## TTU Math2450 Calculus3 Sec 11.1 and 11.2

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MAGDALENA TODA:
We have any people
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who finished the
extra credit and are
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willing to give it to me today?
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I mean, you don't have to.
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That's why it's
called extra credit.
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But I think it's good
for extra practice
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and for the extra points.
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So hold on to it if you cannot
give it to me right now.
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And I'll collect it at
the end of the class.
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Today's a big day.
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We are starting a new
chapter, Chapter 11.
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So practically, we are
going to discuss all
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through this chapter functions
of several variables.
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And you are going to
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why do we need functions
in more than one variable?
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Well, we are all functions
of many variables.
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I was freezing outside,
and I was thinking,
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I'm a function of
everything I eat.
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I'm a function of the
temperature outside.
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Almost everything in
our body is a function
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of hundreds of factors,
actually, thousands.
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But we don't have the time
and the precise information
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to analyze all the
parameters that
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affect our physical
condition every day.
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We are getting there.
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I'm going to give you
just the simple case.
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So instead of y equals f of x
type of function, one variable,
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we are going to look at
functions of the types z
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equals f of xy.
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Can I have many more?
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Absolutely I can.
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And that's kind of
the idea, that I
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can have a function
in an-- let me
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count-- n plus 1
dimensional space
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as being of the type xn plus
1 equals f of x1, x2, x3, x4.
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Somebody stop me. xn.
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Right.
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I have many variables.
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And that is a problem
that affects everything.
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Our physical world is
affected by many parameters.
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In engineering
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seen some of these parameters.
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Can you give me some
examples of parameters you've
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seen in engineering classes?
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x1, x2, x3 could be the
Euclidean coordinates, right,
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for the three [? space. ?] But
besides those, there was an x4.
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It could be?
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Time.
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Excellent, [INAUDIBLE].
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More than that.
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I want more.
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I want x5.
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Who can think of
another parameter
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that affects physical processes
or chemical reactions?
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Yes, sir?
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STUDENT: Temperature.
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MAGDALENA TODA: Temperature.
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Excellent.
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Another very good idea.
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I'm running out of imagination.
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But you have a lot more
information than me.
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Pressure.
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Maybe I'm studying a process of
somewhere up in the atmosphere.
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Maybe I'm in an
airplane, and then it
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becomes a little bit
more complicated,
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because I hate the way
cabins are pressurized.
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I can feel very uneasy.
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My ears pop and so on.
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We can be in the
bottom of the ocean.
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There are very many
physical parameters
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that affect physical
processes, chemical processes,
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biological processes.
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I don't know if this is
fortunate or unfortunate,
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but I think that was
the key to the existence
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of the universe in the first
place-- all these parameters.
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OK.
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Let me give you a simple
example of a function that
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looks like a graph.
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This is a graph.
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And you say, wait a
minute, wait a minute.
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Can I have functions of several
variables that cannot be
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represented as graphs?
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Yeah.
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Absolutely.
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a little bit later.
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So if I were to give you
an example that you've
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seen before, and I would say,
give me a good approximation
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to a valley that is actually
a quadric that we love and we
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studied before for
the first time.
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beautiful object, a valley.
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Any imagination,
recognition, recollection?
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I know I scared
you enough for you
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to know the equations of those
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told me we watched
all the videos,
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book like never before.
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That was kind of the idea.
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I didn't want to scare you away.
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I wanted to scare you
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and watch the videos.
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And I'm talking about a valley
that you've seen before.
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Many of you told me you like
the University of Minnesota
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website that has the
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So you've met this guy before.
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They show the general equation.
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But I said I like the
circular paraboloid.
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elliptic paraboloid.
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Which one do you think I prefer?
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The circular paraboloid.
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Give me an example of
a circular paraboloid.
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STUDENT: A flashlight?
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Inside.
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MAGDALENA TODA: The expression,
the mathematical equation.
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STUDENT: Oh, sorry.
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So it would be x
squred plus y squared.
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MAGDALENA TODA: Very good.
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That's exactly
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Of course, it could be
over something, over r.
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All right.
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That's my favorite.
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Now, if I put the flashlight
in here just like one of you
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said, or the sign on
top of the z-axis.
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Then I'm going to look at
the various-- we discussed
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that a little bit before.
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So various horizontal
planes, they're going to cut.
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They're going to cut the
surface in different circles,
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upon different circles.
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We love them, and we use them.
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And what did we do
with them last time?
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We projected them on the floor.
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And by floor, I mean the what?
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By floor, I mean the xy plane.
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Plus this xy plane.
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I label it like you like it.
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You said you like
it when I label it,
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so you have the
imagination of a table.
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This is x and y and z.
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And so I gave you an example
of a graph cut in with z equals
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constant positive or negative?
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Well, it better be positive,
because for negative, I
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have no solutions.
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Positive or zero.
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Well, for zero I have
a degenerate conic.
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A degenerate conic
could be a point,
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or it could be a bunch of lines.
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In this case, all those
circles-- doo-doo-doo-doo-doo--
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a family of one parameter,
family of circles.
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Like the ones that
is-- a dolphin
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is now doing that
in San Antonio,
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San Diego-- to take
those old circles
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from the bottom of the sea,
and bring them different sizes,
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and put them together.
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So they are very smart.
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I love dolphins.
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So we'll see 0
[INAUDIBLE] get a point.
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That's still a conic.
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It's a degenerate circle.
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Do you realize
that's a limit case?
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It's really beautiful.
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You know what I mean?
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Circle on top of a circle
on top of a circle,
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smaller and smaller.
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All right.
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So good.
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because that's
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why you guys wanted
the source of light
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on top-- of the projections
of these circles,
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I'm going to have them
at the same color.
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But dotted lines because I
think the book doesn't show them
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dotted.
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But on my way here,
I was thinking,
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I think it's more beautiful
if I draw them dotted.
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And how big is this circle?
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Well, god knows.
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I'm going to make a purple
circle that is, of course,
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equal in size, equal in radius
with the original purple
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circle.
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So the dotted purple circle,
that's on the ground--
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is just the projection of
the continuous purple circle.
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So for the family of
circles on the surface,
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I have a family of projections
on the ground in the xy plane.
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And such a family of
projections represents
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a bunch of level curves.
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We call this family
of level curves.
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OK?
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All right.
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So if you think about it,
what are level curves?
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You view them as being in plane.
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Oh, my god.
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So I should view them as a bunch
of points, a set of points.
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If I make it like
that, that means
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I view this as an
element of what?
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Element of the xy plane,
right, with the property
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that f of x and y is a constant.
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OK?
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In my case, I have a
[INAUDIBLE] constant.
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In general, I have an
arbitrary real constant.
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That's a level curve for
level C, for the level
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C called the level, or altitude
would be the same thing.
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So have you seen these
guys in geography?
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What in the world are these
level curves in geography?
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STUDENT: [INAUDIBLE]
show the slope
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of a-- the steepness of a hill.
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MAGDALENA TODA: You've
seen topographical maps.
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And I'm going to try
and draw one of them.
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I don't know, guys,
how-- excuse me.
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I'm not very good
today at drawing.
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But I'll do my best.
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It could be a temperature
map or pressure map.
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[INAUDIBLE] or whatever.
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Now I'll say, this is
going to go-- well,
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I cannot draw the
infinite family.
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I have a one-parameter family.
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And then I'll-- I'm dreaming of
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You see what I'm doing.
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Do you know what I'm doing?
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That means I'm dreaming of the
different depths of the sea.
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line, I have the same depth.
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The same altitude for
every continuous rule.
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The same depth
for every-- so OK.
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I'm not going to swim
too far, because that's
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where the sharks are.
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And I cannot draw the sharks,
but I ask you to imagine them.
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It's fundamental in
a calculus class.
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So somewhere here
I'm going to have--
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what's the deepest--
guys, what's
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the deepest point in that?
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[? STUDENT: 11,300. ?]
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MAGDALENA TODA: And
do you know the name?
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I know the--
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STUDENT: Mariana Trench.
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MAGDALENA TODA: Mariana Trench.
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STUDENT: Trench.
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MAGDALENA TODA: All right.
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So these topographical
are full of curves.
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These are level curves.
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So you didn't know,
but there is a lot
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of mathematics in geography.
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And there is a
lot of mathematics
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in-- oh, you knew it.
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When you watch the
weather report,
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that's all mathematics, right?
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It shows you the distribution
of temperatures everyday.
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That is what we can
[INAUDIBLE] also
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of several parameters, right?
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And those functions could
be pressure, wind, whatever.
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OK.
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Speed of the wind.
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Something like that.
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I did not dare to
look at the prediction
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of the weather for this place.
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This place used to
be a beautiful place.
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300 days of the
year of sunshine.
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Not anymore.
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So there is something
fishy in Denmark
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and also something
fishy in [INAUDIBLE].
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The world is changing.
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So if you don't believe in
global warming, think again,
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and global cooling, think again.
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All right.
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So unfortunately,
I am afraid still
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to look at the temperatures
for the next few days.
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But--
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STUDENT: It's going to
be 80 degrees on Tuesday.
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MAGDALENA TODA: Really? [?
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Well, see, I should
have looked at it. ?]
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[LAUGHTER]
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I should gather the
courage, because I
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knew-- when I was
interviewed here
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for assistant professor,
gosh, I was young.
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2001.
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And my interview
was in mid-February.
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And birds were chirping, it was
blue skies, beautiful flowers
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everywhere on campus.
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And I love the campus.
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OK.
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Give me an example of a surface
that cannot be represented
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as a graph in its
entirety as a whole graph.
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You gave me that before,
and I was so proud of you.
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It was a--
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[LAUGHS]
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What kind of surface
am I trying to mimic?
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MAGDALENA TODA: That
can be actually a graph.
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That's a good
example of a graph.
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But give me an example
of a non-graph that
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is given as an implicit form.
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So graph or explicit
is the same thing.
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z equals f of xy.
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Give me a non-graph.
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One of you said it.
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x squared plus y squared
plus z squared equals 1.
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Why is this not a graph?
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Not a graph.
• 15:13 - 15:14
Why is this not a graph?
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STUDENT: [INAUDIBLE].
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When you move it over to
1, you can't actually--
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MAGDALENA TODA: You
cannot but you can cut it.
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You can take a
sword and-- I'm OK.
• 15:31 - 15:34
I don't want to think about it.
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So z is going to be two graphs.
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So I can split this surface
even in a parametric form
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as two different graphs.
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Different graphs.
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If I cut along-- I have this
orange, or sphere, globe.
• 15:51 - 15:54
And I cut it along
a great circle.
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It doesn't have
to be the equator.
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But you have to
imagine something
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like the world and the equator.
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This is kind of in
the unit sphere.
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Today I drank enough
coffee to draw better.
• 16:09 - 16:10
Why don't I draw better?
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I have no idea.
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OK.
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So that's the unit sphere.
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What does it mean?
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It means it has radius how much?
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STUDENT: 1.
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MAGDALENA TODA: 1.
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And it has two graphs.
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It's not one graph,
it's two graphs.
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So this is called
implicit equation.
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from-- I was chatting
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studying last night,
• 16:39 - 16:41
I was chatting with
you at midnight.
• 16:41 - 16:45
And one of you said, if I had
something I hated in calculus,
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it was the implicit
differentiation.
• 16:48 - 16:50
And I know this is
• 16:50 - 16:53
So we'll do a lot of
implicit differentiation,
• 16:53 - 16:55
so you become more comfortable.
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Usually we have one exercise in
this differentiation at least
• 16:59 - 17:02
on the final.
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So this is an implicit equation.
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And z is going to be
two graphs-- 1 minus x
• 17:10 - 17:11
squared minus y squared.
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So I have, like, two charts,
two different charts.
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OK.
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The upper hemisphere--
I'm talking geography,
• 17:20 - 17:23
but that's how we talk
in geometry as well.
• 17:23 - 17:26
So geography right
now is like geometry.
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I have a north pole.
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Somebody quickly give me the
coordinates of the north pole.
• 17:32 - 17:33
STUDENT: 0, 0, 1.
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MAGDALENA TODA: 0, 0, 1.
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Thank you, Brian.
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0, 0, 1.
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STUDENT: 0, 0, minus 1.
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MAGDALENA TODA: 0, 0, minus 1.
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And write yourself a
note, because as you know,
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I'm very absent-minded
and I forget
• 17:49 - 17:52
what I eat for lunch and so on.
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Remind me to talk
to you sometime
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at the end of the chapter
• 17:58 - 18:01
It's a very important
mathematical notion
• 18:01 - 18:04
that also has to do a
little bit with geography.
• 18:04 - 18:06
But it's a one-to-one
correspondence
• 18:06 - 18:09
between a certain
part of a sphere
• 18:09 - 18:12
and a certain huge
part of a plane.
• 18:12 - 18:14
Now, we're not going
• 18:14 - 18:16
because that's not [INAUDIBLE].
• 18:16 - 18:18
That's a little bit
harder [INAUDIBLE].
• 18:18 - 18:21
You guys should now
see this line, right?
• 18:21 - 18:24
This should be beyond--
in the twilight zone,
• 18:24 - 18:26
behind the sphere.
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OK?
• 18:27 - 18:29
So you don't see it.
• 18:29 - 18:31
And who is this? z equals 0.
• 18:31 - 18:35
And so this green
fellow should be
• 18:35 - 18:39
the circle x squared
plus y squared equals 1
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in the xy plane.
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• 18:43 - 18:45
Good.
• 18:45 - 18:47
So I have two graphs.
• 18:47 - 18:55
Now, if I were to ask
you, what is the domain
• 18:55 - 18:59
and the range of the function?
• 18:59 - 19:03
I'm going to erase
the whole thing.
• 19:03 - 19:10
What is the domain and the range
of the related function, z,
• 19:10 - 19:14
which gives the
upper hemisphere?
• 19:14 - 19:15
Upper hemisphere.
• 19:15 - 19:17
It's a graph.
• 19:17 - 19:21
And square root of 1 minus
x squared minus y squared.
• 19:21 - 19:23
You may stare at
it until tomorrow.
• 19:23 - 19:28
It's not hard to figure
out what I mean by domain
• 19:28 - 19:31
and range of such a function.
• 19:31 - 19:33
You are familiar
with domain and range
• 19:33 - 19:37
for a function of one variable.
• 19:37 - 19:40
For most of you,
that's a piece of cake.
• 19:40 - 19:42
That was even
pre-calc wasn't it?
• 19:42 - 19:44
It was in Calc 1.
• 19:44 - 19:47
algebra and pre-calc.
• 19:47 - 19:52
Now, what is the domain
of such a function?
• 19:52 - 19:57
Domain of definition has to be a
set of points, x and y in plane
• 19:57 - 20:01
for which the
function is defined.
• 20:01 - 20:03
If the function is
impossible to be defined
• 20:03 - 20:06
for a certain pair, x, y,
you kick that couple out
• 20:06 - 20:08
and you say, never come back.
• 20:08 - 20:09
Right?
• 20:09 - 20:15
So what I mean by domain is
those couples that we hate.
• 20:15 - 20:16
Who we hate?
• 20:16 - 20:21
The couples x, y for which x
squared plus y squared is how?
• 20:21 - 20:24
• 20:24 - 20:25
What existence condition do I--
• 20:25 - 20:26
STUDENT: [INAUDIBLE].
• 20:26 - 20:27
MAGDALENA TODA: Yeah.
• 20:27 - 20:30
You see this guy
under the square root
• 20:30 - 20:34
has to be positive or 0.
• 20:34 - 20:35
Right?
• 20:35 - 20:38
Otherwise, there is no
square root in real numbers.
• 20:38 - 20:40
That's going to be
in imaginary numbers,
• 20:40 - 20:42
and you can take
a walk, because we
• 20:42 - 20:45
are in real calculus
in real time as well.
• 20:45 - 20:49
So x squared plus y
squared must be how?
• 20:49 - 20:51
Less than or equal to 1.
• 20:51 - 20:54
We call that a certain name.
• 20:54 - 20:59
This is called a
closed unit disk.
• 20:59 - 21:03
you a little bit more
• 21:03 - 21:06
than a regular Calc 3 class.
• 21:06 - 21:09
They will never
make a distinction.
• 21:09 - 21:10
What's closing with this?
• 21:10 - 21:12
What's opening with this?
• 21:12 - 21:15
Everything will come
into place when you
• 21:15 - 21:20
• 21:20 - 21:25
If I don't take the boundary--
so everything inside the disk
• 21:25 - 21:28
except for the boundary, I have
to put strictly less than 1.
• 21:28 - 21:31
That's called open unit disk.
• 21:31 - 21:35
this is [INAUDIBLE].
• 21:35 - 21:36
All right.
• 21:36 - 21:37
This is just a parentheses.
• 21:37 - 21:40
My domain is the closed one.
• 21:40 - 21:43
What is the range?
• 21:43 - 21:46
The range is going to be--
• 21:46 - 21:47
STUDENT: [INAUDIBLE].
• 21:47 - 21:50
MAGDALENA TODA: The altitude
starts having values from--
• 21:50 - 21:51
STUDENT: Negative 1 to 1.
• 21:51 - 21:52
STUDENT: 0 to 1.
• 21:52 - 21:53
MAGDALENA TODA: So I'm 0 to 1.
• 21:53 - 21:55
the upper hemisphere.
• 21:55 - 21:58
I should even erase,
because I don't want it.
• 21:58 - 21:59
So say it again, guys.
• 21:59 - 22:00
STUDENT: 0 to 1.
• 22:00 - 22:01
MAGDALENA TODA: 0.
• 22:01 - 22:02
Open or closed?
• 22:02 - 22:03
STUDENT: Open.
• 22:03 - 22:03
STUDENT: Closed.
• 22:03 - 22:06
STUDENT: Closed, closed.
• 22:06 - 22:08
MAGDALENA TODA: Closed to?
• 22:08 - 22:08
STUDENT: 1 closed.
• 22:08 - 22:10
MAGDALENA TODA: 1 closed.
• 22:10 - 22:10
Yes.
• 22:10 - 22:14
Because that is the north pole.
• 22:14 - 22:19
I've been meaning to
give you this example.
• 22:19 - 22:22
And give me the other example
for the lower hemisphere.
• 22:22 - 22:23
What's different?
• 22:23 - 22:25
The same domain?
• 22:25 - 22:26
STUDENT: It ranges from--
• 22:26 - 22:27
STUDENT: Negative 1.
• 22:27 - 22:29
STUDENT: Negative 1 to 0.
• 22:29 - 22:30
MAGDALENA TODA: Closed
internal, right?
• 22:30 - 22:34
When we include the endpoints,
we call that closed interval.
• 22:34 - 22:36
It has a certain
topological sense.
• 22:36 - 22:39
You haven't taken
topology, but very soon,
• 22:39 - 22:44
if you are a math major, or
you are a double major, or some
• 22:44 - 22:48
of you even-- they want to
• 22:48 - 22:52
you will learn what an open
set is versus a closed set.
• 22:52 - 22:54
Remember we called this closed.
• 22:54 - 22:56
This is open.
• 22:56 - 23:00
And if it's closed here and
open there, it's neither.
• 23:00 - 23:00
OK?
• 23:00 - 23:03
• 23:03 - 23:03
OK.
• 23:03 - 23:08
To be closed, it has to be
containing both endpoints.
• 23:08 - 23:09
I'm going to erase this.
• 23:09 - 23:12
• 23:12 - 23:20
And this was, of course, 11.1.
• 23:20 - 23:23
We are in the middle of it.
• 23:23 - 23:28
In 11.1, one of you gave me a
• 23:28 - 23:31
And I'm going to give
you something to do,
• 23:31 - 23:33
because I don't want
you to get lazy.
• 23:33 - 23:36
I'm very happy you came
• 23:36 - 23:39
• 23:39 - 23:39
All right.
• 23:39 - 23:41
• 23:41 - 23:44
• 23:44 - 23:47
And I did my best,
but it's not hard.
• 23:47 - 23:50
It's not easy to draw saddle.
• 23:50 - 23:55
When I am looking at the
coordinates, x, y, z,
• 23:55 - 24:02
I have z equals minus y
squared will look down.
• 24:02 - 24:05
• 24:05 - 24:07
Maybe I made it too fat.
• 24:07 - 24:09
I'm really sorry.
• 24:09 - 24:11
And down.
• 24:11 - 24:12
This continues.
• 24:12 - 24:21
• 24:21 - 24:22
OK?
• 24:22 - 24:27
And then what other thing
did I want to point out?
• 24:27 - 24:31
I want to point out--
do you see this?
• 24:31 - 24:34
This should look a
little bit more round.
• 24:34 - 24:37
It doesn't look
round enough here.
• 24:37 - 24:38
STUDENT: Your'e drawing
• 24:38 - 24:40
MAGDALENA TODA: No, I'm
drawing just the section
• 24:40 - 24:42
z equals minus y squared.
• 24:42 - 24:44
So I took x to be 0.
• 24:44 - 24:48
And the purple line
should be on this wall.
• 24:48 - 24:50
I know you guys have
enough imagination.
• 24:50 - 24:54
So this is going to
be z equals minus y
• 24:54 - 24:59
squared drawn on yz wall.
• 24:59 - 25:03
• 25:03 - 25:06
I've done this before,
but I'm just reviewing.
• 25:06 - 25:08
What if it's y0?
• 25:08 - 25:11
Then I have to
draw on that wall.
• 25:11 - 25:14
And I have to draw beautifully,
which I am not-- don't always--
• 25:14 - 25:16
I can't always do.
• 25:16 - 25:17
But I'll try.
• 25:17 - 25:24
I have z equals x squared
drawn on that wall.
• 25:24 - 25:27
If I start drawing,
I'll get fired.
• 25:27 - 25:29
That I have this branch.
• 25:29 - 25:33
I should go through that
corner and go out of the room
• 25:33 - 25:35
and continue with that branch.
• 25:35 - 25:36
All right?
• 25:36 - 25:39
• 25:39 - 25:43
This is curved like
that in this direction.
• 25:43 - 25:45
And this other is
curved like this.
• 25:45 - 25:50
So if the guy is
going to put his feet,
• 25:50 - 25:55
where is the butt of
the writer going to sit?
• 25:55 - 25:57
He is here.
• 25:57 - 26:00
And these are his legs.
• 26:00 - 26:02
• 26:02 - 26:06
And these are his cowboy boots.
• 26:06 - 26:07
OK.
• 26:07 - 26:08
Do they look like cowboy boots?
• 26:08 - 26:11
No, I apologize.
• 26:11 - 26:12
STUDENT: Looks like socks.
• 26:12 - 26:13
MAGDALENA TODA: Yeah.
• 26:13 - 26:16
They look more like
Christmas socks.
• 26:16 - 26:18
But anyway, it's a poor cowboy.
• 26:18 - 26:23
• 26:23 - 26:25
Let's lower the
• 26:25 - 26:27
He cannot see the horse, OK?
• 26:27 - 26:30
• 26:30 - 26:35
• 26:35 - 26:38
And these are his hands.
• 26:38 - 26:41
And he is holding his hat.
• 26:41 - 26:42
This is [INAUDIBLE].
• 26:42 - 26:46
And with one hand
is on the horse.
• 26:46 - 26:47
I don't know.
• 26:47 - 26:48
It's very [INAUDIBLE].
• 26:48 - 26:57
So what I'm trying to draw
looks something like this.
• 26:57 - 26:58
Right?
• 26:58 - 26:59
Eh.
• 26:59 - 27:02
Sorry.
• 27:02 - 27:02
More or less.
• 27:02 - 27:04
It's an abstract picture.
• 27:04 - 27:06
Very abstract picture.
• 27:06 - 27:15
So with this in mind, if I were
to look at the level curves,
• 27:15 - 27:19
what are the level curves?
• 27:19 - 27:22
Oh, my god, what are
the level curves?
• 27:22 - 27:25
• 27:25 - 27:28
• 27:28 - 27:30
But for one point
extra credit, I
• 27:30 - 27:34
want you to draw
them on the floor.
• 27:34 - 27:38
Draw the level curves.
• 27:38 - 27:39
Remember what those were?
• 27:39 - 27:43
They were projections of
the curves on the surface
• 27:43 - 27:47
at the intersection
with z equals c planes.
• 27:47 - 27:49
You project them on the ground.
• 27:49 - 27:50
What do you think they are?
• 27:50 - 27:51
• 27:51 - 27:53
What are these?
• 27:53 - 27:58
If I take c, what
if c is positive?
• 27:58 - 28:02
• 28:02 - 28:05
What if c is 0?
• 28:05 - 28:12
What if c is less than 0?
• 28:12 - 28:14
What am I going to have?
• 28:14 - 28:18
you c equals 1, Magdalena.
• 28:18 - 28:20
Let's draw that.
• 28:20 - 28:20
OK.
• 28:20 - 28:22
Well, I'll try.
• 28:22 - 28:24
a and b would be 1, right, guys?
• 28:24 - 28:26
So a and b would be 1.
• 28:26 - 28:27
This is a square.
• 28:27 - 28:30
These would be the asymptotes.
• 28:30 - 28:36
So very, very
briefly, the hyperbola
• 28:36 - 28:41
would be this one-- x squared
minus y squared equals 1,
• 28:41 - 28:42
right?
• 28:42 - 28:45
If I have the last
case for c equals 1,
• 28:45 - 28:47
I'm going to have-- c
equals negative 1-- I'm
• 28:47 - 28:49
going to have the conjugate.
• 28:49 - 28:51
Are you guys with me?
• 28:51 - 28:58
So I'll have an a squared,
asymptotes, conjugate.
• 28:58 - 29:01
• 29:01 - 29:05
What if I have different level
c? c equals 1/2. c equals 2.
• 29:05 - 29:08
c equals pi. c
equals-- what are they?
• 29:08 - 29:12
I'm going to get
families of hyperbolas,
• 29:12 - 29:15
trenches that look like that.
• 29:15 - 29:16
Standard trenches and
conjugate trenches.
• 29:16 - 29:21
A multitude of them, an infinite
family of such hyperbolas,
• 29:21 - 29:22
an infinite family
of such hyperbolas.
• 29:22 - 29:25
I wanted to draw it.
• 29:25 - 29:29
What do I get when c is 0?
• 29:29 - 29:30
What are those?
• 29:30 - 29:32
STUDENT: Don't you
get, like, [INAUDIBLE]?
• 29:32 - 29:35
• 29:35 - 29:37
MAGDALENA TODA: They
get-- very good.
• 29:37 - 29:37
Why?
• 29:37 - 29:41
x squared minus y squared
• 29:41 - 29:45
me to y equals plus/minus 1.
• 29:45 - 29:48
And who are those y
equals plus/minus 1?
• 29:48 - 29:50
Exactly.
• 29:50 - 29:54
But exactly the first
bisector, which is y equals x.
• 29:54 - 29:56
They are [? then the ?]
function.
• 29:56 - 30:00
And the other one, y equals
negative [? x. ?] So these
• 30:00 - 30:01
are the asymptotes.
• 30:01 - 30:06
So I'm going to get a-- you guys
have to do this better than me.
• 30:06 - 30:07
Sorry.
• 30:07 - 30:09
These are all
hyperbolic trenches.
• 30:09 - 30:12
They are all going to
infinity like that.
• 30:12 - 30:15
And I'm sorry that
I'm giving you
• 30:15 - 30:17
a little bit too many hints.
• 30:17 - 30:19
This is part of your
• 30:19 - 30:21
I shouldn't talk
• 30:21 - 30:24
• 30:24 - 30:25
Any questions so far?
• 30:25 - 30:26
Is this hard?
• 30:26 - 30:28
Yes, sir?
• 30:28 - 30:28
No.
• 30:28 - 30:30
STUDENT: So [? spherically, ?]
• 30:30 - 30:32
equals y squared
minus x squared,
• 30:32 - 30:34
it's that same
picture, just flipped?
• 30:34 - 30:40
• 30:40 - 30:41
MAGDALENA TODA:
What would it be?
• 30:41 - 30:43
It would be the poor
• 30:43 - 30:44
STUDENT: Would be upside down.
• 30:44 - 30:46
MAGDALENA TODA:
--would be upside down.
• 30:46 - 30:50
Or projected in
something like a mirror.
• 30:50 - 30:51
I don't know how to say.
• 30:51 - 30:53
It would be exactly upside down.
• 30:53 - 30:56
So the reflection of that.
• 30:56 - 30:59
So you take all the points.
• 30:59 - 31:01
If you have-- I don't know.
• 31:01 - 31:03
It's hard to draw a reflection
in three dimensions.
• 31:03 - 31:04
But--
• 31:04 - 31:05
STUDENT: No, I understand.
• 31:05 - 31:09
MAGDALENA TODA:
Practically every curve
• 31:09 - 31:15
would be upside down with
respect to the floor.
• 31:15 - 31:16
OK.
• 31:16 - 31:17
All right.
• 31:17 - 31:21
I'm going to erase in one.
• 31:21 - 31:24
And you say, well, you've
• 31:24 - 31:26
like the domain and range.
• 31:26 - 31:31
like continuity and stuff?
• 31:31 - 31:33
• 31:33 - 31:51
Let me move on to 11.2.
• 31:51 - 32:00
Limits of functions of
the type z equals f of xy.
• 32:00 - 32:15
• 32:15 - 32:20
So what do you remember
• 32:20 - 32:23
of a function of one variable?
• 32:23 - 32:24
Comparison.
• 32:24 - 32:28
• 32:28 - 32:36
What about the limit if you
take [? z's, ?] I don't know.
• 32:36 - 32:38
I should look stunned.
• 32:38 - 32:39
And I should be stunned.
• 32:39 - 32:49
Of a function of y equals
f of x of one variable.
• 32:49 - 32:57
• 32:57 - 33:11
When do we say that
f has a limit at a?
• 33:11 - 33:12
• 33:12 - 33:15
STUDENT: When the [INAUDIBLE]
approaches from the right
• 33:15 - 33:17
and the left to the same value.
• 33:17 - 33:23
MAGDALENA TODA: Actually, that
was the simpler definition.
• 33:23 - 33:26
Let's think a little bit deeper.
• 33:26 - 33:35
We say that f has a
limit L at x equals a.
• 33:35 - 33:41
That's kind of the idea,
left and right limits.
• 33:41 - 33:44
But not both of them
have to exist, you see.
• 33:44 - 33:46
Maybe only the limit
from the left or limit
• 33:46 - 33:47
from the right only exists.
• 33:47 - 33:50
• 33:50 - 34:04
If, for any choice of values
of x, closer and closer, closer
• 34:04 - 34:24
and closer to a, we get that
F gets closer and closer to L.
• 34:24 - 34:27
And this "any" I put in.
• 34:27 - 34:34
My god, I put it in
a red circle thing,
• 34:34 - 34:40
because one could get
subsequencies of a sequence.
• 34:40 - 34:42
And for that subsequence
thing, things
• 34:42 - 34:45
look like I would have a limit.
• 34:45 - 34:48
And then you say,
well, but in the end,
• 34:48 - 34:51
I don't have a limit, because
I can get another subsequence
• 34:51 - 34:52
of the sequence.
• 34:52 - 34:59
And for that one, I'm not
going to have a limit.
• 34:59 - 35:04
Can you give me an example
of some crazy function that
• 35:04 - 35:09
does not have a limit at 0?
• 35:09 - 35:12
Example of a crazy function.
• 35:12 - 35:13
No.
• 35:13 - 35:15
No, don't write "crazy."
• 35:15 - 35:26
Of a function f of x
that is not defined at 0
• 35:26 - 35:44
and does not have
limit at 0, although it
• 35:44 - 35:53
is defined for values
arbitrarily close to 0.
• 35:53 - 36:00
• 36:00 - 36:08
Moreover, I want that function
to be drawn without-- I
• 36:08 - 36:23
want the function to be
drawn without leaving
• 36:23 - 36:26
the paper when I draw.
• 36:26 - 36:31
• 36:31 - 36:31
[INAUDIBLE]
• 36:31 - 36:35
• 36:35 - 36:44
So something that would
be defined on the whole 0
• 36:44 - 36:59
infinity except for 0 that
I can draw continuously
• 36:59 - 37:03
except when I get to 0, I
• 37:03 - 37:08
I don't have a limit
for that function.
• 37:08 - 37:09
You are close to that.
• 37:09 - 37:11
Sine of 1/x.
• 37:11 - 37:13
STUDENT: I said y equals 1/x.
• 37:13 - 37:15
MAGDALENA TODA: y equals 1/x.
• 37:15 - 37:16
Very good.
• 37:16 - 37:19
Let's see.
• 37:19 - 37:20
STUDENT: Oh, yeah. [INAUDIBLE].
• 37:20 - 37:21
MAGDALENA TODA: Yeah, yeah.
• 37:21 - 37:22
Both are excellent examples.
• 37:22 - 37:25
So let's see.
• 37:25 - 37:29
This guy is a very
nice function.
• 37:29 - 37:31
How do we draw him, or her?
• 37:31 - 37:33
Well, it's a her, right?
• 37:33 - 37:33
It's a she.
• 37:33 - 37:34
It's a function.
• 37:34 - 37:35
No, no.
• 37:35 - 37:36
In English, it doesn't
make any sense,
• 37:36 - 37:41
but if I think French, Italian,
Spanish, Romanian-- now
• 37:41 - 37:44
I speak both Italian
and Romanian--
• 37:44 - 37:47
we say it's a she,
it's a feminine.
• 37:47 - 37:52
So as I approach with values
closer and closer and closer
• 37:52 - 37:56
to 0, what happens
to my poor function?
• 37:56 - 37:59
It blows up.
• 37:59 - 37:59
OK.
• 37:59 - 38:05
So I have limit of 1/x from
the right and from the left.
• 38:05 - 38:08
If I take it from the
left, I don't care.
• 38:08 - 38:11
Let's take it only
from the right.
• 38:11 - 38:12
OK?
• 38:12 - 38:18
• 38:18 - 38:20
It's close to 0.
• 38:20 - 38:21
That's going to blow up, right?
• 38:21 - 38:25
• 38:25 - 38:25
And I restrict it.
• 38:25 - 38:30
So let's say, if I want
the domain to be containing
• 38:30 - 38:33
[? both, ?] that's also fine.
• 38:33 - 38:36
So if you guys want, we
can draw the other one.
• 38:36 - 38:37
• 38:37 - 38:40
The other one, I'm not
going to say where it goes.
• 38:40 - 38:43
But it's the same idea,
that as you approach 0
• 38:43 - 38:46
with closer and closer
and closer values,
• 38:46 - 38:48
it's going to blow up.
• 38:48 - 38:51
It's going to explode.
• 38:51 - 38:54
This is a beautiful function.
• 38:54 - 38:55
How beautiful [INAUDIBLE].
• 38:55 - 38:58
Beautiful with a
• 38:58 - 39:00
So I'm not going
to have a limit.
• 39:00 - 39:01
No limit.
• 39:01 - 39:03
Some people say, limit
exists and is infinity.
• 39:03 - 39:05
But does infinity exist?
• 39:05 - 39:07
Well, this is a
really philosophical,
• 39:07 - 39:11
religious notion, so I
don't want to get into it.
• 39:11 - 39:13
But in mathematics, we consider
that unless the limit is
• 39:13 - 39:17
finite, you cannot have a limit.
• 39:17 - 39:21
So if the limit is plus/minus
infinity, there is no limit.
• 39:21 - 39:25
Could the limit be different
or different subsequences?
• 39:25 - 39:28
This is what I
wanted to point out.
• 39:28 - 39:34
If you try this guy, you are
in real trouble on that guy.
• 39:34 - 39:35
Why?
• 39:35 - 39:37
You can have two.
• 39:37 - 39:39
If you have a graphing
calculator, which
• 39:39 - 39:44
I'm going to be opposed to you
being used in the classroom,
• 39:44 - 39:46
you would probably
see what happens.
• 39:46 - 39:52
Sine is defined on
all the real numbers.
• 39:52 - 39:54
But you cannot
have a value at 0,
• 39:54 - 39:58
because the 1/x is
not defined at 0.
• 39:58 - 40:02
Imagine you get closer and
closer to 0 from both sides.
• 40:02 - 40:05
I cannot draw very beautifully.
• 40:05 - 40:10
But as 1, this is plus
1 and this is minus 1.
• 40:10 - 40:12
I'm going to have some behavior.
• 40:12 - 40:15
And how many of you have seen
that on a computer screen
• 40:15 - 40:16
or calculator?
• 40:16 - 40:16
You've seen.
• 40:16 - 40:18
Yeah, you've seen.
• 40:18 - 40:20
By the way, did you
see the Lubbuck High?
• 40:20 - 40:23
Was it in high school you saw
it the first time in Calc 1
• 40:23 - 40:25
or pre-calc?
• 40:25 - 40:28
STUDENT: [INAUDIBLE]
Algebra 1 with Mr. West.
• 40:28 - 40:29
[INAUDIBLE]
• 40:29 - 40:33
MAGDALENA TODA: So I'll
try-- oh, guys, you
• 40:33 - 40:34
have to be patient with me.
• 40:34 - 40:38
I'm not leaving the poor board
with the tip of my pencil.
• 40:38 - 40:39
I'm not leaving him.
• 40:39 - 40:42
I have continuity.
• 40:42 - 40:46
As I got closer to this, I still
have the [INAUDIBLE] property.
• 40:46 - 40:47
Anyway, it's OK.
• 40:47 - 40:48
I'm not leaving this.
• 40:48 - 40:52
I am taking all the values
possible between minus 1 and 1.
• 40:52 - 40:55
So on intervals that
are smaller, smaller,
• 40:55 - 40:58
I'm really taking all the
values between minus 1 and 1,
• 40:58 - 41:01
and really rapidly--
[INAUDIBLE].
• 41:01 - 41:08
When I'm getting closer to 0,
I'm not going to have a limit.
• 41:08 - 41:10
But as somebody
may say, but wait.
• 41:10 - 41:12
When I have a sequence
of values that
• 41:12 - 41:14
is getting closer
and closer to 0,
• 41:14 - 41:19
is that no guarantee that
I'm going to have a limit?
• 41:19 - 41:20
Nope.
• 41:20 - 41:21
It depends.
• 41:21 - 41:25
If you say "any," it has to
be for any choice of points,
• 41:25 - 41:28
any choice of points
that you go closer to 0.
• 41:28 - 41:30
Not for one sequence
of points that
• 41:30 - 41:32
is getting closer
and closer to 0.
• 41:32 - 41:35
For example, if your
choice of points is this,
• 41:35 - 41:36
choice of points.
• 41:36 - 41:40
• 41:40 - 41:44
Getting closer to 0.
• 41:44 - 41:50
[INAUDIBLE] xn
equals 1 over 2 pi n.
• 41:50 - 41:52
Isn't this going to 0?
• 41:52 - 41:52
Yeah.
• 41:52 - 41:54
It then goes to infinity.
• 41:54 - 41:56
This sequence goes to 0.
• 41:56 - 41:56
What is it?
• 41:56 - 41:57
1 over 2 pi?
• 41:57 - 41:58
1 over 4 pi?
• 41:58 - 41:59
1 over 8 pi?
• 41:59 - 42:00
1 over 16 pi?
• 42:00 - 42:01
1 over 32 pi?
• 42:01 - 42:03
1 over 64 pi?
• 42:03 - 42:04
This is what my
son is doing to me.
• 42:04 - 42:06
• 42:06 - 42:07
OK?
• 42:07 - 42:08
He's 10 years old.
• 42:08 - 42:10
He's so funny.
• 42:10 - 42:13
Now, another choice of points.
• 42:13 - 42:20
• 42:20 - 42:21
Ah.
• 42:21 - 42:26
Somebody-- all of you are
smart enough to do this.
• 42:26 - 42:29
What do you think
I'm going to pick?
• 42:29 - 42:31
1 over what?
• 42:31 - 42:34
And when [? other ?]
something that goes to 0
• 42:34 - 42:35
then goes to infinity.
• 42:35 - 42:42
And I know that your
professor showed you that.
• 42:42 - 42:45
pi over 2 plus 2 pi n.
• 42:45 - 42:46
Doesn't this go to 0?
• 42:46 - 42:46
Yes.
• 42:46 - 42:50
As n gets bigger and
bigger, this is going to 0.
• 42:50 - 42:51
However, there is no limit.
• 42:51 - 42:52
Why?
• 42:52 - 42:59
Well, for the first sequence,
as xn goes to 0, f of xn
• 42:59 - 43:04
goes to-- what is
sine of-- OK, I
• 43:04 - 43:06
am too lazy to write this down.
• 43:06 - 43:11
Sine of 1 over 1 over--
of 1 over 1 over 2 pi?
• 43:11 - 43:15
• 43:15 - 43:17
STUDENT: It's the
sine over 2 pi.
• 43:17 - 43:21
MAGDALENA TODA: This
is sine of 2 pi n.
• 43:21 - 43:22
And how much is that?
• 43:22 - 43:23
STUDENT: 0.
• 43:23 - 43:24
MAGDALENA TODA: 0.
• 43:24 - 43:26
So this is a 0.
• 43:26 - 43:29
And this is a--
this converges to 0.
• 43:29 - 43:31
So I say, oh, so maybe I have
a limit, and that'll be 0.
• 43:31 - 43:33
Wrong.
• 43:33 - 43:36
That would be the rapid,
stupid conclusion.
• 43:36 - 43:39
If somebody jumps [? up, ?]
I picked some points,
• 43:39 - 43:42
I formed the sequence that
gets closer and closer to 0.
• 43:42 - 43:44
I'm sure that the limit exists.
• 43:44 - 43:46
I've got a 0.
• 43:46 - 43:49
Well, did you think of
any possible choice?
• 43:49 - 43:50
That's the problem.
• 43:50 - 43:52
You have to have
any possible choice.
• 43:52 - 44:03
F of yn sine of 1 over
pi over 2 plus 1 over 1
• 44:03 - 44:09
over-- Magdalena-- pi
over 2 plus 2 pi n.
• 44:09 - 44:11
So we saw that this was 0.
• 44:11 - 44:15
What happens to sine of
1 over 1 over sine of pi
• 44:15 - 44:19
over 2 plus 2 pi n?
• 44:19 - 44:20
And where does this go?
• 44:20 - 44:21
It then goes to infinity.
• 44:21 - 44:27
• 44:27 - 44:29
This sequence goes to 0.
• 44:29 - 44:33
What is f of the
sequence going to?
• 44:33 - 44:34
To another limit.
• 44:34 - 44:36
So there is no limit.
• 44:36 - 44:39
What's the limit of
this subsequence?
• 44:39 - 44:41
It's a constant one, right?
• 44:41 - 44:46
Because look, what does it
mean pi over 2 plus 2 pi n?
• 44:46 - 44:49
Where am I on the unit
trigonometric circle?
• 44:49 - 44:51
[INTERPOSING VOICES]
• 44:51 - 44:54
Always here, right?
• 44:54 - 44:56
Always on the sort of
like the north pole.
• 44:56 - 44:59
So what is the sine
of this north pole?
• 44:59 - 45:00
STUDENT: 1.
• 45:00 - 45:01
MAGDALENA TODA: Always 1.
• 45:01 - 45:02
So I get the limit 1.
• 45:02 - 45:06
So I'm done because there
are two different limits.
• 45:06 - 45:09
So pay attention to
this type of problem.
• 45:09 - 45:18
Somebody can get you in trouble
with this kind of thing.
• 45:18 - 45:20
On the other hand,
• 45:20 - 45:23
what if I want to make this
a function of two variables?
• 45:23 - 45:28
• 45:28 - 45:31
So I'll say, one
point extra credit.
• 45:31 - 45:34
I'm giving you too
much extra credit.
• 45:34 - 45:36
Maybe I give you
too much-- it's OK.
• 45:36 - 45:40
One point extra credit--
put them together.
• 45:40 - 45:43
• 45:43 - 45:48
Does f-- do you
like to do the f?
• 45:48 - 45:51
I used big F, and then I
changed it to little f.
• 45:51 - 45:54
This time I have a function
of two variables-- little
• 45:54 - 46:01
f with xy-- to be sine of 1
over x squared plus y squared.
• 46:01 - 46:09
Does this function have a
limit at the point 0, 0?
• 46:09 - 46:12
• 46:12 - 46:16
So when I approach 0,
0, do I have a limit?
• 46:16 - 46:17
OK.
• 46:17 - 46:20
And you say, well, it depends
how I approach that 0, 0.
• 46:20 - 46:21
That's exactly the thing.
• 46:21 - 46:23
Yes, sir.
• 46:23 - 46:25
Oh, you didn't want to ask me.
• 46:25 - 46:28
• 46:28 - 46:37
And does f of xy
equals-- let me give you
• 46:37 - 46:41
another one, a
really sexy one. x
• 46:41 - 46:45
squared plus y squared
times sine of 1
• 46:45 - 46:48
over x squared plus y squared.
• 46:48 - 46:55
Have a limit at 0, 0?
• 46:55 - 47:00
• 47:00 - 47:01
I don't know.
• 47:01 - 47:04
Continuous it cannot be,
because it's not defined there.
• 47:04 - 47:05
Right?
• 47:05 - 47:08
For a function to be
continuous at a point,
• 47:08 - 47:11
the function has to
satisfy three conditions.
• 47:11 - 47:15
The function has to be
defined there at that point.
• 47:15 - 47:17
The function has to
have a limit there
• 47:17 - 47:19
at that point of the domain.
• 47:19 - 47:23
And the limit and the function
value have to coincide.
• 47:23 - 47:25
Three conditions.
• 47:25 - 47:28
continuity later.
• 47:28 - 47:30
Hint.
• 47:30 - 47:32
Magdalena, too many hints.
• 47:32 - 47:34
This should remind
you of somebody
• 47:34 - 47:36
from the first
variable calculus.
• 47:36 - 47:38
It's a more challenging problem.
• 47:38 - 47:40
That's why I gave
it to extra credit.
• 47:40 - 47:46
If I had x sine of 1/x,
what would that look like?
• 47:46 - 47:47
STUDENT: x times--
• 47:47 - 47:50
MAGDALENA TODA: x
times sine of 1/x.
• 47:50 - 47:55
When I approach 0
with-- so if I have-- I
• 47:55 - 47:57
• 47:57 - 47:59
You go home, you think about it.
• 47:59 - 48:00
You take the calculator.
• 48:00 - 48:06
But keep in mind that your
calculator can fool you.
• 48:06 - 48:11
Sometimes it can show an
image that misguides you.
• 48:11 - 48:15
So you have to think
how to do that.
• 48:15 - 48:19
sine of 1/x when--
• 48:19 - 48:22
does it have a limit
when x goes to 0?
• 48:22 - 48:24
Is there such a limit?
• 48:24 - 48:25
Does it exist?
• 48:25 - 48:28
• 48:28 - 48:31
So if such a limit
would exist, maybe we
• 48:31 - 48:36
can extend by continuity the
function x times sine over x.
• 48:36 - 48:37
What does it mean?
• 48:37 - 48:39
Like, extend it, prolong it.
• 48:39 - 48:44
And say, it's this 4x equals
0 and this if x is not 0.
• 48:44 - 48:48
So this is obviously x is
different from 0, right?
• 48:48 - 48:50
Can we extend it by continuity?
• 48:50 - 48:51
• 48:51 - 48:54
• 48:54 - 48:58
And I think it's time for me
to keep the promise I made
• 48:58 - 49:05
to [? Aaron, ?]
because I see no way.
• 49:05 - 49:08
Oh, my god, [? Aaron, ?]
I see no way out.
• 49:08 - 49:10
• 49:10 - 49:14
The epsilon delta
definition of limit.
• 49:14 - 49:17
[? Right? ?] OK.
• 49:17 - 49:21
So what does it mean for a
real mathematician or somebody
• 49:21 - 49:25
with a strong mathematical
foundation and education
• 49:25 - 49:27
that they know the
true definition
• 49:27 - 49:31
of a limit of a function
of, let's say, one variable?
• 49:31 - 49:35
The epsilon delta, the one your
• 49:35 - 49:40
try to fool you when avoid it
• 49:40 - 49:42
People try to avoid
the epsilon delta,
• 49:42 - 49:46
because they think the students
will never, never understand
• 49:46 - 49:50
it, because it's
such an abstract one.
• 49:50 - 49:52
• 49:52 - 49:53
I wasn't smart enough.
• 49:53 - 49:58
I think I was 16 when I was
• 49:58 - 49:59
competitions.
• 49:59 - 50:03
And one professor taught me
the epsilon delta and said,
• 50:03 - 50:05
do you understand it?
• 50:05 - 50:07
My 16-year-old mind said, no.
• 50:07 - 50:09
But guess what?
• 50:09 - 50:11
Some other people
smarter than me,
• 50:11 - 50:12
they told me, when
you first see it,
• 50:12 - 50:17
you don't understand
it in any case.
• 50:17 - 50:20
So it takes a little bit
more time to sink in.
• 50:20 - 50:22
So the same idea.
• 50:22 - 50:25
As I'm getting closer and
closer and closer and closer
• 50:25 - 50:30
to an x0 with my x values
from anywhere around-- left,
• 50:30 - 50:35
right-- I have to pick an
arbitrary choice of points
• 50:35 - 50:40
going towards x0, I have to
be sure that at the same time,
• 50:40 - 50:45
the corresponding sequence
of values is going to L,
• 50:45 - 50:47
I can express that
in epsilon delta.
• 50:47 - 50:51
• 50:51 - 50:52
So we say that.
• 50:52 - 51:00
• 51:00 - 51:13
f of x has limit L
at x equals x0 if.
• 51:13 - 51:17
• 51:17 - 51:24
For every epsilon positive, any
choice of an epsilon positive,
• 51:24 - 51:25
there is a delta.
• 51:25 - 51:27
There exists-- oh, OK, guys.
• 51:27 - 51:28
You don't know the symbols.
• 51:28 - 51:31
I'll write it in English.
• 51:31 - 51:36
For every epsilon
positive, no matter
• 51:36 - 51:41
how small-- put
parentheses, because you
• 51:41 - 51:47
are just [? tired-- ?]
no matter how small,
• 51:47 - 51:56
there exists a delta number
that depends on epsilon.
• 51:56 - 52:02
• 52:02 - 52:16
So that whenever x minus
x0 is less than delta,
• 52:16 - 52:34
this would imply
that f of x minus L,
• 52:34 - 52:37
that limit I taught you
• 52:37 - 52:39
is less than epsilon.
• 52:39 - 52:48
• 52:48 - 52:50
What does this mean?
• 52:50 - 52:55
I'm going to try
and draw something
• 52:55 - 52:58
that happens on a line.
• 52:58 - 53:00
So this is x0.
• 53:00 - 53:04
And these are my values of x.
• 53:04 - 53:05
They can come from anywhere.
• 53:05 - 53:09
• 53:09 - 53:12
And this is f of x.
• 53:12 - 53:17
And this is L. So it
says, no matter-- this
• 53:17 - 53:19
says-- this is an
abstract way of saying,
• 53:19 - 53:24
no matter how close, you see,
for every epsilon positive,
• 53:24 - 53:27
no matter how close
you get to the L.
• 53:27 - 53:31
I decide to be in this
interval, very tiny epsilon.
• 53:31 - 53:32
L minus epsilon.
• 53:32 - 53:36
L plus epsilon L. You give
• 53:36 - 53:39
You say, Magdalena, pick
something really small.
• 53:39 - 53:42
Big epsilon to be 0.00001.
• 53:42 - 53:44
• 53:44 - 53:48
Well, if I really
have a limit there,
• 53:48 - 53:54
an L at x0, that means that
no matter how much you shrink
• 53:54 - 53:58
this interval for me, you
can be mean and shrink it
• 53:58 - 53:59
as much as you want.
• 53:59 - 54:03
I will still find a
small interval around x0.
• 54:03 - 54:07
• 54:07 - 54:09
[? But ?] I will
still find the smaller
• 54:09 - 54:13
interval around x0, which is--
this would be x0 minus delta.
• 54:13 - 54:16
This would be x0 plus delta.
• 54:16 - 54:21
So that the image of this
purple interval fits inside.
• 54:21 - 54:22
You say, what?
• 54:22 - 54:26
So that the image of this
purple interval fits inside.
• 54:26 - 54:30
So f of x minus L, the distance
is still that, less than xy.
• 54:30 - 54:31
Yes, sir?
• 54:31 - 54:33
STUDENT: Where'd you
get epsilon [INAUDIBLE]?
• 54:33 - 54:34
MAGDALENA TODA: So
epsilon has to be
• 54:34 - 54:38
chose no matter how small.
• 54:38 - 54:40
STUDENT: [INAUDIBLE].
• 54:40 - 54:41
MAGDALENA TODA: Huh?
• 54:41 - 54:42
Real number.
• 54:42 - 54:46
So I'm saying, you should not
set the epsilon to be 0.0001.
• 54:46 - 54:48
That would be a mistake.
• 54:48 - 54:51
You have to think of that number
as being as small as you want,
• 54:51 - 54:55
infinitesimally small, smaller
than any particle in physics
• 54:55 - 54:57
• 54:57 - 55:00
And this is what I had the
problem understanding--
• 55:00 - 55:04
that notion of-- not
the notion of, hey, not
• 55:04 - 55:06
matter how close
I am, I can still
• 55:06 - 55:12
get something even smaller
around x0 that fits in this.
• 55:12 - 55:14
That's not what I
• 55:14 - 55:18
The notion is to perceive
an infinitesimal.
• 55:18 - 55:22
Our mind is too limited
to understand infinity.
• 55:22 - 55:24
It's like trying
to understand God.
• 55:24 - 55:30
And the same limitation comes
with microscopic problems.
• 55:30 - 55:31
Yeah, we can see some
things on the microscope,
• 55:31 - 55:32
and we understand.
• 55:32 - 55:35
Ah, I understand I
have this bacteria.
• 55:35 - 55:36
This is staph.
• 55:36 - 55:37
Oh, my god.
• 55:37 - 55:43
But then there are molecules,
atoms, subatomic particles
• 55:43 - 55:47
that we don't understand,
because our mind is really
• 55:47 - 55:49
[? small. ?] Imagine
something smaller
• 55:49 - 55:51
than the subatomic particles.
• 55:51 - 55:55
That's the abstract notion
of infinitesimally small.
• 55:55 - 55:59
So I'm saying, if I really
have a limit L there,
• 55:59 - 56:03
that means no matter how small
I have this ball around it,
• 56:03 - 56:07
I can still find a
smaller ball that
• 56:07 - 56:10
fits-- whose image fits inside.
• 56:10 - 56:10
All right?
• 56:10 - 56:15
The same kind of definition--
I will try to generalize this.
• 56:15 - 56:20
Can you guys help me
generalize this limit notion
• 56:20 - 56:25
to the notion of function
of two variables?
• 56:25 - 56:29
• 56:29 - 56:41
So we say, that f of xy
has the limit L at x0y0.
• 56:41 - 56:45
• 56:45 - 56:51
What was x0y0 when I
• 56:51 - 56:53
example did I give you guys?
• 56:53 - 56:55
Sine of 1 over x squared
plus y squared, right?
• 56:55 - 56:56
Something like that.
• 56:56 - 56:57
I don't know.
• 56:57 - 57:00
I said, think of 0, 0.
• 57:00 - 57:02
That was the given point.
• 57:02 - 57:04
It has to be a fixed couple.
• 57:04 - 57:08
So you think of the origin, 0,
0, as being as a fixed couple.
• 57:08 - 57:12
Or you think of the point 1,
0 as being as a fixed couple
• 57:12 - 57:15
in that plane you look at.
• 57:15 - 57:18
That is the fixed couple.
• 57:18 - 57:21
If-- now somebody
has to help me.
• 57:21 - 57:28
For every epsilon positive,
no matter how small,
• 57:28 - 57:31
that's where I have a problem
imagining infinitesimally
• 57:31 - 57:32
small.
• 57:32 - 57:35
There exists-- I no
longer have this problem.
• 57:35 - 57:37
when I was in my 20s.
• 57:37 - 57:40
I don't want to go back to
my 20s and have-- I mean,
• 57:40 - 57:41
I would love to.
• 57:41 - 57:43
[LAUGHTER]
• 57:43 - 57:46
To go having vacations
with no worries and so on.
• 57:46 - 57:49
But I wouldn't like
to go back to my 20s
• 57:49 - 57:50
and have to relearn
all the mathematics.
• 57:50 - 57:51
Now way.
• 57:51 - 57:53
That was too much of a struggle.
• 57:53 - 58:00
There exists a delta positive
that depends on epsilon.
• 58:00 - 58:03
What does it mean,
depends on epsilon?
• 58:03 - 58:05
Because guys, imagine
you make this epsilon
• 58:05 - 58:06
smaller and smaller.
• 58:06 - 58:08
You have to make delta
smaller and smaller,
• 58:08 - 58:12
so that you can fit that
little ball in the big ball.
• 58:12 - 58:13
OK?
• 58:13 - 58:20
That depends on epsilon,
so that whenever-- now,
• 58:20 - 58:22
that is a big problem.
• 58:22 - 58:28
How do I say, distance between
the point xy and the point
• 58:28 - 58:29
x0y0?
• 58:29 - 58:32
Oh, my god.
• 58:32 - 58:37
This is distance between xy
and x0y0 is less than delta.
• 58:37 - 58:48
This would imply
that-- well, this
• 58:48 - 58:54
is a function with values in
R. This is in R. Real number.
• 58:54 - 58:55
So I don't have a problem.
• 58:55 - 58:57
I can use absolute value here.
• 58:57 - 59:11
Absolute value of f of
the couple xy minus L
• 59:11 - 59:15
is less than epsilon.
• 59:15 - 59:19
The thing is, can you
visualize that little ball,
• 59:19 - 59:21
that little disk?
• 59:21 - 59:22
What do I mean?
• 59:22 - 59:26
Being close, xy is me, right?
• 59:26 - 59:28
But I'm moving.
• 59:28 - 59:29
I'm the moving point.
• 59:29 - 59:30
I'm dancing around.
• 59:30 - 59:33
And [? Nateesh ?] is x0y0.
• 59:33 - 59:38
How do I say that I have
to be close enough to him?
• 59:38 - 59:39
I cannot touch him.
• 59:39 - 59:40
That's against the rules.
• 59:40 - 59:42
That's considered
[INAUDIBLE] harassment.
• 59:42 - 59:46
But I can come as
close as I want.
• 59:46 - 59:49
So I say, the
distance between me--
• 59:49 - 59:52
I'm xy-- and
[? Nateesh, ?] who is
• 59:52 - 59:58
fixed x0y0, has to be smaller
than that small delta.
• 59:58 - 60:01
How do I represent that
in plane mathematics?
• 60:01 - 60:02
STUDENT: Doesn't [INAUDIBLE]?
• 60:02 - 60:06
• 60:06 - 60:07
MAGDALENA TODA: Exactly.
• 60:07 - 60:09
So that delta has to
be small enough so
• 60:09 - 60:17
that the image of f at me minus
the limit is less than epsilon.
• 60:17 - 60:21
Now you understand why all
the other teachers avoid
• 60:21 - 60:23
[? one. ?] So I
• 60:23 - 60:28
want to get small enough-- not
too close-- but close enough
• 60:28 - 60:40
to him, so that my value--
I'm f of xy-- minus the limit,
• 60:40 - 60:42
the limit-- I have
a preset limit.
• 60:42 - 60:45
All around [? Nateesh, ?] I
can have different values,
• 60:45 - 60:47
no matter where I go.
• 60:47 - 60:51
My value at all these points
around [? Nateesh ?] have
• 60:51 - 60:55
to be close enough
to L. So I say,
• 60:55 - 60:58
well, you have to get
close enough to L.
• 60:58 - 60:59
Somebody presents me an epsilon.
• 60:59 - 61:02
Then I have to reduce my
distance to [? Nateesh ?]
• 61:02 - 61:04
depending to that epsilon.
• 61:04 - 61:08
Because otherwise,
the image doesn't fit.
• 61:08 - 61:09
It's a little bit tricky.
• 61:09 - 61:11
STUDENT: So is this like
the squeeze theorem kind of?
• 61:11 - 61:12
MAGDALENA TODA: It is
the squeeze theorem.
• 61:12 - 61:13
STUDENT: Oh, all right.
• 61:13 - 61:14
MAGDALENA TODA: OK?
• 61:14 - 61:19
So the squeezing-- I ball into
another [? ball ?] [? limit. ?]
• 61:19 - 61:21
This is why-- it's not
a ball, but it's a--
• 61:21 - 61:22
STUDENT: A circle.
• 61:22 - 61:23
MAGDALENA TODA: Disk.
• 61:23 - 61:24
A circle, right?
• 61:24 - 61:29
So how do we express
that in Calc 3 in plain?
• 61:29 - 61:31
This is the
[? ingredient, ?] distance d.
• 61:31 - 61:34
So Seth, can you tell me what is
the distance between these two
• 61:34 - 61:35
points?
• 61:35 - 61:36
Square root of--
• 61:36 - 61:37
STUDENT: [INAUDIBLE].
• 61:37 - 61:42
MAGDALENA TODA: x minus
x0 squared plus y minus y0
• 61:42 - 61:43
squared.
• 61:43 - 61:45
Now shut up. [? And I ?]
am talking to myself.
• 61:45 - 61:46
STUDENT: Must be
less than delta.
• 61:46 - 61:47
[LAUGHTER]
• 61:47 - 61:48
MAGDALENA TODA: Less than delta.
• 61:48 - 61:52
writing this, I need
• 61:52 - 61:54
to write that I can
do that in my mind.
• 61:54 - 61:58
• 61:58 - 62:00
OK?
• 62:00 - 62:01
All right.
• 62:01 - 62:02
This is hard.
• 62:02 - 62:03
We need to sleep on that.
• 62:03 - 62:09
I have one or two more problems
that are less hard-- nah,
• 62:09 - 62:11
they are still hard, but
they are more intuitive,
• 62:11 - 62:15
that I would like to
• 62:15 - 62:17
I'm going to give
you a function.
• 62:17 - 62:21
And we would have to visualize
as I get closer to a point
• 62:21 - 62:25
where I am actually going.
• 62:25 - 62:30
So I have this nasty
function, f of xy
• 62:30 - 62:35
equals xy over z
squared plus y squared.
• 62:35 - 62:39
• 62:39 - 62:45
And I'm saying, [INAUDIBLE]
the point is the origin.
• 62:45 - 62:47
I choose the origin.
• 62:47 - 62:48
Question.
• 62:48 - 62:53
Do I have a limit that's--
do I have a limit?
• 62:53 - 62:55
Not [? really ?] for me.
• 62:55 - 63:02
Does f have a limit
at the origin?
• 63:02 - 63:06
• 63:06 - 63:10
You would have to imagine
that you'd draw this function.
• 63:10 - 63:13
And except you cannot draw, and
you really don't care to draw
• 63:13 - 63:14
it.
• 63:14 - 63:17
You only have to imagine that
you have some abstract graph--
• 63:17 - 63:19
z equals f of xy.
• 63:19 - 63:21
You don't care
what it looks like.
• 63:21 - 63:24
But then you take
points on the floor,
• 63:24 - 63:27
just like I did the exercise
with [? Nateesh ?] before.
• 63:27 - 63:31
And you get closer and
closer to the origin.
• 63:31 - 63:34
But no attention-- no
matter what path I take,
• 63:34 - 63:37
I have to get the same limit.
• 63:37 - 63:38
What?
• 63:38 - 63:47
No matter what path I take
towards [? Nateesh-- ?]
• 63:47 - 63:53
don't write that down-- towards
[? z0y0, ?] I have to get
• 63:53 - 63:54
the same limit.
• 63:54 - 63:57
• 63:57 - 63:59
Do I?
• 63:59 - 64:04
Let's imagine with the
• 64:04 - 64:07
And [? Nateesh ?]
is the point 0, 0.
• 64:07 - 64:11
And you are aspiring to get
closer and closer to him
• 64:11 - 64:13
without touching him.
• 64:13 - 64:15
Because otherwise,
he's going to sue you.
• 64:15 - 64:18
So what do we have here?
• 64:18 - 64:19
We have different paths?
• 64:19 - 64:21
How can I get closer?
• 64:21 - 64:26
Either on this path
or maybe on this path.
• 64:26 - 64:28
Or maybe on this path.
• 64:28 - 64:32
Or maybe, if I had something
to drink last night-- which
• 64:32 - 64:35
I did not, because
after the age of 35,
• 64:35 - 64:37
I stopped drinking completely.
• 64:37 - 64:41
• 64:41 - 64:45
That's when I decided
I want to be a mom,
• 64:45 - 64:47
and I didn't want to
• 64:47 - 64:50
So no matter what path you
take, you can make it wiggly,
• 64:50 - 64:52
you can make it
any way you want.
• 64:52 - 64:54
We are still approaching 0, 0.
• 64:54 - 64:56
You still have to
get the same limit.
• 64:56 - 65:00
Oh, that's tricky, because
it's also the same in life.
• 65:00 - 65:02
Depending on the path
you take in life,
• 65:02 - 65:05
you have different
results, different limits.
• 65:05 - 65:11
Now, what if I take the path
number one, number two, number
• 65:11 - 65:13
three possibility.
• 65:13 - 65:17
And number [? blooie ?]
is the drunken variant.
• 65:17 - 65:22
That is hard to
implement in an exercise.
• 65:22 - 65:27
Imagine that I have
limit along the path one.
• 65:27 - 65:28
Path one.
• 65:28 - 65:35
xy goes to 0, 0 of xy over
x squared plus y squared.
• 65:35 - 65:37
Do you guys see what's
going to happen?
• 65:37 - 65:41
So I'm along the--
OK, here it is.
• 65:41 - 65:47
This line, right, this is
the x-axis, y-axis, z-axis.
• 65:47 - 65:49
What's special for the x-axis?
• 65:49 - 65:50
Who is 0?
• 65:50 - 65:53
STUDENT: x.
• 65:53 - 65:53
STUDENT: yz.
• 65:53 - 65:54
MAGDALENA TODA: y is 0.
• 65:54 - 65:57
So y is 0.
• 65:57 - 65:59
So y is 0.
• 65:59 - 66:00
Don't laugh at me.
• 66:00 - 66:03
I'm going to write like
that because it's easier.
• 66:03 - 66:07
And it's going to be
something like limit
• 66:07 - 66:14
when x approaches 0
of x over x squared.
• 66:14 - 66:16
STUDENT: It's 1/x.
• 66:16 - 66:18
MAGDALENA TODA: Times 0 up.
• 66:18 - 66:19
Oh, my god.
• 66:19 - 66:21
Is that-- how much is that?
• 66:21 - 66:21
STUDENT: 0.
• 66:21 - 66:21
STUDENT: 0.
• 66:21 - 66:22
MAGDALENA TODA: 0!
• 66:22 - 66:23
I'm happy.
• 66:23 - 66:24
I say, maybe I have the limit.
• 66:24 - 66:25
I have the limit 0.
• 66:25 - 66:27
No, never rush in life.
• 66:27 - 66:28
Check.
• 66:28 - 66:31
Experiment any other paths.
• 66:31 - 66:35
And it's actually very easy
to see where I can go wrong.
• 66:35 - 66:40
If I take the path number two,
I will get the same result.
• 66:40 - 66:41
You don't need a
lot of imagination
• 66:41 - 66:44
to realize, hey, whether
she does it for x
• 66:44 - 66:48
or does it for y, if she
goes along the 2, what
• 66:48 - 66:50
the heck is going to happen?
• 66:50 - 66:51
y is going to shrink.
• 66:51 - 66:53
x will always be 0.
• 66:53 - 66:57
Because this means
a point's like what?
• 66:57 - 66:59
0,1.
• 66:59 - 67:01
0, 1/2.
• 67:01 - 67:03
0, 1/n, and so on.
• 67:03 - 67:08
But plug them all in here,
I get 0, 1/n times 0.
• 67:08 - 67:09
It's still 0.
• 67:09 - 67:10
So I still get 0.
• 67:10 - 67:12
Path two.
• 67:12 - 67:15
When I approach my--
xt goes to 0, 0.
• 67:15 - 67:19
The poor [? Nateesh ?]
• 67:19 - 67:21
I still get 0.
• 67:21 - 67:24
Let's take not the
drunken path, because I
• 67:24 - 67:26
don't know [? it unless ?]
the sine function.
• 67:26 - 67:27
That is really crazy.
• 67:27 - 67:29
I'll take this one.
• 67:29 - 67:31
What is this one,
• 67:31 - 67:33
Is that going to help me?
• 67:33 - 67:36
I don't know, but I
need some intuition.
• 67:36 - 67:40
Mathematicians need intuition
and a lot of patience.
• 67:40 - 67:42
• 67:42 - 67:45
The one in the middle, I'm going
to start walking on that, OK,
• 67:45 - 67:47
until you tell me what it is.
• 67:47 - 67:48
STUDENT: y [INAUDIBLE].
• 67:48 - 67:50
MAGDALENA TODA: y equals
x is the first bisector
• 67:50 - 67:51
• 67:51 - 67:55
And I'm very happy
I can go both ways.
• 67:55 - 67:56
y equals x.
• 67:56 - 67:56
x [INAUDIBLE].
• 67:56 - 68:07
So limit when x equals y,
but the pair xy goes to 0,0.
• 68:07 - 68:08
I'm silly.
• 68:08 - 68:11
I can say that,
well, Magdalena, this
• 68:11 - 68:16
is the pair xx,
because x equals what?
• 68:16 - 68:17
Let me plug them in.
• 68:17 - 68:19
So it's like two people.
• 68:19 - 68:21
x and y are married.
• 68:21 - 68:22
They are a couple, a pair.
• 68:22 - 68:24
They look identical.
• 68:24 - 68:26
Sometimes it happens.
• 68:26 - 68:28
Like twins, they
start looking alike,
• 68:28 - 68:31
dressing alike, and so on.
• 68:31 - 68:37
The x and the y have to
• 68:37 - 68:41
And you have to tell me what
in the world the limit will be.
• 68:41 - 68:44
• 68:44 - 68:44
STUDENT: 1/2.
• 68:44 - 68:46
MAGDALENA TODA: 1/2.
• 68:46 - 68:47
Oh, my god.
• 68:47 - 68:48
So now I'm deflated.
• 68:48 - 68:52
So now I realize that
taking two different paths,
• 68:52 - 68:58
I show that I have-- on
this path, I have 1/2.
• 68:58 - 69:00
On this path, I have 0.
• 69:00 - 69:01
I don't match.
• 69:01 - 69:03
I don't have an overall limit.
• 69:03 - 69:10
no overall limit.
• 69:10 - 69:11
Oh, my god.
• 69:11 - 69:15
So what you need to
• 69:15 - 69:18
section 11.1 and section 11.2.
• 69:18 - 69:21
And I will ask you next
time-- and you can lie,
• 69:21 - 69:23
you can do whatever.
• 69:23 - 69:26
Did the book explain
better than me,
• 69:26 - 69:29
or I explain better
than the book?
• 69:29 - 69:32
This type of example when
the limit does not exist.
• 69:32 - 69:33
We are going to
see more examples.
• 69:33 - 69:38
You are going to see examples
where the limit does exist.
• 69:38 - 69:40
Now, one last thing.
• 69:40 - 69:47
When you have to compute limits
of compositions of functions
• 69:47 - 69:49
whose limit exist--
for example, you
• 69:49 - 69:58
know that limit is
xy goes to x0y0 of f
• 69:58 - 70:10
of xy [INAUDIBLE] limit
of xy go to x0y0 of gxy
• 70:10 - 70:14
is L-- L-- L-- M-- M.
• 70:14 - 70:25
How are you going to compute the
limit of alpha f plus beta g?
• 70:25 - 70:27
This is in the book.
• 70:27 - 70:33
But you don't need the
book to understand that.
• 70:33 - 70:34
• 70:34 - 70:39
because this is the equivalent
thing to the function of one
• 70:39 - 70:41
variable thing in Calc 1.
• 70:41 - 70:44
So if you would only
have f of x or g of x,
• 70:44 - 70:45
it would be piece of cake.
• 70:45 - 70:47
What would you say?
• 70:47 - 70:47
STUDENT: [INAUDIBLE].
• 70:47 - 70:48
MAGDALENA TODA: Right.
• 70:48 - 70:54
Alpha times L plus beta
times M. Can you also
• 70:54 - 70:55
multiply functions.
• 70:55 - 70:56
Yes, you can.
• 70:56 - 71:07
Limit of fg as xy goes
to x0 or y0-- will be LM.
• 71:07 - 71:10
How about-- now I'm going to
• 71:10 - 71:13
that you are going to catch me.
• 71:13 - 71:16
You are going to catch
me, and shout at me,
• 71:16 - 71:18
and say, ooh, pay
attention, Magdalena,
• 71:18 - 71:22
you can make a mistake there.
• 71:22 - 71:26
I say it's L/M when I do
the division rule, right?
• 71:26 - 71:28
Where should I pay attention?
• 71:28 - 71:30
STUDENT: M [INAUDIBLE].
• 71:30 - 71:31
MAGDALENA TODA: Pay attention.
• 71:31 - 71:39
Sometimes you can
have the-- right?
• 71:39 - 71:45
And this also has
to exist as well.
• 71:45 - 71:47
STUDENT: [INAUDIBLE].
• 71:47 - 71:51
MAGDALENA TODA: So one
last-- how many minutes
• 71:51 - 71:54
have I spent with you?
• 71:54 - 71:58
I've spent with you a long
number of hours of my life.
• 71:58 - 71:59
No, I'm just kidding.
• 71:59 - 72:04
So you have one hour and
15, a little bit more.
• 72:04 - 72:05
Do I have a little bit more?
• 72:05 - 72:06
Yes.
• 72:06 - 72:08
I have 15 minutes.
• 72:08 - 72:08
I have--
• 72:08 - 72:10
STUDENT: So we get out at--
• 72:10 - 72:10
[INTERPOSING VOICES]
• 72:10 - 72:11
MAGDALENA TODA: 50.
• 72:11 - 72:13
Five more minutes.
• 72:13 - 72:15
OK.
• 72:15 - 72:21
So I want to ask you what
• 72:21 - 72:25
functions involved in limits.
• 72:25 - 72:28
• 72:28 - 72:32
Why did we study
limits at the point
• 72:32 - 72:34
where the function's
not defined?
• 72:34 - 72:35
Well, to heck with it.
• 72:35 - 72:36
We don't care.
• 72:36 - 72:38
The function is
not defined at 0.
• 72:38 - 72:40
But the limit is.
• 72:40 - 72:43
And nobody showed you how
to do the epsilon delta
• 72:43 - 72:44
to show anything like that.
• 72:44 - 72:49
• 72:49 - 72:50
OK.
• 72:50 - 72:52
Can you do that
with epsilon delta?
• 72:52 - 72:58
• 72:58 - 73:00
Actually, you can do
everything with epsilon delta.
• 73:00 - 73:02
But I'm not going to give
you any extra credit.
• 73:02 - 73:08
So I trust you that
you remember that.
• 73:08 - 73:09
1!
• 73:09 - 73:11
• 73:11 - 73:12
I am so proud of you.
• 73:12 - 73:13
Let me challenge you more.
• 73:13 - 73:15
Let me challenge you more.
• 73:15 - 73:18
Tangent of ax over bx.
• 73:18 - 73:20
x go to 0.
• 73:20 - 73:22
I asked this to a girl
from Lubbock High.
• 73:22 - 73:23
She was in high school.
• 73:23 - 73:25
• 73:25 - 73:28
STUDENT: Oh, I can't disappoint
everybody in getting this.
• 73:28 - 73:31
STUDENT: Is it 1/a?
• 73:31 - 73:32
Oh, I can't remember.
• 73:32 - 73:34
MAGDALENA TODA: Tell me
what to do to be smart.
• 73:34 - 73:34
Right?
• 73:34 - 73:38
I have to be doing
something smart.
• 73:38 - 73:40
She-- can you give me hint?
• 73:40 - 73:42
and you say, well--
• 73:42 - 73:42
STUDENT: ba--
• 73:42 - 73:44
STUDENT: It's 0.
• 73:44 - 73:45
STUDENT: It's [INAUDIBLE].
• 73:45 - 73:47
MAGDALENA TODA: Um, it's a what?
• 73:47 - 73:48
STUDENT: b/a?
• 73:48 - 73:50
MAGDALENA TODA: I'm
not [INAUDIBLE].
• 73:50 - 73:51
I don't think so.
• 73:51 - 73:53
So what should I do?
• 73:53 - 73:58
bx-- that drives me nuts.
• 73:58 - 74:00
This goes on my nerves-- bx.
• 74:00 - 74:04
Like, maybe I go on your
nerves. bx is ax, right?
• 74:04 - 74:07
If it were ax, I would
be more constructive,
• 74:07 - 74:09
and I knew what to do.
• 74:09 - 74:13
I say replace bx with
ax, compensate for it,
• 74:13 - 74:15
and divide by bx.
• 74:15 - 74:18
And I was trying to
explain that to my son,
• 74:18 - 74:24
that if you have a fraction
a/b, and then you write a/n
• 74:24 - 74:27
times n/b, it's the same thing.
• 74:27 - 74:29
problem with him.
• 74:29 - 74:33
And then I realized that he
didn't do simplifications
• 74:33 - 74:35
in school.
• 74:35 - 74:41
So it took a little more
hours to explain these things.
• 74:41 - 74:43
• 74:43 - 74:45
I think I remember doing
• 74:45 - 74:47
• 74:47 - 74:50
So these two guys disappear.
• 74:50 - 74:54
I haven't changed
my problem at all.
• 74:54 - 74:58
But I've changed the status,
the shape of my problem
• 74:58 - 75:01
to something I can mold,
because this goes to somebody,
• 75:01 - 75:03
and this goes to somebody else.
• 75:03 - 75:05
Who is this fellow?
• 75:05 - 75:08
It's a limit that's
a constant-- a/b.
• 75:08 - 75:09
Who is this fellow?
• 75:09 - 75:10
STUDENT: 1.
• 75:10 - 75:11
MAGDALENA TODA: 1.
• 75:11 - 75:16
Because tangent of x/x as x
goes to 0 goes to 1 exactly
• 75:16 - 75:16
like that.
• 75:16 - 75:22
So limit of sine x over cosine
x, that's tangent, right?
• 75:22 - 75:23
Over x.
• 75:23 - 75:25
You do it exactly the same.
• 75:25 - 75:32
It's limit of sine x/x
times 1 over cosine x.
• 75:32 - 75:34
That's how we did
it in high school.
• 75:34 - 75:35
This goes to 1.
• 75:35 - 75:37
This goes to 1.
• 75:37 - 75:37
So it's 1.
• 75:37 - 75:39
So thank you, this is 1.
• 75:39 - 75:43
I know I took a little more time
to explain than I wanted to.
• 75:43 - 75:46
But now you are grown up.
• 75:46 - 75:49
In two minutes, you are
going to be finishing
• 75:49 - 75:51
this section, more or less.
• 75:51 - 75:55
What if I put a function
of two variables,
• 75:55 - 75:58
the limit will be,
• 75:58 - 76:01
if it's the same
type of function.
• 76:01 - 76:03
So you say, oh, Magdalena,
what you doing to us?
• 76:03 - 76:05
OK, we'll see it's fun.
• 76:05 - 76:06
This one's fun.
• 76:06 - 76:08
It's not like the one before.
• 76:08 - 76:11
This one is pretty beautiful.
• 76:11 - 76:13
It's nice to you.
• 76:13 - 76:15
It exists.
• 76:15 - 76:17
xy goes to 0, 0.
• 76:17 - 76:20
So you have to imagine
some preferable function
• 76:20 - 76:22
in abstract thinking.
• 76:22 - 76:25
And you want it in
a little disk here.
• 76:25 - 76:32
And xy, these are all points
xy close enough to 0, 0,
• 76:32 - 76:34
in the neighborhood of 0, 0.
• 76:34 - 76:35
OK.
• 76:35 - 76:37
What's going to happen as
you get closer and closer
• 76:37 - 76:40
and closer and closer with
tinier and tinier and tinier
• 76:40 - 76:44
disks around 0, 0?
• 76:44 - 76:48
You're going to shrink so much.
• 76:48 - 76:49
What do you think
this will going to be,
• 76:49 - 76:51
and how do I prove it?
• 76:51 - 76:52
STUDENT: [INAUDIBLE].
• 76:52 - 76:54
MAGDALENA TODA: Who said it?
• 76:54 - 76:56
You, sir? [INAUDIBLE]
going to go to 1.
• 76:56 - 76:58
And he's right.
• 76:58 - 77:01
He has the intuition.
• 77:01 - 77:03
A mathematician will
tell you, prove it.
• 77:03 - 77:05
STUDENT: Um, well,
let's see here.
• 77:05 - 77:06
MAGDALENA TODA: Can you prove?
• 77:06 - 77:10
STUDENT: You could use
the right triangle proof,
• 77:10 - 77:12
but that would probably
take way more [INAUDIBLE].
• 77:12 - 77:13
MAGDALENA TODA: x and
y are independent.
• 77:13 - 77:14
That's the problem.
• 77:14 - 77:16
They are married, but they
are still independent.
• 77:16 - 77:17
It's a couple.
• 77:17 - 77:21
However, we can use
polar coordinates.
• 77:21 - 77:23
Why is polar coordinates?
• 77:23 - 77:29
Well, in general, if we
are in xy, it's a pair.
• 77:29 - 77:31
This is r, right?
• 77:31 - 77:34
So rx is r cosine theta.
• 77:34 - 77:35
y is r sine theta.
• 77:35 - 77:37
And I can get closer and
closer to the original.
• 77:37 - 77:39
I don't care.
• 77:39 - 77:41
squared plus y squared,
• 77:41 - 77:43
this is r squared.
• 77:43 - 77:44
And r is a real number.
• 77:44 - 77:47
And as you walk closer
and closer to the original
• 77:47 - 77:53
without touching it,
that r goes to 0.
• 77:53 - 77:54
It shrinks to 0.
• 77:54 - 77:58
So that r squared goes
to 0 but never touches 0.
• 77:58 - 78:04
So this becomes limit as r goes
to 0, the radius of that disk
• 78:04 - 78:06
goes to 0.
• 78:06 - 78:11
Sine of r squared
over r squared.
• 78:11 - 78:14
But r squared could be replaced
by the real function, t,
• 78:14 - 78:17
by the real parameter,
lambda, by whatever you want.
• 78:17 - 78:19
So then it's 1.
• 78:19 - 78:23
And then Alexander was right.
• 78:23 - 78:26
He based it on, like,
observation, intuition,
• 78:26 - 78:27
everything you want.
• 78:27 - 78:29
It was not a proof.
• 78:29 - 78:32
On a multiple-choice exam,
he would be a lucky guy.
• 78:32 - 78:34
I don't want you to prove it.
• 78:34 - 78:37
But if I want you to
prove it, you have to say,
• 78:37 - 78:40
Magdalena, I know
polar coordinates,
• 78:40 - 78:42
and so I can do it.
• 78:42 - 78:45
And one last question for today.
• 78:45 - 78:50
limit xy goes to 0, 0.
• 78:50 - 78:54
You will see some of these in
• 78:54 - 78:57
that's waiting for
you, homework 3.
• 78:57 - 79:03
Tangent of 2 x squared
plus y squared over 3
• 79:03 - 79:06
x squared plus y squared.
• 79:06 - 79:09
What is that?
• 79:09 - 79:10
2/3.
• 79:10 - 79:11
STUDENT: 2/3.
• 79:11 - 79:13
MAGDALENA TODA: Am
• 79:13 - 79:14
No, enough.
• 79:14 - 79:14
OK.
• 79:14 - 79:17
[INAUDIBLE] I gave
you everything
• 79:17 - 79:21
you need to show that.
• 79:21 - 79:24
x squared plus y squared,
again, is Mr. r squared.
• 79:24 - 79:25
It's OK.
• 79:25 - 79:29
I taught you that.
a/b. a is 2, b is 3.
• 79:29 - 79:30
Is it hard?
• 79:30 - 79:32
It is not easy, for sure.
• 79:32 - 79:35
Calc 3 is really difficult
compared to other topics
• 79:35 - 79:38
you are probably taking.
• 79:38 - 79:41
But I hope that I
can convince you
• 79:41 - 79:45
that math, although
difficult, [INAUDIBLE] Calc 3,
• 79:45 - 79:48
is also fun.
• 79:48 - 79:50
OK?
• 79:50 - 79:51
All right.
• 79:51 - 79:55
So I need attendance and
I need the extra credit.
• 79:55 - 79:56
STUDENT: Yeah, [INAUDIBLE].
• 79:56 - 79:59
• 79:59 - 80:02
MAGDALENA TODA: Before
you go, you need to sign.
• 80:02 - 80:04
Title:
TTU Math2450 Calculus3 Sec 11.1 and 11.2
Description:

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

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 11.1 and 11.2