Multivariable Calculus, MATH 241-02

Prof. Gareth Roberts

Exam #3

Thursday, April 25, 7:15 - 8:45 pm

Location: Smith Labs 154

The third exam covers Sections 9.7, 11.8, 12.1 - 12.4, 12.7, 12.8, 13.1, and 13.2. It is recommended that you go over homework problems (HW #7 - 9, both written and WebAssign), class notes, and worksheets. Many of the problems and questions we discuss in class are excellent examples of test questions. The solutions to your WebAssign problems on HW #7 - 9 can be seen by clicking "View Key" near the top of each assignment. You can also click on "Practice Another Version" to redo certain homework problems.

In addition, some review problems from the Chapter 9, 11, 12, and 13 Review Exercises are listed below. The odd answers are in the back of the book while the evens are listed here. The Concept Check problems at the end of each chapter (before the Exercises) are also a good source of questions. The exam will be designed to take roughly one hour although you will have 90 minutes if needed.

Note: You will be allowed a scientific calculator for the exam which does NOT have graphing capabilities. Please bring your own calculator.

Exam Review: We will review for the exam during class on Tuesday, April 23. Please come prepared with specific questions.

Chapter 9 Review, p. 689 - 690
Problems:   37, 39, 40, 41, 42

The answers to the evens are:
40.   (a) The half-plane y = x, with x ≥ 0. (b) A cone facing upwards with equation z = sqrt{x2 + y2}.
42.   Cylindrical coordinates: r = 2. Spherical coordinates: ρ sin φ = 2.

Chapter 11 Review, p. 823 - 826
Problems:   59, 60, 61, 64

The answers to the evens are:
60.   The absolute maximum is sqrt{2} and it occurs at (sqrt{2}, sqrt{2}); the absolute minimum is -sqrt{2} and it occurs at (-sqrt{2}, -sqrt{2}).
64.   Length 36, width and height both 18 (do this problem using Lagrange multipliers).

Chapter 12 Review, p. 900 - 902
Problems:   1, 3, 5, 7, 9, 10, 13, 14, 15, 17, 19, 21, 22, 23, 28, 29, 30, 32, 34, 41, 42

The answers to the evens are:
10.   y - 4 ≤ x ≤ 4 - y, 0 ≤ y ≤ 4
14.   (e - 1)/4. The region is 0 ≤ y ≤ x2, 0 ≤ x ≤ 1.
22.   (1/3)(23/2 - 1)
28.   π/14
30.   53/20
32.   12π
34.   π/6 (use cylindrical coordinates)
42.   64π/9 (use the identity sin2θ = ½(1 - cos(2θ)) to help compute the integral)

Chapter 13 Review, p. 975 - 976
Problems:   1a, 4, 6, 7, 9

The answers to the evens are:
4.   0 (use x = 3 cos t, y = 2 sin t as the parametrization)
6.   e - 9/70