Problem 1

Given a function , which has the following properties , , and .

(a). Suppose Represents the Height of a Mountain at Location . What is the Direction of Steepest Increase of Height at the Location ? At what Rate Does the Height Change, in the Direction of Steepest Increase

(b). Find the Directional Derivative at in the North-west Direction (ie. in the Direction of )

(c). Let . Determine

Problem 2

(a). Determine All the Critical Points of the Function with and

There is a single critical point at .

(b). Using the Second Derivative Test, Determine whether the Critical Point(s) You Found is a Local Maximum, a Local Minimum, a Saddle, or None of These

The critical point at is a saddle point.

Problem 4

(a). Sketch the Graph of , in Polar Coordinates

(b). Find the Mass of the the Region Bounded by the Graph in part (a), and whose Density is given by . Hint: You May Find the following Identity Useful:

Problem 5

Consider the integral

(a). Sketch the Domain of Integration

(b). Interchange the order of Integration in the Integral