Interpreting graphs with slices | Multivariable calculus | Khan Academy

So in the last video, I described how to interpret three-dimensional graphs. I have another three-dimensional graph here; it's a very bumpy guy. This happens to be the graph of the function ( f(x,y) = \cos(x) \cdot \sin(y) ). You know, 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.

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. For example, let's say that we took a slice with this plane. This plane represents the value ( x = 0 ). You can kind of see that because this is the ( x )-axis. So when you're at ( 0 ) on the ( x )-axis, you pass through the origin, and then the values of ( y ) and ( z ) can go freely.

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 is that the red line looks like a sinusoidal wave. In fact, it looks exactly like the sine function itself. You know, it passes through the origin, and it starts by going up.

This makes sense if we start to plug things into the original form here. Because if you take, you know, if you take ( f ) and you plug in ( x = 0 ), but then we still let ( y ) range freely, what it means is you're looking at ( \cos(0) \cdot \sin(y) ). What is ( \cos(0) )? ( \cos(0) ) evaluates to one, so this whole function should look just like ( \sin(y) ). 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.

Let's try this at a different point. Let's see what would happen if instead of plugging in ( x = 0 ), let's imagine that we plugged in ( y = 0 ). 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 = 0 ).

So now I'm going to write over on the other side. We have ( f(x) ), where ( x ) will still run freely, and ( y ) is going to be fixed at zero. What this means is we have ( \cos(x) ). You might expect to see something that looks kind of like a cosine graph, and then ( \sin(0) )—except what is ( \sin(0) )? ( \sin(0) ) cancels out and just becomes zero, which multiplied by ( \cos(x) ) means everything cancels out and becomes zero.

What you'd expect is that this is going to look like a constant function that's constantly equal to zero. Let's see if that's what we get. So I'm going to slice it with ( y = 0 ) here. We look at the ( y )-axis; we see when it's zero, and ( x ) and ( z ) both run freely. Then I'm going to chop off my graph at that point, and indeed it chops 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 = 0 ). So rather than ( y = 0 ), let's erase all of this; let's say that I cut things at some other value. In this case, what I've chosen is ( y = \frac{\pi}{2} ). It looks like we've got a wave here, like a cosine wave, and you can probably see where this is going.

You know, 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 ( \cos(x) ) and then ( y ) is held at a constant ( \sin\left(\frac{\pi}{2}\right) ). ( \sin\left(\frac{\pi}{2}\right) ) just always equals one, so we could replace this with one, which means the function as a whole should look like ( \cos(x) ).

So again, the multivariable function—we've frozen ( y ) and we're letting ( x ) range freely, and it ends up looking like a cosine function. I think a really good way to understand a given three-dimensional graph when you see it—let's say you look back at the original graph and we don't have anything going on. Get rid of that little line.

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 always boils down into a normal two-dimensional graph. You can even think about, as you're letting planes kind of slide back and forth, what that means for the amplitude of the wave that you see, and things like that. This becomes especially important, by the way, when we introduce a notion of partial derivatives.