Calculus Chapter 14 Practice Exam

(Answers tomorrow! Some of these questions are harder than will be on the actual exam, but practice exams work better that way)

1. The material for the bottom of a rectangular box costs twice as much per square inch as the material for the sides and top. If the volume is fixed, find the relative dimensions that minimize the cost.
2. Minimize f(x,y,z) = xyz, subject to the constraint x² + 4y² + 2z² = 8.

3. If

   _{f(x,y) = (x}2 _{+ 3y}2 _)(e-(x² + y²)_),

   find all the critial points, and classify them as local maximums, minimums, or saddle points. Then find all of the global maximums or minimums.

4. Find the maximum and minimum values of f on R, if f(x,y) = x² + 4² -x + 2y, and R is the region bounded by the ellipse x² + 4y² = 1.
5. If f(x,y) = x² + y² and g(x,y) = 2x-3y+5, use the contours of f and the graph of g to come up with a rough minimum value of f subject to g, sketch the gradients of both f and g at the optimal point, and give a rough approximation of the gradient at that point. (Before you decide to just to do this algebraically and circumvent the geometry altogether, note that what I would like to do is just give you a contour diagram of f and the graph of g without equations, so try to do this geometrically!)