,,0000,0000,0000,,
Dialogue: 0,0:45:02.94,0:45:04.12,Default,,0000,0000,0000,,So I'll try to draw it.
Dialogue: 0,0:45:04.12,0:45:05.12,Default,,0000,0000,0000,,It's not easy.
Dialogue: 0,0:45:05.12,0:45:10.58,Default,,0000,0000,0000,,
Dialogue: 0,0:45:10.58,0:45:16.41,Default,,0000,0000,0000,,This is x and y and z, and you\Nwant your x0 to be a constant.
Dialogue: 0,0:45:16.41,0:45:17.46,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,0:45:17.46,0:45:19.54,Default,,0000,0000,0000,,MAGDALENA TODA: It's a\Nso-called coordinate curve.
Dialogue: 0,0:45:19.54,0:45:20.13,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:45:20.13,0:45:23.08,Default,,0000,0000,0000,,It's a curve, but I want to\Nbe good enough to draw it.
Dialogue: 0,0:45:23.08,0:45:25.14,Default,,0000,0000,0000,,So you guys have\Nto wish me luck,
Dialogue: 0,0:45:25.14,0:45:28.30,Default,,0000,0000,0000,,because I don't-- didn't have\Nenough coffee and I don't feel
Dialogue: 0,0:45:28.30,0:45:30.06,Default,,0000,0000,0000,,like I can draw very well.
Dialogue: 0,0:45:30.06,0:45:33.53,Default,,0000,0000,0000,,x0 is here.
Dialogue: 0,0:45:33.53,0:45:39.58,Default,,0000,0000,0000,,So x is there, so you\Ncut with this board-- are
Dialogue: 0,0:45:39.58,0:45:40.88,Default,,0000,0000,0000,,you guys with me?
Dialogue: 0,0:45:40.88,0:45:44.12,Default,,0000,0000,0000,,You cut with this board\Nat the level x0 over here.
Dialogue: 0,0:45:44.12,0:45:45.92,Default,,0000,0000,0000,,You cut.
Dialogue: 0,0:45:45.92,0:45:49.29,Default,,0000,0000,0000,,When you cut with\Nthis board-- you
Dialogue: 0,0:45:49.29,0:45:52.56,Default,,0000,0000,0000,,cut your surface\Nwith this board--
Dialogue: 0,0:45:52.56,0:45:54.88,Default,,0000,0000,0000,,you get a curve like that.
Dialogue: 0,0:45:54.88,0:46:00.35,Default,,0000,0000,0000,,And we call that a\Ncurve f of x0, y.
Dialogue: 0,0:46:00.35,0:46:05.84,Default,,0000,0000,0000,,Some people who are a little bit\Nin a hurry and smarter than me,
Dialogue: 0,0:46:05.84,0:46:07.78,Default,,0000,0000,0000,,they say x equals x0.
Dialogue: 0,0:46:07.78,0:46:09.73,Default,,0000,0000,0000,,That's called coordinate curve.
Dialogue: 0,0:46:09.73,0:46:16.55,Default,,0000,0000,0000,,
Dialogue: 0,0:46:16.55,0:46:20.27,Default,,0000,0000,0000,,So, the thing is, this--\Nit's a curve in plane.
Dialogue: 0,0:46:20.27,0:46:21.32,Default,,0000,0000,0000,,This is the blue plane.
Dialogue: 0,0:46:21.32,0:46:22.49,Default,,0000,0000,0000,,I don't know how to call it.
Dialogue: 0,0:46:22.49,0:46:23.42,Default,,0000,0000,0000,,Pi.
Dialogue: 0,0:46:23.42,0:46:25.84,Default,,0000,0000,0000,,You know I love to call it pi.
Dialogue: 0,0:46:25.84,0:46:28.18,Default,,0000,0000,0000,,Since I'm in plane with\Na point in a curve--
Dialogue: 0,0:46:28.18,0:46:33.57,Default,,0000,0000,0000,,a plane curve-- this curve\Nhas a slope at x0, y0.
Dialogue: 0,0:46:33.57,0:46:35.18,Default,,0000,0000,0000,,Can I draw that slope?
Dialogue: 0,0:46:35.18,0:46:36.73,Default,,0000,0000,0000,,I'll try.
Dialogue: 0,0:46:36.73,0:46:38.54,Default,,0000,0000,0000,,The slope of the\Nblue line, though.
Dialogue: 0,0:46:38.54,0:46:39.98,Default,,0000,0000,0000,,Let me make it red.
Dialogue: 0,0:46:39.98,0:46:43.95,Default,,0000,0000,0000,,The slope of the red line--\Nnow, if you don't have colors
Dialogue: 0,0:46:43.95,0:46:47.21,Default,,0000,0000,0000,,you can make it a dotted line.
Dialogue: 0,0:46:47.21,0:46:57.81,Default,,0000,0000,0000,,The slope of the dotted line\Nis-- who the heck is that?
Dialogue: 0,0:46:57.81,0:47:07.14,Default,,0000,0000,0000,,The derivative of f with respect\Nto y, because x0 is a constant.
Dialogue: 0,0:47:07.14,0:47:09.49,Default,,0000,0000,0000,,So how do we write that?
Dialogue: 0,0:47:09.49,0:47:12.80,Default,,0000,0000,0000,,Because x0 is sort of in\Nour way, driving us crazy.
Dialogue: 0,0:47:12.80,0:47:15.00,Default,,0000,0000,0000,,Although he was fixed.
Dialogue: 0,0:47:15.00,0:47:18.41,Default,,0000,0000,0000,,We keep him fixed by\Nkeeping him in this plane.
Dialogue: 0,0:47:18.41,0:47:20.13,Default,,0000,0000,0000,,x0 is fixed.
Dialogue: 0,0:47:20.13,0:47:21.98,Default,,0000,0000,0000,,We have to write\Nanother notation.
Dialogue: 0,0:47:21.98,0:47:24.21,Default,,0000,0000,0000,,We cannot say f prime.
Dialogue: 0,0:47:24.21,0:47:27.49,Default,,0000,0000,0000,,Because f depends\Non two variables.
Dialogue: 0,0:47:27.49,0:47:31.60,Default,,0000,0000,0000,,f prime were for when we\Nwere babies in calculus 1.
Dialogue: 0,0:47:31.60,0:47:32.93,Default,,0000,0000,0000,,We cannot use f prime anymore.
Dialogue: 0,0:47:32.93,0:47:33.85,Default,,0000,0000,0000,,We have two variables.
Dialogue: 0,0:47:33.85,0:47:36.31,Default,,0000,0000,0000,,Life became too complicated.
Dialogue: 0,0:47:36.31,0:47:37.39,Default,,0000,0000,0000,,So we have to say--
Dialogue: 0,0:47:37.39,0:47:38.18,Default,,0000,0000,0000,,STUDENT: Professor?
Dialogue: 0,0:47:38.18,0:47:40.70,Default,,0000,0000,0000,,MAGDALENA TODA: --instead\Nof df dy-- yes, sir.
Dialogue: 0,0:47:40.70,0:47:41.70,Default,,0000,0000,0000,,May you use a subscript?
Dialogue: 0,0:47:41.70,0:47:45.34,Default,,0000,0000,0000,,MAGDALENA TODA: You use--\Nyeah, you can do that as well.
Dialogue: 0,0:47:45.34,0:47:47.30,Default,,0000,0000,0000,,That's what I do.
Dialogue: 0,0:47:47.30,0:47:49.07,Default,,0000,0000,0000,,Let me do both.
Dialogue: 0,0:47:49.07,0:47:55.75,Default,,0000,0000,0000,,f sub y at-- who\Nwas fixed? x0 and y.
Dialogue: 0,0:47:55.75,0:47:58.75,Default,,0000,0000,0000,,But this is my\Nfavorite notation.
Dialogue: 0,0:47:58.75,0:48:01.19,Default,,0000,0000,0000,,I'm going to make a\Nface because I love it.
Dialogue: 0,0:48:01.19,0:48:02.76,Default,,0000,0000,0000,,This is what engineers love.
Dialogue: 0,0:48:02.76,0:48:04.82,Default,,0000,0000,0000,,This is what we physicists love.
Dialogue: 0,0:48:04.82,0:48:07.32,Default,,0000,0000,0000,,Mathematicians, though,\Nare crazy people.
Dialogue: 0,0:48:07.32,0:48:08.24,Default,,0000,0000,0000,,They are.
Dialogue: 0,0:48:08.24,0:48:09.33,Default,,0000,0000,0000,,All of them.
Dialogue: 0,0:48:09.33,0:48:12.93,Default,,0000,0000,0000,,And they invented\Nanother notation.
Dialogue: 0,0:48:12.93,0:48:15.06,Default,,0000,0000,0000,,Do you remember\Nthat Mr. Leibniz,
Dialogue: 0,0:48:15.06,0:48:19.03,Default,,0000,0000,0000,,because he had nothing better to\Ndo, when he invented calculus,
Dialogue: 0,0:48:19.03,0:48:23.25,Default,,0000,0000,0000,,he did df dy, or df dx?
Dialogue: 0,0:48:23.25,0:48:24.25,Default,,0000,0000,0000,,What is that?
Dialogue: 0,0:48:24.25,0:48:27.14,Default,,0000,0000,0000,,That was the limit of\Ndelta f, delta y, right?
Dialogue: 0,0:48:27.14,0:48:28.55,Default,,0000,0000,0000,,That's what Leibniz did.
Dialogue: 0,0:48:28.55,0:48:30.84,Default,,0000,0000,0000,,He introduced this\Ndelta notation,
Dialogue: 0,0:48:30.84,0:48:34.51,Default,,0000,0000,0000,,and then he said if you have\Ndelta space over delta time,
Dialogue: 0,0:48:34.51,0:48:38.08,Default,,0000,0000,0000,,then shrink both, and you\Nmake a ratio in the limit,
Dialogue: 0,0:48:38.08,0:48:40.62,Default,,0000,0000,0000,,you should read-- you\Nshould write it df dy.
Dialogue: 0,0:48:40.62,0:48:44.11,Default,,0000,0000,0000,,And that's the so-called\NLeibniz notation, right?
Dialogue: 0,0:48:44.11,0:48:46.76,Default,,0000,0000,0000,,That was in calc 1.
Dialogue: 0,0:48:46.76,0:48:49.63,Default,,0000,0000,0000,,But I erased it because\Nthat was calc 1.
Dialogue: 0,0:48:49.63,0:48:54.18,Default,,0000,0000,0000,,Now, mathematicians, to\Nimitate the Leibniz notation,
Dialogue: 0,0:48:54.18,0:48:57.87,Default,,0000,0000,0000,,they said, I cannot use df dy.
Dialogue: 0,0:48:57.87,0:49:00.61,Default,,0000,0000,0000,,So what the heck shall I use?
Dialogue: 0,0:49:00.61,0:49:02.65,Default,,0000,0000,0000,,After they thought\Nfor about a year,
Dialogue: 0,0:49:02.65,0:49:05.27,Default,,0000,0000,0000,,and I was reading through\Nthe history about how
Dialogue: 0,0:49:05.27,0:49:07.14,Default,,0000,0000,0000,,they invented this,\Nthey said, let's take
Dialogue: 0,0:49:07.14,0:49:09.86,Default,,0000,0000,0000,,the Greek-- the Greek d.
Dialogue: 0,0:49:09.86,0:49:12.14,Default,,0000,0000,0000,,Which is the del.
Dialogue: 0,0:49:12.14,0:49:13.99,Default,,0000,0000,0000,,That's partial.
Dialogue: 0,0:49:13.99,0:49:19.40,Default,,0000,0000,0000,,The del f, del y, at x0, y.
Dialogue: 0,0:49:19.40,0:49:22.40,Default,,0000,0000,0000,,When I was 20--\Nno, I was 18 when
Dialogue: 0,0:49:22.40,0:49:27.10,Default,,0000,0000,0000,,I saw this the first time--\NI had the hardest time making
Dialogue: 0,0:49:27.10,0:49:27.92,Default,,0000,0000,0000,,this sign.
Dialogue: 0,0:49:27.92,0:49:29.59,Default,,0000,0000,0000,,It's all in the wrist.
Dialogue: 0,0:49:29.59,0:49:32.16,Default,,0000,0000,0000,,It's very-- OK.
Dialogue: 0,0:49:32.16,0:49:32.95,Default,,0000,0000,0000,,Now.
Dialogue: 0,0:49:32.95,0:49:33.59,Default,,0000,0000,0000,,df dy.
Dialogue: 0,0:49:33.59,0:49:35.90,Default,,0000,0000,0000,,If you don't like it,\Nthen what do you do?
Dialogue: 0,0:49:35.90,0:49:38.86,Default,,0000,0000,0000,,You can adopt this notation.
Dialogue: 0,0:49:38.86,0:49:41.71,Default,,0000,0000,0000,,And what is the meaning\Nof this by definition?
Dialogue: 0,0:49:41.71,0:49:45.22,Default,,0000,0000,0000,,You say, you haven't even\Ndefined it, Magdalena.
Dialogue: 0,0:49:45.22,0:49:47.98,Default,,0000,0000,0000,,It has to be limit of\Na difference quotient,
Dialogue: 0,0:49:47.98,0:49:49.07,Default,,0000,0000,0000,,just like here.
Dialogue: 0,0:49:49.07,0:49:53.32,Default,,0000,0000,0000,,But we have to be happy\Nand think of that.
Dialogue: 0,0:49:53.32,0:49:57.36,Default,,0000,0000,0000,,What is the delta f\Nversus the delta y?
Dialogue: 0,0:49:57.36,0:49:59.32,Default,,0000,0000,0000,,It has to be like that.
Dialogue: 0,0:49:59.32,0:50:02.71,Default,,0000,0000,0000,,f of Mr. x0 is fixed.
Dialogue: 0,0:50:02.71,0:50:07.06,Default,,0000,0000,0000,,x0, comma, y.
Dialogue: 0,0:50:07.06,0:50:09.85,Default,,0000,0000,0000,,We have an increment in y.
Dialogue: 0,0:50:09.85,0:50:16.31,Default,,0000,0000,0000,,y plus delta y. y plus\Ndelta y minus-- that's
Dialogue: 0,0:50:16.31,0:50:18.22,Default,,0000,0000,0000,,the difference quotient.
Dialogue: 0,0:50:18.22,0:50:22.96,Default,,0000,0000,0000,,f of what-- the original\Npoint was, well--
Dialogue: 0,0:50:22.96,0:50:24.45,Default,,0000,0000,0000,,STUDENT: x0, y0.
Dialogue: 0,0:50:24.45,0:50:26.80,Default,,0000,0000,0000,,MAGDALENA TODA:\Nx0-- let me put y0
Dialogue: 0,0:50:26.80,0:50:29.85,Default,,0000,0000,0000,,because our original\Npoint was x0, y0.
Dialogue: 0,0:50:29.85,0:50:37.81,Default,,0000,0000,0000,,x0, y0 over-- over delta y.
Dialogue: 0,0:50:37.81,0:50:43.38,Default,,0000,0000,0000,,But if I am at x0, y0, I better\Nput x0, y0 fixed point here.
Dialogue: 0,0:50:43.38,0:50:46.80,Default,,0000,0000,0000,,
Dialogue: 0,0:50:46.80,0:50:51.58,Default,,0000,0000,0000,,And I would like you to\Nphotograph or put this thing--
Dialogue: 0,0:50:51.58,0:50:54.52,Default,,0000,0000,0000,,STUDENT: So is that a delta\Nthat's in front of the f?
Dialogue: 0,0:50:54.52,0:50:56.40,Default,,0000,0000,0000,,MAGDALENA TODA: Let me\Nreview the whole thing
Dialogue: 0,0:50:56.40,0:50:58.83,Default,,0000,0000,0000,,because it's very important.
Dialogue: 0,0:50:58.83,0:51:00.81,Default,,0000,0000,0000,,Where shall I start,\Nhere, or here?
Dialogue: 0,0:51:00.81,0:51:01.86,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:51:01.86,0:51:02.96,Default,,0000,0000,0000,,So the limit--
Dialogue: 0,0:51:02.96,0:51:05.33,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] start at m.
Dialogue: 0,0:51:05.33,0:51:06.20,Default,,0000,0000,0000,,MAGDALENA TODA: At m?
Dialogue: 0,0:51:06.20,0:51:06.70,Default,,0000,0000,0000,,At m.
Dialogue: 0,0:51:06.70,0:51:07.93,Default,,0000,0000,0000,,OK, I'll start at m.
Dialogue: 0,0:51:07.93,0:51:13.07,Default,,0000,0000,0000,,The slopes of this line at\Nx0, y0, right at my point,
Dialogue: 0,0:51:13.07,0:51:18.64,Default,,0000,0000,0000,,will be, my favorite\Nnotation is f sub y at x0,
Dialogue: 0,0:51:18.64,0:51:22.03,Default,,0000,0000,0000,,y0, which means partial\Nderivative of f with respect
Dialogue: 0,0:51:22.03,0:51:26.19,Default,,0000,0000,0000,,to y at the point--\Nfixed point x0, y0.
Dialogue: 0,0:51:26.19,0:51:30.67,Default,,0000,0000,0000,,Or, for most mathematicians,\Ndf-- of del-- del f,
Dialogue: 0,0:51:30.67,0:51:34.31,Default,,0000,0000,0000,,del y at x0, y0.
Dialogue: 0,0:51:34.31,0:51:38.88,Default,,0000,0000,0000,,Which is by definition the limit\Nof this difference quotient.
Dialogue: 0,0:51:38.88,0:51:42.27,Default,,0000,0000,0000,,So x0 is held fixed\Nin both cases.
Dialogue: 0,0:51:42.27,0:51:45.48,Default,,0000,0000,0000,,y0 is allowed to\Ndeviate a little bit.
Dialogue: 0,0:51:45.48,0:51:50.40,Default,,0000,0000,0000,,So y0 is fixed, but you\Ndisplace it by a little delta,
Dialogue: 0,0:51:50.40,0:51:53.63,Default,,0000,0000,0000,,or by a little-- how did we\Ndenote that in calc 1, h?
Dialogue: 0,0:51:53.63,0:51:54.37,Default,,0000,0000,0000,,Little h?
Dialogue: 0,0:51:54.37,0:51:54.95,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,0:51:54.95,0:51:56.62,Default,,0000,0000,0000,,MAGDALENA TODA: So\Ndelta y, sometimes it
Dialogue: 0,0:51:56.62,0:51:58.44,Default,,0000,0000,0000,,was called little h.
Dialogue: 0,0:51:58.44,0:52:00.95,Default,,0000,0000,0000,,And this is the\Nsame as little h.
Dialogue: 0,0:52:00.95,0:52:03.84,Default,,0000,0000,0000,,Over that h.
Dialogue: 0,0:52:03.84,0:52:07.37,Default,,0000,0000,0000,,Now you, without my\Nhelp, because you
Dialogue: 0,0:52:07.37,0:52:10.87,Default,,0000,0000,0000,,have all the knowledge\Nand you're smart,
Dialogue: 0,0:52:10.87,0:52:17.43,Default,,0000,0000,0000,,you should tell me how I\Ndefine f sub x at x0, y0,
Dialogue: 0,0:52:17.43,0:52:22.19,Default,,0000,0000,0000,,and shut up, Magdalena,\Nlet people talk.
Dialogue: 0,0:52:22.19,0:52:23.95,Default,,0000,0000,0000,,This is hard.
Dialogue: 0,0:52:23.95,0:52:24.78,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:52:24.78,0:52:26.09,Default,,0000,0000,0000,,MAGDALENA TODA: No.
Dialogue: 0,0:52:26.09,0:52:28.25,Default,,0000,0000,0000,,I hope not.
Dialogue: 0,0:52:28.25,0:52:31.52,Default,,0000,0000,0000,,As a limit of a\Ndifference quotient,
Dialogue: 0,0:52:31.52,0:52:34.57,Default,,0000,0000,0000,,so it's gonna be an\Ninstantaneous rate of change.
Dialogue: 0,0:52:34.57,0:52:37.06,Default,,0000,0000,0000,,That's the limit of a\Ndifference quotient.
Dialogue: 0,0:52:37.06,0:52:38.77,Default,,0000,0000,0000,,Limit of what?
Dialogue: 0,0:52:38.77,0:52:39.27,Default,,0000,0000,0000,,Shut up.
Dialogue: 0,0:52:39.27,0:52:40.53,Default,,0000,0000,0000,,I will zip my lips.
Dialogue: 0,0:52:40.53,0:52:41.38,Default,,0000,0000,0000,,STUDENT: Delta x
Dialogue: 0,0:52:41.38,0:52:42.86,Default,,0000,0000,0000,,MAGDALENA TODA:\NDelta x, excellent.
Dialogue: 0,0:52:42.86,0:52:44.49,Default,,0000,0000,0000,,Delta x going to 0.
Dialogue: 0,0:52:44.49,0:52:47.99,Default,,0000,0000,0000,,So you shrink-- you displace\Nby a small displacement
Dialogue: 0,0:52:47.99,0:52:50.25,Default,,0000,0000,0000,,only in the direction of x.
Dialogue: 0,0:52:50.25,0:52:52.22,Default,,0000,0000,0000,,STUDENT: So f.
Dialogue: 0,0:52:52.22,0:52:53.45,Default,,0000,0000,0000,,MAGDALENA TODA: f.
Dialogue: 0,0:52:53.45,0:52:57.26,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] this\Ntime, x is changing, so--
Dialogue: 0,0:52:57.26,0:52:58.58,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:52:58.58,0:53:04.89,Default,,0000,0000,0000,,MAGDALENA TODA: X0 plus\Ndelta x, y0 is still fixed,
Dialogue: 0,0:53:04.89,0:53:11.18,Default,,0000,0000,0000,,minus f of x0, y0.
Dialogue: 0,0:53:11.18,0:53:13.12,Default,,0000,0000,0000,,Thank God this is always fixed.
Dialogue: 0,0:53:13.12,0:53:14.08,Default,,0000,0000,0000,,I love this guy.
Dialogue: 0,0:53:14.08,0:53:16.50,Default,,0000,0000,0000,,STUDENT: Delta--
Dialogue: 0,0:53:16.50,0:53:19.91,Default,,0000,0000,0000,,MAGDALENA TODA:\NDelta x, which is
Dialogue: 0,0:53:19.91,0:53:23.20,Default,,0000,0000,0000,,like the h we were\Ntalking about.
Dialogue: 0,0:53:23.20,0:53:24.68,Default,,0000,0000,0000,,Now in reality,\Nyou never do that.
Dialogue: 0,0:53:24.68,0:53:28.77,Default,,0000,0000,0000,,You would die if for every\Nexercise, derivation exercise,
Dialogue: 0,0:53:28.77,0:53:31.50,Default,,0000,0000,0000,,you would have to compute a\Nlimit of a difference quotient.
Dialogue: 0,0:53:31.50,0:53:33.05,Default,,0000,0000,0000,,You will go bananas.
Dialogue: 0,0:53:33.05,0:53:34.51,Default,,0000,0000,0000,,What we do?
Dialogue: 0,0:53:34.51,0:53:37.40,Default,,0000,0000,0000,,We do exactly the same thing.
Dialogue: 0,0:53:37.40,0:53:38.71,Default,,0000,0000,0000,,How can I draw?
Dialogue: 0,0:53:38.71,0:53:40.62,Default,,0000,0000,0000,,Can anybody help me draw?
Dialogue: 0,0:53:40.62,0:53:46.28,Default,,0000,0000,0000,,For y0, I would need to take\Nthis other plane through y0.
Dialogue: 0,0:53:46.28,0:53:47.33,Default,,0000,0000,0000,,Where is y0?
Dialogue: 0,0:53:47.33,0:53:48.94,Default,,0000,0000,0000,,Here.
Dialogue: 0,0:53:48.94,0:53:50.20,Default,,0000,0000,0000,,Is my drawing good enough?
Dialogue: 0,0:53:50.20,0:53:51.60,Default,,0000,0000,0000,,I hope so.
Dialogue: 0,0:53:51.60,0:53:56.47,Default,,0000,0000,0000,,So it's something like\NI have this plane with,
Dialogue: 0,0:53:56.47,0:53:57.92,Default,,0000,0000,0000,,oh, do you see that, guys?
Dialogue: 0,0:53:57.92,0:53:58.57,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:53:58.57,0:54:02.84,Default,,0000,0000,0000,,So what is that, the other\Ncurve, coordinate curve, look
Dialogue: 0,0:54:02.84,0:54:03.34,Default,,0000,0000,0000,,like?
Dialogue: 0,0:54:03.34,0:54:06.68,Default,,0000,0000,0000,,
Dialogue: 0,0:54:06.68,0:54:08.70,Default,,0000,0000,0000,,Oh my God.
Dialogue: 0,0:54:08.70,0:54:10.26,Default,,0000,0000,0000,,Looks like that.
Dialogue: 0,0:54:10.26,0:54:13.07,Default,,0000,0000,0000,,Through the same point,\Nand then the slope
Dialogue: 0,0:54:13.07,0:54:18.37,Default,,0000,0000,0000,,of the line will be a\Nblue slope and the slope
Dialogue: 0,0:54:18.37,0:54:23.71,Default,,0000,0000,0000,,will be f sub-- well OK.
Dialogue: 0,0:54:23.71,0:54:28.13,Default,,0000,0000,0000,,So here I have in the red\None, which was the blue one,
Dialogue: 0,0:54:28.13,0:54:33.67,Default,,0000,0000,0000,,this is f sub y, and for\Nthis one, this is f sub x.
Dialogue: 0,0:54:33.67,0:54:34.89,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:54:34.89,0:54:42.24,Default,,0000,0000,0000,,So guys, don't look\Nat the picture.
Dialogue: 0,0:54:42.24,0:54:43.24,Default,,0000,0000,0000,,The picture's confusing.
Dialogue: 0,0:54:43.24,0:54:46.51,Default,,0000,0000,0000,,This is x coming\Ntowards me, right?
Dialogue: 0,0:54:46.51,0:54:49.63,Default,,0000,0000,0000,,And y going there\Nand z is going up.
Dialogue: 0,0:54:49.63,0:54:52.15,Default,,0000,0000,0000,,This is the graph.
Dialogue: 0,0:54:52.15,0:54:54.65,Default,,0000,0000,0000,,When I do the\Nderivative with respect
Dialogue: 0,0:54:54.65,0:55:00.83,Default,,0000,0000,0000,,to what is this, y, the\Nderivative with respect to y,
Dialogue: 0,0:55:00.83,0:55:04.07,Default,,0000,0000,0000,,with respect to y, y\Nis my only variable,
Dialogue: 0,0:55:04.07,0:55:06.64,Default,,0000,0000,0000,,so the curve will be like that.
Dialogue: 0,0:55:06.64,0:55:11.48,Default,,0000,0000,0000,,And the slope will be for a\Ncurve that depends on y only.
Dialogue: 0,0:55:11.48,0:55:14.36,Default,,0000,0000,0000,,When I do derivative\Nwith respect to x,
Dialogue: 0,0:55:14.36,0:55:19.46,Default,,0000,0000,0000,,it's like I'm on top of a hill\Nand I decide to go skiing.
Dialogue: 0,0:55:19.46,0:55:22.31,Default,,0000,0000,0000,,And I'm-- and I point\Nmy skis like that,
Dialogue: 0,0:55:22.31,0:55:27.10,Default,,0000,0000,0000,,and the slope is going down,\Nand that's the x direction.
Dialogue: 0,0:55:27.10,0:55:27.75,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:55:27.75,0:55:30.91,Default,,0000,0000,0000,,And what I'm going to\Ndescribe as a skier
Dialogue: 0,0:55:30.91,0:55:34.13,Default,,0000,0000,0000,,will be a plane curve going\Ndown in this direction.
Dialogue: 0,0:55:34.13,0:55:35.85,Default,,0000,0000,0000,,Zzzzsssshh, like that.
Dialogue: 0,0:55:35.85,0:55:40.90,Default,,0000,0000,0000,,And the slope at every\Npoint, the slope of the line,
Dialogue: 0,0:55:40.90,0:55:44.72,Default,,0000,0000,0000,,of y trajectory, will\Nbe the derivative.
Dialogue: 0,0:55:44.72,0:55:47.46,Default,,0000,0000,0000,,So I have a curve like\Nthat, and a curve like this.
Dialogue: 0,0:55:47.46,0:55:50.14,Default,,0000,0000,0000,,And they're called\Ncoordinate curves.
Dialogue: 0,0:55:50.14,0:55:51.48,Default,,0000,0000,0000,,Now this is hard.
Dialogue: 0,0:55:51.48,0:55:53.26,Default,,0000,0000,0000,,You'll see how\Nbeautiful and easy
Dialogue: 0,0:55:53.26,0:55:57.16,Default,,0000,0000,0000,,it is when you actually\Ncompute the partial derivatives
Dialogue: 0,0:55:57.16,0:55:59.89,Default,,0000,0000,0000,,of functions by hand.
Dialogue: 0,0:55:59.89,0:56:01.93,Default,,0000,0000,0000,,Examples?
Dialogue: 0,0:56:01.93,0:56:08.10,Default,,0000,0000,0000,,Let's take f of x, y to be\Nx squared plus y squared.
Dialogue: 0,0:56:08.10,0:56:12.55,Default,,0000,0000,0000,,I'm asking you, who\Nis f sub x at x, y?
Dialogue: 0,0:56:12.55,0:56:17.56,Default,,0000,0000,0000,,Who is f sub x at 1\Nminus 1, 1, 0, OK.
Dialogue: 0,0:56:17.56,0:56:20.63,Default,,0000,0000,0000,,Who is f sub y at x, y?
Dialogue: 0,0:56:20.63,0:56:26.13,Default,,0000,0000,0000,,And who is f sub y at 3 and 2.
Dialogue: 0,0:56:26.13,0:56:28.24,Default,,0000,0000,0000,,Since I make up my\Nexample-- I don't
Dialogue: 0,0:56:28.24,0:56:30.46,Default,,0000,0000,0000,,want to copy the\Nexamples from the book,
Dialogue: 0,0:56:30.46,0:56:35.02,Default,,0000,0000,0000,,because you are supposedly\Ngoing to read the book.
Dialogue: 0,0:56:35.02,0:56:40.23,Default,,0000,0000,0000,,This is-- should be another\Nexample, just for you.
Dialogue: 0,0:56:40.23,0:56:44.37,Default,,0000,0000,0000,,
Dialogue: 0,0:56:44.37,0:56:49.08,Default,,0000,0000,0000,,So who's gonna help me-- I'm\Npausing a little bit-- who's
Dialogue: 0,0:56:49.08,0:56:50.76,Default,,0000,0000,0000,,gonna help me here?
Dialogue: 0,0:56:50.76,0:56:53.88,Default,,0000,0000,0000,,What's the answer here?
Dialogue: 0,0:56:53.88,0:56:55.81,Default,,0000,0000,0000,,So how do I think?
Dialogue: 0,0:56:55.81,0:57:00.06,Default,,0000,0000,0000,,I think I got-- when I\Nprime with respect to x, y
Dialogue: 0,0:57:00.06,0:57:01.46,Default,,0000,0000,0000,,is like a held constant.
Dialogue: 0,0:57:01.46,0:57:02.95,Default,,0000,0000,0000,,He's held prisoner.
Dialogue: 0,0:57:02.95,0:57:05.20,Default,,0000,0000,0000,,Poor guy cannot leave his cell.
Dialogue: 0,0:57:05.20,0:57:06.24,Default,,0000,0000,0000,,That's awful.
Dialogue: 0,0:57:06.24,0:57:09.45,Default,,0000,0000,0000,,So you prime with respect to x.
Dialogue: 0,0:57:09.45,0:57:11.35,Default,,0000,0000,0000,,Because x is the only variable.
Dialogue: 0,0:57:11.35,0:57:12.18,Default,,0000,0000,0000,,And he is--
Dialogue: 0,0:57:12.18,0:57:14.52,Default,,0000,0000,0000,,STUDENT: So then it's 2x plus y?
Dialogue: 0,0:57:14.52,0:57:17.07,Default,,0000,0000,0000,,MAGDALENA TODA: 2x plus 0.
Dialogue: 0,0:57:17.07,0:57:17.76,Default,,0000,0000,0000,,Plus 0.
Dialogue: 0,0:57:17.76,0:57:20.32,Default,,0000,0000,0000,,Because y is a constant and\Nwhen you prime a constant,
Dialogue: 0,0:57:20.32,0:57:22.07,Default,,0000,0000,0000,,you get 0.
Dialogue: 0,0:57:22.07,0:57:23.99,Default,,0000,0000,0000,,STUDENT: So when you\Ntake partial derivatives,
Dialogue: 0,0:57:23.99,0:57:25.66,Default,,0000,0000,0000,,you-- when you're\Ntaking it with respect
Dialogue: 0,0:57:25.66,0:57:28.61,Default,,0000,0000,0000,,to the first derivative, the\Nfirst variable [INAUDIBLE]
Dialogue: 0,0:57:28.61,0:57:30.11,Default,,0000,0000,0000,,MAGDALENA TODA: You\Ndon't completely
Dialogue: 0,0:57:30.11,0:57:31.96,Default,,0000,0000,0000,,know because it\Nmight be multiplied.
Dialogue: 0,0:57:31.96,0:57:33.44,Default,,0000,0000,0000,,But you view it as a constant.
Dialogue: 0,0:57:33.44,0:57:35.32,Default,,0000,0000,0000,,So for you-- very good, Ryan.
Dialogue: 0,0:57:35.32,0:57:38.09,Default,,0000,0000,0000,,So for you, it's like,\Nas if y would be 7.
Dialogue: 0,0:57:38.09,0:57:39.97,Default,,0000,0000,0000,,Imagine that y would be 7.
Dialogue: 0,0:57:39.97,0:57:44.15,Default,,0000,0000,0000,,And then you have x squared plus\N7 squared prime is u, right?
Dialogue: 0,0:57:44.15,0:57:47.74,Default,,0000,0000,0000,,STUDENT: So then that means\Nf of 1-- or f x of 1,0
Dialogue: 0,0:57:47.74,0:57:48.48,Default,,0000,0000,0000,,is [INAUDIBLE]
Dialogue: 0,0:57:48.48,0:57:49.56,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:57:49.56,0:57:50.87,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,0:57:50.87,0:57:54.67,Default,,0000,0000,0000,,And in this case, f sub y,\Nwhat do you think it is?
Dialogue: 0,0:57:54.67,0:57:56.44,Default,,0000,0000,0000,,STUDENT: 2y.
Dialogue: 0,0:57:56.44,0:57:57.23,Default,,0000,0000,0000,,MAGDALENA TODA: 2y.
Dialogue: 0,0:57:57.23,0:57:59.96,Default,,0000,0000,0000,,And what is f y of 3, 2?
Dialogue: 0,0:57:59.96,0:58:01.09,Default,,0000,0000,0000,,STUDENT: 4.
Dialogue: 0,0:58:01.09,0:58:02.05,Default,,0000,0000,0000,,MAGDALENA TODA: It's 4.
Dialogue: 0,0:58:02.05,0:58:04.56,Default,,0000,0000,0000,,And you say, OK, that\Nmakes sense, that was easy.
Dialogue: 0,0:58:04.56,0:58:06.58,Default,,0000,0000,0000,,Let's try something hard.
Dialogue: 0,0:58:06.58,0:58:09.38,Default,,0000,0000,0000,,I'm going to build them\Non so many examples
Dialogue: 0,0:58:09.38,0:58:12.67,Default,,0000,0000,0000,,that you say, stop,\NMagdalena, because I became
Dialogue: 0,0:58:12.67,0:58:15.58,Default,,0000,0000,0000,,an expert in partial\Ndifferentiation
Dialogue: 0,0:58:15.58,0:58:19.29,Default,,0000,0000,0000,,and I-- now everything is so\Ntrivial that you have to stop.
Dialogue: 0,0:58:19.29,0:58:38.38,Default,,0000,0000,0000,,So example A, example B. A was f\Nof x, y [INAUDIBLE] x, y plus y
Dialogue: 0,0:58:38.38,0:58:39.93,Default,,0000,0000,0000,,sine x.
Dialogue: 0,0:58:39.93,0:58:41.47,Default,,0000,0000,0000,,And you say, wait,\Nwait, wait, you're
Dialogue: 0,0:58:41.47,0:58:44.24,Default,,0000,0000,0000,,giving me a little\Nbit of trouble.
Dialogue: 0,0:58:44.24,0:58:45.62,Default,,0000,0000,0000,,No, I don't mean to.
Dialogue: 0,0:58:45.62,0:58:47.02,Default,,0000,0000,0000,,It's very easy.
Dialogue: 0,0:58:47.02,0:58:50.26,Default,,0000,0000,0000,,Believe me guys,\Nvery, very easy.
Dialogue: 0,0:58:50.26,0:58:55.35,Default,,0000,0000,0000,,We just have to\Nthink how we do this.
Dialogue: 0,0:58:55.35,0:59:02.17,Default,,0000,0000,0000,,f sub x at 1 and 2, f\Nsub y at x, y in general,
Dialogue: 0,0:59:02.17,0:59:06.98,Default,,0000,0000,0000,,f sub y at 1 and\N2, for God's sake.
Dialogue: 0,0:59:06.98,0:59:08.27,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:59:08.27,0:59:09.88,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:59:09.88,0:59:19.38,Default,,0000,0000,0000,,And now, while you're\Nstaring at that,
Dialogue: 0,0:59:19.38,0:59:23.60,Default,,0000,0000,0000,,I take out my beautiful\Ncolors that I paid $6 for.
Dialogue: 0,0:59:23.60,0:59:26.14,Default,,0000,0000,0000,,
Dialogue: 0,0:59:26.14,0:59:31.62,Default,,0000,0000,0000,,The department told me that\Nthey don't buy different colors,
Dialogue: 0,0:59:31.62,0:59:35.49,Default,,0000,0000,0000,,just two or three basic ones.
Dialogue: 0,0:59:35.49,0:59:35.99,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:59:35.99,0:59:38.17,Default,,0000,0000,0000,,So what do we do?
Dialogue: 0,0:59:38.17,0:59:40.62,Default,,0000,0000,0000,,STUDENT: First\None will be the y.
Dialogue: 0,0:59:40.62,0:59:43.32,Default,,0000,0000,0000,,MAGDALENA TODA: It's like y\Nwould be a constant 7, right,
Dialogue: 0,0:59:43.32,0:59:46.71,Default,,0000,0000,0000,,but you have to keep in\Nmind it's mister called y.
Dialogue: 0,0:59:46.71,0:59:48.58,Default,,0000,0000,0000,,Which for you is a constant.
Dialogue: 0,0:59:48.58,0:59:52.73,Default,,0000,0000,0000,,So you go, I'm priming this\Nwith respect to x only--
Dialogue: 0,0:59:52.73,0:59:54.51,Default,,0000,0000,0000,,STUDENT: Then you get y.
Dialogue: 0,0:59:54.51,0:59:56.32,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:59:56.32,0:59:56.82,Default,,0000,0000,0000,,Plus--
Dialogue: 0,0:59:56.82,0:59:59.91,Default,,0000,0000,0000,,
Dialogue: 0,0:59:59.91,1:00:01.11,Default,,0000,0000,0000,,STUDENT: y cosine x.
Dialogue: 0,1:00:01.11,1:00:01.60,Default,,0000,0000,0000,,MAGDALENA TODA: y cosine x.
Dialogue: 0,1:00:01.60,1:00:02.10,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,1:00:02.10,1:00:03.52,Default,,0000,0000,0000,,And stop.
Dialogue: 0,1:00:03.52,1:00:04.97,Default,,0000,0000,0000,,And stop.
Dialogue: 0,1:00:04.97,1:00:06.18,Default,,0000,0000,0000,,Because that's all I have.
Dialogue: 0,1:00:06.18,1:00:08.76,Default,,0000,0000,0000,,You see, it's not hard.
Dialogue: 0,1:00:08.76,1:00:11.71,Default,,0000,0000,0000,,Let me put here a y.
Dialogue: 0,1:00:11.71,1:00:13.44,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:00:13.44,1:00:19.04,Default,,0000,0000,0000,,And then, I plug\Na different color.
Dialogue: 0,1:00:19.04,1:00:21.52,Default,,0000,0000,0000,,I'm a girl, of course I\Nlike different colors.
Dialogue: 0,1:00:21.52,1:00:26.70,Default,,0000,0000,0000,,So 1, 2. x is 1, and y is 2.
Dialogue: 0,1:00:26.70,1:00:30.27,Default,,0000,0000,0000,,2 plus 2 cosine 1.
Dialogue: 0,1:00:30.27,1:00:33.31,Default,,0000,0000,0000,,And you say, oh, wait a minute,\Nwhat is that cosine of 1?
Dialogue: 0,1:00:33.31,1:00:33.81,Default,,0000,0000,0000,,Never mind.
Dialogue: 0,1:00:33.81,1:00:34.79,Default,,0000,0000,0000,,Don't worry about it.
Dialogue: 0,1:00:34.79,1:00:37.23,Default,,0000,0000,0000,,It's like cosine\Nof 1, [INAUDIBLE]
Dialogue: 0,1:00:37.23,1:00:41.13,Default,,0000,0000,0000,,plug it in the\Ncalculator, nobody cares.
Dialogue: 0,1:00:41.13,1:00:44.55,Default,,0000,0000,0000,,Well, in the final, you\Ndon't have a calculator,
Dialogue: 0,1:00:44.55,1:00:47.97,Default,,0000,0000,0000,,so you leave it like that.
Dialogue: 0,1:00:47.97,1:00:49.43,Default,,0000,0000,0000,,Who cares?
Dialogue: 0,1:00:49.43,1:00:52.75,Default,,0000,0000,0000,,It's just the perfect--\NI would actually hate it
Dialogue: 0,1:00:52.75,1:00:54.23,Default,,0000,0000,0000,,that you gave me--\Nbecause all you
Dialogue: 0,1:00:54.23,1:00:56.48,Default,,0000,0000,0000,,could give me would be an\Napproximation, a truncation,
Dialogue: 0,1:00:56.48,1:00:58.22,Default,,0000,0000,0000,,with two decimals.
Dialogue: 0,1:00:58.22,1:01:01.33,Default,,0000,0000,0000,,I prefer you give me the\Nprecise answer, which
Dialogue: 0,1:01:01.33,1:01:03.96,Default,,0000,0000,0000,,is an exact answer like that.
Dialogue: 0,1:01:03.96,1:01:04.75,Default,,0000,0000,0000,,f sub y.
Dialogue: 0,1:01:04.75,1:01:07.50,Default,,0000,0000,0000,,Now, Mr. x is held prisoner.
Dialogue: 0,1:01:07.50,1:01:09.01,Default,,0000,0000,0000,,He is a constant.
Dialogue: 0,1:01:09.01,1:01:10.62,Default,,0000,0000,0000,,He cannot move.
Dialogue: 0,1:01:10.62,1:01:11.80,Default,,0000,0000,0000,,Mr. y can move.
Dialogue: 0,1:01:11.80,1:01:13.28,Default,,0000,0000,0000,,He has all the freedom.
Dialogue: 0,1:01:13.28,1:01:16.60,Default,,0000,0000,0000,,So prime with respect\Nto y, what do you have?
Dialogue: 0,1:01:16.60,1:01:17.10,Default,,0000,0000,0000,,STUDENT: x--
Dialogue: 0,1:01:17.10,1:01:18.03,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:01:18.03,1:01:21.95,Default,,0000,0000,0000,,MAGDALENA TODA: x plus\Nsine x is a constant.
Dialogue: 0,1:01:21.95,1:01:25.20,Default,,0000,0000,0000,,So for God's sake,\NI'll write it.
Dialogue: 0,1:01:25.20,1:01:30.68,Default,,0000,0000,0000,,So then I get 1,\Nplug in x equals 1. y
Dialogue: 0,1:01:30.68,1:01:31.96,Default,,0000,0000,0000,,doesn't appear in the picture.
Dialogue: 0,1:01:31.96,1:01:33.11,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,1:01:33.11,1:01:35.13,Default,,0000,0000,0000,,1 plus sine 1.
Dialogue: 0,1:01:35.13,1:01:38.34,Default,,0000,0000,0000,,
Dialogue: 0,1:01:38.34,1:01:39.82,Default,,0000,0000,0000,,And now comes-- don't erase.
Dialogue: 0,1:01:39.82,1:01:42.24,Default,,0000,0000,0000,,Now comes the-- I mean,\Nyou cannot erase it.
Dialogue: 0,1:01:42.24,1:01:44.75,Default,,0000,0000,0000,,I can erase it.
Dialogue: 0,1:01:44.75,1:01:48.77,Default,,0000,0000,0000,,Comes this mean professor\Nwho says, wait a minute,
Dialogue: 0,1:01:48.77,1:01:50.95,Default,,0000,0000,0000,,I want more.
Dialogue: 0,1:01:50.95,1:01:53.60,Default,,0000,0000,0000,,Mathematicians always want more.
Dialogue: 0,1:01:53.60,1:01:57.33,Default,,0000,0000,0000,,He goes, I want the\Nsecond derivative.
Dialogue: 0,1:01:57.33,1:02:01.04,Default,,0000,0000,0000,,f sub x x of x, y.
Dialogue: 0,1:02:01.04,1:02:03.53,Default,,0000,0000,0000,,And you say, what in\Nthe world is that?
Dialogue: 0,1:02:03.53,1:02:06.29,Default,,0000,0000,0000,,Even some mathematicians,\Nthey denote it
Dialogue: 0,1:02:06.29,1:02:13.30,Default,,0000,0000,0000,,as del 2 f dx 2, which\Nis d of-- d with respect
Dialogue: 0,1:02:13.30,1:02:16.52,Default,,0000,0000,0000,,to x sub d u with respect to x.
Dialogue: 0,1:02:16.52,1:02:17.81,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,1:02:17.81,1:02:20.64,Default,,0000,0000,0000,,You take the first derivative\Nand you derive it again.
Dialogue: 0,1:02:20.64,1:02:23.18,Default,,0000,0000,0000,,And don't drink and derive\Nbecause you'll be in trouble.
Dialogue: 0,1:02:23.18,1:02:23.68,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:02:23.68,1:02:28.04,Default,,0000,0000,0000,,So you have d of dx primed\Nagain, with-- differentiated
Dialogue: 0,1:02:28.04,1:02:30.41,Default,,0000,0000,0000,,again with respect to x.
Dialogue: 0,1:02:30.41,1:02:31.37,Default,,0000,0000,0000,,Is that hard?
Dialogue: 0,1:02:31.37,1:02:31.87,Default,,0000,0000,0000,,Uh-uh.
Dialogue: 0,1:02:31.87,1:02:32.95,Default,,0000,0000,0000,,What you do?
Dialogue: 0,1:02:32.95,1:02:36.06,Default,,0000,0000,0000,,In the-- don't do it here.
Dialogue: 0,1:02:36.06,1:02:37.43,Default,,0000,0000,0000,,You do it in general, right?
Dialogue: 0,1:02:37.43,1:02:43.00,Default,,0000,0000,0000,,With respect to x as a variable,\Ny is again held as a prisoner,
Dialogue: 0,1:02:43.00,1:02:44.57,Default,,0000,0000,0000,,constant.
Dialogue: 0,1:02:44.57,1:02:47.68,Default,,0000,0000,0000,,So when you prime\Nthat y goes away.
Dialogue: 0,1:02:47.68,1:02:50.71,Default,,0000,0000,0000,,You're gonna get 0.
Dialogue: 0,1:02:50.71,1:02:54.58,Default,,0000,0000,0000,,I'll write 0 like a silly\Nbecause we are just starters.
Dialogue: 0,1:02:54.58,1:02:56.21,Default,,0000,0000,0000,,And what else?
Dialogue: 0,1:02:56.21,1:02:57.70,Default,,0000,0000,0000,,STUDENT: Negative y sine of x.
Dialogue: 0,1:02:57.70,1:02:59.90,Default,,0000,0000,0000,,MAGDALENA TODA:\NMinus y sine of x.
Dialogue: 0,1:02:59.90,1:03:02.34,Default,,0000,0000,0000,,And I know you've gonna\Nlove this process.
Dialogue: 0,1:03:02.34,1:03:04.94,Default,,0000,0000,0000,,You are becoming\Nexperts in that.
Dialogue: 0,1:03:04.94,1:03:10.05,Default,,0000,0000,0000,,And in a way I'm a little\Nbit sorry it's so easy,
Dialogue: 0,1:03:10.05,1:03:13.18,Default,,0000,0000,0000,,but I guess not\Neverybody gets it.
Dialogue: 0,1:03:13.18,1:03:16.29,Default,,0000,0000,0000,,There are students who\Ndon't get it the first time.
Dialogue: 0,1:03:16.29,1:03:17.98,Default,,0000,0000,0000,,So what do we get here?
Dialogue: 0,1:03:17.98,1:03:18.91,Default,,0000,0000,0000,,Minus--
Dialogue: 0,1:03:18.91,1:03:21.77,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:03:21.77,1:03:25.82,Default,,0000,0000,0000,,MAGDALENA TODA: Please\Ntell me-- sine 1, 0.
Dialogue: 0,1:03:25.82,1:03:26.32,Default,,0000,0000,0000,,Good.
Dialogue: 0,1:03:26.32,1:03:30.00,Default,,0000,0000,0000,,I could do the same\Nthing for f y y.
Dialogue: 0,1:03:30.00,1:03:34.80,Default,,0000,0000,0000,,I could do this thing--\Nwhat is f sub x y?
Dialogue: 0,1:03:34.80,1:03:37.25,Default,,0000,0000,0000,,By definition f sub x y--
Dialogue: 0,1:03:37.25,1:03:39.98,Default,,0000,0000,0000,,STUDENT: Is that taking the\Nderivative of the derivative
Dialogue: 0,1:03:39.98,1:03:42.10,Default,,0000,0000,0000,,with respect-- is that\Ntaking the second derivative
Dialogue: 0,1:03:42.10,1:03:44.06,Default,,0000,0000,0000,,with respect to y after\Nyou take the derivative
Dialogue: 0,1:03:44.06,1:03:46.11,Default,,0000,0000,0000,,of the-- first derivative\Nwith respect to x?
Dialogue: 0,1:03:46.11,1:03:47.03,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,1:03:47.03,1:03:49.51,Default,,0000,0000,0000,,So when I write like that,\Nbecause that's a little bit
Dialogue: 0,1:03:49.51,1:03:54.20,Default,,0000,0000,0000,,confusing, when students\Nask me, which one is first?
Dialogue: 0,1:03:54.20,1:03:57.83,Default,,0000,0000,0000,,First you do f sub\Nx, and then you do y.
Dialogue: 0,1:03:57.83,1:04:02.77,Default,,0000,0000,0000,,And then f sub y x would be the\Nderivative with respect to y
Dialogue: 0,1:04:02.77,1:04:04.65,Default,,0000,0000,0000,,primed again with respect to x.
Dialogue: 0,1:04:04.65,1:04:07.48,Default,,0000,0000,0000,,Now, let me tell\Nyou the good news.
Dialogue: 0,1:04:07.48,1:04:13.39,Default,,0000,0000,0000,,They-- the book doesn't call\Nit any name, because we don't
Dialogue: 0,1:04:13.39,1:04:14.89,Default,,0000,0000,0000,,like to call anybody names.
Dialogue: 0,1:04:14.89,1:04:15.84,Default,,0000,0000,0000,,I'm just kidding.
Dialogue: 0,1:04:15.84,1:04:23.71,Default,,0000,0000,0000,,It's called the\NSchwartz principle,
Dialogue: 0,1:04:23.71,1:04:27.47,Default,,0000,0000,0000,,or the theorem of Schwartz.
Dialogue: 0,1:04:27.47,1:04:30.97,Default,,0000,0000,0000,,When I told my co-authors,\Nthey said, who cares?
Dialogue: 0,1:04:30.97,1:04:34.95,Default,,0000,0000,0000,,Well I care, because I was a\Nstudent when my professors told
Dialogue: 0,1:04:34.95,1:04:38.29,Default,,0000,0000,0000,,me that this German\Nmathematician made
Dialogue: 0,1:04:38.29,1:04:41.04,Default,,0000,0000,0000,,this discovery, which\Nis so beautiful.
Dialogue: 0,1:04:41.04,1:04:55.38,Default,,0000,0000,0000,,If f is twice differentiable\Nwith respect to x and y,
Dialogue: 0,1:04:55.38,1:04:58.82,Default,,0000,0000,0000,,and the partial derivatives--\Nthe second partial
Dialogue: 0,1:04:58.82,1:05:14.75,Default,,0000,0000,0000,,derivatives-- are continuous,\Nthen, now in English
Dialogue: 0,1:05:14.75,1:05:17.48,Default,,0000,0000,0000,,it would say it doesn't\Nmatter in which order
Dialogue: 0,1:05:17.48,1:05:18.53,Default,,0000,0000,0000,,you differentiate.
Dialogue: 0,1:05:18.53,1:05:20.92,Default,,0000,0000,0000,,The mixed ones are\Nalways the same.
Dialogue: 0,1:05:20.92,1:05:22.19,Default,,0000,0000,0000,,Say what?
Dialogue: 0,1:05:22.19,1:05:26.76,Default,,0000,0000,0000,,f sub x y equals f sub\Ny x for every point.
Dialogue: 0,1:05:26.76,1:05:31.59,Default,,0000,0000,0000,,For every-- do you remember\Nwhat I taught you for every x, y
Dialogue: 0,1:05:31.59,1:05:32.36,Default,,0000,0000,0000,,in the domain.
Dialogue: 0,1:05:32.36,1:05:36.11,Default,,0000,0000,0000,,Or for every x, y\Nwhere this happens.
Dialogue: 0,1:05:36.11,1:05:38.41,Default,,0000,0000,0000,,So what does this mean?
Dialogue: 0,1:05:38.41,1:05:41.22,Default,,0000,0000,0000,,That means that whether\Nyou differentiate
Dialogue: 0,1:05:41.22,1:05:43.76,Default,,0000,0000,0000,,first with respect to x and then\Nwith respect to, y, or first
Dialogue: 0,1:05:43.76,1:05:46.10,Default,,0000,0000,0000,,with respect to y and\Nthen with respect to x,
Dialogue: 0,1:05:46.10,1:05:48.25,Default,,0000,0000,0000,,it doesn't matter.
Dialogue: 0,1:05:48.25,1:05:50.90,Default,,0000,0000,0000,,The mixed partial\Nderivatives are the same.
Dialogue: 0,1:05:50.90,1:05:52.14,Default,,0000,0000,0000,,Which is wonderful.
Dialogue: 0,1:05:52.14,1:05:55.11,Default,,0000,0000,0000,,I mean, this is one\Nof the best things
Dialogue: 0,1:05:55.11,1:05:58.10,Default,,0000,0000,0000,,that ever happened to us.
Dialogue: 0,1:05:58.10,1:06:01.31,Default,,0000,0000,0000,,Let's see if this\Nis true in our case.
Dialogue: 0,1:06:01.31,1:06:03.86,Default,,0000,0000,0000,,I mean, of course it's true\Nbecause it's a theorem,
Dialogue: 0,1:06:03.86,1:06:06.15,Default,,0000,0000,0000,,if it weren't true\NI wouldn't teach it,
Dialogue: 0,1:06:06.15,1:06:11.05,Default,,0000,0000,0000,,but let's verify it on a baby.
Dialogue: 0,1:06:11.05,1:06:14.34,Default,,0000,0000,0000,,Not on a real baby,\Non a baby example.
Dialogue: 0,1:06:14.34,1:06:15.23,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:06:15.23,1:06:21.04,Default,,0000,0000,0000,,So, f sub x is y plus y\Nequals sine x primed again
Dialogue: 0,1:06:21.04,1:06:22.78,Default,,0000,0000,0000,,with respect to y.
Dialogue: 0,1:06:22.78,1:06:27.63,Default,,0000,0000,0000,,And what do we get out of it?
Dialogue: 0,1:06:27.63,1:06:29.25,Default,,0000,0000,0000,,Cosine of x.
Dialogue: 0,1:06:29.25,1:06:31.08,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:06:31.08,1:06:35.13,Default,,0000,0000,0000,,So f sub x was y\Nplus y equals sine x.
Dialogue: 0,1:06:35.13,1:06:39.08,Default,,0000,0000,0000,,Take this guy\Nagain, put it here,
Dialogue: 0,1:06:39.08,1:06:42.67,Default,,0000,0000,0000,,squeeze them up a little\Nbit, divide by-- no.
Dialogue: 0,1:06:42.67,1:06:47.78,Default,,0000,0000,0000,,Time with respect to y, x is\Na constant, what do you think?
Dialogue: 0,1:06:47.78,1:06:48.79,Default,,0000,0000,0000,,Cosine of x, am I right?
Dialogue: 0,1:06:48.79,1:06:49.96,Default,,0000,0000,0000,,STUDENT: 1 plus [INAUDIBLE].
Dialogue: 0,1:06:49.96,1:06:52.39,Default,,0000,0000,0000,,
Dialogue: 0,1:06:52.39,1:06:54.18,Default,,0000,0000,0000,,MAGDALENA TODA: That's\Nwhat it starts with.
Dialogue: 0,1:06:54.18,1:06:56.70,Default,,0000,0000,0000,,Plus [INAUDIBLE].
Dialogue: 0,1:06:56.70,1:07:02.03,Default,,0000,0000,0000,,So cosine of x, [INAUDIBLE]\Na constant, plus 1.
Dialogue: 0,1:07:02.03,1:07:04.96,Default,,0000,0000,0000,,Another way to have done\Nit is, like, wait a minute,
Dialogue: 0,1:07:04.96,1:07:10.81,Default,,0000,0000,0000,,at this point I go, constant\Nout-- are you with me?--
Dialogue: 0,1:07:10.81,1:07:14.63,Default,,0000,0000,0000,,constant out, prime with respect\Nto y, equals sine x plus 1.
Dialogue: 0,1:07:14.63,1:07:16.88,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:07:16.88,1:07:17.38,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:07:17.38,1:07:20.82,Default,,0000,0000,0000,,
Dialogue: 0,1:07:20.82,1:07:26.37,Default,,0000,0000,0000,,F sub yx is going to be f sub y.
Dialogue: 0,1:07:26.37,1:07:32.34,Default,,0000,0000,0000,,x plus sine x, but I have\Nto take it from here,
Dialogue: 0,1:07:32.34,1:07:38.42,Default,,0000,0000,0000,,and I prime again with respect\Nto x, and I get the same thing.
Dialogue: 0,1:07:38.42,1:07:39.79,Default,,0000,0000,0000,,I don't know,\Nmaybe I'm dyslexic,
Dialogue: 0,1:07:39.79,1:07:43.45,Default,,0000,0000,0000,,I go from the right to the\Nleft, what's the matter with me.
Dialogue: 0,1:07:43.45,1:07:47.30,Default,,0000,0000,0000,,Instead of saying 1 plus,\NI go cosine of x plus 1.
Dialogue: 0,1:07:47.30,1:07:53.02,Default,,0000,0000,0000,,
Dialogue: 0,1:07:53.02,1:07:54.48,Default,,0000,0000,0000,,So it's the same thing.
Dialogue: 0,1:07:54.48,1:07:55.44,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:07:55.44,1:07:58.61,Default,,0000,0000,0000,,STUDENT:I'm looking at\Nthe f of xy from the--
Dialogue: 0,1:07:58.61,1:08:00.49,Default,,0000,0000,0000,,MAGDALENA TODA: Which\None are you looking at?
Dialogue: 0,1:08:00.49,1:08:01.46,Default,,0000,0000,0000,,Show me.
Dialogue: 0,1:08:01.46,1:08:03.52,Default,,0000,0000,0000,,STUDENT: It's in the purple.
Dialogue: 0,1:08:03.52,1:08:05.15,Default,,0000,0000,0000,,MAGDALENA TODA: It\Nis in the purple.
Dialogue: 0,1:08:05.15,1:08:05.68,Default,,0000,0000,0000,,STUDENT: It's that\None right there.
Dialogue: 0,1:08:05.68,1:08:06.14,Default,,0000,0000,0000,,So--
Dialogue: 0,1:08:06.14,1:08:06.72,Default,,0000,0000,0000,,MAGDALENA TODA: This one?
Dialogue: 0,1:08:06.72,1:08:07.31,Default,,0000,0000,0000,,STUDENT: Mmhm.
Dialogue: 0,1:08:07.31,1:08:10.55,Default,,0000,0000,0000,,So, I'm looking at\Nthe y plus y cosine x.
Dialogue: 0,1:08:10.55,1:08:12.50,Default,,0000,0000,0000,,You got that from f of x.
Dialogue: 0,1:08:12.50,1:08:14.20,Default,,0000,0000,0000,,MAGDALENA TODA: I\Ngot this from f of x,
Dialogue: 0,1:08:14.20,1:08:16.64,Default,,0000,0000,0000,,and I prime it again,\Nwith respect to y.
Dialogue: 0,1:08:16.64,1:08:18.92,Default,,0000,0000,0000,,The whole thing.
Dialogue: 0,1:08:18.92,1:08:21.51,Default,,0000,0000,0000,,STUDENT: OK, so you're not\Nwriting that as a derivative?
Dialogue: 0,1:08:21.51,1:08:25.27,Default,,0000,0000,0000,,You're just substituting\Nthat in for f of x?
Dialogue: 0,1:08:25.27,1:08:27.48,Default,,0000,0000,0000,,MAGDALENA TODA: So,\Nlet me write it better,
Dialogue: 0,1:08:27.48,1:08:30.78,Default,,0000,0000,0000,,because I was a little bit\Nrushed, and I don't know,
Dialogue: 0,1:08:30.78,1:08:32.17,Default,,0000,0000,0000,,silly or something.
Dialogue: 0,1:08:32.17,1:08:35.14,Default,,0000,0000,0000,,When I prime this\Nwith respect to y--
Dialogue: 0,1:08:35.14,1:08:38.27,Default,,0000,0000,0000,,STUDENT: Then you get\Nthe cosine of x plus 1.
Dialogue: 0,1:08:38.27,1:08:39.15,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:08:39.15,1:08:42.43,Default,,0000,0000,0000,,I could say, I can take\Nout all the constants.
Dialogue: 0,1:08:42.43,1:08:43.16,Default,,0000,0000,0000,,STUDENT: OK.
Dialogue: 0,1:08:43.16,1:08:46.21,Default,,0000,0000,0000,,MAGDALENA TODA: And that\Nconstant is this plus 1.
Dialogue: 0,1:08:46.21,1:08:47.46,Default,,0000,0000,0000,,And that's all I'm left with.
Dialogue: 0,1:08:47.46,1:08:47.96,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:08:47.96,1:08:51.61,Default,,0000,0000,0000,,It's the same thing\Nas 1 plus cosine x,
Dialogue: 0,1:08:51.61,1:08:53.88,Default,,0000,0000,0000,,which is a constant times y.
Dialogue: 0,1:08:53.88,1:08:57.24,Default,,0000,0000,0000,,Prime this with respect\Nto y, I get the constant.
Dialogue: 0,1:08:57.24,1:09:04.02,Default,,0000,0000,0000,,It's the same principal as when\Nyou have bdy of 7y equals 7.
Dialogue: 0,1:09:04.02,1:09:06.75,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:09:06.75,1:09:08.70,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:09:08.70,1:09:10.13,Default,,0000,0000,0000,,Is this too easy?
Dialogue: 0,1:09:10.13,1:09:13.42,Default,,0000,0000,0000,,I'll give you a nicer function.
Dialogue: 0,1:09:13.42,1:09:28.76,Default,,0000,0000,0000,,I'm imitating the one\Nin WeBWorK [INAUDIBLE]
Dialogue: 0,1:09:28.76,1:09:31.44,Default,,0000,0000,0000,,To make it harder for you.
Dialogue: 0,1:09:31.44,1:09:34.35,Default,,0000,0000,0000,,Nothing I can make at this\Npoint is hard for you,
Dialogue: 0,1:09:34.35,1:09:39.25,Default,,0000,0000,0000,,because you're becoming experts\Nin partial differentiation,
Dialogue: 0,1:09:39.25,1:09:41.72,Default,,0000,0000,0000,,and I cannot\Nchallenge you on that.
Dialogue: 0,1:09:41.72,1:09:54.11,Default,,0000,0000,0000,,
Dialogue: 0,1:09:54.11,1:09:57.05,Default,,0000,0000,0000,,I'm just trying to\Nmake it harder for you.
Dialogue: 0,1:09:57.05,1:09:59.00,Default,,0000,0000,0000,,And I'm trying to\Nlook up something.
Dialogue: 0,1:09:59.00,1:10:02.93,Default,,0000,0000,0000,,
Dialogue: 0,1:10:02.93,1:10:03.97,Default,,0000,0000,0000,,OK, how about that?
Dialogue: 0,1:10:03.97,1:10:06.67,Default,,0000,0000,0000,,
Dialogue: 0,1:10:06.67,1:10:09.11,Default,,0000,0000,0000,,This is harder than the\Nones you have in WeBWorK.
Dialogue: 0,1:10:09.11,1:10:11.97,Default,,0000,0000,0000,,But that was kind of\Nthe idea-- that when
Dialogue: 0,1:10:11.97,1:10:15.62,Default,,0000,0000,0000,,you go home, and open\Nthose WeBWorK problem sets,
Dialogue: 0,1:10:15.62,1:10:17.43,Default,,0000,0000,0000,,that's a piece of cake.
Dialogue: 0,1:10:17.43,1:10:20.60,Default,,0000,0000,0000,,What we did in class was harder.
Dialogue: 0,1:10:20.60,1:10:23.96,Default,,0000,0000,0000,,When I was a graduate\Nstudent, one professor said,
Dialogue: 0,1:10:23.96,1:10:27.32,Default,,0000,0000,0000,,the easy examples are the\Nones that the professor's
Dialogue: 0,1:10:27.32,1:10:29.74,Default,,0000,0000,0000,,supposed to write in\Nclass, on the board.
Dialogue: 0,1:10:29.74,1:10:31.44,Default,,0000,0000,0000,,The hard examples\Nare the ones that
Dialogue: 0,1:10:31.44,1:10:34.29,Default,,0000,0000,0000,,are left for the\Nstudents' homework.
Dialogue: 0,1:10:34.29,1:10:35.75,Default,,0000,0000,0000,,I disagree.
Dialogue: 0,1:10:35.75,1:10:37.76,Default,,0000,0000,0000,,I think it should be\Nthe other way around.
Dialogue: 0,1:10:37.76,1:10:40.26,Default,,0000,0000,0000,,So f sub x.
Dialogue: 0,1:10:40.26,1:10:43.38,Default,,0000,0000,0000,,
Dialogue: 0,1:10:43.38,1:10:50.55,Default,,0000,0000,0000,,That means bfdx for\Nthe pair xy, any xy.
Dialogue: 0,1:10:50.55,1:10:53.56,Default,,0000,0000,0000,,I'm not specifying an x and a y.
Dialogue: 0,1:10:53.56,1:10:56.18,Default,,0000,0000,0000,,I'm not making them a constant.
Dialogue: 0,1:10:56.18,1:10:58.97,Default,,0000,0000,0000,,What am I going to\Nhave in this case?
Dialogue: 0,1:10:58.97,1:11:03.85,Default,,0000,0000,0000,,Chain -- if I catch you\Nnot knowing the chain rule,
Dialogue: 0,1:11:03.85,1:11:05.37,Default,,0000,0000,0000,,you fail the final.
Dialogue: 0,1:11:05.37,1:11:12.59,Default,,0000,0000,0000,,Not really, but, OK,\Nyou get some penalty.
Dialogue: 0,1:11:12.59,1:11:13.73,Default,,0000,0000,0000,,You know it.
Dialogue: 0,1:11:13.73,1:11:16.11,Default,,0000,0000,0000,,Just pay attention\Nto what you do.
Dialogue: 0,1:11:16.11,1:11:18.34,Default,,0000,0000,0000,,I make my own\Nmistakes sometimes.
Dialogue: 0,1:11:18.34,1:11:21.16,Default,,0000,0000,0000,,So 1 over.
Dialogue: 0,1:11:21.16,1:11:23.59,Default,,0000,0000,0000,,What do you do here\Nwhen you differentiate
Dialogue: 0,1:11:23.59,1:11:24.35,Default,,0000,0000,0000,,with respect to x?
Dialogue: 0,1:11:24.35,1:11:31.60,Default,,0000,0000,0000,,You think, OK, from the outside\Nto the inside, one at a time.
Dialogue: 0,1:11:31.60,1:11:36.13,Default,,0000,0000,0000,,1 over the variable\Nsquared plus 1, right?
Dialogue: 0,1:11:36.13,1:11:42.22,Default,,0000,0000,0000,,Whatever that variable,\Nit's like you call variable
Dialogue: 0,1:11:42.22,1:11:44.89,Default,,0000,0000,0000,,of the argument xy, right?
Dialogue: 0,1:11:44.89,1:11:47.24,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:11:47.24,1:11:49.56,Default,,0000,0000,0000,,MAGDALENA TODA: Square plus 1.
Dialogue: 0,1:11:49.56,1:11:56.71,Default,,0000,0000,0000,,Times-- cover it with your\Nhand-- prime with respect to x.
Dialogue: 0,1:11:56.71,1:11:59.00,Default,,0000,0000,0000,,y, right?
Dialogue: 0,1:11:59.00,1:12:00.40,Default,,0000,0000,0000,,Good!
Dialogue: 0,1:12:00.40,1:12:01.32,Default,,0000,0000,0000,,And you're done.
Dialogue: 0,1:12:01.32,1:12:02.72,Default,,0000,0000,0000,,You see how easy it was.
Dialogue: 0,1:12:02.72,1:12:07.51,Default,,0000,0000,0000,,Just don't forget something\Nbecause it can cost you points.
Dialogue: 0,1:12:07.51,1:12:09.64,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:12:09.64,1:12:13.38,Default,,0000,0000,0000,,So, once we are done with\Nsaying, 1 over argument
Dialogue: 0,1:12:13.38,1:12:16.13,Default,,0000,0000,0000,,squared plus 1, I cover\Nthis with my hand,
Dialogue: 0,1:12:16.13,1:12:20.11,Default,,0000,0000,0000,,xy prime with\Nrespect to 2x is y.
Dialogue: 0,1:12:20.11,1:12:22.38,Default,,0000,0000,0000,,And I'm done.
Dialogue: 0,1:12:22.38,1:12:23.25,Default,,0000,0000,0000,,And I'm done.
Dialogue: 0,1:12:23.25,1:12:26.25,Default,,0000,0000,0000,,And here, pause.
Dialogue: 0,1:12:26.25,1:12:29.68,Default,,0000,0000,0000,,What's the easiest\Nway to do that?
Dialogue: 0,1:12:29.68,1:12:32.01,Default,,0000,0000,0000,,You look at it like,\Nshe wants me to get
Dialogue: 0,1:12:32.01,1:12:34.31,Default,,0000,0000,0000,,caught in the quotient rule.
Dialogue: 0,1:12:34.31,1:12:37.31,Default,,0000,0000,0000,,She wants to catch me\Nnot knowing this rule,
Dialogue: 0,1:12:37.31,1:12:40.25,Default,,0000,0000,0000,,while I can do better.
Dialogue: 0,1:12:40.25,1:12:43.46,Default,,0000,0000,0000,,One way to do it would\Nbe numerator prime plus
Dialogue: 0,1:12:43.46,1:12:47.74,Default,,0000,0000,0000,,denominator, minus\Nnumerator [INAUDIBLE] What's
Dialogue: 0,1:12:47.74,1:12:50.41,Default,,0000,0000,0000,,the easier way to do it?
Dialogue: 0,1:12:50.41,1:12:52.87,Default,,0000,0000,0000,,STUDENT: x squared plus\Ny squared, all of it
Dialogue: 0,1:12:52.87,1:12:53.86,Default,,0000,0000,0000,,to the negative one.
Dialogue: 0,1:12:53.86,1:12:54.77,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,1:12:54.77,1:12:56.89,Default,,0000,0000,0000,,So you say, hey,\Nyou cannot catch me,
Dialogue: 0,1:12:56.89,1:13:00.68,Default,,0000,0000,0000,,I'm the gingerbread man.
Dialogue: 0,1:13:00.68,1:13:01.28,Default,,0000,0000,0000,,Good!
Dialogue: 0,1:13:01.28,1:13:03.10,Default,,0000,0000,0000,,That was a good idea.
Dialogue: 0,1:13:03.10,1:13:10.33,Default,,0000,0000,0000,,Chain rule, and\Nminus 1/2, times--
Dialogue: 0,1:13:10.33,1:13:11.76,Default,,0000,0000,0000,,who tells me what's next?
Dialogue: 0,1:13:11.76,1:13:13.21,Default,,0000,0000,0000,,I'm not going to say a word.
Dialogue: 0,1:13:13.21,1:13:15.26,Default,,0000,0000,0000,,STUDENT: 2x plus y squared.
Dialogue: 0,1:13:15.26,1:13:19.17,Default,,0000,0000,0000,,No, it's 2x.
Dialogue: 0,1:13:19.17,1:13:20.64,Default,,0000,0000,0000,,x squared plus y squared.
Dialogue: 0,1:13:20.64,1:13:22.60,Default,,0000,0000,0000,,MAGDALENA TODA: From the\Noutside to the inside.
Dialogue: 0,1:13:22.60,1:13:25.34,Default,,0000,0000,0000,,From the outside-- to the what?
Dialogue: 0,1:13:25.34,1:13:27.19,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,1:13:27.19,1:13:28.06,Default,,0000,0000,0000,,MAGDALENA TODA: Good.
Dialogue: 0,1:13:28.06,1:13:28.97,Default,,0000,0000,0000,,And now I'm done.
Dialogue: 0,1:13:28.97,1:13:31.22,Default,,0000,0000,0000,,I don't see that anymore.
Dialogue: 0,1:13:31.22,1:13:33.86,Default,,0000,0000,0000,,I focus to the core.
Dialogue: 0,1:13:33.86,1:13:35.65,Default,,0000,0000,0000,,2x.
Dialogue: 0,1:13:35.65,1:13:38.64,Default,,0000,0000,0000,,Times 2x.
Dialogue: 0,1:13:38.64,1:13:42.39,Default,,0000,0000,0000,,And that is plenty.
Dialogue: 0,1:13:42.39,1:13:45.25,Default,,0000,0000,0000,,OK, now, let me\Nask you a question.
Dialogue: 0,1:13:45.25,1:13:51.34,Default,,0000,0000,0000,,What if you would\Nask a smart kid,
Dialogue: 0,1:13:51.34,1:13:56.76,Default,,0000,0000,0000,,I don't know, somebody\Nwho knows that,
Dialogue: 0,1:13:56.76,1:14:01.62,Default,,0000,0000,0000,,can you pose the f sub y of xy\Nwithout doing the whole thing
Dialogue: 0,1:14:01.62,1:14:03.37,Default,,0000,0000,0000,,all over again?
Dialogue: 0,1:14:03.37,1:14:06.35,Default,,0000,0000,0000,,Can you sort of figure\Nout what it would be?
Dialogue: 0,1:14:06.35,1:14:08.70,Default,,0000,0000,0000,,The beautiful\Nthing about x and y
Dialogue: 0,1:14:08.70,1:14:11.04,Default,,0000,0000,0000,,is that these are\Nsymmetric polynomials.
Dialogue: 0,1:14:11.04,1:14:12.75,Default,,0000,0000,0000,,What does it mean,\Nsymmetric polynomials?
Dialogue: 0,1:14:12.75,1:14:19.26,Default,,0000,0000,0000,,That means, if you swap x\Nand y, and you swap x and y,
Dialogue: 0,1:14:19.26,1:14:20.81,Default,,0000,0000,0000,,it's the same thing.
Dialogue: 0,1:14:20.81,1:14:23.30,Default,,0000,0000,0000,,Just think of that--\Nswapping x and y.
Dialogue: 0,1:14:23.30,1:14:25.23,Default,,0000,0000,0000,,Swapping the roles of x and y.
Dialogue: 0,1:14:25.23,1:14:28.25,Default,,0000,0000,0000,,So what do you think\Nyou're going to get?
Dialogue: 0,1:14:28.25,1:14:31.13,Default,,0000,0000,0000,,OK, one student said,\Nthis is for smart people,
Dialogue: 0,1:14:31.13,1:14:32.56,Default,,0000,0000,0000,,not for people like me.
Dialogue: 0,1:14:32.56,1:14:34.89,Default,,0000,0000,0000,,And I said, well, OK,\Nwhat's the matter with you?
Dialogue: 0,1:14:34.89,1:14:36.24,Default,,0000,0000,0000,,I'm a hard worker.
Dialogue: 0,1:14:36.24,1:14:39.91,Default,,0000,0000,0000,,I'm the kind of guy who takes\Nthe whole thing again, and does
Dialogue: 0,1:14:39.91,1:14:42.18,Default,,0000,0000,0000,,the derivation from scratch.
Dialogue: 0,1:14:42.18,1:14:45.32,Default,,0000,0000,0000,,And thinking back in high\Nschool, I think, even
Dialogue: 0,1:14:45.32,1:14:47.81,Default,,0000,0000,0000,,for symmetric polynomials,
Dialogue: 0,1:14:47.81,1:14:49.57,Default,,0000,0000,0000,,I'm sure that being\Nsmart and being
Dialogue: 0,1:14:49.57,1:14:53.51,Default,,0000,0000,0000,,able to guess the\Nwhole thing-- but I
Dialogue: 0,1:14:53.51,1:14:56.00,Default,,0000,0000,0000,,did the computation\Nmany times mechanically,
Dialogue: 0,1:14:56.00,1:14:59.44,Default,,0000,0000,0000,,just in the same way,\Nbecause I was a hard worker.
Dialogue: 0,1:14:59.44,1:15:01.22,Default,,0000,0000,0000,,So what do you\Nhave in that case?
Dialogue: 0,1:15:01.22,1:15:09.81,Default,,0000,0000,0000,,1/xy squared plus 1 times x\Nplus-- the same kind of thing.
Dialogue: 0,1:15:09.81,1:15:14.34,Default,,0000,0000,0000,,Attention, this is the symmetric\Npolynomial, and I go to that.
Dialogue: 0,1:15:14.34,1:15:17.28,Default,,0000,0000,0000,,And then times 2y.
Dialogue: 0,1:15:17.28,1:15:20.61,Default,,0000,0000,0000,,So, see-- that kind\Nof easy, fast thing.
Dialogue: 0,1:15:20.61,1:15:24.34,Default,,0000,0000,0000,,Why is this a good\Nobservation when
Dialogue: 0,1:15:24.34,1:15:26.02,Default,,0000,0000,0000,,you have symmetric polynomials?
Dialogue: 0,1:15:26.02,1:15:28.71,Default,,0000,0000,0000,,If you are on the final and\Nyou don't have that much time,
Dialogue: 0,1:15:28.71,1:15:33.54,Default,,0000,0000,0000,,or on any kind of exam when\Nyou are in a time-crunch.
Dialogue: 0,1:15:33.54,1:15:36.12,Default,,0000,0000,0000,,Now, we want those\Nexams so you are not
Dialogue: 0,1:15:36.12,1:15:38.02,Default,,0000,0000,0000,,going to be in a time-crunch.
Dialogue: 0,1:15:38.02,1:15:41.66,Default,,0000,0000,0000,,If there is something I hate,\NI hate a final of 2 hours
Dialogue: 0,1:15:41.66,1:15:44.64,Default,,0000,0000,0000,,and a half with 25\Nserious problems,
Dialogue: 0,1:15:44.64,1:15:48.26,Default,,0000,0000,0000,,and you know nobody can do that.
Dialogue: 0,1:15:48.26,1:15:50.71,Default,,0000,0000,0000,,So, it happens a lot.
Dialogue: 0,1:15:50.71,1:15:55.88,Default,,0000,0000,0000,,I see that-- one of my jobs\Nis also to look at the finals
Dialogue: 0,1:15:55.88,1:15:58.59,Default,,0000,0000,0000,,after people wrote\Nthem, and I still
Dialogue: 0,1:15:58.59,1:16:05.26,Default,,0000,0000,0000,,do that every semester-- I see\Ntoo many people making finals.
Dialogue: 0,1:16:05.26,1:16:06.88,Default,,0000,0000,0000,,The finals are not\Nsupposed to be long.
Dialogue: 0,1:16:06.88,1:16:10.99,Default,,0000,0000,0000,,The finals are supposed\Nto be comprehensive, cover
Dialogue: 0,1:16:10.99,1:16:16.30,Default,,0000,0000,0000,,everything, but not extensive.
Dialogue: 0,1:16:16.30,1:16:21.06,Default,,0000,0000,0000,,So maybe you'll have 15\Nproblems that cover practically
Dialogue: 0,1:16:21.06,1:16:22.91,Default,,0000,0000,0000,,the material entirely.
Dialogue: 0,1:16:22.91,1:16:23.41,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:16:23.41,1:16:29.08,Default,,0000,0000,0000,,Because every little problem\Ncan have two short questions.
Dialogue: 0,1:16:29.08,1:16:30.53,Default,,0000,0000,0000,,You were done with\Na section, you
Dialogue: 0,1:16:30.53,1:16:34.69,Default,,0000,0000,0000,,shot half of a chapter\Nonly one question.
Dialogue: 0,1:16:34.69,1:16:39.66,Default,,0000,0000,0000,,This is one example just--\Nnot involving [INAUDIBLE]
Dialogue: 0,1:16:39.66,1:16:41.21,Default,,0000,0000,0000,,of an expression like that, no.
Dialogue: 0,1:16:41.21,1:16:43.31,Default,,0000,0000,0000,,That's too time-consuming.
Dialogue: 0,1:16:43.31,1:16:47.60,Default,,0000,0000,0000,,But maybe just tangent of\Nx-squared plus y-squared,
Dialogue: 0,1:16:47.60,1:16:49.96,Default,,0000,0000,0000,,find the partial derivatives.
Dialogue: 0,1:16:49.96,1:16:53.38,Default,,0000,0000,0000,,That's a good exam\Nquestion, and that's enough
Dialogue: 0,1:16:53.38,1:16:55.90,Default,,0000,0000,0000,,when it comes to\Ntesting partials.
Dialogue: 0,1:16:55.90,1:16:58.13,Default,,0000,0000,0000,,By the way, how\Nmuch-- what is that?
Dialogue: 0,1:16:58.13,1:17:00.83,Default,,0000,0000,0000,,And I'm going to let\Nyou go right now.
Dialogue: 0,1:17:00.83,1:17:01.82,Default,,0000,0000,0000,,Use the bathroom.
Dialogue: 0,1:17:01.82,1:17:05.24,Default,,0000,0000,0000,,And when you come back from the\Nbathroom, we'll fill in this.
Dialogue: 0,1:17:05.24,1:17:10.52,Default,,0000,0000,0000,,You know I am horrible in the\Nsense that I want-- I'm greedy.
Dialogue: 0,1:17:10.52,1:17:12.06,Default,,0000,0000,0000,,I need extra time.
Dialogue: 0,1:17:12.06,1:17:15.34,Default,,0000,0000,0000,,I want to use more time.
Dialogue: 0,1:17:15.34,1:17:17.85,Default,,0000,0000,0000,,I will do your\Nproblems from now on,
Dialogue: 0,1:17:17.85,1:17:22.06,Default,,0000,0000,0000,,and you can use the bathroom,\Neat something, wash your hands.
Dialogue: 0,1:17:22.06,1:17:26.50,Default,,0000,0000,0000,,
Dialogue: 0,1:17:26.50,1:17:28.47,Default,,0000,0000,0000,,I'll start in\Nabout five minutes.
Dialogue: 0,1:17:28.47,1:17:29.46,Default,,0000,0000,0000,,Don't worry.
Dialogue: 0,1:17:29.46,1:17:32.91,Default,,0000,0000,0000,,
Dialogue: 0,1:17:32.91,1:17:33.89,Default,,0000,0000,0000,,Alexander?
Dialogue: 0,1:17:33.89,1:17:35.37,Default,,0000,0000,0000,,Are you here?
Dialogue: 0,1:17:35.37,1:17:37.84,Default,,0000,0000,0000,,Come get this.
Dialogue: 0,1:17:37.84,1:17:40.35,Default,,0000,0000,0000,,I apologize.
Dialogue: 0,1:17:40.35,1:17:42.43,Default,,0000,0000,0000,,This is long due back to you.
Dialogue: 0,1:17:42.43,1:17:43.42,Default,,0000,0000,0000,,STUDENT: Oh.
Dialogue: 0,1:17:43.42,1:17:43.92,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:17:43.92,1:17:47.41,Default,,0000,0000,0000,,
Dialogue: 0,1:17:47.41,1:17:49.90,Default,,0000,0000,0000,,STUDENT: Is there an\Nattendance sheet today?
Dialogue: 0,1:17:49.90,1:17:53.11,Default,,0000,0000,0000,,MAGDALENA TODA: I will--\NI'm making up one.
Dialogue: 0,1:17:53.11,1:17:56.75,Default,,0000,0000,0000,,There is already on\None side attendance.
Dialogue: 0,1:17:56.75,1:17:58.65,Default,,0000,0000,0000,,Let's use the other side.
Dialogue: 0,1:17:58.65,1:18:01.57,Default,,0000,0000,0000,,Put today's date.
Dialogue: 0,1:18:01.57,1:18:02.07,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,1:18:02.07,1:18:44.40,Default,,0000,0000,0000,,
Dialogue: 0,1:18:44.40,1:18:48.38,Default,,0000,0000,0000,,[SIDE CONVERSATIONS]
Dialogue: 0,1:18:48.38,1:18:56.35,Default,,0000,0000,0000,,
Dialogue: 0,1:18:56.35,1:18:58.34,Default,,0000,0000,0000,,MAGDALENA TODA: They\Nare spoiling me.
Dialogue: 0,1:18:58.34,1:19:03.33,Default,,0000,0000,0000,,They give me new\Nsprays every week.
Dialogue: 0,1:19:03.33,1:19:05.30,Default,,0000,0000,0000,,[INAUDIBLE] take care of this.
Dialogue: 0,1:19:05.30,1:19:09.23,Default,,0000,0000,0000,,[SIDE CONVERSATIONS]
Dialogue: 0,1:19:09.23,1:19:14.14,Default,,0000,0000,0000,,
Dialogue: 0,1:19:14.14,1:19:17.08,Default,,0000,0000,0000,,MAGDALENA TODA: So I'm\Ngoing to ask you something.
Dialogue: 0,1:19:17.08,1:19:20.53,Default,,0000,0000,0000,,And you respond honestly.
Dialogue: 0,1:19:20.53,1:19:24.81,Default,,0000,0000,0000,,Which chapter-- we already\Nbrowsed through three chapters.
Dialogue: 0,1:19:24.81,1:19:26.84,Default,,0000,0000,0000,,I mean, Chapter 9\Nwas vector spaces,
Dialogue: 0,1:19:26.84,1:19:29.54,Default,,0000,0000,0000,,and it was all review\Nfrom-- from what?
Dialogue: 0,1:19:29.54,1:19:30.54,Default,,0000,0000,0000,,From Calc 2.
Dialogue: 0,1:19:30.54,1:19:34.61,Default,,0000,0000,0000,,Chapter 10 was curves in\N[INAUDIBLE] and curves
Dialogue: 0,1:19:34.61,1:19:36.60,Default,,0000,0000,0000,,in space, practically.
Dialogue: 0,1:19:36.60,1:19:40.54,Default,,0000,0000,0000,,
Dialogue: 0,1:19:40.54,1:19:46.63,Default,,0000,0000,0000,,And Chapter 11 is functions\Nof several variables.
Dialogue: 0,1:19:46.63,1:19:48.64,Default,,0000,0000,0000,,Now you have a flavor\Nof all of them,
Dialogue: 0,1:19:48.64,1:19:50.48,Default,,0000,0000,0000,,which one was hardest for you?
Dialogue: 0,1:19:50.48,1:19:51.48,Default,,0000,0000,0000,,STUDENT: 9 and 10, both.
Dialogue: 0,1:19:51.48,1:19:52.72,Default,,0000,0000,0000,,MAGDALENA TODA: 9 and 10 both.
Dialogue: 0,1:19:52.72,1:19:53.71,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:19:53.71,1:19:56.42,Default,,0000,0000,0000,,MAGDALENA TODA: This is so\Nmuch better than the other.
Dialogue: 0,1:19:56.42,1:20:00.66,Default,,0000,0000,0000,,No, I think you guys\Nactually-- it looks better,
Dialogue: 0,1:20:00.66,1:20:06.70,Default,,0000,0000,0000,,because you've seen a lot more\Nvectors and vector functions.
Dialogue: 0,1:20:06.70,1:20:08.59,Default,,0000,0000,0000,,STUDENT: I didn't\Nunderstand any of 9 or 10.
Dialogue: 0,1:20:08.59,1:20:09.54,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:20:09.54,1:20:10.37,Default,,0000,0000,0000,,MAGDALENA TODA: Yes, ma'am.
Dialogue: 0,1:20:10.37,1:20:12.37,Default,,0000,0000,0000,,STUDENT: Could you go over\Nparametrization [INAUDIBLE]?
Dialogue: 0,1:20:12.37,1:20:14.37,Default,,0000,0000,0000,,MAGDALENA TODA: I will\Ngo over that again.
Dialogue: 0,1:20:14.37,1:20:18.37,Default,,0000,0000,0000,,And I will go over some\Nother parametrizations today.
Dialogue: 0,1:20:18.37,1:20:24.50,Default,,0000,0000,0000,,And I promised that at the\Nend, in those 20 minutes,
Dialogue: 0,1:20:24.50,1:20:28.38,Default,,0000,0000,0000,,I will do that problem that\Ngave a few of you trouble.
Dialogue: 0,1:20:28.38,1:20:29.18,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,1:20:29.18,1:20:30.80,Default,,0000,0000,0000,,STUDENT: Do we take\Nthe same final exam
Dialogue: 0,1:20:30.80,1:20:33.23,Default,,0000,0000,0000,,as all the other [INAUDIBLE]\Nclasses? [INAUDIBLE]?
Dialogue: 0,1:20:33.23,1:20:36.50,Default,,0000,0000,0000,,MAGDALENA TODA: Well, that's\Nwhat I was asked yesterday.
Dialogue: 0,1:20:36.50,1:20:43.15,Default,,0000,0000,0000,,So practically, it's at the\Nlatitude of the instructor who
Dialogue: 0,1:20:43.15,1:20:45.32,Default,,0000,0000,0000,,teaches honors if they\Nwrite their own final,
Dialogue: 0,1:20:45.32,1:20:48.62,Default,,0000,0000,0000,,and in general make\Nit harder, or they
Dialogue: 0,1:20:48.62,1:20:51.06,Default,,0000,0000,0000,,take the general final\Nlike everybody else.
Dialogue: 0,1:20:51.06,1:20:53.99,Default,,0000,0000,0000,,For your formative\Npurposes, and as a study,
Dialogue: 0,1:20:53.99,1:20:57.89,Default,,0000,0000,0000,,I would like you to\Ntake the general final,
Dialogue: 0,1:20:57.89,1:21:01.35,Default,,0000,0000,0000,,because I want to see\Nwhere you stand compared
Dialogue: 0,1:21:01.35,1:21:02.78,Default,,0000,0000,0000,,to the rest of the population.
Dialogue: 0,1:21:02.78,1:21:06.85,Default,,0000,0000,0000,,So you are my sample, and\Nthey are the entire student
Dialogue: 0,1:21:06.85,1:21:08.39,Default,,0000,0000,0000,,population of Calc\N3, I want to make
Dialogue: 0,1:21:08.39,1:21:14.49,Default,,0000,0000,0000,,the statistical analysis of your\Nperformance compared to them.
Dialogue: 0,1:21:14.49,1:21:16.45,Default,,0000,0000,0000,,STUDENT: So we'll\Ntake the regular one?
Dialogue: 0,1:21:16.45,1:21:17.42,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:21:17.42,1:21:19.00,Default,,0000,0000,0000,,For this one, I just\Nhave to make sure
Dialogue: 0,1:21:19.00,1:21:22.30,Default,,0000,0000,0000,,that they also have that\Nextra credit added in.
Dialogue: 0,1:21:22.30,1:21:25.84,Default,,0000,0000,0000,,Because if I have too much\Nextra credit in there,
Dialogue: 0,1:21:25.84,1:21:27.31,Default,,0000,0000,0000,,well they also count that.
Dialogue: 0,1:21:27.31,1:21:28.78,Default,,0000,0000,0000,,So that's what that means.
Dialogue: 0,1:21:28.78,1:21:30.25,Default,,0000,0000,0000,,So we can [INAUDIBLE].
Dialogue: 0,1:21:30.25,1:21:34.18,Default,,0000,0000,0000,,
Dialogue: 0,1:21:34.18,1:21:35.69,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:21:35.69,1:21:37.02,Default,,0000,0000,0000,,Let me finish this exercise.
Dialogue: 0,1:21:37.02,1:21:41.00,Default,,0000,0000,0000,,And then [? stop ?]\N[INAUDIBLE] and go
Dialogue: 0,1:21:41.00,1:21:45.64,Default,,0000,0000,0000,,over some homework problems and\Nsome parametrization problems.
Dialogue: 0,1:21:45.64,1:21:48.60,Default,,0000,0000,0000,,And I will see what else.
Dialogue: 0,1:21:48.60,1:21:55.54,Default,,0000,0000,0000,,So tangent of [INAUDIBLE].
Dialogue: 0,1:21:55.54,1:21:59.50,Default,,0000,0000,0000,,
Dialogue: 0,1:21:59.50,1:22:00.48,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,1:22:00.48,1:22:01.97,Default,,0000,0000,0000,,No, it's [INAUDIBLE].
Dialogue: 0,1:22:01.97,1:22:05.93,Default,,0000,0000,0000,,But you have to\Nremind me, because I
Dialogue: 0,1:22:05.93,1:22:09.42,Default,,0000,0000,0000,,pretend that I\Nforgot-- let me pretend
Dialogue: 0,1:22:09.42,1:22:14.25,Default,,0000,0000,0000,,that I forgot what the\Nderivative [INAUDIBLE] notation
Dialogue: 0,1:22:14.25,1:22:17.72,Default,,0000,0000,0000,,of tangent of t was.
Dialogue: 0,1:22:17.72,1:22:19.79,Default,,0000,0000,0000,,STUDENT: Secant squared.
Dialogue: 0,1:22:19.79,1:22:23.24,Default,,0000,0000,0000,,MAGDALENA TODA: You guys love\Nthat secant squared thingy.
Dialogue: 0,1:22:23.24,1:22:26.20,Default,,0000,0000,0000,,
Dialogue: 0,1:22:26.20,1:22:30.63,Default,,0000,0000,0000,,Why do you like secant squared?
Dialogue: 0,1:22:30.63,1:22:34.08,Default,,0000,0000,0000,,I, as a student, I didn't\Nlike expressing it like that.
Dialogue: 0,1:22:34.08,1:22:35.56,Default,,0000,0000,0000,,I liked [INAUDIBLE].
Dialogue: 0,1:22:35.56,1:22:37.04,Default,,0000,0000,0000,,Of course, it's the same thing.
Dialogue: 0,1:22:37.04,1:22:40.49,Default,,0000,0000,0000,,But I always like it like\N1 over cosine [INAUDIBLE].
Dialogue: 0,1:22:40.49,1:22:45.42,Default,,0000,0000,0000,,
Dialogue: 0,1:22:45.42,1:22:47.72,Default,,0000,0000,0000,,And of course, I have\Nto ask you something,
Dialogue: 0,1:22:47.72,1:22:52.22,Default,,0000,0000,0000,,because I'm curious to\Nsee what you remember.
Dialogue: 0,1:22:52.22,1:22:55.21,Default,,0000,0000,0000,,And you say yeah,\Ncuriosity killed the cat.
Dialogue: 0,1:22:55.21,1:23:00.19,Default,,0000,0000,0000,,But where did the\Nderivative exist?
Dialogue: 0,1:23:00.19,1:23:06.72,Default,,0000,0000,0000,,Because maybe was\Nthat tangent of T--
Dialogue: 0,1:23:06.72,1:23:08.30,Default,,0000,0000,0000,,STUDENT: Wasn't\Nit a quotient rule
Dialogue: 0,1:23:08.30,1:23:10.26,Default,,0000,0000,0000,,of sine and [? cosine x? ?]
Dialogue: 0,1:23:10.26,1:23:11.73,Default,,0000,0000,0000,,MAGDALENA TODA: Good.
Dialogue: 0,1:23:11.73,1:23:15.17,Default,,0000,0000,0000,,I'm proud of you.
Dialogue: 0,1:23:15.17,1:23:17.43,Default,,0000,0000,0000,,That is the answer.
Dialogue: 0,1:23:17.43,1:23:23.17,Default,,0000,0000,0000,,So [? my ?] [? have ?] this\Nblowing up, this blows up--
Dialogue: 0,1:23:23.17,1:23:29.52,Default,,0000,0000,0000,,blows up where cosine\NT was zero, right?
Dialogue: 0,1:23:29.52,1:23:32.38,Default,,0000,0000,0000,,So where did that blow up?
Dialogue: 0,1:23:32.38,1:23:36.78,Default,,0000,0000,0000,,[INAUDIBLE] blow up of\Ncosine and zero [INAUDIBLE].
Dialogue: 0,1:23:36.78,1:23:40.74,Default,,0000,0000,0000,,The cosine was the\Nshadow on the x-axis.
Dialogue: 0,1:23:40.74,1:23:43.99,Default,,0000,0000,0000,,So here you blow up here, you\Nblow up here, you blow up here,
Dialogue: 0,1:23:43.99,1:23:44.70,Default,,0000,0000,0000,,you blow up here.
Dialogue: 0,1:23:44.70,1:23:49.16,Default,,0000,0000,0000,,
Dialogue: 0,1:23:49.16,1:23:51.63,Default,,0000,0000,0000,,So [? what does ?] [INAUDIBLE].
Dialogue: 0,1:23:51.63,1:23:53.12,Default,,0000,0000,0000,,It should not be what?
Dialogue: 0,1:23:53.12,1:23:55.50,Default,,0000,0000,0000,,STUDENT: Pi over 2.
Dialogue: 0,1:23:55.50,1:23:56.37,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,1:23:56.37,1:23:59.63,Default,,0000,0000,0000,,And can we express\Nthat OK, among 0pi,
Dialogue: 0,1:23:59.63,1:24:03.32,Default,,0000,0000,0000,,let's say you go in\Nbetween 0 and 2pi only.
Dialogue: 0,1:24:03.32,1:24:08.30,Default,,0000,0000,0000,,I get rid of pi over\N2 and 3pi over 2.
Dialogue: 0,1:24:08.30,1:24:11.94,Default,,0000,0000,0000,,But if I express that in\Ngeneral for [INAUDIBLE] T
Dialogue: 0,1:24:11.94,1:24:15.00,Default,,0000,0000,0000,,not restricted to 0\Nto T, what do I say?
Dialogue: 0,1:24:15.00,1:24:15.96,Default,,0000,0000,0000,,STUDENT: It's k.
Dialogue: 0,1:24:15.96,1:24:18.82,Default,,0000,0000,0000,,STUDENT: So it can\N[? never be ?] pi over 2
Dialogue: 0,1:24:18.82,1:24:19.32,Default,,0000,0000,0000,,plus pi?
Dialogue: 0,1:24:19.32,1:24:21.24,Default,,0000,0000,0000,,MAGDALENA TODA: 2k plus 1.
Dialogue: 0,1:24:21.24,1:24:23.95,Default,,0000,0000,0000,,2k plus 1.
Dialogue: 0,1:24:23.95,1:24:25.40,Default,,0000,0000,0000,,Odd number over--
Dialogue: 0,1:24:25.40,1:24:26.24,Default,,0000,0000,0000,,STUDENT: Pi over 2.
Dialogue: 0,1:24:26.24,1:24:27.33,Default,,0000,0000,0000,,MAGDALENA TODA: Pi over 2.
Dialogue: 0,1:24:27.33,1:24:28.29,Default,,0000,0000,0000,,Odd number, pi over 2.
Dialogue: 0,1:24:28.29,1:24:30.20,Default,,0000,0000,0000,,And all the odd\Nnumbers are 2k plus 1.
Dialogue: 0,1:24:30.20,1:24:30.70,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:24:30.70,1:24:32.15,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:24:32.15,1:24:38.99,Default,,0000,0000,0000,,So you have a not\Nexistence and-- OK.
Dialogue: 0,1:24:38.99,1:24:39.64,Default,,0000,0000,0000,,Coming back.
Dialogue: 0,1:24:39.64,1:24:42.41,Default,,0000,0000,0000,,I'm just playing, because\Nwe are still in the break.
Dialogue: 0,1:24:42.41,1:24:44.29,Default,,0000,0000,0000,,Now we are ready.
Dialogue: 0,1:24:44.29,1:24:50.20,Default,,0000,0000,0000,,What is dfdx, del f, del x, xy.
Dialogue: 0,1:24:50.20,1:24:51.92,Default,,0000,0000,0000,,And what is del f, del y?
Dialogue: 0,1:24:51.92,1:24:55.60,Default,,0000,0000,0000,,I'm not going to ask you for\Nthe second partial derivative.
Dialogue: 0,1:24:55.60,1:24:57.38,Default,,0000,0000,0000,,We've had enough of that.
Dialogue: 0,1:24:57.38,1:25:05.23,Default,,0000,0000,0000,,We also agreed that we have\Nimportant results in that.
Dialogue: 0,1:25:05.23,1:25:08.98,Default,,0000,0000,0000,,What is the final answer here?
Dialogue: 0,1:25:08.98,1:25:13.94,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE] plus\Nx-squared [INAUDIBLE].
Dialogue: 0,1:25:13.94,1:25:15.49,Default,,0000,0000,0000,,MAGDALENA TODA: 1\Nover [INAUDIBLE].
Dialogue: 0,1:25:15.49,1:25:17.98,Default,,0000,0000,0000,,I love this one, OK?
Dialogue: 0,1:25:17.98,1:25:20.48,Default,,0000,0000,0000,,Don't tell me what I\Nwant to [INAUDIBLE].
Dialogue: 0,1:25:20.48,1:25:22.38,Default,,0000,0000,0000,,I'm just kidding.
Dialogue: 0,1:25:22.38,1:25:23.94,Default,,0000,0000,0000,,[INAUDIBLE] squared times--
Dialogue: 0,1:25:23.94,1:25:24.80,Default,,0000,0000,0000,,STUDENT: 2x.
Dialogue: 0,1:25:24.80,1:25:26.45,Default,,0000,0000,0000,,MAGDALENA TODA: 2x, good.
Dialogue: 0,1:25:26.45,1:25:27.70,Default,,0000,0000,0000,,How about the other one?
Dialogue: 0,1:25:27.70,1:25:28.44,Default,,0000,0000,0000,,The same thing.
Dialogue: 0,1:25:28.44,1:25:34.32,Default,,0000,0000,0000,,
Dialogue: 0,1:25:34.32,1:25:36.67,Default,,0000,0000,0000,,Times 2y.
Dialogue: 0,1:25:36.67,1:25:41.84,Default,,0000,0000,0000,,
Dialogue: 0,1:25:41.84,1:25:43.30,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:25:43.30,1:25:46.28,Default,,0000,0000,0000,,I want to tell you something\Nthat I will repeat.
Dialogue: 0,1:25:46.28,1:25:49.36,Default,,0000,0000,0000,,But you will see it\Nall through the course.
Dialogue: 0,1:25:49.36,1:25:52.18,Default,,0000,0000,0000,,There is a certain\Nnotion that Alexander,
Dialogue: 0,1:25:52.18,1:25:54.44,Default,,0000,0000,0000,,who is not talking--\NI'm just kidding,
Dialogue: 0,1:25:54.44,1:25:58.40,Default,,0000,0000,0000,,you can talk-- he\Nreminded me of gradient.
Dialogue: 0,1:25:58.40,1:26:02.96,Default,,0000,0000,0000,,We don't talk about gradient\Nuntil a few sections from now.
Dialogue: 0,1:26:02.96,1:26:05.12,Default,,0000,0000,0000,,But I'd like to\Nanticipate a little bit.
Dialogue: 0,1:26:05.12,1:26:08.44,Default,,0000,0000,0000,,So the gradient of\Na function, wherever
Dialogue: 0,1:26:08.44,1:26:15.08,Default,,0000,0000,0000,,the partial derivatives exist,\Nwith the partial derivative--
Dialogue: 0,1:26:15.08,1:26:21.22,Default,,0000,0000,0000,,that is, f sub x\Nand f sub y exist--
Dialogue: 0,1:26:21.22,1:26:26.57,Default,,0000,0000,0000,,I'm going to have that\Ndelta f-- nabla f.
Dialogue: 0,1:26:26.57,1:26:29.49,Default,,0000,0000,0000,,nabla is a [INAUDIBLE].
Dialogue: 0,1:26:29.49,1:26:34.34,Default,,0000,0000,0000,,Nable f at xy represents what?
Dialogue: 0,1:26:34.34,1:26:34.85,Default,,0000,0000,0000,,The vector.
Dialogue: 0,1:26:34.85,1:26:37.40,Default,,0000,0000,0000,,
Dialogue: 0,1:26:37.40,1:26:39.02,Default,,0000,0000,0000,,And I know you love vectors.
Dialogue: 0,1:26:39.02,1:26:45.58,Default,,0000,0000,0000,,And that's why I'm going back\Nto the vector notation f sub x
Dialogue: 0,1:26:45.58,1:26:51.51,Default,,0000,0000,0000,,at xy times i, i being\Nthe standard vector i
Dialogue: 0,1:26:51.51,1:26:59.13,Default,,0000,0000,0000,,unit along the x axis,\Nf sub y at xy times j.
Dialogue: 0,1:26:59.13,1:27:03.45,Default,,0000,0000,0000,,STUDENT: So it's just like\Nthe notation of [INAUDIBLE]?
Dialogue: 0,1:27:03.45,1:27:05.32,Default,,0000,0000,0000,,MAGDALENA TODA: Just\Nthe vector notation.
Dialogue: 0,1:27:05.32,1:27:08.01,Default,,0000,0000,0000,,How else could I write it?
Dialogue: 0,1:27:08.01,1:27:13.28,Default,,0000,0000,0000,,Angular bracket, f sub x x\Nat xy, comma, f sub y at xy.
Dialogue: 0,1:27:13.28,1:27:16.94,Default,,0000,0000,0000,,And you know-- people who\Nsaw my videos, colleagues
Dialogue: 0,1:27:16.94,1:27:19.77,Default,,0000,0000,0000,,who teach Calc 3\Nat the same time
Dialogue: 0,1:27:19.77,1:27:25.11,Default,,0000,0000,0000,,said I have a tendency of not\Ngoing by the book notations
Dialogue: 0,1:27:25.11,1:27:28.26,Default,,0000,0000,0000,,all the time, and just give you\Nthe [? round ?] parentheses.
Dialogue: 0,1:27:28.26,1:27:28.96,Default,,0000,0000,0000,,It's OK.
Dialogue: 0,1:27:28.96,1:27:31.34,Default,,0000,0000,0000,,I mean, different books,\Ndifferent notations.
Dialogue: 0,1:27:31.34,1:27:35.46,Default,,0000,0000,0000,,But what I mean is to represent\Nthe vector in the standard way
Dialogue: 0,1:27:35.46,1:27:37.43,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,1:27:37.43,1:27:38.01,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:27:38.01,1:27:38.51,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:27:38.51,1:27:41.76,Default,,0000,0000,0000,,Can you have this\Nnotion for something
Dialogue: 0,1:27:41.76,1:27:44.84,Default,,0000,0000,0000,,like a function of\Nthree variables?
Dialogue: 0,1:27:44.84,1:27:45.87,Default,,0000,0000,0000,,Absolutely.
Dialogue: 0,1:27:45.87,1:27:48.43,Default,,0000,0000,0000,,Now I'll give you an easy one.
Dialogue: 0,1:27:48.43,1:27:50.92,Default,,0000,0000,0000,,Suppose that you have\Nx-squared plus y-squared
Dialogue: 0,1:27:50.92,1:27:54.40,Default,,0000,0000,0000,,plus z-squared equals 1.
Dialogue: 0,1:27:54.40,1:28:00.13,Default,,0000,0000,0000,,And that is called-- let's\Ncall it names-- f of x, y, z.
Dialogue: 0,1:28:00.13,1:28:16.80,Default,,0000,0000,0000,,Compute the gradient nabla f\Nat any point x, y, z for f.
Dialogue: 0,1:28:16.80,1:28:20.70,Default,,0000,0000,0000,,Find the meaning of that\Ngradient-- of that-- find
Dialogue: 0,1:28:20.70,1:28:29.46,Default,,0000,0000,0000,,the geometric meaning of it.
Dialogue: 0,1:28:29.46,1:28:34.21,Default,,0000,0000,0000,,For this case, not in\Ngeneral, for this case.
Dialogue: 0,1:28:34.21,1:28:35.86,Default,,0000,0000,0000,,So you say, wait,\Nwait, Magdalena.
Dialogue: 0,1:28:35.86,1:28:38.85,Default,,0000,0000,0000,,A-dah-dah, you're confusing me.
Dialogue: 0,1:28:38.85,1:28:39.94,Default,,0000,0000,0000,,This is the gradient.
Dialogue: 0,1:28:39.94,1:28:40.44,Default,,0000,0000,0000,,Hmm.
Dialogue: 0,1:28:40.44,1:28:43.36,Default,,0000,0000,0000,,Depends on how many\Nvariables you have.
Dialogue: 0,1:28:43.36,1:28:47.68,Default,,0000,0000,0000,,So you have to show a vector\Nwhose coordinates represent
Dialogue: 0,1:28:47.68,1:28:50.80,Default,,0000,0000,0000,,the partial derivatives with\Nrespect to all the variables.
Dialogue: 0,1:28:50.80,1:28:56.06,Default,,0000,0000,0000,,If I have n variables, I have\Nf sub x1 comma f sub x2 comma
Dialogue: 0,1:28:56.06,1:28:58.77,Default,,0000,0000,0000,,f sub x3 comma f\Nsub xn, and stop.
Dialogue: 0,1:28:58.77,1:28:59.57,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,1:28:59.57,1:29:03.61,Default,,0000,0000,0000,,STUDENT: If the formula\Nwas just f of xy,
Dialogue: 0,1:29:03.61,1:29:05.16,Default,,0000,0000,0000,,wouldn't that be implicit?
Dialogue: 0,1:29:05.16,1:29:06.64,Default,,0000,0000,0000,,MAGDALENA TODA:\NThat is implicit.
Dialogue: 0,1:29:06.64,1:29:08.51,Default,,0000,0000,0000,,That's exactly what I meant.
Dialogue: 0,1:29:08.51,1:29:12.22,Default,,0000,0000,0000,,What's the geometric\Nmeaning of this animal?
Dialogue: 0,1:29:12.22,1:29:13.92,Default,,0000,0000,0000,,Forget about the left hand side.
Dialogue: 0,1:29:13.92,1:29:15.51,Default,,0000,0000,0000,,I'm going to clean it quickly.
Dialogue: 0,1:29:15.51,1:29:16.74,Default,,0000,0000,0000,,What is that animal?
Dialogue: 0,1:29:16.74,1:29:19.64,Default,,0000,0000,0000,,That is a hippopotamus.
Dialogue: 0,1:29:19.64,1:29:20.50,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:29:20.50,1:29:22.15,Default,,0000,0000,0000,,STUDENT: It's a sphere.
Dialogue: 0,1:29:22.15,1:29:23.40,Default,,0000,0000,0000,,MAGDALENA TODA: It's a sphere.
Dialogue: 0,1:29:23.40,1:29:24.85,Default,,0000,0000,0000,,But what kind of sphere?
Dialogue: 0,1:29:24.85,1:29:27.95,Default,,0000,0000,0000,,Center 0, 0, 0 with radius 1.
Dialogue: 0,1:29:27.95,1:29:30.15,Default,,0000,0000,0000,,What do we call that?
Dialogue: 0,1:29:30.15,1:29:30.67,Default,,0000,0000,0000,,Unit sphere.
Dialogue: 0,1:29:30.67,1:29:33.59,Default,,0000,0000,0000,,Do you know what notation\Nthat mathematicians
Dialogue: 0,1:29:33.59,1:29:36.52,Default,,0000,0000,0000,,use for that object?
Dialogue: 0,1:29:36.52,1:29:40.38,Default,,0000,0000,0000,,You don't know but I'll\Ntell you. s1 is the sphere.
Dialogue: 0,1:29:40.38,1:29:42.34,Default,,0000,0000,0000,,We have s2, I'm\Nsorry, the sphere
Dialogue: 0,1:29:42.34,1:29:45.19,Default,,0000,0000,0000,,of dimension 2, which\Nmeans the surface.
Dialogue: 0,1:29:45.19,1:29:47.13,Default,,0000,0000,0000,,s1 is the circle.
Dialogue: 0,1:29:47.13,1:29:49.12,Default,,0000,0000,0000,,s1 is a circle.
Dialogue: 0,1:29:49.12,1:29:51.58,Default,,0000,0000,0000,,s2 is a sphere.
Dialogue: 0,1:29:51.58,1:29:54.86,Default,,0000,0000,0000,,So what is this number\Nhere for a mathematician?
Dialogue: 0,1:29:54.86,1:29:59.05,Default,,0000,0000,0000,,That's the dimension of\Nthat kind of manifold.
Dialogue: 0,1:29:59.05,1:30:02.31,Default,,0000,0000,0000,,So if I have just a\Ncircle, we call it s1
Dialogue: 0,1:30:02.31,1:30:05.51,Default,,0000,0000,0000,,because there is only a one\Nindependent variable, which
Dialogue: 0,1:30:05.51,1:30:08.00,Default,,0000,0000,0000,,is time, and we parameterize.
Dialogue: 0,1:30:08.00,1:30:09.14,Default,,0000,0000,0000,,Why go clockwise?
Dialogue: 0,1:30:09.14,1:30:09.74,Default,,0000,0000,0000,,Shame on me.
Dialogue: 0,1:30:09.74,1:30:12.19,Default,,0000,0000,0000,,Go counterclockwise.
Dialogue: 0,1:30:12.19,1:30:13.03,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:30:13.03,1:30:14.11,Default,,0000,0000,0000,,That's s1.
Dialogue: 0,1:30:14.11,1:30:16.26,Default,,0000,0000,0000,,For s2, I have two\Ndegrees of freedom.
Dialogue: 0,1:30:16.26,1:30:18.77,Default,,0000,0000,0000,,It's a surface.
Dialogue: 0,1:30:18.77,1:30:23.00,Default,,0000,0000,0000,,On earth, what are those\Ntwo degrees of freedom?
Dialogue: 0,1:30:23.00,1:30:25.98,Default,,0000,0000,0000,,It's a riddle.
Dialogue: 0,1:30:25.98,1:30:27.06,Default,,0000,0000,0000,,No extra credit.
Dialogue: 0,1:30:27.06,1:30:30.32,Default,,0000,0000,0000,,STUDENT: The latitude\Nand longitude?
Dialogue: 0,1:30:30.32,1:30:31.88,Default,,0000,0000,0000,,MAGDALENA TODA: Who said it?
Dialogue: 0,1:30:31.88,1:30:33.53,Default,,0000,0000,0000,,Who said it first?
Dialogue: 0,1:30:33.53,1:30:35.30,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:30:35.30,1:30:40.18,Default,,0000,0000,0000,,MAGDALENA TODA: How many of\Nyou said it at the same time?
Dialogue: 0,1:30:40.18,1:30:40.94,Default,,0000,0000,0000,,Alexander said it.
Dialogue: 0,1:30:40.94,1:30:42.73,Default,,0000,0000,0000,,STUDENT: I know there\Nwas one other person.
Dialogue: 0,1:30:42.73,1:30:43.91,Default,,0000,0000,0000,,I wasn't the only one.
Dialogue: 0,1:30:43.91,1:30:44.66,Default,,0000,0000,0000,,STUDENT: I didn't.
Dialogue: 0,1:30:44.66,1:30:47.49,Default,,0000,0000,0000,,
Dialogue: 0,1:30:47.49,1:30:48.88,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE], sorry.
Dialogue: 0,1:30:48.88,1:30:49.72,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:30:49.72,1:30:52.42,Default,,0000,0000,0000,,MAGDALENA TODA: I\Ndon't have enough.
Dialogue: 0,1:30:52.42,1:30:54.34,Default,,0000,0000,0000,,STUDENT: I'll take\Nthe credit for it.
Dialogue: 0,1:30:54.34,1:30:56.17,Default,,0000,0000,0000,,MAGDALENA TODA:\N[INAUDIBLE] extra credit.
Dialogue: 0,1:30:56.17,1:30:58.62,Default,,0000,0000,0000,,OK, you choose.
Dialogue: 0,1:30:58.62,1:30:59.56,Default,,0000,0000,0000,,These are good.
Dialogue: 0,1:30:59.56,1:31:01.95,Default,,0000,0000,0000,,They are Valentine's hearts,\Nchocolate [INAUDIBLE].
Dialogue: 0,1:31:01.95,1:31:04.79,Default,,0000,0000,0000,,
Dialogue: 0,1:31:04.79,1:31:05.29,Default,,0000,0000,0000,,Wilson.
Dialogue: 0,1:31:05.29,1:31:09.38,Default,,0000,0000,0000,,
Dialogue: 0,1:31:09.38,1:31:12.04,Default,,0000,0000,0000,,I heard you saying Wilson.
Dialogue: 0,1:31:12.04,1:31:13.01,Default,,0000,0000,0000,,I have more.
Dialogue: 0,1:31:13.01,1:31:13.98,Default,,0000,0000,0000,,I have more.
Dialogue: 0,1:31:13.98,1:31:17.36,Default,,0000,0000,0000,,These are cough drops,\Nso I'm [INAUDIBLE].
Dialogue: 0,1:31:17.36,1:31:20.49,Default,,0000,0000,0000,,You set it right\Nnext time, Alexander.
Dialogue: 0,1:31:20.49,1:31:21.97,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:31:21.97,1:31:22.95,Default,,0000,0000,0000,,MAGDALENA TODA: OK.
Dialogue: 0,1:31:22.95,1:31:23.94,Default,,0000,0000,0000,,Anybody else?
Dialogue: 0,1:31:23.94,1:31:26.40,Default,,0000,0000,0000,,Anybody needing cough drops?
Dialogue: 0,1:31:26.40,1:31:26.90,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:31:26.90,1:31:27.88,Default,,0000,0000,0000,,I'll leave them here.
Dialogue: 0,1:31:27.88,1:31:29.36,Default,,0000,0000,0000,,Just let me see.
Dialogue: 0,1:31:29.36,1:31:31.83,Default,,0000,0000,0000,,Do I have more chocolate?
Dialogue: 0,1:31:31.83,1:31:32.99,Default,,0000,0000,0000,,Eh, next time.
Dialogue: 0,1:31:32.99,1:31:35.16,Default,,0000,0000,0000,,I'm going to get some\Nbefore-- we have-- we
Dialogue: 0,1:31:35.16,1:31:37.09,Default,,0000,0000,0000,,need before Valentine's, right?
Dialogue: 0,1:31:37.09,1:31:37.97,Default,,0000,0000,0000,,So it's Thursday.
Dialogue: 0,1:31:37.97,1:31:41.06,Default,,0000,0000,0000,,I'm going to bring\Nyou a lot more.
Dialogue: 0,1:31:41.06,1:31:46.65,Default,,0000,0000,0000,,So in that case, what\Nis the gradient of f?
Dialogue: 0,1:31:46.65,1:31:47.97,Default,,0000,0000,0000,,An x, y, z.
Dialogue: 0,1:31:47.97,1:31:48.47,Default,,0000,0000,0000,,Aha.
Dialogue: 0,1:31:48.47,1:31:50.17,Default,,0000,0000,0000,,I have three variables.
Dialogue: 0,1:31:50.17,1:31:52.52,Default,,0000,0000,0000,,What's the gradient?
Dialogue: 0,1:31:52.52,1:31:56.05,Default,,0000,0000,0000,,I can write it as a\Nbracket, angular notation.
Dialogue: 0,1:31:56.05,1:31:58.07,Default,,0000,0000,0000,,Am I right?
Dialogue: 0,1:31:58.07,1:32:02.79,Default,,0000,0000,0000,,Or I can write it 2xi\Nplus 2ij plus 2zk.
Dialogue: 0,1:32:02.79,1:32:06.59,Default,,0000,0000,0000,,Can anybody tell me why?
Dialogue: 0,1:32:06.59,1:32:09.64,Default,,0000,0000,0000,,What in the world are\Nthese, 2x, 2y, 2z?
Dialogue: 0,1:32:09.64,1:32:11.70,Default,,0000,0000,0000,,STUDENT: Those are the\Npartial derivatives.
Dialogue: 0,1:32:11.70,1:32:14.03,Default,,0000,0000,0000,,MAGDALENA TODA: They are\Nexactly the partial derivatives
Dialogue: 0,1:32:14.03,1:32:17.81,Default,,0000,0000,0000,,with respect to x, with respect\Nto y, with respect to z.
Dialogue: 0,1:32:17.81,1:32:19.41,Default,,0000,0000,0000,,Does this have a\Ngeometric meaning?
Dialogue: 0,1:32:19.41,1:32:20.54,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:32:20.54,1:32:21.87,Default,,0000,0000,0000,,I have to draw.
Dialogue: 0,1:32:21.87,1:32:24.21,Default,,0000,0000,0000,,And maybe when I\Ndraw, I get an idea.
Dialogue: 0,1:32:24.21,1:32:29.10,Default,,0000,0000,0000,,
Dialogue: 0,1:32:29.10,1:32:31.53,Default,,0000,0000,0000,,Is this a unit vector?
Dialogue: 0,1:32:31.53,1:32:32.48,Default,,0000,0000,0000,,Uh-uh.
Dialogue: 0,1:32:32.48,1:32:33.98,Default,,0000,0000,0000,,It's not.
Dialogue: 0,1:32:33.98,1:32:35.77,Default,,0000,0000,0000,,Nabla s, right.
Dialogue: 0,1:32:35.77,1:32:36.49,Default,,0000,0000,0000,,In a way it is.
Dialogue: 0,1:32:36.49,1:32:37.80,Default,,0000,0000,0000,,It's not a unit vector.
Dialogue: 0,1:32:37.80,1:32:41.04,Default,,0000,0000,0000,,But if I were to\N[? uniterize ?] it--
Dialogue: 0,1:32:41.04,1:32:43.71,Default,,0000,0000,0000,,and you know very well what it\Nmeans to [? uniterize it ?].
Dialogue: 0,1:32:43.71,1:32:44.73,Default,,0000,0000,0000,,It means to--
Dialogue: 0,1:32:44.73,1:32:45.69,Default,,0000,0000,0000,,STUDENT: Divide it by--
Dialogue: 0,1:32:45.69,1:32:47.44,Default,,0000,0000,0000,,MAGDALENA TODA: Divide\Nit by its magnitude
Dialogue: 0,1:32:47.44,1:32:51.08,Default,,0000,0000,0000,,and make it a unit vector\Nthat would have a meaning.
Dialogue: 0,1:32:51.08,1:32:52.18,Default,,0000,0000,0000,,This is the sphere.
Dialogue: 0,1:32:52.18,1:32:56.02,Default,,0000,0000,0000,,
Dialogue: 0,1:32:56.02,1:32:57.67,Default,,0000,0000,0000,,What if I make like this?
Dialogue: 0,1:32:57.67,1:33:04.14,Default,,0000,0000,0000,,n equals nabla f over\Na magnitude of f.
Dialogue: 0,1:33:04.14,1:33:10.05,Default,,0000,0000,0000,,And what is the meaning\Nof that going to be?
Dialogue: 0,1:33:10.05,1:33:12.07,Default,,0000,0000,0000,,Can you tell me what\NI'm going to get here?
Dialogue: 0,1:33:12.07,1:33:18.83,Default,,0000,0000,0000,,
Dialogue: 0,1:33:18.83,1:33:24.87,Default,,0000,0000,0000,,In your head,\Ncompute the magnitude
Dialogue: 0,1:33:24.87,1:33:29.43,Default,,0000,0000,0000,,and divide by the magnitude,\Nand you have exactly 15 seconds
Dialogue: 0,1:33:29.43,1:33:31.42,Default,,0000,0000,0000,,to tell me what it is.
Dialogue: 0,1:33:31.42,1:33:32.84,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:33:32.84,1:33:34.47,Default,,0000,0000,0000,,MAGDALENA TODA:\N[? Ryan, ?] [? Ryan, ?]
Dialogue: 0,1:33:34.47,1:33:36.17,Default,,0000,0000,0000,,you are in a Twilight Zone.
Dialogue: 0,1:33:36.17,1:33:39.80,Default,,0000,0000,0000,,But I'm sure once I tell you,\Nonce I tell you, [INAUDIBLE].
Dialogue: 0,1:33:39.80,1:33:41.55,Default,,0000,0000,0000,,STUDENT: 1 divided by\Nthe square root of 2
Dialogue: 0,1:33:41.55,1:33:42.92,Default,,0000,0000,0000,,for the [? i controller. ?]
Dialogue: 0,1:33:42.92,1:33:43.80,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:33:43.80,1:33:47.61,Default,,0000,0000,0000,,
Dialogue: 0,1:33:47.61,1:33:49.73,Default,,0000,0000,0000,,MAGDALENA TODA: Well, OK.
Dialogue: 0,1:33:49.73,1:33:51.12,Default,,0000,0000,0000,,Say it again, somebody.
Dialogue: 0,1:33:51.12,1:33:52.99,Default,,0000,0000,0000,,STUDENT: x plus y plus z.
Dialogue: 0,1:33:52.99,1:33:58.11,Default,,0000,0000,0000,,MAGDALENA TODA: xi plus yj\Nplus zk, not x plus x, y,
Dialogue: 0,1:33:58.11,1:33:59.82,Default,,0000,0000,0000,,z because that\Nwould be a mistake.
Dialogue: 0,1:33:59.82,1:34:03.32,Default,,0000,0000,0000,,It would be a scalar function.\N[INAUDIBLE] has to be a vector.
Dialogue: 0,1:34:03.32,1:34:07.19,Default,,0000,0000,0000,,If I am to draw this vector,\Nhow am I going to draw it?
Dialogue: 0,1:34:07.19,1:34:10.03,Default,,0000,0000,0000,,Well, this is the\Nposition vector.
Dialogue: 0,1:34:10.03,1:34:11.22,Default,,0000,0000,0000,,Say it again.
Dialogue: 0,1:34:11.22,1:34:12.72,Default,,0000,0000,0000,,This is the position vector.
Dialogue: 0,1:34:12.72,1:34:15.88,Default,,0000,0000,0000,,When I have a point on this\Nstinking earth, whatever
Dialogue: 0,1:34:15.88,1:34:21.10,Default,,0000,0000,0000,,it is, x, y, z, the\Nposition vector is x, y, z.
Dialogue: 0,1:34:21.10,1:34:26.26,Default,,0000,0000,0000,,It's xi plus yj plus zk.
Dialogue: 0,1:34:26.26,1:34:28.69,Default,,0000,0000,0000,,I have this identification\Nbetween the point
Dialogue: 0,1:34:28.69,1:34:29.64,Default,,0000,0000,0000,,and the vector.
Dialogue: 0,1:34:29.64,1:34:30.43,Default,,0000,0000,0000,,This is our vector.
Dialogue: 0,1:34:30.43,1:34:33.30,Default,,0000,0000,0000,,So I'm going to draw these\Nneedles, all these needles,
Dialogue: 0,1:34:33.30,1:34:41.88,Default,,0000,0000,0000,,all these vectors whose tips\Nare exactly on the sphere.
Dialogue: 0,1:34:41.88,1:34:42.96,Default,,0000,0000,0000,,So why?
Dialogue: 0,1:34:42.96,1:34:43.87,Default,,0000,0000,0000,,You say, OK.
Dialogue: 0,1:34:43.87,1:34:46.47,Default,,0000,0000,0000,,I understand that is\Nthe position vector,
Dialogue: 0,1:34:46.47,1:34:48.74,Default,,0000,0000,0000,,but why did you put an n here?
Dialogue: 0,1:34:48.74,1:34:52.90,Default,,0000,0000,0000,,And anybody who answers\Nthat gets a cough drops.
Dialogue: 0,1:34:52.90,1:34:54.74,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:34:54.74,1:34:56.07,Default,,0000,0000,0000,,MAGDALENA TODA: Because that is?
Dialogue: 0,1:34:56.07,1:34:58.24,Default,,0000,0000,0000,,STUDENT: The normal\Nto the surface.
Dialogue: 0,1:34:58.24,1:34:59.36,Default,,0000,0000,0000,,MAGDALENA TODA: You get a--
Dialogue: 0,1:34:59.36,1:35:00.82,Default,,0000,0000,0000,,STUDENT: Yeah, cough drop.
Dialogue: 0,1:35:00.82,1:35:02.29,Default,,0000,0000,0000,,MAGDALENA TODA: Two of them.
Dialogue: 0,1:35:02.29,1:35:03.16,Default,,0000,0000,0000,,STUDENT: Aw, yeah.
Dialogue: 0,1:35:03.16,1:35:04.24,Default,,0000,0000,0000,,MAGDALENA TODA: All right.
Dialogue: 0,1:35:04.24,1:35:07.84,Default,,0000,0000,0000,,So that's the normal\Nto the surface, which
Dialogue: 0,1:35:07.84,1:35:11.09,Default,,0000,0000,0000,,would be a continuation\Nof the position vector.
Dialogue: 0,1:35:11.09,1:35:11.67,Default,,0000,0000,0000,,You see, guys?
Dialogue: 0,1:35:11.67,1:35:14.45,Default,,0000,0000,0000,,So imagine you take\Nyour position vector.
Dialogue: 0,1:35:14.45,1:35:15.84,Default,,0000,0000,0000,,This is the sphere.
Dialogue: 0,1:35:15.84,1:35:17.58,Default,,0000,0000,0000,,It's like an egg.
Dialogue: 0,1:35:17.58,1:35:20.55,Default,,0000,0000,0000,,And these tips\Nare on the sphere.
Dialogue: 0,1:35:20.55,1:35:24.95,Default,,0000,0000,0000,,If you continue from\Nsitting on the sphere,
Dialogue: 0,1:35:24.95,1:35:29.24,Default,,0000,0000,0000,,another radius vector\Ncolinear to that,
Dialogue: 0,1:35:29.24,1:35:31.28,Default,,0000,0000,0000,,that would be the\Nnormal to the sphere.
Dialogue: 0,1:35:31.28,1:35:36.48,Default,,0000,0000,0000,,So in topology, we\Nhave a name for that.
Dialogue: 0,1:35:36.48,1:35:38.91,Default,,0000,0000,0000,,We call that the hairy ball.
Dialogue: 0,1:35:38.91,1:35:41.97,Default,,0000,0000,0000,,The hairy ball in\Nmathematics, I'm not kidding,
Dialogue: 0,1:35:41.97,1:35:44.41,Default,,0000,0000,0000,,it's a concentrated notations.
Dialogue: 0,1:35:44.41,1:35:47.62,Default,,0000,0000,0000,,You see it in graduate\Ncourses, if you're
Dialogue: 0,1:35:47.62,1:35:50.16,Default,,0000,0000,0000,,going to become a graduate\Nstudent in mathematics,
Dialogue: 0,1:35:50.16,1:35:51.99,Default,,0000,0000,0000,,or you want to do a\Ndual degree or whatever,
Dialogue: 0,1:35:51.99,1:35:55.69,Default,,0000,0000,0000,,you're going to see the hairy\Nball, all those normal vectors
Dialogue: 0,1:35:55.69,1:35:58.82,Default,,0000,0000,0000,,of length 1.
Dialogue: 0,1:35:58.82,1:36:01.62,Default,,0000,0000,0000,,It's also called\Nthe normal field.
Dialogue: 0,1:36:01.62,1:36:04.53,Default,,0000,0000,0000,,So if you ask Dr.\NIbragimov, because he
Dialogue: 0,1:36:04.53,1:36:08.90,Default,,0000,0000,0000,,is in this kind of field\Ntheory, [INAUDIBLE] normal field
Dialogue: 0,1:36:08.90,1:36:10.08,Default,,0000,0000,0000,,to a surface.
Dialogue: 0,1:36:10.08,1:36:13.01,Default,,0000,0000,0000,,But for the topologists\Nor geometers,
Dialogue: 0,1:36:13.01,1:36:15.30,Default,,0000,0000,0000,,they say, oh, that's\Nthe hairy ball.
Dialogue: 0,1:36:15.30,1:36:18.86,Default,,0000,0000,0000,,So if you ask him what the\Nhairy ball is, he will say,
Dialogue: 0,1:36:18.86,1:36:21.80,Default,,0000,0000,0000,,why are you talking\Nnonsense to me?
Dialogue: 0,1:36:21.80,1:36:22.78,Default,,0000,0000,0000,,Right.
Dialogue: 0,1:36:22.78,1:36:24.25,Default,,0000,0000,0000,,Exactly.
Dialogue: 0,1:36:24.25,1:36:30.74,Default,,0000,0000,0000,,So here's where we stopped\Nour intrusion in chapter 11.
Dialogue: 0,1:36:30.74,1:36:33.13,Default,,0000,0000,0000,,It's going to be as\Nfun as it was today
Dialogue: 0,1:36:33.13,1:36:34.42,Default,,0000,0000,0000,,with these partial derivatives.
Dialogue: 0,1:36:34.42,1:36:35.59,Default,,0000,0000,0000,,You're going to love them.
Dialogue: 0,1:36:35.59,1:36:39.86,Default,,0000,0000,0000,,You have a lot of computations\Nlike the ones we did today.
Dialogue: 0,1:36:39.86,1:36:42.59,Default,,0000,0000,0000,,Let's go back to\Nsomething you hated,
Dialogue: 0,1:36:42.59,1:36:45.63,Default,,0000,0000,0000,,which is the parameterizations.
Dialogue: 0,1:36:45.63,1:36:48.59,Default,,0000,0000,0000,,So one of you--\Nno, three of you--
Dialogue: 0,1:36:48.59,1:36:51.61,Default,,0000,0000,0000,,asked me to redo one\Nproblem like the one
Dialogue: 0,1:36:51.61,1:36:54.37,Default,,0000,0000,0000,,with the parameterization\Nof a circle.
Dialogue: 0,1:36:54.37,1:36:58.28,Default,,0000,0000,0000,,But now I have to pay\Nattention to the data
Dialogue: 0,1:36:58.28,1:36:59.92,Default,,0000,0000,0000,,that I come up with.
Dialogue: 0,1:36:59.92,1:37:14.26,Default,,0000,0000,0000,,So write the parameterization\Nof a circle of radius.
Dialogue: 0,1:37:14.26,1:37:17.24,Default,,0000,0000,0000,,
Dialogue: 0,1:37:17.24,1:37:20.89,Default,,0000,0000,0000,,Do you want specific\Ndata or you want letters?
Dialogue: 0,1:37:20.89,1:37:21.76,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:37:21.76,1:37:25.56,Default,,0000,0000,0000,,
Dialogue: 0,1:37:25.56,1:37:26.35,Default,,0000,0000,0000,,MAGDALENA TODA: OK.
Dialogue: 0,1:37:26.35,1:37:30.48,Default,,0000,0000,0000,,Let's do it [INAUDIBLE] r,\Nand then I'll give an example.
Dialogue: 0,1:37:30.48,1:37:43.23,Default,,0000,0000,0000,,And center x0, y0 in plane\Nwhere-- what is the point?
Dialogue: 0,1:37:43.23,1:37:57.20,Default,,0000,0000,0000,,Where is the particle\Nmoving for time t equals 0?
Dialogue: 0,1:37:57.20,1:37:59.26,Default,,0000,0000,0000,,Where is it located?
Dialogue: 0,1:37:59.26,1:38:00.31,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:38:00.31,1:38:02.75,Default,,0000,0000,0000,,So review.
Dialogue: 0,1:38:02.75,1:38:15.62,Default,,0000,0000,0000,,We had frame that we always\Npicked at the origin.
Dialogue: 0,1:38:15.62,1:38:23.49,Default,,0000,0000,0000,,That was bad because we could\Npick x0, y0 as a center,
Dialogue: 0,1:38:23.49,1:38:25.12,Default,,0000,0000,0000,,and that has a separate radius.
Dialogue: 0,1:38:25.12,1:38:31.90,Default,,0000,0000,0000,,
Dialogue: 0,1:38:31.90,1:38:39.05,Default,,0000,0000,0000,,And now, they want me to write\Na parameterization of a circle.
Dialogue: 0,1:38:39.05,1:38:41.02,Default,,0000,0000,0000,,How do you achieve it?
Dialogue: 0,1:38:41.02,1:38:49.40,Default,,0000,0000,0000,,You say the circle is x minus x0\Nsquared plus y minus y0 squared
Dialogue: 0,1:38:49.40,1:38:50.98,Default,,0000,0000,0000,,equals r squared.
Dialogue: 0,1:38:50.98,1:38:53.74,Default,,0000,0000,0000,,And one of you\Nasked me by email--
Dialogue: 0,1:38:53.74,1:38:56.65,Default,,0000,0000,0000,,and that was a good\Nquestion-- you said, come on.
Dialogue: 0,1:38:56.65,1:38:58.92,Default,,0000,0000,0000,,Look, it was [INAUDIBLE].
Dialogue: 0,1:38:58.92,1:39:02.76,Default,,0000,0000,0000,,So you said, I was\Nquite good in math.
Dialogue: 0,1:39:02.76,1:39:04.05,Default,,0000,0000,0000,,I was smart.
Dialogue: 0,1:39:04.05,1:39:09.55,Default,,0000,0000,0000,,Why didn't I know the equations,\Nthe parametric equations,
Dialogue: 0,1:39:09.55,1:39:11.49,Default,,0000,0000,0000,,or even this?
Dialogue: 0,1:39:11.49,1:39:13.73,Default,,0000,0000,0000,,I'll tell you why.
Dialogue: 0,1:39:13.73,1:39:15.85,Default,,0000,0000,0000,,This used to be\Ncovered in high school.
Dialogue: 0,1:39:15.85,1:39:18.06,Default,,0000,0000,0000,,It's something called\Ncollege algebra.
Dialogue: 0,1:39:18.06,1:39:21.46,Default,,0000,0000,0000,,We had a chapter,\Neither trigonometry
Dialogue: 0,1:39:21.46,1:39:22.27,Default,,0000,0000,0000,,or college algebra.
Dialogue: 0,1:39:22.27,1:39:24.52,Default,,0000,0000,0000,,We had a chapter called\Nanalytic geometry.
Dialogue: 0,1:39:24.52,1:39:26.34,Default,,0000,0000,0000,,This is analytic geometry.
Dialogue: 0,1:39:26.34,1:39:28.53,Default,,0000,0000,0000,,It's the same chapter\Nin which you guys
Dialogue: 0,1:39:28.53,1:39:33.51,Default,,0000,0000,0000,,covered conics, [INAUDIBLE],\Nellipse, [INAUDIBLE], parabola.
Dialogue: 0,1:39:33.51,1:39:36.12,Default,,0000,0000,0000,,It's no longer covered\Nin most high schools.
Dialogue: 0,1:39:36.12,1:39:37.03,Default,,0000,0000,0000,,I asked around.
Dialogue: 0,1:39:37.03,1:39:39.92,Default,,0000,0000,0000,,The teachers told\Nme that we reduced
Dialogue: 0,1:39:39.92,1:39:41.81,Default,,0000,0000,0000,,the geometric\Napplications a lot,
Dialogue: 0,1:39:41.81,1:39:47.92,Default,,0000,0000,0000,,according to the general\Nstandards that are imposed.
Dialogue: 0,1:39:47.92,1:39:51.60,Default,,0000,0000,0000,,That's a pity, because you\Nreally need this in college.
Dialogue: 0,1:39:51.60,1:39:52.59,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:39:52.59,1:39:55.52,Default,,0000,0000,0000,,So how do you come up\Nwith a parameterization?
Dialogue: 0,1:39:55.52,1:40:01.06,Default,,0000,0000,0000,,You say, I would like to\Nparameterize in such way
Dialogue: 0,1:40:01.06,1:40:03.49,Default,,0000,0000,0000,,that this would be\Neasy to understand
Dialogue: 0,1:40:03.49,1:40:06.30,Default,,0000,0000,0000,,this for Pythagorean theorem.
Dialogue: 0,1:40:06.30,1:40:07.45,Default,,0000,0000,0000,,Oh, OK.
Dialogue: 0,1:40:07.45,1:40:10.40,Default,,0000,0000,0000,,So what is the Pythagorean\Ntheorem telling me?
Dialogue: 0,1:40:10.40,1:40:14.24,Default,,0000,0000,0000,,It's telling you that if you are\Nin a unit circle practically,
Dialogue: 0,1:40:14.24,1:40:19.00,Default,,0000,0000,0000,,then this is cosine and\Ntheta and this is sine theta,
Dialogue: 0,1:40:19.00,1:40:21.64,Default,,0000,0000,0000,,and the sum of\Ncosine theta squared
Dialogue: 0,1:40:21.64,1:40:24.05,Default,,0000,0000,0000,,plus sine theta squared is 1.
Dialogue: 0,1:40:24.05,1:40:26.78,Default,,0000,0000,0000,,This is 1, so that is\Nthe Pythagorean theorem
Dialogue: 0,1:40:26.78,1:40:28.72,Default,,0000,0000,0000,,[INAUDIBLE].
Dialogue: 0,1:40:28.72,1:40:38.23,Default,,0000,0000,0000,,So xy plus x0 should be cosine\Nof theta times an R. Why an R?
Dialogue: 0,1:40:38.23,1:40:41.92,Default,,0000,0000,0000,,Because I want, when I square,\NI want the R squared up.
Dialogue: 0,1:40:41.92,1:40:46.23,Default,,0000,0000,0000,,And here, this guy inside\Nwill be our sine [? thing. ?]
Dialogue: 0,1:40:46.23,1:40:47.55,Default,,0000,0000,0000,,Am I going to be in good shape?
Dialogue: 0,1:40:47.55,1:40:51.45,Default,,0000,0000,0000,,Yes, because when I\Nsquare this fellow squared
Dialogue: 0,1:40:51.45,1:40:54.66,Default,,0000,0000,0000,,plus this fellow squared will\Ngive me exactly R squared.
Dialogue: 0,1:40:54.66,1:40:58.30,Default,,0000,0000,0000,,And here is my\N[INAUDIBLE] smiley face.
Dialogue: 0,1:40:58.30,1:41:01.26,Default,,0000,0000,0000,,So I want to understand\Nwhat I'm doing.
Dialogue: 0,1:41:01.26,1:41:05.44,Default,,0000,0000,0000,,x minus x0 must\Nbe R cosine theta.
Dialogue: 0,1:41:05.44,1:41:09.11,Default,,0000,0000,0000,,y minus y0 is R sine theta.
Dialogue: 0,1:41:09.11,1:41:13.86,Default,,0000,0000,0000,,Theta in general is an\Nangular velocity, [INAUDIBLE].
Dialogue: 0,1:41:13.86,1:41:17.25,Default,,0000,0000,0000,,But it's also time, right?
Dialogue: 0,1:41:17.25,1:41:19.29,Default,,0000,0000,0000,,It has the meaning\Nof time parameter.
Dialogue: 0,1:41:19.29,1:41:22.98,Default,,0000,0000,0000,,So when we wrote those--\Nand some of you are bored,
Dialogue: 0,1:41:22.98,1:41:25.62,Default,,0000,0000,0000,,but I think it's not\Ngoing to harm anybody
Dialogue: 0,1:41:25.62,1:41:27.24,Default,,0000,0000,0000,,that I do this again.
Dialogue: 0,1:41:27.24,1:41:36.41,Default,,0000,0000,0000,,R cosine of t plus x0 y is R\Nsine t plus x0, or plus y0.
Dialogue: 0,1:41:36.41,1:41:41.49,Default,,0000,0000,0000,,Now note, all those\Nexamples in web work,
Dialogue: 0,1:41:41.49,1:41:43.84,Default,,0000,0000,0000,,they were not very imaginative.
Dialogue: 0,1:41:43.84,1:41:47.58,Default,,0000,0000,0000,,They didn't mean for\Nyou to try other things.
Dialogue: 0,1:41:47.58,1:41:53.63,Default,,0000,0000,0000,,Like if one would put here\Ncosine of 5t or sine of 5t,
Dialogue: 0,1:41:53.63,1:41:56.83,Default,,0000,0000,0000,,that person would move five\Ntimes faster on the circle.
Dialogue: 0,1:41:56.83,1:42:00.24,Default,,0000,0000,0000,,And instead of being back\Nat 2 pi, in time 2 pi,
Dialogue: 0,1:42:00.24,1:42:02.97,Default,,0000,0000,0000,,they would be there\Nin time 2 pi over 5.
Dialogue: 0,1:42:02.97,1:42:06.86,Default,,0000,0000,0000,,All the examples-- and each of\Nyou, it was randomized somehow.
Dialogue: 0,1:42:06.86,1:42:09.73,Default,,0000,0000,0000,,Each of you has a\Ndifferent data set.
Dialogue: 0,1:42:09.73,1:42:11.97,Default,,0000,0000,0000,,Different R,\Ndifferent x0 with 0,
Dialogue: 0,1:42:11.97,1:42:15.57,Default,,0000,0000,0000,,and a different place where\Nthe particle is moving.
Dialogue: 0,1:42:15.57,1:42:18.58,Default,,0000,0000,0000,,But no matter what\Nthey gave you,
Dialogue: 0,1:42:18.58,1:42:21.91,Default,,0000,0000,0000,,it's a response to\Nthe same problem.
Dialogue: 0,1:42:21.91,1:42:26.93,Default,,0000,0000,0000,,And at time t equals\N0, you have M. Do
Dialogue: 0,1:42:26.93,1:42:28.65,Default,,0000,0000,0000,,you want me to call it M0?
Dialogue: 0,1:42:28.65,1:42:33.09,Default,,0000,0000,0000,,Yes, from my initial-- M0.
Dialogue: 0,1:42:33.09,1:42:41.04,Default,,0000,0000,0000,,For t equals 0, you're\Ngoing to have R plus x0.
Dialogue: 0,1:42:41.04,1:42:44.67,Default,,0000,0000,0000,,And for t equals 0, you have y0.
Dialogue: 0,1:42:44.67,1:42:50.42,Default,,0000,0000,0000,,So for example, Ryan had-- Ryan,\NI don't remember what you had.
Dialogue: 0,1:42:50.42,1:42:54.23,Default,,0000,0000,0000,,You had some where theta R was--
Dialogue: 0,1:42:54.23,1:42:54.94,Default,,0000,0000,0000,,STUDENT: 4 and 8.
Dialogue: 0,1:42:54.94,1:42:57.06,Default,,0000,0000,0000,,MAGDALENA TODA: 7.
Dialogue: 0,1:42:57.06,1:42:58.02,Default,,0000,0000,0000,,You, what did you have?
Dialogue: 0,1:42:58.02,1:43:00.40,Default,,0000,0000,0000,,STUDENT: No, R was 7\Nand x was 3, y was 1.
Dialogue: 0,1:43:00.40,1:43:03.51,Default,,0000,0000,0000,,MAGDALENA TODA: R\Nwas 7 and x0 was--
Dialogue: 0,1:43:03.51,1:43:05.74,Default,,0000,0000,0000,,STUDENT: 3, 1.
Dialogue: 0,1:43:05.74,1:43:11.57,Default,,0000,0000,0000,,MAGDALENA TODA: 3, 1 was x0, y0\Nso in that case, the point they
Dialogue: 0,1:43:11.57,1:43:15.82,Default,,0000,0000,0000,,gave here was 7 plus 3.
Dialogue: 0,1:43:15.82,1:43:16.82,Default,,0000,0000,0000,,Am I right, Ryan?
Dialogue: 0,1:43:16.82,1:43:17.70,Default,,0000,0000,0000,,You can always check.
Dialogue: 0,1:43:17.70,1:43:18.20,Default,,0000,0000,0000,,I remember.
Dialogue: 0,1:43:18.20,1:43:22.01,Default,,0000,0000,0000,,It was 10 and God\Nknows, and 10 and 1.
Dialogue: 0,1:43:22.01,1:43:25.81,Default,,0000,0000,0000,,So all of the data that\Nyou had in that problem
Dialogue: 0,1:43:25.81,1:43:30.48,Default,,0000,0000,0000,,was created so that you\Nhave these equations.
Dialogue: 0,1:43:30.48,1:43:36.39,Default,,0000,0000,0000,,And at time 0, you were exactly\Nat the time t equals 0 replaced
Dialogue: 0,1:43:36.39,1:43:37.18,Default,,0000,0000,0000,,the t.
Dialogue: 0,1:43:37.18,1:43:38.13,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:43:38.13,1:43:39.23,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:43:39.23,1:43:40.32,Default,,0000,0000,0000,,STUDENT: What's the M0?
Dialogue: 0,1:43:40.32,1:43:42.28,Default,,0000,0000,0000,,What is--
Dialogue: 0,1:43:42.28,1:43:45.24,Default,,0000,0000,0000,,MAGDALENA TODA: M0\Nis Magdalena times 0.
Dialogue: 0,1:43:45.24,1:43:46.82,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,1:43:46.82,1:43:51.04,Default,,0000,0000,0000,,I mean, it's the\Npoint where you are.
Dialogue: 0,1:43:51.04,1:43:55.14,Default,,0000,0000,0000,,I couldn't come up\Nwith a better name.
Dialogue: 0,1:43:55.14,1:44:01.57,Default,,0000,0000,0000,,So I'm going to\Nerase here and I'll
Dialogue: 0,1:44:01.57,1:44:08.28,Default,,0000,0000,0000,,get to another problem, which\Ngave you guys a big headache.
Dialogue: 0,1:44:08.28,1:44:16.57,Default,,0000,0000,0000,,And it's not so hard, but this\Nis the computational problem,
Dialogue: 0,1:44:16.57,1:44:18.05,Default,,0000,0000,0000,,very pretty in itself.
Dialogue: 0,1:44:18.05,1:44:24.93,Default,,0000,0000,0000,,
Dialogue: 0,1:44:24.93,1:44:35.71,Default,,0000,0000,0000,,[INAUDIBLE] cosine t i plus\Ne to the 3t sine t j plus e
Dialogue: 0,1:44:35.71,1:44:36.54,Default,,0000,0000,0000,,to the 3tk.
Dialogue: 0,1:44:36.54,1:44:40.04,Default,,0000,0000,0000,,
Dialogue: 0,1:44:40.04,1:44:43.85,Default,,0000,0000,0000,,And I think this was more or\Nless in everybody's homework
Dialogue: 0,1:44:43.85,1:44:45.48,Default,,0000,0000,0000,,the same.
Dialogue: 0,1:44:45.48,1:44:51.68,Default,,0000,0000,0000,,There's a position vector\Ngiven as parameterized form.
Dialogue: 0,1:44:51.68,1:44:54.33,Default,,0000,0000,0000,,So since you love\Nparameterization so much,
Dialogue: 0,1:44:54.33,1:45:00.52,Default,,0000,0000,0000,,I'm going to remind you what\Nthat means for x and y and zr.
Dialogue: 0,1:45:00.52,1:45:03.48,Default,,0000,0000,0000,,And what did they want from you?
Dialogue: 0,1:45:03.48,1:45:07.92,Default,,0000,0000,0000,,I forget what number of\Nthe problem that was.
Dialogue: 0,1:45:07.92,1:45:16.27,Default,,0000,0000,0000,,They wanted the length of\Nthe arc of a curve from t
Dialogue: 0,1:45:16.27,1:45:18.60,Default,,0000,0000,0000,,equals-- I don't know.
Dialogue: 0,1:45:18.60,1:45:19.58,Default,,0000,0000,0000,,STUDENT: 2 to 5.
Dialogue: 0,1:45:19.58,1:45:21.04,Default,,0000,0000,0000,,MAGDALENA TODA: 2 to 5.
Dialogue: 0,1:45:21.04,1:45:22.02,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,1:45:22.02,1:45:23.97,Default,,0000,0000,0000,,[INAUDIBLE] t equals 5.
Dialogue: 0,1:45:23.97,1:45:30.32,Default,,0000,0000,0000,,So this is the beginning and the\Nend of the curve, the beginning
Dialogue: 0,1:45:30.32,1:45:32.27,Default,,0000,0000,0000,,and the end of a curve.
Dialogue: 0,1:45:32.27,1:45:35.63,Default,,0000,0000,0000,,So what is that going\Nto be [INAUDIBLE]?
Dialogue: 0,1:45:35.63,1:45:40.50,Default,,0000,0000,0000,,How does [INAUDIBLE],\Nwhich we have
Dialogue: 0,1:45:40.50,1:45:46.83,Default,,0000,0000,0000,,to write down 2 to 5\Nmagnitude of r prime at t, dt.
Dialogue: 0,1:45:46.83,1:45:50.14,Default,,0000,0000,0000,,
Dialogue: 0,1:45:50.14,1:45:53.00,Default,,0000,0000,0000,,And I don't know.
Dialogue: 0,1:45:53.00,1:45:56.98,Default,,0000,0000,0000,,But I want to review this\Nbecause-- so what in the world?
Dialogue: 0,1:45:56.98,1:45:59.54,Default,,0000,0000,0000,,Maybe I put this\Non the midterm or I
Dialogue: 0,1:45:59.54,1:46:03.53,Default,,0000,0000,0000,,make it a little bit easier,\Nbut the same what I don't like,
Dialogue: 0,1:46:03.53,1:46:05.07,Default,,0000,0000,0000,,it's time consuming.
Dialogue: 0,1:46:05.07,1:46:07.78,Default,,0000,0000,0000,,But I can give you\Nsomething a lot easier
Dialogue: 0,1:46:07.78,1:46:10.57,Default,,0000,0000,0000,,that tests the\Nconcept, the idea, not
Dialogue: 0,1:46:10.57,1:46:13.09,Default,,0000,0000,0000,,the computational power.
Dialogue: 0,1:46:13.09,1:46:20.21,Default,,0000,0000,0000,,So r prime of t here with\Na little bit of attention,
Dialogue: 0,1:46:20.21,1:46:25.10,Default,,0000,0000,0000,,of course, most of you\Ncomputing this correctly.
Dialogue: 0,1:46:25.10,1:46:28.01,Default,,0000,0000,0000,,You are just a little bit scared\Nof what happened after that,
Dialogue: 0,1:46:28.01,1:46:30.47,Default,,0000,0000,0000,,and you should not be scared\Nbecause now I'll tell you
Dialogue: 0,1:46:30.47,1:46:32.86,Default,,0000,0000,0000,,why you shouldn't be scared.
Dialogue: 0,1:46:32.86,1:46:34.85,Default,,0000,0000,0000,,Chain rule, product rule.
Dialogue: 0,1:46:34.85,1:46:37.89,Default,,0000,0000,0000,,So I have first prime--
Dialogue: 0,1:46:37.89,1:46:38.90,Default,,0000,0000,0000,,STUDENT: 3.
Dialogue: 0,1:46:38.90,1:46:42.49,Default,,0000,0000,0000,,MAGDALENA TODA: 3 into the\N3e second and [? time ?]
Dialogue: 0,1:46:42.49,1:46:46.73,Default,,0000,0000,0000,,cosine t plus-- I'm\Ngoing to do that later.
Dialogue: 0,1:46:46.73,1:46:48.32,Default,,0000,0000,0000,,I know what you're thinking.
Dialogue: 0,1:46:48.32,1:46:49.79,Default,,0000,0000,0000,,STUDENT: e 3t.
Dialogue: 0,1:46:49.79,1:46:53.26,Default,,0000,0000,0000,,MAGDALENA TODA: e to\Nthe 3t minus sine.
Dialogue: 0,1:46:53.26,1:46:55.60,Default,,0000,0000,0000,,I'm not worried\Nabout this minus now.
Dialogue: 0,1:46:55.60,1:46:57.43,Default,,0000,0000,0000,,I'll take care of that later.
Dialogue: 0,1:46:57.43,1:46:58.01,Default,,0000,0000,0000,,Times i.
Dialogue: 0,1:46:58.01,1:47:00.71,Default,,0000,0000,0000,,
Dialogue: 0,1:47:00.71,1:47:03.39,Default,,0000,0000,0000,,Now with your\Npermission-- when you
Dialogue: 0,1:47:03.39,1:47:08.15,Default,,0000,0000,0000,,say, why is she not writing the\Nwhole thing in continuation?
Dialogue: 0,1:47:08.15,1:47:09.47,Default,,0000,0000,0000,,Because I don't want to.
Dialogue: 0,1:47:09.47,1:47:09.97,Default,,0000,0000,0000,,No.
Dialogue: 0,1:47:09.97,1:47:13.17,Default,,0000,0000,0000,,Because I want to help\Nyou see what's going on.
Dialogue: 0,1:47:13.17,1:47:16.31,Default,,0000,0000,0000,,You do the same kind of stuff\Nfor this individual one.
Dialogue: 0,1:47:16.31,1:47:17.80,Default,,0000,0000,0000,,I want to put it\Nright underneath.
Dialogue: 0,1:47:17.80,1:47:21.29,Default,,0000,0000,0000,,If I put it right underneath,\Nit's going to [? agree ?].
Dialogue: 0,1:47:21.29,1:47:23.97,Default,,0000,0000,0000,,Otherwise it's not\Ngoing to [? agree ?].
Dialogue: 0,1:47:23.97,1:47:32.37,Default,,0000,0000,0000,,E to the 3t times sine t\Nplus e to the 3t cosine t.
Dialogue: 0,1:47:32.37,1:47:34.11,Default,,0000,0000,0000,,You didn't have a\Nproblem because you
Dialogue: 0,1:47:34.11,1:47:36.00,Default,,0000,0000,0000,,know how to differentiate.
Dialogue: 0,1:47:36.00,1:47:40.78,Default,,0000,0000,0000,,You started having the\Nproblem from this point on.
Dialogue: 0,1:47:40.78,1:47:44.41,Default,,0000,0000,0000,,3 into the 3tk.
Dialogue: 0,1:47:44.41,1:47:47.01,Default,,0000,0000,0000,,The problem came when\Nyou were supposed
Dialogue: 0,1:47:47.01,1:47:55.51,Default,,0000,0000,0000,,to identify the coordinates and\Nsquare them and squeeze them
Dialogue: 0,1:47:55.51,1:47:57.28,Default,,0000,0000,0000,,under the same square root.
Dialogue: 0,1:47:57.28,1:48:01.25,Default,,0000,0000,0000,,And that drove you crazy\Nwhen you have enough.
Dialogue: 0,1:48:01.25,1:48:04.21,Default,,0000,0000,0000,,Let me put the minus here to\Nmake it more obvious what's
Dialogue: 0,1:48:04.21,1:48:06.36,Default,,0000,0000,0000,,going to happen.
Dialogue: 0,1:48:06.36,1:48:08.12,Default,,0000,0000,0000,,When you're going\Nto have problems
Dialogue: 0,1:48:08.12,1:48:09.77,Default,,0000,0000,0000,,like that in\Ndifferential equations,
Dialogue: 0,1:48:09.77,1:48:14.67,Default,,0000,0000,0000,,you better have the eye\Nfor it, [INAUDIBLE].
Dialogue: 0,1:48:14.67,1:48:18.87,Default,,0000,0000,0000,,You should be able to recognize\Nthis is like a pattern.
Dialogue: 0,1:48:18.87,1:48:26.58,Default,,0000,0000,0000,,Have you seen the\Nmovie A Beautiful Mind?
Dialogue: 0,1:48:26.58,1:48:27.21,Default,,0000,0000,0000,,STUDENT: Yeah.
Dialogue: 0,1:48:27.21,1:48:28.62,Default,,0000,0000,0000,,MAGDALENA TODA:\NOK, so Nash, when
Dialogue: 0,1:48:28.62,1:48:34.18,Default,,0000,0000,0000,,he was writing with the finger\Non everything, on the walls
Dialogue: 0,1:48:34.18,1:48:39.84,Default,,0000,0000,0000,,at Princeton, on the window,\Nhe was thinking of patterns.
Dialogue: 0,1:48:39.84,1:48:42.27,Default,,0000,0000,0000,,He's actually\Ntrying to-- and it's
Dialogue: 0,1:48:42.27,1:48:44.13,Default,,0000,0000,0000,,hard to visualize\Nwithout drawing,
Dialogue: 0,1:48:44.13,1:48:48.37,Default,,0000,0000,0000,,but this is what most of us\Nrecognize all the time when
Dialogue: 0,1:48:48.37,1:48:51.21,Default,,0000,0000,0000,,a mathematician writes\Ndown some computations
Dialogue: 0,1:48:51.21,1:48:52.90,Default,,0000,0000,0000,,in a different way.
Dialogue: 0,1:48:52.90,1:48:58.42,Default,,0000,0000,0000,,All we hope for is to get a\Nfew steps behind that board
Dialogue: 0,1:48:58.42,1:48:59.85,Default,,0000,0000,0000,,and see a pattern.
Dialogue: 0,1:48:59.85,1:49:02.34,Default,,0000,0000,0000,,And when you do that,\Nyou see the pattern.
Dialogue: 0,1:49:02.34,1:49:05.58,Default,,0000,0000,0000,,This is an a minus b\Nand that's an a plus b.
Dialogue: 0,1:49:05.58,1:49:08.68,Default,,0000,0000,0000,,And then you say, OK, if\NI'm going to square them,
Dialogue: 0,1:49:08.68,1:49:10.63,Default,,0000,0000,0000,,what's going to happen?
Dialogue: 0,1:49:10.63,1:49:15.33,Default,,0000,0000,0000,,When you square an a minus\Nb and you square an a plus b
Dialogue: 0,1:49:15.33,1:49:18.81,Default,,0000,0000,0000,,and you have this giggly\Nguy there-- leave him there.
Dialogue: 0,1:49:18.81,1:49:21.98,Default,,0000,0000,0000,,He's having too much fun.
Dialogue: 0,1:49:21.98,1:49:28.19,Default,,0000,0000,0000,,You actually develop these\Nguys and you put them one
Dialogue: 0,1:49:28.19,1:49:31.53,Default,,0000,0000,0000,,under the other\Nand say wow, what
Dialogue: 0,1:49:31.53,1:49:34.28,Default,,0000,0000,0000,,a beautiful simplification.
Dialogue: 0,1:49:34.28,1:49:36.76,Default,,0000,0000,0000,,When I'm going to\Nadd these guys,
Dialogue: 0,1:49:36.76,1:49:40.34,Default,,0000,0000,0000,,this thing in the middle\Nwill simply will cancel out,
Dialogue: 0,1:49:40.34,1:49:44.70,Default,,0000,0000,0000,,but the a squared will double\Nand the b squared will double.
Dialogue: 0,1:49:44.70,1:49:46.67,Default,,0000,0000,0000,,And that's the beauty\Nof seeing pattern.
Dialogue: 0,1:49:46.67,1:49:50.97,Default,,0000,0000,0000,,You see how there is\Nsomething symmetric and magic
Dialogue: 0,1:49:50.97,1:49:56.47,Default,,0000,0000,0000,,in mathematics that make\Nthe answer simplified.
Dialogue: 0,1:49:56.47,1:50:01.36,Default,,0000,0000,0000,,And that allows you to compress\Nyour equations that originally
Dialogue: 0,1:50:01.36,1:50:05.74,Default,,0000,0000,0000,,seemed to be a mess\Ninto something that's
Dialogue: 0,1:50:05.74,1:50:08.60,Default,,0000,0000,0000,,more easily expressed.
Dialogue: 0,1:50:08.60,1:50:11.45,Default,,0000,0000,0000,,So when you're going\Nto compute this r
Dialogue: 0,1:50:11.45,1:50:17.79,Default,,0000,0000,0000,,prime of t magic absolute\Nvalue of the magnitude, that's
Dialogue: 0,1:50:17.79,1:50:21.66,Default,,0000,0000,0000,,going to be square root of--\Ninstead of writing all the
Dialogue: 0,1:50:21.66,1:50:25.19,Default,,0000,0000,0000,,[INAUDIBLE], I hate writing\Nand rewriting the whole thing
Dialogue: 0,1:50:25.19,1:50:28.80,Default,,0000,0000,0000,,squared plus the whole thing\Nsquared plus this squared.
Dialogue: 0,1:50:28.80,1:50:32.97,Default,,0000,0000,0000,,If I love to write so much,\NI'd be in humanities and not
Dialogue: 0,1:50:32.97,1:50:34.78,Default,,0000,0000,0000,,in mathematics.
Dialogue: 0,1:50:34.78,1:50:41.18,Default,,0000,0000,0000,,So as a mathematician, how\Nam I going to write that?
Dialogue: 0,1:50:41.18,1:50:44.38,Default,,0000,0000,0000,,As a mathematician, I'm going\Nto use some sort of-- like the U
Dialogue: 0,1:50:44.38,1:50:44.92,Default,,0000,0000,0000,,substitution.
Dialogue: 0,1:50:44.92,1:50:48.91,Default,,0000,0000,0000,,So I say, I call this Mr.\NA, and I call this Mr. B.
Dialogue: 0,1:50:48.91,1:50:50.97,Default,,0000,0000,0000,,And that's A minus B,\Nand that's A plus B.
Dialogue: 0,1:50:50.97,1:50:53.81,Default,,0000,0000,0000,,And that's somebody else.
Dialogue: 0,1:50:53.81,1:50:57.47,Default,,0000,0000,0000,,So when I square\Nthe first guy, and I
Dialogue: 0,1:50:57.47,1:51:00.67,Default,,0000,0000,0000,,square the second component, and\NI square the third component,
Dialogue: 0,1:51:00.67,1:51:09.78,Default,,0000,0000,0000,,and I add them together,\NI'm going to get what?
Dialogue: 0,1:51:09.78,1:51:15.81,Default,,0000,0000,0000,,Square root of 2A\Nsquared plus 2B squared.
Dialogue: 0,1:51:15.81,1:51:19.01,Default,,0000,0000,0000,,Because I know that\Nthese are the first two.
Dialogue: 0,1:51:19.01,1:51:21.29,Default,,0000,0000,0000,,This guy squared\Nplus this guy squared
Dialogue: 0,1:51:21.29,1:51:23.43,Default,,0000,0000,0000,,is going to be\Nexactly 2A squared
Dialogue: 0,1:51:23.43,1:51:25.97,Default,,0000,0000,0000,,plus 2B squared,\Nnothing in the middle.
Dialogue: 0,1:51:25.97,1:51:28.83,Default,,0000,0000,0000,,These guys cancel out.
Dialogue: 0,1:51:28.83,1:51:30.64,Default,,0000,0000,0000,,STUDENT: A and B\Nare not the same.
Dialogue: 0,1:51:30.64,1:51:34.21,Default,,0000,0000,0000,,
Dialogue: 0,1:51:34.21,1:51:42.03,Default,,0000,0000,0000,,MAGDALENA TODA: Well,\Nyeah, you're right.
Dialogue: 0,1:51:42.03,1:51:45.87,Default,,0000,0000,0000,,Let me call-- you're\Nright, this is the same,
Dialogue: 0,1:51:45.87,1:51:47.63,Default,,0000,0000,0000,,but these are different.
Dialogue: 0,1:51:47.63,1:51:51.100,Default,,0000,0000,0000,,So let me call them\NA prime plus B prime.
Dialogue: 0,1:51:51.100,1:51:53.68,Default,,0000,0000,0000,,No, that's derivative.
Dialogue: 0,1:51:53.68,1:51:56.24,Default,,0000,0000,0000,,Let me call them C\Nand D-- very good,
Dialogue: 0,1:51:56.24,1:52:03.81,Default,,0000,0000,0000,,thank you-- C squared\Nplus 2CD plus D squared.
Dialogue: 0,1:52:03.81,1:52:06.37,Default,,0000,0000,0000,,
Dialogue: 0,1:52:06.37,1:52:08.13,Default,,0000,0000,0000,,But the principle is the same.
Dialogue: 0,1:52:08.13,1:52:11.49,Default,,0000,0000,0000,,So I'm going to have A\Nsquared plus C squared.
Dialogue: 0,1:52:11.49,1:52:12.94,Default,,0000,0000,0000,,This goes away.
Dialogue: 0,1:52:12.94,1:52:14.66,Default,,0000,0000,0000,,Why?
Dialogue: 0,1:52:14.66,1:52:18.53,Default,,0000,0000,0000,,Because this times that is\Nthe same as this times that.
Dialogue: 0,1:52:18.53,1:52:19.84,Default,,0000,0000,0000,,Say it again.
Dialogue: 0,1:52:19.84,1:52:22.69,Default,,0000,0000,0000,,If we look in the\Nmiddle, the middle term
Dialogue: 0,1:52:22.69,1:52:28.30,Default,,0000,0000,0000,,will have 3e to the 3t cosine\Nt times e to the 3t sine t.
Dialogue: 0,1:52:28.30,1:52:33.03,Default,,0000,0000,0000,,Middle term here is 3e to the\N3t e to the 3t sine and cosine.
Dialogue: 0,1:52:33.03,1:52:36.44,Default,,0000,0000,0000,,So they will cancel\Nout, this and that.
Dialogue: 0,1:52:36.44,1:52:40.19,Default,,0000,0000,0000,,So here I have the\Nsum of the square of A
Dialogue: 0,1:52:40.19,1:52:45.91,Default,,0000,0000,0000,,plus the square\Nof C. And here I'm
Dialogue: 0,1:52:45.91,1:52:50.68,Default,,0000,0000,0000,,going to have the square\Nof B plus the square of D.
Dialogue: 0,1:52:50.68,1:52:54.39,Default,,0000,0000,0000,,OK, now when I square this\Nand that, what do I get?
Dialogue: 0,1:52:54.39,1:52:57.22,Default,,0000,0000,0000,,
Dialogue: 0,1:52:57.22,1:53:00.97,Default,,0000,0000,0000,,The beauty of that-- let me\Nwrite it down then explicitly.
Dialogue: 0,1:53:00.97,1:53:06.91,Default,,0000,0000,0000,,9e to the 3t cosine squared\Nt remains from this guy.
Dialogue: 0,1:53:06.91,1:53:08.79,Default,,0000,0000,0000,,Plus from the square\Nof that, we'll
Dialogue: 0,1:53:08.79,1:53:19.59,Default,,0000,0000,0000,,have 9e to the 3t-- no, just 3,\N9 to the 6t, 9 to the 6t sine
Dialogue: 0,1:53:19.59,1:53:21.54,Default,,0000,0000,0000,,squared.
Dialogue: 0,1:53:21.54,1:53:22.87,Default,,0000,0000,0000,,So I take this guy.
Dialogue: 0,1:53:22.87,1:53:23.54,Default,,0000,0000,0000,,I square it.
Dialogue: 0,1:53:23.54,1:53:24.71,Default,,0000,0000,0000,,I take this guy.
Dialogue: 0,1:53:24.71,1:53:26.92,Default,,0000,0000,0000,,I square it.
Dialogue: 0,1:53:26.92,1:53:30.05,Default,,0000,0000,0000,,The middle terms will\Ndisappear, thank god.
Dialogue: 0,1:53:30.05,1:53:33.49,Default,,0000,0000,0000,,Then I have this guy, I square\Nit, that guy, I square it,
Dialogue: 0,1:53:33.49,1:53:34.84,Default,,0000,0000,0000,,good.
Dialogue: 0,1:53:34.84,1:53:41.29,Default,,0000,0000,0000,,Plus another parenthesis-- e\Nto the 6t sine squared t plus e
Dialogue: 0,1:53:41.29,1:53:44.27,Default,,0000,0000,0000,,to the 6t cosine squared t.
Dialogue: 0,1:53:44.27,1:53:47.11,Default,,0000,0000,0000,,
Dialogue: 0,1:53:47.11,1:53:50.34,Default,,0000,0000,0000,,So even if they don't\Ndouble because they're not
Dialogue: 0,1:53:50.34,1:53:52.82,Default,,0000,0000,0000,,the same thing, what\Nis the principle
Dialogue: 0,1:53:52.82,1:53:54.39,Default,,0000,0000,0000,,that will make my life easier?
Dialogue: 0,1:53:54.39,1:53:58.84,Default,,0000,0000,0000,,The same pattern\Nof simplification.
Dialogue: 0,1:53:58.84,1:54:00.73,Default,,0000,0000,0000,,What is that same pattern\Nof simplification?
Dialogue: 0,1:54:00.73,1:54:03.72,Default,,0000,0000,0000,,Look at the beauty\Nof this guy and look
Dialogue: 0,1:54:03.72,1:54:05.11,Default,,0000,0000,0000,,at the beauty of this guy.
Dialogue: 0,1:54:05.11,1:54:06.79,Default,,0000,0000,0000,,And then there is\Nsomething missing,
Dialogue: 0,1:54:06.79,1:54:12.56,Default,,0000,0000,0000,,the happy guy that was quiet\Nbecause I told him to be quiet.
Dialogue: 0,1:54:12.56,1:54:17.16,Default,,0000,0000,0000,,That's 9e to the 6t.
Dialogue: 0,1:54:17.16,1:54:18.49,Default,,0000,0000,0000,,He was there in the corner.
Dialogue: 0,1:54:18.49,1:54:22.33,Default,,0000,0000,0000,,And you had to square this\Nguy and square this guy
Dialogue: 0,1:54:22.33,1:54:26.19,Default,,0000,0000,0000,,and square this guy and\Nadd them on top together.
Dialogue: 0,1:54:26.19,1:54:27.65,Default,,0000,0000,0000,,Now what is the pattern?
Dialogue: 0,1:54:27.65,1:54:35.44,Default,,0000,0000,0000,,The pattern is 9e to the 6t\Nwith 9e to the 6t, same guy.
Dialogue: 0,1:54:35.44,1:54:38.19,Default,,0000,0000,0000,,The orange guys-- that's\Nwhy I love the colors.
Dialogue: 0,1:54:38.19,1:54:40.60,Default,,0000,0000,0000,,Cosine squared cosine\Nsquared will be 1.
Dialogue: 0,1:54:40.60,1:54:47.40,Default,,0000,0000,0000,,Another pattern like that, I\Nhave e to the 6t, to the 6t,
Dialogue: 0,1:54:47.40,1:54:52.26,Default,,0000,0000,0000,,and the same happy guys sine\Nsquared t, sine squared t,
Dialogue: 0,1:54:52.26,1:54:54.66,Default,,0000,0000,0000,,add them together is 1.
Dialogue: 0,1:54:54.66,1:55:00.50,Default,,0000,0000,0000,,So all in all, this mess\Nis not a mess anymore.
Dialogue: 0,1:55:00.50,1:55:11.29,Default,,0000,0000,0000,,So it becomes 9e to the 6t plus\Ne to the 6t plus 9e to the 6t.
Dialogue: 0,1:55:11.29,1:55:12.63,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,1:55:12.63,1:55:17.96,Default,,0000,0000,0000,,All right, now how many\Ne to the 6t's do we have?
Dialogue: 0,1:55:17.96,1:55:25.85,Default,,0000,0000,0000,,9 plus 9 plus 1, 19, square\Nroot of 19 e to the 6t.
Dialogue: 0,1:55:25.85,1:55:29.90,Default,,0000,0000,0000,,So when we integrate,\Nwe go integral
Dialogue: 0,1:55:29.90,1:55:33.41,Default,,0000,0000,0000,,from 2 to 5 square root of 19.
Dialogue: 0,1:55:33.41,1:55:34.85,Default,,0000,0000,0000,,Kick him out of your life.
Dialogue: 0,1:55:34.85,1:55:36.99,Default,,0000,0000,0000,,He's just making\Nyour life harder.
Dialogue: 0,1:55:36.99,1:55:40.06,Default,,0000,0000,0000,,And then you have square root\Nof e to the 6t e to the 3t.
Dialogue: 0,1:55:40.06,1:55:42.91,Default,,0000,0000,0000,,
Dialogue: 0,1:55:42.91,1:55:47.93,Default,,0000,0000,0000,,So after you kick\Nthe guy out, you
Dialogue: 0,1:55:47.93,1:55:55.06,Default,,0000,0000,0000,,have e to the 3t divided\Nby 3 between t equals 2
Dialogue: 0,1:55:55.06,1:55:58.17,Default,,0000,0000,0000,,and t equals 5.
Dialogue: 0,1:55:58.17,1:56:03.23,Default,,0000,0000,0000,,Actually, I took it right off\Nthe WeBWorK problem you had.
Dialogue: 0,1:56:03.23,1:56:06.10,Default,,0000,0000,0000,,So if you type this\Nin your WeBWorK--
Dialogue: 0,1:56:06.10,1:56:12.00,Default,,0000,0000,0000,,you probably already did-- you\Nshould get exactly the answer
Dialogue: 0,1:56:12.00,1:56:13.20,Default,,0000,0000,0000,,as being correct.
Dialogue: 0,1:56:13.20,1:56:17.80,Default,,0000,0000,0000,,
Dialogue: 0,1:56:17.80,1:56:24.16,Default,,0000,0000,0000,,On the exam, do not\Nexpect anything that long.
Dialogue: 0,1:56:24.16,1:56:26.72,Default,,0000,0000,0000,,The idea of simplifying\Nthese patterns
Dialogue: 0,1:56:26.72,1:56:31.78,Default,,0000,0000,0000,,by finding the sine cosine, sine\Nsquared plus cosine squared is
Dialogue: 0,1:56:31.78,1:56:33.11,Default,,0000,0000,0000,,1, is still going to be there.
Dialogue: 0,1:56:33.11,1:56:35.69,Default,,0000,0000,0000,,But don't expect\Nanything that long.
Dialogue: 0,1:56:35.69,1:56:43.37,Default,,0000,0000,0000,,Also, don't expect-- once\Nyou get to this state,
Dialogue: 0,1:56:43.37,1:56:44.82,Default,,0000,0000,0000,,I don't want an answer.
Dialogue: 0,1:56:44.82,1:56:46.27,Default,,0000,0000,0000,,This is the answer.
Dialogue: 0,1:56:46.27,1:56:48.20,Default,,0000,0000,0000,,That's the precise answer.
Dialogue: 0,1:56:48.20,1:56:52.56,Default,,0000,0000,0000,,I don't want any approximation\Nor anything like that.
Dialogue: 0,1:56:52.56,1:56:54.27,Default,,0000,0000,0000,,A few of you did this\Nwith a calculator.
Dialogue: 0,1:56:54.27,1:56:57.66,Default,,0000,0000,0000,,Well, you will not have\Ncalculators in the final.
Dialogue: 0,1:56:57.66,1:56:59.28,Default,,0000,0000,0000,,You are going to\Nhave easy problems.
Dialogue: 0,1:56:59.28,1:57:03.17,Default,,0000,0000,0000,,If you did that\Nwith a calculator,
Dialogue: 0,1:57:03.17,1:57:05.23,Default,,0000,0000,0000,,and you truncated\Nyour answer later,
Dialogue: 0,1:57:05.23,1:57:11.27,Default,,0000,0000,0000,,and if you were within\N0.01 of the correct answer,
Dialogue: 0,1:57:11.27,1:57:12.34,Default,,0000,0000,0000,,you were fine.
Dialogue: 0,1:57:12.34,1:57:14.86,Default,,0000,0000,0000,,But some people\Napproximated too much.
Dialogue: 0,1:57:14.86,1:57:16.78,Default,,0000,0000,0000,,And that's always a problem.
Dialogue: 0,1:57:16.78,1:57:19.49,Default,,0000,0000,0000,,So it's always a good\Nidea to enter something
Dialogue: 0,1:57:19.49,1:57:23.86,Default,,0000,0000,0000,,like that in WeBWorK.
Dialogue: 0,1:57:23.86,1:57:27.47,Default,,0000,0000,0000,,I said I wouldn't do it\Nexcept in the last 20 minutes.
Dialogue: 0,1:57:27.47,1:57:31.19,Default,,0000,0000,0000,,But I wanted to do\Nsomething like that.
Dialogue: 0,1:57:31.19,1:57:34.50,Default,,0000,0000,0000,,I want to give you another\Nexample, because you love
Dialogue: 0,1:57:34.50,1:57:39.22,Default,,0000,0000,0000,,parametrization so much it just\Noccurred to me that it would
Dialogue: 0,1:57:39.22,1:57:41.94,Default,,0000,0000,0000,,be very, very helpful--\Nmaybe, I don't
Dialogue: 0,1:57:41.94,1:57:47.06,Default,,0000,0000,0000,,know-- to give you another\Nproblem similar to this one.
Dialogue: 0,1:57:47.06,1:57:50.25,Default,,0000,0000,0000,,It's not in the book,\Nbut it was cooked up
Dialogue: 0,1:57:50.25,1:57:53.70,Default,,0000,0000,0000,,by one of my colleagues\Nfor his homework.
Dialogue: 0,1:57:53.70,1:58:02.55,Default,,0000,0000,0000,,So I'd like to show it to you.
Dialogue: 0,1:58:02.55,1:58:06.49,Default,,0000,0000,0000,,
Dialogue: 0,1:58:06.49,1:58:09.58,Default,,0000,0000,0000,,e to the t i is\Na parametrization
Dialogue: 0,1:58:09.58,1:58:13.24,Default,,0000,0000,0000,,of a [INAUDIBLE] space.
Dialogue: 0,1:58:13.24,1:58:28.14,Default,,0000,0000,0000,,Plus e to the minus t j\Nplus square root of 2 tk.
Dialogue: 0,1:58:28.14,1:58:36.03,Default,,0000,0000,0000,,
Dialogue: 0,1:58:36.03,1:58:37.47,Default,,0000,0000,0000,,And how do I know?
Dialogue: 0,1:58:37.47,1:58:41.10,Default,,0000,0000,0000,,Well, one of his\Nstudents came to me
Dialogue: 0,1:58:41.10,1:58:43.66,Default,,0000,0000,0000,,and asked for help\Nwith homework.
Dialogue: 0,1:58:43.66,1:58:51.45,Default,,0000,0000,0000,,Well, we don't give help when\Nit comes from another colleague.
Dialogue: 0,1:58:51.45,1:58:55.79,Default,,0000,0000,0000,,So in the end, the student\Nwent to the tutoring center.
Dialogue: 0,1:58:55.79,1:58:58.71,Default,,0000,0000,0000,,And the tutoring center\Nhelped only in parts.
Dialogue: 0,1:58:58.71,1:59:00.52,Default,,0000,0000,0000,,She came back to me.
Dialogue: 0,1:59:00.52,1:59:03.86,Default,,0000,0000,0000,,So what was the deal here?
Dialogue: 0,1:59:03.86,1:59:13.66,Default,,0000,0000,0000,,Find f prime of t in\Nthe most simplified form
Dialogue: 0,1:59:13.66,1:59:16.44,Default,,0000,0000,0000,,and find the absolute\Nvalue r prime of t
Dialogue: 0,1:59:16.44,1:59:17.83,Default,,0000,0000,0000,,in the most simplified form.
Dialogue: 0,1:59:17.83,1:59:22.83,Default,,0000,0000,0000,,
Dialogue: 0,1:59:22.83,1:59:31.83,Default,,0000,0000,0000,,And find the length of the\Narc of this curve between t
Dialogue: 0,1:59:31.83,1:59:33.82,Default,,0000,0000,0000,,equals 0 and t equals 1.
Dialogue: 0,1:59:33.82,1:59:36.63,Default,,0000,0000,0000,,If this were given\Nby a physicist,
Dialogue: 0,1:59:36.63,1:59:39.76,Default,,0000,0000,0000,,how would that physicist\Nreformulate the problem?
Dialogue: 0,1:59:39.76,1:59:47.90,Default,,0000,0000,0000,,He would say-- he or she--\Nwhat is the distance travelled
Dialogue: 0,1:59:47.90,1:59:54.45,Default,,0000,0000,0000,,by the particle between\N0 seconds and 1 second?
Dialogue: 0,1:59:54.45,1:59:56.12,Default,,0000,0000,0000,,So how do you write that?
Dialogue: 0,1:59:56.12,2:00:03.55,Default,,0000,0000,0000,,Integral from 0 to 1 of\Nr prime of t [INAUDIBLE].
Dialogue: 0,2:00:03.55,2:00:05.53,Default,,0000,0000,0000,,And you have to do the rest.
Dialogue: 0,2:00:05.53,2:00:08.51,Default,,0000,0000,0000,,
Dialogue: 0,2:00:08.51,2:00:13.04,Default,,0000,0000,0000,,So arguably, this is\Nthe Chapter 10 review.
Dialogue: 0,2:00:13.04,2:00:15.07,Default,,0000,0000,0000,,It's very useful for\Nthe midterm exam.
Dialogue: 0,2:00:15.07,2:00:17.57,Default,,0000,0000,0000,,So although we are\Njust doing this review,
Dialogue: 0,2:00:17.57,2:00:20.69,Default,,0000,0000,0000,,you should not erase\Nit from your memory.
Dialogue: 0,2:00:20.69,2:00:24.38,Default,,0000,0000,0000,,Because I don't like to\Nput surprise problems
Dialogue: 0,2:00:24.38,2:00:25.25,Default,,0000,0000,0000,,on the midterm.
Dialogue: 0,2:00:25.25,2:00:28.95,Default,,0000,0000,0000,,But if you worked a\Ncertain type of problem,
Dialogue: 0,2:00:28.95,2:00:31.32,Default,,0000,0000,0000,,you may expect\Nsomething like that.
Dialogue: 0,2:00:31.32,2:00:33.72,Default,,0000,0000,0000,,Maybe it's different\Nbut in the same spirit.
Dialogue: 0,2:00:33.72,2:00:37.69,Default,,0000,0000,0000,,r prime of t, who's going to\Nhelp me with r prime of t?
Dialogue: 0,2:00:37.69,2:00:40.72,Default,,0000,0000,0000,,
Dialogue: 0,2:00:40.72,2:00:44.14,Default,,0000,0000,0000,,This fellow-- e to the t.
Dialogue: 0,2:00:44.14,2:00:46.86,Default,,0000,0000,0000,,And how about that?
Dialogue: 0,2:00:46.86,2:00:50.30,Default,,0000,0000,0000,,Negative e to the negative t.
Dialogue: 0,2:00:50.30,2:00:53.25,Default,,0000,0000,0000,,STUDENT: I thought the arc\Nlength was the square root of 1
Dialogue: 0,2:00:53.25,2:00:56.19,Default,,0000,0000,0000,,plus f prime of t squared.
Dialogue: 0,2:00:56.19,2:00:58.73,Default,,0000,0000,0000,,
Dialogue: 0,2:00:58.73,2:01:02.36,Default,,0000,0000,0000,,MAGDALENA TODA:\NFor a plane curve.
Dialogue: 0,2:01:02.36,2:01:04.44,Default,,0000,0000,0000,,OK, let me remind you.
Dialogue: 0,2:01:04.44,2:01:05.98,Default,,0000,0000,0000,,If you have a plane\Ncurve y equals
Dialogue: 0,2:01:05.98,2:01:12.47,Default,,0000,0000,0000,,f of x, then this thing\Nwould become integral from A
Dialogue: 0,2:01:12.47,2:01:17.74,Default,,0000,0000,0000,,to B square root of 1\Nplus f prime of x dx.
Dialogue: 0,2:01:17.74,2:01:22.01,Default,,0000,0000,0000,,And that, did you do that\Nwith your Calc II instructor?
Dialogue: 0,2:01:22.01,2:01:25.74,Default,,0000,0000,0000,,How many of you\Nhad Dr. Williams?
Dialogue: 0,2:01:25.74,2:01:28.00,Default,,0000,0000,0000,,That was a wonderful\Nclass, wasn't it?
Dialogue: 0,2:01:28.00,2:01:29.38,Default,,0000,0000,0000,,And he taught that.
Dialogue: 0,2:01:29.38,2:01:31.46,Default,,0000,0000,0000,,And of course he\Nwas not supposed
Dialogue: 0,2:01:31.46,2:01:36.12,Default,,0000,0000,0000,,to tell you that was the\Nspeed of a parametric curve.
Dialogue: 0,2:01:36.12,2:01:39.02,Default,,0000,0000,0000,,If you were to\Nparametrize here, x of t
Dialogue: 0,2:01:39.02,2:01:44.00,Default,,0000,0000,0000,,was t and y of t\Nwould be f of t.
Dialogue: 0,2:01:44.00,2:01:45.45,Default,,0000,0000,0000,,He could have told you.
Dialogue: 0,2:01:45.45,2:01:46.32,Default,,0000,0000,0000,,Maybe he told you.
Dialogue: 0,2:01:46.32,2:01:47.47,Default,,0000,0000,0000,,Maybe you don't remember.
Dialogue: 0,2:01:47.47,2:01:48.99,Default,,0000,0000,0000,,OK, let's forget about it.
Dialogue: 0,2:01:48.99,2:01:50.34,Default,,0000,0000,0000,,That was Calc II.
Dialogue: 0,2:01:50.34,2:01:54.12,Default,,0000,0000,0000,,Now, coming back here,\NI have to list what?
Dialogue: 0,2:01:54.12,2:01:57.92,Default,,0000,0000,0000,,Square root of 2 times\Nt prime is one k.
Dialogue: 0,2:01:57.92,2:01:59.58,Default,,0000,0000,0000,,Who's going to help\Nme compute the speed
Dialogue: 0,2:01:59.58,2:02:02.38,Default,,0000,0000,0000,,and put it in a nice formula?
Dialogue: 0,2:02:02.38,2:02:04.16,Default,,0000,0000,0000,,Well, my god--
Dialogue: 0,2:02:04.16,2:02:04.100,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:02:04.100,2:02:08.23,Default,,0000,0000,0000,,
Dialogue: 0,2:02:08.23,2:02:10.79,Default,,0000,0000,0000,,MAGDALENA TODA: Ahh,\Nyou are too smart.
Dialogue: 0,2:02:10.79,2:02:15.15,Default,,0000,0000,0000,,Today you had some what is\Nthat called with caffeine
Dialogue: 0,2:02:15.15,2:02:17.04,Default,,0000,0000,0000,,and vitamins and--
Dialogue: 0,2:02:17.04,2:02:18.92,Default,,0000,0000,0000,,STUDENT: You're\Nthinking of Red Bull.
Dialogue: 0,2:02:18.92,2:02:20.34,Default,,0000,0000,0000,,MAGDALENA TODA: I know.
Dialogue: 0,2:02:20.34,2:02:22.66,Default,,0000,0000,0000,,That was very nice.
Dialogue: 0,2:02:22.66,2:02:23.74,Default,,0000,0000,0000,,I try to stay away.
Dialogue: 0,2:02:23.74,2:02:28.22,Default,,0000,0000,0000,,What is that called\Nwith the energy booster?
Dialogue: 0,2:02:28.22,2:02:29.26,Default,,0000,0000,0000,,STUDENT: I wouldn't know.
Dialogue: 0,2:02:29.26,2:02:30.49,Default,,0000,0000,0000,,STUDENT: 5-Hour Energy.
Dialogue: 0,2:02:30.49,2:02:31.72,Default,,0000,0000,0000,,MAGDALENA TODA: 5-Hour, OK.
Dialogue: 0,2:02:31.72,2:02:33.19,Default,,0000,0000,0000,,I used to have that.
Dialogue: 0,2:02:33.19,2:02:36.67,Default,,0000,0000,0000,,When I had that, I could\Nanticipate two steps computing.
Dialogue: 0,2:02:36.67,2:02:39.81,Default,,0000,0000,0000,,Just a joke, Alex,\Ndon't take it up.
Dialogue: 0,2:02:39.81,2:02:40.72,Default,,0000,0000,0000,,Very good observation.
Dialogue: 0,2:02:40.72,2:02:43.46,Default,,0000,0000,0000,,So Alex saw.
Dialogue: 0,2:02:43.46,2:02:45.65,Default,,0000,0000,0000,,He has a premonition.
Dialogue: 0,2:02:45.65,2:02:48.82,Default,,0000,0000,0000,,He can see two steps in advance.
Dialogue: 0,2:02:48.82,2:02:50.92,Default,,0000,0000,0000,,He said, OK, square that.
Dialogue: 0,2:02:50.92,2:02:52.71,Default,,0000,0000,0000,,You have e to the 2t.
Dialogue: 0,2:02:52.71,2:02:53.40,Default,,0000,0000,0000,,Square this.
Dialogue: 0,2:02:53.40,2:02:55.61,Default,,0000,0000,0000,,The minus doesn't matter.
Dialogue: 0,2:02:55.61,2:03:00.33,Default,,0000,0000,0000,,Plus e to the minus\N2t, and square that.
Dialogue: 0,2:03:00.33,2:03:02.56,Default,,0000,0000,0000,,Then he saw patterns.
Dialogue: 0,2:03:02.56,2:03:06.13,Default,,0000,0000,0000,,Because he is the\Nwizard 101 today.
Dialogue: 0,2:03:06.13,2:03:09.09,Default,,0000,0000,0000,,So what is the\Nwitchcraft he performed?
Dialogue: 0,2:03:09.09,2:03:10.47,Default,,0000,0000,0000,,Do you see?
Dialogue: 0,2:03:10.47,2:03:13.35,Default,,0000,0000,0000,,Does anybody else\Nsee the pattern?
Dialogue: 0,2:03:13.35,2:03:15.36,Default,,0000,0000,0000,,[? Nateesh ?] sees the pattern.
Dialogue: 0,2:03:15.36,2:03:16.72,Default,,0000,0000,0000,,Anybody illuminated?
Dialogue: 0,2:03:16.72,2:03:18.01,Default,,0000,0000,0000,,I didn't see it from the start.
Dialogue: 0,2:03:18.01,2:03:19.66,Default,,0000,0000,0000,,You guys saw it faster than me.
Dialogue: 0,2:03:19.66,2:03:23.19,Default,,0000,0000,0000,,It took me about a\Nminute and a half
Dialogue: 0,2:03:23.19,2:03:26.71,Default,,0000,0000,0000,,when I saw this\Nfor the first time.
Dialogue: 0,2:03:26.71,2:03:29.93,Default,,0000,0000,0000,,Is this a perfect square?
Dialogue: 0,2:03:29.93,2:03:32.06,Default,,0000,0000,0000,,Of who?
Dialogue: 0,2:03:32.06,2:03:36.35,Default,,0000,0000,0000,,e to the t plus e to\Nthe minus 2 squared
Dialogue: 0,2:03:36.35,2:03:40.39,Default,,0000,0000,0000,,is-- anybody else sees the\Npattern I don't have candy.
Dialogue: 0,2:03:40.39,2:03:44.21,Default,,0000,0000,0000,,Next time-- Alex,\N[INAUDIBLE], anybody else?
Dialogue: 0,2:03:44.21,2:03:47.00,Default,,0000,0000,0000,,Do you now see the\Npattern, e to the 2t plus
Dialogue: 0,2:03:47.00,2:03:51.34,Default,,0000,0000,0000,,e to the minus 2t plus\Ntwice the product?
Dialogue: 0,2:03:51.34,2:03:54.47,Default,,0000,0000,0000,,And that's where the student\Nwas having the problem.
Dialogue: 0,2:03:54.47,2:03:56.55,Default,,0000,0000,0000,,Where do you see the product?
Dialogue: 0,2:03:56.55,2:03:58.47,Default,,0000,0000,0000,,The product is 1.
Dialogue: 0,2:03:58.47,2:03:59.92,Default,,0000,0000,0000,,The product is 1 doubled.
Dialogue: 0,2:03:59.92,2:04:02.10,Default,,0000,0000,0000,,So you get 2.
Dialogue: 0,2:04:02.10,2:04:06.69,Default,,0000,0000,0000,,So it's indeed exactly\Nthe perfect square.
Dialogue: 0,2:04:06.69,2:04:09.43,Default,,0000,0000,0000,,So once-- it was a she.
Dialogue: 0,2:04:09.43,2:04:14.49,Default,,0000,0000,0000,,Once she saw the perfect\Nsquare, she was so happy.
Dialogue: 0,2:04:14.49,2:04:16.85,Default,,0000,0000,0000,,Because you get square\Nroot of the square.
Dialogue: 0,2:04:16.85,2:04:19.56,Default,,0000,0000,0000,,You get e to the t\Nplus e to the minus t.
Dialogue: 0,2:04:19.56,2:04:22.69,Default,,0000,0000,0000,,And that's a trivial thing\Nto integrate that you
Dialogue: 0,2:04:22.69,2:04:23.86,Default,,0000,0000,0000,,have no problem integrating.
Dialogue: 0,2:04:23.86,2:04:26.98,Default,,0000,0000,0000,,It's a positive\Nfunction, very beautiful.
Dialogue: 0,2:04:26.98,2:04:31.88,Default,,0000,0000,0000,,The professor who gave this was\NDr. [INAUDIBLE] from Denmark.
Dialogue: 0,2:04:31.88,2:04:34.73,Default,,0000,0000,0000,,He's one of the best\Nteachers we have.
Dialogue: 0,2:04:34.73,2:04:40.69,Default,,0000,0000,0000,,But he makes up his\Nhomework as far as I know.
Dialogue: 0,2:04:40.69,2:04:43.20,Default,,0000,0000,0000,,I think in the sixth\Nedition, this edition,
Dialogue: 0,2:04:43.20,2:04:48.77,Default,,0000,0000,0000,,we actually stole his idea,\Nand we made a problem like that
Dialogue: 0,2:04:48.77,2:04:51.49,Default,,0000,0000,0000,,in the book somewhere.
Dialogue: 0,2:04:51.49,2:04:55.19,Default,,0000,0000,0000,,We doubled the number of\Nproblems more or less.
Dialogue: 0,2:04:55.19,2:05:00.90,Default,,0000,0000,0000,,So if you are to compute\N0 to 1 of the speed,
Dialogue: 0,2:05:00.90,2:05:03.07,Default,,0000,0000,0000,,what is the speed?
Dialogue: 0,2:05:03.07,2:05:05.53,Default,,0000,0000,0000,,The speed is this\Nbeautiful thing.
Dialogue: 0,2:05:05.53,2:05:09.97,Default,,0000,0000,0000,,Because you were able\Nto see the pattern.
Dialogue: 0,2:05:09.97,2:05:12.76,Default,,0000,0000,0000,,If you're not able\Nto see that, do you
Dialogue: 0,2:05:12.76,2:05:15.44,Default,,0000,0000,0000,,realize it's\Nimpossible, practically,
Dialogue: 0,2:05:15.44,2:05:17.94,Default,,0000,0000,0000,,for you to integrate by hand?
Dialogue: 0,2:05:17.94,2:05:22.70,Default,,0000,0000,0000,,You have to go to a\Ncalculator, Matlab, whatever.
Dialogue: 0,2:05:22.70,2:05:23.83,Default,,0000,0000,0000,,So this is easy.
Dialogue: 0,2:05:23.83,2:05:29.17,Default,,0000,0000,0000,,Why is that easy? e to the t\Nminus e to the minus t at 1
Dialogue: 0,2:05:29.17,2:05:32.04,Default,,0000,0000,0000,,and at 0-- you compare them.
Dialogue: 0,2:05:32.04,2:05:36.42,Default,,0000,0000,0000,,You get at 1 e minus\Ne to the minus 1
Dialogue: 0,2:05:36.42,2:05:41.09,Default,,0000,0000,0000,,minus the fundamental theorem\Nof calc e to the 0 minus
Dialogue: 0,2:05:41.09,2:05:42.50,Default,,0000,0000,0000,,e to the 0.
Dialogue: 0,2:05:42.50,2:05:43.62,Default,,0000,0000,0000,,Well, that's silly.
Dialogue: 0,2:05:43.62,2:05:45.44,Default,,0000,0000,0000,,Why is that silly?
Dialogue: 0,2:05:45.44,2:05:49.17,Default,,0000,0000,0000,,Because I'm going to give it up.
Dialogue: 0,2:05:49.17,2:05:52.11,Default,,0000,0000,0000,,So the answer was\Ne to the minus 1/e.
Dialogue: 0,2:05:52.11,2:05:54.57,Default,,0000,0000,0000,,And she knew what\Nthe answer would be.
Dialogue: 0,2:05:54.57,2:05:57.03,Default,,0000,0000,0000,,But she didn't know why.
Dialogue: 0,2:05:57.03,2:05:58.43,Default,,0000,0000,0000,,So she came back to me.
Dialogue: 0,2:05:58.43,2:06:02.61,Default,,0000,0000,0000,,I don't know how the tutoring\Ncenter helped her figure
Dialogue: 0,2:06:02.61,2:06:03.47,Default,,0000,0000,0000,,out the answer.
Dialogue: 0,2:06:03.47,2:06:06.20,Default,,0000,0000,0000,,But she did not\Nunderstand the solution.
Dialogue: 0,2:06:06.20,2:06:08.95,Default,,0000,0000,0000,,So I said, I'm not going to\Ntake anymore people coming
Dialogue: 0,2:06:08.95,2:06:11.02,Default,,0000,0000,0000,,from Professor [INAUDIBLE].
Dialogue: 0,2:06:11.02,2:06:12.83,Default,,0000,0000,0000,,I was also told it's not OK.
Dialogue: 0,2:06:12.83,2:06:16.67,Default,,0000,0000,0000,,So don't go to another\Nprofessor with homework coming
Dialogue: 0,2:06:16.67,2:06:18.41,Default,,0000,0000,0000,,for me or the other way around.
Dialogue: 0,2:06:18.41,2:06:20.60,Default,,0000,0000,0000,,Because it's not OK.
Dialogue: 0,2:06:20.60,2:06:25.31,Default,,0000,0000,0000,,But you can go to the tutoring\Ncenter asking them for hints.
Dialogue: 0,2:06:25.31,2:06:30.22,Default,,0000,0000,0000,,They're open starting 9:00\NAM and until around when?
Dialogue: 0,2:06:30.22,2:06:31.56,Default,,0000,0000,0000,,Do you know?
Dialogue: 0,2:06:31.56,2:06:32.99,Default,,0000,0000,0000,,They used to have until 4:00.
Dialogue: 0,2:06:32.99,2:06:35.87,Default,,0000,0000,0000,,But now they're going to\Nwork on an extended schedule
Dialogue: 0,2:06:35.87,2:06:37.85,Default,,0000,0000,0000,,until 8:00 PM.
Dialogue: 0,2:06:37.85,2:06:40.32,Default,,0000,0000,0000,,It's going to be\Nsomething crazy.
Dialogue: 0,2:06:40.32,2:06:43.79,Default,,0000,0000,0000,,Now, the thing is, we want\Nthe students to be better,
Dialogue: 0,2:06:43.79,2:06:48.62,Default,,0000,0000,0000,,to do better, to not give\Nup, to be successful,
Dialogue: 0,2:06:48.62,2:06:51.73,Default,,0000,0000,0000,,top one, two, three.
Dialogue: 0,2:06:51.73,2:06:54.29,Default,,0000,0000,0000,,I'm a little bit\Nconcerned, but maybe I
Dialogue: 0,2:06:54.29,2:06:56.57,Default,,0000,0000,0000,,shouldn't be, about those hours.
Dialogue: 0,2:06:56.57,2:06:59.94,Default,,0000,0000,0000,,So I don't know if they managed\Nto put a security camera
Dialogue: 0,2:06:59.94,2:07:00.90,Default,,0000,0000,0000,,or not.
Dialogue: 0,2:07:00.90,2:07:04.52,Default,,0000,0000,0000,,But having extended\Nhours may be a problem.
Dialogue: 0,2:07:04.52,2:07:09.78,Default,,0000,0000,0000,,Take advantage of\Nthose afternoon hours,
Dialogue: 0,2:07:09.78,2:07:11.73,Default,,0000,0000,0000,,especially if you are busy.
Dialogue: 0,2:07:11.73,2:07:18.70,Default,,0000,0000,0000,,Those late hours will\Nbe a big help for you.
Dialogue: 0,2:07:18.70,2:07:21.27,Default,,0000,0000,0000,,Do you know where it is?
Dialogue: 0,2:07:21.27,2:07:23.66,Default,,0000,0000,0000,,Room 106 over there.
Dialogue: 0,2:07:23.66,2:07:26.53,Default,,0000,0000,0000,,
Dialogue: 0,2:07:26.53,2:07:29.80,Default,,0000,0000,0000,,Any other questions related\Nto this type of problem
Dialogue: 0,2:07:29.80,2:07:35.24,Default,,0000,0000,0000,,or related to anything\Nelse in the material
Dialogue: 0,2:07:35.24,2:07:38.87,Default,,0000,0000,0000,,that maybe I can\Ngive you hints on,
Dialogue: 0,2:07:38.87,2:07:40.97,Default,,0000,0000,0000,,at least the hint I'm\Ngoing to give you?
Dialogue: 0,2:07:40.97,2:07:44.86,Default,,0000,0000,0000,,Sometimes I cannot stop, and\NI just give the problem away.
Dialogue: 0,2:07:44.86,2:07:46.33,Default,,0000,0000,0000,,I'm not supposed to do that.
Dialogue: 0,2:07:46.33,2:07:50.75,Default,,0000,0000,0000,,
Dialogue: 0,2:07:50.75,2:07:54.30,Default,,0000,0000,0000,,Look at your WeBWorK, see what\Nkind of help I can give you.
Dialogue: 0,2:07:54.30,2:07:56.43,Default,,0000,0000,0000,,You still have a\Nlittle bit of time.
Dialogue: 0,2:07:56.43,2:07:57.26,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:07:57.26,2:08:00.71,Default,,0000,0000,0000,,
Dialogue: 0,2:08:00.71,2:08:05.14,Default,,0000,0000,0000,,MAGDALENA TODA: That's\Nthe maximum of what?
Dialogue: 0,2:08:05.14,2:08:06.68,Default,,0000,0000,0000,,It was--
Dialogue: 0,2:08:06.68,2:08:07.51,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:08:07.51,2:08:11.11,Default,,0000,0000,0000,,
Dialogue: 0,2:08:11.11,2:08:12.68,Default,,0000,0000,0000,,MAGDALENA TODA: Was\Nthis the problem?
Dialogue: 0,2:08:12.68,2:08:14.60,Default,,0000,0000,0000,,STUDENT: e to the 2x\Nor something like that.
Dialogue: 0,2:08:14.60,2:08:15.56,Default,,0000,0000,0000,,MAGDALENA TODA:\NSomething like that?
Dialogue: 0,2:08:15.56,2:08:16.06,Default,,0000,0000,0000,,I erased it.
Dialogue: 0,2:08:16.06,2:08:19.40,Default,,0000,0000,0000,,STUDENT: You erased\Nthat? [INAUDIBLE].
Dialogue: 0,2:08:19.40,2:08:21.33,Default,,0000,0000,0000,,I found an answer.
Dialogue: 0,2:08:21.33,2:08:23.40,Default,,0000,0000,0000,,MAGDALENA TODA: It's\Nvery computational I saw.
Dialogue: 0,2:08:23.40,2:08:26.75,Default,,0000,0000,0000,,But before that, I\Nsaw that seven of you
Dialogue: 0,2:08:26.75,2:08:28.99,Default,,0000,0000,0000,,guys-- you two also did it.
Dialogue: 0,2:08:28.99,2:08:33.71,Default,,0000,0000,0000,,So I wrote-- you have a\Nbrownie waiting for that.
Dialogue: 0,2:08:33.71,2:08:35.16,Default,,0000,0000,0000,,But then I erased it.
Dialogue: 0,2:08:35.16,2:08:39.51,Default,,0000,0000,0000,,STUDENT: You erased the previous\None too in the homework one.
Dialogue: 0,2:08:39.51,2:08:42.04,Default,,0000,0000,0000,,MAGDALENA TODA: Because\Nthat had a bug in it.
Dialogue: 0,2:08:42.04,2:08:45.40,Default,,0000,0000,0000,,That one, the one in the\Nhomework one, had a bug in it.
Dialogue: 0,2:08:45.40,2:08:46.96,Default,,0000,0000,0000,,It only worked for some data.
Dialogue: 0,2:08:46.96,2:08:50.09,Default,,0000,0000,0000,,And for other data\Nit didn't work.
Dialogue: 0,2:08:50.09,2:08:53.58,Default,,0000,0000,0000,,So every time you find\Na bug, you tell me,
Dialogue: 0,2:08:53.58,2:08:56.20,Default,,0000,0000,0000,,and I will tell the programmer\Nof those problems, who's
Dialogue: 0,2:08:56.20,2:08:57.01,Default,,0000,0000,0000,,really careful.
Dialogue: 0,2:08:57.01,2:09:02.42,Default,,0000,0000,0000,,But one in 1,000 you\Nare bound to find a bug.
Dialogue: 0,2:09:02.42,2:09:06.21,Default,,0000,0000,0000,,And I'm going to\Ngive you a chocolate
Dialogue: 0,2:09:06.21,2:09:08.09,Default,,0000,0000,0000,,or something for every bug.
Dialogue: 0,2:09:08.09,2:09:09.82,Default,,0000,0000,0000,,And any other questions?
Dialogue: 0,2:09:09.82,2:09:14.70,Default,,0000,0000,0000,,
Dialogue: 0,2:09:14.70,2:09:17.66,Default,,0000,0000,0000,,STUDENT: So are you\Nsaying this is too long?
Dialogue: 0,2:09:17.66,2:09:20.14,Default,,0000,0000,0000,,MAGDALENA TODA: Actually,\Nit's very beautiful.
Dialogue: 0,2:09:20.14,2:09:23.60,Default,,0000,0000,0000,,If you have a calculator,\Nit's easier to solve it.
Dialogue: 0,2:09:23.60,2:09:25.58,Default,,0000,0000,0000,,You can do it by hand,\Nwrite it by hand, also.
Dialogue: 0,2:09:25.58,2:09:27.26,Default,,0000,0000,0000,,But it's a long--
Dialogue: 0,2:09:27.26,2:09:28.09,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]
Dialogue: 0,2:09:28.09,2:09:30.78,Default,,0000,0000,0000,,
Dialogue: 0,2:09:30.78,2:09:34.26,Default,,0000,0000,0000,,MAGDALENA TODA: Right,\Nso let's do it now
Dialogue: 0,2:09:34.26,2:09:36.75,Default,,0000,0000,0000,,for anybody who wants to stay.
Dialogue: 0,2:09:36.75,2:09:37.75,Default,,0000,0000,0000,,You don't have to stay.
Dialogue: 0,2:09:37.75,2:09:39.74,Default,,0000,0000,0000,,So practicing what you do--
Dialogue: 0,2:09:39.74,2:09:44.97,Default,,0000,0000,0000,,[SIDE CONVERSATIONS]
Dialogue: 0,2:09:44.97,2:11:55.45,Default,,0000,0000,0000,,