Functions of two variables. Worked examples
In this next example, we will sketch the graphs of three-dimensional functions. Sketch a graph of each function a)z=sqrt(x^2+y^2) b)z=x^2 So z here as a function of x and y. On this second one, z is also a function of x and y, it just happens not to depend on y. When you graph these, we’d suggest considering the x axis and the y axis separately. So what happens if you consider x equals 0 or if you consider z equals 0? As you graph these, let’s see what you can do. So why don’t we start by examining this function, z-squared equals x squared plus y squared. So x is pointing toward us, y to the right, and z up. A nice way to get started with these problems is to just try setting the variables x and y variously equal to zero and then see what happens when we do so; this should give us a surface instead of a curve and we’ll see what curve we get. For instance, if we set x equal to zero, then we just get z is the square root of y squared. So we see that z is the absolute value of v. This means that whatever the surface looks like, we know what it would look like if we sliced it in the plane of the blackboard. We know that it just looks like this is just the graph of y equals 0, z equals 0. So, now if you think about it, what we just said works just as well for × instead of for y. So if were to graph this in the xz-plane, where we set y equals 0, then we would get. Let’s draw the graph of y equals x^3 + y^2? in blue and the graph of y equals 4 – x^2? in white. If we set z equal to 0, there is one solution, which is this point here. But what will be interesting is if we set z equal to some positive value such as 2. So if we said that z was equal to 2, then we have 2 = x^3 + y^2?. Solving this is equivalent to saying that x^3 + y^2? = 4. [Graph] So. at the height of 2, we will have circle with radius 2. This is just the equation for is a circle with radius 2. And so at height 2, we just have a circle. And actually as you can see, there’s nothing special about 2; at every height, we’re just going to have a circle. So this is what’s called a cone. a Now for B. we can expect something funny to happen when we gO over here to b because it doesn’t depend on y. Let’s see if we can see how the fact that z doesn’t depend on y enters into our picture. So we’ll just walk over here and we’ll consider z equals x squared. So again, we have our x-axis, X-, y- and z-axes. Now, let’s consider what this looks like when we intersect the xz- plane. When y equals 0, the equation does not change. Therefore, z equals X squared. We know what that looks like: it is a parabola. It lies in the xz- plane and goes in and out of the board. But now if you think about it, what it means it to say that this function does not depend on y is that we have the exact same picture at every value of y. So if we go out here, then we are going to have the same picture. If we go over here, we are going to have the same picture. And in fact what you will get is a prism. The graph of a parabola can be made into a rectangular prism by stretching out its shape. We could call this a prism of a parabola. Now let’s see if we can get more insight from these two pictures. So look what happened in this instance: as you vary y, the picture had to be unchanged. So here are the function z; it obviously did not depend on y. [Graph] The fact that the equation of a cone is z equals x squared plus y squared and the fact that this function is not dependent on y are one in the same fact. Now, if we go over to the prism. Here our function z very much depends on both x and y. But notice that it depends on x and y only in the sense that z equals the radius r which is equal to x squared plus y squared. Therefore, it is because the dependence of z on x and y can be rewritten as a dependence on r that this surface has radial symmetry. As in translation symmetry of the prism, we can expect that if the dependence of z on x and y can be rewritten as a dependence on r, then we will get radial symmetry.
