Multivariable Functions

• Domain:
• Range:

What does it look like?
For side view 1, set , then , so

For side view 2, set

For the top view, we set , , which is the equation of a circle with radius .

With this information, we can see that the function takes the shape of a cone

• Domain:
• Range:

When we set and to , we get the same deal as above

But our level curve looks different

Which ends up looking like

What is the domain?

This forms a circle with radius 1

What is the range?

Now we set , and obtain , which is the equation of a circle with radius .

We get the same thing setting .

Now we draw contours:

And the result is:

Natural logarithm must take a positive argument, but the product of two negative numbers is also positive, so we can use quadrants 1 and 3. We also cannot include the boundaries.

Domain: .

Range:

Since is not in the domain, slicing at or would give us nothing. So we should slice at instead. So

Slicing at would be the same.

For contours, we have , where
Since is a positive constant, we are dealing with a rational function.

As
As