Question

In Exercises \(19-26,\) use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ x^{2}+y^{2}=e^{-z} $$

Short answer

The surface referred to by the equation \(x^{2}+y^{2}=e^{-z}\) can be graphed as a downward-opening funnel.

Step-by-step solution

Rewrite the equation into a compatible format

The given equation is \(x^{2}+y^{2}=e^{-z}\). One way to solve this problem is by plotting the equation directly, assuming the computational software can handle the implicit definition of z. Most systems, like Wolfram Mathematica or MatLab, should be able to handle this type of equation definition.

Plot the surface

Input the equation into the graphing software as it is. Since there is only one equation, there is no need to separate it into two different equations. Tweak the viewing window as necessary to ensure a sufficient perspective on the graph. It is expected that a sort of 'funnel' surface is obtained, opening downwards.

Validate the graph

Verify the graph forms the correct shape and correctly represents the given equation. The surface should visually match the equation. In this case, the graph should represent a funnel-like structure opening downwards, given the exponential decay associated with negative z values.