[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:01.47,Default,,0000,0000,0000,,MAGDALENA TODA:\NWe have any people
Dialogue: 0,0:00:01.47,0:00:04.89,Default,,0000,0000,0000,,who finished the\Nextra credit and are
Dialogue: 0,0:00:04.89,0:00:08.31,Default,,0000,0000,0000,,willing to give it to me today?
Dialogue: 0,0:00:08.31,0:00:10.76,Default,,0000,0000,0000,,I mean, you don't have to.
Dialogue: 0,0:00:10.76,0:00:12.72,Default,,0000,0000,0000,,That's why it's\Ncalled extra credit.
Dialogue: 0,0:00:12.72,0:00:18.38,Default,,0000,0000,0000,,But I think it's good\Nfor extra practice
Dialogue: 0,0:00:18.38,0:00:21.35,Default,,0000,0000,0000,,and for the extra points.
Dialogue: 0,0:00:21.35,0:00:25.32,Default,,0000,0000,0000,,So hold on to it if you cannot\Ngive it to me right now.
Dialogue: 0,0:00:25.32,0:00:29.29,Default,,0000,0000,0000,,And I'll collect it at\Nthe end of the class.
Dialogue: 0,0:00:29.29,0:00:30.28,Default,,0000,0000,0000,,Today's a big day.
Dialogue: 0,0:00:30.28,0:00:33.26,Default,,0000,0000,0000,,We are starting a new\Nchapter, Chapter 11.
Dialogue: 0,0:00:33.26,0:00:44.17,Default,,0000,0000,0000,,
Dialogue: 0,0:00:44.17,0:00:47.39,Default,,0000,0000,0000,,So practically, we are\Ngoing to discuss all
Dialogue: 0,0:00:47.39,0:00:50.62,Default,,0000,0000,0000,,through this chapter functions\Nof several variables.
Dialogue: 0,0:00:50.62,0:01:02.03,Default,,0000,0000,0000,,
Dialogue: 0,0:01:02.03,0:01:05.09,Default,,0000,0000,0000,,And you are going to\Nask me, wait a minute,
Dialogue: 0,0:01:05.09,0:01:12.33,Default,,0000,0000,0000,,why do we need functions\Nin more than one variable?
Dialogue: 0,0:01:12.33,0:01:15.77,Default,,0000,0000,0000,,Well, we are all functions\Nof many variables.
Dialogue: 0,0:01:15.77,0:01:18.76,Default,,0000,0000,0000,,I was freezing outside,\Nand I was thinking,
Dialogue: 0,0:01:18.76,0:01:20.62,Default,,0000,0000,0000,,I'm a function of\Neverything I eat.
Dialogue: 0,0:01:20.62,0:01:24.07,Default,,0000,0000,0000,,I'm a function of the\Ntemperature outside.
Dialogue: 0,0:01:24.07,0:01:26.50,Default,,0000,0000,0000,,Almost everything in\Nour body is a function
Dialogue: 0,0:01:26.50,0:01:29.20,Default,,0000,0000,0000,,of hundreds of factors,\Nactually, thousands.
Dialogue: 0,0:01:29.20,0:01:34.35,Default,,0000,0000,0000,,But we don't have the time\Nand the precise information
Dialogue: 0,0:01:34.35,0:01:37.42,Default,,0000,0000,0000,,to analyze all the\Nparameters that
Dialogue: 0,0:01:37.42,0:01:42.38,Default,,0000,0000,0000,,affect our physical\Ncondition every day.
Dialogue: 0,0:01:42.38,0:01:44.99,Default,,0000,0000,0000,,We are getting there.
Dialogue: 0,0:01:44.99,0:01:48.36,Default,,0000,0000,0000,,I'm going to give you\Njust the simple case.
Dialogue: 0,0:01:48.36,0:01:53.31,Default,,0000,0000,0000,,So instead of y equals f of x\Ntype of function, one variable,
Dialogue: 0,0:01:53.31,0:01:57.42,Default,,0000,0000,0000,,we are going to look at\Nfunctions of the types z
Dialogue: 0,0:01:57.42,0:01:58.58,Default,,0000,0000,0000,,equals f of xy.
Dialogue: 0,0:01:58.58,0:02:01.76,Default,,0000,0000,0000,,
Dialogue: 0,0:02:01.76,0:02:03.75,Default,,0000,0000,0000,,Can I have many more?
Dialogue: 0,0:02:03.75,0:02:05.56,Default,,0000,0000,0000,,Absolutely I can.
Dialogue: 0,0:02:05.56,0:02:08.88,Default,,0000,0000,0000,,And that's kind of\Nthe idea, that I
Dialogue: 0,0:02:08.88,0:02:14.64,Default,,0000,0000,0000,,can have a function\Nin an-- let me
Dialogue: 0,0:02:14.64,0:02:18.58,Default,,0000,0000,0000,,count-- n plus 1\Ndimensional space
Dialogue: 0,0:02:18.58,0:02:30.35,Default,,0000,0000,0000,,as being of the type xn plus\N1 equals f of x1, x2, x3, x4.
Dialogue: 0,0:02:30.35,0:02:32.82,Default,,0000,0000,0000,,Somebody stop me. xn.
Dialogue: 0,0:02:32.82,0:02:33.32,Default,,0000,0000,0000,,Right.
Dialogue: 0,0:02:33.32,0:02:37.88,Default,,0000,0000,0000,,I have many variables.
Dialogue: 0,0:02:37.88,0:02:41.79,Default,,0000,0000,0000,,And that is a problem\Nthat affects everything.
Dialogue: 0,0:02:41.79,0:02:44.60,Default,,0000,0000,0000,,Our physical world is\Naffected by many parameters.
Dialogue: 0,0:02:44.60,0:02:47.63,Default,,0000,0000,0000,,
Dialogue: 0,0:02:47.63,0:02:49.85,Default,,0000,0000,0000,,In engineering\Nproblems, you've already
Dialogue: 0,0:02:49.85,0:02:51.40,Default,,0000,0000,0000,,seen some of these parameters.
Dialogue: 0,0:02:51.40,0:02:55.04,Default,,0000,0000,0000,,Can you give me some\Nexamples of parameters you've
Dialogue: 0,0:02:55.04,0:02:57.19,Default,,0000,0000,0000,,seen in engineering classes?
Dialogue: 0,0:02:57.19,0:03:00.74,Default,,0000,0000,0000,,x1, x2, x3 could be the\NEuclidean coordinates, right,
Dialogue: 0,0:03:00.74,0:03:03.63,Default,,0000,0000,0000,,for the three [? space. ?] But\Nbesides those, there was an x4.
Dialogue: 0,0:03:03.63,0:03:05.60,Default,,0000,0000,0000,,It could be?
Dialogue: 0,0:03:05.60,0:03:06.10,Default,,0000,0000,0000,,Time.
Dialogue: 0,0:03:06.10,0:03:07.13,Default,,0000,0000,0000,,Excellent, [INAUDIBLE].
Dialogue: 0,0:03:07.13,0:03:08.40,Default,,0000,0000,0000,,More than that.
Dialogue: 0,0:03:08.40,0:03:09.69,Default,,0000,0000,0000,,I want more.
Dialogue: 0,0:03:09.69,0:03:11.23,Default,,0000,0000,0000,,I want x5.
Dialogue: 0,0:03:11.23,0:03:12.65,Default,,0000,0000,0000,,Who can think of\Nanother parameter
Dialogue: 0,0:03:12.65,0:03:17.08,Default,,0000,0000,0000,,that affects physical processes\Nor chemical reactions?
Dialogue: 0,0:03:17.08,0:03:17.58,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:03:17.58,0:03:17.94,Default,,0000,0000,0000,,STUDENT: Temperature.
Dialogue: 0,0:03:17.94,0:03:19.10,Default,,0000,0000,0000,,MAGDALENA TODA: Temperature.
Dialogue: 0,0:03:19.10,0:03:19.91,Default,,0000,0000,0000,,Excellent.
Dialogue: 0,0:03:19.91,0:03:21.31,Default,,0000,0000,0000,,Another very good idea.
Dialogue: 0,0:03:21.31,0:03:23.10,Default,,0000,0000,0000,,How about x6?
Dialogue: 0,0:03:23.10,0:03:24.59,Default,,0000,0000,0000,,I'm running out of imagination.
Dialogue: 0,0:03:24.59,0:03:28.91,Default,,0000,0000,0000,,But you have a lot more\Ninformation than me.
Dialogue: 0,0:03:28.91,0:03:29.89,Default,,0000,0000,0000,,Pressure.
Dialogue: 0,0:03:29.89,0:03:35.62,Default,,0000,0000,0000,,Maybe I'm studying a process of\Nsomewhere up in the atmosphere.
Dialogue: 0,0:03:35.62,0:03:37.68,Default,,0000,0000,0000,,Maybe I'm in an\Nairplane, and then it
Dialogue: 0,0:03:37.68,0:03:39.26,Default,,0000,0000,0000,,becomes a little bit\Nmore complicated,
Dialogue: 0,0:03:39.26,0:03:41.89,Default,,0000,0000,0000,,because I hate the way\Ncabins are pressurized.
Dialogue: 0,0:03:41.89,0:03:45.06,Default,,0000,0000,0000,,I can feel very uneasy.
Dialogue: 0,0:03:45.06,0:03:46.99,Default,,0000,0000,0000,,My ears pop and so on.
Dialogue: 0,0:03:46.99,0:03:48.85,Default,,0000,0000,0000,,We can be in the\Nbottom of the ocean.
Dialogue: 0,0:03:48.85,0:03:52.90,Default,,0000,0000,0000,,There are very many\Nphysical parameters
Dialogue: 0,0:03:52.90,0:03:56.25,Default,,0000,0000,0000,,that affect physical\Nprocesses, chemical processes,
Dialogue: 0,0:03:56.25,0:03:57.96,Default,,0000,0000,0000,,biological processes.
Dialogue: 0,0:03:57.96,0:04:00.41,Default,,0000,0000,0000,,I don't know if this is\Nfortunate or unfortunate,
Dialogue: 0,0:04:00.41,0:04:03.12,Default,,0000,0000,0000,,but I think that was\Nthe key to the existence
Dialogue: 0,0:04:03.12,0:04:07.40,Default,,0000,0000,0000,,of the universe in the first\Nplace-- all these parameters.
Dialogue: 0,0:04:07.40,0:04:08.03,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:04:08.03,0:04:11.70,Default,,0000,0000,0000,,Let me give you a simple\Nexample of a function that
Dialogue: 0,0:04:11.70,0:04:13.40,Default,,0000,0000,0000,,looks like a graph.
Dialogue: 0,0:04:13.40,0:04:14.46,Default,,0000,0000,0000,,This is a graph.
Dialogue: 0,0:04:14.46,0:04:17.18,Default,,0000,0000,0000,,
Dialogue: 0,0:04:17.18,0:04:19.47,Default,,0000,0000,0000,,And you say, wait a\Nminute, wait a minute.
Dialogue: 0,0:04:19.47,0:04:22.47,Default,,0000,0000,0000,,Can I have functions of several\Nvariables that cannot be
Dialogue: 0,0:04:22.47,0:04:26.26,Default,,0000,0000,0000,,represented as graphs?
Dialogue: 0,0:04:26.26,0:04:28.10,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:04:28.10,0:04:29.02,Default,,0000,0000,0000,,Absolutely.
Dialogue: 0,0:04:29.02,0:04:31.30,Default,,0000,0000,0000,,We will talk about that\Na little bit later.
Dialogue: 0,0:04:31.30,0:04:35.20,Default,,0000,0000,0000,,So if I were to give you\Nan example that you've
Dialogue: 0,0:04:35.20,0:04:42.27,Default,,0000,0000,0000,,seen before, and I would say,\Ngive me a good approximation
Dialogue: 0,0:04:42.27,0:04:47.00,Default,,0000,0000,0000,,to a valley that is actually\Na quadric that we love and we
Dialogue: 0,0:04:47.00,0:04:51.75,Default,,0000,0000,0000,,studied before for\Nthe first time.
Dialogue: 0,0:04:51.75,0:04:59.21,Default,,0000,0000,0000,,That quadric is a\Nbeautiful object, a valley.
Dialogue: 0,0:04:59.21,0:05:02.84,Default,,0000,0000,0000,,Any imagination,\Nrecognition, recollection?
Dialogue: 0,0:05:02.84,0:05:04.93,Default,,0000,0000,0000,,I know I scared\Nyou enough for you
Dialogue: 0,0:05:04.93,0:05:08.13,Default,,0000,0000,0000,,to know the equations of those\Nquadrics since some of you
Dialogue: 0,0:05:08.13,0:05:10.00,Default,,0000,0000,0000,,told me we watched\Nall the videos,
Dialogue: 0,0:05:10.00,0:05:13.44,Default,,0000,0000,0000,,we read all the stinking\Nbook like never before.
Dialogue: 0,0:05:13.44,0:05:14.77,Default,,0000,0000,0000,,That was kind of the idea.
Dialogue: 0,0:05:14.77,0:05:16.97,Default,,0000,0000,0000,,I didn't want to scare you away.
Dialogue: 0,0:05:16.97,0:05:19.33,Default,,0000,0000,0000,,I wanted to scare you\Nenough to read the book
Dialogue: 0,0:05:19.33,0:05:20.62,Default,,0000,0000,0000,,and watch the videos.
Dialogue: 0,0:05:20.62,0:05:26.24,Default,,0000,0000,0000,,And I'm talking about a valley\Nthat you've seen before.
Dialogue: 0,0:05:26.24,0:05:31.07,Default,,0000,0000,0000,,Many of you told me you like\Nthe University of Minnesota
Dialogue: 0,0:05:31.07,0:05:34.90,Default,,0000,0000,0000,,website that has the\Nquadric gallery of quadrics.
Dialogue: 0,0:05:34.90,0:05:40.14,Default,,0000,0000,0000,,
Dialogue: 0,0:05:40.14,0:05:43.55,Default,,0000,0000,0000,,So you've met this guy before.
Dialogue: 0,0:05:43.55,0:05:46.43,Default,,0000,0000,0000,,They show the general equation.
Dialogue: 0,0:05:46.43,0:05:50.01,Default,,0000,0000,0000,,But I said I like the\Ncircular paraboloid.
Dialogue: 0,0:05:50.01,0:05:53.81,Default,,0000,0000,0000,,So they talk about\Nelliptic paraboloid.
Dialogue: 0,0:05:53.81,0:05:56.67,Default,,0000,0000,0000,,Which one do you think I prefer?
Dialogue: 0,0:05:56.67,0:05:59.06,Default,,0000,0000,0000,,The circular paraboloid.
Dialogue: 0,0:05:59.06,0:06:02.88,Default,,0000,0000,0000,,Give me an example of\Na circular paraboloid.
Dialogue: 0,0:06:02.88,0:06:05.27,Default,,0000,0000,0000,,STUDENT: A flashlight?
Dialogue: 0,0:06:05.27,0:06:06.72,Default,,0000,0000,0000,,Inside.
Dialogue: 0,0:06:06.72,0:06:09.30,Default,,0000,0000,0000,,MAGDALENA TODA: The expression,\Nthe mathematical equation.
Dialogue: 0,0:06:09.30,0:06:10.09,Default,,0000,0000,0000,,STUDENT: Oh, sorry.
Dialogue: 0,0:06:10.09,0:06:11.74,Default,,0000,0000,0000,,So it would be x\Nsqured plus y squared.
Dialogue: 0,0:06:11.74,0:06:12.82,Default,,0000,0000,0000,,MAGDALENA TODA: Very good.
Dialogue: 0,0:06:12.82,0:06:14.47,Default,,0000,0000,0000,,That's exactly\Nwhat I had in mind.
Dialogue: 0,0:06:14.47,0:06:18.11,Default,,0000,0000,0000,,Of course, it could be\Nover something, over r.
Dialogue: 0,0:06:18.11,0:06:18.61,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:06:18.61,0:06:19.93,Default,,0000,0000,0000,,That's my favorite.
Dialogue: 0,0:06:19.93,0:06:23.59,Default,,0000,0000,0000,,Now, if I put the flashlight\Nin here just like one of you
Dialogue: 0,0:06:23.59,0:06:29.58,Default,,0000,0000,0000,,said, or the sign on\Ntop of the z-axis.
Dialogue: 0,0:06:29.58,0:06:36.78,Default,,0000,0000,0000,,Then I'm going to look at\Nthe various-- we discussed
Dialogue: 0,0:06:36.78,0:06:38.11,Default,,0000,0000,0000,,that a little bit before.
Dialogue: 0,0:06:38.11,0:06:42.90,Default,,0000,0000,0000,,So various horizontal\Nplanes, they're going to cut.
Dialogue: 0,0:06:42.90,0:06:48.45,Default,,0000,0000,0000,,They're going to cut the\Nsurface in different circles,
Dialogue: 0,0:06:48.45,0:06:50.39,Default,,0000,0000,0000,,upon different circles.
Dialogue: 0,0:06:50.39,0:06:52.78,Default,,0000,0000,0000,,We love them, and we use them.
Dialogue: 0,0:06:52.78,0:06:54.94,Default,,0000,0000,0000,,And what did we do\Nwith them last time?
Dialogue: 0,0:06:54.94,0:06:59.83,Default,,0000,0000,0000,,We projected them on the floor.
Dialogue: 0,0:06:59.83,0:07:03.39,Default,,0000,0000,0000,,And by floor, I mean the what?
Dialogue: 0,0:07:03.39,0:07:10.08,Default,,0000,0000,0000,,By floor, I mean the xy plane.
Dialogue: 0,0:07:10.08,0:07:12.42,Default,,0000,0000,0000,,Plus this xy plane.
Dialogue: 0,0:07:12.42,0:07:14.45,Default,,0000,0000,0000,,I label it like you like it.
Dialogue: 0,0:07:14.45,0:07:17.29,Default,,0000,0000,0000,,You said you like\Nit when I label it,
Dialogue: 0,0:07:17.29,0:07:19.99,Default,,0000,0000,0000,,so you have the\Nimagination of a table.
Dialogue: 0,0:07:19.99,0:07:23.64,Default,,0000,0000,0000,,This is x and y and z.
Dialogue: 0,0:07:23.64,0:07:31.04,Default,,0000,0000,0000,,And so I gave you an example\Nof a graph cut in with z equals
Dialogue: 0,0:07:31.04,0:07:33.40,Default,,0000,0000,0000,,constant positive or negative?
Dialogue: 0,0:07:33.40,0:07:36.08,Default,,0000,0000,0000,,Well, it better be positive,\Nbecause for negative, I
Dialogue: 0,0:07:36.08,0:07:37.77,Default,,0000,0000,0000,,have no solutions.
Dialogue: 0,0:07:37.77,0:07:39.02,Default,,0000,0000,0000,,Positive or zero.
Dialogue: 0,0:07:39.02,0:07:42.36,Default,,0000,0000,0000,,Well, for zero I have\Na degenerate conic.
Dialogue: 0,0:07:42.36,0:07:46.18,Default,,0000,0000,0000,,A degenerate conic\Ncould be a point,
Dialogue: 0,0:07:46.18,0:07:48.82,Default,,0000,0000,0000,,or it could be a bunch of lines.
Dialogue: 0,0:07:48.82,0:07:52.43,Default,,0000,0000,0000,,In this case, all those\Ncircles-- doo-doo-doo-doo-doo--
Dialogue: 0,0:07:52.43,0:07:55.75,Default,,0000,0000,0000,,a family of one parameter,\Nfamily of circles.
Dialogue: 0,0:07:55.75,0:07:58.17,Default,,0000,0000,0000,,Like the ones that\Nis-- a dolphin
Dialogue: 0,0:07:58.17,0:08:01.73,Default,,0000,0000,0000,,is now doing that\Nin San Antonio,
Dialogue: 0,0:08:01.73,0:08:04.49,Default,,0000,0000,0000,,San Diego-- to take\Nthose old circles
Dialogue: 0,0:08:04.49,0:08:07.76,Default,,0000,0000,0000,,from the bottom of the sea,\Nand bring them different sizes,
Dialogue: 0,0:08:07.76,0:08:09.03,Default,,0000,0000,0000,,and put them together.
Dialogue: 0,0:08:09.03,0:08:10.40,Default,,0000,0000,0000,,So they are very smart.
Dialogue: 0,0:08:10.40,0:08:11.82,Default,,0000,0000,0000,,I love dolphins.
Dialogue: 0,0:08:11.82,0:08:15.54,Default,,0000,0000,0000,,So we'll see 0\N[INAUDIBLE] get a point.
Dialogue: 0,0:08:15.54,0:08:17.29,Default,,0000,0000,0000,,That's still a conic.
Dialogue: 0,0:08:17.29,0:08:18.96,Default,,0000,0000,0000,,It's a degenerate circle.
Dialogue: 0,0:08:18.96,0:08:21.58,Default,,0000,0000,0000,,Do you realize\Nthat's a limit case?
Dialogue: 0,0:08:21.58,0:08:22.54,Default,,0000,0000,0000,,It's really beautiful.
Dialogue: 0,0:08:22.54,0:08:23.41,Default,,0000,0000,0000,,You know what I mean?
Dialogue: 0,0:08:23.41,0:08:25.28,Default,,0000,0000,0000,,Circle on top of a circle\Non top of a circle,
Dialogue: 0,0:08:25.28,0:08:26.18,Default,,0000,0000,0000,,smaller and smaller.
Dialogue: 0,0:08:26.18,0:08:27.57,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:08:27.57,0:08:30.47,Default,,0000,0000,0000,,So good.
Dialogue: 0,0:08:30.47,0:08:33.27,Default,,0000,0000,0000,,If I create shadows--\Nbecause that's
Dialogue: 0,0:08:33.27,0:08:35.35,Default,,0000,0000,0000,,why you guys wanted\Nthe source of light
Dialogue: 0,0:08:35.35,0:08:39.36,Default,,0000,0000,0000,,on top-- of the projections\Nof these circles,
Dialogue: 0,0:08:39.36,0:08:42.63,Default,,0000,0000,0000,,I'm going to have them\Nat the same color.
Dialogue: 0,0:08:42.63,0:08:47.99,Default,,0000,0000,0000,,But dotted lines because I\Nthink the book doesn't show them
Dialogue: 0,0:08:47.99,0:08:48.49,Default,,0000,0000,0000,,dotted.
Dialogue: 0,0:08:48.49,0:08:51.21,Default,,0000,0000,0000,,But on my way here,\NI was thinking,
Dialogue: 0,0:08:51.21,0:08:55.04,Default,,0000,0000,0000,,I think it's more beautiful\Nif I draw them dotted.
Dialogue: 0,0:08:55.04,0:08:56.64,Default,,0000,0000,0000,,And how big is this circle?
Dialogue: 0,0:08:56.64,0:08:57.44,Default,,0000,0000,0000,,Well, god knows.
Dialogue: 0,0:08:57.44,0:09:01.99,Default,,0000,0000,0000,,I'm going to make a purple\Ncircle that is, of course,
Dialogue: 0,0:09:01.99,0:09:05.94,Default,,0000,0000,0000,,equal in size, equal in radius\Nwith the original purple
Dialogue: 0,0:09:05.94,0:09:07.08,Default,,0000,0000,0000,,circle.
Dialogue: 0,0:09:07.08,0:09:10.29,Default,,0000,0000,0000,,So the dotted purple circle,\Nthat's on the ground--
Dialogue: 0,0:09:10.29,0:09:13.52,Default,,0000,0000,0000,,is just the projection of\Nthe continuous purple circle.
Dialogue: 0,0:09:13.52,0:09:16.45,Default,,0000,0000,0000,,It's identical in radius.
Dialogue: 0,0:09:16.45,0:09:26.81,Default,,0000,0000,0000,,So for the family of\Ncircles on the surface,
Dialogue: 0,0:09:26.81,0:09:37.83,Default,,0000,0000,0000,,I have a family of projections\Non the ground in the xy plane.
Dialogue: 0,0:09:37.83,0:09:42.42,Default,,0000,0000,0000,,And such a family of\Nprojections represents
Dialogue: 0,0:09:42.42,0:09:44.91,Default,,0000,0000,0000,,a bunch of level curves.
Dialogue: 0,0:09:44.91,0:09:47.11,Default,,0000,0000,0000,,We call this family\Nof level curves.
Dialogue: 0,0:09:47.11,0:09:54.40,Default,,0000,0000,0000,,
Dialogue: 0,0:09:54.40,0:09:55.86,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:09:55.86,0:09:56.83,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:09:56.83,0:09:59.07,Default,,0000,0000,0000,,So if you think about it,\Nwhat are level curves?
Dialogue: 0,0:09:59.07,0:10:02.36,Default,,0000,0000,0000,,You view them as being in plane.
Dialogue: 0,0:10:02.36,0:10:02.91,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,0:10:02.91,0:10:07.60,Default,,0000,0000,0000,,So I should view them as a bunch\Nof points, a set of points.
Dialogue: 0,0:10:07.60,0:10:10.66,Default,,0000,0000,0000,,If I make it like\Nthat, that means
Dialogue: 0,0:10:10.66,0:10:14.10,Default,,0000,0000,0000,,I view this as an\Nelement of what?
Dialogue: 0,0:10:14.10,0:10:18.82,Default,,0000,0000,0000,,Element of the xy plane,\Nright, with the property
Dialogue: 0,0:10:18.82,0:10:22.92,Default,,0000,0000,0000,,that f of x and y is a constant.
Dialogue: 0,0:10:22.92,0:10:25.85,Default,,0000,0000,0000,,
Dialogue: 0,0:10:25.85,0:10:28.36,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:10:28.36,0:10:30.69,Default,,0000,0000,0000,,In my case, I have a\N[INAUDIBLE] constant.
Dialogue: 0,0:10:30.69,0:10:34.27,Default,,0000,0000,0000,,In general, I have an\Narbitrary real constant.
Dialogue: 0,0:10:34.27,0:10:40.02,Default,,0000,0000,0000,,That's a level curve for\Nlevel C, for the level
Dialogue: 0,0:10:40.02,0:10:46.40,Default,,0000,0000,0000,,C called the level, or altitude\Nwould be the same thing.
Dialogue: 0,0:10:46.40,0:10:50.11,Default,,0000,0000,0000,,So have you seen these\Nguys in geography?
Dialogue: 0,0:10:50.11,0:10:54.12,Default,,0000,0000,0000,,What in the world are these\Nlevel curves in geography?
Dialogue: 0,0:10:54.12,0:10:56.52,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\Nshow the slope
Dialogue: 0,0:10:56.52,0:10:59.40,Default,,0000,0000,0000,,of a-- the steepness of a hill.
Dialogue: 0,0:10:59.40,0:11:02.20,Default,,0000,0000,0000,,MAGDALENA TODA: You've\Nseen topographical maps.
Dialogue: 0,0:11:02.20,0:11:06.14,Default,,0000,0000,0000,,And I'm going to try\Nand draw one of them.
Dialogue: 0,0:11:06.14,0:11:08.31,Default,,0000,0000,0000,,I don't know, guys,\Nhow-- excuse me.
Dialogue: 0,0:11:08.31,0:11:10.28,Default,,0000,0000,0000,,I'm not very good\Ntoday at drawing.
Dialogue: 0,0:11:10.28,0:11:13.18,Default,,0000,0000,0000,,But I'll do my best.
Dialogue: 0,0:11:13.18,0:11:19.86,Default,,0000,0000,0000,,It could be a temperature\Nmap or pressure map.
Dialogue: 0,0:11:19.86,0:11:22.61,Default,,0000,0000,0000,,[INAUDIBLE] or whatever.
Dialogue: 0,0:11:22.61,0:11:29.09,Default,,0000,0000,0000,,Now I'll say, this is\Ngoing to go-- well,
Dialogue: 0,0:11:29.09,0:11:31.61,Default,,0000,0000,0000,,I cannot draw the\Ninfinite family.
Dialogue: 0,0:11:31.61,0:11:34.42,Default,,0000,0000,0000,,I have a one-parameter family.
Dialogue: 0,0:11:34.42,0:11:42.84,Default,,0000,0000,0000,,And then I'll-- I'm dreaming of\Nthe sea, summer break already.
Dialogue: 0,0:11:42.84,0:11:44.60,Default,,0000,0000,0000,,You see what I'm doing.
Dialogue: 0,0:11:44.60,0:11:47.58,Default,,0000,0000,0000,,Do you know what I'm doing?
Dialogue: 0,0:11:47.58,0:11:51.61,Default,,0000,0000,0000,,That means I'm dreaming of the\Ndifferent depths of the sea.
Dialogue: 0,0:11:51.61,0:11:55.90,Default,,0000,0000,0000,,So for every such broad\Nline, I have the same depth.
Dialogue: 0,0:11:55.90,0:11:59.38,Default,,0000,0000,0000,,The same altitude for\Nevery continuous rule.
Dialogue: 0,0:11:59.38,0:12:02.53,Default,,0000,0000,0000,,The same depth\Nfor every-- so OK.
Dialogue: 0,0:12:02.53,0:12:05.11,Default,,0000,0000,0000,,I'm not going to swim\Ntoo far, because that's
Dialogue: 0,0:12:05.11,0:12:06.27,Default,,0000,0000,0000,,where the sharks are.
Dialogue: 0,0:12:06.27,0:12:10.10,Default,,0000,0000,0000,,And I cannot draw the sharks,\Nbut I ask you to imagine them.
Dialogue: 0,0:12:10.10,0:12:12.71,Default,,0000,0000,0000,,It's fundamental in\Na calculus class.
Dialogue: 0,0:12:12.71,0:12:18.49,Default,,0000,0000,0000,,So somewhere here\NI'm going to have--
Dialogue: 0,0:12:18.49,0:12:21.33,Default,,0000,0000,0000,,what's the deepest--\Nguys, what's
Dialogue: 0,0:12:21.33,0:12:23.07,Default,,0000,0000,0000,,the deepest point in that?
Dialogue: 0,0:12:23.07,0:12:24.17,Default,,0000,0000,0000,,[? STUDENT: 11,300. ?]
Dialogue: 0,0:12:24.17,0:12:25.88,Default,,0000,0000,0000,,MAGDALENA TODA: And\Ndo you know the name?
Dialogue: 0,0:12:25.88,0:12:26.38,Default,,0000,0000,0000,,I know the--
Dialogue: 0,0:12:26.38,0:12:27.38,Default,,0000,0000,0000,,STUDENT: Mariana Trench.
Dialogue: 0,0:12:27.38,0:12:28.70,Default,,0000,0000,0000,,MAGDALENA TODA: Mariana Trench.
Dialogue: 0,0:12:28.70,0:12:30.76,Default,,0000,0000,0000,,STUDENT: Trench.
Dialogue: 0,0:12:30.76,0:12:31.84,Default,,0000,0000,0000,,MAGDALENA TODA: All right.
Dialogue: 0,0:12:31.84,0:12:34.49,Default,,0000,0000,0000,,So these topographical\Nare full of curves.
Dialogue: 0,0:12:34.49,0:12:37.30,Default,,0000,0000,0000,,These are level curves.
Dialogue: 0,0:12:37.30,0:12:39.58,Default,,0000,0000,0000,,So you didn't know,\Nbut there is a lot
Dialogue: 0,0:12:39.58,0:12:42.06,Default,,0000,0000,0000,,of mathematics in geography.
Dialogue: 0,0:12:42.06,0:12:43.86,Default,,0000,0000,0000,,And there is a\Nlot of mathematics
Dialogue: 0,0:12:43.86,0:12:45.60,Default,,0000,0000,0000,,in-- oh, you knew it.
Dialogue: 0,0:12:45.60,0:12:47.48,Default,,0000,0000,0000,,When you watch the\Nweather report,
Dialogue: 0,0:12:47.48,0:12:49.74,Default,,0000,0000,0000,,that's all mathematics, right?
Dialogue: 0,0:12:49.74,0:12:53.03,Default,,0000,0000,0000,,It shows you the distribution\Nof temperatures everyday.
Dialogue: 0,0:12:53.03,0:12:55.49,Default,,0000,0000,0000,,That is what we can\N[INAUDIBLE] also
Dialogue: 0,0:12:55.49,0:13:00.55,Default,,0000,0000,0000,,care about other functions\Nof several parameters, right?
Dialogue: 0,0:13:00.55,0:13:04.43,Default,,0000,0000,0000,,And those functions could\Nbe pressure, wind, whatever.
Dialogue: 0,0:13:04.43,0:13:05.65,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:13:05.65,0:13:06.77,Default,,0000,0000,0000,,Speed of the wind.
Dialogue: 0,0:13:06.77,0:13:09.04,Default,,0000,0000,0000,,Something like that.
Dialogue: 0,0:13:09.04,0:13:12.35,Default,,0000,0000,0000,,I did not dare to\Nlook at the prediction
Dialogue: 0,0:13:12.35,0:13:14.67,Default,,0000,0000,0000,,of the weather for this place.
Dialogue: 0,0:13:14.67,0:13:16.67,Default,,0000,0000,0000,,This place used to\Nbe a beautiful place.
Dialogue: 0,0:13:16.67,0:13:22.53,Default,,0000,0000,0000,,300 days of the\Nyear of sunshine.
Dialogue: 0,0:13:22.53,0:13:23.76,Default,,0000,0000,0000,,Not anymore.
Dialogue: 0,0:13:23.76,0:13:26.55,Default,,0000,0000,0000,,So there is something\Nfishy in Denmark
Dialogue: 0,0:13:26.55,0:13:29.01,Default,,0000,0000,0000,,and also something\Nfishy in [INAUDIBLE].
Dialogue: 0,0:13:29.01,0:13:30.25,Default,,0000,0000,0000,,The world is changing.
Dialogue: 0,0:13:30.25,0:13:34.72,Default,,0000,0000,0000,,So if you don't believe in\Nglobal warming, think again,
Dialogue: 0,0:13:34.72,0:13:38.18,Default,,0000,0000,0000,,and global cooling, think again.
Dialogue: 0,0:13:38.18,0:13:39.13,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:13:39.13,0:13:42.96,Default,,0000,0000,0000,,So unfortunately,\NI am afraid still
Dialogue: 0,0:13:42.96,0:13:45.62,Default,,0000,0000,0000,,to look at the temperatures\Nfor the next few days.
Dialogue: 0,0:13:45.62,0:13:46.12,Default,,0000,0000,0000,,But--
Dialogue: 0,0:13:46.12,0:13:48.60,Default,,0000,0000,0000,,STUDENT: It's going to\Nbe 80 degrees on Tuesday.
Dialogue: 0,0:13:48.60,0:13:49.15,Default,,0000,0000,0000,,MAGDALENA TODA: Really? [?
Dialogue: 0,0:13:49.15,0:13:50.86,Default,,0000,0000,0000,,Well, see, I should\Nhave looked at it. ?]
Dialogue: 0,0:13:50.86,0:13:51.58,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:13:51.58,0:13:54.71,Default,,0000,0000,0000,,I should gather the\Ncourage, because I
Dialogue: 0,0:13:54.71,0:13:57.14,Default,,0000,0000,0000,,knew-- when I was\Ninterviewed here
Dialogue: 0,0:13:57.14,0:14:00.48,Default,,0000,0000,0000,,for assistant professor,\Ngosh, I was young.
Dialogue: 0,0:14:00.48,0:14:02.23,Default,,0000,0000,0000,,2001.
Dialogue: 0,0:14:02.23,0:14:03.83,Default,,0000,0000,0000,,And my interview\Nwas in mid-February.
Dialogue: 0,0:14:03.83,0:14:08.18,Default,,0000,0000,0000,,And birds were chirping, it was\Nblue skies, beautiful flowers
Dialogue: 0,0:14:08.18,0:14:09.70,Default,,0000,0000,0000,,everywhere on campus.
Dialogue: 0,0:14:09.70,0:14:11.38,Default,,0000,0000,0000,,And I love the campus.
Dialogue: 0,0:14:11.38,0:14:12.91,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:14:12.91,0:14:18.52,Default,,0000,0000,0000,,Give me an example of a surface\Nthat cannot be represented
Dialogue: 0,0:14:18.52,0:14:23.81,Default,,0000,0000,0000,,as a graph in its\Nentirety as a whole graph.
Dialogue: 0,0:14:23.81,0:14:27.07,Default,,0000,0000,0000,,You gave me that before,\Nand I was so proud of you.
Dialogue: 0,0:14:27.07,0:14:27.94,Default,,0000,0000,0000,,It was a--
Dialogue: 0,0:14:27.94,0:14:30.84,Default,,0000,0000,0000,,
Dialogue: 0,0:14:30.84,0:14:32.32,Default,,0000,0000,0000,,[LAUGHS]
Dialogue: 0,0:14:32.32,0:14:33.30,Default,,0000,0000,0000,,
Dialogue: 0,0:14:33.30,0:14:35.73,Default,,0000,0000,0000,,What kind of surface\Nam I trying to mimic?
Dialogue: 0,0:14:35.73,0:14:36.69,Default,,0000,0000,0000,,STUDENT: A saddle.
Dialogue: 0,0:14:36.69,0:14:39.59,Default,,0000,0000,0000,,
Dialogue: 0,0:14:39.59,0:14:42.12,Default,,0000,0000,0000,,MAGDALENA TODA: That\Ncan be actually a graph.
Dialogue: 0,0:14:42.12,0:14:44.63,Default,,0000,0000,0000,,That's a good\Nexample of a graph.
Dialogue: 0,0:14:44.63,0:14:45.92,Default,,0000,0000,0000,,A saddle.
Dialogue: 0,0:14:45.92,0:14:49.03,Default,,0000,0000,0000,,But give me an example\Nof a non-graph that
Dialogue: 0,0:14:49.03,0:14:51.72,Default,,0000,0000,0000,,is given as an implicit form.
Dialogue: 0,0:14:51.72,0:14:58.16,Default,,0000,0000,0000,,So graph or explicit\Nis the same thing.
Dialogue: 0,0:14:58.16,0:15:00.88,Default,,0000,0000,0000,,z equals f of xy.
Dialogue: 0,0:15:00.88,0:15:02.59,Default,,0000,0000,0000,,Give me a non-graph.
Dialogue: 0,0:15:02.59,0:15:05.14,Default,,0000,0000,0000,,One of you said it.
Dialogue: 0,0:15:05.14,0:15:08.60,Default,,0000,0000,0000,,x squared plus y squared\Nplus z squared equals 1.
Dialogue: 0,0:15:08.60,0:15:10.82,Default,,0000,0000,0000,,Why is this not a graph?
Dialogue: 0,0:15:10.82,0:15:12.94,Default,,0000,0000,0000,,Not a graph.
Dialogue: 0,0:15:12.94,0:15:14.16,Default,,0000,0000,0000,,Why is this not a graph?
Dialogue: 0,0:15:14.16,0:15:18.99,Default,,0000,0000,0000,,
Dialogue: 0,0:15:18.99,0:15:22.37,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:15:22.37,0:15:26.26,Default,,0000,0000,0000,,When you move it over to\N1, you can't actually--
Dialogue: 0,0:15:26.26,0:15:28.87,Default,,0000,0000,0000,,MAGDALENA TODA: You\Ncannot but you can cut it.
Dialogue: 0,0:15:28.87,0:15:31.46,Default,,0000,0000,0000,,You can take a\Nsword and-- I'm OK.
Dialogue: 0,0:15:31.46,0:15:33.72,Default,,0000,0000,0000,,I don't want to think about it.
Dialogue: 0,0:15:33.72,0:15:37.18,Default,,0000,0000,0000,,So z is going to be two graphs.
Dialogue: 0,0:15:37.18,0:15:41.97,Default,,0000,0000,0000,,So I can split this surface\Neven in a parametric form
Dialogue: 0,0:15:41.97,0:15:44.86,Default,,0000,0000,0000,,as two different graphs.
Dialogue: 0,0:15:44.86,0:15:46.63,Default,,0000,0000,0000,,Different graphs.
Dialogue: 0,0:15:46.63,0:15:51.49,Default,,0000,0000,0000,,If I cut along-- I have this\Norange, or sphere, globe.
Dialogue: 0,0:15:51.49,0:15:54.07,Default,,0000,0000,0000,,And I cut it along\Na great circle.
Dialogue: 0,0:15:54.07,0:15:57.58,Default,,0000,0000,0000,,It doesn't have\Nto be the equator.
Dialogue: 0,0:15:57.58,0:15:59.71,Default,,0000,0000,0000,,But you have to\Nimagine something
Dialogue: 0,0:15:59.71,0:16:02.54,Default,,0000,0000,0000,,like the world and the equator.
Dialogue: 0,0:16:02.54,0:16:06.03,Default,,0000,0000,0000,,This is kind of in\Nthe unit sphere.
Dialogue: 0,0:16:06.03,0:16:09.11,Default,,0000,0000,0000,,Today I drank enough\Ncoffee to draw better.
Dialogue: 0,0:16:09.11,0:16:10.16,Default,,0000,0000,0000,,Why don't I draw better?
Dialogue: 0,0:16:10.16,0:16:12.08,Default,,0000,0000,0000,,I have no idea.
Dialogue: 0,0:16:12.08,0:16:15.91,Default,,0000,0000,0000,,So that's not bad, though.
Dialogue: 0,0:16:15.91,0:16:16.76,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:16:16.76,0:16:18.09,Default,,0000,0000,0000,,So that's the unit sphere.
Dialogue: 0,0:16:18.09,0:16:18.95,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:16:18.95,0:16:21.77,Default,,0000,0000,0000,,It means it has radius how much?
Dialogue: 0,0:16:21.77,0:16:22.27,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:16:22.27,0:16:23.02,Default,,0000,0000,0000,,MAGDALENA TODA: 1.
Dialogue: 0,0:16:23.02,0:16:25.67,Default,,0000,0000,0000,,Radius 1, and we\Nare happy about it.
Dialogue: 0,0:16:25.67,0:16:29.27,Default,,0000,0000,0000,,And it has two graphs.
Dialogue: 0,0:16:29.27,0:16:31.77,Default,,0000,0000,0000,,It's not one graph,\Nit's two graphs.
Dialogue: 0,0:16:31.77,0:16:34.18,Default,,0000,0000,0000,,So this is called\Nimplicit equation.
Dialogue: 0,0:16:34.18,0:16:36.72,Default,,0000,0000,0000,,This is your lab\Nfrom-- I was chatting
Dialogue: 0,0:16:36.72,0:16:38.67,Default,,0000,0000,0000,,with-- instead of\Nstudying last night,
Dialogue: 0,0:16:38.67,0:16:40.96,Default,,0000,0000,0000,,I was chatting with\Nyou at midnight.
Dialogue: 0,0:16:40.96,0:16:45.08,Default,,0000,0000,0000,,And one of you said, if I had\Nsomething I hated in calculus,
Dialogue: 0,0:16:45.08,0:16:48.16,Default,,0000,0000,0000,,it was the implicit\Ndifferentiation.
Dialogue: 0,0:16:48.16,0:16:50.40,Default,,0000,0000,0000,,And I know this is\Nyour weak point.
Dialogue: 0,0:16:50.40,0:16:52.57,Default,,0000,0000,0000,,So we'll do a lot of\Nimplicit differentiation,
Dialogue: 0,0:16:52.57,0:16:54.77,Default,,0000,0000,0000,,so you become more comfortable.
Dialogue: 0,0:16:54.77,0:16:59.12,Default,,0000,0000,0000,,Usually we have one exercise in\Nthis differentiation at least
Dialogue: 0,0:16:59.12,0:17:01.50,Default,,0000,0000,0000,,on the final.
Dialogue: 0,0:17:01.50,0:17:04.58,Default,,0000,0000,0000,,So this is an implicit equation.
Dialogue: 0,0:17:04.58,0:17:09.69,Default,,0000,0000,0000,,And z is going to be\Ntwo graphs-- 1 minus x
Dialogue: 0,0:17:09.69,0:17:11.12,Default,,0000,0000,0000,,squared minus y squared.
Dialogue: 0,0:17:11.12,0:17:13.86,Default,,0000,0000,0000,,So I have, like, two charts,\Ntwo different charts.
Dialogue: 0,0:17:13.86,0:17:14.36,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:17:14.36,0:17:17.15,Default,,0000,0000,0000,,
Dialogue: 0,0:17:17.15,0:17:20.09,Default,,0000,0000,0000,,The upper hemisphere--\NI'm talking geography,
Dialogue: 0,0:17:20.09,0:17:23.19,Default,,0000,0000,0000,,but that's how we talk\Nin geometry as well.
Dialogue: 0,0:17:23.19,0:17:26.06,Default,,0000,0000,0000,,So geography right\Nnow is like geometry.
Dialogue: 0,0:17:26.06,0:17:28.15,Default,,0000,0000,0000,,I have a north pole.
Dialogue: 0,0:17:28.15,0:17:31.62,Default,,0000,0000,0000,,Somebody quickly give me the\Ncoordinates of the north pole.
Dialogue: 0,0:17:31.62,0:17:32.53,Default,,0000,0000,0000,,STUDENT: 0, 0, 1.
Dialogue: 0,0:17:32.53,0:17:33.53,Default,,0000,0000,0000,,MAGDALENA TODA: 0, 0, 1.
Dialogue: 0,0:17:33.53,0:17:34.78,Default,,0000,0000,0000,,Thank you, Brian.
Dialogue: 0,0:17:34.78,0:17:35.64,Default,,0000,0000,0000,,0, 0, 1.
Dialogue: 0,0:17:35.64,0:17:37.51,Default,,0000,0000,0000,,How about the south pole?
Dialogue: 0,0:17:37.51,0:17:38.72,Default,,0000,0000,0000,,STUDENT: 0, 0, minus 1.
Dialogue: 0,0:17:38.72,0:17:41.29,Default,,0000,0000,0000,,MAGDALENA TODA: 0, 0, minus 1.
Dialogue: 0,0:17:41.29,0:17:45.64,Default,,0000,0000,0000,,And write yourself a\Nnote, because as you know,
Dialogue: 0,0:17:45.64,0:17:48.57,Default,,0000,0000,0000,,I'm very absent-minded\Nand I forget
Dialogue: 0,0:17:48.57,0:17:52.49,Default,,0000,0000,0000,,what I eat for lunch and so on.
Dialogue: 0,0:17:52.49,0:17:55.56,Default,,0000,0000,0000,,Remind me to talk\Nto you sometime
Dialogue: 0,0:17:55.56,0:17:58.28,Default,,0000,0000,0000,,at the end of the chapter\Nabout stereographic projection.
Dialogue: 0,0:17:58.28,0:18:01.08,Default,,0000,0000,0000,,It's a very important\Nmathematical notion
Dialogue: 0,0:18:01.08,0:18:03.78,Default,,0000,0000,0000,,that also has to do a\Nlittle bit with geography.
Dialogue: 0,0:18:03.78,0:18:06.06,Default,,0000,0000,0000,,But it's a one-to-one\Ncorrespondence
Dialogue: 0,0:18:06.06,0:18:08.96,Default,,0000,0000,0000,,between a certain\Npart of a sphere
Dialogue: 0,0:18:08.96,0:18:11.88,Default,,0000,0000,0000,,and a certain huge\Npart of a plane.
Dialogue: 0,0:18:11.88,0:18:14.05,Default,,0000,0000,0000,,Now, we're not going\Nto talk about that now,
Dialogue: 0,0:18:14.05,0:18:16.03,Default,,0000,0000,0000,,because that's not [INAUDIBLE].
Dialogue: 0,0:18:16.03,0:18:18.36,Default,,0000,0000,0000,,That's a little bit\Nharder [INAUDIBLE].
Dialogue: 0,0:18:18.36,0:18:20.69,Default,,0000,0000,0000,,You guys should now\Nsee this line, right?
Dialogue: 0,0:18:20.69,0:18:24.04,Default,,0000,0000,0000,,This should be beyond--\Nin the twilight zone,
Dialogue: 0,0:18:24.04,0:18:25.74,Default,,0000,0000,0000,,behind the sphere.
Dialogue: 0,0:18:25.74,0:18:27.28,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:18:27.28,0:18:28.66,Default,,0000,0000,0000,,So you don't see it.
Dialogue: 0,0:18:28.66,0:18:31.41,Default,,0000,0000,0000,,And who is this? z equals 0.
Dialogue: 0,0:18:31.41,0:18:34.75,Default,,0000,0000,0000,,And so this green\Nfellow should be
Dialogue: 0,0:18:34.75,0:18:39.15,Default,,0000,0000,0000,,the circle x squared\Nplus y squared equals 1
Dialogue: 0,0:18:39.15,0:18:40.62,Default,,0000,0000,0000,,in the xy plane.
Dialogue: 0,0:18:40.62,0:18:43.28,Default,,0000,0000,0000,,
Dialogue: 0,0:18:43.28,0:18:44.70,Default,,0000,0000,0000,,Good.
Dialogue: 0,0:18:44.70,0:18:47.03,Default,,0000,0000,0000,,So I have two graphs.
Dialogue: 0,0:18:47.03,0:18:54.87,Default,,0000,0000,0000,,Now, if I were to ask\Nyou, what is the domain
Dialogue: 0,0:18:54.87,0:18:59.25,Default,,0000,0000,0000,,and the range of the function?
Dialogue: 0,0:18:59.25,0:19:02.63,Default,,0000,0000,0000,,I'm going to erase\Nthe whole thing.
Dialogue: 0,0:19:02.63,0:19:10.04,Default,,0000,0000,0000,,What is the domain and the range\Nof the related function, z,
Dialogue: 0,0:19:10.04,0:19:13.90,Default,,0000,0000,0000,,which gives the\Nupper hemisphere?
Dialogue: 0,0:19:13.90,0:19:15.37,Default,,0000,0000,0000,,Upper hemisphere.
Dialogue: 0,0:19:15.37,0:19:17.32,Default,,0000,0000,0000,,It's a graph.
Dialogue: 0,0:19:17.32,0:19:20.81,Default,,0000,0000,0000,,And square root of 1 minus\Nx squared minus y squared.
Dialogue: 0,0:19:20.81,0:19:23.22,Default,,0000,0000,0000,,You may stare at\Nit until tomorrow.
Dialogue: 0,0:19:23.22,0:19:27.94,Default,,0000,0000,0000,,It's not hard to figure\Nout what I mean by domain
Dialogue: 0,0:19:27.94,0:19:30.57,Default,,0000,0000,0000,,and range of such a function.
Dialogue: 0,0:19:30.57,0:19:33.30,Default,,0000,0000,0000,,You are familiar\Nwith domain and range
Dialogue: 0,0:19:33.30,0:19:37.33,Default,,0000,0000,0000,,for a function of one variable.
Dialogue: 0,0:19:37.33,0:19:39.89,Default,,0000,0000,0000,,For most of you,\Nthat's a piece of cake.
Dialogue: 0,0:19:39.89,0:19:41.80,Default,,0000,0000,0000,,That was even\Npre-calc wasn't it?
Dialogue: 0,0:19:41.80,0:19:44.34,Default,,0000,0000,0000,,It was in Calc 1.
Dialogue: 0,0:19:44.34,0:19:47.36,Default,,0000,0000,0000,,So most of you had\Nalgebra and pre-calc.
Dialogue: 0,0:19:47.36,0:19:51.98,Default,,0000,0000,0000,,Now, what is the domain\Nof such a function?
Dialogue: 0,0:19:51.98,0:19:57.39,Default,,0000,0000,0000,,Domain of definition has to be a\Nset of points, x and y in plane
Dialogue: 0,0:19:57.39,0:20:00.59,Default,,0000,0000,0000,,for which the\Nfunction is defined.
Dialogue: 0,0:20:00.59,0:20:03.04,Default,,0000,0000,0000,,If the function is\Nimpossible to be defined
Dialogue: 0,0:20:03.04,0:20:05.100,Default,,0000,0000,0000,,for a certain pair, x, y,\Nyou kick that couple out
Dialogue: 0,0:20:05.100,0:20:07.92,Default,,0000,0000,0000,,and you say, never come back.
Dialogue: 0,0:20:07.92,0:20:09.14,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:20:09.14,0:20:14.66,Default,,0000,0000,0000,,So what I mean by domain is\Nthose couples that we hate.
Dialogue: 0,0:20:14.66,0:20:16.43,Default,,0000,0000,0000,,Who we hate?
Dialogue: 0,0:20:16.43,0:20:20.97,Default,,0000,0000,0000,,The couples x, y for which x\Nsquared plus y squared is how?
Dialogue: 0,0:20:20.97,0:20:24.13,Default,,0000,0000,0000,,
Dialogue: 0,0:20:24.13,0:20:25.42,Default,,0000,0000,0000,,What existence condition do I--
Dialogue: 0,0:20:25.42,0:20:26.30,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:20:26.30,0:20:27.44,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,0:20:27.44,0:20:30.23,Default,,0000,0000,0000,,You see this guy\Nunder the square root
Dialogue: 0,0:20:30.23,0:20:33.79,Default,,0000,0000,0000,,has to be positive or 0.
Dialogue: 0,0:20:33.79,0:20:35.28,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:20:35.28,0:20:37.89,Default,,0000,0000,0000,,Otherwise, there is no\Nsquare root in real numbers.
Dialogue: 0,0:20:37.89,0:20:39.92,Default,,0000,0000,0000,,That's going to be\Nin imaginary numbers,
Dialogue: 0,0:20:39.92,0:20:41.63,Default,,0000,0000,0000,,and you can take\Na walk, because we
Dialogue: 0,0:20:41.63,0:20:45.22,Default,,0000,0000,0000,,are in real calculus\Nin real time as well.
Dialogue: 0,0:20:45.22,0:20:48.67,Default,,0000,0000,0000,,So x squared plus y\Nsquared must be how?
Dialogue: 0,0:20:48.67,0:20:50.66,Default,,0000,0000,0000,,Less than or equal to 1.
Dialogue: 0,0:20:50.66,0:20:54.04,Default,,0000,0000,0000,,We call that a certain name.
Dialogue: 0,0:20:54.04,0:20:59.48,Default,,0000,0000,0000,,This is called a\Nclosed unit disk.
Dialogue: 0,0:20:59.48,0:21:03.16,Default,,0000,0000,0000,,Please remember, I'm teaching\Nyou a little bit more
Dialogue: 0,0:21:03.16,0:21:06.20,Default,,0000,0000,0000,,than a regular Calc 3 class.
Dialogue: 0,0:21:06.20,0:21:09.32,Default,,0000,0000,0000,,They will never\Nmake a distinction.
Dialogue: 0,0:21:09.32,0:21:10.36,Default,,0000,0000,0000,,What's closing with this?
Dialogue: 0,0:21:10.36,0:21:11.90,Default,,0000,0000,0000,,What's opening with this?
Dialogue: 0,0:21:11.90,0:21:14.67,Default,,0000,0000,0000,,Everything will come\Ninto place when you
Dialogue: 0,0:21:14.67,0:21:19.70,Default,,0000,0000,0000,,move on to advanced calculus.
Dialogue: 0,0:21:19.70,0:21:24.64,Default,,0000,0000,0000,,If I don't take the boundary--\Nso everything inside the disk
Dialogue: 0,0:21:24.64,0:21:28.46,Default,,0000,0000,0000,,except for the boundary, I have\Nto put strictly less than 1.
Dialogue: 0,0:21:28.46,0:21:30.61,Default,,0000,0000,0000,,That's called open unit disk.
Dialogue: 0,0:21:30.61,0:21:35.08,Default,,0000,0000,0000,,For advanced calculus,\Nthis is [INAUDIBLE].
Dialogue: 0,0:21:35.08,0:21:35.58,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:21:35.58,0:21:37.32,Default,,0000,0000,0000,,This is just a parentheses.
Dialogue: 0,0:21:37.32,0:21:40.39,Default,,0000,0000,0000,,My domain is the closed one.
Dialogue: 0,0:21:40.39,0:21:43.14,Default,,0000,0000,0000,,What is the range?
Dialogue: 0,0:21:43.14,0:21:45.69,Default,,0000,0000,0000,,The range is going to be--
Dialogue: 0,0:21:45.69,0:21:47.00,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:21:47.00,0:21:49.90,Default,,0000,0000,0000,,MAGDALENA TODA: The altitude\Nstarts having values from--
Dialogue: 0,0:21:49.90,0:21:51.09,Default,,0000,0000,0000,,STUDENT: Negative 1 to 1.
Dialogue: 0,0:21:51.09,0:21:51.76,Default,,0000,0000,0000,,STUDENT: 0 to 1.
Dialogue: 0,0:21:51.76,0:21:53.01,Default,,0000,0000,0000,,MAGDALENA TODA: So I'm 0 to 1.
Dialogue: 0,0:21:53.01,0:21:55.09,Default,,0000,0000,0000,,I'll only talk about\Nthe upper hemisphere.
Dialogue: 0,0:21:55.09,0:21:58.31,Default,,0000,0000,0000,,I should even erase,\Nbecause I don't want it.
Dialogue: 0,0:21:58.31,0:21:59.33,Default,,0000,0000,0000,,So say it again, guys.
Dialogue: 0,0:21:59.33,0:22:00.35,Default,,0000,0000,0000,,STUDENT: 0 to 1.
Dialogue: 0,0:22:00.35,0:22:01.10,Default,,0000,0000,0000,,MAGDALENA TODA: 0.
Dialogue: 0,0:22:01.10,0:22:01.86,Default,,0000,0000,0000,,Open or closed?
Dialogue: 0,0:22:01.86,0:22:02.64,Default,,0000,0000,0000,,STUDENT: Open.
Dialogue: 0,0:22:02.64,0:22:03.30,Default,,0000,0000,0000,,STUDENT: Closed.
Dialogue: 0,0:22:03.30,0:22:05.74,Default,,0000,0000,0000,,STUDENT: Closed, closed.
Dialogue: 0,0:22:05.74,0:22:07.63,Default,,0000,0000,0000,,MAGDALENA TODA: Closed to?
Dialogue: 0,0:22:07.63,0:22:08.45,Default,,0000,0000,0000,,STUDENT: 1 closed.
Dialogue: 0,0:22:08.45,0:22:09.65,Default,,0000,0000,0000,,MAGDALENA TODA: 1 closed.
Dialogue: 0,0:22:09.65,0:22:10.15,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:22:10.15,0:22:13.87,Default,,0000,0000,0000,,Because that is the north pole.
Dialogue: 0,0:22:13.87,0:22:19.16,Default,,0000,0000,0000,,I've been meaning to\Ngive you this example.
Dialogue: 0,0:22:19.16,0:22:22.27,Default,,0000,0000,0000,,And give me the other example\Nfor the lower hemisphere.
Dialogue: 0,0:22:22.27,0:22:23.40,Default,,0000,0000,0000,,What's different?
Dialogue: 0,0:22:23.40,0:22:24.89,Default,,0000,0000,0000,,The same domain?
Dialogue: 0,0:22:24.89,0:22:25.94,Default,,0000,0000,0000,,STUDENT: It ranges from--
Dialogue: 0,0:22:25.94,0:22:27.04,Default,,0000,0000,0000,,STUDENT: Negative 1.
Dialogue: 0,0:22:27.04,0:22:28.71,Default,,0000,0000,0000,,STUDENT: Negative 1 to 0.
Dialogue: 0,0:22:28.71,0:22:30.49,Default,,0000,0000,0000,,MAGDALENA TODA: Closed\Ninternal, right?
Dialogue: 0,0:22:30.49,0:22:33.86,Default,,0000,0000,0000,,When we include the endpoints,\Nwe call that closed interval.
Dialogue: 0,0:22:33.86,0:22:36.20,Default,,0000,0000,0000,,It has a certain\Ntopological sense.
Dialogue: 0,0:22:36.20,0:22:39.11,Default,,0000,0000,0000,,You haven't taken\Ntopology, but very soon,
Dialogue: 0,0:22:39.11,0:22:43.94,Default,,0000,0000,0000,,if you are a math major, or\Nyou are a double major, or some
Dialogue: 0,0:22:43.94,0:22:48.09,Default,,0000,0000,0000,,of you even-- they want to\Nlearn more about topology,
Dialogue: 0,0:22:48.09,0:22:51.74,Default,,0000,0000,0000,,you will learn what an open\Nset is versus a closed set.
Dialogue: 0,0:22:51.74,0:22:53.63,Default,,0000,0000,0000,,Remember we called this closed.
Dialogue: 0,0:22:53.63,0:22:56.23,Default,,0000,0000,0000,,This is open.
Dialogue: 0,0:22:56.23,0:22:59.66,Default,,0000,0000,0000,,And if it's closed here and\Nopen there, it's neither.
Dialogue: 0,0:22:59.66,0:23:00.16,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:23:00.16,0:23:02.94,Default,,0000,0000,0000,,Don't say anything about that.
Dialogue: 0,0:23:02.94,0:23:03.44,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:23:03.44,0:23:08.11,Default,,0000,0000,0000,,To be closed, it has to be\Ncontaining both endpoints.
Dialogue: 0,0:23:08.11,0:23:09.30,Default,,0000,0000,0000,,I'm going to erase this.
Dialogue: 0,0:23:09.30,0:23:12.26,Default,,0000,0000,0000,,
Dialogue: 0,0:23:12.26,0:23:19.73,Default,,0000,0000,0000,,And this was, of course, 11.1.
Dialogue: 0,0:23:19.73,0:23:22.63,Default,,0000,0000,0000,,We are in the middle of it.
Dialogue: 0,0:23:22.63,0:23:28.44,Default,,0000,0000,0000,,In 11.1, one of you gave me a\Nbeautiful graph to think about.
Dialogue: 0,0:23:28.44,0:23:30.52,Default,,0000,0000,0000,,And I'm going to give\Nyou something to do,
Dialogue: 0,0:23:30.52,0:23:32.69,Default,,0000,0000,0000,,because I don't want\Nyou to get lazy.
Dialogue: 0,0:23:32.69,0:23:36.04,Default,,0000,0000,0000,,I'm very happy you came\Nup with the saddle.
Dialogue: 0,0:23:36.04,0:23:38.91,Default,,0000,0000,0000,,
Dialogue: 0,0:23:38.91,0:23:39.41,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:23:39.41,0:23:41.48,Default,,0000,0000,0000,,We drew such a saddle.
Dialogue: 0,0:23:41.48,0:23:44.46,Default,,0000,0000,0000,,
Dialogue: 0,0:23:44.46,0:23:46.88,Default,,0000,0000,0000,,And I did my best,\Nbut it's not hard.
Dialogue: 0,0:23:46.88,0:23:50.36,Default,,0000,0000,0000,,It's not easy to draw saddle.
Dialogue: 0,0:23:50.36,0:23:54.54,Default,,0000,0000,0000,,When I am looking at the\Ncoordinates, x, y, z,
Dialogue: 0,0:23:54.54,0:24:01.58,Default,,0000,0000,0000,,I have z equals minus y\Nsquared will look down.
Dialogue: 0,0:24:01.58,0:24:05.02,Default,,0000,0000,0000,,
Dialogue: 0,0:24:05.02,0:24:06.81,Default,,0000,0000,0000,,Maybe I made it too fat.
Dialogue: 0,0:24:06.81,0:24:08.98,Default,,0000,0000,0000,,I'm really sorry.
Dialogue: 0,0:24:08.98,0:24:11.43,Default,,0000,0000,0000,,And down.
Dialogue: 0,0:24:11.43,0:24:12.41,Default,,0000,0000,0000,,This continues.
Dialogue: 0,0:24:12.41,0:24:21.23,Default,,0000,0000,0000,,
Dialogue: 0,0:24:21.23,0:24:22.21,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:24:22.21,0:24:26.92,Default,,0000,0000,0000,,And then what other thing\Ndid I want to point out?
Dialogue: 0,0:24:26.92,0:24:31.10,Default,,0000,0000,0000,,I want to point out--\Ndo you see this?
Dialogue: 0,0:24:31.10,0:24:33.66,Default,,0000,0000,0000,,This should look a\Nlittle bit more round.
Dialogue: 0,0:24:33.66,0:24:36.56,Default,,0000,0000,0000,,It doesn't look\Nround enough here.
Dialogue: 0,0:24:36.56,0:24:38.29,Default,,0000,0000,0000,,STUDENT: Your'e drawing\Na saddle, right?
Dialogue: 0,0:24:38.29,0:24:40.29,Default,,0000,0000,0000,,MAGDALENA TODA: No, I'm\Ndrawing just the section
Dialogue: 0,0:24:40.29,0:24:41.97,Default,,0000,0000,0000,,z equals minus y squared.
Dialogue: 0,0:24:41.97,0:24:44.33,Default,,0000,0000,0000,,So I took x to be 0.
Dialogue: 0,0:24:44.33,0:24:47.88,Default,,0000,0000,0000,,And the purple line\Nshould be on this wall.
Dialogue: 0,0:24:47.88,0:24:50.16,Default,,0000,0000,0000,,I know you guys have\Nenough imagination.
Dialogue: 0,0:24:50.16,0:24:54.04,Default,,0000,0000,0000,,So this is going to\Nbe z equals minus y
Dialogue: 0,0:24:54.04,0:24:58.70,Default,,0000,0000,0000,,squared drawn on yz wall.
Dialogue: 0,0:24:58.70,0:25:03.17,Default,,0000,0000,0000,,
Dialogue: 0,0:25:03.17,0:25:05.59,Default,,0000,0000,0000,,I've done this before,\Nbut I'm just reviewing.
Dialogue: 0,0:25:05.59,0:25:08.36,Default,,0000,0000,0000,,What if it's y0?
Dialogue: 0,0:25:08.36,0:25:10.92,Default,,0000,0000,0000,,Then I have to\Ndraw on that wall.
Dialogue: 0,0:25:10.92,0:25:14.33,Default,,0000,0000,0000,,And I have to draw beautifully,\Nwhich I am not-- don't always--
Dialogue: 0,0:25:14.33,0:25:15.72,Default,,0000,0000,0000,,I can't always do.
Dialogue: 0,0:25:15.72,0:25:17.11,Default,,0000,0000,0000,,But I'll try.
Dialogue: 0,0:25:17.11,0:25:23.75,Default,,0000,0000,0000,,I have z equals x squared\Ndrawn on that wall.
Dialogue: 0,0:25:23.75,0:25:26.58,Default,,0000,0000,0000,,If I start drawing,\NI'll get fired.
Dialogue: 0,0:25:26.58,0:25:29.02,Default,,0000,0000,0000,,That I have this branch.
Dialogue: 0,0:25:29.02,0:25:32.85,Default,,0000,0000,0000,,I should go through that\Ncorner and go out of the room
Dialogue: 0,0:25:32.85,0:25:35.31,Default,,0000,0000,0000,,and continue with that branch.
Dialogue: 0,0:25:35.31,0:25:36.01,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:25:36.01,0:25:39.48,Default,,0000,0000,0000,,
Dialogue: 0,0:25:39.48,0:25:43.05,Default,,0000,0000,0000,,This is curved like\Nthat in this direction.
Dialogue: 0,0:25:43.05,0:25:45.46,Default,,0000,0000,0000,,And this other is\Ncurved like this.
Dialogue: 0,0:25:45.46,0:25:50.30,Default,,0000,0000,0000,,So if the guy is\Ngoing to put his feet,
Dialogue: 0,0:25:50.30,0:25:54.59,Default,,0000,0000,0000,,where is the butt of\Nthe writer going to sit?
Dialogue: 0,0:25:54.59,0:25:57.45,Default,,0000,0000,0000,,He is here.
Dialogue: 0,0:25:57.45,0:25:59.60,Default,,0000,0000,0000,,And these are his legs.
Dialogue: 0,0:25:59.60,0:26:02.49,Default,,0000,0000,0000,,
Dialogue: 0,0:26:02.49,0:26:06.14,Default,,0000,0000,0000,,And these are his cowboy boots.
Dialogue: 0,0:26:06.14,0:26:06.64,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:26:06.64,0:26:08.12,Default,,0000,0000,0000,,Do they look like cowboy boots?
Dialogue: 0,0:26:08.12,0:26:10.57,Default,,0000,0000,0000,,No, I apologize.
Dialogue: 0,0:26:10.57,0:26:12.10,Default,,0000,0000,0000,,STUDENT: Looks like socks.
Dialogue: 0,0:26:12.10,0:26:12.98,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah.
Dialogue: 0,0:26:12.98,0:26:15.77,Default,,0000,0000,0000,,They look more like\NChristmas socks.
Dialogue: 0,0:26:15.77,0:26:17.75,Default,,0000,0000,0000,,But anyway, it's a poor cowboy.
Dialogue: 0,0:26:17.75,0:26:22.88,Default,,0000,0000,0000,,
Dialogue: 0,0:26:22.88,0:26:24.71,Default,,0000,0000,0000,,Let's lower the\Nsaddle a little bit.
Dialogue: 0,0:26:24.71,0:26:27.20,Default,,0000,0000,0000,,He cannot see the horse, OK?
Dialogue: 0,0:26:27.20,0:26:30.29,Default,,0000,0000,0000,,So the saddle.
Dialogue: 0,0:26:30.29,0:26:35.18,Default,,0000,0000,0000,,If I cross the saddle,\Nthis is the saddle.
Dialogue: 0,0:26:35.18,0:26:38.16,Default,,0000,0000,0000,,And these are his hands.
Dialogue: 0,0:26:38.16,0:26:41.12,Default,,0000,0000,0000,,And he is holding his hat.
Dialogue: 0,0:26:41.12,0:26:42.06,Default,,0000,0000,0000,,This is [INAUDIBLE].
Dialogue: 0,0:26:42.06,0:26:45.86,Default,,0000,0000,0000,,And with one hand\Nis on the horse.
Dialogue: 0,0:26:45.86,0:26:46.82,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:26:46.82,0:26:48.32,Default,,0000,0000,0000,,It's very [INAUDIBLE].
Dialogue: 0,0:26:48.32,0:26:56.76,Default,,0000,0000,0000,,So what I'm trying to draw\Nlooks something like this.
Dialogue: 0,0:26:56.76,0:26:57.66,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:26:57.66,0:26:59.10,Default,,0000,0000,0000,,Eh.
Dialogue: 0,0:26:59.10,0:27:01.50,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,0:27:01.50,0:27:02.24,Default,,0000,0000,0000,,More or less.
Dialogue: 0,0:27:02.24,0:27:03.94,Default,,0000,0000,0000,,It's an abstract picture.
Dialogue: 0,0:27:03.94,0:27:05.75,Default,,0000,0000,0000,,Very abstract picture.
Dialogue: 0,0:27:05.75,0:27:14.95,Default,,0000,0000,0000,,So with this in mind, if I were\Nto look at the level curves,
Dialogue: 0,0:27:14.95,0:27:18.80,Default,,0000,0000,0000,,I'm going to ask you,\Nwhat are the level curves?
Dialogue: 0,0:27:18.80,0:27:22.20,Default,,0000,0000,0000,,Oh, my god, what are\Nthe level curves?
Dialogue: 0,0:27:22.20,0:27:25.39,Default,,0000,0000,0000,,
Dialogue: 0,0:27:25.39,0:27:27.81,Default,,0000,0000,0000,,You already have them in\Nyour WeBWorK homework.
Dialogue: 0,0:27:27.81,0:27:30.30,Default,,0000,0000,0000,,But for one point\Nextra credit, I
Dialogue: 0,0:27:30.30,0:27:34.07,Default,,0000,0000,0000,,want you to draw\Nthem on the floor.
Dialogue: 0,0:27:34.07,0:27:37.69,Default,,0000,0000,0000,,Draw the level curves.
Dialogue: 0,0:27:37.69,0:27:39.25,Default,,0000,0000,0000,,Remember what those were?
Dialogue: 0,0:27:39.25,0:27:43.29,Default,,0000,0000,0000,,They were projections of\Nthe curves on the surface
Dialogue: 0,0:27:43.29,0:27:46.57,Default,,0000,0000,0000,,at the intersection\Nwith z equals c planes.
Dialogue: 0,0:27:46.57,0:27:48.84,Default,,0000,0000,0000,,You project them on the ground.
Dialogue: 0,0:27:48.84,0:27:50.28,Default,,0000,0000,0000,,What do you think they are?
Dialogue: 0,0:27:50.28,0:27:51.27,Default,,0000,0000,0000,,Think about it.
Dialogue: 0,0:27:51.27,0:27:53.28,Default,,0000,0000,0000,,What are these?
Dialogue: 0,0:27:53.28,0:27:58.46,Default,,0000,0000,0000,,If I take c, what\Nif c is positive?
Dialogue: 0,0:27:58.46,0:28:01.54,Default,,0000,0000,0000,,
Dialogue: 0,0:28:01.54,0:28:04.73,Default,,0000,0000,0000,,What if c is 0?
Dialogue: 0,0:28:04.73,0:28:12.26,Default,,0000,0000,0000,,What if c is less than 0?
Dialogue: 0,0:28:12.26,0:28:14.50,Default,,0000,0000,0000,,What am I going to have?
Dialogue: 0,0:28:14.50,0:28:18.46,Default,,0000,0000,0000,,Your imagination gives\Nyou c equals 1, Magdalena.
Dialogue: 0,0:28:18.46,0:28:19.87,Default,,0000,0000,0000,,Let's draw that.
Dialogue: 0,0:28:19.87,0:28:20.37,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:28:20.37,0:28:21.94,Default,,0000,0000,0000,,Well, I'll try.
Dialogue: 0,0:28:21.94,0:28:23.65,Default,,0000,0000,0000,,a and b would be 1, right, guys?
Dialogue: 0,0:28:23.65,0:28:26.28,Default,,0000,0000,0000,,So a and b would be 1.
Dialogue: 0,0:28:26.28,0:28:27.10,Default,,0000,0000,0000,,This is a square.
Dialogue: 0,0:28:27.10,0:28:29.97,Default,,0000,0000,0000,,These would be the asymptotes.
Dialogue: 0,0:28:29.97,0:28:36.11,Default,,0000,0000,0000,,So very, very\Nbriefly, the hyperbola
Dialogue: 0,0:28:36.11,0:28:41.05,Default,,0000,0000,0000,,would be this one-- x squared\Nminus y squared equals 1,
Dialogue: 0,0:28:41.05,0:28:42.22,Default,,0000,0000,0000,,right?
Dialogue: 0,0:28:42.22,0:28:45.13,Default,,0000,0000,0000,,If I have the last\Ncase for c equals 1,
Dialogue: 0,0:28:45.13,0:28:47.46,Default,,0000,0000,0000,,I'm going to have-- c\Nequals negative 1-- I'm
Dialogue: 0,0:28:47.46,0:28:49.22,Default,,0000,0000,0000,,going to have the conjugate.
Dialogue: 0,0:28:49.22,0:28:50.66,Default,,0000,0000,0000,,Are you guys with me?
Dialogue: 0,0:28:50.66,0:28:57.79,Default,,0000,0000,0000,,So I'll have an a squared,\Nasymptotes, conjugate.
Dialogue: 0,0:28:57.79,0:29:01.23,Default,,0000,0000,0000,,
Dialogue: 0,0:29:01.23,0:29:05.06,Default,,0000,0000,0000,,What if I have different level\Nc? c equals 1/2. c equals 2.
Dialogue: 0,0:29:05.06,0:29:08.00,Default,,0000,0000,0000,,c equals pi. c\Nequals-- what are they?
Dialogue: 0,0:29:08.00,0:29:12.33,Default,,0000,0000,0000,,I'm going to get\Nfamilies of hyperbolas,
Dialogue: 0,0:29:12.33,0:29:14.72,Default,,0000,0000,0000,,trenches that look like that.
Dialogue: 0,0:29:14.72,0:29:16.47,Default,,0000,0000,0000,,Standard trenches and\Nconjugate trenches.
Dialogue: 0,0:29:16.47,0:29:20.60,Default,,0000,0000,0000,,A multitude of them, an infinite\Nfamily of such hyperbolas,
Dialogue: 0,0:29:20.60,0:29:22.48,Default,,0000,0000,0000,,an infinite family\Nof such hyperbolas.
Dialogue: 0,0:29:22.48,0:29:24.54,Default,,0000,0000,0000,,I wanted to draw it.
Dialogue: 0,0:29:24.54,0:29:28.88,Default,,0000,0000,0000,,What do I get when c is 0?
Dialogue: 0,0:29:28.88,0:29:30.02,Default,,0000,0000,0000,,What are those?
Dialogue: 0,0:29:30.02,0:29:31.77,Default,,0000,0000,0000,,STUDENT: Don't you\Nget, like, [INAUDIBLE]?
Dialogue: 0,0:29:31.77,0:29:34.98,Default,,0000,0000,0000,,
Dialogue: 0,0:29:34.98,0:29:36.73,Default,,0000,0000,0000,,MAGDALENA TODA: They\Nget-- very good.
Dialogue: 0,0:29:36.73,0:29:37.23,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:29:37.23,0:29:40.65,Default,,0000,0000,0000,,x squared minus y squared\Nequals 0 would lead
Dialogue: 0,0:29:40.65,0:29:44.92,Default,,0000,0000,0000,,me to y equals plus/minus 1.
Dialogue: 0,0:29:44.92,0:29:48.47,Default,,0000,0000,0000,,And who are those y\Nequals plus/minus 1?
Dialogue: 0,0:29:48.47,0:29:49.59,Default,,0000,0000,0000,,Exactly.
Dialogue: 0,0:29:49.59,0:29:54.50,Default,,0000,0000,0000,,But exactly the first\Nbisector, which is y equals x.
Dialogue: 0,0:29:54.50,0:29:56.41,Default,,0000,0000,0000,,They are [? then the ?]\Nfunction.
Dialogue: 0,0:29:56.41,0:29:59.82,Default,,0000,0000,0000,,And the other one, y equals\Nnegative [? x. ?] So these
Dialogue: 0,0:29:59.82,0:30:01.16,Default,,0000,0000,0000,,are the asymptotes.
Dialogue: 0,0:30:01.16,0:30:05.62,Default,,0000,0000,0000,,So I'm going to get a-- you guys\Nhave to do this better than me.
Dialogue: 0,0:30:05.62,0:30:06.88,Default,,0000,0000,0000,,Sorry.
Dialogue: 0,0:30:06.88,0:30:08.79,Default,,0000,0000,0000,,These are all\Nhyperbolic trenches.
Dialogue: 0,0:30:08.79,0:30:11.73,Default,,0000,0000,0000,,They are all going to\Ninfinity like that.
Dialogue: 0,0:30:11.73,0:30:15.33,Default,,0000,0000,0000,,And I'm sorry that\NI'm giving you
Dialogue: 0,0:30:15.33,0:30:17.09,Default,,0000,0000,0000,,a little bit too many hints.
Dialogue: 0,0:30:17.09,0:30:19.28,Default,,0000,0000,0000,,This is part of your\Nhomework, your WeBWorK.
Dialogue: 0,0:30:19.28,0:30:20.74,Default,,0000,0000,0000,,I shouldn't talk\Ntoo much about it.
Dialogue: 0,0:30:20.74,0:30:23.82,Default,,0000,0000,0000,,
Dialogue: 0,0:30:23.82,0:30:25.35,Default,,0000,0000,0000,,Any questions so far?
Dialogue: 0,0:30:25.35,0:30:26.50,Default,,0000,0000,0000,,Is this hard?
Dialogue: 0,0:30:26.50,0:30:28.00,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:30:28.00,0:30:28.50,Default,,0000,0000,0000,,No.
Dialogue: 0,0:30:28.50,0:30:30.13,Default,,0000,0000,0000,,STUDENT: So [? spherically, ?]\Nif you had z
Dialogue: 0,0:30:30.13,0:30:31.51,Default,,0000,0000,0000,,equals y squared\Nminus x squared,
Dialogue: 0,0:30:31.51,0:30:33.89,Default,,0000,0000,0000,,it's that same\Npicture, just flipped?
Dialogue: 0,0:30:33.89,0:30:39.96,Default,,0000,0000,0000,,
Dialogue: 0,0:30:39.96,0:30:41.45,Default,,0000,0000,0000,,MAGDALENA TODA:\NWhat would it be?
Dialogue: 0,0:30:41.45,0:30:43.24,Default,,0000,0000,0000,,It would be the poor\Nsaddle-- or cowboy--
Dialogue: 0,0:30:43.24,0:30:44.49,Default,,0000,0000,0000,,STUDENT: Would be upside down.
Dialogue: 0,0:30:44.49,0:30:46.34,Default,,0000,0000,0000,,MAGDALENA TODA:\N--would be upside down.
Dialogue: 0,0:30:46.34,0:30:49.65,Default,,0000,0000,0000,,Or projected in\Nsomething like a mirror.
Dialogue: 0,0:30:49.65,0:30:51.10,Default,,0000,0000,0000,,I don't know how to say.
Dialogue: 0,0:30:51.10,0:30:52.85,Default,,0000,0000,0000,,It would be exactly upside down.
Dialogue: 0,0:30:52.85,0:30:55.53,Default,,0000,0000,0000,,So the reflection of that.
Dialogue: 0,0:30:55.53,0:30:59.23,Default,,0000,0000,0000,,So you take all the points.
Dialogue: 0,0:30:59.23,0:31:01.08,Default,,0000,0000,0000,,If you have-- I don't know.
Dialogue: 0,0:31:01.08,0:31:03.38,Default,,0000,0000,0000,,It's hard to draw a reflection\Nin three dimensions.
Dialogue: 0,0:31:03.38,0:31:03.88,Default,,0000,0000,0000,,But--
Dialogue: 0,0:31:03.88,0:31:04.96,Default,,0000,0000,0000,,STUDENT: No, I understand.
Dialogue: 0,0:31:04.96,0:31:09.43,Default,,0000,0000,0000,,MAGDALENA TODA:\NPractically every curve
Dialogue: 0,0:31:09.43,0:31:14.69,Default,,0000,0000,0000,,would be upside down with\Nrespect to the floor.
Dialogue: 0,0:31:14.69,0:31:15.65,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:31:15.65,0:31:16.56,Default,,0000,0000,0000,,All right.
Dialogue: 0,0:31:16.56,0:31:20.60,Default,,0000,0000,0000,,I'm going to erase in one.
Dialogue: 0,0:31:20.60,0:31:24.06,Default,,0000,0000,0000,,And you say, well, you've\Ntaught us about these things,
Dialogue: 0,0:31:24.06,0:31:26.15,Default,,0000,0000,0000,,like the domain and range.
Dialogue: 0,0:31:26.15,0:31:30.53,Default,,0000,0000,0000,,But what about other notions,\Nlike continuity and stuff?
Dialogue: 0,0:31:30.53,0:31:33.30,Default,,0000,0000,0000,,
Dialogue: 0,0:31:33.30,0:31:50.73,Default,,0000,0000,0000,,Let me move on to 11.2.
Dialogue: 0,0:31:50.73,0:32:00.19,Default,,0000,0000,0000,,Limits of functions of\Nthe type z equals f of xy.
Dialogue: 0,0:32:00.19,0:32:14.64,Default,,0000,0000,0000,,
Dialogue: 0,0:32:14.64,0:32:20.03,Default,,0000,0000,0000,,So what do you remember\Nabout the limit
Dialogue: 0,0:32:20.03,0:32:23.13,Default,,0000,0000,0000,,of a function of one variable?
Dialogue: 0,0:32:23.13,0:32:23.63,Default,,0000,0000,0000,,Comparison.
Dialogue: 0,0:32:23.63,0:32:27.96,Default,,0000,0000,0000,,
Dialogue: 0,0:32:27.96,0:32:36.26,Default,,0000,0000,0000,,What about the limit if you\Ntake [? z's, ?] I don't know.
Dialogue: 0,0:32:36.26,0:32:37.70,Default,,0000,0000,0000,,I should look stunned.
Dialogue: 0,0:32:37.70,0:32:38.70,Default,,0000,0000,0000,,And I should be stunned.
Dialogue: 0,0:32:38.70,0:32:49.43,Default,,0000,0000,0000,,Of a function of y equals\Nf of x of one variable.
Dialogue: 0,0:32:49.43,0:32:56.73,Default,,0000,0000,0000,,
Dialogue: 0,0:32:56.73,0:33:10.51,Default,,0000,0000,0000,,When do we say that\Nf has a limit at a?
Dialogue: 0,0:33:10.51,0:33:12.47,Default,,0000,0000,0000,,
Dialogue: 0,0:33:12.47,0:33:14.76,Default,,0000,0000,0000,,STUDENT: When the [INAUDIBLE]\Napproaches from the right
Dialogue: 0,0:33:14.76,0:33:16.74,Default,,0000,0000,0000,,and the left to the same value.
Dialogue: 0,0:33:16.74,0:33:22.61,Default,,0000,0000,0000,,MAGDALENA TODA: Actually, that\Nwas the simpler definition.
Dialogue: 0,0:33:22.61,0:33:25.56,Default,,0000,0000,0000,,Let's think a little bit deeper.
Dialogue: 0,0:33:25.56,0:33:35.33,Default,,0000,0000,0000,,We say that f has a\Nlimit L at x equals a.
Dialogue: 0,0:33:35.33,0:33:40.55,Default,,0000,0000,0000,,That's kind of the idea,\Nleft and right limits.
Dialogue: 0,0:33:40.55,0:33:43.68,Default,,0000,0000,0000,,But not both of them\Nhave to exist, you see.
Dialogue: 0,0:33:43.68,0:33:45.53,Default,,0000,0000,0000,,Maybe only the limit\Nfrom the left or limit
Dialogue: 0,0:33:45.53,0:33:46.98,Default,,0000,0000,0000,,from the right only exists.
Dialogue: 0,0:33:46.98,0:33:49.87,Default,,0000,0000,0000,,
Dialogue: 0,0:33:49.87,0:34:04.37,Default,,0000,0000,0000,,If, for any choice of values\Nof x, closer and closer, closer
Dialogue: 0,0:34:04.37,0:34:23.91,Default,,0000,0000,0000,,and closer to a, we get that\NF gets closer and closer to L.
Dialogue: 0,0:34:23.91,0:34:27.16,Default,,0000,0000,0000,,And this "any" I put in.
Dialogue: 0,0:34:27.16,0:34:33.50,Default,,0000,0000,0000,,My god, I put it in\Na red circle thing,
Dialogue: 0,0:34:33.50,0:34:40.03,Default,,0000,0000,0000,,because one could get\Nsubsequencies of a sequence.
Dialogue: 0,0:34:40.03,0:34:42.40,Default,,0000,0000,0000,,And for that subsequence\Nthing, things
Dialogue: 0,0:34:42.40,0:34:44.99,Default,,0000,0000,0000,,look like I would have a limit.
Dialogue: 0,0:34:44.99,0:34:47.83,Default,,0000,0000,0000,,And then you say,\Nwell, but in the end,
Dialogue: 0,0:34:47.83,0:34:50.89,Default,,0000,0000,0000,,I don't have a limit, because\NI can get another subsequence
Dialogue: 0,0:34:50.89,0:34:52.35,Default,,0000,0000,0000,,of the sequence.
Dialogue: 0,0:34:52.35,0:34:59.03,Default,,0000,0000,0000,,And for that one, I'm not\Ngoing to have a limit.
Dialogue: 0,0:34:59.03,0:35:04.15,Default,,0000,0000,0000,,Can you give me an example\Nof some crazy function that
Dialogue: 0,0:35:04.15,0:35:08.67,Default,,0000,0000,0000,,does not have a limit at 0?
Dialogue: 0,0:35:08.67,0:35:12.13,Default,,0000,0000,0000,,Example of a crazy function.
Dialogue: 0,0:35:12.13,0:35:12.63,Default,,0000,0000,0000,,No.
Dialogue: 0,0:35:12.63,0:35:14.56,Default,,0000,0000,0000,,No, don't write "crazy."
Dialogue: 0,0:35:14.56,0:35:26.27,Default,,0000,0000,0000,,Of a function f of x\Nthat is not defined at 0
Dialogue: 0,0:35:26.27,0:35:43.86,Default,,0000,0000,0000,,and does not have\Nlimit at 0, although it
Dialogue: 0,0:35:43.86,0:35:53.41,Default,,0000,0000,0000,,is defined for values\Narbitrarily close to 0.
Dialogue: 0,0:35:53.41,0:35:59.71,Default,,0000,0000,0000,,
Dialogue: 0,0:35:59.71,0:36:07.90,Default,,0000,0000,0000,,Moreover, I want that function\Nto be drawn without-- I
Dialogue: 0,0:36:07.90,0:36:22.96,Default,,0000,0000,0000,,want the function to be\Ndrawn without leaving
Dialogue: 0,0:36:22.96,0:36:26.30,Default,,0000,0000,0000,,the paper when I draw.
Dialogue: 0,0:36:26.30,0:36:30.75,Default,,0000,0000,0000,,
Dialogue: 0,0:36:30.75,0:36:31.25,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:36:31.25,0:36:34.57,Default,,0000,0000,0000,,
Dialogue: 0,0:36:34.57,0:36:43.53,Default,,0000,0000,0000,,So something that would\Nbe defined on the whole 0
Dialogue: 0,0:36:43.53,0:36:58.66,Default,,0000,0000,0000,,infinity except for 0 that\NI can draw continuously
Dialogue: 0,0:36:58.66,0:37:03.40,Default,,0000,0000,0000,,except when I get to 0, I\Nget some really bad behavior.
Dialogue: 0,0:37:03.40,0:37:07.56,Default,,0000,0000,0000,,I don't have a limit\Nfor that function.
Dialogue: 0,0:37:07.56,0:37:08.91,Default,,0000,0000,0000,,You are close to that.
Dialogue: 0,0:37:08.91,0:37:10.72,Default,,0000,0000,0000,,Sine of 1/x.
Dialogue: 0,0:37:10.72,0:37:13.11,Default,,0000,0000,0000,,STUDENT: I said y equals 1/x.
Dialogue: 0,0:37:13.11,0:37:15.32,Default,,0000,0000,0000,,MAGDALENA TODA: y equals 1/x.
Dialogue: 0,0:37:15.32,0:37:16.26,Default,,0000,0000,0000,,Very good.
Dialogue: 0,0:37:16.26,0:37:18.60,Default,,0000,0000,0000,,Let's see.
Dialogue: 0,0:37:18.60,0:37:20.00,Default,,0000,0000,0000,,STUDENT: Oh, yeah. [INAUDIBLE].
Dialogue: 0,0:37:20.00,0:37:21.13,Default,,0000,0000,0000,,MAGDALENA TODA: Yeah, yeah.
Dialogue: 0,0:37:21.13,0:37:22.39,Default,,0000,0000,0000,,Both are excellent examples.
Dialogue: 0,0:37:22.39,0:37:24.53,Default,,0000,0000,0000,,So let's see.
Dialogue: 0,0:37:24.53,0:37:29.08,Default,,0000,0000,0000,,This guy is a very\Nnice function.
Dialogue: 0,0:37:29.08,0:37:31.37,Default,,0000,0000,0000,,How do we draw him, or her?
Dialogue: 0,0:37:31.37,0:37:32.54,Default,,0000,0000,0000,,Well, it's a her, right?
Dialogue: 0,0:37:32.54,0:37:33.04,Default,,0000,0000,0000,,It's a she.
Dialogue: 0,0:37:33.04,0:37:33.99,Default,,0000,0000,0000,,It's a function.
Dialogue: 0,0:37:33.99,0:37:34.66,Default,,0000,0000,0000,,No, no.
Dialogue: 0,0:37:34.66,0:37:36.36,Default,,0000,0000,0000,,In English, it doesn't\Nmake any sense,
Dialogue: 0,0:37:36.36,0:37:40.73,Default,,0000,0000,0000,,but if I think French, Italian,\NSpanish, Romanian-- now
Dialogue: 0,0:37:40.73,0:37:44.14,Default,,0000,0000,0000,,I speak both Italian\Nand Romanian--
Dialogue: 0,0:37:44.14,0:37:47.22,Default,,0000,0000,0000,,we say it's a she,\Nit's a feminine.
Dialogue: 0,0:37:47.22,0:37:52.34,Default,,0000,0000,0000,,So as I approach with values\Ncloser and closer and closer
Dialogue: 0,0:37:52.34,0:37:56.34,Default,,0000,0000,0000,,to 0, what happens\Nto my poor function?
Dialogue: 0,0:37:56.34,0:37:58.91,Default,,0000,0000,0000,,It blows up.
Dialogue: 0,0:37:58.91,0:37:59.41,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:37:59.41,0:38:05.04,Default,,0000,0000,0000,,So I have limit of 1/x from\Nthe right and from the left.
Dialogue: 0,0:38:05.04,0:38:08.31,Default,,0000,0000,0000,,If I take it from the\Nleft, I don't care.
Dialogue: 0,0:38:08.31,0:38:11.17,Default,,0000,0000,0000,,Let's take it only\Nfrom the right.
Dialogue: 0,0:38:11.17,0:38:11.67,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:38:11.67,0:38:17.91,Default,,0000,0000,0000,,
Dialogue: 0,0:38:17.91,0:38:19.63,Default,,0000,0000,0000,,It's close to 0.
Dialogue: 0,0:38:19.63,0:38:21.21,Default,,0000,0000,0000,,That's going to blow up, right?
Dialogue: 0,0:38:21.21,0:38:24.56,Default,,0000,0000,0000,,
Dialogue: 0,0:38:24.56,0:38:25.49,Default,,0000,0000,0000,,And I restrict it.
Dialogue: 0,0:38:25.49,0:38:30.46,Default,,0000,0000,0000,,So let's say, if I want\Nthe domain to be containing
Dialogue: 0,0:38:30.46,0:38:32.74,Default,,0000,0000,0000,,[? both, ?] that's also fine.
Dialogue: 0,0:38:32.74,0:38:35.73,Default,,0000,0000,0000,,So if you guys want, we\Ncan draw the other one.
Dialogue: 0,0:38:35.73,0:38:37.01,Default,,0000,0000,0000,,This goes to paradise.
Dialogue: 0,0:38:37.01,0:38:39.66,Default,,0000,0000,0000,,The other one, I'm not\Ngoing to say where it goes.
Dialogue: 0,0:38:39.66,0:38:43.27,Default,,0000,0000,0000,,But it's the same idea,\Nthat as you approach 0
Dialogue: 0,0:38:43.27,0:38:45.71,Default,,0000,0000,0000,,with closer and closer\Nand closer values,
Dialogue: 0,0:38:45.71,0:38:47.66,Default,,0000,0000,0000,,it's going to blow up.
Dialogue: 0,0:38:47.66,0:38:51.02,Default,,0000,0000,0000,,It's going to explode.
Dialogue: 0,0:38:51.02,0:38:54.01,Default,,0000,0000,0000,,This is a beautiful function.
Dialogue: 0,0:38:54.01,0:38:55.16,Default,,0000,0000,0000,,How beautiful [INAUDIBLE].
Dialogue: 0,0:38:55.16,0:38:57.94,Default,,0000,0000,0000,,Beautiful with a\Nbad behavior near 0.
Dialogue: 0,0:38:57.94,0:38:59.73,Default,,0000,0000,0000,,So I'm not going\Nto have a limit.
Dialogue: 0,0:38:59.73,0:39:00.61,Default,,0000,0000,0000,,No limit.
Dialogue: 0,0:39:00.61,0:39:02.59,Default,,0000,0000,0000,,Some people say, limit\Nexists and is infinity.
Dialogue: 0,0:39:02.59,0:39:05.26,Default,,0000,0000,0000,,But does infinity exist?
Dialogue: 0,0:39:05.26,0:39:07.38,Default,,0000,0000,0000,,Well, this is a\Nreally philosophical,
Dialogue: 0,0:39:07.38,0:39:10.57,Default,,0000,0000,0000,,religious notion, so I\Ndon't want to get into it.
Dialogue: 0,0:39:10.57,0:39:13.48,Default,,0000,0000,0000,,But in mathematics, we consider\Nthat unless the limit is
Dialogue: 0,0:39:13.48,0:39:16.73,Default,,0000,0000,0000,,finite, you cannot have a limit.
Dialogue: 0,0:39:16.73,0:39:20.85,Default,,0000,0000,0000,,So if the limit is plus/minus\Ninfinity, there is no limit.
Dialogue: 0,0:39:20.85,0:39:25.24,Default,,0000,0000,0000,,Could the limit be different\Nor different subsequences?
Dialogue: 0,0:39:25.24,0:39:28.19,Default,,0000,0000,0000,,This is what I\Nwanted to point out.
Dialogue: 0,0:39:28.19,0:39:34.27,Default,,0000,0000,0000,,If you try this guy, you are\Nin real trouble on that guy.
Dialogue: 0,0:39:34.27,0:39:35.15,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:39:35.15,0:39:36.57,Default,,0000,0000,0000,,You can have two.
Dialogue: 0,0:39:36.57,0:39:38.93,Default,,0000,0000,0000,,If you have a graphing\Ncalculator, which
Dialogue: 0,0:39:38.93,0:39:43.99,Default,,0000,0000,0000,,I'm going to be opposed to you\Nbeing used in the classroom,
Dialogue: 0,0:39:43.99,0:39:46.16,Default,,0000,0000,0000,,you would probably\Nsee what happens.
Dialogue: 0,0:39:46.16,0:39:51.54,Default,,0000,0000,0000,,Sine is defined on\Nall the real numbers.
Dialogue: 0,0:39:51.54,0:39:54.24,Default,,0000,0000,0000,,But you cannot\Nhave a value at 0,
Dialogue: 0,0:39:54.24,0:39:57.51,Default,,0000,0000,0000,,because the 1/x is\Nnot defined at 0.
Dialogue: 0,0:39:57.51,0:40:02.33,Default,,0000,0000,0000,,Imagine you get closer and\Ncloser to 0 from both sides.
Dialogue: 0,0:40:02.33,0:40:05.28,Default,,0000,0000,0000,,I cannot draw very beautifully.
Dialogue: 0,0:40:05.28,0:40:09.59,Default,,0000,0000,0000,,But as 1, this is plus\N1 and this is minus 1.
Dialogue: 0,0:40:09.59,0:40:11.84,Default,,0000,0000,0000,,I'm going to have some behavior.
Dialogue: 0,0:40:11.84,0:40:15.16,Default,,0000,0000,0000,,And how many of you have seen\Nthat on a computer screen
Dialogue: 0,0:40:15.16,0:40:15.94,Default,,0000,0000,0000,,or calculator?
Dialogue: 0,0:40:15.94,0:40:16.44,Default,,0000,0000,0000,,You've seen.
Dialogue: 0,0:40:16.44,0:40:17.58,Default,,0000,0000,0000,,Yeah, you've seen.
Dialogue: 0,0:40:17.58,0:40:20.45,Default,,0000,0000,0000,,By the way, did you\Nsee the Lubbuck High?
Dialogue: 0,0:40:20.45,0:40:23.47,Default,,0000,0000,0000,,Was it in high school you saw\Nit the first time in Calc 1
Dialogue: 0,0:40:23.47,0:40:25.14,Default,,0000,0000,0000,,or pre-calc?
Dialogue: 0,0:40:25.14,0:40:28.44,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE]\NAlgebra 1 with Mr. West.
Dialogue: 0,0:40:28.44,0:40:28.94,Default,,0000,0000,0000,,[INAUDIBLE]
Dialogue: 0,0:40:28.94,0:40:32.51,Default,,0000,0000,0000,,MAGDALENA TODA: So I'll\Ntry-- oh, guys, you
Dialogue: 0,0:40:32.51,0:40:34.24,Default,,0000,0000,0000,,have to be patient with me.
Dialogue: 0,0:40:34.24,0:40:38.15,Default,,0000,0000,0000,,I'm not leaving the poor board\Nwith the tip of my pencil.
Dialogue: 0,0:40:38.15,0:40:39.38,Default,,0000,0000,0000,,I'm not leaving him.
Dialogue: 0,0:40:39.38,0:40:42.36,Default,,0000,0000,0000,,I have continuity.
Dialogue: 0,0:40:42.36,0:40:45.81,Default,,0000,0000,0000,,As I got closer to this, I still\Nhave the [INAUDIBLE] property.
Dialogue: 0,0:40:45.81,0:40:47.15,Default,,0000,0000,0000,,Anyway, it's OK.
Dialogue: 0,0:40:47.15,0:40:48.41,Default,,0000,0000,0000,,I'm not leaving this.
Dialogue: 0,0:40:48.41,0:40:52.10,Default,,0000,0000,0000,,I am taking all the values\Npossible between minus 1 and 1.
Dialogue: 0,0:40:52.10,0:40:54.82,Default,,0000,0000,0000,,So on intervals that\Nare smaller, smaller,
Dialogue: 0,0:40:54.82,0:40:57.76,Default,,0000,0000,0000,,I'm really taking all the\Nvalues between minus 1 and 1,
Dialogue: 0,0:40:57.76,0:41:01.39,Default,,0000,0000,0000,,and really rapidly--\N[INAUDIBLE].
Dialogue: 0,0:41:01.39,0:41:07.52,Default,,0000,0000,0000,,When I'm getting closer to 0,\NI'm not going to have a limit.
Dialogue: 0,0:41:07.52,0:41:10.06,Default,,0000,0000,0000,,But as somebody\Nmay say, but wait.
Dialogue: 0,0:41:10.06,0:41:12.27,Default,,0000,0000,0000,,When I have a sequence\Nof values that
Dialogue: 0,0:41:12.27,0:41:14.36,Default,,0000,0000,0000,,is getting closer\Nand closer to 0,
Dialogue: 0,0:41:14.36,0:41:18.61,Default,,0000,0000,0000,,is that no guarantee that\NI'm going to have a limit?
Dialogue: 0,0:41:18.61,0:41:20.12,Default,,0000,0000,0000,,Nope.
Dialogue: 0,0:41:20.12,0:41:20.96,Default,,0000,0000,0000,,It depends.
Dialogue: 0,0:41:20.96,0:41:25.48,Default,,0000,0000,0000,,If you say "any," it has to\Nbe for any choice of points,
Dialogue: 0,0:41:25.48,0:41:27.96,Default,,0000,0000,0000,,any choice of points\Nthat you go closer to 0.
Dialogue: 0,0:41:27.96,0:41:30.15,Default,,0000,0000,0000,,Not for one sequence\Nof points that
Dialogue: 0,0:41:30.15,0:41:32.29,Default,,0000,0000,0000,,is getting closer\Nand closer to 0.
Dialogue: 0,0:41:32.29,0:41:35.30,Default,,0000,0000,0000,,For example, if your\Nchoice of points is this,
Dialogue: 0,0:41:35.30,0:41:36.28,Default,,0000,0000,0000,,choice of points.
Dialogue: 0,0:41:36.28,0:41:39.74,Default,,0000,0000,0000,,
Dialogue: 0,0:41:39.74,0:41:43.80,Default,,0000,0000,0000,,Getting closer to 0.
Dialogue: 0,0:41:43.80,0:41:49.54,Default,,0000,0000,0000,,[INAUDIBLE] xn\Nequals 1 over 2 pi n.
Dialogue: 0,0:41:49.54,0:41:51.82,Default,,0000,0000,0000,,Isn't this going to 0?
Dialogue: 0,0:41:51.82,0:41:52.32,Default,,0000,0000,0000,,Yeah.
Dialogue: 0,0:41:52.32,0:41:54.09,Default,,0000,0000,0000,,It then goes to infinity.
Dialogue: 0,0:41:54.09,0:41:55.69,Default,,0000,0000,0000,,This sequence goes to 0.
Dialogue: 0,0:41:55.69,0:41:56.30,Default,,0000,0000,0000,,What is it?
Dialogue: 0,0:41:56.30,0:41:57.02,Default,,0000,0000,0000,,1 over 2 pi?
Dialogue: 0,0:41:57.02,0:41:58.07,Default,,0000,0000,0000,,1 over 4 pi?
Dialogue: 0,0:41:58.07,0:41:58.89,Default,,0000,0000,0000,,1 over 8 pi?
Dialogue: 0,0:41:58.89,0:41:59.98,Default,,0000,0000,0000,,1 over 16 pi?
Dialogue: 0,0:41:59.98,0:42:01.10,Default,,0000,0000,0000,,1 over 32 pi?
Dialogue: 0,0:42:01.10,0:42:02.55,Default,,0000,0000,0000,,1 over 64 pi?
Dialogue: 0,0:42:02.55,0:42:04.45,Default,,0000,0000,0000,,This is what my\Nson is doing to me.
Dialogue: 0,0:42:04.45,0:42:06.18,Default,,0000,0000,0000,,And I say, please stop.
Dialogue: 0,0:42:06.18,0:42:07.13,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:42:07.13,0:42:08.24,Default,,0000,0000,0000,,He's 10 years old.
Dialogue: 0,0:42:08.24,0:42:09.59,Default,,0000,0000,0000,,He's so funny.
Dialogue: 0,0:42:09.59,0:42:12.96,Default,,0000,0000,0000,,Now, another choice of points.
Dialogue: 0,0:42:12.96,0:42:20.49,Default,,0000,0000,0000,,
Dialogue: 0,0:42:20.49,0:42:21.17,Default,,0000,0000,0000,,Ah.
Dialogue: 0,0:42:21.17,0:42:26.45,Default,,0000,0000,0000,,Somebody-- all of you are\Nsmart enough to do this.
Dialogue: 0,0:42:26.45,0:42:29.23,Default,,0000,0000,0000,,What do you think\NI'm going to pick?
Dialogue: 0,0:42:29.23,0:42:30.78,Default,,0000,0000,0000,,1 over what?
Dialogue: 0,0:42:30.78,0:42:33.60,Default,,0000,0000,0000,,And when [? other ?]\Nsomething that goes to 0
Dialogue: 0,0:42:33.60,0:42:34.52,Default,,0000,0000,0000,,then goes to infinity.
Dialogue: 0,0:42:34.52,0:42:41.66,Default,,0000,0000,0000,,And I know that your\Nprofessor showed you that.
Dialogue: 0,0:42:41.66,0:42:44.100,Default,,0000,0000,0000,,pi over 2 plus 2 pi n.
Dialogue: 0,0:42:44.100,0:42:45.87,Default,,0000,0000,0000,,Doesn't this go to 0?
Dialogue: 0,0:42:45.87,0:42:46.37,Default,,0000,0000,0000,,Yes.
Dialogue: 0,0:42:46.37,0:42:49.74,Default,,0000,0000,0000,,As n gets bigger and\Nbigger, this is going to 0.
Dialogue: 0,0:42:49.74,0:42:51.04,Default,,0000,0000,0000,,However, there is no limit.
Dialogue: 0,0:42:51.04,0:42:51.81,Default,,0000,0000,0000,,Why?
Dialogue: 0,0:42:51.81,0:42:59.28,Default,,0000,0000,0000,,Well, for the first sequence,\Nas xn goes to 0, f of xn
Dialogue: 0,0:42:59.28,0:43:04.06,Default,,0000,0000,0000,,goes to-- what is\Nsine of-- OK, I
Dialogue: 0,0:43:04.06,0:43:05.87,Default,,0000,0000,0000,,am too lazy to write this down.
Dialogue: 0,0:43:05.87,0:43:11.02,Default,,0000,0000,0000,,Sine of 1 over 1 over--\Nof 1 over 1 over 2 pi?
Dialogue: 0,0:43:11.02,0:43:14.81,Default,,0000,0000,0000,,
Dialogue: 0,0:43:14.81,0:43:16.78,Default,,0000,0000,0000,,STUDENT: It's the\Nsine over 2 pi.
Dialogue: 0,0:43:16.78,0:43:20.65,Default,,0000,0000,0000,,MAGDALENA TODA: This\Nis sine of 2 pi n.
Dialogue: 0,0:43:20.65,0:43:22.36,Default,,0000,0000,0000,,And how much is that?
Dialogue: 0,0:43:22.36,0:43:22.86,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,0:43:22.86,0:43:24.21,Default,,0000,0000,0000,,MAGDALENA TODA: 0.
Dialogue: 0,0:43:24.21,0:43:25.59,Default,,0000,0000,0000,,So this is a 0.
Dialogue: 0,0:43:25.59,0:43:28.73,Default,,0000,0000,0000,,And this is a--\Nthis converges to 0.
Dialogue: 0,0:43:28.73,0:43:31.40,Default,,0000,0000,0000,,So I say, oh, so maybe I have\Na limit, and that'll be 0.
Dialogue: 0,0:43:31.40,0:43:32.52,Default,,0000,0000,0000,,Wrong.
Dialogue: 0,0:43:32.52,0:43:35.73,Default,,0000,0000,0000,,That would be the rapid,\Nstupid conclusion.
Dialogue: 0,0:43:35.73,0:43:38.69,Default,,0000,0000,0000,,If somebody jumps [? up, ?]\NI picked some points,
Dialogue: 0,0:43:38.69,0:43:41.82,Default,,0000,0000,0000,,I formed the sequence that\Ngets closer and closer to 0.
Dialogue: 0,0:43:41.82,0:43:43.70,Default,,0000,0000,0000,,I'm sure that the limit exists.
Dialogue: 0,0:43:43.70,0:43:45.50,Default,,0000,0000,0000,,I've got a 0.
Dialogue: 0,0:43:45.50,0:43:48.85,Default,,0000,0000,0000,,Well, did you think of\Nany possible choice?
Dialogue: 0,0:43:48.85,0:43:49.67,Default,,0000,0000,0000,,That's the problem.
Dialogue: 0,0:43:49.67,0:43:51.94,Default,,0000,0000,0000,,You have to have\Nany possible choice.
Dialogue: 0,0:43:51.94,0:44:02.58,Default,,0000,0000,0000,,F of yn sine of 1 over\Npi over 2 plus 1 over 1
Dialogue: 0,0:44:02.58,0:44:09.03,Default,,0000,0000,0000,,over-- Magdalena-- pi\Nover 2 plus 2 pi n.
Dialogue: 0,0:44:09.03,0:44:10.86,Default,,0000,0000,0000,,So we saw that this was 0.
Dialogue: 0,0:44:10.86,0:44:14.85,Default,,0000,0000,0000,,What happens to sine of\N1 over 1 over sine of pi
Dialogue: 0,0:44:14.85,0:44:18.72,Default,,0000,0000,0000,,over 2 plus 2 pi n?
Dialogue: 0,0:44:18.72,0:44:19.89,Default,,0000,0000,0000,,And where does this go?
Dialogue: 0,0:44:19.89,0:44:21.13,Default,,0000,0000,0000,,It then goes to infinity.
Dialogue: 0,0:44:21.13,0:44:26.97,Default,,0000,0000,0000,,
Dialogue: 0,0:44:26.97,0:44:29.25,Default,,0000,0000,0000,,This sequence goes to 0.
Dialogue: 0,0:44:29.25,0:44:32.63,Default,,0000,0000,0000,,What is f of the\Nsequence going to?
Dialogue: 0,0:44:32.63,0:44:33.52,Default,,0000,0000,0000,,To another limit.
Dialogue: 0,0:44:33.52,0:44:36.03,Default,,0000,0000,0000,,So there is no limit.
Dialogue: 0,0:44:36.03,0:44:38.58,Default,,0000,0000,0000,,What's the limit of\Nthis subsequence?
Dialogue: 0,0:44:38.58,0:44:41.40,Default,,0000,0000,0000,,It's a constant one, right?
Dialogue: 0,0:44:41.40,0:44:45.93,Default,,0000,0000,0000,,Because look, what does it\Nmean pi over 2 plus 2 pi n?
Dialogue: 0,0:44:45.93,0:44:49.24,Default,,0000,0000,0000,,Where am I on the unit\Ntrigonometric circle?
Dialogue: 0,0:44:49.24,0:44:50.69,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,0:44:50.69,0:44:53.93,Default,,0000,0000,0000,,Always here, right?
Dialogue: 0,0:44:53.93,0:44:56.45,Default,,0000,0000,0000,,Always on the sort of\Nlike the north pole.
Dialogue: 0,0:44:56.45,0:44:59.22,Default,,0000,0000,0000,,So what is the sine\Nof this north pole?
Dialogue: 0,0:44:59.22,0:44:59.90,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,0:44:59.90,0:45:00.97,Default,,0000,0000,0000,,MAGDALENA TODA: Always 1.
Dialogue: 0,0:45:00.97,0:45:02.40,Default,,0000,0000,0000,,So I get the limit 1.
Dialogue: 0,0:45:02.40,0:45:06.48,Default,,0000,0000,0000,,So I'm done because there\Nare two different limits.
Dialogue: 0,0:45:06.48,0:45:09.36,Default,,0000,0000,0000,,So pay attention to\Nthis type of problem.
Dialogue: 0,0:45:09.36,0:45:17.53,Default,,0000,0000,0000,,Somebody can get you in trouble\Nwith this kind of thing.
Dialogue: 0,0:45:17.53,0:45:20.22,Default,,0000,0000,0000,,On the other hand,\NI'm asking you,
Dialogue: 0,0:45:20.22,0:45:22.100,Default,,0000,0000,0000,,what if I want to make this\Na function of two variables?
Dialogue: 0,0:45:22.100,0:45:27.66,Default,,0000,0000,0000,,
Dialogue: 0,0:45:27.66,0:45:30.96,Default,,0000,0000,0000,,So I'll say, one\Npoint extra credit.
Dialogue: 0,0:45:30.96,0:45:34.07,Default,,0000,0000,0000,,I'm giving you too\Nmuch extra credit.
Dialogue: 0,0:45:34.07,0:45:36.21,Default,,0000,0000,0000,,Maybe I give you\Ntoo much-- it's OK.
Dialogue: 0,0:45:36.21,0:45:39.93,Default,,0000,0000,0000,,One point extra credit--\Nput them together.
Dialogue: 0,0:45:39.93,0:45:43.34,Default,,0000,0000,0000,,
Dialogue: 0,0:45:43.34,0:45:47.52,Default,,0000,0000,0000,,Does f-- do you\Nlike to do the f?
Dialogue: 0,0:45:47.52,0:45:51.40,Default,,0000,0000,0000,,I used big F, and then I\Nchanged it to little f.
Dialogue: 0,0:45:51.40,0:45:54.07,Default,,0000,0000,0000,,This time I have a function\Nof two variables-- little
Dialogue: 0,0:45:54.07,0:46:01.18,Default,,0000,0000,0000,,f with xy-- to be sine of 1\Nover x squared plus y squared.
Dialogue: 0,0:46:01.18,0:46:09.44,Default,,0000,0000,0000,,Does this function have a\Nlimit at the point 0, 0?
Dialogue: 0,0:46:09.44,0:46:12.35,Default,,0000,0000,0000,,
Dialogue: 0,0:46:12.35,0:46:15.54,Default,,0000,0000,0000,,So when I approach 0,\N0, do I have a limit?
Dialogue: 0,0:46:15.54,0:46:16.68,Default,,0000,0000,0000,,OK.
Dialogue: 0,0:46:16.68,0:46:19.81,Default,,0000,0000,0000,,And you say, well, it depends\Nhow I approach that 0, 0.
Dialogue: 0,0:46:19.81,0:46:21.49,Default,,0000,0000,0000,,That's exactly the thing.
Dialogue: 0,0:46:21.49,0:46:23.21,Default,,0000,0000,0000,,Yes, sir.
Dialogue: 0,0:46:23.21,0:46:25.28,Default,,0000,0000,0000,,Oh, you didn't want to ask me.
Dialogue: 0,0:46:25.28,0:46:28.48,Default,,0000,0000,0000,,
Dialogue: 0,0:46:28.48,0:46:37.12,Default,,0000,0000,0000,,And does f of xy\Nequals-- let me give you
Dialogue: 0,0:46:37.12,0:46:41.08,Default,,0000,0000,0000,,another one, a\Nreally sexy one. x
Dialogue: 0,0:46:41.08,0:46:44.78,Default,,0000,0000,0000,,squared plus y squared\Ntimes sine of 1
Dialogue: 0,0:46:44.78,0:46:48.46,Default,,0000,0000,0000,,over x squared plus y squared.
Dialogue: 0,0:46:48.46,0:46:55.05,Default,,0000,0000,0000,,Have a limit at 0, 0?
Dialogue: 0,0:46:55.05,0:47:00.03,Default,,0000,0000,0000,,
Dialogue: 0,0:47:00.03,0:47:01.49,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:47:01.49,0:47:04.22,Default,,0000,0000,0000,,Continuous it cannot be,\Nbecause it's not defined there.
Dialogue: 0,0:47:04.22,0:47:04.72,Default,,0000,0000,0000,,Right?
Dialogue: 0,0:47:04.72,0:47:07.67,Default,,0000,0000,0000,,For a function to be\Ncontinuous at a point,
Dialogue: 0,0:47:07.67,0:47:11.36,Default,,0000,0000,0000,,the function has to\Nsatisfy three conditions.
Dialogue: 0,0:47:11.36,0:47:14.59,Default,,0000,0000,0000,,The function has to be\Ndefined there at that point.
Dialogue: 0,0:47:14.59,0:47:16.56,Default,,0000,0000,0000,,The function has to\Nhave a limit there
Dialogue: 0,0:47:16.56,0:47:18.89,Default,,0000,0000,0000,,at that point of the domain.
Dialogue: 0,0:47:18.89,0:47:23.14,Default,,0000,0000,0000,,And the limit and the function\Nvalue have to coincide.
Dialogue: 0,0:47:23.14,0:47:24.75,Default,,0000,0000,0000,,Three conditions.
Dialogue: 0,0:47:24.75,0:47:28.47,Default,,0000,0000,0000,,We will talk about\Ncontinuity later.
Dialogue: 0,0:47:28.47,0:47:29.50,Default,,0000,0000,0000,,Hint.
Dialogue: 0,0:47:29.50,0:47:31.62,Default,,0000,0000,0000,,Magdalena, too many hints.
Dialogue: 0,0:47:31.62,0:47:33.68,Default,,0000,0000,0000,,This should remind\Nyou of somebody
Dialogue: 0,0:47:33.68,0:47:36.08,Default,,0000,0000,0000,,from the first\Nvariable calculus.
Dialogue: 0,0:47:36.08,0:47:37.97,Default,,0000,0000,0000,,It's a more challenging problem.
Dialogue: 0,0:47:37.97,0:47:40.49,Default,,0000,0000,0000,,That's why I gave\Nit to extra credit.
Dialogue: 0,0:47:40.49,0:47:45.66,Default,,0000,0000,0000,,If I had x sine of 1/x,\Nwhat would that look like?
Dialogue: 0,0:47:45.66,0:47:46.89,Default,,0000,0000,0000,,STUDENT: x times--
Dialogue: 0,0:47:46.89,0:47:50.26,Default,,0000,0000,0000,,MAGDALENA TODA: x\Ntimes sine of 1/x.
Dialogue: 0,0:47:50.26,0:47:55.46,Default,,0000,0000,0000,,When I approach 0\Nwith-- so if I have-- I
Dialogue: 0,0:47:55.46,0:47:57.16,Default,,0000,0000,0000,,don't ask for an answer now.
Dialogue: 0,0:47:57.16,0:47:58.80,Default,,0000,0000,0000,,You go home, you think about it.
Dialogue: 0,0:47:58.80,0:48:00.31,Default,,0000,0000,0000,,You take the calculator.
Dialogue: 0,0:48:00.31,0:48:05.91,Default,,0000,0000,0000,,But keep in mind that your\Ncalculator can fool you.
Dialogue: 0,0:48:05.91,0:48:11.30,Default,,0000,0000,0000,,Sometimes it can show an\Nimage that misguides you.
Dialogue: 0,0:48:11.30,0:48:14.58,Default,,0000,0000,0000,,So you have to think\Nhow to do that.
Dialogue: 0,0:48:14.58,0:48:18.92,Default,,0000,0000,0000,,How about x times\Nsine of 1/x when--
Dialogue: 0,0:48:18.92,0:48:22.13,Default,,0000,0000,0000,,does it have a limit\Nwhen x goes to 0?
Dialogue: 0,0:48:22.13,0:48:23.62,Default,,0000,0000,0000,,Is there such a limit?
Dialogue: 0,0:48:23.62,0:48:24.60,Default,,0000,0000,0000,,Does it exist?
Dialogue: 0,0:48:24.60,0:48:28.17,Default,,0000,0000,0000,,
Dialogue: 0,0:48:28.17,0:48:31.42,Default,,0000,0000,0000,,So if such a limit\Nwould exist, maybe we
Dialogue: 0,0:48:31.42,0:48:35.60,Default,,0000,0000,0000,,can extend by continuity the\Nfunction x times sine over x.
Dialogue: 0,0:48:35.60,0:48:36.61,Default,,0000,0000,0000,,What does it mean?
Dialogue: 0,0:48:36.61,0:48:38.67,Default,,0000,0000,0000,,Like, extend it, prolong it.
Dialogue: 0,0:48:38.67,0:48:44.42,Default,,0000,0000,0000,,And say, it's this 4x equals\N0 and this if x is not 0.
Dialogue: 0,0:48:44.42,0:48:47.61,Default,,0000,0000,0000,,So this is obviously x is\Ndifferent from 0, right?
Dialogue: 0,0:48:47.61,0:48:49.71,Default,,0000,0000,0000,,Can we extend it by continuity?
Dialogue: 0,0:48:49.71,0:48:51.23,Default,,0000,0000,0000,,Think about the drawing.
Dialogue: 0,0:48:51.23,0:48:54.50,Default,,0000,0000,0000,,Think about the arguments.
Dialogue: 0,0:48:54.50,0:48:58.35,Default,,0000,0000,0000,,And I think it's time for me\Nto keep the promise I made
Dialogue: 0,0:48:58.35,0:49:04.95,Default,,0000,0000,0000,,to [? Aaron, ?]\Nbecause I see no way.
Dialogue: 0,0:49:04.95,0:49:07.86,Default,,0000,0000,0000,,Oh, my god, [? Aaron, ?]\NI see no way out.
Dialogue: 0,0:49:07.86,0:49:10.38,Default,,0000,0000,0000,,
Dialogue: 0,0:49:10.38,0:49:14.49,Default,,0000,0000,0000,,The epsilon delta\Ndefinition of limit.
Dialogue: 0,0:49:14.49,0:49:17.46,Default,,0000,0000,0000,,[? Right? ?] OK.
Dialogue: 0,0:49:17.46,0:49:21.07,Default,,0000,0000,0000,,So what does it mean for a\Nreal mathematician or somebody
Dialogue: 0,0:49:21.07,0:49:25.35,Default,,0000,0000,0000,,with a strong mathematical\Nfoundation and education
Dialogue: 0,0:49:25.35,0:49:27.43,Default,,0000,0000,0000,,that they know the\Ntrue definition
Dialogue: 0,0:49:27.43,0:49:31.31,Default,,0000,0000,0000,,of a limit of a function\Nof, let's say, one variable?
Dialogue: 0,0:49:31.31,0:49:34.91,Default,,0000,0000,0000,,The epsilon delta, the one your\Ndad told you about. [INAUDIBLE]
Dialogue: 0,0:49:34.91,0:49:39.98,Default,,0000,0000,0000,,try to fool you when avoid it\Nin undergraduate education.
Dialogue: 0,0:49:39.98,0:49:41.74,Default,,0000,0000,0000,,People try to avoid\Nthe epsilon delta,
Dialogue: 0,0:49:41.74,0:49:45.62,Default,,0000,0000,0000,,because they think the students\Nwill never, never understand
Dialogue: 0,0:49:45.62,0:49:50.47,Default,,0000,0000,0000,,it, because it's\Nsuch an abstract one.
Dialogue: 0,0:49:50.47,0:49:51.84,Default,,0000,0000,0000,,I think I wasn't ready.
Dialogue: 0,0:49:51.84,0:49:52.76,Default,,0000,0000,0000,,I wasn't smart enough.
Dialogue: 0,0:49:52.76,0:49:58.02,Default,,0000,0000,0000,,I think I was 16 when I was\Ngetting ready for some math
Dialogue: 0,0:49:58.02,0:49:58.72,Default,,0000,0000,0000,,competitions.
Dialogue: 0,0:49:58.72,0:50:02.81,Default,,0000,0000,0000,,And one professor taught me\Nthe epsilon delta and said,
Dialogue: 0,0:50:02.81,0:50:04.70,Default,,0000,0000,0000,,do you understand it?
Dialogue: 0,0:50:04.70,0:50:07.31,Default,,0000,0000,0000,,My 16-year-old mind said, no.
Dialogue: 0,0:50:07.31,0:50:08.86,Default,,0000,0000,0000,,But guess what?
Dialogue: 0,0:50:08.86,0:50:10.55,Default,,0000,0000,0000,,Some other people\Nsmarter than me,
Dialogue: 0,0:50:10.55,0:50:12.41,Default,,0000,0000,0000,,they told me, when\Nyou first see it,
Dialogue: 0,0:50:12.41,0:50:16.69,Default,,0000,0000,0000,,you don't understand\Nit in any case.
Dialogue: 0,0:50:16.69,0:50:19.95,Default,,0000,0000,0000,,So it takes a little bit\Nmore time to sink in.
Dialogue: 0,0:50:19.95,0:50:21.88,Default,,0000,0000,0000,,So the same idea.
Dialogue: 0,0:50:21.88,0:50:25.14,Default,,0000,0000,0000,,As I'm getting closer and\Ncloser and closer and closer
Dialogue: 0,0:50:25.14,0:50:30.28,Default,,0000,0000,0000,,to an x0 with my x values\Nfrom anywhere around-- left,
Dialogue: 0,0:50:30.28,0:50:35.48,Default,,0000,0000,0000,,right-- I have to pick an\Narbitrary choice of points
Dialogue: 0,0:50:35.48,0:50:39.94,Default,,0000,0000,0000,,going towards x0, I have to\Nbe sure that at the same time,
Dialogue: 0,0:50:39.94,0:50:45.21,Default,,0000,0000,0000,,the corresponding sequence\Nof values is going to L,
Dialogue: 0,0:50:45.21,0:50:47.01,Default,,0000,0000,0000,,I can express that\Nin epsilon delta.
Dialogue: 0,0:50:47.01,0:50:50.86,Default,,0000,0000,0000,,
Dialogue: 0,0:50:50.86,0:50:51.94,Default,,0000,0000,0000,,So we say that.
Dialogue: 0,0:50:51.94,0:50:59.70,Default,,0000,0000,0000,,
Dialogue: 0,0:50:59.70,0:51:13.18,Default,,0000,0000,0000,,f of x has limit L\Nat x equals x0 if.
Dialogue: 0,0:51:13.18,0:51:17.18,Default,,0000,0000,0000,,
Dialogue: 0,0:51:17.18,0:51:24.10,Default,,0000,0000,0000,,For every epsilon positive, any\Nchoice of an epsilon positive,
Dialogue: 0,0:51:24.10,0:51:25.23,Default,,0000,0000,0000,,there is a delta.
Dialogue: 0,0:51:25.23,0:51:27.11,Default,,0000,0000,0000,,There exists-- oh, OK, guys.
Dialogue: 0,0:51:27.11,0:51:28.43,Default,,0000,0000,0000,,You don't know the symbols.
Dialogue: 0,0:51:28.43,0:51:31.29,Default,,0000,0000,0000,,I'll write it in English.
Dialogue: 0,0:51:31.29,0:51:36.45,Default,,0000,0000,0000,,For every epsilon\Npositive, no matter
Dialogue: 0,0:51:36.45,0:51:40.73,Default,,0000,0000,0000,,how small-- put\Nparentheses, because you
Dialogue: 0,0:51:40.73,0:51:47.15,Default,,0000,0000,0000,,are just [? tired-- ?]\Nno matter how small,
Dialogue: 0,0:51:47.15,0:51:55.51,Default,,0000,0000,0000,,there exists a delta number\Nthat depends on epsilon.
Dialogue: 0,0:51:55.51,0:52:02.36,Default,,0000,0000,0000,,
Dialogue: 0,0:52:02.36,0:52:16.19,Default,,0000,0000,0000,,So that whenever x minus\Nx0 is less than delta,
Dialogue: 0,0:52:16.19,0:52:33.80,Default,,0000,0000,0000,,this would imply\Nthat f of x minus L,
Dialogue: 0,0:52:33.80,0:52:37.10,Default,,0000,0000,0000,,that limit I taught you\Nabout in absolute value,
Dialogue: 0,0:52:37.10,0:52:38.56,Default,,0000,0000,0000,,is less than epsilon.
Dialogue: 0,0:52:38.56,0:52:48.34,Default,,0000,0000,0000,,
Dialogue: 0,0:52:48.34,0:52:50.37,Default,,0000,0000,0000,,What does this mean?
Dialogue: 0,0:52:50.37,0:52:55.20,Default,,0000,0000,0000,,I'm going to try\Nand draw something
Dialogue: 0,0:52:55.20,0:52:58.44,Default,,0000,0000,0000,,that happens on a line.
Dialogue: 0,0:52:58.44,0:53:00.37,Default,,0000,0000,0000,,So this is x0.
Dialogue: 0,0:53:00.37,0:53:03.89,Default,,0000,0000,0000,,And these are my values of x.
Dialogue: 0,0:53:03.89,0:53:05.06,Default,,0000,0000,0000,,They can come from anywhere.
Dialogue: 0,0:53:05.06,0:53:08.76,Default,,0000,0000,0000,,
Dialogue: 0,0:53:08.76,0:53:12.03,Default,,0000,0000,0000,,And this is f of x.
Dialogue: 0,0:53:12.03,0:53:16.70,Default,,0000,0000,0000,,And this is L. So it\Nsays, no matter-- this
Dialogue: 0,0:53:16.70,0:53:19.15,Default,,0000,0000,0000,,says-- this is an\Nabstract way of saying,
Dialogue: 0,0:53:19.15,0:53:23.81,Default,,0000,0000,0000,,no matter how close, you see,\Nfor every epsilon positive,
Dialogue: 0,0:53:23.81,0:53:27.28,Default,,0000,0000,0000,,no matter how close\Nyou get to the L.
Dialogue: 0,0:53:27.28,0:53:30.60,Default,,0000,0000,0000,,I decide to be in this\Ninterval, very tiny epsilon.
Dialogue: 0,0:53:30.60,0:53:32.26,Default,,0000,0000,0000,,L minus epsilon.
Dialogue: 0,0:53:32.26,0:53:35.84,Default,,0000,0000,0000,,L plus epsilon L. You give\Nme your favorite epsilon.
Dialogue: 0,0:53:35.84,0:53:38.64,Default,,0000,0000,0000,,You say, Magdalena, pick\Nsomething really small.
Dialogue: 0,0:53:38.64,0:53:42.18,Default,,0000,0000,0000,,Big epsilon to be 0.00001.
Dialogue: 0,0:53:42.18,0:53:43.95,Default,,0000,0000,0000,,How about that?
Dialogue: 0,0:53:43.95,0:53:47.66,Default,,0000,0000,0000,,Well, if I really\Nhave a limit there,
Dialogue: 0,0:53:47.66,0:53:54.29,Default,,0000,0000,0000,,an L at x0, that means that\Nno matter how much you shrink
Dialogue: 0,0:53:54.29,0:53:57.54,Default,,0000,0000,0000,,this interval for me, you\Ncan be mean and shrink it
Dialogue: 0,0:53:57.54,0:53:59.32,Default,,0000,0000,0000,,as much as you want.
Dialogue: 0,0:53:59.32,0:54:03.40,Default,,0000,0000,0000,,I will still find a\Nsmall interval around x0.
Dialogue: 0,0:54:03.40,0:54:06.66,Default,,0000,0000,0000,,
Dialogue: 0,0:54:06.66,0:54:08.62,Default,,0000,0000,0000,,[? But ?] I will\Nstill find the smaller
Dialogue: 0,0:54:08.62,0:54:13.23,Default,,0000,0000,0000,,interval around x0, which is--\Nthis would be x0 minus delta.
Dialogue: 0,0:54:13.23,0:54:15.76,Default,,0000,0000,0000,,This would be x0 plus delta.
Dialogue: 0,0:54:15.76,0:54:20.96,Default,,0000,0000,0000,,So that the image of this\Npurple interval fits inside.
Dialogue: 0,0:54:20.96,0:54:22.15,Default,,0000,0000,0000,,You say, what?
Dialogue: 0,0:54:22.15,0:54:25.84,Default,,0000,0000,0000,,So that the image of this\Npurple interval fits inside.
Dialogue: 0,0:54:25.84,0:54:30.04,Default,,0000,0000,0000,,So f of x minus L, the distance\Nis still that, less than xy.
Dialogue: 0,0:54:30.04,0:54:30.55,Default,,0000,0000,0000,,Yes, sir?
Dialogue: 0,0:54:30.55,0:54:32.71,Default,,0000,0000,0000,,STUDENT: Where'd you\Nget epsilon [INAUDIBLE]?
Dialogue: 0,0:54:32.71,0:54:34.44,Default,,0000,0000,0000,,MAGDALENA TODA: So\Nepsilon has to be
Dialogue: 0,0:54:34.44,0:54:38.25,Default,,0000,0000,0000,,chose no matter how small.
Dialogue: 0,0:54:38.25,0:54:39.84,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,0:54:39.84,0:54:40.67,Default,,0000,0000,0000,,MAGDALENA TODA: Huh?
Dialogue: 0,0:54:40.67,0:54:42.16,Default,,0000,0000,0000,,Real number.
Dialogue: 0,0:54:42.16,0:54:46.47,Default,,0000,0000,0000,,So I'm saying, you should not\Nset the epsilon to be 0.0001.
Dialogue: 0,0:54:46.47,0:54:47.77,Default,,0000,0000,0000,,That would be a mistake.
Dialogue: 0,0:54:47.77,0:54:50.89,Default,,0000,0000,0000,,You have to think of that number\Nas being as small as you want,
Dialogue: 0,0:54:50.89,0:54:54.59,Default,,0000,0000,0000,,infinitesimally small, smaller\Nthan any particle in physics
Dialogue: 0,0:54:54.59,0:54:57.33,Default,,0000,0000,0000,,that you are aware about.
Dialogue: 0,0:54:57.33,0:54:59.88,Default,,0000,0000,0000,,And this is what I had the\Nproblem understanding--
Dialogue: 0,0:54:59.88,0:55:03.53,Default,,0000,0000,0000,,that notion of-- not\Nthe notion of, hey, not
Dialogue: 0,0:55:03.53,0:55:06.00,Default,,0000,0000,0000,,matter how close\NI am, I can still
Dialogue: 0,0:55:06.00,0:55:12.43,Default,,0000,0000,0000,,get something even smaller\Naround x0 that fits in this.
Dialogue: 0,0:55:12.43,0:55:14.36,Default,,0000,0000,0000,,That's not what I\Nhad the problem with.
Dialogue: 0,0:55:14.36,0:55:18.32,Default,,0000,0000,0000,,The notion is to perceive\Nan infinitesimal.
Dialogue: 0,0:55:18.32,0:55:21.52,Default,,0000,0000,0000,,Our mind is too limited\Nto understand infinity.
Dialogue: 0,0:55:21.52,0:55:24.41,Default,,0000,0000,0000,,It's like trying\Nto understand God.
Dialogue: 0,0:55:24.41,0:55:29.51,Default,,0000,0000,0000,,And the same limitation comes\Nwith microscopic problems.
Dialogue: 0,0:55:29.51,0:55:31.47,Default,,0000,0000,0000,,Yeah, we can see some\Nthings on the microscope,
Dialogue: 0,0:55:31.47,0:55:32.22,Default,,0000,0000,0000,,and we understand.
Dialogue: 0,0:55:32.22,0:55:34.58,Default,,0000,0000,0000,,Ah, I understand I\Nhave this bacteria.
Dialogue: 0,0:55:34.58,0:55:36.04,Default,,0000,0000,0000,,This is staph.
Dialogue: 0,0:55:36.04,0:55:37.46,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,0:55:37.46,0:55:43.35,Default,,0000,0000,0000,,But then there are molecules,\Natoms, subatomic particles
Dialogue: 0,0:55:43.35,0:55:46.88,Default,,0000,0000,0000,,that we don't understand,\Nbecause our mind is really
Dialogue: 0,0:55:46.88,0:55:49.06,Default,,0000,0000,0000,,[? small. ?] Imagine\Nsomething smaller
Dialogue: 0,0:55:49.06,0:55:50.90,Default,,0000,0000,0000,,than the subatomic particles.
Dialogue: 0,0:55:50.90,0:55:54.52,Default,,0000,0000,0000,,That's the abstract notion\Nof infinitesimally small.
Dialogue: 0,0:55:54.52,0:55:58.87,Default,,0000,0000,0000,,So I'm saying, if I really\Nhave a limit L there,
Dialogue: 0,0:55:58.87,0:56:03.47,Default,,0000,0000,0000,,that means no matter how small\NI have this ball around it,
Dialogue: 0,0:56:03.47,0:56:06.63,Default,,0000,0000,0000,,I can still find a\Nsmaller ball that
Dialogue: 0,0:56:06.63,0:56:10.00,Default,,0000,0000,0000,,fits-- whose image fits inside.
Dialogue: 0,0:56:10.00,0:56:10.50,Default,,0000,0000,0000,,All right?
Dialogue: 0,0:56:10.50,0:56:15.00,Default,,0000,0000,0000,,The same kind of definition--\NI will try to generalize this.
Dialogue: 0,0:56:15.00,0:56:19.66,Default,,0000,0000,0000,,Can you guys help me\Ngeneralize this limit notion
Dialogue: 0,0:56:19.66,0:56:24.79,Default,,0000,0000,0000,,to the notion of function\Nof two variables?
Dialogue: 0,0:56:24.79,0:56:29.04,Default,,0000,0000,0000,,
Dialogue: 0,0:56:29.04,0:56:41.40,Default,,0000,0000,0000,,So we say, that f of xy\Nhas the limit L at x0y0.
Dialogue: 0,0:56:41.40,0:56:44.97,Default,,0000,0000,0000,,
Dialogue: 0,0:56:44.97,0:56:50.71,Default,,0000,0000,0000,,What was x0y0 when I\Ntalked about-- what
Dialogue: 0,0:56:50.71,0:56:52.55,Default,,0000,0000,0000,,example did I give you guys?
Dialogue: 0,0:56:52.55,0:56:55.05,Default,,0000,0000,0000,,Sine of 1 over x squared\Nplus y squared, right?
Dialogue: 0,0:56:55.05,0:56:56.22,Default,,0000,0000,0000,,Something like that.
Dialogue: 0,0:56:56.22,0:56:56.85,Default,,0000,0000,0000,,I don't know.
Dialogue: 0,0:56:56.85,0:56:59.73,Default,,0000,0000,0000,,I said, think of 0, 0.
Dialogue: 0,0:56:59.73,0:57:01.81,Default,,0000,0000,0000,,That was the given point.
Dialogue: 0,0:57:01.81,0:57:03.54,Default,,0000,0000,0000,,It has to be a fixed couple.
Dialogue: 0,0:57:03.54,0:57:07.87,Default,,0000,0000,0000,,So you think of the origin, 0,\N0, as being as a fixed couple.
Dialogue: 0,0:57:07.87,0:57:12.39,Default,,0000,0000,0000,,Or you think of the point 1,\N0 as being as a fixed couple
Dialogue: 0,0:57:12.39,0:57:14.76,Default,,0000,0000,0000,,in that plane you look at.
Dialogue: 0,0:57:14.76,0:57:18.42,Default,,0000,0000,0000,,That is the fixed couple.
Dialogue: 0,0:57:18.42,0:57:21.46,Default,,0000,0000,0000,,If-- now somebody\Nhas to help me.
Dialogue: 0,0:57:21.46,0:57:27.54,Default,,0000,0000,0000,,For every epsilon positive,\Nno matter how small,
Dialogue: 0,0:57:27.54,0:57:30.69,Default,,0000,0000,0000,,that's where I have a problem\Nimagining infinitesimally
Dialogue: 0,0:57:30.69,0:57:31.62,Default,,0000,0000,0000,,small.
Dialogue: 0,0:57:31.62,0:57:34.65,Default,,0000,0000,0000,,There exists-- I no\Nlonger have this problem.
Dialogue: 0,0:57:34.65,0:57:37.31,Default,,0000,0000,0000,,But I had it enough\Nwhen I was in my 20s.
Dialogue: 0,0:57:37.31,0:57:40.16,Default,,0000,0000,0000,,I don't want to go back to\Nmy 20s and have-- I mean,
Dialogue: 0,0:57:40.16,0:57:41.06,Default,,0000,0000,0000,,I would love to.
Dialogue: 0,0:57:41.06,0:57:42.89,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,0:57:42.89,0:57:46.42,Default,,0000,0000,0000,,To go having vacations\Nwith no worries and so on.
Dialogue: 0,0:57:46.42,0:57:48.57,Default,,0000,0000,0000,,But I wouldn't like\Nto go back to my 20s
Dialogue: 0,0:57:48.57,0:57:50.47,Default,,0000,0000,0000,,and have to relearn\Nall the mathematics.
Dialogue: 0,0:57:50.47,0:57:50.97,Default,,0000,0000,0000,,Now way.
Dialogue: 0,0:57:50.97,0:57:53.19,Default,,0000,0000,0000,,That was too much of a struggle.
Dialogue: 0,0:57:53.19,0:58:00.31,Default,,0000,0000,0000,,There exists a delta positive\Nthat depends on epsilon.
Dialogue: 0,0:58:00.31,0:58:02.82,Default,,0000,0000,0000,,What does it mean,\Ndepends on epsilon?
Dialogue: 0,0:58:02.82,0:58:04.65,Default,,0000,0000,0000,,Because guys, imagine\Nyou make this epsilon
Dialogue: 0,0:58:04.65,0:58:05.86,Default,,0000,0000,0000,,smaller and smaller.
Dialogue: 0,0:58:05.86,0:58:08.10,Default,,0000,0000,0000,,You have to make delta\Nsmaller and smaller,
Dialogue: 0,0:58:08.10,0:58:12.32,Default,,0000,0000,0000,,so that you can fit that\Nlittle ball in the big ball.
Dialogue: 0,0:58:12.32,0:58:12.82,Default,,0000,0000,0000,,OK?
Dialogue: 0,0:58:12.82,0:58:19.77,Default,,0000,0000,0000,,That depends on epsilon,\Nso that whenever-- now,
Dialogue: 0,0:58:19.77,0:58:21.93,Default,,0000,0000,0000,,that is a big problem.
Dialogue: 0,0:58:21.93,0:58:28.30,Default,,0000,0000,0000,,How do I say, distance between\Nthe point xy and the point
Dialogue: 0,0:58:28.30,0:58:29.31,Default,,0000,0000,0000,,x0y0?
Dialogue: 0,0:58:29.31,0:58:32.13,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,0:58:32.13,0:58:37.31,Default,,0000,0000,0000,,This is distance between xy\Nand x0y0 is less than delta.
Dialogue: 0,0:58:37.31,0:58:48.48,Default,,0000,0000,0000,,This would imply\Nthat-- well, this
Dialogue: 0,0:58:48.48,0:58:54.09,Default,,0000,0000,0000,,is a function with values in\NR. This is in R. Real number.
Dialogue: 0,0:58:54.09,0:58:55.47,Default,,0000,0000,0000,,So I don't have a problem.
Dialogue: 0,0:58:55.47,0:58:57.46,Default,,0000,0000,0000,,I can use absolute value here.
Dialogue: 0,0:58:57.46,0:59:11.33,Default,,0000,0000,0000,,Absolute value of f of\Nthe couple xy minus L
Dialogue: 0,0:59:11.33,0:59:14.58,Default,,0000,0000,0000,,is less than epsilon.
Dialogue: 0,0:59:14.58,0:59:19.08,Default,,0000,0000,0000,,The thing is, can you\Nvisualize that little ball,
Dialogue: 0,0:59:19.08,0:59:20.99,Default,,0000,0000,0000,,that little disk?
Dialogue: 0,0:59:20.99,0:59:22.31,Default,,0000,0000,0000,,What do I mean?
Dialogue: 0,0:59:22.31,0:59:26.44,Default,,0000,0000,0000,,Being close, xy is me, right?
Dialogue: 0,0:59:26.44,0:59:27.52,Default,,0000,0000,0000,,But I'm moving.
Dialogue: 0,0:59:27.52,0:59:28.76,Default,,0000,0000,0000,,I'm the moving point.
Dialogue: 0,0:59:28.76,0:59:30.35,Default,,0000,0000,0000,,I'm dancing around.
Dialogue: 0,0:59:30.35,0:59:33.44,Default,,0000,0000,0000,,And [? Nateesh ?] is x0y0.
Dialogue: 0,0:59:33.44,0:59:37.57,Default,,0000,0000,0000,,How do I say that I have\Nto be close enough to him?
Dialogue: 0,0:59:37.57,0:59:38.60,Default,,0000,0000,0000,,I cannot touch him.
Dialogue: 0,0:59:38.60,0:59:39.71,Default,,0000,0000,0000,,That's against the rules.
Dialogue: 0,0:59:39.71,0:59:42.27,Default,,0000,0000,0000,,That's considered\N[INAUDIBLE] harassment.
Dialogue: 0,0:59:42.27,0:59:45.80,Default,,0000,0000,0000,,But I can come as\Nclose as I want.
Dialogue: 0,0:59:45.80,0:59:49.21,Default,,0000,0000,0000,,So I say, the\Ndistance between me--
Dialogue: 0,0:59:49.21,0:59:52.10,Default,,0000,0000,0000,,I'm xy-- and\N[? Nateesh, ?] who is
Dialogue: 0,0:59:52.10,0:59:58.26,Default,,0000,0000,0000,,fixed x0y0, has to be smaller\Nthan that small delta.
Dialogue: 0,0:59:58.26,1:00:00.80,Default,,0000,0000,0000,,How do I represent that\Nin plane mathematics?
Dialogue: 0,1:00:00.80,1:00:02.00,Default,,0000,0000,0000,,STUDENT: Doesn't [INAUDIBLE]?
Dialogue: 0,1:00:02.00,1:00:05.52,Default,,0000,0000,0000,,
Dialogue: 0,1:00:05.52,1:00:06.52,Default,,0000,0000,0000,,MAGDALENA TODA: Exactly.
Dialogue: 0,1:00:06.52,1:00:09.48,Default,,0000,0000,0000,,So that delta has to\Nbe small enough so
Dialogue: 0,1:00:09.48,1:00:16.97,Default,,0000,0000,0000,,that the image of f at me minus\Nthe limit is less than epsilon.
Dialogue: 0,1:00:16.97,1:00:20.96,Default,,0000,0000,0000,,Now you understand why all\Nthe other teachers avoid
Dialogue: 0,1:00:20.96,1:00:22.82,Default,,0000,0000,0000,,talking about this\N[? one. ?] So I
Dialogue: 0,1:00:22.82,1:00:27.84,Default,,0000,0000,0000,,want to get small enough-- not\Ntoo close-- but close enough
Dialogue: 0,1:00:27.84,1:00:39.52,Default,,0000,0000,0000,,to him, so that my value--\NI'm f of xy-- minus the limit,
Dialogue: 0,1:00:39.52,1:00:42.17,Default,,0000,0000,0000,,the limit-- I have\Na preset limit.
Dialogue: 0,1:00:42.17,1:00:45.14,Default,,0000,0000,0000,,All around [? Nateesh, ?] I\Ncan have different values,
Dialogue: 0,1:00:45.14,1:00:47.49,Default,,0000,0000,0000,,no matter where I go.
Dialogue: 0,1:00:47.49,1:00:51.06,Default,,0000,0000,0000,,My value at all these points\Naround [? Nateesh ?] have
Dialogue: 0,1:00:51.06,1:00:54.51,Default,,0000,0000,0000,,to be close enough\Nto L. So I say,
Dialogue: 0,1:00:54.51,1:00:57.55,Default,,0000,0000,0000,,well, you have to get\Nclose enough to L.
Dialogue: 0,1:00:57.55,1:00:59.49,Default,,0000,0000,0000,,Somebody presents me an epsilon.
Dialogue: 0,1:00:59.49,1:01:02.35,Default,,0000,0000,0000,,Then I have to reduce my\Ndistance to [? Nateesh ?]
Dialogue: 0,1:01:02.35,1:01:03.99,Default,,0000,0000,0000,,depending to that epsilon.
Dialogue: 0,1:01:03.99,1:01:07.96,Default,,0000,0000,0000,,Because otherwise,\Nthe image doesn't fit.
Dialogue: 0,1:01:07.96,1:01:09.00,Default,,0000,0000,0000,,It's a little bit tricky.
Dialogue: 0,1:01:09.00,1:01:10.96,Default,,0000,0000,0000,,STUDENT: So is this like\Nthe squeeze theorem kind of?
Dialogue: 0,1:01:10.96,1:01:12.43,Default,,0000,0000,0000,,MAGDALENA TODA: It is\Nthe squeeze theorem.
Dialogue: 0,1:01:12.43,1:01:12.93,Default,,0000,0000,0000,,STUDENT: Oh, all right.
Dialogue: 0,1:01:12.93,1:01:13.72,Default,,0000,0000,0000,,MAGDALENA TODA: OK?
Dialogue: 0,1:01:13.72,1:01:18.58,Default,,0000,0000,0000,,So the squeezing-- I ball into\Nanother [? ball ?] [? limit. ?]
Dialogue: 0,1:01:18.58,1:01:21.03,Default,,0000,0000,0000,,This is why-- it's not\Na ball, but it's a--
Dialogue: 0,1:01:21.03,1:01:21.78,Default,,0000,0000,0000,,STUDENT: A circle.
Dialogue: 0,1:01:21.78,1:01:22.66,Default,,0000,0000,0000,,MAGDALENA TODA: Disk.
Dialogue: 0,1:01:22.66,1:01:23.72,Default,,0000,0000,0000,,A circle, right?
Dialogue: 0,1:01:23.72,1:01:28.83,Default,,0000,0000,0000,,So how do we express\Nthat in Calc 3 in plain?
Dialogue: 0,1:01:28.83,1:01:31.00,Default,,0000,0000,0000,,This is the\N[? ingredient, ?] distance d.
Dialogue: 0,1:01:31.00,1:01:33.63,Default,,0000,0000,0000,,So Seth, can you tell me what is\Nthe distance between these two
Dialogue: 0,1:01:33.63,1:01:34.79,Default,,0000,0000,0000,,points?
Dialogue: 0,1:01:34.79,1:01:36.01,Default,,0000,0000,0000,,Square root of--
Dialogue: 0,1:01:36.01,1:01:37.36,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:01:37.36,1:01:41.75,Default,,0000,0000,0000,,MAGDALENA TODA: x minus\Nx0 squared plus y minus y0
Dialogue: 0,1:01:41.75,1:01:43.15,Default,,0000,0000,0000,,squared.
Dialogue: 0,1:01:43.15,1:01:45.38,Default,,0000,0000,0000,,Now shut up. [? And I ?]\Nam talking to myself.
Dialogue: 0,1:01:45.38,1:01:45.86,Default,,0000,0000,0000,,STUDENT: Must be\Nless than delta.
Dialogue: 0,1:01:45.86,1:01:46.70,Default,,0000,0000,0000,,[LAUGHTER]
Dialogue: 0,1:01:46.70,1:01:48.28,Default,,0000,0000,0000,,MAGDALENA TODA: Less than delta.
Dialogue: 0,1:01:48.28,1:01:51.53,Default,,0000,0000,0000,,So instead of\Nwriting this, I need
Dialogue: 0,1:01:51.53,1:01:53.73,Default,,0000,0000,0000,,to write that I can\Ndo that in my mind.
Dialogue: 0,1:01:53.73,1:01:58.36,Default,,0000,0000,0000,,
Dialogue: 0,1:01:58.36,1:02:00.02,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:02:00.02,1:02:00.99,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:02:00.99,1:02:01.85,Default,,0000,0000,0000,,This is hard.
Dialogue: 0,1:02:01.85,1:02:03.47,Default,,0000,0000,0000,,We need to sleep on that.
Dialogue: 0,1:02:03.47,1:02:09.45,Default,,0000,0000,0000,,I have one or two more problems\Nthat are less hard-- nah,
Dialogue: 0,1:02:09.45,1:02:11.49,Default,,0000,0000,0000,,they are still hard, but\Nthey are more intuitive,
Dialogue: 0,1:02:11.49,1:02:14.57,Default,,0000,0000,0000,,that I would like to\Nask you about the limit.
Dialogue: 0,1:02:14.57,1:02:16.94,Default,,0000,0000,0000,,I'm going to give\Nyou a function.
Dialogue: 0,1:02:16.94,1:02:21.48,Default,,0000,0000,0000,,And we would have to visualize\Nas I get closer to a point
Dialogue: 0,1:02:21.48,1:02:24.57,Default,,0000,0000,0000,,where I am actually going.
Dialogue: 0,1:02:24.57,1:02:30.17,Default,,0000,0000,0000,,So I have this nasty\Nfunction, f of xy
Dialogue: 0,1:02:30.17,1:02:34.96,Default,,0000,0000,0000,,equals xy over z\Nsquared plus y squared.
Dialogue: 0,1:02:34.96,1:02:39.02,Default,,0000,0000,0000,,
Dialogue: 0,1:02:39.02,1:02:44.60,Default,,0000,0000,0000,,And I'm saying, [INAUDIBLE]\Nthe point is the origin.
Dialogue: 0,1:02:44.60,1:02:47.03,Default,,0000,0000,0000,,I choose the origin.
Dialogue: 0,1:02:47.03,1:02:47.59,Default,,0000,0000,0000,,Question.
Dialogue: 0,1:02:47.59,1:02:53.02,Default,,0000,0000,0000,,Do I have a limit that's--\Ndo I have a limit?
Dialogue: 0,1:02:53.02,1:02:54.62,Default,,0000,0000,0000,,Not [? really ?] for me.
Dialogue: 0,1:02:54.62,1:03:02.48,Default,,0000,0000,0000,,Does f have a limit\Nat the origin?
Dialogue: 0,1:03:02.48,1:03:06.40,Default,,0000,0000,0000,,
Dialogue: 0,1:03:06.40,1:03:09.82,Default,,0000,0000,0000,,You would have to imagine\Nthat you'd draw this function.
Dialogue: 0,1:03:09.82,1:03:13.14,Default,,0000,0000,0000,,And except you cannot draw, and\Nyou really don't care to draw
Dialogue: 0,1:03:13.14,1:03:13.78,Default,,0000,0000,0000,,it.
Dialogue: 0,1:03:13.78,1:03:17.03,Default,,0000,0000,0000,,You only have to imagine that\Nyou have some abstract graph--
Dialogue: 0,1:03:17.03,1:03:19.05,Default,,0000,0000,0000,,z equals f of xy.
Dialogue: 0,1:03:19.05,1:03:20.88,Default,,0000,0000,0000,,You don't care\Nwhat it looks like.
Dialogue: 0,1:03:20.88,1:03:24.10,Default,,0000,0000,0000,,But then you take\Npoints on the floor,
Dialogue: 0,1:03:24.10,1:03:27.45,Default,,0000,0000,0000,,just like I did the exercise\Nwith [? Nateesh ?] before.
Dialogue: 0,1:03:27.45,1:03:30.82,Default,,0000,0000,0000,,And you get closer and\Ncloser to the origin.
Dialogue: 0,1:03:30.82,1:03:34.42,Default,,0000,0000,0000,,But no attention-- no\Nmatter what path I take,
Dialogue: 0,1:03:34.42,1:03:36.85,Default,,0000,0000,0000,,I have to get the same limit.
Dialogue: 0,1:03:36.85,1:03:37.55,Default,,0000,0000,0000,,What?
Dialogue: 0,1:03:37.55,1:03:46.68,Default,,0000,0000,0000,,No matter what path I take\Ntowards [? Nateesh-- ?]
Dialogue: 0,1:03:46.68,1:03:52.86,Default,,0000,0000,0000,,don't write that down-- towards\N[? z0y0, ?] I have to get
Dialogue: 0,1:03:52.86,1:03:53.79,Default,,0000,0000,0000,,the same limit.
Dialogue: 0,1:03:53.79,1:03:56.97,Default,,0000,0000,0000,,
Dialogue: 0,1:03:56.97,1:03:59.03,Default,,0000,0000,0000,,Do I?
Dialogue: 0,1:03:59.03,1:04:04.03,Default,,0000,0000,0000,,Let's imagine with the\Neyes of your imaginations.
Dialogue: 0,1:04:04.03,1:04:07.41,Default,,0000,0000,0000,,And [? Nateesh ?]\Nis the point 0, 0.
Dialogue: 0,1:04:07.41,1:04:10.65,Default,,0000,0000,0000,,And you are aspiring to get\Ncloser and closer to him
Dialogue: 0,1:04:10.65,1:04:12.75,Default,,0000,0000,0000,,without touching him.
Dialogue: 0,1:04:12.75,1:04:15.26,Default,,0000,0000,0000,,Because otherwise,\Nhe's going to sue you.
Dialogue: 0,1:04:15.26,1:04:18.12,Default,,0000,0000,0000,,So what do we have here?
Dialogue: 0,1:04:18.12,1:04:19.35,Default,,0000,0000,0000,,We have different paths?
Dialogue: 0,1:04:19.35,1:04:21.39,Default,,0000,0000,0000,,How can I get closer?
Dialogue: 0,1:04:21.39,1:04:25.55,Default,,0000,0000,0000,,Either on this path\Nor maybe on this path.
Dialogue: 0,1:04:25.55,1:04:28.21,Default,,0000,0000,0000,,Or maybe on this path.
Dialogue: 0,1:04:28.21,1:04:31.59,Default,,0000,0000,0000,,Or maybe, if I had something\Nto drink last night-- which
Dialogue: 0,1:04:31.59,1:04:35.12,Default,,0000,0000,0000,,I did not, because\Nafter the age of 35,
Dialogue: 0,1:04:35.12,1:04:37.03,Default,,0000,0000,0000,,I stopped drinking completely.
Dialogue: 0,1:04:37.03,1:04:40.99,Default,,0000,0000,0000,,
Dialogue: 0,1:04:40.99,1:04:44.82,Default,,0000,0000,0000,,That's when I decided\NI want to be a mom,
Dialogue: 0,1:04:44.82,1:04:47.27,Default,,0000,0000,0000,,and I didn't want to\Nmake a bad example.
Dialogue: 0,1:04:47.27,1:04:50.45,Default,,0000,0000,0000,,So no matter what path you\Ntake, you can make it wiggly,
Dialogue: 0,1:04:50.45,1:04:52.03,Default,,0000,0000,0000,,you can make it\Nany way you want.
Dialogue: 0,1:04:52.03,1:04:53.70,Default,,0000,0000,0000,,We are still approaching 0, 0.
Dialogue: 0,1:04:53.70,1:04:55.92,Default,,0000,0000,0000,,You still have to\Nget the same limit.
Dialogue: 0,1:04:55.92,1:05:00.17,Default,,0000,0000,0000,,Oh, that's tricky, because\Nit's also the same in life.
Dialogue: 0,1:05:00.17,1:05:02.43,Default,,0000,0000,0000,,Depending on the path\Nyou take in life,
Dialogue: 0,1:05:02.43,1:05:05.45,Default,,0000,0000,0000,,you have different\Nresults, different limits.
Dialogue: 0,1:05:05.45,1:05:11.24,Default,,0000,0000,0000,,Now, what if I take the path\Nnumber one, number two, number
Dialogue: 0,1:05:11.24,1:05:12.72,Default,,0000,0000,0000,,three possibility.
Dialogue: 0,1:05:12.72,1:05:16.90,Default,,0000,0000,0000,,And number [? blooie ?]\Nis the drunken variant.
Dialogue: 0,1:05:16.90,1:05:22.12,Default,,0000,0000,0000,,That is hard to\Nimplement in an exercise.
Dialogue: 0,1:05:22.12,1:05:26.60,Default,,0000,0000,0000,,Imagine that I have\Nlimit along the path one.
Dialogue: 0,1:05:26.60,1:05:28.29,Default,,0000,0000,0000,,Path one.
Dialogue: 0,1:05:28.29,1:05:34.68,Default,,0000,0000,0000,,xy goes to 0, 0 of xy over\Nx squared plus y squared.
Dialogue: 0,1:05:34.68,1:05:36.97,Default,,0000,0000,0000,,Do you guys see what's\Ngoing to happen?
Dialogue: 0,1:05:36.97,1:05:40.66,Default,,0000,0000,0000,,So I'm along the--\NOK, here it is.
Dialogue: 0,1:05:40.66,1:05:46.59,Default,,0000,0000,0000,,This line, right, this is\Nthe x-axis, y-axis, z-axis.
Dialogue: 0,1:05:46.59,1:05:48.53,Default,,0000,0000,0000,,What's special for the x-axis?
Dialogue: 0,1:05:48.53,1:05:50.47,Default,,0000,0000,0000,,Who is 0?
Dialogue: 0,1:05:50.47,1:05:52.70,Default,,0000,0000,0000,,STUDENT: x.
Dialogue: 0,1:05:52.70,1:05:53.33,Default,,0000,0000,0000,,STUDENT: yz.
Dialogue: 0,1:05:53.33,1:05:54.42,Default,,0000,0000,0000,,MAGDALENA TODA: y is 0.
Dialogue: 0,1:05:54.42,1:05:57.40,Default,,0000,0000,0000,,So y is 0.
Dialogue: 0,1:05:57.40,1:05:59.41,Default,,0000,0000,0000,,So y is 0.
Dialogue: 0,1:05:59.41,1:06:00.36,Default,,0000,0000,0000,,Don't laugh at me.
Dialogue: 0,1:06:00.36,1:06:03.47,Default,,0000,0000,0000,,I'm going to write like\Nthat because it's easier.
Dialogue: 0,1:06:03.47,1:06:06.77,Default,,0000,0000,0000,,And it's going to be\Nsomething like limit
Dialogue: 0,1:06:06.77,1:06:13.74,Default,,0000,0000,0000,,when x approaches 0\Nof x over x squared.
Dialogue: 0,1:06:13.74,1:06:15.69,Default,,0000,0000,0000,,STUDENT: It's 1/x.
Dialogue: 0,1:06:15.69,1:06:17.95,Default,,0000,0000,0000,,MAGDALENA TODA: Times 0 up.
Dialogue: 0,1:06:17.95,1:06:18.95,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,1:06:18.95,1:06:20.67,Default,,0000,0000,0000,,Is that-- how much is that?
Dialogue: 0,1:06:20.67,1:06:21.17,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:06:21.17,1:06:21.29,Default,,0000,0000,0000,,STUDENT: 0.
Dialogue: 0,1:06:21.29,1:06:22.04,Default,,0000,0000,0000,,MAGDALENA TODA: 0!
Dialogue: 0,1:06:22.04,1:06:22.57,Default,,0000,0000,0000,,I'm happy.
Dialogue: 0,1:06:22.57,1:06:23.98,Default,,0000,0000,0000,,I say, maybe I have the limit.
Dialogue: 0,1:06:23.98,1:06:24.82,Default,,0000,0000,0000,,I have the limit 0.
Dialogue: 0,1:06:24.82,1:06:27.12,Default,,0000,0000,0000,,No, never rush in life.
Dialogue: 0,1:06:27.12,1:06:28.04,Default,,0000,0000,0000,,Check.
Dialogue: 0,1:06:28.04,1:06:30.78,Default,,0000,0000,0000,,Experiment any other paths.
Dialogue: 0,1:06:30.78,1:06:35.12,Default,,0000,0000,0000,,And it's actually very easy\Nto see where I can go wrong.
Dialogue: 0,1:06:35.12,1:06:39.63,Default,,0000,0000,0000,,If I take the path number two,\NI will get the same result.
Dialogue: 0,1:06:39.63,1:06:41.47,Default,,0000,0000,0000,,You don't need a\Nlot of imagination
Dialogue: 0,1:06:41.47,1:06:44.36,Default,,0000,0000,0000,,to realize, hey, whether\Nshe does it for x
Dialogue: 0,1:06:44.36,1:06:48.36,Default,,0000,0000,0000,,or does it for y, if she\Ngoes along the 2, what
Dialogue: 0,1:06:48.36,1:06:49.79,Default,,0000,0000,0000,,the heck is going to happen?
Dialogue: 0,1:06:49.79,1:06:51.03,Default,,0000,0000,0000,,y is going to shrink.
Dialogue: 0,1:06:51.03,1:06:53.15,Default,,0000,0000,0000,,x will always be 0.
Dialogue: 0,1:06:53.15,1:06:56.85,Default,,0000,0000,0000,,Because this means\Na point's like what?
Dialogue: 0,1:06:56.85,1:06:58.85,Default,,0000,0000,0000,,0,1.
Dialogue: 0,1:06:58.85,1:07:00.84,Default,,0000,0000,0000,,0, 1/2.
Dialogue: 0,1:07:00.84,1:07:03.18,Default,,0000,0000,0000,,0, 1/n, and so on.
Dialogue: 0,1:07:03.18,1:07:07.82,Default,,0000,0000,0000,,But plug them all in here,\NI get 0, 1/n times 0.
Dialogue: 0,1:07:07.82,1:07:08.67,Default,,0000,0000,0000,,It's still 0.
Dialogue: 0,1:07:08.67,1:07:10.46,Default,,0000,0000,0000,,So I still get 0.
Dialogue: 0,1:07:10.46,1:07:12.31,Default,,0000,0000,0000,,Path two.
Dialogue: 0,1:07:12.31,1:07:15.47,Default,,0000,0000,0000,,When I approach my--\Nxt goes to 0, 0.
Dialogue: 0,1:07:15.47,1:07:18.79,Default,,0000,0000,0000,,The poor [? Nateesh ?]\Nis waiting for an answer.
Dialogue: 0,1:07:18.79,1:07:20.95,Default,,0000,0000,0000,,I still get 0.
Dialogue: 0,1:07:20.95,1:07:23.64,Default,,0000,0000,0000,,Let's take not the\Ndrunken path, because I
Dialogue: 0,1:07:23.64,1:07:25.52,Default,,0000,0000,0000,,don't know [? it unless ?]\Nthe sine function.
Dialogue: 0,1:07:25.52,1:07:26.97,Default,,0000,0000,0000,,That is really crazy.
Dialogue: 0,1:07:26.97,1:07:29.01,Default,,0000,0000,0000,,I'll take this one.
Dialogue: 0,1:07:29.01,1:07:31.42,Default,,0000,0000,0000,,What is this one,\Nin your opinion?
Dialogue: 0,1:07:31.42,1:07:32.97,Default,,0000,0000,0000,,Is that going to help me?
Dialogue: 0,1:07:32.97,1:07:35.72,Default,,0000,0000,0000,,I don't know, but I\Nneed some intuition.
Dialogue: 0,1:07:35.72,1:07:39.55,Default,,0000,0000,0000,,Mathematicians need intuition\Nand a lot of patience.
Dialogue: 0,1:07:39.55,1:07:42.11,Default,,0000,0000,0000,,So what is your intuition?
Dialogue: 0,1:07:42.11,1:07:45.12,Default,,0000,0000,0000,,The one in the middle, I'm going\Nto start walking on that, OK,
Dialogue: 0,1:07:45.12,1:07:46.54,Default,,0000,0000,0000,,until you tell me what it is.
Dialogue: 0,1:07:46.54,1:07:47.50,Default,,0000,0000,0000,,STUDENT: y [INAUDIBLE].
Dialogue: 0,1:07:47.50,1:07:49.54,Default,,0000,0000,0000,,MAGDALENA TODA: y equals\Nx is the first bisector
Dialogue: 0,1:07:49.54,1:07:51.04,Default,,0000,0000,0000,,or the first quadrant.
Dialogue: 0,1:07:51.04,1:07:54.58,Default,,0000,0000,0000,,And I'm very happy\NI can go both ways.
Dialogue: 0,1:07:54.58,1:07:55.85,Default,,0000,0000,0000,,y equals x.
Dialogue: 0,1:07:55.85,1:07:56.46,Default,,0000,0000,0000,,x [INAUDIBLE].
Dialogue: 0,1:07:56.46,1:08:07.13,Default,,0000,0000,0000,,So limit when x equals y,\Nbut the pair xy goes to 0,0.
Dialogue: 0,1:08:07.13,1:08:07.98,Default,,0000,0000,0000,,I'm silly.
Dialogue: 0,1:08:07.98,1:08:10.98,Default,,0000,0000,0000,,I can say that,\Nwell, Magdalena, this
Dialogue: 0,1:08:10.98,1:08:15.54,Default,,0000,0000,0000,,is the pair xx,\Nbecause x equals what?
Dialogue: 0,1:08:15.54,1:08:16.90,Default,,0000,0000,0000,,Let me plug them in.
Dialogue: 0,1:08:16.90,1:08:19.08,Default,,0000,0000,0000,,So it's like two people.
Dialogue: 0,1:08:19.08,1:08:20.56,Default,,0000,0000,0000,,x and y are married.
Dialogue: 0,1:08:20.56,1:08:22.18,Default,,0000,0000,0000,,They are a couple, a pair.
Dialogue: 0,1:08:22.18,1:08:24.33,Default,,0000,0000,0000,,They look identical.
Dialogue: 0,1:08:24.33,1:08:26.38,Default,,0000,0000,0000,,Sometimes it happens.
Dialogue: 0,1:08:26.38,1:08:28.44,Default,,0000,0000,0000,,Like twins, they\Nstart looking alike,
Dialogue: 0,1:08:28.44,1:08:30.82,Default,,0000,0000,0000,,dressing alike, and so on.
Dialogue: 0,1:08:30.82,1:08:36.53,Default,,0000,0000,0000,,The x and the y have to\Nreceive the same letter.
Dialogue: 0,1:08:36.53,1:08:41.03,Default,,0000,0000,0000,,And you have to tell me what\Nin the world the limit will be.
Dialogue: 0,1:08:41.03,1:08:43.82,Default,,0000,0000,0000,,
Dialogue: 0,1:08:43.82,1:08:44.36,Default,,0000,0000,0000,,STUDENT: 1/2.
Dialogue: 0,1:08:44.36,1:08:45.52,Default,,0000,0000,0000,,MAGDALENA TODA: 1/2.
Dialogue: 0,1:08:45.52,1:08:46.64,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,1:08:46.64,1:08:48.48,Default,,0000,0000,0000,,So now I'm deflated.
Dialogue: 0,1:08:48.48,1:08:52.47,Default,,0000,0000,0000,,So now I realize that\Ntaking two different paths,
Dialogue: 0,1:08:52.47,1:08:57.58,Default,,0000,0000,0000,,I show that I have-- on\Nthis path, I have 1/2.
Dialogue: 0,1:08:57.58,1:09:00.07,Default,,0000,0000,0000,,On this path, I have 0.
Dialogue: 0,1:09:00.07,1:09:01.15,Default,,0000,0000,0000,,I don't match.
Dialogue: 0,1:09:01.15,1:09:02.71,Default,,0000,0000,0000,,I don't have an overall limit.
Dialogue: 0,1:09:02.71,1:09:10.14,Default,,0000,0000,0000,,So the answer is,\Nno overall limit.
Dialogue: 0,1:09:10.14,1:09:10.96,Default,,0000,0000,0000,,Oh, my god.
Dialogue: 0,1:09:10.96,1:09:14.64,Default,,0000,0000,0000,,So what you need to\Ndo, guys, is read
Dialogue: 0,1:09:14.64,1:09:18.34,Default,,0000,0000,0000,,section 11.1 and section 11.2.
Dialogue: 0,1:09:18.34,1:09:21.10,Default,,0000,0000,0000,,And I will ask you next\Ntime-- and you can lie,
Dialogue: 0,1:09:21.10,1:09:22.89,Default,,0000,0000,0000,,you can do whatever.
Dialogue: 0,1:09:22.89,1:09:26.37,Default,,0000,0000,0000,,Did the book explain\Nbetter than me,
Dialogue: 0,1:09:26.37,1:09:28.89,Default,,0000,0000,0000,,or I explain better\Nthan the book?
Dialogue: 0,1:09:28.89,1:09:31.79,Default,,0000,0000,0000,,This type of example when\Nthe limit does not exist.
Dialogue: 0,1:09:31.79,1:09:33.43,Default,,0000,0000,0000,,We are going to\Nsee more examples.
Dialogue: 0,1:09:33.43,1:09:37.78,Default,,0000,0000,0000,,You are going to see examples\Nwhere the limit does exist.
Dialogue: 0,1:09:37.78,1:09:40.16,Default,,0000,0000,0000,,Now, one last thing.
Dialogue: 0,1:09:40.16,1:09:46.64,Default,,0000,0000,0000,,When you have to compute limits\Nof compositions of functions
Dialogue: 0,1:09:46.64,1:09:48.53,Default,,0000,0000,0000,,whose limit exist--\Nfor example, you
Dialogue: 0,1:09:48.53,1:09:58.29,Default,,0000,0000,0000,,know that limit is\Nxy goes to x0y0 of f
Dialogue: 0,1:09:58.29,1:10:10.41,Default,,0000,0000,0000,,of xy [INAUDIBLE] limit\Nof xy go to x0y0 of gxy
Dialogue: 0,1:10:10.41,1:10:13.73,Default,,0000,0000,0000,,is L-- L-- L-- M-- M.
Dialogue: 0,1:10:13.73,1:10:25.20,Default,,0000,0000,0000,,How are you going to compute the\Nlimit of alpha f plus beta g?
Dialogue: 0,1:10:25.20,1:10:27.10,Default,,0000,0000,0000,,This is in the book.
Dialogue: 0,1:10:27.10,1:10:32.65,Default,,0000,0000,0000,,But you don't need the\Nbook to understand that.
Dialogue: 0,1:10:32.65,1:10:34.32,Default,,0000,0000,0000,,You will already\Ngive me the answer,
Dialogue: 0,1:10:34.32,1:10:39.21,Default,,0000,0000,0000,,because this is the equivalent\Nthing to the function of one
Dialogue: 0,1:10:39.21,1:10:41.32,Default,,0000,0000,0000,,variable thing in Calc 1.
Dialogue: 0,1:10:41.32,1:10:43.67,Default,,0000,0000,0000,,So if you would only\Nhave f of x or g of x,
Dialogue: 0,1:10:43.67,1:10:45.41,Default,,0000,0000,0000,,it would be piece of cake.
Dialogue: 0,1:10:45.41,1:10:46.50,Default,,0000,0000,0000,,What would you say?
Dialogue: 0,1:10:46.50,1:10:47.38,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:10:47.38,1:10:48.29,Default,,0000,0000,0000,,MAGDALENA TODA: Right.
Dialogue: 0,1:10:48.29,1:10:54.08,Default,,0000,0000,0000,,Alpha times L plus beta\Ntimes M. Can you also
Dialogue: 0,1:10:54.08,1:10:55.38,Default,,0000,0000,0000,,multiply functions.
Dialogue: 0,1:10:55.38,1:10:55.92,Default,,0000,0000,0000,,Yes, you can.
Dialogue: 0,1:10:55.92,1:11:07.16,Default,,0000,0000,0000,,Limit of fg as xy goes\Nto x0 or y0-- will be LM.
Dialogue: 0,1:11:07.16,1:11:09.75,Default,,0000,0000,0000,,How about-- now I'm going to\Njump to conclusion, hoping
Dialogue: 0,1:11:09.75,1:11:13.17,Default,,0000,0000,0000,,that you are going to catch me.
Dialogue: 0,1:11:13.17,1:11:15.65,Default,,0000,0000,0000,,You are going to catch\Nme, and shout at me,
Dialogue: 0,1:11:15.65,1:11:18.41,Default,,0000,0000,0000,,and say, ooh, pay\Nattention, Magdalena,
Dialogue: 0,1:11:18.41,1:11:21.56,Default,,0000,0000,0000,,you can make a mistake there.
Dialogue: 0,1:11:21.56,1:11:26.43,Default,,0000,0000,0000,,I say it's L/M when I do\Nthe division rule, right?
Dialogue: 0,1:11:26.43,1:11:28.22,Default,,0000,0000,0000,,Where should I pay attention?
Dialogue: 0,1:11:28.22,1:11:29.76,Default,,0000,0000,0000,,STUDENT: M [INAUDIBLE].
Dialogue: 0,1:11:29.76,1:11:31.01,Default,,0000,0000,0000,,MAGDALENA TODA: Pay attention.
Dialogue: 0,1:11:31.01,1:11:38.74,Default,,0000,0000,0000,,Sometimes you can\Nhave the-- right?
Dialogue: 0,1:11:38.74,1:11:45.12,Default,,0000,0000,0000,,And this also has\Nto exist as well.
Dialogue: 0,1:11:45.12,1:11:46.56,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:11:46.56,1:11:51.07,Default,,0000,0000,0000,,MAGDALENA TODA: So one\Nlast-- how many minutes
Dialogue: 0,1:11:51.07,1:11:53.81,Default,,0000,0000,0000,,have I spent with you?
Dialogue: 0,1:11:53.81,1:11:57.93,Default,,0000,0000,0000,,I've spent with you a long\Nnumber of hours of my life.
Dialogue: 0,1:11:57.93,1:11:58.80,Default,,0000,0000,0000,,No, I'm just kidding.
Dialogue: 0,1:11:58.80,1:12:04.23,Default,,0000,0000,0000,,So you have one hour and\N15, a little bit more.
Dialogue: 0,1:12:04.23,1:12:05.40,Default,,0000,0000,0000,,Do I have a little bit more?
Dialogue: 0,1:12:05.40,1:12:05.90,Default,,0000,0000,0000,,Yes.
Dialogue: 0,1:12:05.90,1:12:07.99,Default,,0000,0000,0000,,I have 15 minutes.
Dialogue: 0,1:12:07.99,1:12:08.49,Default,,0000,0000,0000,,I have--
Dialogue: 0,1:12:08.49,1:12:09.62,Default,,0000,0000,0000,,STUDENT: So we get out at--
Dialogue: 0,1:12:09.62,1:12:10.45,Default,,0000,0000,0000,,[INTERPOSING VOICES]
Dialogue: 0,1:12:10.45,1:12:11.24,Default,,0000,0000,0000,,MAGDALENA TODA: 50.
Dialogue: 0,1:12:11.24,1:12:12.51,Default,,0000,0000,0000,,Five more minutes.
Dialogue: 0,1:12:12.51,1:12:15.36,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:12:15.36,1:12:20.66,Default,,0000,0000,0000,,So I want to ask you what\Nyou remember about some
Dialogue: 0,1:12:20.66,1:12:25.09,Default,,0000,0000,0000,,of your friends, the trig\Nfunctions involved in limits.
Dialogue: 0,1:12:25.09,1:12:28.03,Default,,0000,0000,0000,,
Dialogue: 0,1:12:28.03,1:12:32.00,Default,,0000,0000,0000,,Why did we study\Nlimits at the point
Dialogue: 0,1:12:32.00,1:12:34.29,Default,,0000,0000,0000,,where the function's\Nnot defined?
Dialogue: 0,1:12:34.29,1:12:35.43,Default,,0000,0000,0000,,Well, to heck with it.
Dialogue: 0,1:12:35.43,1:12:36.03,Default,,0000,0000,0000,,We don't care.
Dialogue: 0,1:12:36.03,1:12:37.95,Default,,0000,0000,0000,,The function is\Nnot defined at 0.
Dialogue: 0,1:12:37.95,1:12:40.08,Default,,0000,0000,0000,,But the limit is.
Dialogue: 0,1:12:40.08,1:12:42.92,Default,,0000,0000,0000,,And nobody showed you how\Nto do the epsilon delta
Dialogue: 0,1:12:42.92,1:12:44.26,Default,,0000,0000,0000,,to show anything like that.
Dialogue: 0,1:12:44.26,1:12:48.53,Default,,0000,0000,0000,,
Dialogue: 0,1:12:48.53,1:12:49.99,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:12:49.99,1:12:52.42,Default,,0000,0000,0000,,Can you do that\Nwith epsilon delta?
Dialogue: 0,1:12:52.42,1:12:57.79,Default,,0000,0000,0000,,
Dialogue: 0,1:12:57.79,1:13:00.35,Default,,0000,0000,0000,,Actually, you can do\Neverything with epsilon delta.
Dialogue: 0,1:13:00.35,1:13:02.31,Default,,0000,0000,0000,,But I'm not going to give\Nyou any extra credit.
Dialogue: 0,1:13:02.31,1:13:07.54,Default,,0000,0000,0000,,So I trust you that\Nyou remember that.
Dialogue: 0,1:13:07.54,1:13:08.90,Default,,0000,0000,0000,,1!
Dialogue: 0,1:13:08.90,1:13:10.82,Default,,0000,0000,0000,,How about-- let me-- OK.
Dialogue: 0,1:13:10.82,1:13:11.77,Default,,0000,0000,0000,,I am so proud of you.
Dialogue: 0,1:13:11.77,1:13:12.85,Default,,0000,0000,0000,,Let me challenge you more.
Dialogue: 0,1:13:12.85,1:13:14.69,Default,,0000,0000,0000,,Let me challenge you more.
Dialogue: 0,1:13:14.69,1:13:17.95,Default,,0000,0000,0000,,Tangent of ax over bx.
Dialogue: 0,1:13:17.95,1:13:19.50,Default,,0000,0000,0000,,x go to 0.
Dialogue: 0,1:13:19.50,1:13:22.11,Default,,0000,0000,0000,,I asked this to a girl\Nfrom Lubbock High.
Dialogue: 0,1:13:22.11,1:13:23.45,Default,,0000,0000,0000,,She was in high school.
Dialogue: 0,1:13:23.45,1:13:25.39,Default,,0000,0000,0000,,She knew the answer.
Dialogue: 0,1:13:25.39,1:13:28.30,Default,,0000,0000,0000,,STUDENT: Oh, I can't disappoint\Neverybody in getting this.
Dialogue: 0,1:13:28.30,1:13:31.48,Default,,0000,0000,0000,,STUDENT: Is it 1/a?
Dialogue: 0,1:13:31.48,1:13:32.35,Default,,0000,0000,0000,,Oh, I can't remember.
Dialogue: 0,1:13:32.35,1:13:33.86,Default,,0000,0000,0000,,MAGDALENA TODA: Tell me\Nwhat to do to be smart.
Dialogue: 0,1:13:33.86,1:13:34.36,Default,,0000,0000,0000,,Right?
Dialogue: 0,1:13:34.36,1:13:37.53,Default,,0000,0000,0000,,I have to be doing\Nsomething smart.
Dialogue: 0,1:13:37.53,1:13:40.05,Default,,0000,0000,0000,,She-- can you give me hint?
Dialogue: 0,1:13:40.05,1:13:41.78,Default,,0000,0000,0000,,I'm your student\Nand you say, well--
Dialogue: 0,1:13:41.78,1:13:42.32,Default,,0000,0000,0000,,STUDENT: ba--
Dialogue: 0,1:13:42.32,1:13:44.06,Default,,0000,0000,0000,,STUDENT: It's 0.
Dialogue: 0,1:13:44.06,1:13:45.45,Default,,0000,0000,0000,,STUDENT: It's [INAUDIBLE].
Dialogue: 0,1:13:45.45,1:13:46.78,Default,,0000,0000,0000,,MAGDALENA TODA: Um, it's a what?
Dialogue: 0,1:13:46.78,1:13:48.09,Default,,0000,0000,0000,,STUDENT: b/a?
Dialogue: 0,1:13:48.09,1:13:49.59,Default,,0000,0000,0000,,MAGDALENA TODA: I'm\Nnot [INAUDIBLE].
Dialogue: 0,1:13:49.59,1:13:51.08,Default,,0000,0000,0000,,I don't think so.
Dialogue: 0,1:13:51.08,1:13:52.73,Default,,0000,0000,0000,,So what should I do?
Dialogue: 0,1:13:52.73,1:13:58.07,Default,,0000,0000,0000,,I should say, instead of\Nbx-- that drives me nuts.
Dialogue: 0,1:13:58.07,1:14:00.39,Default,,0000,0000,0000,,This goes on my nerves-- bx.
Dialogue: 0,1:14:00.39,1:14:03.55,Default,,0000,0000,0000,,Like, maybe I go on your\Nnerves. bx is ax, right?
Dialogue: 0,1:14:03.55,1:14:06.74,Default,,0000,0000,0000,,If it were ax, I would\Nbe more constructive,
Dialogue: 0,1:14:06.74,1:14:09.27,Default,,0000,0000,0000,,and I knew what to do.
Dialogue: 0,1:14:09.27,1:14:13.19,Default,,0000,0000,0000,,I say replace bx with\Nax, compensate for it,
Dialogue: 0,1:14:13.19,1:14:15.12,Default,,0000,0000,0000,,and divide by bx.
Dialogue: 0,1:14:15.12,1:14:17.88,Default,,0000,0000,0000,,And I was trying to\Nexplain that to my son,
Dialogue: 0,1:14:17.88,1:14:23.55,Default,,0000,0000,0000,,that if you have a fraction\Na/b, and then you write a/n
Dialogue: 0,1:14:23.55,1:14:26.70,Default,,0000,0000,0000,,times n/b, it's the same thing.
Dialogue: 0,1:14:26.70,1:14:28.56,Default,,0000,0000,0000,,Gosh, I had the\Nproblem with him.
Dialogue: 0,1:14:28.56,1:14:33.31,Default,,0000,0000,0000,,And then I realized that he\Ndidn't do simplifications
Dialogue: 0,1:14:33.31,1:14:34.87,Default,,0000,0000,0000,,in school.
Dialogue: 0,1:14:34.87,1:14:41.18,Default,,0000,0000,0000,,So it took a little more\Nhours to explain these things.
Dialogue: 0,1:14:41.18,1:14:42.59,Default,,0000,0000,0000,,This is fourth grade.
Dialogue: 0,1:14:42.59,1:14:45.15,Default,,0000,0000,0000,,I think I remember doing\Nthat in fourth grade.
Dialogue: 0,1:14:45.15,1:14:47.43,Default,,0000,0000,0000,,Third grade, actually.
Dialogue: 0,1:14:47.43,1:14:50.36,Default,,0000,0000,0000,,So these two guys disappear.
Dialogue: 0,1:14:50.36,1:14:53.79,Default,,0000,0000,0000,,I haven't changed\Nmy problem at all.
Dialogue: 0,1:14:53.79,1:14:57.92,Default,,0000,0000,0000,,But I've changed the status,\Nthe shape of my problem
Dialogue: 0,1:14:57.92,1:15:01.34,Default,,0000,0000,0000,,to something I can mold,\Nbecause this goes to somebody,
Dialogue: 0,1:15:01.34,1:15:02.68,Default,,0000,0000,0000,,and this goes to somebody else.
Dialogue: 0,1:15:02.68,1:15:05.17,Default,,0000,0000,0000,,Who is this fellow?
Dialogue: 0,1:15:05.17,1:15:07.62,Default,,0000,0000,0000,,It's a limit that's\Na constant-- a/b.
Dialogue: 0,1:15:07.62,1:15:09.26,Default,,0000,0000,0000,,Who is this fellow?
Dialogue: 0,1:15:09.26,1:15:09.76,Default,,0000,0000,0000,,STUDENT: 1.
Dialogue: 0,1:15:09.76,1:15:10.84,Default,,0000,0000,0000,,MAGDALENA TODA: 1.
Dialogue: 0,1:15:10.84,1:15:15.66,Default,,0000,0000,0000,,Because tangent of x/x as x\Ngoes to 0 goes to 1 exactly
Dialogue: 0,1:15:15.66,1:15:16.16,Default,,0000,0000,0000,,like that.
Dialogue: 0,1:15:16.16,1:15:22.14,Default,,0000,0000,0000,,So limit of sine x over cosine\Nx, that's tangent, right?
Dialogue: 0,1:15:22.14,1:15:23.29,Default,,0000,0000,0000,,Over x.
Dialogue: 0,1:15:23.29,1:15:25.02,Default,,0000,0000,0000,,You do it exactly the same.
Dialogue: 0,1:15:25.02,1:15:32.41,Default,,0000,0000,0000,,It's limit of sine x/x\Ntimes 1 over cosine x.
Dialogue: 0,1:15:32.41,1:15:34.40,Default,,0000,0000,0000,,That's how we did\Nit in high school.
Dialogue: 0,1:15:34.40,1:15:35.02,Default,,0000,0000,0000,,This goes to 1.
Dialogue: 0,1:15:35.02,1:15:36.64,Default,,0000,0000,0000,,This goes to 1.
Dialogue: 0,1:15:36.64,1:15:37.39,Default,,0000,0000,0000,,So it's 1.
Dialogue: 0,1:15:37.39,1:15:39.19,Default,,0000,0000,0000,,So thank you, this is 1.
Dialogue: 0,1:15:39.19,1:15:43.29,Default,,0000,0000,0000,,I know I took a little more time\Nto explain than I wanted to.
Dialogue: 0,1:15:43.29,1:15:46.13,Default,,0000,0000,0000,,But now you are grown up.
Dialogue: 0,1:15:46.13,1:15:49.26,Default,,0000,0000,0000,,In two minutes, you are\Ngoing to be finishing
Dialogue: 0,1:15:49.26,1:15:50.77,Default,,0000,0000,0000,,this section, more or less.
Dialogue: 0,1:15:50.77,1:15:54.64,Default,,0000,0000,0000,,What if I put a function\Nof two variables,
Dialogue: 0,1:15:54.64,1:15:57.79,Default,,0000,0000,0000,,and I ask you what\Nthe limit will be,
Dialogue: 0,1:15:57.79,1:16:01.29,Default,,0000,0000,0000,,if it's the same\Ntype of function.
Dialogue: 0,1:16:01.29,1:16:03.49,Default,,0000,0000,0000,,So you say, oh, Magdalena,\Nwhat you doing to us?
Dialogue: 0,1:16:03.49,1:16:05.24,Default,,0000,0000,0000,,OK, we'll see it's fun.
Dialogue: 0,1:16:05.24,1:16:06.01,Default,,0000,0000,0000,,This one's fun.
Dialogue: 0,1:16:06.01,1:16:07.85,Default,,0000,0000,0000,,It's not like the one before.
Dialogue: 0,1:16:07.85,1:16:11.08,Default,,0000,0000,0000,,This one is pretty beautiful.
Dialogue: 0,1:16:11.08,1:16:12.77,Default,,0000,0000,0000,,It's nice to you.
Dialogue: 0,1:16:12.77,1:16:14.95,Default,,0000,0000,0000,,It exists.
Dialogue: 0,1:16:14.95,1:16:16.59,Default,,0000,0000,0000,,xy goes to 0, 0.
Dialogue: 0,1:16:16.59,1:16:19.74,Default,,0000,0000,0000,,So you have to imagine\Nsome preferable function
Dialogue: 0,1:16:19.74,1:16:22.07,Default,,0000,0000,0000,,in abstract thinking.
Dialogue: 0,1:16:22.07,1:16:24.72,Default,,0000,0000,0000,,And you want it in\Na little disk here.
Dialogue: 0,1:16:24.72,1:16:31.58,Default,,0000,0000,0000,,And xy, these are all points\Nxy close enough to 0, 0,
Dialogue: 0,1:16:31.58,1:16:34.03,Default,,0000,0000,0000,,in the neighborhood of 0, 0.
Dialogue: 0,1:16:34.03,1:16:34.81,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:16:34.81,1:16:37.48,Default,,0000,0000,0000,,What's going to happen as\Nyou get closer and closer
Dialogue: 0,1:16:37.48,1:16:40.38,Default,,0000,0000,0000,,and closer and closer with\Ntinier and tinier and tinier
Dialogue: 0,1:16:40.38,1:16:43.88,Default,,0000,0000,0000,,disks around 0, 0?
Dialogue: 0,1:16:43.88,1:16:47.59,Default,,0000,0000,0000,,You're going to shrink so much.
Dialogue: 0,1:16:47.59,1:16:49.29,Default,,0000,0000,0000,,What do you think\Nthis will going to be,
Dialogue: 0,1:16:49.29,1:16:50.75,Default,,0000,0000,0000,,and how do I prove it?
Dialogue: 0,1:16:50.75,1:16:52.13,Default,,0000,0000,0000,,STUDENT: [INAUDIBLE].
Dialogue: 0,1:16:52.13,1:16:53.97,Default,,0000,0000,0000,,MAGDALENA TODA: Who said it?
Dialogue: 0,1:16:53.97,1:16:55.95,Default,,0000,0000,0000,,You, sir? [INAUDIBLE]\Ngoing to go to 1.
Dialogue: 0,1:16:55.95,1:16:57.77,Default,,0000,0000,0000,,And he's right.
Dialogue: 0,1:16:57.77,1:17:00.94,Default,,0000,0000,0000,,He has the intuition.
Dialogue: 0,1:17:00.94,1:17:03.20,Default,,0000,0000,0000,,A mathematician will\Ntell you, prove it.
Dialogue: 0,1:17:03.20,1:17:04.62,Default,,0000,0000,0000,,STUDENT: Um, well,\Nlet's see here.
Dialogue: 0,1:17:04.62,1:17:06.43,Default,,0000,0000,0000,,MAGDALENA TODA: Can you prove?
Dialogue: 0,1:17:06.43,1:17:09.90,Default,,0000,0000,0000,,STUDENT: You could use\Nthe right triangle proof,
Dialogue: 0,1:17:09.90,1:17:11.98,Default,,0000,0000,0000,,but that would probably\Ntake way more [INAUDIBLE].
Dialogue: 0,1:17:11.98,1:17:12.94,Default,,0000,0000,0000,,MAGDALENA TODA: x and\Ny are independent.
Dialogue: 0,1:17:12.94,1:17:13.68,Default,,0000,0000,0000,,That's the problem.
Dialogue: 0,1:17:13.68,1:17:15.72,Default,,0000,0000,0000,,They are married, but they\Nare still independent.
Dialogue: 0,1:17:15.72,1:17:17.22,Default,,0000,0000,0000,,It's a couple.
Dialogue: 0,1:17:17.22,1:17:20.92,Default,,0000,0000,0000,,However, we can use\Npolar coordinates.
Dialogue: 0,1:17:20.92,1:17:22.78,Default,,0000,0000,0000,,Why is polar coordinates?
Dialogue: 0,1:17:22.78,1:17:28.95,Default,,0000,0000,0000,,Well, in general, if we\Nare in xy, it's a pair.
Dialogue: 0,1:17:28.95,1:17:30.58,Default,,0000,0000,0000,,This is r, right?
Dialogue: 0,1:17:30.58,1:17:33.59,Default,,0000,0000,0000,,So rx is r cosine theta.
Dialogue: 0,1:17:33.59,1:17:35.44,Default,,0000,0000,0000,,y is r sine theta.
Dialogue: 0,1:17:35.44,1:17:37.44,Default,,0000,0000,0000,,And I can get closer and\Ncloser to the original.
Dialogue: 0,1:17:37.44,1:17:38.68,Default,,0000,0000,0000,,I don't care.
Dialogue: 0,1:17:38.68,1:17:41.31,Default,,0000,0000,0000,,What happens about x\Nsquared plus y squared,
Dialogue: 0,1:17:41.31,1:17:43.04,Default,,0000,0000,0000,,this is r squared.
Dialogue: 0,1:17:43.04,1:17:44.28,Default,,0000,0000,0000,,And r is a real number.
Dialogue: 0,1:17:44.28,1:17:47.44,Default,,0000,0000,0000,,And as you walk closer\Nand closer to the original
Dialogue: 0,1:17:47.44,1:17:52.80,Default,,0000,0000,0000,,without touching it,\Nthat r goes to 0.
Dialogue: 0,1:17:52.80,1:17:53.85,Default,,0000,0000,0000,,It shrinks to 0.
Dialogue: 0,1:17:53.85,1:17:58.26,Default,,0000,0000,0000,,So that r squared goes\Nto 0 but never touches 0.
Dialogue: 0,1:17:58.26,1:18:04.31,Default,,0000,0000,0000,,So this becomes limit as r goes\Nto 0, the radius of that disk
Dialogue: 0,1:18:04.31,1:18:06.29,Default,,0000,0000,0000,,goes to 0.
Dialogue: 0,1:18:06.29,1:18:10.54,Default,,0000,0000,0000,,Sine of r squared\Nover r squared.
Dialogue: 0,1:18:10.54,1:18:13.87,Default,,0000,0000,0000,,But r squared could be replaced\Nby the real function, t,
Dialogue: 0,1:18:13.87,1:18:17.42,Default,,0000,0000,0000,,by the real parameter,\Nlambda, by whatever you want.
Dialogue: 0,1:18:17.42,1:18:19.44,Default,,0000,0000,0000,,So then it's 1.
Dialogue: 0,1:18:19.44,1:18:23.39,Default,,0000,0000,0000,,And then Alexander was right.
Dialogue: 0,1:18:23.39,1:18:26.21,Default,,0000,0000,0000,,He based it on, like,\Nobservation, intuition,
Dialogue: 0,1:18:26.21,1:18:27.04,Default,,0000,0000,0000,,everything you want.
Dialogue: 0,1:18:27.04,1:18:28.83,Default,,0000,0000,0000,,It was not a proof.
Dialogue: 0,1:18:28.83,1:18:32.47,Default,,0000,0000,0000,,On a multiple-choice exam,\Nhe would be a lucky guy.
Dialogue: 0,1:18:32.47,1:18:34.30,Default,,0000,0000,0000,,I don't want you to prove it.
Dialogue: 0,1:18:34.30,1:18:36.95,Default,,0000,0000,0000,,But if I want you to\Nprove it, you have to say,
Dialogue: 0,1:18:36.95,1:18:39.53,Default,,0000,0000,0000,,Magdalena, I know\Npolar coordinates,
Dialogue: 0,1:18:39.53,1:18:41.93,Default,,0000,0000,0000,,and so I can do it.
Dialogue: 0,1:18:41.93,1:18:45.46,Default,,0000,0000,0000,,And one last question for today.
Dialogue: 0,1:18:45.46,1:18:49.57,Default,,0000,0000,0000,,Guys, I'm asking you,\Nlimit xy goes to 0, 0.
Dialogue: 0,1:18:49.57,1:18:53.60,Default,,0000,0000,0000,,You will see some of these in\Nyour WeBWorK for Chapter 11
Dialogue: 0,1:18:53.60,1:18:56.89,Default,,0000,0000,0000,,that's waiting for\Nyou, homework 3.
Dialogue: 0,1:18:56.89,1:19:03.20,Default,,0000,0000,0000,,Tangent of 2 x squared\Nplus y squared over 3
Dialogue: 0,1:19:03.20,1:19:06.45,Default,,0000,0000,0000,,x squared plus y squared.
Dialogue: 0,1:19:06.45,1:19:09.32,Default,,0000,0000,0000,,What is that?
Dialogue: 0,1:19:09.32,1:19:10.28,Default,,0000,0000,0000,,2/3.
Dialogue: 0,1:19:10.28,1:19:11.02,Default,,0000,0000,0000,,STUDENT: 2/3.
Dialogue: 0,1:19:11.02,1:19:12.52,Default,,0000,0000,0000,,MAGDALENA TODA: Am\NI asking you why?
Dialogue: 0,1:19:12.52,1:19:13.54,Default,,0000,0000,0000,,No, enough.
Dialogue: 0,1:19:13.54,1:19:14.44,Default,,0000,0000,0000,,OK.
Dialogue: 0,1:19:14.44,1:19:17.03,Default,,0000,0000,0000,,[INAUDIBLE] I gave\Nyou everything
Dialogue: 0,1:19:17.03,1:19:20.92,Default,,0000,0000,0000,,you need to show that.
Dialogue: 0,1:19:20.92,1:19:23.62,Default,,0000,0000,0000,,x squared plus y squared,\Nagain, is Mr. r squared.
Dialogue: 0,1:19:23.62,1:19:24.59,Default,,0000,0000,0000,,It's OK.
Dialogue: 0,1:19:24.59,1:19:29.47,Default,,0000,0000,0000,,I taught you that.\Na/b. a is 2, b is 3.
Dialogue: 0,1:19:29.47,1:19:30.45,Default,,0000,0000,0000,,Is it hard?
Dialogue: 0,1:19:30.45,1:19:31.91,Default,,0000,0000,0000,,It is not easy, for sure.
Dialogue: 0,1:19:31.91,1:19:35.33,Default,,0000,0000,0000,,Calc 3 is really difficult\Ncompared to other topics
Dialogue: 0,1:19:35.33,1:19:37.78,Default,,0000,0000,0000,,you are probably taking.
Dialogue: 0,1:19:37.78,1:19:40.96,Default,,0000,0000,0000,,But I hope that I\Ncan convince you
Dialogue: 0,1:19:40.96,1:19:45.44,Default,,0000,0000,0000,,that math, although\Ndifficult, [INAUDIBLE] Calc 3,
Dialogue: 0,1:19:45.44,1:19:48.23,Default,,0000,0000,0000,,is also fun.
Dialogue: 0,1:19:48.23,1:19:49.93,Default,,0000,0000,0000,,OK?
Dialogue: 0,1:19:49.93,1:19:50.76,Default,,0000,0000,0000,,All right.
Dialogue: 0,1:19:50.76,1:19:55.08,Default,,0000,0000,0000,,So I need attendance and\NI need the extra credit.
Dialogue: 0,1:19:55.08,1:19:56.21,Default,,0000,0000,0000,,STUDENT: Yeah, [INAUDIBLE].
Dialogue: 0,1:19:56.21,1:19:59.08,Default,,0000,0000,0000,,
Dialogue: 0,1:19:59.08,1:20:01.92,Default,,0000,0000,0000,,MAGDALENA TODA: Before\Nyou go, you need to sign.
Dialogue: 0,1:20:01.92,1:20:04.44,Default,,0000,0000,0000,,