pg. 722, #15 (moderate)

Find the traces of the surface in the planes x = k, y = k, z = k. Then identify and

sketch it.

Solution:

The easiest way to see this surface is by taking fixed z values. For example, when z = 0, we have

y = -x^2; when z = 1, y = 1 - x^2; when z = 2, y = 4-x^2. These are parabolas opening to the left.

Since y = z^2 - x^2, the maximium y value will occur when x = 0. That is, the tips of the parabolas

will be on the parabola y = z^2.

The total will be a snowplow: