Multivariable Calculus, MATH 241-02

Prof. Gareth Roberts

Exam #1

Wednesday, Feb. 20, 7:00 - 8:30 pm

Location: Smith Labs 154

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, 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 #1 - 3 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 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 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 or symbolic capabilities. Please bring your own calculator with you to the exam.

Exam Review: We will review for the exam during class on Tuesday, February 19th. 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, 31, 32, 33, 34, 35, 36

Note: For #31-36, 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°
26.   (a) x + 3y + z = 6, (c) 43°, (d) x = 2 + t, y = -t, z = 4 + 2t (other answers are possible)
32.   The graph is the top half of an ellipsoid. The traces for x = k and z = k are ellipses, while if y = k, the traces are circles.
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, the hyperbolas are degenerate and form two intersecting lines.

Chapter 10 Review, p. 734
Problems:   1, 2, 3, 4 (no graph required), 5, 6 (a, b), 11a

The answers to the evens are:
2.   (a) (-1,0) ∪ (0,2], (b) < sqrt(2), 1, 0 >, (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