## Math 221 - Chapter 15 Study Guide

1. Sketch the domain of the function f. Look at problems 15.1.15-18.
2. Sketch the level surface. Look at problems 15.1.37-40.
3. Find the limit, if the limit exists. Two parts. Look at problems 15.2.25-32.
4. Find fx(x,y) and fy(x,y). Look at problems 15.3.17-22.
5. Find fxx, fyy, fxy, and fyx. Look at problems 15.3.43.
6. Find dz/dt using the chain rule. Look at problems 15.4.1-6.
7. Related rate application problem. Look at problems 15.4.25-29.
8. Find equations for the tangent plane and normal line to the given surface at the point P. Look at problems 15.5.1-8.
9. Find the gradient of f at the indicated point. Look at problems 15.6.7-10.
10. Find the directional derivative of f at P in the direction of a. Look at problems 15.6.15-20.
11. Find a unit vector in the direction in which f most increases or decreases most rapidly at P; and find the rate of change of f at P in that direction. Look at problems 15.6.29-36.
12. Find the differential dw. Look at problems 15.7.39-42.
13. Use a tree diagram to construct the formulas for the chain rule. Look at figures 15.7.7-10.
14. Find all relative maxima, relative minima, and saddle points. Look at problems 15.8.9-20.
15. 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 15.9.3-10.

### Notes:

 # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Total Pts 4 4 8 6 12 6 8 6 6 6 6 6 6 8 8 100