WEBVTT
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DAVID JORDAN: Hello, and
welcome back to recitation.
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In this problem, what
I'd like us to do
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is I'd like us to sketch the
graphs, in three dimensions,
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of these functions.
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So z here is a
function of x and y.
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On this second one, z is
also a function of x and y.
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It just happens
not to depend on y.
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When you graph these, I'd
suggest to consider slices,
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so what happens if you
consider x equals 0
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or if you consider z equals 0.
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As you graph these, let's
see what you can do.
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So why don't you
pause the video,
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and I'll check back
with you in a moment,
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and I can show you
how I solved these.
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OK, welcome back.
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So why don't we start by
looking at this function:
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z is the square root of
x squared plus y squared.
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OK, I'll try to always draw
my axes in the same way
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as we do in lecture.
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So x is pointing towards us,
y to the right, and z up.
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So as I suggested,
I think a nice way
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to get started
with these problems
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is to just try setting the
variables x and y variously
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equal to 0, and then
seeing-- instead
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of a surface in that case,
then we'll get a curve,
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and we'll see what curve we get.
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So for instance, if
we set x equals to 0,
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then we just get z is the
square root of y squared,
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so we just get that z is
the absolute value of y.
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So what that means is that
whatever this surface looks
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like, we know what it
looks like if we slice it
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in the blackboard, in the
plane of the blackboard.
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We know that it just
looks like-- this
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is just the graph of the
function absolute value of y, z
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equals absolute value of y.
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So now, if you think about it,
what I just said works just
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as well for x instead of for y.
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So if we were to graph
this in the xz-plane
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where we set y equals to
0, then we would get-- OK,
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I'm going to try to draw this.
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So let me draw that
in blue, actually.
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So the blue is in the
xz-plane and the white
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is in the yz-plane.
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OK, now I think what's going
to be really illustrative
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is if we think about what
happens as we fix values of z.
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Well, obviously, if
we set z equals to 0,
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then there's just one solution,
which is this point here.
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But what's going
to be interesting
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is if we set z to be
some positive value.
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So, for instance,
let's take z to be 2.
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So, for instance,
we set z equals 2,
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then we get 2 equals
the square root
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of x squared plus y squared.
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Solving this, this is the
same as saying x squared
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plus y squared equals 4.
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So what that tells us is that
at the height z equals 2,
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we're just going to have
a circle of radius 2.
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This is just the equation
for a circle of radius 2,
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and so at height 2,
we just have a circle.
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And actually, as
you can see, there's
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nothing special about 2.
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At every height, we're just
going to have a circle,
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and so this is
what's called a cone.
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OK.
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Now for b, we can expect,
when we go over here
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to b, that something
funny is going to happen
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because it doesn't depend on y.
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So let's see if we can
see how the fact that z
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doesn't depend on y, how
this enters into our picture.
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So I'll just walk over
here, and we'll consider z
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equals x squared.
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OK.
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So again, we have our
x-axis, x, y, and z axes.
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Now, let's consider
what this looks
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like when we intersect
with the xz-plane,
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so when we set y equals to 0.
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Well, setting y
equal to 0 actually
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doesn't change the equation,
and we get z equals x squared.
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So we know what that looks like.
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It's a parabola.
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And this parabola, I want you
to think that it's, you know,
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coming out at us, so
it's in the xz-plane,
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going in and out of the board.
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But now if you
think about it, what
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it means to say that this
function doesn't depend on y,
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what that means is that we
have the exact same picture
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at every value of y.
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So if we go out here, then we're
going to have the same picture.
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And if we go over here, we're
going to have the same picture.
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And, in fact, what
you're going to get
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is you're going to get a prism.
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Oh, that's really hard to read.
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Let's see if we
can-- so let me--
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since that's a bit hard
to read on the axes,
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let's draw this again.
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What we'll get is
we're going to have
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a prism, which looks
like a parabola,
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looks like a sheet
that's just stretched out
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in the shape of a parabola.
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And so, this we could call
a prism of a parabola.
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Now let's see if we can get
any more insight from these two
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pictures.
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So look what happened
in this instance.
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So here, the function z, it
obviously didn't depend on y.
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And we can see that by
looking at the graph,
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because, you know,
as you vary y,
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the picture had to be unchanged.
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So the fact that this was
a prism in y and the fact
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that the function didn't depend
on y are one and the same fact.
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Now if we go over
to the cone-- OK,
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so here, our function z very
much depended on both x and y.
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But you notice that it depended
on x and y only in the sense--
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so z is actually equal
just to the radius r, which
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is x squared plus y squared.
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So the fact that this cone--
thank you-- the square root
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of x squared plus y squared.
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So the fact that
this only depends
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on the radius and not the
relative angle of x and y
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is why we got what--
this is an example
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of a surface of revolution.
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So we can always expect
that if the dependence
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of z on the variables x
and y, if you can actually
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just rewrite that as a
dependence on r, then
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you'll get this nice
radial symmetry,
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just like we had translational
symmetry for the prism.
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And I think I'll
leave it at that.