TTU Math2450 Calculus3 Sec 11.1 and 11.2

Intoduction to functions of several varibles; Limit of a function of several variables

MAGDALENA TODA:
We have any people

who finished the
extra credit and are

willing to give it to me today?

I mean, you don't have to.

That's why it's
called extra credit.

But I think it's good
for extra practice

and for the extra points.

So hold on to it if you cannot
give it to me right now.

And I'll collect it at
the end of the class.

Today's a big day.

We are starting a new
chapter, Chapter 11.

So practically, we are
going to discuss all

through this chapter functions
of several variables.

And you are going to
ask me, wait a minute,

why do we need functions
in more than one variable?

Well, we are all functions
of many variables.

I was freezing outside,
and I was thinking,

I'm a function of
everything I eat.

I'm a function of the
temperature outside.

Almost everything in
our body is a function

of hundreds of factors,
actually, thousands.

But we don't have the time
and the precise information

to analyze all the
parameters that

affect our physical
condition every day.

We are getting there.

I'm going to give you
just the simple case.

So instead of y equals f of x
type of function, one variable,

we are going to look at
functions of the types z

equals f of xy.

Can I have many more?

Absolutely I can.

And that's kind of
the idea, that I

can have a function
in an-- let me

count-- n plus 1
dimensional space

as being of the type xn plus
1 equals f of x1, x2, x3, x4.

Somebody stop me. xn.

Right.

I have many variables.

And that is a problem
that affects everything.

Our physical world is
affected by many parameters.

In engineering
problems, you've already

seen some of these parameters.

Can you give me some
examples of parameters you've

seen in engineering classes?

x1, x2, x3 could be the
Euclidean coordinates, right,

for the three [? space. ?] But
besides those, there was an x4.

It could be?

Time.

Excellent, [INAUDIBLE].

More than that.

I want more.

I want x5.

Who can think of
another parameter

that affects physical processes
or chemical reactions?

Yes, sir?

STUDENT: Temperature.

MAGDALENA TODA: Temperature.

Excellent.

Another very good idea.

How about x6?

I'm running out of imagination.

But you have a lot more
information than me.

Pressure.

Maybe I'm studying a process of
somewhere up in the atmosphere.

Maybe I'm in an
airplane, and then it

becomes a little bit
more complicated,

because I hate the way
cabins are pressurized.

I can feel very uneasy.

My ears pop and so on.

We can be in the
bottom of the ocean.

There are very many
physical parameters

that affect physical
processes, chemical processes,

biological processes.

I don't know if this is
fortunate or unfortunate,

but I think that was
the key to the existence

of the universe in the first
place-- all these parameters.

OK.

Let me give you a simple
example of a function that

looks like a graph.

This is a graph.

And you say, wait a
minute, wait a minute.

Can I have functions of several
variables that cannot be

represented as graphs?

Yeah.

Absolutely.

We will talk about that
a little bit later.

So if I were to give you
an example that you've

seen before, and I would say,
give me a good approximation

to a valley that is actually
a quadric that we love and we

studied before for
the first time.

That quadric is a
beautiful object, a valley.

Any imagination,
recognition, recollection?

I know I scared
you enough for you

to know the equations of those
quadrics since some of you

told me we watched
all the videos,

we read all the stinking
book like never before.

That was kind of the idea.

I didn't want to scare you away.

I wanted to scare you
enough to read the book

and watch the videos.

And I'm talking about a valley
that you've seen before.

Many of you told me you like
the University of Minnesota

website that has the
quadric gallery of quadrics.

So you've met this guy before.

They show the general equation.

But I said I like the
circular paraboloid.

So they talk about
elliptic paraboloid.

Which one do you think I prefer?

The circular paraboloid.

Give me an example of
a circular paraboloid.

STUDENT: A flashlight?

Inside.

MAGDALENA TODA: The expression,
the mathematical equation.

STUDENT: Oh, sorry.

So it would be x
squred plus y squared.

MAGDALENA TODA: Very good.

That's exactly
what I had in mind.

Of course, it could be
over something, over r.

All right.

That's my favorite.

Now, if I put the flashlight
in here just like one of you

said, or the sign on
top of the z-axis.

Then I'm going to look at
the various-- we discussed

that a little bit before.

So various horizontal
planes, they're going to cut.

They're going to cut the
surface in different circles,

upon different circles.

We love them, and we use them.

And what did we do
with them last time?

We projected them on the floor.

And by floor, I mean the what?

By floor, I mean the xy plane.

Plus this xy plane.

I label it like you like it.

You said you like
it when I label it,

so you have the
imagination of a table.

This is x and y and z.

And so I gave you an example
of a graph cut in with z equals

constant positive or negative?

Well, it better be positive,
because for negative, I

have no solutions.

Positive or zero.

Well, for zero I have
a degenerate conic.

A degenerate conic
could be a point,

or it could be a bunch of lines.

In this case, all those
circles-- doo-doo-doo-doo-doo--

a family of one parameter,
family of circles.

Like the ones that
is-- a dolphin

is now doing that
in San Antonio,

San Diego-- to take
those old circles

from the bottom of the sea,
and bring them different sizes,

and put them together.

So they are very smart.

I love dolphins.

So we'll see 0
[INAUDIBLE] get a point.

That's still a conic.

It's a degenerate circle.

Do you realize
that's a limit case?

It's really beautiful.

You know what I mean?

Circle on top of a circle
on top of a circle,

smaller and smaller.

All right.

So good.

If I create shadows--
because that's

why you guys wanted
the source of light

on top-- of the projections
of these circles,

I'm going to have them
at the same color.

But dotted lines because I
think the book doesn't show them

dotted.

But on my way here,
I was thinking,

I think it's more beautiful
if I draw them dotted.

And how big is this circle?

Well, god knows.

I'm going to make a purple
circle that is, of course,

equal in size, equal in radius
with the original purple

circle.

So the dotted purple circle,
that's on the ground--

is just the projection of
the continuous purple circle.

It's identical in radius.

So for the family of
circles on the surface,

I have a family of projections
on the ground in the xy plane.

And such a family of
projections represents

a bunch of level curves.

We call this family
of level curves.

OK?

All right.

So if you think about it,
what are level curves?

You view them as being in plane.

Oh, my god.

So I should view them as a bunch
of points, a set of points.

If I make it like
that, that means

I view this as an
element of what?

Element of the xy plane,
right, with the property

that f of x and y is a constant.

OK?

In my case, I have a
[INAUDIBLE] constant.

In general, I have an
arbitrary real constant.

That's a level curve for
level C, for the level

C called the level, or altitude
would be the same thing.

So have you seen these
guys in geography?

What in the world are these
level curves in geography?

STUDENT: [INAUDIBLE]
show the slope

of a-- the steepness of a hill.

MAGDALENA TODA: You've
seen topographical maps.

And I'm going to try
and draw one of them.

I don't know, guys,
how-- excuse me.

I'm not very good
today at drawing.

But I'll do my best.

It could be a temperature
map or pressure map.

[INAUDIBLE] or whatever.

Now I'll say, this is
going to go-- well,

I cannot draw the
infinite family.

I have a one-parameter family.

And then I'll-- I'm dreaming of
the sea, summer break already.

You see what I'm doing.

Do you know what I'm doing?

That means I'm dreaming of the
different depths of the sea.

So for every such broad
line, I have the same depth.

The same altitude for
every continuous rule.

The same depth
for every-- so OK.

I'm not going to swim
too far, because that's

where the sharks are.

And I cannot draw the sharks,
but I ask you to imagine them.

It's fundamental in
a calculus class.

So somewhere here
I'm going to have--

what's the deepest--
guys, what's

the deepest point in that?

[? STUDENT: 11,300. ?]

MAGDALENA TODA: And
do you know the name?

I know the--

STUDENT: Mariana Trench.

MAGDALENA TODA: Mariana Trench.

STUDENT: Trench.

MAGDALENA TODA: All right.

So these topographical
are full of curves.

These are level curves.

So you didn't know,
but there is a lot

of mathematics in geography.

And there is a
lot of mathematics

in-- oh, you knew it.

When you watch the
weather report,

that's all mathematics, right?

It shows you the distribution
of temperatures everyday.

That is what we can
[INAUDIBLE] also

care about other functions
of several parameters, right?

And those functions could
be pressure, wind, whatever.

OK.

Speed of the wind.

Something like that.

I did not dare to
look at the prediction

of the weather for this place.

This place used to
be a beautiful place.

300 days of the
year of sunshine.

Not anymore.

So there is something
fishy in Denmark

and also something
fishy in [INAUDIBLE].

The world is changing.

So if you don't believe in
global warming, think again,

and global cooling, think again.

All right.

So unfortunately,
I am afraid still

to look at the temperatures
for the next few days.

But--

STUDENT: It's going to
be 80 degrees on Tuesday.

MAGDALENA TODA: Really? [?

Well, see, I should
have looked at it. ?]

[LAUGHTER]

I should gather the
courage, because I

knew-- when I was
interviewed here

for assistant professor,
gosh, I was young.

2001.

And my interview
was in mid-February.

And birds were chirping, it was
blue skies, beautiful flowers

everywhere on campus.

And I love the campus.

OK.

Give me an example of a surface
that cannot be represented

as a graph in its
entirety as a whole graph.

You gave me that before,
and I was so proud of you.

It was a--

[LAUGHS]

What kind of surface
am I trying to mimic?

STUDENT: A saddle.

MAGDALENA TODA: That
can be actually a graph.

That's a good
example of a graph.

A saddle.

But give me an example
of a non-graph that

is given as an implicit form.

So graph or explicit
is the same thing.

z equals f of xy.

Give me a non-graph.

One of you said it.

x squared plus y squared
plus z squared equals 1.

Why is this not a graph?

Not a graph.

Why is this not a graph?

STUDENT: [INAUDIBLE].

When you move it over to
1, you can't actually--

MAGDALENA TODA: You
cannot but you can cut it.

You can take a
sword and-- I'm OK.

I don't want to think about it.

So z is going to be two graphs.

So I can split this surface
even in a parametric form

as two different graphs.

Different graphs.

If I cut along-- I have this
orange, or sphere, globe.

And I cut it along
a great circle.

It doesn't have
to be the equator.

But you have to
imagine something

like the world and the equator.

This is kind of in
the unit sphere.

Today I drank enough
coffee to draw better.

Why don't I draw better?

I have no idea.

So that's not bad, though.

OK.

So that's the unit sphere.

What does it mean?

It means it has radius how much?

STUDENT: 1.

MAGDALENA TODA: 1.

Radius 1, and we
are happy about it.

And it has two graphs.

It's not one graph,
it's two graphs.

So this is called
implicit equation.

This is your lab
from-- I was chatting

with-- instead of
studying last night,

I was chatting with
you at midnight.

And one of you said, if I had
something I hated in calculus,

it was the implicit
differentiation.

And I know this is
your weak point.

So we'll do a lot of
implicit differentiation,

so you become more comfortable.

Usually we have one exercise in
this differentiation at least

on the final.

So this is an implicit equation.

And z is going to be
two graphs-- 1 minus x

squared minus y squared.

So I have, like, two charts,
two different charts.

OK.

The upper hemisphere--
I'm talking geography,

but that's how we talk
in geometry as well.

So geography right
now is like geometry.

I have a north pole.

Somebody quickly give me the
coordinates of the north pole.

STUDENT: 0, 0, 1.

MAGDALENA TODA: 0, 0, 1.

Thank you, Brian.

0, 0, 1.

How about the south pole?

STUDENT: 0, 0, minus 1.

MAGDALENA TODA: 0, 0, minus 1.

And write yourself a
note, because as you know,

I'm very absent-minded
and I forget

what I eat for lunch and so on.

Remind me to talk
to you sometime

at the end of the chapter
about stereographic projection.

It's a very important
mathematical notion

that also has to do a
little bit with geography.

But it's a one-to-one
correspondence

between a certain
part of a sphere

and a certain huge
part of a plane.

Now, we're not going
to talk about that now,

because that's not [INAUDIBLE].

That's a little bit
harder [INAUDIBLE].

You guys should now
see this line, right?

This should be beyond--
in the twilight zone,

behind the sphere.

OK?

So you don't see it.

And who is this? z equals 0.

And so this green
fellow should be

the circle x squared
plus y squared equals 1

in the xy plane.

Good.

So I have two graphs.

Now, if I were to ask
you, what is the domain

and the range of the function?

I'm going to erase
the whole thing.

What is the domain and the range
of the related function, z,

which gives the
upper hemisphere?

Upper hemisphere.

It's a graph.

And square root of 1 minus
x squared minus y squared.

You may stare at
it until tomorrow.

It's not hard to figure
out what I mean by domain

and range of such a function.

You are familiar
with domain and range

for a function of one variable.

For most of you,
that's a piece of cake.

That was even
pre-calc wasn't it?

It was in Calc 1.

So most of you had
algebra and pre-calc.

Now, what is the domain
of such a function?

Domain of definition has to be a
set of points, x and y in plane

for which the
function is defined.

If the function is
impossible to be defined

for a certain pair, x, y,
you kick that couple out

and you say, never come back.

Right?

So what I mean by domain is
those couples that we hate.

Who we hate?

The couples x, y for which x
squared plus y squared is how?

What existence condition do I--

STUDENT: [INAUDIBLE].

MAGDALENA TODA: Yeah.

You see this guy
under the square root

has to be positive or 0.

Right?

Otherwise, there is no
square root in real numbers.

That's going to be
in imaginary numbers,

and you can take
a walk, because we

are in real calculus
in real time as well.

So x squared plus y
squared must be how?

Less than or equal to 1.

We call that a certain name.

This is called a
closed unit disk.

Please remember, I'm teaching
you a little bit more

than a regular Calc 3 class.

They will never
make a distinction.

What's closing with this?

What's opening with this?

Everything will come
into place when you

move on to advanced calculus.

If I don't take the boundary--
so everything inside the disk

except for the boundary, I have
to put strictly less than 1.

That's called open unit disk.

For advanced calculus,
this is [INAUDIBLE].

All right.

This is just a parentheses.

My domain is the closed one.

What is the range?

The range is going to be--

STUDENT: [INAUDIBLE].

MAGDALENA TODA: The altitude
starts having values from--

STUDENT: Negative 1 to 1.

STUDENT: 0 to 1.

MAGDALENA TODA: So I'm 0 to 1.

I'll only talk about
the upper hemisphere.

I should even erase,
because I don't want it.

So say it again, guys.

STUDENT: 0 to 1.

MAGDALENA TODA: 0.

Open or closed?

STUDENT: Open.

STUDENT: Closed.

STUDENT: Closed, closed.

MAGDALENA TODA: Closed to?

STUDENT: 1 closed.

MAGDALENA TODA: 1 closed.

Yes.

Because that is the north pole.

I've been meaning to
give you this example.

And give me the other example
for the lower hemisphere.

What's different?

The same domain?

STUDENT: It ranges from--

STUDENT: Negative 1.

STUDENT: Negative 1 to 0.

MAGDALENA TODA: Closed
internal, right?

When we include the endpoints,
we call that closed interval.

It has a certain
topological sense.

You haven't taken
topology, but very soon,

if you are a math major, or
you are a double major, or some

of you even-- they want to
learn more about topology,

you will learn what an open
set is versus a closed set.

Remember we called this closed.

This is open.

And if it's closed here and
open there, it's neither.

OK?

Don't say anything about that.

OK.

To be closed, it has to be
containing both endpoints.

I'm going to erase this.

And this was, of course, 11.1.

We are in the middle of it.

In 11.1, one of you gave me a
beautiful graph to think about.

And I'm going to give
you something to do,

because I don't want
you to get lazy.

I'm very happy you came
up with the saddle.

All right.

We drew such a saddle.

And I did my best,
but it's not hard.

It's not easy to draw saddle.

When I am looking at the
coordinates, x, y, z,

I have z equals minus y
squared will look down.

Maybe I made it too fat.

I'm really sorry.

And down.

This continues.

OK?

And then what other thing
did I want to point out?

I want to point out--
do you see this?

This should look a
little bit more round.

It doesn't look
round enough here.

STUDENT: Your'e drawing
a saddle, right?

MAGDALENA TODA: No, I'm
drawing just the section

z equals minus y squared.

So I took x to be 0.

And the purple line
should be on this wall.

I know you guys have
enough imagination.

So this is going to
be z equals minus y

squared drawn on yz wall.

I've done this before,
but I'm just reviewing.

What if it's y0?

Then I have to
draw on that wall.

And I have to draw beautifully,
which I am not-- don't always--

I can't always do.

But I'll try.

I have z equals x squared
drawn on that wall.

If I start drawing,
I'll get fired.

That I have this branch.

I should go through that
corner and go out of the room

and continue with that branch.

All right?

This is curved like
that in this direction.

And this other is
curved like this.

So if the guy is
going to put his feet,

where is the butt of
the writer going to sit?

He is here.

And these are his legs.

And these are his cowboy boots.

OK.

Do they look like cowboy boots?

No, I apologize.

STUDENT: Looks like socks.

MAGDALENA TODA: Yeah.

They look more like
Christmas socks.

But anyway, it's a poor cowboy.

Let's lower the
saddle a little bit.

He cannot see the horse, OK?

So the saddle.

If I cross the saddle,
this is the saddle.

And these are his hands.

And he is holding his hat.

This is [INAUDIBLE].

And with one hand
is on the horse.

I don't know.

It's very [INAUDIBLE].

So what I'm trying to draw
looks something like this.

Right?

Eh.

Sorry.

More or less.

It's an abstract picture.

Very abstract picture.

So with this in mind, if I were
to look at the level curves,

I'm going to ask you,
what are the level curves?

Oh, my god, what are
the level curves?

You already have them in
your WeBWorK homework.

But for one point
extra credit, I

want you to draw
them on the floor.

Draw the level curves.

Remember what those were?

They were projections of
the curves on the surface

at the intersection
with z equals c planes.

You project them on the ground.

What do you think they are?

Think about it.

What are these?

If I take c, what
if c is positive?

What if c is 0?

What if c is less than 0?

What am I going to have?

Your imagination gives
you c equals 1, Magdalena.

Let's draw that.

OK.

Well, I'll try.

a and b would be 1, right, guys?

So a and b would be 1.

This is a square.

These would be the asymptotes.

So very, very
briefly, the hyperbola

would be this one-- x squared
minus y squared equals 1,

right?

If I have the last
case for c equals 1,

I'm going to have-- c
equals negative 1-- I'm

going to have the conjugate.

Are you guys with me?

So I'll have an a squared,
asymptotes, conjugate.

What if I have different level
c? c equals 1/2. c equals 2.

c equals pi. c
equals-- what are they?

I'm going to get
families of hyperbolas,

trenches that look like that.

Standard trenches and
conjugate trenches.

A multitude of them, an infinite
family of such hyperbolas,

an infinite family
of such hyperbolas.

I wanted to draw it.

What do I get when c is 0?

What are those?

STUDENT: Don't you
get, like, [INAUDIBLE]?

MAGDALENA TODA: They
get-- very good.

Why?

x squared minus y squared
equals 0 would lead

me to y equals plus/minus 1.

And who are those y
equals plus/minus 1?

Exactly.

But exactly the first
bisector, which is y equals x.

They are [? then the ?]
function.

And the other one, y equals
negative [? x. ?] So these

are the asymptotes.

So I'm going to get a-- you guys
have to do this better than me.

Sorry.

These are all
hyperbolic trenches.

They are all going to
infinity like that.

And I'm sorry that
I'm giving you

a little bit too many hints.

This is part of your
homework, your WeBWorK.

I shouldn't talk
too much about it.

Any questions so far?

Is this hard?

Yes, sir?

No.

STUDENT: So [? spherically, ?]
if you had z

equals y squared
minus x squared,

it's that same
picture, just flipped?

MAGDALENA TODA:
What would it be?

It would be the poor
saddle-- or cowboy--

STUDENT: Would be upside down.

MAGDALENA TODA:
--would be upside down.

Or projected in
something like a mirror.

I don't know how to say.

It would be exactly upside down.

So the reflection of that.

So you take all the points.

If you have-- I don't know.

It's hard to draw a reflection
in three dimensions.

But--

STUDENT: No, I understand.

MAGDALENA TODA:
Practically every curve

would be upside down with
respect to the floor.

OK.

All right.

I'm going to erase in one.

And you say, well, you've
taught us about these things,

like the domain and range.

But what about other notions,
like continuity and stuff?

Let me move on to 11.2.

Limits of functions of
the type z equals f of xy.

So what do you remember
about the limit

of a function of one variable?

Comparison.

What about the limit if you
take [? z's, ?] I don't know.

I should look stunned.

And I should be stunned.

Of a function of y equals
f of x of one variable.

When do we say that
f has a limit at a?

STUDENT: When the [INAUDIBLE]
approaches from the right

and the left to the same value.

MAGDALENA TODA: Actually, that
was the simpler definition.

Let's think a little bit deeper.

We say that f has a
limit L at x equals a.

That's kind of the idea,
left and right limits.

But not both of them
have to exist, you see.

Maybe only the limit
from the left or limit

from the right only exists.

If, for any choice of values
of x, closer and closer, closer

and closer to a, we get that
F gets closer and closer to L.

And this "any" I put in.

My god, I put it in
a red circle thing,

because one could get
subsequencies of a sequence.

And for that subsequence
thing, things

look like I would have a limit.

And then you say,
well, but in the end,

I don't have a limit, because
I can get another subsequence

of the sequence.

And for that one, I'm not
going to have a limit.

Can you give me an example
of some crazy function that

does not have a limit at 0?

Example of a crazy function.

No.

No, don't write "crazy."

Of a function f of x
that is not defined at 0

and does not have
limit at 0, although it

is defined for values
arbitrarily close to 0.

Moreover, I want that function
to be drawn without-- I

want the function to be
drawn without leaving

the paper when I draw.

[INAUDIBLE]

So something that would
be defined on the whole 0

infinity except for 0 that
I can draw continuously

except when I get to 0, I
get some really bad behavior.

I don't have a limit
for that function.

You are close to that.

Sine of 1/x.

STUDENT: I said y equals 1/x.

MAGDALENA TODA: y equals 1/x.

Very good.

Let's see.

STUDENT: Oh, yeah. [INAUDIBLE].

MAGDALENA TODA: Yeah, yeah.

Both are excellent examples.

So let's see.

This guy is a very
nice function.

How do we draw him, or her?

Well, it's a her, right?

It's a she.

It's a function.

No, no.

In English, it doesn't
make any sense,

but if I think French, Italian,
Spanish, Romanian-- now

I speak both Italian
and Romanian--

we say it's a she,
it's a feminine.

So as I approach with values
closer and closer and closer

to 0, what happens
to my poor function?

It blows up.

OK.

So I have limit of 1/x from
the right and from the left.

If I take it from the
left, I don't care.

Let's take it only
from the right.

OK?

It's close to 0.

That's going to blow up, right?

And I restrict it.

So let's say, if I want
the domain to be containing

[? both, ?] that's also fine.

So if you guys want, we
can draw the other one.

This goes to paradise.

The other one, I'm not
going to say where it goes.

But it's the same idea,
that as you approach 0

with closer and closer
and closer values,

it's going to blow up.

It's going to explode.

This is a beautiful function.

How beautiful [INAUDIBLE].

Beautiful with a
bad behavior near 0.

So I'm not going
to have a limit.

No limit.

Some people say, limit
exists and is infinity.

But does infinity exist?

Well, this is a
really philosophical,

religious notion, so I
don't want to get into it.

But in mathematics, we consider
that unless the limit is

finite, you cannot have a limit.

So if the limit is plus/minus
infinity, there is no limit.

Could the limit be different
or different subsequences?

This is what I
wanted to point out.

If you try this guy, you are
in real trouble on that guy.

Why?

You can have two.

If you have a graphing
calculator, which

I'm going to be opposed to you
being used in the classroom,

you would probably
see what happens.

Sine is defined on
all the real numbers.

But you cannot
have a value at 0,

because the 1/x is
not defined at 0.

Imagine you get closer and
closer to 0 from both sides.

I cannot draw very beautifully.

But as 1, this is plus
1 and this is minus 1.

I'm going to have some behavior.

And how many of you have seen
that on a computer screen

or calculator?

You've seen.

Yeah, you've seen.

By the way, did you
see the Lubbuck High?

Was it in high school you saw
it the first time in Calc 1

or pre-calc?

STUDENT: [INAUDIBLE]
Algebra 1 with Mr. West.

[INAUDIBLE]

MAGDALENA TODA: So I'll
try-- oh, guys, you

have to be patient with me.

I'm not leaving the poor board
with the tip of my pencil.

I'm not leaving him.

I have continuity.

As I got closer to this, I still
have the [INAUDIBLE] property.

Anyway, it's OK.

I'm not leaving this.

I am taking all the values
possible between minus 1 and 1.

So on intervals that
are smaller, smaller,

I'm really taking all the
values between minus 1 and 1,

and really rapidly--
[INAUDIBLE].

When I'm getting closer to 0,
I'm not going to have a limit.

But as somebody
may say, but wait.

When I have a sequence
of values that

is getting closer
and closer to 0,

is that no guarantee that
I'm going to have a limit?

Nope.

It depends.

If you say "any," it has to
be for any choice of points,

any choice of points
that you go closer to 0.

Not for one sequence
of points that

is getting closer
and closer to 0.

For example, if your
choice of points is this,

choice of points.

Getting closer to 0.

[INAUDIBLE] xn
equals 1 over 2 pi n.

Isn't this going to 0?

Yeah.

It then goes to infinity.

This sequence goes to 0.

What is it?

1 over 2 pi?

1 over 4 pi?

1 over 8 pi?

1 over 16 pi?

1 over 32 pi?

1 over 64 pi?

This is what my
son is doing to me.

And I say, please stop.

OK?

He's 10 years old.

He's so funny.

Now, another choice of points.

Ah.

Somebody-- all of you are
smart enough to do this.

What do you think
I'm going to pick?

1 over what?

And when [? other ?]
something that goes to 0

then goes to infinity.

And I know that your
professor showed you that.

pi over 2 plus 2 pi n.

Doesn't this go to 0?

Yes.

As n gets bigger and
bigger, this is going to 0.

However, there is no limit.

Why?

Well, for the first sequence,
as xn goes to 0, f of xn

goes to-- what is
sine of-- OK, I

am too lazy to write this down.

Sine of 1 over 1 over--
of 1 over 1 over 2 pi?

STUDENT: It's the
sine over 2 pi.

MAGDALENA TODA: This
is sine of 2 pi n.

And how much is that?

STUDENT: 0.

MAGDALENA TODA: 0.

So this is a 0.

And this is a--
this converges to 0.

So I say, oh, so maybe I have
a limit, and that'll be 0.

Wrong.

That would be the rapid,
stupid conclusion.

If somebody jumps [? up, ?]
I picked some points,

I formed the sequence that
gets closer and closer to 0.

I'm sure that the limit exists.

I've got a 0.

Well, did you think of
any possible choice?

That's the problem.

You have to have
any possible choice.

F of yn sine of 1 over
pi over 2 plus 1 over 1

over-- Magdalena-- pi
over 2 plus 2 pi n.

So we saw that this was 0.

What happens to sine of
1 over 1 over sine of pi

over 2 plus 2 pi n?

And where does this go?

It then goes to infinity.

This sequence goes to 0.

What is f of the
sequence going to?

To another limit.

So there is no limit.

What's the limit of
this subsequence?

It's a constant one, right?

Because look, what does it
mean pi over 2 plus 2 pi n?

Where am I on the unit
trigonometric circle?

[INTERPOSING VOICES]

Always here, right?

Always on the sort of
like the north pole.

So what is the sine
of this north pole?

STUDENT: 1.

MAGDALENA TODA: Always 1.

So I get the limit 1.

So I'm done because there
are two different limits.

So pay attention to
this type of problem.

Somebody can get you in trouble
with this kind of thing.

On the other hand,
I'm asking you,

what if I want to make this
a function of two variables?

So I'll say, one
point extra credit.

I'm giving you too
much extra credit.

Maybe I give you
too much-- it's OK.

One point extra credit--
put them together.

Does f-- do you
like to do the f?

I used big F, and then I
changed it to little f.

This time I have a function
of two variables-- little

f with xy-- to be sine of 1
over x squared plus y squared.

Does this function have a
limit at the point 0, 0?

So when I approach 0,
0, do I have a limit?

OK.

And you say, well, it depends
how I approach that 0, 0.

That's exactly the thing.

Yes, sir.

Oh, you didn't want to ask me.

And does f of xy
equals-- let me give you

another one, a
really sexy one. x

squared plus y squared
times sine of 1

over x squared plus y squared.

Have a limit at 0, 0?

I don't know.

Continuous it cannot be,
because it's not defined there.

Right?

For a function to be
continuous at a point,

the function has to
satisfy three conditions.

The function has to be
defined there at that point.

The function has to
have a limit there

at that point of the domain.

And the limit and the function
value have to coincide.

Three conditions.

We will talk about
continuity later.

Hint.

Magdalena, too many hints.

This should remind
you of somebody

from the first
variable calculus.

It's a more challenging problem.

That's why I gave
it to extra credit.

If I had x sine of 1/x,
what would that look like?

STUDENT: x times--

MAGDALENA TODA: x
times sine of 1/x.

When I approach 0
with-- so if I have-- I

don't ask for an answer now.

You go home, you think about it.

You take the calculator.

But keep in mind that your
calculator can fool you.

Sometimes it can show an
image that misguides you.

So you have to think
how to do that.

How about x times
sine of 1/x when--

does it have a limit
when x goes to 0?

Is there such a limit?

Does it exist?

So if such a limit
would exist, maybe we

can extend by continuity the
function x times sine over x.

What does it mean?

Like, extend it, prolong it.

And say, it's this 4x equals
0 and this if x is not 0.

So this is obviously x is
different from 0, right?

Can we extend it by continuity?

Think about the drawing.

Think about the arguments.

And I think it's time for me
to keep the promise I made

to [? Aaron, ?]
because I see no way.

Oh, my god, [? Aaron, ?]
I see no way out.

The epsilon delta
definition of limit.

[? Right? ?] OK.

So what does it mean for a
real mathematician or somebody

with a strong mathematical
foundation and education

that they know the
true definition

of a limit of a function
of, let's say, one variable?

The epsilon delta, the one your
dad told you about. [INAUDIBLE]

try to fool you when avoid it
in undergraduate education.

People try to avoid
the epsilon delta,

because they think the students
will never, never understand

it, because it's
such an abstract one.

I think I wasn't ready.

I wasn't smart enough.

I think I was 16 when I was
getting ready for some math

competitions.

And one professor taught me
the epsilon delta and said,

do you understand it?

My 16-year-old mind said, no.

But guess what?

Some other people
smarter than me,

they told me, when
you first see it,

you don't understand
it in any case.

So it takes a little bit
more time to sink in.

So the same idea.

As I'm getting closer and
closer and closer and closer

to an x0 with my x values
from anywhere around-- left,

right-- I have to pick an
arbitrary choice of points

going towards x0, I have to
be sure that at the same time,

the corresponding sequence
of values is going to L,

I can express that
in epsilon delta.

So we say that.

f of x has limit L
at x equals x0 if.

For every epsilon positive, any
choice of an epsilon positive,

there is a delta.

There exists-- oh, OK, guys.

You don't know the symbols.

I'll write it in English.

For every epsilon
positive, no matter

how small-- put
parentheses, because you

are just [? tired-- ?]
no matter how small,

there exists a delta number
that depends on epsilon.

So that whenever x minus
x0 is less than delta,

this would imply
that f of x minus L,

that limit I taught you
about in absolute value,

is less than epsilon.

What does this mean?

I'm going to try
and draw something

that happens on a line.

So this is x0.

And these are my values of x.

They can come from anywhere.

And this is f of x.

And this is L. So it
says, no matter-- this

says-- this is an
abstract way of saying,

no matter how close, you see,
for every epsilon positive,

no matter how close
you get to the L.

I decide to be in this
interval, very tiny epsilon.

L minus epsilon.

L plus epsilon L. You give
me your favorite epsilon.

You say, Magdalena, pick
something really small.

Big epsilon to be 0.00001.

How about that?

Well, if I really
have a limit there,

an L at x0, that means that
no matter how much you shrink

this interval for me, you
can be mean and shrink it

as much as you want.

I will still find a
small interval around x0.

[? But ?] I will
still find the smaller

interval around x0, which is--
this would be x0 minus delta.

This would be x0 plus delta.

So that the image of this
purple interval fits inside.

You say, what?

So that the image of this
purple interval fits inside.

So f of x minus L, the distance
is still that, less than xy.

Yes, sir?

STUDENT: Where'd you
get epsilon [INAUDIBLE]?

MAGDALENA TODA: So
epsilon has to be

chose no matter how small.

STUDENT: [INAUDIBLE].

MAGDALENA TODA: Huh?

Real number.

So I'm saying, you should not
set the epsilon to be 0.0001.

That would be a mistake.

You have to think of that number
as being as small as you want,

infinitesimally small, smaller
than any particle in physics

that you are aware about.

And this is what I had the
problem understanding--

that notion of-- not
the notion of, hey, not

matter how close
I am, I can still

get something even smaller
around x0 that fits in this.

That's not what I
had the problem with.

The notion is to perceive
an infinitesimal.

Our mind is too limited
to understand infinity.

It's like trying
to understand God.

And the same limitation comes
with microscopic problems.

Yeah, we can see some
things on the microscope,

and we understand.

Ah, I understand I
have this bacteria.

This is staph.

Oh, my god.

But then there are molecules,
atoms, subatomic particles

that we don't understand,
because our mind is really

[? small. ?] Imagine
something smaller

than the subatomic particles.

That's the abstract notion
of infinitesimally small.

So I'm saying, if I really
have a limit L there,

that means no matter how small
I have this ball around it,

I can still find a
smaller ball that

fits-- whose image fits inside.

All right?

The same kind of definition--
I will try to generalize this.

Can you guys help me
generalize this limit notion

to the notion of function
of two variables?

So we say, that f of xy
has the limit L at x0y0.

What was x0y0 when I
talked about-- what

example did I give you guys?

Sine of 1 over x squared
plus y squared, right?

Something like that.

I don't know.

I said, think of 0, 0.

That was the given point.

It has to be a fixed couple.

So you think of the origin, 0,
0, as being as a fixed couple.

Or you think of the point 1,
0 as being as a fixed couple

in that plane you look at.

That is the fixed couple.

If-- now somebody
has to help me.

For every epsilon positive,
no matter how small,

that's where I have a problem
imagining infinitesimally

small.

There exists-- I no
longer have this problem.

But I had it enough
when I was in my 20s.

I don't want to go back to
my 20s and have-- I mean,

I would love to.

[LAUGHTER]

To go having vacations
with no worries and so on.

But I wouldn't like
to go back to my 20s

and have to relearn
all the mathematics.

Now way.

That was too much of a struggle.

There exists a delta positive
that depends on epsilon.

What does it mean,
depends on epsilon?

Because guys, imagine
you make this epsilon

smaller and smaller.

You have to make delta
smaller and smaller,

so that you can fit that
little ball in the big ball.

OK?

That depends on epsilon,
so that whenever-- now,

that is a big problem.

How do I say, distance between
the point xy and the point

x0y0?

Oh, my god.

This is distance between xy
and x0y0 is less than delta.

This would imply
that-- well, this

is a function with values in
R. This is in R. Real number.

So I don't have a problem.

I can use absolute value here.

Absolute value of f of
the couple xy minus L

is less than epsilon.

The thing is, can you
visualize that little ball,

that little disk?

What do I mean?

Being close, xy is me, right?

But I'm moving.

I'm the moving point.

I'm dancing around.

And [? Nateesh ?] is x0y0.

How do I say that I have
to be close enough to him?

I cannot touch him.

That's against the rules.

That's considered
[INAUDIBLE] harassment.

But I can come as
close as I want.

So I say, the
distance between me--

I'm xy-- and
[? Nateesh, ?] who is

fixed x0y0, has to be smaller
than that small delta.

How do I represent that
in plane mathematics?

STUDENT: Doesn't [INAUDIBLE]?

MAGDALENA TODA: Exactly.

So that delta has to
be small enough so

that the image of f at me minus
the limit is less than epsilon.

Now you understand why all
the other teachers avoid

talking about this
[? one. ?] So I

want to get small enough-- not
too close-- but close enough

to him, so that my value--
I'm f of xy-- minus the limit,

the limit-- I have
a preset limit.

All around [? Nateesh, ?] I
can have different values,

no matter where I go.

My value at all these points
around [? Nateesh ?] have

to be close enough
to L. So I say,

well, you have to get
close enough to L.

Somebody presents me an epsilon.

Then I have to reduce my
distance to [? Nateesh ?]

depending to that epsilon.

Because otherwise,
the image doesn't fit.

It's a little bit tricky.

STUDENT: So is this like
the squeeze theorem kind of?

MAGDALENA TODA: It is
the squeeze theorem.

STUDENT: Oh, all right.

MAGDALENA TODA: OK?

So the squeezing-- I ball into
another [? ball ?] [? limit. ?]

This is why-- it's not
a ball, but it's a--

STUDENT: A circle.

MAGDALENA TODA: Disk.

A circle, right?

So how do we express
that in Calc 3 in plain?

This is the
[? ingredient, ?] distance d.

So Seth, can you tell me what is
the distance between these two

points?

Square root of--

STUDENT: [INAUDIBLE].

MAGDALENA TODA: x minus
x0 squared plus y minus y0

squared.

Now shut up. [? And I ?]
am talking to myself.

STUDENT: Must be
less than delta.

[LAUGHTER]

MAGDALENA TODA: Less than delta.

So instead of
writing this, I need

to write that I can
do that in my mind.

OK?

All right.

This is hard.

We need to sleep on that.

I have one or two more problems
that are less hard-- nah,

they are still hard, but
they are more intuitive,

that I would like to
ask you about the limit.

I'm going to give
you a function.

And we would have to visualize
as I get closer to a point

where I am actually going.

So I have this nasty
function, f of xy

equals xy over z
squared plus y squared.

And I'm saying, [INAUDIBLE]
the point is the origin.

I choose the origin.

Question.

Do I have a limit that's--
do I have a limit?

Not [? really ?] for me.

Does f have a limit
at the origin?

You would have to imagine
that you'd draw this function.

And except you cannot draw, and
you really don't care to draw

it.

You only have to imagine that
you have some abstract graph--

z equals f of xy.

You don't care
what it looks like.

But then you take
points on the floor,

just like I did the exercise
with [? Nateesh ?] before.

And you get closer and
closer to the origin.

But no attention-- no
matter what path I take,

I have to get the same limit.

What?

No matter what path I take
towards [? Nateesh-- ?]

don't write that down-- towards
[? z0y0, ?] I have to get

the same limit.

Do I?

Let's imagine with the
eyes of your imaginations.

And [? Nateesh ?]
is the point 0, 0.

And you are aspiring to get
closer and closer to him

without touching him.

Because otherwise,
he's going to sue you.

So what do we have here?

We have different paths?

How can I get closer?

Either on this path
or maybe on this path.

Or maybe on this path.

Or maybe, if I had something
to drink last night-- which

I did not, because
after the age of 35,

I stopped drinking completely.

That's when I decided
I want to be a mom,

and I didn't want to
make a bad example.

So no matter what path you
take, you can make it wiggly,

you can make it
any way you want.

We are still approaching 0, 0.

You still have to
get the same limit.

Oh, that's tricky, because
it's also the same in life.

Depending on the path
you take in life,

you have different
results, different limits.

Now, what if I take the path
number one, number two, number

three possibility.

And number [? blooie ?]
is the drunken variant.

That is hard to
implement in an exercise.

Imagine that I have
limit along the path one.

Path one.

xy goes to 0, 0 of xy over
x squared plus y squared.

Do you guys see what's
going to happen?

So I'm along the--
OK, here it is.

This line, right, this is
the x-axis, y-axis, z-axis.

What's special for the x-axis?

Who is 0?

STUDENT: x.

STUDENT: yz.

MAGDALENA TODA: y is 0.

So y is 0.

So y is 0.

Don't laugh at me.

I'm going to write like
that because it's easier.

And it's going to be
something like limit

when x approaches 0
of x over x squared.

STUDENT: It's 1/x.

MAGDALENA TODA: Times 0 up.

Oh, my god.

Is that-- how much is that?

STUDENT: 0.

STUDENT: 0.

MAGDALENA TODA: 0!

I'm happy.

I say, maybe I have the limit.

I have the limit 0.

No, never rush in life.

Check.

Experiment any other paths.

And it's actually very easy
to see where I can go wrong.

If I take the path number two,
I will get the same result.

You don't need a
lot of imagination

to realize, hey, whether
she does it for x

or does it for y, if she
goes along the 2, what

the heck is going to happen?

y is going to shrink.

x will always be 0.

Because this means
a point's like what?

0,1.

0, 1/2.

0, 1/n, and so on.

But plug them all in here,
I get 0, 1/n times 0.

It's still 0.

So I still get 0.

Path two.

When I approach my--
xt goes to 0, 0.

The poor [? Nateesh ?]
is waiting for an answer.

I still get 0.

Let's take not the
drunken path, because I

don't know [? it unless ?]
the sine function.

That is really crazy.

I'll take this one.

What is this one,
in your opinion?

Is that going to help me?

I don't know, but I
need some intuition.

Mathematicians need intuition
and a lot of patience.

So what is your intuition?

The one in the middle, I'm going
to start walking on that, OK,

until you tell me what it is.

STUDENT: y [INAUDIBLE].

MAGDALENA TODA: y equals
x is the first bisector

or the first quadrant.

And I'm very happy
I can go both ways.

y equals x.

x [INAUDIBLE].

So limit when x equals y,
but the pair xy goes to 0,0.

I'm silly.

I can say that,
well, Magdalena, this

is the pair xx,
because x equals what?

Let me plug them in.

So it's like two people.

x and y are married.

They are a couple, a pair.

They look identical.

Sometimes it happens.

Like twins, they
start looking alike,

dressing alike, and so on.

The x and the y have to
receive the same letter.

And you have to tell me what
in the world the limit will be.

STUDENT: 1/2.

MAGDALENA TODA: 1/2.

Oh, my god.

So now I'm deflated.

So now I realize that
taking two different paths,

I show that I have-- on
this path, I have 1/2.

On this path, I have 0.

I don't match.

I don't have an overall limit.

So the answer is,
no overall limit.

Oh, my god.

So what you need to
do, guys, is read

section 11.1 and section 11.2.

And I will ask you next
time-- and you can lie,

you can do whatever.

Did the book explain
better than me,

or I explain better
than the book?

This type of example when
the limit does not exist.

We are going to
see more examples.

You are going to see examples
where the limit does exist.

Now, one last thing.

When you have to compute limits
of compositions of functions

whose limit exist--
for example, you

know that limit is
xy goes to x0y0 of f

of xy [INAUDIBLE] limit
of xy go to x0y0 of gxy

is L-- L-- L-- M-- M.

How are you going to compute the
limit of alpha f plus beta g?

This is in the book.

But you don't need the
book to understand that.

You will already
give me the answer,

because this is the equivalent
thing to the function of one

variable thing in Calc 1.

So if you would only
have f of x or g of x,

it would be piece of cake.

What would you say?

STUDENT: [INAUDIBLE].

MAGDALENA TODA: Right.

Alpha times L plus beta
times M. Can you also

multiply functions.

Yes, you can.

Limit of fg as xy goes
to x0 or y0-- will be LM.

How about-- now I'm going to
jump to conclusion, hoping

that you are going to catch me.

You are going to catch
me, and shout at me,

and say, ooh, pay
attention, Magdalena,

you can make a mistake there.

I say it's L/M when I do
the division rule, right?

Where should I pay attention?

STUDENT: M [INAUDIBLE].

MAGDALENA TODA: Pay attention.

Sometimes you can
have the-- right?

And this also has
to exist as well.

STUDENT: [INAUDIBLE].

MAGDALENA TODA: So one
last-- how many minutes

have I spent with you?

I've spent with you a long
number of hours of my life.

No, I'm just kidding.

So you have one hour and
15, a little bit more.

Do I have a little bit more?

Yes.

I have 15 minutes.

I have--

STUDENT: So we get out at--

[INTERPOSING VOICES]

MAGDALENA TODA: 50.

Five more minutes.

OK.

So I want to ask you what
you remember about some

of your friends, the trig
functions involved in limits.

Why did we study
limits at the point

where the function's
not defined?

Well, to heck with it.

We don't care.

The function is
not defined at 0.

But the limit is.

And nobody showed you how
to do the epsilon delta

to show anything like that.

OK.

Can you do that
with epsilon delta?

Actually, you can do
everything with epsilon delta.

But I'm not going to give
you any extra credit.

So I trust you that
you remember that.

1!

How about-- let me-- OK.

I am so proud of you.

Let me challenge you more.

Let me challenge you more.

Tangent of ax over bx.

x go to 0.

I asked this to a girl
from Lubbock High.

She was in high school.

She knew the answer.

STUDENT: Oh, I can't disappoint
everybody in getting this.

STUDENT: Is it 1/a?

Oh, I can't remember.

MAGDALENA TODA: Tell me
what to do to be smart.

Right?

I have to be doing
something smart.

She-- can you give me hint?

I'm your student
and you say, well--

STUDENT: ba--

STUDENT: It's 0.

STUDENT: It's [INAUDIBLE].

MAGDALENA TODA: Um, it's a what?

STUDENT: b/a?

MAGDALENA TODA: I'm
not [INAUDIBLE].

I don't think so.

So what should I do?

I should say, instead of
bx-- that drives me nuts.

This goes on my nerves-- bx.

Like, maybe I go on your
nerves. bx is ax, right?

If it were ax, I would
be more constructive,

and I knew what to do.

I say replace bx with
ax, compensate for it,

and divide by bx.

And I was trying to
explain that to my son,

that if you have a fraction
a/b, and then you write a/n

times n/b, it's the same thing.

Gosh, I had the
problem with him.

And then I realized that he
didn't do simplifications

in school.

So it took a little more
hours to explain these things.

This is fourth grade.

I think I remember doing
that in fourth grade.

Third grade, actually.

So these two guys disappear.

I haven't changed
my problem at all.

But I've changed the status,
the shape of my problem

to something I can mold,
because this goes to somebody,

and this goes to somebody else.

Who is this fellow?

It's a limit that's
a constant-- a/b.

Who is this fellow?

STUDENT: 1.

MAGDALENA TODA: 1.

Because tangent of x/x as x
goes to 0 goes to 1 exactly

like that.

So limit of sine x over cosine
x, that's tangent, right?

Over x.

You do it exactly the same.

It's limit of sine x/x
times 1 over cosine x.

That's how we did
it in high school.

This goes to 1.

This goes to 1.

So it's 1.

So thank you, this is 1.

I know I took a little more time
to explain than I wanted to.

But now you are grown up.

In two minutes, you are
going to be finishing

this section, more or less.

What if I put a function
of two variables,

and I ask you what
the limit will be,

if it's the same
type of function.

So you say, oh, Magdalena,
what you doing to us?

OK, we'll see it's fun.

This one's fun.

It's not like the one before.

This one is pretty beautiful.

It's nice to you.

It exists.

xy goes to 0, 0.

So you have to imagine
some preferable function

in abstract thinking.

And you want it in
a little disk here.

And xy, these are all points
xy close enough to 0, 0,

in the neighborhood of 0, 0.

OK.

What's going to happen as
you get closer and closer

and closer and closer with
tinier and tinier and tinier

disks around 0, 0?

You're going to shrink so much.

What do you think
this will going to be,

and how do I prove it?

STUDENT: [INAUDIBLE].

MAGDALENA TODA: Who said it?

You, sir? [INAUDIBLE]
going to go to 1.

And he's right.

He has the intuition.

A mathematician will
tell you, prove it.

STUDENT: Um, well,
let's see here.

MAGDALENA TODA: Can you prove?

STUDENT: You could use
the right triangle proof,

but that would probably
take way more [INAUDIBLE].

MAGDALENA TODA: x and
y are independent.

That's the problem.

They are married, but they
are still independent.

It's a couple.

However, we can use
polar coordinates.

Why is polar coordinates?

Well, in general, if we
are in xy, it's a pair.

This is r, right?

So rx is r cosine theta.

y is r sine theta.

And I can get closer and
closer to the original.

I don't care.

What happens about x
squared plus y squared,

this is r squared.

And r is a real number.

And as you walk closer
and closer to the original

without touching it,
that r goes to 0.

It shrinks to 0.

So that r squared goes
to 0 but never touches 0.

So this becomes limit as r goes
to 0, the radius of that disk

goes to 0.

Sine of r squared
over r squared.

But r squared could be replaced
by the real function, t,

by the real parameter,
lambda, by whatever you want.

So then it's 1.

And then Alexander was right.

He based it on, like,
observation, intuition,

everything you want.

It was not a proof.

On a multiple-choice exam,
he would be a lucky guy.

I don't want you to prove it.

But if I want you to
prove it, you have to say,

Magdalena, I know
polar coordinates,

and so I can do it.

And one last question for today.

Guys, I'm asking you,
limit xy goes to 0, 0.

You will see some of these in
your WeBWorK for Chapter 11

that's waiting for
you, homework 3.

Tangent of 2 x squared
plus y squared over 3

x squared plus y squared.

What is that?

2/3.

STUDENT: 2/3.

MAGDALENA TODA: Am
I asking you why?

No, enough.

OK.

[INAUDIBLE] I gave
you everything

you need to show that.

x squared plus y squared,
again, is Mr. r squared.

It's OK.

I taught you that.
a/b. a is 2, b is 3.

Is it hard?

It is not easy, for sure.

Calc 3 is really difficult
compared to other topics

you are probably taking.

But I hope that I
can convince you

that math, although
difficult, [INAUDIBLE] Calc 3,

is also fun.

OK?

All right.

So I need attendance and
I need the extra credit.

STUDENT: Yeah, [INAUDIBLE].

MAGDALENA TODA: Before
you go, you need to sign.