WEBVTT
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DAVID JORDAN: Hello, and
welcome back to recitation.
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Today what we want to work
on is drawing level curves.
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This is for all the artists
out there in the audience.
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We have three functions
here: z is 2x plus y,
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z is x squared plus
y squared, and z
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is x squared minus
y squared, and we
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want to get some practice
drawing their level curves.
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Now, just to remind
you, the level curves
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are not drawn in
three dimensions.
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They're drawn in the
xy-plane and they're
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constructed by setting
z to be a constant
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and then graphing the
curve that we get,
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so we can think
about a relief map
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that we might use
if we were hiking.
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So why don't you
get started on that.
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Pause the video, and
we'll check back,
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and I'll show you
how I solve this.
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Welcome back.
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So over here, we've
got the equation
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for part a already set up.
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So z is 2x plus y.
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So now, what we need
to do to get started
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is just draw the xy-axis.
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And, you know, there's
really not a precise science
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for drawing these
level curves out.
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We just need to choose
some values of z
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that we feel are representative
and then just draw them in.
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So one thing we
notice about this
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is that if we choose
z to be a constant,
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then the equations
that we're going to get
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is 2x plus y equals
some constant, right?
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So, you know, these are
just going to be lines.
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The level curves in this case
are just going to be lines.
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So, for instance, if we take
the level curve at z equals 0,
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then we have just the
equation 2x plus y equals 0.
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And so that has intercept--
so we're looking at-- so 0
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equals 2x plus y, so that's
just y equals minus 2x.
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So that's this level curve.
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That's the level
curve at z equals 0.
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Now, if you think about it,
all the other level curves,
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we're just going to be
varying the constant here,
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and so we're just going
to be shifting this line.
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So all of our level
curves in this case
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are just straight lines.
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So let's see if we can
make some sense out of that
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by thinking about
the graph in three
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dimensions of this function.
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So over here, I'm going
to draw-- So this function
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z equals 2x plus y,
if we draw its graph,
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it's just a plane, right?
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So it's just a plane, which I'll
just kind of draw in cartoon
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form, something like that.
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And now when we do level
curves, what we're doing
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is we're slicing this plane
with another plane, which
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is the horizontal values
where z is a constant.
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And so, for instance, if we
take the level curve here,
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then we're just intersecting
these two planes,
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and their intersection
is just a line,
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and that's exactly the lines
that we're drawing here.
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So it's not surprising that we
were graphing a linear function
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and that our contour
lines, our level curves,
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were just straight lines.
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So let's go on to a slightly
more interesting example, which
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is part b, which I
have written up here.
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So this is the function z
equals x squared plus y squared.
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Actually, this is even
easier to get started
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drawing the level curves for.
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Well, if you think about
it, if I fix the value of z,
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then this is
exactly the equation
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for the circle with
radius square root of z.
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So level curves, level
curves for the function z
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equals x squared plus y
squared, these are just
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circles in the xy-plane.
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And if we're being careful
and if we take the convention
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that our level curves are
evenly spaced in the z-plane,
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then these are going to get
closer and closer together,
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and we'll see in a minute
where that's coming from.
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So let's draw what's going
on in three dimensions.
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So if we graph z
equals x squared
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plus y squared in
three dimensions,
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this is just a
paraboloid opening up.
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And now what you can
see is that if we
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slice this through the
constant-- through z
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equals constant
planes, then we're
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just getting these circles,
and those are precisely
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the circles that we're
drawing on the level curve.
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And because the parabola
gets steeper and steeper,
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that's telling us
that these circles,
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if we keep incrementing z in a
constant way, that's telling us
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that the circles, which
are the shadows below here,
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are going to get closer
and closer and closer.
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This reflects the fact that this
is getting steeper and steeper.
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In fact, this is generally true.
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If you're looking
at a contour plot
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where the intervals
between level curves
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are at regular distances,
then very close contour lines
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means that the function
is very steep there.
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So that's something
to keep in mind.
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Let's look at one more example.
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This is z equals x
squared minus y squared.
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So to get started with
this, well, again,
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if we start choosing
constant values of z,
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this is just giving
us hyperbola,
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hyperbolas of two sheets.
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So, for instance, if
we take-- so let's
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see what happens if
we take z equals 0.
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So if we take z equals 0, then
something a little special
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happens.
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This becomes x plus y
times x minus y equals 0.
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We can factorize x
squared minus y squared
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as x plus y times x minus
y, and if this is 0,
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then that means either plus
y is 0 or x minus y is 0.
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So that tells is that the zero
level curves for this graph
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are the lines y equals minus
x and the lines y equals x.
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OK.
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And now, if we move z away from
that, then what we're getting
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are hyperbolas, and these
hyperbolas will approach
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this asymptotic line y equals
minus x or this line-- sorry,
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this line y equals x or this
line equals y equals minus x.
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They'll approach
this as they go down,
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but they'll never
quite reach it.
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So the level curves here
are just hyperbolas.
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So now let's see, what
is this telling us
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about the three-dimensional
graph of this function?
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OK, so, first of all, we
have these level curves
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when y equals x and
when y equals minus
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x, and so those level
curves we can kind of draw.
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OK, so I want you to think
that that sits in the xy-plane.
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It's kind of hard to
draw in three dimensions.
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And so this is where our
function is going to be zero.
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Now, if we take x to
be positive-- sorry.
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If we take x to be greater
than y and both positive,
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then this is a positive number
and this is a positive number.
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So if we look in
the region where
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x and y are both
positive, that's in here,
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and where x is greater than
y, then our function comes up.
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So it looks like
this, and then it
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dips down and goes down, comes
back up, and goes back down.
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And now at the middle here,
it has to dip down to zero,
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so we have something like this.
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So what we end up getting in
the end, this is a saddle,
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so it's a bit hard to draw.
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It's a bit hard to see on this
so let me draw a sketch of it
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off of the axes for you.
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So we have a rise,
and then a drop,
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and then a rise in the back,
and then a drop, and then
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down the middle it dips
down in this direction
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and it rises up
in this direction,
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so it's a saddle like you
could put this on a horse
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and ride it.
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And so we can see that the
three-dimensional contours
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of the saddle, when we
look at their projection
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down onto the contour plot,
become these hyperbolas.
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So a saddle is sort
of like a hyperbola
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stretched out into
three dimensions.
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And I think I'll
leave it at that.