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- [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.