Math 221: Chapter 13 Exam Study Guide

1. A contour plot is given with a point. Sketch the direction of steepest increase / decrease.
2. Find the limits (if they exist). Three parts.
3. Find the total differential.
4. 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. For example, z is a function of u and v, u is a function of x and y, v is a function of y; find the partial of z with respect to y.
5. 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.
6. A function of two variables and a point are given. Find the direction of greatest increase at the point, the directional derivative in the indicated direction at the point, a unit normal vector to the level curve containing the point, the equation of the tangent plane at the point, symmetric equations of the normal line, the angle of inclination of the tangent plane containing the point, and the places on the surface where the tangent plane is horizontal.
7. Examine the function for relative extrema and saddle points and use them to label the contour plot. The contour plot is given.
8. Find the relative maximum or minimum for an applied problem. Although this is technically out of 13.9, it's really just an application of the concepts in 13.8.
9. Use Lagrange multipliers to locate and classify any extrema of the function.

Notes

• This exam is made up of two parts.
• Questions 1-6 are to be done without computer technology (calculator is okay) and turned in before starting the second part.
• Maxima can be used for questions 7-9.

Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 Total 4 12 6 12 12 28 10 8 8 100