Multivariable Calculus, MATH 241-02

Prof. Gareth Roberts

Exam #2

Wednesday, March 27, 7:00 - 8:30 pm

Location: Smith Labs 154

The second exam covers Sections 10.3, 10.4 and 11.1 - 11.7. It is recommended that you go over homework problems (HW #4 - 6, 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 #4 - 6 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 10 and 11 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, March 26. Please come prepared with specific questions.

Chapter 10 Review, p. 734
Problems:   8, 9, 10, 11, 12, 17, 18, 20

The answers to the evens are:
8.   (2/27)(133/2 - 8)
10.   r(t(s)) = <1 + (1/sqrt{3})s, (1 + (1/sqrt{3})s) sin( ln(1 + (1/sqrt{3})s)), (1 + (1/sqrt{3})s) cos( ln(1 + (1/sqrt{3})s))>
12.   At (3,0), the curvature is 3/16. At (0,4), the curvature is 4/9.
18.   r(t) = < t3 + t, t4 - t, 3t - t3 >
20.   aT = 4t/sqrt{4 t2 + 5}, aN = 2sqrt{5}/sqrt{4 t2 + 5}

Chapter 11 Review, p. 823-826
Problems:   1, 2, 5, 6, 9, 10, 11a, 11b, 13, 15, 16, 17, 18, 19, 20, 22, 24, 25a, 26a, 27a, 35, 36, 37, 43, 44, 45, 46, 47, 48, 51, 52, 53, 55, 64

The answers to the evens are:
2.   The domain is the set of points on and inside the circle x2 + y2 = 4 satisfying -1 ≤ x ≤ 1.
6.   The level curves are translations of the exponential function y = -ex.
10.   The limit does not exist.
16.   wx = 1/(y-z), wy = -x/(y-z)^2, wz = x/(y-z)^2
18.   CT ≈ 3.587, which means the speed of sound increases by about 3.6 meters per second if the temperature increases by one degree Celsius. CS = 1.24, which means the speed of sound increases by about 1.24 meters per second if the salinity of the water increases by one part per thousand. CD = 0.016, which means the speed of sound increases by about 0.016 meters per second if the depth of the water is increased by one meter.
20.   zxx = 0, zyy = 4x e-2y, zxy = zyx = -2 e-2y
22.   vrr = 0, vss = -r cos(s + 2t), vtt = -4r cos(s + 2t), vrs = vsr = -sin(s + 2t), vrt = vtr = -2sin(s + 2t), vst = vts = -2r cos(s + 2t).
24.   Just compute the four different partial derivatives and show the left-hand side and right-hand sides of the equation are identical.
26a.   z = x + 1
36.   vs = 5, vt = 0.
44.   (a) Max occurs when u points in the same direction as the gradient vector.
(b) Min occurs when u points in the opposite directon of the gradient vector.
(c) The directional derivative is 0 along a contour or level curve (surface). This is the same as when u is orthogonal to the gradient vector.
(d) Half the maximum value occurs when the angle between u and the gradient vector is 600.
46.   25/6
48.   Direction: <2/sqrt{5}, 0, 1/sqrt{5}>; Maximum rate of increase is sqrt{5}.
52.   (0,0) is a saddle point; (1,1/2) is a local minimum
64.   Length 36, width and height both 18.