- [Voiceover] So in the last video

I described how to interpret three dimensional graphs

and I have another three dimensional graph here,

it's a very bumpy guy.

And this happens to be the graph with the function f of x, y

is equal to cosine of x

multiplied by the sine of y

and you know I could also say like,

that this graph represents

I could also say that this graph represents

z is equal to that whole value

because we think about the output of the function

as the z coordinate of each point

and what I want to do here is describe

how you can interpret the relationship between this graph

and these functions that you know

by taking slices of it.

So for example let's say that we took a slice with

this plane here

and what this plane here is

is it represents the value x equals zero

and you can kinda see that because this is the x-axis.

So when you're at zero on the x-axis

you know, you pass through the origin

and then the values of y and z can go freely

so you end up with

this plane.

And let's say you want to just consider

where this cuts through the graph, okay,

so we'll limit our graph

just down to the point where it cuts it

and I'm going to draw a little red line over that spot.

Now what you might notice here

that red line looks like a sinusoidal wave

in fact it looks exactly like the sine function itself

you know, passes through the origin

it starts by going up and this makes sense

if we start to plug things into

the original form here.

Because if you take

f and you plug in x equals zero

but then we still let y range freely

what it means is that you're looking at cosine of zero

multiplied by sine of y

And what is cosine of zero?

cosine of zero evaluates to one

so this whole function should look just like sine of y

in that when we let y run freely

the output, which is still represented by the z coordinate,

will give us this graph that's just a

normal two-dimensional graph

that we're probably familiar with.

And let's try this at a different point.

Let's see what would happen if

instead of plugging in x equals zero

let's imagine that we plugged in y equals zero

and this time before I graph it

and before I show everything that goes on

let's just try to figure out

purely from the formula here

what it's going to look like when we plug in y equals zero.

So now I'm going to write over on the other side.

We have

f of x will still run freely

y is going to be fixed as zero

and what this means is we have cosine of x

so maybe expect to see something

that looks kinda like a cosine graph

and then sine of zero.

Except, what is sine of zero?

Sine of zero cancels out and just becomes zero

which multiplied by cosine of x

means everything cancels out and becomes zero

so what you'd expect is that this

is going to look like a constant function

that's constantly equal to zero.

And let's see if that's what we get.

So I'm going to slice it with

y equals zero here

and you look at the y axis

we see when it's zero

and x and z both run freely.

I'm going to chop off my graph at that point,

and indeed it chops it just at this straight line,

the straight line that goes right along the x-axis.

But let's say that we did a different constant value of y.

Rather than y equals zero,

and we'll erase all of this,

let's say that I cut things

at some other value.

So in this case what I've chosen

is y is equal to pi halves

and it looks

kinda like we've got a wave here

and it looks like a cosine wave

and you can probably see where this is going.

This is when x is running freely

and if we start to imagine plugging this in

I'll just actually write it out.

We've got cosine of x

and then y is held at a constant sine of pi halves.

Sine of pi halves is just,

this just always equals one

so we could replace this with one

which means the function as a whole

should look like cosine x.

So again the multi-variable function

we've frozen y and we're letting x range freely

and it ends up looking like a cosine function

and I think a really good way to understand

a given three-dimensional graph when you see it,

let's say you,

you look back at the original graph

when we don't have anything going on.

Get rid of that little line.

So you've got this graph

and it looks wavy and bumpy

and a little bit hard to understand at first

but if you just think in terms of

holding one variable constant

it boils down always into

a normal two-dimensional graph

and you can even think about,

as you're lighting planes, kinda slide back and forth

what that means for the

amplitude of the wave that you see

and things like that.

This become especially important, by the way,

when we introduce a notion of partial derivatives.