Math 221: Chapter 14 Study Guide

1. Sketch the domain of the function f. Look at problems 14.1.19-22.
2. Sketch the level surface. Look at problems 14.1.45-48.
3. Find the limit, if the limit exists. Two parts. Look at problems 14.2.13-20.
4. Find fx(x,y) and fy(x,y). Look at problems 14.3.17-22.
5. Find the differential. Look at problems 14.4.25-36.
6. Find the local linear approximation for the given function at the indicated point. Then use the local linear approximation to approximate the function at another point near the given point. Look at problems 14.4.17-24.
7. Find all first and second order partial derivatives for a function and evaluate at a given point. Then determine whether that point is a critical point, and if so, whether there is a relative maximum, relative minimum, or saddle point there. Look at problems 14.3.67-74 and 14.8.9-20.
8. Use the chain rule to find the partial derivatives and the derivative. Three parts. Look at problems 14.5.7-10 and 13-18.
9. Use a tree diagram to construct the formulas for the chain rule. Look at figures 14.5.2-6.
10. Find the directional derivative of f at P in the direction of a. Look at problems 14.6.9-18.
11. Find the gradient of f at the indicated point. Look at problems 14.6.37-40.
12. Find equations for the tangent plane and normal line to the given surface at the point P. Look at problems 14.7.1-8.
13. Find unit vectors in the directions in which f increases and decreases most rapidly at P; and find the rate of change of f at P in those directions. Look at problems 14.6.47-60.
14. Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraint. Also, find the points at which these extreme values occur. Look at problems 14.9.5-12.

Notes:

• Some of the problems may be directly from the text.

Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Total 4 4 8 6 4 8 10 9 9 8 4 8 10 8 100