Question

Let be the surface defined by z=x ²+y ².

a.Find the traces of S in the coordinate planes.

b.Find the traces of S in the plane z=k,where k is a constant.

c.Sketch the surface S.

Answer 1

Solution

a.Find the traces of S in the coordinate planes.

Setting z=0 gives x ²+y²=0,from which we see that the xy-trace is the origin (0,0). (See Figure 3a.) Next,setting x=0 gives z=y ²,from which we see that the yz-trace is a parabola. (See Figure 3b.) Finally,setting y=0 gives z=x ²,so the xz-trace is also a parabola. (See Figure 3c.)

b.Find the traces of S in the plane z=k,where k is a constant.

Setting z=k,we obtain x ²+y²=k ,from which we see that the trace of S in the plane z=k is a circle of radius (k)^1/2 centered at the point of intersection of the plane and the z-axis,provided that k>0. (See Figure 3d.) Observe that if k=0,the trace is the point (0,0) (degenerate circle) obtained in part (a).

c.Sketch the surface S.

The graph of z=x ²+y² sketched in Figure 3e is called a circular paraboloid because its traces in planes parallel to the coordinate planes are either circles or parabolas.