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