Multivariable Calculus, MATH 241

Exam #1

Friday, Feb. 20, In Class

The first exam covers Chapter 9 (except for Section 9.7) and Sections 10.1 and 10.2. It is recommended that you go over homework problems (HW #1 - 3) and your class notes. Many of the problems and questions we discuss in class are excellent examples of test questions. The solutions to your WebAssign problems on HW #1 - 3 can be seen by clicking "View Key" near the top of each assignment.

In addition, some review problems from the Chapter 9 and 10 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 the full class period (45-50 minutes).

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 on Wednesday, Feb. 18, 6:00 - 7:30 pm in O'Neil 112. Please come prepared with specific questions.

Chapter 9 Review, pp. 689 - 690
Problems:   1, 2, 3, 4, 5, 6, 9, 11, 15, 17, 18, 19, 20, 22, 24, 26 (a, c, d), 27, 28, 33, 34, 35, 36

Note: For #33-36, focus more on identifying the surface and its traces. The sketch is less important.

The answers to the evens are:
4.   (a) 11i - 4j - k; (b) sqrt(14); (c) -1; (d) -3i - 7j - 5k; (e) 3 sqrt(35); (f) 18; (g) 0; (h) 33i - 21j + 6k; (i) -1/sqrt(6); (j) -1/6i - 1/6j + 1/3k; (k) 96 degrees
6.   +- 1/(3 sqrt(6)) (7i + 2j - k)
18.   x + 4y - 3z = 6
20.   6x + 9y - z = 26
22.   (1, 4, 4)
24.   (a) the normal vectors are neither parallel nor perpendicular; (b) 58 degrees
26.   (a) x + 3y + z = 6; (c) 43 degrees; (d) x = 2 + t, y = -t, z = 4 + 2t (other answers are possible)
28.   {(x, y): 2n ≤ x2 + y2 ≤ 2n + 1, where n is an integer}
34.   The surface is an elliptic (or circular) paraboloid opening in the positive x-direction. Traces for x = k are circles; for y = k and z = k they are parabolas opening up.
36.   The surface is a hyperboloid of one sheet with symmetry axis equal to the x-axis. Traces for x = k are circles; for y = k and z = k they are hyperbolas that flip their orientation once |k| > 1; if |k| = 1, then these hyperbolas are degenerate and form two intersecting lines.

Chapter 10 Review, p. 734
Problems:   1, 2, 3, 4, 5, 6

The answers to the evens are:
2.   (a) (-1,0) ∪ (0,2]; (b) ; (c) <-1/(2 sqrt(2-t)), (t et - et + 1)/t2, 1/(t+1)>
4.   x = 1 + sqrt(3) t, y = sqrt(3) + 2t, z = 2
6.   (a) (15/8, 0, -ln(2)); (b) x = 1 - 3t, y = 1 + 2t, z = t; (c) 3x - 2y - z = 1