The Final will cover the material we did through Monday, December 1: Chapter 11.1-11.4, Chapter 13.1-13.7, Chapter 14.1-14.4 (only part of 14.3), Chapter 15.1-15.7, and Chapter 16.2-16.4 plus 16.7. The test is cumulative, i.e. you are responsible for everything done since the beginning of the class. Each chapter has quite a few review problems; I have suggested some to work on. I have included a few from the "Problems Plus" sections at the end of the review sections, since these are a little harder. I have also added a few additional problems of my own. I will post solutions to the even-numbered problems and the additional problems. You might find the "concept check" sections of each chapter review, as well as true-false quizzes, to be helpful.

Chapter 11 (pp. 733-735). True-false quiz.

Exercises 7, 10, 14, 16, 17 (by hand), 20, 24, 29, 38.

Problems Plus: 1, 5 (except for parts c and h).

Chapter 13 (pp. 881-883). True-false quiz.

Exercises 2, 4, 9, 10, 11, 17, 21, 24, 33, 34, 39, 46. 47.

Problems Plus: 2, 4. If you have trouble with 2, try the analogous problem for a rectangle in the plane.

Chapter 14 (pp. 918-919). True-false quiz (1-6).

Exercises 3, 4 (no graph), 6, 19.

Problems Plus: 1, 3 (you'll have to read example 5 in 14.4), 5(a).

Chapter 15. True-false quiz.

Exercises 1, 3, 6, 12, 13, 14, 15, 19, 20, 21, 23, 26, 34, 35, 39, 40, 41, 43, 47, 50 (Hint: Use problem M), 51--54, 63.

Problems Plus: 4 (try first with just two variables, ie (x+y)^{r}/(x^{2}+y^{2}), and 7--but only if you really have nothing better to do over Thanksgiving.

Chapter 16. True-false quiz (1-7).

Exercises 3, 4, 5, 7, 9, 11, 14, 15, 17, 20, 22, 29, 31, 32.

Problems Plus: 2. (5, 6 and 7 are really amazing, but we didn't get quite far enough to do this kind of substitution.)

**A few additional problems, with numbering continued from the second midterm review**

- Express the area enclosed by the curve r
^{2}=9 cos(5θ) as a double integral, and carry out the integral. (Compare review problem chapter 11, #29). Do the same for the area in problem 34. - Suppose that f(x,y,z) is a differentiable function which satisfies ∇ f(x,y,z)=0 for all (x,y,z). What can you say about f? Justify your answer.
- Suppose that the vector ∇ f(x,y) = (g(x,y),h(x,y)) where the partial derivatives of g and h are continuous. Show that g
_{y}= h_{x}. - Consider the curve of intersection of two surfaces, given by equations F(x,y,z) =0 and G(x,y,z) = 0. Assuming that the normal vectors are not parallel, show that the tangent to this curve is orthogonal to both normal vectors.