Math 221: Chapter 13 Exam Study Guide

  1. Sketch the level curves to a surface.
  2. Find the limits (if they exist).
  3. Find all first partial derivatives.
  4. Find all second partial derivatives and show the second mixed partials are equal.
  5. Find the total differential.
  6. Find the deriviative two ways. By using the chain rule and then rewriting in terms of t and by rewriting in terms of t and then finding the derivative.
  7. The relationships between functions are given. Use them to write the derivatives using the chain rule. There are no actual functions here, just the relationship between them.
  8. Find the gradient and the directional derivative in the indicated direction, both at the indicated point.
  9. Find the direction of greatest increase in a function and the maximum value of the directional derivatives at the indicated point.
  10. Given a surface and a point, find a unit normal vector to the level curve at the point, the equation of the tangent plane at that the point, and the symmetric equations of the normal line at that point.
  11. Examine the function for relative extrema and saddle points and use them to label the contour plot.
  12. Use Lagrange multipliers to locate and classify any extrema of the function.

Notes

Points per problem

# 1 2 3 4 5 6 7 8 9 10 11 12 Total
Pts 5 12 10 8 4 8 9 10 6 12 9 7 100