# Solutions to Problem Set 15

Math 311-01/02

2-19-2018

[Functions of several variables problems]

1. A function is defined by

Fill in the table to evaluate the function at the given points.

2. Find the points for which is undefined. What is the domain of f?

is undefined when the denominator is 0 --- that is, when , or .

You could also say that f is undefined at all points of the form .

The domain is the set of points such that .

3. Find the points for which is undefined. What is the domain of f?

is undefined when . Now can't be negative, but at .

So f is undefined at .

The domain is the set of points .

4. Describe the image (or the range) of the function .

For any x and y,

I know that attains every value between -1 and 1 (inclusive), and if I set I have .

Therefore, the image of f is .

5. (a) Plot the surface .

(b) Sketch contour lines for .

(a)

6. (a) Plot the surface .

(b) Sketch contour lines for .

(a)

7. (a) Plot the surface .

(b) Sketch contour lines for .

(a)

8. Describe some level surfaces for the function .

The level surfaces are paraboloids.

It is difficult to begin without borrowing. - Henry David Thoreau

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