Exercise 30

Consider the function $h$ defined by $h(x,y) = 8 - \sqrt{4 - x^2 - y^2}$.

1. What is the domain of $h$? (Hint: describe a set of ordered pairs in the plane by explaining their relationship relative to a key circle.)

2. The range of a function is the set of all outputs the function generates. Given that the range of the square root function $g(t) = \sqrt{t}$ is the set of all nonnegative real numbers, what do you think is the range of $h$? Why?

3. Choose 4 different values from the range of $h$ and plot the corresponding level curves in the plane. What is the shape of a typical level curve?

4. Choose 5 different values of $x$ (including at least one negative value and zero), and sketch the corresponding traces of the function $h$.

5. Choose 5 different values of $y$ (including at least one negative value and zero), and sketch the corresponding traces of the function $h$.

6. Sketch an overall picture of the surface generated by $h$ and write at least one sentence to describe how the surface appears visually. Does the surface remind you of a familiar physical structure in nature?