Final Review Problems

The Final will cover the material we did through Tuesday, April 19: Chapter 11.1-11.4, Chapter 13.1-13.7, Chapter 14.1-14.3 (only part of 14.3), Chapter 15.1-15.7, Chapter 16.1-16.8, and Chapter 17.1-17.2. 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; these are pretty hard. 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.

I strongly suggest you go over the midterm exams and quizzes, and make sure that you have mastered the problems that gave you difficulty the first time around.

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.(you'll need example 5 from 14.4, which we did in class).
Problems Plus: 1, 3, 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 N), 51--54, 63.
Problems Plus: 4 (try first with just two variables, ie (x+y)r/(x2+y2)), and 7--but only if you really have nothing better to do over Passover.

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.)

Chapter 17.

A few additional problems.

  1. Express the area enclosed by the curve r2=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.
  2. 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.
  3. 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 gy = hx.
  4. 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.