Calculus III

Test 2 Study Guide

Test 2 covers sections 11.1-11.7.

Treat the sample tests and practice problems as additional review, not as a complete representa-tion of the questions youll be given on the exam- you are responsible for all material we covered(i.e., all learning objectives below).

11.1 - Functions of Several Variables

Find the domain and range of a multivariable function. Sketch level curves given an equation for a function z = f(x, y). Use level curves to sketch the graph of a function z = f(x, y).

Supplemental Homework Problems: 5-9 odd, 8, 12, 13-27 oddExtra Practice Problems: 2, 6, 16, 20, 28

11.2 - Limits and Continuity

Use the definition of continuity to find points at which a function is continuous or discon-tinuous.

Be able to show that limit does not exist by showing the limit has two different values fromtwo different directions.

Be able to use the Squeeze Theorem to verify that the limit of a function does exist. Use continuity and rewriting techniques to evaluate limits.

Supplemental Homework Problems: 1, 5-9 odd, 10, 13-17 odd, 14, 16, 35Extra Practice Problems: 6, 13, 15, 36

11.3 - Partial Derivatives

Compute partial derivatives of explicitly defined functions z = f(x, y). Compute partial derivatives of z, a function of x and y, when implicitly defined. Interpret partial derivatives as slopes in the appropriate directions. Recall that if fx and fy are continuous near (a, b), then fxy(x, y) = fyx(x, y) for (x, y) near

(a, b).

Supplemental Homework Problems: 3, 5-8, 9, 11, 15-37 odd, 22, 32, 39, 41-53 odd,48, 57-65 oddExtra Practice Problems: 5-8, 12, 18, 20, 26, 30, 40, 44, 46, 50, 54, 58, 62, 66

Test 2 Study Guide page 2

11.4 - Tangent Planes and Linear Approximations

Find the tangent plane to a surface at a point (x0, y0, z0). Use the tangent plane to approximate L(x, y) (the linearization of f(x, y)).

Supplemental Homework Problems: 1-5 odd, 4, 11-19 odd, 14, 18, 23-33 odd, 30Extra Practice Problems: 2, 6, 12, 16, 19, 28, 29, 32, 33

11.5 - The Chain Rule

Write the definition of the chain rule in terms of independent variables and partial deriva-tives of dependent and intermediate variables. (Use tree diagrams if you like.)

Use the chain rule to compute partial derivatives. Differentiate implicitly to find partial derivatives.

Supplemental Homework Problems: 1-29 odd, 14, 18, 24, 33, 38, 39Extra Practice Problems: 6, 10, 13, 16, 20, 22, 28, 30

11.6 -Directional Derivatives and the Gradient Vector

Find directional derivatives in the direction of any unit vector. (Required skill: Find a unitvector in the direction of a given vector.)

Find the gradient vector (f) of a function. Find the directional derivative in the direction of a unit vector using the gradient. Determine the maximal rate of change of a function and its direction using f .

Supplemental Homework Problems: 4, 5-23 odd, 8, 16, 24Extra Practice Problems: 6, 10, 14, 20, 22

11.7 - Optimization

Locate local maxima, local minima, and saddle points. Locate critical points using the first partials. Classify critical points using the second partials. Find absolute maxima and absolute minima on a closed, bounded set

Supplemental Homework Problems: 1-21 odd, 6, 14, 27-31 odd, 30Extra Practice Problems: 2, 10, 12, 16, 31, 32