0:00:00.000,0:00:01.467
MAGDALENA TODA:[br]We have any people

0:00:01.467,0:00:04.890
who finished the[br]extra credit and are

0:00:04.890,0:00:08.313
willing to give it to me today?

0:00:08.313,0:00:10.758
I mean, you don't have to.

0:00:10.758,0:00:12.720
That's why it's[br]called extra credit.

0:00:12.720,0:00:18.378
But I think it's good[br]for extra practice

0:00:18.378,0:00:21.354
and for the extra points.

0:00:21.354,0:00:25.322
So hold on to it if you cannot[br]give it to me right now.

0:00:25.322,0:00:29.290
And I'll collect it at[br]the end of the class.

0:00:29.290,0:00:30.282
Today's a big day.

0:00:30.282,0:00:33.258
We are starting a new[br]chapter, Chapter 11.

0:00:33.258,0:00:44.170


0:00:44.170,0:00:47.394
So practically, we are[br]going to discuss all

0:00:47.394,0:00:50.618
through this chapter functions[br]of several variables.

0:00:50.618,0:01:02.026


0:01:02.026,0:01:05.089
And you are going to[br]ask me, wait a minute,

0:01:05.089,0:01:12.330
why do we need functions[br]in more than one variable?

0:01:12.330,0:01:15.770
Well, we are all functions[br]of many variables.

0:01:15.770,0:01:18.765
I was freezing outside,[br]and I was thinking,

0:01:18.765,0:01:20.620
I'm a function of[br]everything I eat.

0:01:20.620,0:01:24.070
I'm a function of the[br]temperature outside.

0:01:24.070,0:01:26.500
Almost everything in[br]our body is a function

0:01:26.500,0:01:29.200
of hundreds of factors,[br]actually, thousands.

0:01:29.200,0:01:34.350
But we don't have the time[br]and the precise information

0:01:34.350,0:01:37.420
to analyze all the[br]parameters that

0:01:37.420,0:01:42.380
affect our physical[br]condition every day.

0:01:42.380,0:01:44.990
We are getting there.

0:01:44.990,0:01:48.360
I'm going to give you[br]just the simple case.

0:01:48.360,0:01:53.312
So instead of y equals f of x[br]type of function, one variable,

0:01:53.312,0:01:57.420
we are going to look at[br]functions of the types z

0:01:57.420,0:01:58.583
equals f of xy.

0:01:58.583,0:02:01.760


0:02:01.760,0:02:03.750
Can I have many more?

0:02:03.750,0:02:05.560
Absolutely I can.

0:02:05.560,0:02:08.880
And that's kind of[br]the idea, that I

0:02:08.880,0:02:14.640
can have a function[br]in an-- let me

0:02:14.640,0:02:18.580
count-- n plus 1[br]dimensional space

0:02:18.580,0:02:30.350
as being of the type xn plus[br]1 equals f of x1, x2, x3, x4.

0:02:30.350,0:02:32.820
Somebody stop me. xn.

0:02:32.820,0:02:33.320
Right.

0:02:33.320,0:02:37.885
I have many variables.

0:02:37.885,0:02:41.790
And that is a problem[br]that affects everything.

0:02:41.790,0:02:44.600
Our physical world is[br]affected by many parameters.

0:02:44.600,0:02:47.630


0:02:47.630,0:02:49.850
In engineering[br]problems, you've already

0:02:49.850,0:02:51.400
seen some of these parameters.

0:02:51.400,0:02:55.040
Can you give me some[br]examples of parameters you've

0:02:55.040,0:02:57.190
seen in engineering classes?

0:02:57.190,0:03:00.740
x1, x2, x3 could be the[br]Euclidean coordinates, right,

0:03:00.740,0:03:03.630
for the three [? space. ?] But[br]besides those, there was an x4.

0:03:03.630,0:03:05.600
It could be?

0:03:05.600,0:03:06.100
Time.

0:03:06.100,0:03:07.133
Excellent, [INAUDIBLE].

0:03:07.133,0:03:08.402
More than that.

0:03:08.402,0:03:09.690
I want more.

0:03:09.690,0:03:11.234
I want x5.

0:03:11.234,0:03:12.650
Who can think of[br]another parameter

0:03:12.650,0:03:17.083
that affects physical processes[br]or chemical reactions?

0:03:17.083,0:03:17.583
Yes, sir?

0:03:17.583,0:03:17.936
STUDENT: Temperature.

0:03:17.936,0:03:19.102
MAGDALENA TODA: Temperature.

0:03:19.102,0:03:19.910
Excellent.

0:03:19.910,0:03:21.310
Another very good idea.

0:03:21.310,0:03:23.100
How about x6?

0:03:23.100,0:03:24.590
I'm running out of imagination.

0:03:24.590,0:03:28.910
But you have a lot more[br]information than me.

0:03:28.910,0:03:29.892
Pressure.

0:03:29.892,0:03:35.620
Maybe I'm studying a process of[br]somewhere up in the atmosphere.

0:03:35.620,0:03:37.677
Maybe I'm in an[br]airplane, and then it

0:03:37.677,0:03:39.260
becomes a little bit[br]more complicated,

0:03:39.260,0:03:41.890
because I hate the way[br]cabins are pressurized.

0:03:41.890,0:03:45.060
I can feel very uneasy.

0:03:45.060,0:03:46.990
My ears pop and so on.

0:03:46.990,0:03:48.850
We can be in the[br]bottom of the ocean.

0:03:48.850,0:03:52.900
There are very many[br]physical parameters

0:03:52.900,0:03:56.252
that affect physical[br]processes, chemical processes,

0:03:56.252,0:03:57.960
biological processes.

0:03:57.960,0:04:00.410
I don't know if this is[br]fortunate or unfortunate,

0:04:00.410,0:04:03.120
but I think that was[br]the key to the existence

0:04:03.120,0:04:07.400
of the universe in the first[br]place-- all these parameters.

0:04:07.400,0:04:08.030
OK.

0:04:08.030,0:04:11.700
Let me give you a simple[br]example of a function that

0:04:11.700,0:04:13.395
looks like a graph.

0:04:13.395,0:04:14.460
This is a graph.

0:04:14.460,0:04:17.180


0:04:17.180,0:04:19.470
And you say, wait a[br]minute, wait a minute.

0:04:19.470,0:04:22.470
Can I have functions of several[br]variables that cannot be

0:04:22.470,0:04:26.260
represented as graphs?

0:04:26.260,0:04:28.100
Yeah.

0:04:28.100,0:04:29.020
Absolutely.

0:04:29.020,0:04:31.300
We will talk about that[br]a little bit later.

0:04:31.300,0:04:35.200
So if I were to give you[br]an example that you've

0:04:35.200,0:04:42.270
seen before, and I would say,[br]give me a good approximation

0:04:42.270,0:04:47.000
to a valley that is actually[br]a quadric that we love and we

0:04:47.000,0:04:51.750
studied before for[br]the first time.

0:04:51.750,0:04:59.210
That quadric is a[br]beautiful object, a valley.

0:04:59.210,0:05:02.840
Any imagination,[br]recognition, recollection?

0:05:02.840,0:05:04.930
I know I scared[br]you enough for you

0:05:04.930,0:05:08.130
to know the equations of those[br]quadrics since some of you

0:05:08.130,0:05:10.000
told me we watched[br]all the videos,

0:05:10.000,0:05:13.440
we read all the stinking[br]book like never before.

0:05:13.440,0:05:14.770
That was kind of the idea.

0:05:14.770,0:05:16.970
I didn't want to scare you away.

0:05:16.970,0:05:19.330
I wanted to scare you[br]enough to read the book

0:05:19.330,0:05:20.620
and watch the videos.

0:05:20.620,0:05:26.240
And I'm talking about a valley[br]that you've seen before.

0:05:26.240,0:05:31.070
Many of you told me you like[br]the University of Minnesota

0:05:31.070,0:05:34.900
website that has the[br]quadric gallery of quadrics.

0:05:34.900,0:05:40.140


0:05:40.140,0:05:43.550
So you've met this guy before.

0:05:43.550,0:05:46.430
They show the general equation.

0:05:46.430,0:05:50.010
But I said I like the[br]circular paraboloid.

0:05:50.010,0:05:53.810
So they talk about[br]elliptic paraboloid.

0:05:53.810,0:05:56.666
Which one do you think I prefer?

0:05:56.666,0:05:59.056
The circular paraboloid.

0:05:59.056,0:06:02.880
Give me an example of[br]a circular paraboloid.

0:06:02.880,0:06:05.270
STUDENT: A flashlight?

0:06:05.270,0:06:06.720
Inside.

0:06:06.720,0:06:09.295
MAGDALENA TODA: The expression,[br]the mathematical equation.

0:06:09.295,0:06:10.086
STUDENT: Oh, sorry.

0:06:10.086,0:06:11.737
So it would be x[br]squred plus y squared.

0:06:11.737,0:06:12.820
MAGDALENA TODA: Very good.

0:06:12.820,0:06:14.470
That's exactly[br]what I had in mind.

0:06:14.470,0:06:18.110
Of course, it could be[br]over something, over r.

0:06:18.110,0:06:18.610
All right.

0:06:18.610,0:06:19.930
That's my favorite.

0:06:19.930,0:06:23.590
Now, if I put the flashlight[br]in here just like one of you

0:06:23.590,0:06:29.585
said, or the sign on[br]top of the z-axis.

0:06:29.585,0:06:36.780
Then I'm going to look at[br]the various-- we discussed

0:06:36.780,0:06:38.110
that a little bit before.

0:06:38.110,0:06:42.900
So various horizontal[br]planes, they're going to cut.

0:06:42.900,0:06:48.450
They're going to cut the[br]surface in different circles,

0:06:48.450,0:06:50.390
upon different circles.

0:06:50.390,0:06:52.780
We love them, and we use them.

0:06:52.780,0:06:54.940
And what did we do[br]with them last time?

0:06:54.940,0:06:59.830
We projected them on the floor.

0:06:59.830,0:07:03.390
And by floor, I mean the what?

0:07:03.390,0:07:10.080
By floor, I mean the xy plane.

0:07:10.080,0:07:12.420
Plus this xy plane.

0:07:12.420,0:07:14.450
I label it like you like it.

0:07:14.450,0:07:17.290
You said you like[br]it when I label it,

0:07:17.290,0:07:19.990
so you have the[br]imagination of a table.

0:07:19.990,0:07:23.640
This is x and y and z.

0:07:23.640,0:07:31.040
And so I gave you an example[br]of a graph cut in with z equals

0:07:31.040,0:07:33.400
constant positive or negative?

0:07:33.400,0:07:36.080
Well, it better be positive,[br]because for negative, I

0:07:36.080,0:07:37.770
have no solutions.

0:07:37.770,0:07:39.020
Positive or zero.

0:07:39.020,0:07:42.360
Well, for zero I have[br]a degenerate conic.

0:07:42.360,0:07:46.180
A degenerate conic[br]could be a point,

0:07:46.180,0:07:48.820
or it could be a bunch of lines.

0:07:48.820,0:07:52.432
In this case, all those[br]circles-- doo-doo-doo-doo-doo--

0:07:52.432,0:07:55.750
a family of one parameter,[br]family of circles.

0:07:55.750,0:07:58.170
Like the ones that[br]is-- a dolphin

0:07:58.170,0:08:01.730
is now doing that[br]in San Antonio,

0:08:01.730,0:08:04.490
San Diego-- to take[br]those old circles

0:08:04.490,0:08:07.765
from the bottom of the sea,[br]and bring them different sizes,

0:08:07.765,0:08:09.030
and put them together.

0:08:09.030,0:08:10.400
So they are very smart.

0:08:10.400,0:08:11.820
I love dolphins.

0:08:11.820,0:08:15.540
So we'll see 0[br][INAUDIBLE] get a point.

0:08:15.540,0:08:17.290
That's still a conic.

0:08:17.290,0:08:18.962
It's a degenerate circle.

0:08:18.962,0:08:21.582
Do you realize[br]that's a limit case?

0:08:21.582,0:08:22.536
It's really beautiful.

0:08:22.536,0:08:23.410
You know what I mean?

0:08:23.410,0:08:25.284
Circle on top of a circle[br]on top of a circle,

0:08:25.284,0:08:26.175
smaller and smaller.

0:08:26.175,0:08:27.570
All right.

0:08:27.570,0:08:30.470
So good.

0:08:30.470,0:08:33.270
If I create shadows--[br]because that's

0:08:33.270,0:08:35.350
why you guys wanted[br]the source of light

0:08:35.350,0:08:39.360
on top-- of the projections[br]of these circles,

0:08:39.360,0:08:42.630
I'm going to have them[br]at the same color.

0:08:42.630,0:08:47.991
But dotted lines because I[br]think the book doesn't show them

0:08:47.991,0:08:48.490
dotted.

0:08:48.490,0:08:51.210
But on my way here,[br]I was thinking,

0:08:51.210,0:08:55.040
I think it's more beautiful[br]if I draw them dotted.

0:08:55.040,0:08:56.635
And how big is this circle?

0:08:56.635,0:08:57.440
Well, god knows.

0:08:57.440,0:09:01.990
I'm going to make a purple[br]circle that is, of course,

0:09:01.990,0:09:05.940
equal in size, equal in radius[br]with the original purple

0:09:05.940,0:09:07.080
circle.

0:09:07.080,0:09:10.290
So the dotted purple circle,[br]that's on the ground--

0:09:10.290,0:09:13.520
is just the projection of[br]the continuous purple circle.

0:09:13.520,0:09:16.450
It's identical in radius.

0:09:16.450,0:09:26.810
So for the family of[br]circles on the surface,

0:09:26.810,0:09:37.830
I have a family of projections[br]on the ground in the xy plane.

0:09:37.830,0:09:42.420
And such a family of[br]projections represents

0:09:42.420,0:09:44.910
a bunch of level curves.

0:09:44.910,0:09:47.112
We call this family[br]of level curves.

0:09:47.112,0:09:54.402


0:09:54.402,0:09:55.860
OK?

0:09:55.860,0:09:56.832
All right.

0:09:56.832,0:09:59.070
So if you think about it,[br]what are level curves?

0:09:59.070,0:10:02.360
You view them as being in plane.

0:10:02.360,0:10:02.910
Oh, my god.

0:10:02.910,0:10:07.600
So I should view them as a bunch[br]of points, a set of points.

0:10:07.600,0:10:10.665
If I make it like[br]that, that means

0:10:10.665,0:10:14.100
I view this as an[br]element of what?

0:10:14.100,0:10:18.820
Element of the xy plane,[br]right, with the property

0:10:18.820,0:10:22.920
that f of x and y is a constant.

0:10:22.920,0:10:25.848


0:10:25.848,0:10:28.356
OK?

0:10:28.356,0:10:30.690
In my case, I have a[br][INAUDIBLE] constant.

0:10:30.690,0:10:34.270
In general, I have an[br]arbitrary real constant.

0:10:34.270,0:10:40.020
That's a level curve for[br]level C, for the level

0:10:40.020,0:10:46.400
C called the level, or altitude[br]would be the same thing.

0:10:46.400,0:10:50.110
So have you seen these[br]guys in geography?

0:10:50.110,0:10:54.120
What in the world are these[br]level curves in geography?

0:10:54.120,0:10:56.520
STUDENT: [INAUDIBLE][br]show the slope

0:10:56.520,0:10:59.400
of a-- the steepness of a hill.

0:10:59.400,0:11:02.200
MAGDALENA TODA: You've[br]seen topographical maps.

0:11:02.200,0:11:06.140
And I'm going to try[br]and draw one of them.

0:11:06.140,0:11:08.310
I don't know, guys,[br]how-- excuse me.

0:11:08.310,0:11:10.275
I'm not very good[br]today at drawing.

0:11:10.275,0:11:13.180
But I'll do my best.

0:11:13.180,0:11:19.860
It could be a temperature[br]map or pressure map.

0:11:19.860,0:11:22.610
[INAUDIBLE] or whatever.

0:11:22.610,0:11:29.090
Now I'll say, this is[br]going to go-- well,

0:11:29.090,0:11:31.610
I cannot draw the[br]infinite family.

0:11:31.610,0:11:34.416
I have a one-parameter family.

0:11:34.416,0:11:42.840
And then I'll-- I'm dreaming of[br]the sea, summer break already.

0:11:42.840,0:11:44.596
You see what I'm doing.

0:11:44.596,0:11:47.580
Do you know what I'm doing?

0:11:47.580,0:11:51.610
That means I'm dreaming of the[br]different depths of the sea.

0:11:51.610,0:11:55.896
So for every such broad[br]line, I have the same depth.

0:11:55.896,0:11:59.380
The same altitude for[br]every continuous rule.

0:11:59.380,0:12:02.530
The same depth[br]for every-- so OK.

0:12:02.530,0:12:05.110
I'm not going to swim[br]too far, because that's

0:12:05.110,0:12:06.270
where the sharks are.

0:12:06.270,0:12:10.096
And I cannot draw the sharks,[br]but I ask you to imagine them.

0:12:10.096,0:12:12.712
It's fundamental in[br]a calculus class.

0:12:12.712,0:12:18.490
So somewhere here[br]I'm going to have--

0:12:18.490,0:12:21.326
what's the deepest--[br]guys, what's

0:12:21.326,0:12:23.066
the deepest point in that?

0:12:23.066,0:12:24.173
[? STUDENT: 11,300. ?]

0:12:24.173,0:12:25.881
MAGDALENA TODA: And[br]do you know the name?

0:12:25.881,0:12:26.381
I know the--

0:12:26.381,0:12:27.381
STUDENT: Mariana Trench.

0:12:27.381,0:12:28.695
MAGDALENA TODA: Mariana Trench.

0:12:28.695,0:12:30.757
STUDENT: Trench.

0:12:30.757,0:12:31.840
MAGDALENA TODA: All right.

0:12:31.840,0:12:34.490
So these topographical[br]are full of curves.

0:12:34.490,0:12:37.300
These are level curves.

0:12:37.300,0:12:39.580
So you didn't know,[br]but there is a lot

0:12:39.580,0:12:42.060
of mathematics in geography.

0:12:42.060,0:12:43.860
And there is a[br]lot of mathematics

0:12:43.860,0:12:45.600
in-- oh, you knew it.

0:12:45.600,0:12:47.480
When you watch the[br]weather report,

0:12:47.480,0:12:49.740
that's all mathematics, right?

0:12:49.740,0:12:53.030
It shows you the distribution[br]of temperatures everyday.

0:12:53.030,0:12:55.490
That is what we can[br][INAUDIBLE] also

0:12:55.490,0:13:00.550
care about other functions[br]of several parameters, right?

0:13:00.550,0:13:04.430
And those functions could[br]be pressure, wind, whatever.

0:13:04.430,0:13:05.650
OK.

0:13:05.650,0:13:06.773
Speed of the wind.

0:13:06.773,0:13:09.040
Something like that.

0:13:09.040,0:13:12.350
I did not dare to[br]look at the prediction

0:13:12.350,0:13:14.670
of the weather for this place.

0:13:14.670,0:13:16.670
This place used to[br]be a beautiful place.

0:13:16.670,0:13:22.530
300 days of the[br]year of sunshine.

0:13:22.530,0:13:23.755
Not anymore.

0:13:23.755,0:13:26.550
So there is something[br]fishy in Denmark

0:13:26.550,0:13:29.010
and also something[br]fishy in [INAUDIBLE].

0:13:29.010,0:13:30.250
The world is changing.

0:13:30.250,0:13:34.720
So if you don't believe in[br]global warming, think again,

0:13:34.720,0:13:38.180
and global cooling, think again.

0:13:38.180,0:13:39.130
All right.

0:13:39.130,0:13:42.960
So unfortunately,[br]I am afraid still

0:13:42.960,0:13:45.622
to look at the temperatures[br]for the next few days.

0:13:45.622,0:13:46.122
But--

0:13:46.122,0:13:48.602
STUDENT: It's going to[br]be 80 degrees on Tuesday.

0:13:48.602,0:13:49.147
MAGDALENA TODA: Really? [?

0:13:49.147,0:13:50.855
Well, see, I should[br]have looked at it. ?]

0:13:50.855,0:13:51.580
[LAUGHTER]

0:13:51.580,0:13:54.710
I should gather the[br]courage, because I

0:13:54.710,0:13:57.140
knew-- when I was[br]interviewed here

0:13:57.140,0:14:00.480
for assistant professor,[br]gosh, I was young.

0:14:00.480,0:14:02.230
2001.

0:14:02.230,0:14:03.830
And my interview[br]was in mid-February.

0:14:03.830,0:14:08.180
And birds were chirping, it was[br]blue skies, beautiful flowers

0:14:08.180,0:14:09.700
everywhere on campus.

0:14:09.700,0:14:11.380
And I love the campus.

0:14:11.380,0:14:12.910
OK.

0:14:12.910,0:14:18.520
Give me an example of a surface[br]that cannot be represented

0:14:18.520,0:14:23.810
as a graph in its[br]entirety as a whole graph.

0:14:23.810,0:14:27.070
You gave me that before,[br]and I was so proud of you.

0:14:27.070,0:14:27.940
It was a--

0:14:27.940,0:14:30.840


0:14:30.840,0:14:32.316
[LAUGHS]

0:14:32.316,0:14:33.300


0:14:33.300,0:14:35.730
What kind of surface[br]am I trying to mimic?

0:14:35.730,0:14:36.694
STUDENT: A saddle.

0:14:36.694,0:14:39.590


0:14:39.590,0:14:42.120
MAGDALENA TODA: That[br]can be actually a graph.

0:14:42.120,0:14:44.630
That's a good[br]example of a graph.

0:14:44.630,0:14:45.920
A saddle.

0:14:45.920,0:14:49.030
But give me an example[br]of a non-graph that

0:14:49.030,0:14:51.720
is given as an implicit form.

0:14:51.720,0:14:58.160
So graph or explicit[br]is the same thing.

0:14:58.160,0:15:00.880
z equals f of xy.

0:15:00.880,0:15:02.590
Give me a non-graph.

0:15:02.590,0:15:05.140
One of you said it.

0:15:05.140,0:15:08.602
x squared plus y squared[br]plus z squared equals 1.

0:15:08.602,0:15:10.820
Why is this not a graph?

0:15:10.820,0:15:12.940
Not a graph.

0:15:12.940,0:15:14.163
Why is this not a graph?

0:15:14.163,0:15:18.993


0:15:18.993,0:15:22.374
STUDENT: [INAUDIBLE].

0:15:22.374,0:15:26.260
When you move it over to[br]1, you can't actually--

0:15:26.260,0:15:28.872
MAGDALENA TODA: You[br]cannot but you can cut it.

0:15:28.872,0:15:31.460
You can take a[br]sword and-- I'm OK.

0:15:31.460,0:15:33.720
I don't want to think about it.

0:15:33.720,0:15:37.180
So z is going to be two graphs.

0:15:37.180,0:15:41.970
So I can split this surface[br]even in a parametric form

0:15:41.970,0:15:44.860
as two different graphs.

0:15:44.860,0:15:46.626
Different graphs.

0:15:46.626,0:15:51.490
If I cut along-- I have this[br]orange, or sphere, globe.

0:15:51.490,0:15:54.070
And I cut it along[br]a great circle.

0:15:54.070,0:15:57.580
It doesn't have[br]to be the equator.

0:15:57.580,0:15:59.710
But you have to[br]imagine something

0:15:59.710,0:16:02.540
like the world and the equator.

0:16:02.540,0:16:06.030
This is kind of in[br]the unit sphere.

0:16:06.030,0:16:09.110
Today I drank enough[br]coffee to draw better.

0:16:09.110,0:16:10.160
Why don't I draw better?

0:16:10.160,0:16:12.081
I have no idea.

0:16:12.081,0:16:15.910
So that's not bad, though.

0:16:15.910,0:16:16.760
OK.

0:16:16.760,0:16:18.090
So that's the unit sphere.

0:16:18.090,0:16:18.950
What does it mean?

0:16:18.950,0:16:21.770
It means it has radius how much?

0:16:21.770,0:16:22.270
STUDENT: 1.

0:16:22.270,0:16:23.020
MAGDALENA TODA: 1.

0:16:23.020,0:16:25.670
Radius 1, and we[br]are happy about it.

0:16:25.670,0:16:29.270
And it has two graphs.

0:16:29.270,0:16:31.769
It's not one graph,[br]it's two graphs.

0:16:31.769,0:16:34.184
So this is called[br]implicit equation.

0:16:34.184,0:16:36.715
This is your lab[br]from-- I was chatting

0:16:36.715,0:16:38.670
with-- instead of[br]studying last night,

0:16:38.670,0:16:40.960
I was chatting with[br]you at midnight.

0:16:40.960,0:16:45.080
And one of you said, if I had[br]something I hated in calculus,

0:16:45.080,0:16:48.160
it was the implicit[br]differentiation.

0:16:48.160,0:16:50.400
And I know this is[br]your weak point.

0:16:50.400,0:16:52.570
So we'll do a lot of[br]implicit differentiation,

0:16:52.570,0:16:54.770
so you become more comfortable.

0:16:54.770,0:16:59.120
Usually we have one exercise in[br]this differentiation at least

0:16:59.120,0:17:01.500
on the final.

0:17:01.500,0:17:04.579
So this is an implicit equation.

0:17:04.579,0:17:09.690
And z is going to be[br]two graphs-- 1 minus x

0:17:09.690,0:17:11.125
squared minus y squared.

0:17:11.125,0:17:13.858
So I have, like, two charts,[br]two different charts.

0:17:13.858,0:17:14.358
OK.

0:17:14.358,0:17:17.150


0:17:17.150,0:17:20.089
The upper hemisphere--[br]I'm talking geography,

0:17:20.089,0:17:23.190
but that's how we talk[br]in geometry as well.

0:17:23.190,0:17:26.055
So geography right[br]now is like geometry.

0:17:26.055,0:17:28.150
I have a north pole.

0:17:28.150,0:17:31.620
Somebody quickly give me the[br]coordinates of the north pole.

0:17:31.620,0:17:32.530
STUDENT: 0, 0, 1.

0:17:32.530,0:17:33.530
MAGDALENA TODA: 0, 0, 1.

0:17:33.530,0:17:34.780
Thank you, Brian.

0:17:34.780,0:17:35.640
0, 0, 1.

0:17:35.640,0:17:37.512
How about the south pole?

0:17:37.512,0:17:38.720
STUDENT: 0, 0, minus 1.

0:17:38.720,0:17:41.290
MAGDALENA TODA: 0, 0, minus 1.

0:17:41.290,0:17:45.640
And write yourself a[br]note, because as you know,

0:17:45.640,0:17:48.570
I'm very absent-minded[br]and I forget

0:17:48.570,0:17:52.490
what I eat for lunch and so on.

0:17:52.490,0:17:55.560
Remind me to talk[br]to you sometime

0:17:55.560,0:17:58.280
at the end of the chapter[br]about stereographic projection.

0:17:58.280,0:18:01.080
It's a very important[br]mathematical notion

0:18:01.080,0:18:03.785
that also has to do a[br]little bit with geography.

0:18:03.785,0:18:06.060
But it's a one-to-one[br]correspondence

0:18:06.060,0:18:08.960
between a certain[br]part of a sphere

0:18:08.960,0:18:11.876
and a certain huge[br]part of a plane.

0:18:11.876,0:18:14.050
Now, we're not going[br]to talk about that now,

0:18:14.050,0:18:16.030
because that's not [INAUDIBLE].

0:18:16.030,0:18:18.360
That's a little bit[br]harder [INAUDIBLE].

0:18:18.360,0:18:20.690
You guys should now[br]see this line, right?

0:18:20.690,0:18:24.044
This should be beyond--[br]in the twilight zone,

0:18:24.044,0:18:25.740
behind the sphere.

0:18:25.740,0:18:27.280
OK?

0:18:27.280,0:18:28.660
So you don't see it.

0:18:28.660,0:18:31.410
And who is this? z equals 0.

0:18:31.410,0:18:34.750
And so this green[br]fellow should be

0:18:34.750,0:18:39.150
the circle x squared[br]plus y squared equals 1

0:18:39.150,0:18:40.615
in the xy plane.

0:18:40.615,0:18:43.285


0:18:43.285,0:18:44.700
Good.

0:18:44.700,0:18:47.030
So I have two graphs.

0:18:47.030,0:18:54.870
Now, if I were to ask[br]you, what is the domain

0:18:54.870,0:18:59.250
and the range of the function?

0:18:59.250,0:19:02.630
I'm going to erase[br]the whole thing.

0:19:02.630,0:19:10.040
What is the domain and the range[br]of the related function, z,

0:19:10.040,0:19:13.904
which gives the[br]upper hemisphere?

0:19:13.904,0:19:15.368
Upper hemisphere.

0:19:15.368,0:19:17.320
It's a graph.

0:19:17.320,0:19:20.810
And square root of 1 minus[br]x squared minus y squared.

0:19:20.810,0:19:23.220
You may stare at[br]it until tomorrow.

0:19:23.220,0:19:27.940
It's not hard to figure[br]out what I mean by domain

0:19:27.940,0:19:30.570
and range of such a function.

0:19:30.570,0:19:33.300
You are familiar[br]with domain and range

0:19:33.300,0:19:37.330
for a function of one variable.

0:19:37.330,0:19:39.890
For most of you,[br]that's a piece of cake.

0:19:39.890,0:19:41.800
That was even[br]pre-calc wasn't it?

0:19:41.800,0:19:44.341
It was in Calc 1.

0:19:44.341,0:19:47.360
So most of you had[br]algebra and pre-calc.

0:19:47.360,0:19:51.980
Now, what is the domain[br]of such a function?

0:19:51.980,0:19:57.390
Domain of definition has to be a[br]set of points, x and y in plane

0:19:57.390,0:20:00.590
for which the[br]function is defined.

0:20:00.590,0:20:03.040
If the function is[br]impossible to be defined

0:20:03.040,0:20:05.995
for a certain pair, x, y,[br]you kick that couple out

0:20:05.995,0:20:07.917
and you say, never come back.

0:20:07.917,0:20:09.140
Right?

0:20:09.140,0:20:14.665
So what I mean by domain is[br]those couples that we hate.

0:20:14.665,0:20:16.430
Who we hate?

0:20:16.430,0:20:20.970
The couples x, y for which x[br]squared plus y squared is how?

0:20:20.970,0:20:24.130


0:20:24.130,0:20:25.422
What existence condition do I--

0:20:25.422,0:20:26.296
STUDENT: [INAUDIBLE].

0:20:26.296,0:20:27.440
MAGDALENA TODA: Yeah.

0:20:27.440,0:20:30.230
You see this guy[br]under the square root

0:20:30.230,0:20:33.790
has to be positive or 0.

0:20:33.790,0:20:35.280
Right?

0:20:35.280,0:20:37.890
Otherwise, there is no[br]square root in real numbers.

0:20:37.890,0:20:39.920
That's going to be[br]in imaginary numbers,

0:20:39.920,0:20:41.630
and you can take[br]a walk, because we

0:20:41.630,0:20:45.220
are in real calculus[br]in real time as well.

0:20:45.220,0:20:48.670
So x squared plus y[br]squared must be how?

0:20:48.670,0:20:50.660
Less than or equal to 1.

0:20:50.660,0:20:54.040
We call that a certain name.

0:20:54.040,0:20:59.484
This is called a[br]closed unit disk.

0:20:59.484,0:21:03.160
Please remember, I'm teaching[br]you a little bit more

0:21:03.160,0:21:06.195
than a regular Calc 3 class.

0:21:06.195,0:21:09.324
They will never[br]make a distinction.

0:21:09.324,0:21:10.365
What's closing with this?

0:21:10.365,0:21:11.900
What's opening with this?

0:21:11.900,0:21:14.670
Everything will come[br]into place when you

0:21:14.670,0:21:19.701
move on to advanced calculus.

0:21:19.701,0:21:24.640
If I don't take the boundary--[br]so everything inside the disk

0:21:24.640,0:21:28.460
except for the boundary, I have[br]to put strictly less than 1.

0:21:28.460,0:21:30.610
That's called open unit disk.

0:21:30.610,0:21:35.080
For advanced calculus,[br]this is [INAUDIBLE].

0:21:35.080,0:21:35.580
All right.

0:21:35.580,0:21:37.320
This is just a parentheses.

0:21:37.320,0:21:40.388
My domain is the closed one.

0:21:40.388,0:21:43.145
What is the range?

0:21:43.145,0:21:45.690
The range is going to be--

0:21:45.690,0:21:47.000
STUDENT: [INAUDIBLE].

0:21:47.000,0:21:49.900
MAGDALENA TODA: The altitude[br]starts having values from--

0:21:49.900,0:21:51.092
STUDENT: Negative 1 to 1.

0:21:51.092,0:21:51.758
STUDENT: 0 to 1.

0:21:51.758,0:21:53.008
MAGDALENA TODA: So I'm 0 to 1.

0:21:53.008,0:21:55.090
I'll only talk about[br]the upper hemisphere.

0:21:55.090,0:21:58.310
I should even erase,[br]because I don't want it.

0:21:58.310,0:21:59.333
So say it again, guys.

0:21:59.333,0:22:00.350
STUDENT: 0 to 1.

0:22:00.350,0:22:01.100
MAGDALENA TODA: 0.

0:22:01.100,0:22:01.865
Open or closed?

0:22:01.865,0:22:02.635
STUDENT: Open.

0:22:02.635,0:22:03.301
STUDENT: Closed.

0:22:03.301,0:22:05.740
STUDENT: Closed, closed.

0:22:05.740,0:22:07.630
MAGDALENA TODA: Closed to?

0:22:07.630,0:22:08.450
STUDENT: 1 closed.

0:22:08.450,0:22:09.650
MAGDALENA TODA: 1 closed.

0:22:09.650,0:22:10.150
Yes.

0:22:10.150,0:22:13.874
Because that is the north pole.

0:22:13.874,0:22:19.160
I've been meaning to[br]give you this example.

0:22:19.160,0:22:22.270
And give me the other example[br]for the lower hemisphere.

0:22:22.270,0:22:23.400
What's different?

0:22:23.400,0:22:24.894
The same domain?

0:22:24.894,0:22:25.935
STUDENT: It ranges from--

0:22:25.935,0:22:27.042
STUDENT: Negative 1.

0:22:27.042,0:22:28.710
STUDENT: Negative 1 to 0.

0:22:28.710,0:22:30.490
MAGDALENA TODA: Closed[br]internal, right?

0:22:30.490,0:22:33.860
When we include the endpoints,[br]we call that closed interval.

0:22:33.860,0:22:36.200
It has a certain[br]topological sense.

0:22:36.200,0:22:39.110
You haven't taken[br]topology, but very soon,

0:22:39.110,0:22:43.940
if you are a math major, or[br]you are a double major, or some

0:22:43.940,0:22:48.090
of you even-- they want to[br]learn more about topology,

0:22:48.090,0:22:51.735
you will learn what an open[br]set is versus a closed set.

0:22:51.735,0:22:53.630
Remember we called this closed.

0:22:53.630,0:22:56.230
This is open.

0:22:56.230,0:22:59.660
And if it's closed here and[br]open there, it's neither.

0:22:59.660,0:23:00.160
OK?

0:23:00.160,0:23:02.940
Don't say anything about that.

0:23:02.940,0:23:03.440
OK.

0:23:03.440,0:23:08.110
To be closed, it has to be[br]containing both endpoints.

0:23:08.110,0:23:09.300
I'm going to erase this.

0:23:09.300,0:23:12.260


0:23:12.260,0:23:19.728
And this was, of course, 11.1.

0:23:19.728,0:23:22.632
We are in the middle of it.

0:23:22.632,0:23:28.440
In 11.1, one of you gave me a[br]beautiful graph to think about.

0:23:28.440,0:23:30.525
And I'm going to give[br]you something to do,

0:23:30.525,0:23:32.690
because I don't want[br]you to get lazy.

0:23:32.690,0:23:36.041
I'm very happy you came[br]up with the saddle.

0:23:36.041,0:23:38.908


0:23:38.908,0:23:39.408
All right.

0:23:39.408,0:23:41.480
We drew such a saddle.

0:23:41.480,0:23:44.460


0:23:44.460,0:23:46.881
And I did my best,[br]but it's not hard.

0:23:46.881,0:23:50.360
It's not easy to draw saddle.

0:23:50.360,0:23:54.540
When I am looking at the[br]coordinates, x, y, z,

0:23:54.540,0:24:01.575
I have z equals minus y[br]squared will look down.

0:24:01.575,0:24:05.020


0:24:05.020,0:24:06.810
Maybe I made it too fat.

0:24:06.810,0:24:08.980
I'm really sorry.

0:24:08.980,0:24:11.430
And down.

0:24:11.430,0:24:12.410
This continues.

0:24:12.410,0:24:21.230


0:24:21.230,0:24:22.210
OK?

0:24:22.210,0:24:26.920
And then what other thing[br]did I want to point out?

0:24:26.920,0:24:31.100
I want to point out--[br]do you see this?

0:24:31.100,0:24:33.660
This should look a[br]little bit more round.

0:24:33.660,0:24:36.556
It doesn't look[br]round enough here.

0:24:36.556,0:24:38.290
STUDENT: Your'e drawing[br]a saddle, right?

0:24:38.290,0:24:40.289
MAGDALENA TODA: No, I'm[br]drawing just the section

0:24:40.289,0:24:41.970
z equals minus y squared.

0:24:41.970,0:24:44.330
So I took x to be 0.

0:24:44.330,0:24:47.880
And the purple line[br]should be on this wall.

0:24:47.880,0:24:50.160
I know you guys have[br]enough imagination.

0:24:50.160,0:24:54.045
So this is going to[br]be z equals minus y

0:24:54.045,0:24:58.700
squared drawn on yz wall.

0:24:58.700,0:25:03.170


0:25:03.170,0:25:05.590
I've done this before,[br]but I'm just reviewing.

0:25:05.590,0:25:08.360
What if it's y0?

0:25:08.360,0:25:10.920
Then I have to[br]draw on that wall.

0:25:10.920,0:25:14.326
And I have to draw beautifully,[br]which I am not-- don't always--

0:25:14.326,0:25:15.718
I can't always do.

0:25:15.718,0:25:17.110
But I'll try.

0:25:17.110,0:25:23.750
I have z equals x squared[br]drawn on that wall.

0:25:23.750,0:25:26.580
If I start drawing,[br]I'll get fired.

0:25:26.580,0:25:29.020
That I have this branch.

0:25:29.020,0:25:32.852
I should go through that[br]corner and go out of the room

0:25:32.852,0:25:35.307
and continue with that branch.

0:25:35.307,0:25:36.010
All right?

0:25:36.010,0:25:39.480


0:25:39.480,0:25:43.050
This is curved like[br]that in this direction.

0:25:43.050,0:25:45.460
And this other is[br]curved like this.

0:25:45.460,0:25:50.300
So if the guy is[br]going to put his feet,

0:25:50.300,0:25:54.590
where is the butt of[br]the writer going to sit?

0:25:54.590,0:25:57.450
He is here.

0:25:57.450,0:25:59.605
And these are his legs.

0:25:59.605,0:26:02.490


0:26:02.490,0:26:06.144
And these are his cowboy boots.

0:26:06.144,0:26:06.644
OK.

0:26:06.644,0:26:08.117
Do they look like cowboy boots?

0:26:08.117,0:26:10.572
No, I apologize.

0:26:10.572,0:26:12.105
STUDENT: Looks like socks.

0:26:12.105,0:26:12.980
MAGDALENA TODA: Yeah.

0:26:12.980,0:26:15.770
They look more like[br]Christmas socks.

0:26:15.770,0:26:17.750
But anyway, it's a poor cowboy.

0:26:17.750,0:26:22.880


0:26:22.880,0:26:24.710
Let's lower the[br]saddle a little bit.

0:26:24.710,0:26:27.200
He cannot see the horse, OK?

0:26:27.200,0:26:30.290
So the saddle.

0:26:30.290,0:26:35.182
If I cross the saddle,[br]this is the saddle.

0:26:35.182,0:26:38.160
And these are his hands.

0:26:38.160,0:26:41.115
And he is holding his hat.

0:26:41.115,0:26:42.065
This is [INAUDIBLE].

0:26:42.065,0:26:45.865
And with one hand[br]is on the horse.

0:26:45.865,0:26:46.815
I don't know.

0:26:46.815,0:26:48.320
It's very [INAUDIBLE].

0:26:48.320,0:26:56.760
So what I'm trying to draw[br]looks something like this.

0:26:56.760,0:26:57.660
Right?

0:26:57.660,0:26:59.100
Eh.

0:26:59.100,0:27:01.500
Sorry.

0:27:01.500,0:27:02.240
More or less.

0:27:02.240,0:27:03.943
It's an abstract picture.

0:27:03.943,0:27:05.750
Very abstract picture.

0:27:05.750,0:27:14.946
So with this in mind, if I were[br]to look at the level curves,

0:27:14.946,0:27:18.802
I'm going to ask you,[br]what are the level curves?

0:27:18.802,0:27:22.200
Oh, my god, what are[br]the level curves?

0:27:22.200,0:27:25.390


0:27:25.390,0:27:27.810
You already have them in[br]your WeBWorK homework.

0:27:27.810,0:27:30.300
But for one point[br]extra credit, I

0:27:30.300,0:27:34.072
want you to draw[br]them on the floor.

0:27:34.072,0:27:37.690
Draw the level curves.

0:27:37.690,0:27:39.250
Remember what those were?

0:27:39.250,0:27:43.290
They were projections of[br]the curves on the surface

0:27:43.290,0:27:46.570
at the intersection[br]with z equals c planes.

0:27:46.570,0:27:48.840
You project them on the ground.

0:27:48.840,0:27:50.280
What do you think they are?

0:27:50.280,0:27:51.270
Think about it.

0:27:51.270,0:27:53.280
What are these?

0:27:53.280,0:27:58.460
If I take c, what[br]if c is positive?

0:27:58.460,0:28:01.540


0:28:01.540,0:28:04.730
What if c is 0?

0:28:04.730,0:28:12.255
What if c is less than 0?

0:28:12.255,0:28:14.500
What am I going to have?

0:28:14.500,0:28:18.460
Your imagination gives[br]you c equals 1, Magdalena.

0:28:18.460,0:28:19.870
Let's draw that.

0:28:19.870,0:28:20.370
OK.

0:28:20.370,0:28:21.940
Well, I'll try.

0:28:21.940,0:28:23.650
a and b would be 1, right, guys?

0:28:23.650,0:28:26.275
So a and b would be 1.

0:28:26.275,0:28:27.100
This is a square.

0:28:27.100,0:28:29.970
These would be the asymptotes.

0:28:29.970,0:28:36.110
So very, very[br]briefly, the hyperbola

0:28:36.110,0:28:41.050
would be this one-- x squared[br]minus y squared equals 1,

0:28:41.050,0:28:42.222
right?

0:28:42.222,0:28:45.130
If I have the last[br]case for c equals 1,

0:28:45.130,0:28:47.460
I'm going to have-- c[br]equals negative 1-- I'm

0:28:47.460,0:28:49.215
going to have the conjugate.

0:28:49.215,0:28:50.660
Are you guys with me?

0:28:50.660,0:28:57.790
So I'll have an a squared,[br]asymptotes, conjugate.

0:28:57.790,0:29:01.230


0:29:01.230,0:29:05.060
What if I have different level[br]c? c equals 1/2. c equals 2.

0:29:05.060,0:29:08.000
c equals pi. c[br]equals-- what are they?

0:29:08.000,0:29:12.330
I'm going to get[br]families of hyperbolas,

0:29:12.330,0:29:14.720
trenches that look like that.

0:29:14.720,0:29:16.470
Standard trenches and[br]conjugate trenches.

0:29:16.470,0:29:20.600
A multitude of them, an infinite[br]family of such hyperbolas,

0:29:20.600,0:29:22.480
an infinite family[br]of such hyperbolas.

0:29:22.480,0:29:24.542
I wanted to draw it.

0:29:24.542,0:29:28.880
What do I get when c is 0?

0:29:28.880,0:29:30.022
What are those?

0:29:30.022,0:29:31.772
STUDENT: Don't you[br]get, like, [INAUDIBLE]?

0:29:31.772,0:29:34.985


0:29:34.985,0:29:36.730
MAGDALENA TODA: They[br]get-- very good.

0:29:36.730,0:29:37.230
Why?

0:29:37.230,0:29:40.650
x squared minus y squared[br]equals 0 would lead

0:29:40.650,0:29:44.920
me to y equals plus/minus 1.

0:29:44.920,0:29:48.470
And who are those y[br]equals plus/minus 1?

0:29:48.470,0:29:49.590
Exactly.

0:29:49.590,0:29:54.500
But exactly the first[br]bisector, which is y equals x.

0:29:54.500,0:29:56.410
They are [? then the ?][br]function.

0:29:56.410,0:29:59.820
And the other one, y equals[br]negative [? x. ?] So these

0:29:59.820,0:30:01.160
are the asymptotes.

0:30:01.160,0:30:05.620
So I'm going to get a-- you guys[br]have to do this better than me.

0:30:05.620,0:30:06.880
Sorry.

0:30:06.880,0:30:08.790
These are all[br]hyperbolic trenches.

0:30:08.790,0:30:11.730
They are all going to[br]infinity like that.

0:30:11.730,0:30:15.330
And I'm sorry that[br]I'm giving you

0:30:15.330,0:30:17.090
a little bit too many hints.

0:30:17.090,0:30:19.282
This is part of your[br]homework, your WeBWorK.

0:30:19.282,0:30:20.740
I shouldn't talk[br]too much about it.

0:30:20.740,0:30:23.820


0:30:23.820,0:30:25.350
Any questions so far?

0:30:25.350,0:30:26.495
Is this hard?

0:30:26.495,0:30:28.000
Yes, sir?

0:30:28.000,0:30:28.500
No.

0:30:28.500,0:30:30.133
STUDENT: So [? spherically, ?][br]if you had z

0:30:30.133,0:30:31.508
equals y squared[br]minus x squared,

0:30:31.508,0:30:33.890
it's that same[br]picture, just flipped?

0:30:33.890,0:30:39.962


0:30:39.962,0:30:41.450
MAGDALENA TODA:[br]What would it be?

0:30:41.450,0:30:43.241
It would be the poor[br]saddle-- or cowboy--

0:30:43.241,0:30:44.490
STUDENT: Would be upside down.

0:30:44.490,0:30:46.340
MAGDALENA TODA:[br]--would be upside down.

0:30:46.340,0:30:49.650
Or projected in[br]something like a mirror.

0:30:49.650,0:30:51.100
I don't know how to say.

0:30:51.100,0:30:52.850
It would be exactly upside down.

0:30:52.850,0:30:55.530
So the reflection of that.

0:30:55.530,0:30:59.234
So you take all the points.

0:30:59.234,0:31:01.075
If you have-- I don't know.

0:31:01.075,0:31:03.380
It's hard to draw a reflection[br]in three dimensions.

0:31:03.380,0:31:03.880
But--

0:31:03.880,0:31:04.963
STUDENT: No, I understand.

0:31:04.963,0:31:09.430
MAGDALENA TODA:[br]Practically every curve

0:31:09.430,0:31:14.690
would be upside down with[br]respect to the floor.

0:31:14.690,0:31:15.650
OK.

0:31:15.650,0:31:16.560
All right.

0:31:16.560,0:31:20.600
I'm going to erase in one.

0:31:20.600,0:31:24.060
And you say, well, you've[br]taught us about these things,

0:31:24.060,0:31:26.150
like the domain and range.

0:31:26.150,0:31:30.530
But what about other notions,[br]like continuity and stuff?

0:31:30.530,0:31:33.300


0:31:33.300,0:31:50.730
Let me move on to 11.2.

0:31:50.730,0:32:00.192
Limits of functions of[br]the type z equals f of xy.

0:32:00.192,0:32:14.640


0:32:14.640,0:32:20.030
So what do you remember[br]about the limit

0:32:20.030,0:32:23.130
of a function of one variable?

0:32:23.130,0:32:23.630
Comparison.

0:32:23.630,0:32:27.960


0:32:27.960,0:32:36.255
What about the limit if you[br]take [? z's, ?] I don't know.

0:32:36.255,0:32:37.700
I should look stunned.

0:32:37.700,0:32:38.700
And I should be stunned.

0:32:38.700,0:32:49.430
Of a function of y equals[br]f of x of one variable.

0:32:49.430,0:32:56.730


0:32:56.730,0:33:10.513
When do we say that[br]f has a limit at a?

0:33:10.513,0:33:12.473


0:33:12.473,0:33:14.764
STUDENT: When the [INAUDIBLE][br]approaches from the right

0:33:14.764,0:33:16.740
and the left to the same value.

0:33:16.740,0:33:22.610
MAGDALENA TODA: Actually, that[br]was the simpler definition.

0:33:22.610,0:33:25.555
Let's think a little bit deeper.

0:33:25.555,0:33:35.330
We say that f has a[br]limit L at x equals a.

0:33:35.330,0:33:40.550
That's kind of the idea,[br]left and right limits.

0:33:40.550,0:33:43.680
But not both of them[br]have to exist, you see.

0:33:43.680,0:33:45.532
Maybe only the limit[br]from the left or limit

0:33:45.532,0:33:46.978
from the right only exists.

0:33:46.978,0:33:49.870


0:33:49.870,0:34:04.370
If, for any choice of values[br]of x, closer and closer, closer

0:34:04.370,0:34:23.909
and closer to a, we get that[br]F gets closer and closer to L.

0:34:23.909,0:34:27.159
And this "any" I put in.

0:34:27.159,0:34:33.500
My god, I put it in[br]a red circle thing,

0:34:33.500,0:34:40.030
because one could get[br]subsequencies of a sequence.

0:34:40.030,0:34:42.400
And for that subsequence[br]thing, things

0:34:42.400,0:34:44.994
look like I would have a limit.

0:34:44.994,0:34:47.830
And then you say,[br]well, but in the end,

0:34:47.830,0:34:50.889
I don't have a limit, because[br]I can get another subsequence

0:34:50.889,0:34:52.350
of the sequence.

0:34:52.350,0:34:59.030
And for that one, I'm not[br]going to have a limit.

0:34:59.030,0:35:04.150
Can you give me an example[br]of some crazy function that

0:35:04.150,0:35:08.670
does not have a limit at 0?

0:35:08.670,0:35:12.131
Example of a crazy function.

0:35:12.131,0:35:12.630
No.

0:35:12.630,0:35:14.560
No, don't write "crazy."

0:35:14.560,0:35:26.270
Of a function f of x[br]that is not defined at 0

0:35:26.270,0:35:43.855
and does not have[br]limit at 0, although it

0:35:43.855,0:35:53.414
is defined for values[br]arbitrarily close to 0.

0:35:53.414,0:35:59.706


0:35:59.706,0:36:07.900
Moreover, I want that function[br]to be drawn without-- I

0:36:07.900,0:36:22.960
want the function to be[br]drawn without leaving

0:36:22.960,0:36:26.300
the paper when I draw.

0:36:26.300,0:36:30.748


0:36:30.748,0:36:31.248
[INAUDIBLE]

0:36:31.248,0:36:34.566


0:36:34.566,0:36:43.530
So something that would[br]be defined on the whole 0

0:36:43.530,0:36:58.660
infinity except for 0 that[br]I can draw continuously

0:36:58.660,0:37:03.400
except when I get to 0, I[br]get some really bad behavior.

0:37:03.400,0:37:07.560
I don't have a limit[br]for that function.

0:37:07.560,0:37:08.910
You are close to that.

0:37:08.910,0:37:10.720
Sine of 1/x.

0:37:10.720,0:37:13.110
STUDENT: I said y equals 1/x.

0:37:13.110,0:37:15.324
MAGDALENA TODA: y equals 1/x.

0:37:15.324,0:37:16.260
Very good.

0:37:16.260,0:37:18.600
Let's see.

0:37:18.600,0:37:20.004
STUDENT: Oh, yeah. [INAUDIBLE].

0:37:20.004,0:37:21.129
MAGDALENA TODA: Yeah, yeah.

0:37:21.129,0:37:22.390
Both are excellent examples.

0:37:22.390,0:37:24.530
So let's see.

0:37:24.530,0:37:29.080
This guy is a very[br]nice function.

0:37:29.080,0:37:31.370
How do we draw him, or her?

0:37:31.370,0:37:32.540
Well, it's a her, right?

0:37:32.540,0:37:33.040
It's a she.

0:37:33.040,0:37:33.990
It's a function.

0:37:33.990,0:37:34.660
No, no.

0:37:34.660,0:37:36.360
In English, it doesn't[br]make any sense,

0:37:36.360,0:37:40.732
but if I think French, Italian,[br]Spanish, Romanian-- now

0:37:40.732,0:37:44.143
I speak both Italian[br]and Romanian--

0:37:44.143,0:37:47.220
we say it's a she,[br]it's a feminine.

0:37:47.220,0:37:52.345
So as I approach with values[br]closer and closer and closer

0:37:52.345,0:37:56.345
to 0, what happens[br]to my poor function?

0:37:56.345,0:37:58.911
It blows up.

0:37:58.911,0:37:59.410
OK.

0:37:59.410,0:38:05.040
So I have limit of 1/x from[br]the right and from the left.

0:38:05.040,0:38:08.310
If I take it from the[br]left, I don't care.

0:38:08.310,0:38:11.170
Let's take it only[br]from the right.

0:38:11.170,0:38:11.670
OK?

0:38:11.670,0:38:17.910


0:38:17.910,0:38:19.630
It's close to 0.

0:38:19.630,0:38:21.210
That's going to blow up, right?

0:38:21.210,0:38:24.560


0:38:24.560,0:38:25.490
And I restrict it.

0:38:25.490,0:38:30.455
So let's say, if I want[br]the domain to be containing

0:38:30.455,0:38:32.742
[? both, ?] that's also fine.

0:38:32.742,0:38:35.730
So if you guys want, we[br]can draw the other one.

0:38:35.730,0:38:37.010
This goes to paradise.

0:38:37.010,0:38:39.665
The other one, I'm not[br]going to say where it goes.

0:38:39.665,0:38:43.270
But it's the same idea,[br]that as you approach 0

0:38:43.270,0:38:45.710
with closer and closer[br]and closer values,

0:38:45.710,0:38:47.662
it's going to blow up.

0:38:47.662,0:38:51.020
It's going to explode.

0:38:51.020,0:38:54.010
This is a beautiful function.

0:38:54.010,0:38:55.160
How beautiful [INAUDIBLE].

0:38:55.160,0:38:57.940
Beautiful with a[br]bad behavior near 0.

0:38:57.940,0:38:59.730
So I'm not going[br]to have a limit.

0:38:59.730,0:39:00.610
No limit.

0:39:00.610,0:39:02.590
Some people say, limit[br]exists and is infinity.

0:39:02.590,0:39:05.265
But does infinity exist?

0:39:05.265,0:39:07.382
Well, this is a[br]really philosophical,

0:39:07.382,0:39:10.570
religious notion, so I[br]don't want to get into it.

0:39:10.570,0:39:13.480
But in mathematics, we consider[br]that unless the limit is

0:39:13.480,0:39:16.730
finite, you cannot have a limit.

0:39:16.730,0:39:20.850
So if the limit is plus/minus[br]infinity, there is no limit.

0:39:20.850,0:39:25.240
Could the limit be different[br]or different subsequences?

0:39:25.240,0:39:28.190
This is what I[br]wanted to point out.

0:39:28.190,0:39:34.270
If you try this guy, you are[br]in real trouble on that guy.

0:39:34.270,0:39:35.152
Why?

0:39:35.152,0:39:36.568
You can have two.

0:39:36.568,0:39:38.928
If you have a graphing[br]calculator, which

0:39:38.928,0:39:43.990
I'm going to be opposed to you[br]being used in the classroom,

0:39:43.990,0:39:46.160
you would probably[br]see what happens.

0:39:46.160,0:39:51.540
Sine is defined on[br]all the real numbers.

0:39:51.540,0:39:54.240
But you cannot[br]have a value at 0,

0:39:54.240,0:39:57.510
because the 1/x is[br]not defined at 0.

0:39:57.510,0:40:02.330
Imagine you get closer and[br]closer to 0 from both sides.

0:40:02.330,0:40:05.280
I cannot draw very beautifully.

0:40:05.280,0:40:09.590
But as 1, this is plus[br]1 and this is minus 1.

0:40:09.590,0:40:11.840
I'm going to have some behavior.

0:40:11.840,0:40:15.160
And how many of you have seen[br]that on a computer screen

0:40:15.160,0:40:15.941
or calculator?

0:40:15.941,0:40:16.440
You've seen.

0:40:16.440,0:40:17.585
Yeah, you've seen.

0:40:17.585,0:40:20.450
By the way, did you[br]see the Lubbuck High?

0:40:20.450,0:40:23.472
Was it in high school you saw[br]it the first time in Calc 1

0:40:23.472,0:40:25.140
or pre-calc?

0:40:25.140,0:40:28.440
STUDENT: [INAUDIBLE][br]Algebra 1 with Mr. West.

0:40:28.440,0:40:28.940
[INAUDIBLE]

0:40:28.940,0:40:32.510
MAGDALENA TODA: So I'll[br]try-- oh, guys, you

0:40:32.510,0:40:34.240
have to be patient with me.

0:40:34.240,0:40:38.150
I'm not leaving the poor board[br]with the tip of my pencil.

0:40:38.150,0:40:39.380
I'm not leaving him.

0:40:39.380,0:40:42.358
I have continuity.

0:40:42.358,0:40:45.810
As I got closer to this, I still[br]have the [INAUDIBLE] property.

0:40:45.810,0:40:47.150
Anyway, it's OK.

0:40:47.150,0:40:48.410
I'm not leaving this.

0:40:48.410,0:40:52.100
I am taking all the values[br]possible between minus 1 and 1.

0:40:52.100,0:40:54.820
So on intervals that[br]are smaller, smaller,

0:40:54.820,0:40:57.756
I'm really taking all the[br]values between minus 1 and 1,

0:40:57.756,0:41:01.390
and really rapidly--[br][INAUDIBLE].

0:41:01.390,0:41:07.520
When I'm getting closer to 0,[br]I'm not going to have a limit.

0:41:07.520,0:41:10.065
But as somebody[br]may say, but wait.

0:41:10.065,0:41:12.270
When I have a sequence[br]of values that

0:41:12.270,0:41:14.360
is getting closer[br]and closer to 0,

0:41:14.360,0:41:18.610
is that no guarantee that[br]I'm going to have a limit?

0:41:18.610,0:41:20.120
Nope.

0:41:20.120,0:41:20.965
It depends.

0:41:20.965,0:41:25.475
If you say "any," it has to[br]be for any choice of points,

0:41:25.475,0:41:27.960
any choice of points[br]that you go closer to 0.

0:41:27.960,0:41:30.150
Not for one sequence[br]of points that

0:41:30.150,0:41:32.290
is getting closer[br]and closer to 0.

0:41:32.290,0:41:35.296
For example, if your[br]choice of points is this,

0:41:35.296,0:41:36.282
choice of points.

0:41:36.282,0:41:39.740


0:41:39.740,0:41:43.805
Getting closer to 0.

0:41:43.805,0:41:49.540
[INAUDIBLE] xn[br]equals 1 over 2 pi n.

0:41:49.540,0:41:51.815
Isn't this going to 0?

0:41:51.815,0:41:52.315
Yeah.

0:41:52.315,0:41:54.090
It then goes to infinity.

0:41:54.090,0:41:55.690
This sequence goes to 0.

0:41:55.690,0:41:56.300
What is it?

0:41:56.300,0:41:57.015
1 over 2 pi?

0:41:57.015,0:41:58.070
1 over 4 pi?

0:41:58.070,0:41:58.890
1 over 8 pi?

0:41:58.890,0:41:59.980
1 over 16 pi?

0:41:59.980,0:42:01.100
1 over 32 pi?

0:42:01.100,0:42:02.550
1 over 64 pi?

0:42:02.550,0:42:04.450
This is what my[br]son is doing to me.

0:42:04.450,0:42:06.182
And I say, please stop.

0:42:06.182,0:42:07.130
OK?

0:42:07.130,0:42:08.240
He's 10 years old.

0:42:08.240,0:42:09.590
He's so funny.

0:42:09.590,0:42:12.964
Now, another choice of points.

0:42:12.964,0:42:20.490


0:42:20.490,0:42:21.170
Ah.

0:42:21.170,0:42:26.450
Somebody-- all of you are[br]smart enough to do this.

0:42:26.450,0:42:29.230
What do you think[br]I'm going to pick?

0:42:29.230,0:42:30.780
1 over what?

0:42:30.780,0:42:33.604
And when [? other ?][br]something that goes to 0

0:42:33.604,0:42:34.520
then goes to infinity.

0:42:34.520,0:42:41.660
And I know that your[br]professor showed you that.

0:42:41.660,0:42:44.995
pi over 2 plus 2 pi n.

0:42:44.995,0:42:45.870
Doesn't this go to 0?

0:42:45.870,0:42:46.370
Yes.

0:42:46.370,0:42:49.740
As n gets bigger and[br]bigger, this is going to 0.

0:42:49.740,0:42:51.040
However, there is no limit.

0:42:51.040,0:42:51.810
Why?

0:42:51.810,0:42:59.280
Well, for the first sequence,[br]as xn goes to 0, f of xn

0:42:59.280,0:43:04.060
goes to-- what is[br]sine of-- OK, I

0:43:04.060,0:43:05.870
am too lazy to write this down.

0:43:05.870,0:43:11.015
Sine of 1 over 1 over--[br]of 1 over 1 over 2 pi?

0:43:11.015,0:43:14.810


0:43:14.810,0:43:16.780
STUDENT: It's the[br]sine over 2 pi.

0:43:16.780,0:43:20.650
MAGDALENA TODA: This[br]is sine of 2 pi n.

0:43:20.650,0:43:22.361
And how much is that?

0:43:22.361,0:43:22.860
STUDENT: 0.

0:43:22.860,0:43:24.210
MAGDALENA TODA: 0.

0:43:24.210,0:43:25.590
So this is a 0.

0:43:25.590,0:43:28.730
And this is a--[br]this converges to 0.

0:43:28.730,0:43:31.400
So I say, oh, so maybe I have[br]a limit, and that'll be 0.

0:43:31.400,0:43:32.520
Wrong.

0:43:32.520,0:43:35.730
That would be the rapid,[br]stupid conclusion.

0:43:35.730,0:43:38.690
If somebody jumps [? up, ?][br]I picked some points,

0:43:38.690,0:43:41.820
I formed the sequence that[br]gets closer and closer to 0.

0:43:41.820,0:43:43.700
I'm sure that the limit exists.

0:43:43.700,0:43:45.500
I've got a 0.

0:43:45.500,0:43:48.850
Well, did you think of[br]any possible choice?

0:43:48.850,0:43:49.670
That's the problem.

0:43:49.670,0:43:51.942
You have to have[br]any possible choice.

0:43:51.942,0:44:02.580
F of yn sine of 1 over[br]pi over 2 plus 1 over 1

0:44:02.580,0:44:09.030
over-- Magdalena-- pi[br]over 2 plus 2 pi n.

0:44:09.030,0:44:10.860
So we saw that this was 0.

0:44:10.860,0:44:14.850
What happens to sine of[br]1 over 1 over sine of pi

0:44:14.850,0:44:18.720
over 2 plus 2 pi n?

0:44:18.720,0:44:19.890
And where does this go?

0:44:19.890,0:44:21.132
It then goes to infinity.

0:44:21.132,0:44:26.970


0:44:26.970,0:44:29.250
This sequence goes to 0.

0:44:29.250,0:44:32.630
What is f of the[br]sequence going to?

0:44:32.630,0:44:33.520
To another limit.

0:44:33.520,0:44:36.030
So there is no limit.

0:44:36.030,0:44:38.580
What's the limit of[br]this subsequence?

0:44:38.580,0:44:41.400
It's a constant one, right?

0:44:41.400,0:44:45.930
Because look, what does it[br]mean pi over 2 plus 2 pi n?

0:44:45.930,0:44:49.240
Where am I on the unit[br]trigonometric circle?

0:44:49.240,0:44:50.690
[INTERPOSING VOICES]

0:44:50.690,0:44:53.930
Always here, right?

0:44:53.930,0:44:56.450
Always on the sort of[br]like the north pole.

0:44:56.450,0:44:59.220
So what is the sine[br]of this north pole?

0:44:59.220,0:44:59.900
STUDENT: 1.

0:44:59.900,0:45:00.970
MAGDALENA TODA: Always 1.

0:45:00.970,0:45:02.400
So I get the limit 1.

0:45:02.400,0:45:06.482
So I'm done because there[br]are two different limits.

0:45:06.482,0:45:09.360
So pay attention to[br]this type of problem.

0:45:09.360,0:45:17.530
Somebody can get you in trouble[br]with this kind of thing.

0:45:17.530,0:45:20.220
On the other hand,[br]I'm asking you,

0:45:20.220,0:45:22.995
what if I want to make this[br]a function of two variables?

0:45:22.995,0:45:27.660


0:45:27.660,0:45:30.960
So I'll say, one[br]point extra credit.

0:45:30.960,0:45:34.070
I'm giving you too[br]much extra credit.

0:45:34.070,0:45:36.210
Maybe I give you[br]too much-- it's OK.

0:45:36.210,0:45:39.930
One point extra credit--[br]put them together.

0:45:39.930,0:45:43.340


0:45:43.340,0:45:47.520
Does f-- do you[br]like to do the f?

0:45:47.520,0:45:51.400
I used big F, and then I[br]changed it to little f.

0:45:51.400,0:45:54.073
This time I have a function[br]of two variables-- little

0:45:54.073,0:46:01.178
f with xy-- to be sine of 1[br]over x squared plus y squared.

0:46:01.178,0:46:09.442
Does this function have a[br]limit at the point 0, 0?

0:46:09.442,0:46:12.350


0:46:12.350,0:46:15.540
So when I approach 0,[br]0, do I have a limit?

0:46:15.540,0:46:16.680
OK.

0:46:16.680,0:46:19.810
And you say, well, it depends[br]how I approach that 0, 0.

0:46:19.810,0:46:21.490
That's exactly the thing.

0:46:21.490,0:46:23.210
Yes, sir.

0:46:23.210,0:46:25.278
Oh, you didn't want to ask me.

0:46:25.278,0:46:28.480


0:46:28.480,0:46:37.115
And does f of xy[br]equals-- let me give you

0:46:37.115,0:46:41.080
another one, a[br]really sexy one. x

0:46:41.080,0:46:44.780
squared plus y squared[br]times sine of 1

0:46:44.780,0:46:48.460
over x squared plus y squared.

0:46:48.460,0:46:55.053
Have a limit at 0, 0?

0:46:55.053,0:47:00.030


0:47:00.030,0:47:01.490
I don't know.

0:47:01.490,0:47:04.220
Continuous it cannot be,[br]because it's not defined there.

0:47:04.220,0:47:04.720
Right?

0:47:04.720,0:47:07.670
For a function to be[br]continuous at a point,

0:47:07.670,0:47:11.360
the function has to[br]satisfy three conditions.

0:47:11.360,0:47:14.590
The function has to be[br]defined there at that point.

0:47:14.590,0:47:16.560
The function has to[br]have a limit there

0:47:16.560,0:47:18.890
at that point of the domain.

0:47:18.890,0:47:23.140
And the limit and the function[br]value have to coincide.

0:47:23.140,0:47:24.750
Three conditions.

0:47:24.750,0:47:28.470
We will talk about[br]continuity later.

0:47:28.470,0:47:29.500
Hint.

0:47:29.500,0:47:31.620
Magdalena, too many hints.

0:47:31.620,0:47:33.680
This should remind[br]you of somebody

0:47:33.680,0:47:36.080
from the first[br]variable calculus.

0:47:36.080,0:47:37.970
It's a more challenging problem.

0:47:37.970,0:47:40.490
That's why I gave[br]it to extra credit.

0:47:40.490,0:47:45.660
If I had x sine of 1/x,[br]what would that look like?

0:47:45.660,0:47:46.890
STUDENT: x times--

0:47:46.890,0:47:50.260
MAGDALENA TODA: x[br]times sine of 1/x.

0:47:50.260,0:47:55.458
When I approach 0[br]with-- so if I have-- I

0:47:55.458,0:47:57.160
don't ask for an answer now.

0:47:57.160,0:47:58.800
You go home, you think about it.

0:47:58.800,0:48:00.310
You take the calculator.

0:48:00.310,0:48:05.910
But keep in mind that your[br]calculator can fool you.

0:48:05.910,0:48:11.300
Sometimes it can show an[br]image that misguides you.

0:48:11.300,0:48:14.580
So you have to think[br]how to do that.

0:48:14.580,0:48:18.920
How about x times[br]sine of 1/x when--

0:48:18.920,0:48:22.130
does it have a limit[br]when x goes to 0?

0:48:22.130,0:48:23.620
Is there such a limit?

0:48:23.620,0:48:24.600
Does it exist?

0:48:24.600,0:48:28.170


0:48:28.170,0:48:31.420
So if such a limit[br]would exist, maybe we

0:48:31.420,0:48:35.600
can extend by continuity the[br]function x times sine over x.

0:48:35.600,0:48:36.610
What does it mean?

0:48:36.610,0:48:38.670
Like, extend it, prolong it.

0:48:38.670,0:48:44.415
And say, it's this 4x equals[br]0 and this if x is not 0.

0:48:44.415,0:48:47.610
So this is obviously x is[br]different from 0, right?

0:48:47.610,0:48:49.710
Can we extend it by continuity?

0:48:49.710,0:48:51.230
Think about the drawing.

0:48:51.230,0:48:54.500
Think about the arguments.

0:48:54.500,0:48:58.350
And I think it's time for me[br]to keep the promise I made

0:48:58.350,0:49:04.950
to [? Aaron, ?][br]because I see no way.

0:49:04.950,0:49:07.860
Oh, my god, [? Aaron, ?][br]I see no way out.

0:49:07.860,0:49:10.380


0:49:10.380,0:49:14.490
The epsilon delta[br]definition of limit.

0:49:14.490,0:49:17.460
[? Right? ?] OK.

0:49:17.460,0:49:21.070
So what does it mean for a[br]real mathematician or somebody

0:49:21.070,0:49:25.350
with a strong mathematical[br]foundation and education

0:49:25.350,0:49:27.430
that they know the[br]true definition

0:49:27.430,0:49:31.310
of a limit of a function[br]of, let's say, one variable?

0:49:31.310,0:49:34.910
The epsilon delta, the one your[br]dad told you about. [INAUDIBLE]

0:49:34.910,0:49:39.980
try to fool you when avoid it[br]in undergraduate education.

0:49:39.980,0:49:41.740
People try to avoid[br]the epsilon delta,

0:49:41.740,0:49:45.615
because they think the students[br]will never, never understand

0:49:45.615,0:49:50.470
it, because it's[br]such an abstract one.

0:49:50.470,0:49:51.844
I think I wasn't ready.

0:49:51.844,0:49:52.760
I wasn't smart enough.

0:49:52.760,0:49:58.015
I think I was 16 when I was[br]getting ready for some math

0:49:58.015,0:49:58.725
competitions.

0:49:58.725,0:50:02.810
And one professor taught me[br]the epsilon delta and said,

0:50:02.810,0:50:04.700
do you understand it?

0:50:04.700,0:50:07.310
My 16-year-old mind said, no.

0:50:07.310,0:50:08.860
But guess what?

0:50:08.860,0:50:10.546
Some other people[br]smarter than me,

0:50:10.546,0:50:12.410
they told me, when[br]you first see it,

0:50:12.410,0:50:16.690
you don't understand[br]it in any case.

0:50:16.690,0:50:19.950
So it takes a little bit[br]more time to sink in.

0:50:19.950,0:50:21.876
So the same idea.

0:50:21.876,0:50:25.140
As I'm getting closer and[br]closer and closer and closer

0:50:25.140,0:50:30.280
to an x0 with my x values[br]from anywhere around-- left,

0:50:30.280,0:50:35.480
right-- I have to pick an[br]arbitrary choice of points

0:50:35.480,0:50:39.940
going towards x0, I have to[br]be sure that at the same time,

0:50:39.940,0:50:45.210
the corresponding sequence[br]of values is going to L,

0:50:45.210,0:50:47.010
I can express that[br]in epsilon delta.

0:50:47.010,0:50:50.860


0:50:50.860,0:50:51.935
So we say that.

0:50:51.935,0:50:59.705


0:50:59.705,0:51:13.178
f of x has limit L[br]at x equals x0 if.

0:51:13.178,0:51:17.180


0:51:17.180,0:51:24.100
For every epsilon positive, any[br]choice of an epsilon positive,

0:51:24.100,0:51:25.230
there is a delta.

0:51:25.230,0:51:27.110
There exists-- oh, OK, guys.

0:51:27.110,0:51:28.430
You don't know the symbols.

0:51:28.430,0:51:31.290
I'll write it in English.

0:51:31.290,0:51:36.450
For every epsilon[br]positive, no matter

0:51:36.450,0:51:40.728
how small-- put[br]parentheses, because you

0:51:40.728,0:51:47.150
are just [? tired-- ?][br]no matter how small,

0:51:47.150,0:51:55.510
there exists a delta number[br]that depends on epsilon.

0:51:55.510,0:52:02.356


0:52:02.356,0:52:16.190
So that whenever x minus[br]x0 is less than delta,

0:52:16.190,0:52:33.800
this would imply[br]that f of x minus L,

0:52:33.800,0:52:37.098
that limit I taught you[br]about in absolute value,

0:52:37.098,0:52:38.562
is less than epsilon.

0:52:38.562,0:52:48.340


0:52:48.340,0:52:50.370
What does this mean?

0:52:50.370,0:52:55.200
I'm going to try[br]and draw something

0:52:55.200,0:52:58.441
that happens on a line.

0:52:58.441,0:53:00.370
So this is x0.

0:53:00.370,0:53:03.894
And these are my values of x.

0:53:03.894,0:53:05.060
They can come from anywhere.

0:53:05.060,0:53:08.760


0:53:08.760,0:53:12.032
And this is f of x.

0:53:12.032,0:53:16.700
And this is L. So it[br]says, no matter-- this

0:53:16.700,0:53:19.150
says-- this is an[br]abstract way of saying,

0:53:19.150,0:53:23.810
no matter how close, you see,[br]for every epsilon positive,

0:53:23.810,0:53:27.280
no matter how close[br]you get to the L.

0:53:27.280,0:53:30.600
I decide to be in this[br]interval, very tiny epsilon.

0:53:30.600,0:53:32.260
L minus epsilon.

0:53:32.260,0:53:35.840
L plus epsilon L. You give[br]me your favorite epsilon.

0:53:35.840,0:53:38.640
You say, Magdalena, pick[br]something really small.

0:53:38.640,0:53:42.180
Big epsilon to be 0.00001.

0:53:42.180,0:53:43.950
How about that?

0:53:43.950,0:53:47.660
Well, if I really[br]have a limit there,

0:53:47.660,0:53:54.290
an L at x0, that means that[br]no matter how much you shrink

0:53:54.290,0:53:57.540
this interval for me, you[br]can be mean and shrink it

0:53:57.540,0:53:59.320
as much as you want.

0:53:59.320,0:54:03.400
I will still find a[br]small interval around x0.

0:54:03.400,0:54:06.660


0:54:06.660,0:54:08.620
[? But ?] I will[br]still find the smaller

0:54:08.620,0:54:13.230
interval around x0, which is--[br]this would be x0 minus delta.

0:54:13.230,0:54:15.760
This would be x0 plus delta.

0:54:15.760,0:54:20.961
So that the image of this[br]purple interval fits inside.

0:54:20.961,0:54:22.150
You say, what?

0:54:22.150,0:54:25.840
So that the image of this[br]purple interval fits inside.

0:54:25.840,0:54:30.040
So f of x minus L, the distance[br]is still that, less than xy.

0:54:30.040,0:54:30.550
Yes, sir?

0:54:30.550,0:54:32.710
STUDENT: Where'd you[br]get epsilon [INAUDIBLE]?

0:54:32.710,0:54:34.440
MAGDALENA TODA: So[br]epsilon has to be

0:54:34.440,0:54:38.252
chose no matter how small.

0:54:38.252,0:54:39.837
STUDENT: [INAUDIBLE].

0:54:39.837,0:54:40.670
MAGDALENA TODA: Huh?

0:54:40.670,0:54:42.160
Real number.

0:54:42.160,0:54:46.470
So I'm saying, you should not[br]set the epsilon to be 0.0001.

0:54:46.470,0:54:47.770
That would be a mistake.

0:54:47.770,0:54:50.890
You have to think of that number[br]as being as small as you want,

0:54:50.890,0:54:54.590
infinitesimally small, smaller[br]than any particle in physics

0:54:54.590,0:54:57.330
that you are aware about.

0:54:57.330,0:54:59.880
And this is what I had the[br]problem understanding--

0:54:59.880,0:55:03.530
that notion of-- not[br]the notion of, hey, not

0:55:03.530,0:55:06.000
matter how close[br]I am, I can still

0:55:06.000,0:55:12.430
get something even smaller[br]around x0 that fits in this.

0:55:12.430,0:55:14.360
That's not what I[br]had the problem with.

0:55:14.360,0:55:18.316
The notion is to perceive[br]an infinitesimal.

0:55:18.316,0:55:21.515
Our mind is too limited[br]to understand infinity.

0:55:21.515,0:55:24.408
It's like trying[br]to understand God.

0:55:24.408,0:55:29.512
And the same limitation comes[br]with microscopic problems.

0:55:29.512,0:55:31.470
Yeah, we can see some[br]things on the microscope,

0:55:31.470,0:55:32.219
and we understand.

0:55:32.219,0:55:34.580
Ah, I understand I[br]have this bacteria.

0:55:34.580,0:55:36.040
This is staph.

0:55:36.040,0:55:37.460
Oh, my god.

0:55:37.460,0:55:43.350
But then there are molecules,[br]atoms, subatomic particles

0:55:43.350,0:55:46.875
that we don't understand,[br]because our mind is really

0:55:46.875,0:55:49.060
[? small. ?] Imagine[br]something smaller

0:55:49.060,0:55:50.900
than the subatomic particles.

0:55:50.900,0:55:54.520
That's the abstract notion[br]of infinitesimally small.

0:55:54.520,0:55:58.870
So I'm saying, if I really[br]have a limit L there,

0:55:58.870,0:56:03.470
that means no matter how small[br]I have this ball around it,

0:56:03.470,0:56:06.630
I can still find a[br]smaller ball that

0:56:06.630,0:56:10.001
fits-- whose image fits inside.

0:56:10.001,0:56:10.500
All right?

0:56:10.500,0:56:15.000
The same kind of definition--[br]I will try to generalize this.

0:56:15.000,0:56:19.660
Can you guys help me[br]generalize this limit notion

0:56:19.660,0:56:24.790
to the notion of function[br]of two variables?

0:56:24.790,0:56:29.040


0:56:29.040,0:56:41.400
So we say, that f of xy[br]has the limit L at x0y0.

0:56:41.400,0:56:44.970


0:56:44.970,0:56:50.710
What was x0y0 when I[br]talked about-- what

0:56:50.710,0:56:52.550
example did I give you guys?

0:56:52.550,0:56:55.050
Sine of 1 over x squared[br]plus y squared, right?

0:56:55.050,0:56:56.220
Something like that.

0:56:56.220,0:56:56.850
I don't know.

0:56:56.850,0:56:59.730
I said, think of 0, 0.

0:56:59.730,0:57:01.810
That was the given point.

0:57:01.810,0:57:03.540
It has to be a fixed couple.

0:57:03.540,0:57:07.870
So you think of the origin, 0,[br]0, as being as a fixed couple.

0:57:07.870,0:57:12.390
Or you think of the point 1,[br]0 as being as a fixed couple

0:57:12.390,0:57:14.761
in that plane you look at.

0:57:14.761,0:57:18.420
That is the fixed couple.

0:57:18.420,0:57:21.460
If-- now somebody[br]has to help me.

0:57:21.460,0:57:27.540
For every epsilon positive,[br]no matter how small,

0:57:27.540,0:57:30.690
that's where I have a problem[br]imagining infinitesimally

0:57:30.690,0:57:31.615
small.

0:57:31.615,0:57:34.650
There exists-- I no[br]longer have this problem.

0:57:34.650,0:57:37.310
But I had it enough[br]when I was in my 20s.

0:57:37.310,0:57:40.155
I don't want to go back to[br]my 20s and have-- I mean,

0:57:40.155,0:57:41.065
I would love to.

0:57:41.065,0:57:42.890
[LAUGHTER]

0:57:42.890,0:57:46.420
To go having vacations[br]with no worries and so on.

0:57:46.420,0:57:48.570
But I wouldn't like[br]to go back to my 20s

0:57:48.570,0:57:50.470
and have to relearn[br]all the mathematics.

0:57:50.470,0:57:50.970
Now way.

0:57:50.970,0:57:53.190
That was too much of a struggle.

0:57:53.190,0:58:00.310
There exists a delta positive[br]that depends on epsilon.

0:58:00.310,0:58:02.820
What does it mean,[br]depends on epsilon?

0:58:02.820,0:58:04.650
Because guys, imagine[br]you make this epsilon

0:58:04.650,0:58:05.860
smaller and smaller.

0:58:05.860,0:58:08.100
You have to make delta[br]smaller and smaller,

0:58:08.100,0:58:12.321
so that you can fit that[br]little ball in the big ball.

0:58:12.321,0:58:12.820
OK?

0:58:12.820,0:58:19.768
That depends on epsilon,[br]so that whenever-- now,

0:58:19.768,0:58:21.930
that is a big problem.

0:58:21.930,0:58:28.305
How do I say, distance between[br]the point xy and the point

0:58:28.305,0:58:29.310
x0y0?

0:58:29.310,0:58:32.130
Oh, my god.

0:58:32.130,0:58:37.310
This is distance between xy[br]and x0y0 is less than delta.

0:58:37.310,0:58:48.480
This would imply[br]that-- well, this

0:58:48.480,0:58:54.090
is a function with values in[br]R. This is in R. Real number.

0:58:54.090,0:58:55.470
So I don't have a problem.

0:58:55.470,0:58:57.460
I can use absolute value here.

0:58:57.460,0:59:11.330
Absolute value of f of[br]the couple xy minus L

0:59:11.330,0:59:14.580
is less than epsilon.

0:59:14.580,0:59:19.080
The thing is, can you[br]visualize that little ball,

0:59:19.080,0:59:20.990
that little disk?

0:59:20.990,0:59:22.310
What do I mean?

0:59:22.310,0:59:26.440
Being close, xy is me, right?

0:59:26.440,0:59:27.520
But I'm moving.

0:59:27.520,0:59:28.760
I'm the moving point.

0:59:28.760,0:59:30.350
I'm dancing around.

0:59:30.350,0:59:33.436
And [? Nateesh ?] is x0y0.

0:59:33.436,0:59:37.570
How do I say that I have[br]to be close enough to him?

0:59:37.570,0:59:38.600
I cannot touch him.

0:59:38.600,0:59:39.710
That's against the rules.

0:59:39.710,0:59:42.270
That's considered[br][INAUDIBLE] harassment.

0:59:42.270,0:59:45.800
But I can come as[br]close as I want.

0:59:45.800,0:59:49.210
So I say, the[br]distance between me--

0:59:49.210,0:59:52.100
I'm xy-- and[br][? Nateesh, ?] who is

0:59:52.100,0:59:58.264
fixed x0y0, has to be smaller[br]than that small delta.

0:59:58.264,1:00:00.796
How do I represent that[br]in plane mathematics?

1:00:00.796,1:00:02.004
STUDENT: Doesn't [INAUDIBLE]?

1:00:02.004,1:00:05.520


1:00:05.520,1:00:06.520
MAGDALENA TODA: Exactly.

1:00:06.520,1:00:09.480
So that delta has to[br]be small enough so

1:00:09.480,1:00:16.970
that the image of f at me minus[br]the limit is less than epsilon.

1:00:16.970,1:00:20.960
Now you understand why all[br]the other teachers avoid

1:00:20.960,1:00:22.820
talking about this[br][? one. ?] So I

1:00:22.820,1:00:27.840
want to get small enough-- not[br]too close-- but close enough

1:00:27.840,1:00:39.520
to him, so that my value--[br]I'm f of xy-- minus the limit,

1:00:39.520,1:00:42.170
the limit-- I have[br]a preset limit.

1:00:42.170,1:00:45.140
All around [? Nateesh, ?] I[br]can have different values,

1:00:45.140,1:00:47.490
no matter where I go.

1:00:47.490,1:00:51.060
My value at all these points[br]around [? Nateesh ?] have

1:00:51.060,1:00:54.510
to be close enough[br]to L. So I say,

1:00:54.510,1:00:57.550
well, you have to get[br]close enough to L.

1:00:57.550,1:00:59.490
Somebody presents me an epsilon.

1:00:59.490,1:01:02.350
Then I have to reduce my[br]distance to [? Nateesh ?]

1:01:02.350,1:01:03.990
depending to that epsilon.

1:01:03.990,1:01:07.959
Because otherwise,[br]the image doesn't fit.

1:01:07.959,1:01:09.000
It's a little bit tricky.

1:01:09.000,1:01:10.960
STUDENT: So is this like[br]the squeeze theorem kind of?

1:01:10.960,1:01:12.430
MAGDALENA TODA: It is[br]the squeeze theorem.

1:01:12.430,1:01:12.930
STUDENT: Oh, all right.

1:01:12.930,1:01:13.721
MAGDALENA TODA: OK?

1:01:13.721,1:01:18.580
So the squeezing-- I ball into[br]another [? ball ?] [? limit. ?]

1:01:18.580,1:01:21.030
This is why-- it's not[br]a ball, but it's a--

1:01:21.030,1:01:21.780
STUDENT: A circle.

1:01:21.780,1:01:22.655
MAGDALENA TODA: Disk.

1:01:22.655,1:01:23.720
A circle, right?

1:01:23.720,1:01:28.830
So how do we express[br]that in Calc 3 in plain?

1:01:28.830,1:01:31.005
This is the[br][? ingredient, ?] distance d.

1:01:31.005,1:01:33.630
So Seth, can you tell me what is[br]the distance between these two

1:01:33.630,1:01:34.790
points?

1:01:34.790,1:01:36.010
Square root of--

1:01:36.010,1:01:37.360
STUDENT: [INAUDIBLE].

1:01:37.360,1:01:41.750
MAGDALENA TODA: x minus[br]x0 squared plus y minus y0

1:01:41.750,1:01:43.154
squared.

1:01:43.154,1:01:45.383
Now shut up. [? And I ?][br]am talking to myself.

1:01:45.383,1:01:45.856
STUDENT: Must be[br]less than delta.

1:01:45.856,1:01:46.695
[LAUGHTER]

1:01:46.695,1:01:48.280
MAGDALENA TODA: Less than delta.

1:01:48.280,1:01:51.530
So instead of[br]writing this, I need

1:01:51.530,1:01:53.726
to write that I can[br]do that in my mind.

1:01:53.726,1:01:58.356


1:01:58.356,1:02:00.024
OK?

1:02:00.024,1:02:00.990
All right.

1:02:00.990,1:02:01.850
This is hard.

1:02:01.850,1:02:03.470
We need to sleep on that.

1:02:03.470,1:02:09.449
I have one or two more problems[br]that are less hard-- nah,

1:02:09.449,1:02:11.490
they are still hard, but[br]they are more intuitive,

1:02:11.490,1:02:14.570
that I would like to[br]ask you about the limit.

1:02:14.570,1:02:16.940
I'm going to give[br]you a function.

1:02:16.940,1:02:21.476
And we would have to visualize[br]as I get closer to a point

1:02:21.476,1:02:24.570
where I am actually going.

1:02:24.570,1:02:30.170
So I have this nasty[br]function, f of xy

1:02:30.170,1:02:34.960
equals xy over z[br]squared plus y squared.

1:02:34.960,1:02:39.020


1:02:39.020,1:02:44.600
And I'm saying, [INAUDIBLE][br]the point is the origin.

1:02:44.600,1:02:47.030
I choose the origin.

1:02:47.030,1:02:47.590
Question.

1:02:47.590,1:02:53.020
Do I have a limit that's--[br]do I have a limit?

1:02:53.020,1:02:54.620
Not [? really ?] for me.

1:02:54.620,1:03:02.480
Does f have a limit[br]at the origin?

1:03:02.480,1:03:06.400


1:03:06.400,1:03:09.820
You would have to imagine[br]that you'd draw this function.

1:03:09.820,1:03:13.145
And except you cannot draw, and[br]you really don't care to draw

1:03:13.145,1:03:13.780
it.

1:03:13.780,1:03:17.030
You only have to imagine that[br]you have some abstract graph--

1:03:17.030,1:03:19.050
z equals f of xy.

1:03:19.050,1:03:20.880
You don't care[br]what it looks like.

1:03:20.880,1:03:24.100
But then you take[br]points on the floor,

1:03:24.100,1:03:27.450
just like I did the exercise[br]with [? Nateesh ?] before.

1:03:27.450,1:03:30.820
And you get closer and[br]closer to the origin.

1:03:30.820,1:03:34.415
But no attention-- no[br]matter what path I take,

1:03:34.415,1:03:36.850
I have to get the same limit.

1:03:36.850,1:03:37.548
What?

1:03:37.548,1:03:46.680
No matter what path I take[br]towards [? Nateesh-- ?]

1:03:46.680,1:03:52.860
don't write that down-- towards[br][? z0y0, ?] I have to get

1:03:52.860,1:03:53.790
the same limit.

1:03:53.790,1:03:56.970


1:03:56.970,1:03:59.030
Do I?

1:03:59.030,1:04:04.030
Let's imagine with the[br]eyes of your imaginations.

1:04:04.030,1:04:07.410
And [? Nateesh ?][br]is the point 0, 0.

1:04:07.410,1:04:10.652
And you are aspiring to get[br]closer and closer to him

1:04:10.652,1:04:12.750
without touching him.

1:04:12.750,1:04:15.265
Because otherwise,[br]he's going to sue you.

1:04:15.265,1:04:18.120
So what do we have here?

1:04:18.120,1:04:19.350
We have different paths?

1:04:19.350,1:04:21.390
How can I get closer?

1:04:21.390,1:04:25.550
Either on this path[br]or maybe on this path.

1:04:25.550,1:04:28.210
Or maybe on this path.

1:04:28.210,1:04:31.590
Or maybe, if I had something[br]to drink last night-- which

1:04:31.590,1:04:35.115
I did not, because[br]after the age of 35,

1:04:35.115,1:04:37.032
I stopped drinking completely.

1:04:37.032,1:04:40.990


1:04:40.990,1:04:44.820
That's when I decided[br]I want to be a mom,

1:04:44.820,1:04:47.270
and I didn't want to[br]make a bad example.

1:04:47.270,1:04:50.450
So no matter what path you[br]take, you can make it wiggly,

1:04:50.450,1:04:52.030
you can make it[br]any way you want.

1:04:52.030,1:04:53.700
We are still approaching 0, 0.

1:04:53.700,1:04:55.925
You still have to[br]get the same limit.

1:04:55.925,1:05:00.170
Oh, that's tricky, because[br]it's also the same in life.

1:05:00.170,1:05:02.430
Depending on the path[br]you take in life,

1:05:02.430,1:05:05.450
you have different[br]results, different limits.

1:05:05.450,1:05:11.243
Now, what if I take the path[br]number one, number two, number

1:05:11.243,1:05:12.720
three possibility.

1:05:12.720,1:05:16.900
And number [? blooie ?][br]is the drunken variant.

1:05:16.900,1:05:22.120
That is hard to[br]implement in an exercise.

1:05:22.120,1:05:26.600
Imagine that I have[br]limit along the path one.

1:05:26.600,1:05:28.290
Path one.

1:05:28.290,1:05:34.680
xy goes to 0, 0 of xy over[br]x squared plus y squared.

1:05:34.680,1:05:36.970
Do you guys see what's[br]going to happen?

1:05:36.970,1:05:40.660
So I'm along the--[br]OK, here it is.

1:05:40.660,1:05:46.590
This line, right, this is[br]the x-axis, y-axis, z-axis.

1:05:46.590,1:05:48.530
What's special for the x-axis?

1:05:48.530,1:05:50.470
Who is 0?

1:05:50.470,1:05:52.700
STUDENT: x.

1:05:52.700,1:05:53.330
STUDENT: yz.

1:05:53.330,1:05:54.420
MAGDALENA TODA: y is 0.

1:05:54.420,1:05:57.400
So y is 0.

1:05:57.400,1:05:59.410
So y is 0.

1:05:59.410,1:06:00.360
Don't laugh at me.

1:06:00.360,1:06:03.470
I'm going to write like[br]that because it's easier.

1:06:03.470,1:06:06.770
And it's going to be[br]something like limit

1:06:06.770,1:06:13.738
when x approaches 0[br]of x over x squared.

1:06:13.738,1:06:15.690
STUDENT: It's 1/x.

1:06:15.690,1:06:17.950
MAGDALENA TODA: Times 0 up.

1:06:17.950,1:06:18.950
Oh, my god.

1:06:18.950,1:06:20.671
Is that-- how much is that?

1:06:20.671,1:06:21.170
STUDENT: 0.

1:06:21.170,1:06:21.290
STUDENT: 0.

1:06:21.290,1:06:22.039
MAGDALENA TODA: 0!

1:06:22.039,1:06:22.570
I'm happy.

1:06:22.570,1:06:23.980
I say, maybe I have the limit.

1:06:23.980,1:06:24.820
I have the limit 0.

1:06:24.820,1:06:27.120
No, never rush in life.

1:06:27.120,1:06:28.040
Check.

1:06:28.040,1:06:30.780
Experiment any other paths.

1:06:30.780,1:06:35.120
And it's actually very easy[br]to see where I can go wrong.

1:06:35.120,1:06:39.630
If I take the path number two,[br]I will get the same result.

1:06:39.630,1:06:41.470
You don't need a[br]lot of imagination

1:06:41.470,1:06:44.355
to realize, hey, whether[br]she does it for x

1:06:44.355,1:06:48.362
or does it for y, if she[br]goes along the 2, what

1:06:48.362,1:06:49.790
the heck is going to happen?

1:06:49.790,1:06:51.030
y is going to shrink.

1:06:51.030,1:06:53.150
x will always be 0.

1:06:53.150,1:06:56.850
Because this means[br]a point's like what?

1:06:56.850,1:06:58.850
0,1.

1:06:58.850,1:07:00.835
0, 1/2.

1:07:00.835,1:07:03.180
0, 1/n, and so on.

1:07:03.180,1:07:07.820
But plug them all in here,[br]I get 0, 1/n times 0.

1:07:07.820,1:07:08.670
It's still 0.

1:07:08.670,1:07:10.460
So I still get 0.

1:07:10.460,1:07:12.310
Path two.

1:07:12.310,1:07:15.470
When I approach my--[br]xt goes to 0, 0.

1:07:15.470,1:07:18.790
The poor [? Nateesh ?][br]is waiting for an answer.

1:07:18.790,1:07:20.950
I still get 0.

1:07:20.950,1:07:23.645
Let's take not the[br]drunken path, because I

1:07:23.645,1:07:25.520
don't know [? it unless ?][br]the sine function.

1:07:25.520,1:07:26.970
That is really crazy.

1:07:26.970,1:07:29.010
I'll take this one.

1:07:29.010,1:07:31.420
What is this one,[br]in your opinion?

1:07:31.420,1:07:32.970
Is that going to help me?

1:07:32.970,1:07:35.720
I don't know, but I[br]need some intuition.

1:07:35.720,1:07:39.550
Mathematicians need intuition[br]and a lot of patience.

1:07:39.550,1:07:42.110
So what is your intuition?

1:07:42.110,1:07:45.120
The one in the middle, I'm going[br]to start walking on that, OK,

1:07:45.120,1:07:46.542
until you tell me what it is.

1:07:46.542,1:07:47.500
STUDENT: y [INAUDIBLE].

1:07:47.500,1:07:49.540
MAGDALENA TODA: y equals[br]x is the first bisector

1:07:49.540,1:07:51.040
or the first quadrant.

1:07:51.040,1:07:54.580
And I'm very happy[br]I can go both ways.

1:07:54.580,1:07:55.850
y equals x.

1:07:55.850,1:07:56.460
x [INAUDIBLE].

1:07:56.460,1:08:07.130
So limit when x equals y,[br]but the pair xy goes to 0,0.

1:08:07.130,1:08:07.980
I'm silly.

1:08:07.980,1:08:10.978
I can say that,[br]well, Magdalena, this

1:08:10.978,1:08:15.540
is the pair xx,[br]because x equals what?

1:08:15.540,1:08:16.899
Let me plug them in.

1:08:16.899,1:08:19.085
So it's like two people.

1:08:19.085,1:08:20.555
x and y are married.

1:08:20.555,1:08:22.180
They are a couple, a pair.

1:08:22.180,1:08:24.330
They look identical.

1:08:24.330,1:08:26.380
Sometimes it happens.

1:08:26.380,1:08:28.439
Like twins, they[br]start looking alike,

1:08:28.439,1:08:30.819
dressing alike, and so on.

1:08:30.819,1:08:36.529
The x and the y have to[br]receive the same letter.

1:08:36.529,1:08:41.029
And you have to tell me what[br]in the world the limit will be.

1:08:41.029,1:08:43.818


1:08:43.818,1:08:44.359
STUDENT: 1/2.

1:08:44.359,1:08:45.520
MAGDALENA TODA: 1/2.

1:08:45.520,1:08:46.640
Oh, my god.

1:08:46.640,1:08:48.479
So now I'm deflated.

1:08:48.479,1:08:52.470
So now I realize that[br]taking two different paths,

1:08:52.470,1:08:57.578
I show that I have-- on[br]this path, I have 1/2.

1:08:57.578,1:09:00.069
On this path, I have 0.

1:09:00.069,1:09:01.149
I don't match.

1:09:01.149,1:09:02.710
I don't have an overall limit.

1:09:02.710,1:09:10.140
So the answer is,[br]no overall limit.

1:09:10.140,1:09:10.960
Oh, my god.

1:09:10.960,1:09:14.640
So what you need to[br]do, guys, is read

1:09:14.640,1:09:18.340
section 11.1 and section 11.2.

1:09:18.340,1:09:21.100
And I will ask you next[br]time-- and you can lie,

1:09:21.100,1:09:22.892
you can do whatever.

1:09:22.892,1:09:26.368
Did the book explain[br]better than me,

1:09:26.368,1:09:28.890
or I explain better[br]than the book?

1:09:28.890,1:09:31.790
This type of example when[br]the limit does not exist.

1:09:31.790,1:09:33.430
We are going to[br]see more examples.

1:09:33.430,1:09:37.783
You are going to see examples[br]where the limit does exist.

1:09:37.783,1:09:40.160
Now, one last thing.

1:09:40.160,1:09:46.640
When you have to compute limits[br]of compositions of functions

1:09:46.640,1:09:48.529
whose limit exist--[br]for example, you

1:09:48.529,1:09:58.290
know that limit is[br]xy goes to x0y0 of f

1:09:58.290,1:10:10.410
of xy [INAUDIBLE] limit[br]of xy go to x0y0 of gxy

1:10:10.410,1:10:13.730
is L-- L-- L-- M-- M.

1:10:13.730,1:10:25.200
How are you going to compute the[br]limit of alpha f plus beta g?

1:10:25.200,1:10:27.100
This is in the book.

1:10:27.100,1:10:32.650
But you don't need the[br]book to understand that.

1:10:32.650,1:10:34.320
You will already[br]give me the answer,

1:10:34.320,1:10:39.210
because this is the equivalent[br]thing to the function of one

1:10:39.210,1:10:41.320
variable thing in Calc 1.

1:10:41.320,1:10:43.670
So if you would only[br]have f of x or g of x,

1:10:43.670,1:10:45.410
it would be piece of cake.

1:10:45.410,1:10:46.501
What would you say?

1:10:46.501,1:10:47.375
STUDENT: [INAUDIBLE].

1:10:47.375,1:10:48.291
MAGDALENA TODA: Right.

1:10:48.291,1:10:54.080
Alpha times L plus beta[br]times M. Can you also

1:10:54.080,1:10:55.379
multiply functions.

1:10:55.379,1:10:55.920
Yes, you can.

1:10:55.920,1:11:07.160
Limit of fg as xy goes[br]to x0 or y0-- will be LM.

1:11:07.160,1:11:09.750
How about-- now I'm going to[br]jump to conclusion, hoping

1:11:09.750,1:11:13.170
that you are going to catch me.

1:11:13.170,1:11:15.652
You are going to catch[br]me, and shout at me,

1:11:15.652,1:11:18.410
and say, ooh, pay[br]attention, Magdalena,

1:11:18.410,1:11:21.560
you can make a mistake there.

1:11:21.560,1:11:26.430
I say it's L/M when I do[br]the division rule, right?

1:11:26.430,1:11:28.222
Where should I pay attention?

1:11:28.222,1:11:29.761
STUDENT: M [INAUDIBLE].

1:11:29.761,1:11:31.010
MAGDALENA TODA: Pay attention.

1:11:31.010,1:11:38.740
Sometimes you can[br]have the-- right?

1:11:38.740,1:11:45.120
And this also has[br]to exist as well.

1:11:45.120,1:11:46.560
STUDENT: [INAUDIBLE].

1:11:46.560,1:11:51.070
MAGDALENA TODA: So one[br]last-- how many minutes

1:11:51.070,1:11:53.810
have I spent with you?

1:11:53.810,1:11:57.926
I've spent with you a long[br]number of hours of my life.

1:11:57.926,1:11:58.800
No, I'm just kidding.

1:11:58.800,1:12:04.234
So you have one hour and[br]15, a little bit more.

1:12:04.234,1:12:05.400
Do I have a little bit more?

1:12:05.400,1:12:05.900
Yes.

1:12:05.900,1:12:07.990
I have 15 minutes.

1:12:07.990,1:12:08.490
I have--

1:12:08.490,1:12:09.615
STUDENT: So we get out at--

1:12:09.615,1:12:10.450
[INTERPOSING VOICES]

1:12:10.450,1:12:11.241
MAGDALENA TODA: 50.

1:12:11.241,1:12:12.514
Five more minutes.

1:12:12.514,1:12:15.360
OK.

1:12:15.360,1:12:20.655
So I want to ask you what[br]you remember about some

1:12:20.655,1:12:25.090
of your friends, the trig[br]functions involved in limits.

1:12:25.090,1:12:28.030


1:12:28.030,1:12:32.000
Why did we study[br]limits at the point

1:12:32.000,1:12:34.290
where the function's[br]not defined?

1:12:34.290,1:12:35.430
Well, to heck with it.

1:12:35.430,1:12:36.030
We don't care.

1:12:36.030,1:12:37.950
The function is[br]not defined at 0.

1:12:37.950,1:12:40.076
But the limit is.

1:12:40.076,1:12:42.916
And nobody showed you how[br]to do the epsilon delta

1:12:42.916,1:12:44.255
to show anything like that.

1:12:44.255,1:12:48.526


1:12:48.526,1:12:49.987
OK.

1:12:49.987,1:12:52.422
Can you do that[br]with epsilon delta?

1:12:52.422,1:12:57.790


1:12:57.790,1:13:00.352
Actually, you can do[br]everything with epsilon delta.

1:13:00.352,1:13:02.310
But I'm not going to give[br]you any extra credit.

1:13:02.310,1:13:07.538
So I trust you that[br]you remember that.

1:13:07.538,1:13:08.900
1!

1:13:08.900,1:13:10.815
How about-- let me-- OK.

1:13:10.815,1:13:11.767
I am so proud of you.

1:13:11.767,1:13:12.850
Let me challenge you more.

1:13:12.850,1:13:14.690
Let me challenge you more.

1:13:14.690,1:13:17.950
Tangent of ax over bx.

1:13:17.950,1:13:19.500
x go to 0.

1:13:19.500,1:13:22.110
I asked this to a girl[br]from Lubbock High.

1:13:22.110,1:13:23.450
She was in high school.

1:13:23.450,1:13:25.390
She knew the answer.

1:13:25.390,1:13:28.300
STUDENT: Oh, I can't disappoint[br]everybody in getting this.

1:13:28.300,1:13:31.476
STUDENT: Is it 1/a?

1:13:31.476,1:13:32.350
Oh, I can't remember.

1:13:32.350,1:13:33.860
MAGDALENA TODA: Tell me[br]what to do to be smart.

1:13:33.860,1:13:34.360
Right?

1:13:34.360,1:13:37.530
I have to be doing[br]something smart.

1:13:37.530,1:13:40.050
She-- can you give me hint?

1:13:40.050,1:13:41.779
I'm your student[br]and you say, well--

1:13:41.779,1:13:42.320
STUDENT: ba--

1:13:42.320,1:13:44.062
STUDENT: It's 0.

1:13:44.062,1:13:45.447
STUDENT: It's [INAUDIBLE].

1:13:45.447,1:13:46.780
MAGDALENA TODA: Um, it's a what?

1:13:46.780,1:13:48.090
STUDENT: b/a?

1:13:48.090,1:13:49.590
MAGDALENA TODA: I'm[br]not [INAUDIBLE].

1:13:49.590,1:13:51.080
I don't think so.

1:13:51.080,1:13:52.730
So what should I do?

1:13:52.730,1:13:58.070
I should say, instead of[br]bx-- that drives me nuts.

1:13:58.070,1:14:00.390
This goes on my nerves-- bx.

1:14:00.390,1:14:03.550
Like, maybe I go on your[br]nerves. bx is ax, right?

1:14:03.550,1:14:06.745
If it were ax, I would[br]be more constructive,

1:14:06.745,1:14:09.270
and I knew what to do.

1:14:09.270,1:14:13.190
I say replace bx with[br]ax, compensate for it,

1:14:13.190,1:14:15.120
and divide by bx.

1:14:15.120,1:14:17.880
And I was trying to[br]explain that to my son,

1:14:17.880,1:14:23.550
that if you have a fraction[br]a/b, and then you write a/n

1:14:23.550,1:14:26.695
times n/b, it's the same thing.

1:14:26.695,1:14:28.555
Gosh, I had the[br]problem with him.

1:14:28.555,1:14:33.310
And then I realized that he[br]didn't do simplifications

1:14:33.310,1:14:34.870
in school.

1:14:34.870,1:14:41.180
So it took a little more[br]hours to explain these things.

1:14:41.180,1:14:42.590
This is fourth grade.

1:14:42.590,1:14:45.150
I think I remember doing[br]that in fourth grade.

1:14:45.150,1:14:47.430
Third grade, actually.

1:14:47.430,1:14:50.360
So these two guys disappear.

1:14:50.360,1:14:53.790
I haven't changed[br]my problem at all.

1:14:53.790,1:14:57.920
But I've changed the status,[br]the shape of my problem

1:14:57.920,1:15:01.340
to something I can mold,[br]because this goes to somebody,

1:15:01.340,1:15:02.680
and this goes to somebody else.

1:15:02.680,1:15:05.170
Who is this fellow?

1:15:05.170,1:15:07.620
It's a limit that's[br]a constant-- a/b.

1:15:07.620,1:15:09.260
Who is this fellow?

1:15:09.260,1:15:09.760
STUDENT: 1.

1:15:09.760,1:15:10.840
MAGDALENA TODA: 1.

1:15:10.840,1:15:15.660
Because tangent of x/x as x[br]goes to 0 goes to 1 exactly

1:15:15.660,1:15:16.160
like that.

1:15:16.160,1:15:22.140
So limit of sine x over cosine[br]x, that's tangent, right?

1:15:22.140,1:15:23.290
Over x.

1:15:23.290,1:15:25.020
You do it exactly the same.

1:15:25.020,1:15:32.410
It's limit of sine x/x[br]times 1 over cosine x.

1:15:32.410,1:15:34.396
That's how we did[br]it in high school.

1:15:34.396,1:15:35.025
This goes to 1.

1:15:35.025,1:15:36.640
This goes to 1.

1:15:36.640,1:15:37.390
So it's 1.

1:15:37.390,1:15:39.190
So thank you, this is 1.

1:15:39.190,1:15:43.286
I know I took a little more time[br]to explain than I wanted to.

1:15:43.286,1:15:46.130
But now you are grown up.

1:15:46.130,1:15:49.260
In two minutes, you are[br]going to be finishing

1:15:49.260,1:15:50.766
this section, more or less.

1:15:50.766,1:15:54.640
What if I put a function[br]of two variables,

1:15:54.640,1:15:57.790
and I ask you what[br]the limit will be,

1:15:57.790,1:16:01.290
if it's the same[br]type of function.

1:16:01.290,1:16:03.490
So you say, oh, Magdalena,[br]what you doing to us?

1:16:03.490,1:16:05.240
OK, we'll see it's fun.

1:16:05.240,1:16:06.010
This one's fun.

1:16:06.010,1:16:07.850
It's not like the one before.

1:16:07.850,1:16:11.080
This one is pretty beautiful.

1:16:11.080,1:16:12.770
It's nice to you.

1:16:12.770,1:16:14.950
It exists.

1:16:14.950,1:16:16.590
xy goes to 0, 0.

1:16:16.590,1:16:19.745
So you have to imagine[br]some preferable function

1:16:19.745,1:16:22.070
in abstract thinking.

1:16:22.070,1:16:24.720
And you want it in[br]a little disk here.

1:16:24.720,1:16:31.580
And xy, these are all points[br]xy close enough to 0, 0,

1:16:31.580,1:16:34.030
in the neighborhood of 0, 0.

1:16:34.030,1:16:34.813
OK.

1:16:34.813,1:16:37.480
What's going to happen as[br]you get closer and closer

1:16:37.480,1:16:40.380
and closer and closer with[br]tinier and tinier and tinier

1:16:40.380,1:16:43.880
disks around 0, 0?

1:16:43.880,1:16:47.590
You're going to shrink so much.

1:16:47.590,1:16:49.290
What do you think[br]this will going to be,

1:16:49.290,1:16:50.750
and how do I prove it?

1:16:50.750,1:16:52.130
STUDENT: [INAUDIBLE].

1:16:52.130,1:16:53.966
MAGDALENA TODA: Who said it?

1:16:53.966,1:16:55.950
You, sir? [INAUDIBLE][br]going to go to 1.

1:16:55.950,1:16:57.770
And he's right.

1:16:57.770,1:17:00.940
He has the intuition.

1:17:00.940,1:17:03.204
A mathematician will[br]tell you, prove it.

1:17:03.204,1:17:04.620
STUDENT: Um, well,[br]let's see here.

1:17:04.620,1:17:06.432
MAGDALENA TODA: Can you prove?

1:17:06.432,1:17:09.897
STUDENT: You could use[br]the right triangle proof,

1:17:09.897,1:17:11.980
but that would probably[br]take way more [INAUDIBLE].

1:17:11.980,1:17:12.940
MAGDALENA TODA: x and[br]y are independent.

1:17:12.940,1:17:13.680
That's the problem.

1:17:13.680,1:17:15.721
They are married, but they[br]are still independent.

1:17:15.721,1:17:17.220
It's a couple.

1:17:17.220,1:17:20.920
However, we can use[br]polar coordinates.

1:17:20.920,1:17:22.775
Why is polar coordinates?

1:17:22.775,1:17:28.950
Well, in general, if we[br]are in xy, it's a pair.

1:17:28.950,1:17:30.585
This is r, right?

1:17:30.585,1:17:33.590
So rx is r cosine theta.

1:17:33.590,1:17:35.443
y is r sine theta.

1:17:35.443,1:17:37.442
And I can get closer and[br]closer to the original.

1:17:37.442,1:17:38.680
I don't care.

1:17:38.680,1:17:41.310
What happens about x[br]squared plus y squared,

1:17:41.310,1:17:43.040
this is r squared.

1:17:43.040,1:17:44.275
And r is a real number.

1:17:44.275,1:17:47.440
And as you walk closer[br]and closer to the original

1:17:47.440,1:17:52.800
without touching it,[br]that r goes to 0.

1:17:52.800,1:17:53.850
It shrinks to 0.

1:17:53.850,1:17:58.260
So that r squared goes[br]to 0 but never touches 0.

1:17:58.260,1:18:04.310
So this becomes limit as r goes[br]to 0, the radius of that disk

1:18:04.310,1:18:06.291
goes to 0.

1:18:06.291,1:18:10.540
Sine of r squared[br]over r squared.

1:18:10.540,1:18:13.870
But r squared could be replaced[br]by the real function, t,

1:18:13.870,1:18:17.425
by the real parameter,[br]lambda, by whatever you want.

1:18:17.425,1:18:19.440
So then it's 1.

1:18:19.440,1:18:23.390
And then Alexander was right.

1:18:23.390,1:18:26.207
He based it on, like,[br]observation, intuition,

1:18:26.207,1:18:27.040
everything you want.

1:18:27.040,1:18:28.830
It was not a proof.

1:18:28.830,1:18:32.470
On a multiple-choice exam,[br]he would be a lucky guy.

1:18:32.470,1:18:34.301
I don't want you to prove it.

1:18:34.301,1:18:36.950
But if I want you to[br]prove it, you have to say,

1:18:36.950,1:18:39.530
Magdalena, I know[br]polar coordinates,

1:18:39.530,1:18:41.930
and so I can do it.

1:18:41.930,1:18:45.460
And one last question for today.

1:18:45.460,1:18:49.570
Guys, I'm asking you,[br]limit xy goes to 0, 0.

1:18:49.570,1:18:53.600
You will see some of these in[br]your WeBWorK for Chapter 11

1:18:53.600,1:18:56.890
that's waiting for[br]you, homework 3.

1:18:56.890,1:19:03.200
Tangent of 2 x squared[br]plus y squared over 3

1:19:03.200,1:19:06.446
x squared plus y squared.

1:19:06.446,1:19:09.320
What is that?

1:19:09.320,1:19:10.278
2/3.

1:19:10.278,1:19:11.021
STUDENT: 2/3.

1:19:11.021,1:19:12.520
MAGDALENA TODA: Am[br]I asking you why?

1:19:12.520,1:19:13.540
No, enough.

1:19:13.540,1:19:14.440
OK.

1:19:14.440,1:19:17.030
[INAUDIBLE] I gave[br]you everything

1:19:17.030,1:19:20.920
you need to show that.

1:19:20.920,1:19:23.616
x squared plus y squared,[br]again, is Mr. r squared.

1:19:23.616,1:19:24.592
It's OK.

1:19:24.592,1:19:29.472
I taught you that.[br]a/b. a is 2, b is 3.

1:19:29.472,1:19:30.448
Is it hard?

1:19:30.448,1:19:31.912
It is not easy, for sure.

1:19:31.912,1:19:35.328
Calc 3 is really difficult[br]compared to other topics

1:19:35.328,1:19:37.780
you are probably taking.

1:19:37.780,1:19:40.960
But I hope that I[br]can convince you

1:19:40.960,1:19:45.440
that math, although[br]difficult, [INAUDIBLE] Calc 3,

1:19:45.440,1:19:48.230
is also fun.

1:19:48.230,1:19:49.930
OK?

1:19:49.930,1:19:50.760
All right.

1:19:50.760,1:19:55.083
So I need attendance and[br]I need the extra credit.

1:19:55.083,1:19:56.208
STUDENT: Yeah, [INAUDIBLE].

1:19:56.208,1:19:59.080


1:19:59.080,1:20:01.920
MAGDALENA TODA: Before[br]you go, you need to sign.

1:20:01.920,1:20:04.439