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Chapter 10: Problem 2

Sketch the appropriate traces, and then sketch and identify the surface. $$z=4-y^{2}$$

Short Answer

Expert verified
The surface is a parabolic cylinder defined by \(z=4-y^{2}\), which extends along the x-axis in both directions. It is obtained by rotating the parabola \(z=4-y^{2}\) (which lies on the yz-plane) along the x-axis. The parabola opens downward along the negative y-axis with a vertex at (y,z) = (0,4).

Step by step solution

01

Understanding the Traces

First, try to understand what it means to sketch the traces. Traces are essentially the intersection of the surface with planes parallel to the coordinate planes. Here, we are dealing with a parabolic cylinder defined by \(z=4-y^{2}\). Let's start with the trace in the yz-plane, which is achieved by setting x = 0.
02

Identifying the Equation for Traces

Setting x = 0 in the equation \(z=4-y^{2}\) does not alter the equation, as there's no 'x' term in this equation. Therefore, the trace in the yz-plane is the parabola \(z=4-y^{2}\). This opens upward along the z-axis with vertex (y,z) = (0,4).
03

Sketching the Traces

Draw this parabola on the yz-plane, starting with the vertex at (0,4) and opening downwards along the negative y-axis. This helps to understand the shape and orientation of the ensuing surface.
04

Identifying the Surface

After having the trace, bearing in mind that the initial equation did not depend on x, we realize that the surface is a parabolic cylinder that extends in the positive and negative x directions. The shape and size of the cylinder is dictated by the parabola.
05

Sketching the Surface

Now that we know it's a parabolic cylinder, sketch it extending along the x-axis. The parabola formed at the yz-plane will be replicated along the x-axis in both directions. Make sure that the graph reflects the downward opening of the parabola along the negative y-axis as obtained from the traces.

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