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.