Mathematics 241, section 1 – Multivariable Calculus Final Examination – December 14, 2010

Directions

Do all work in the blue exam booklet. Do not place anything you wish to have considered for credit on the exam sheet. There are 200 regular and 10 Extra Credit points, distributed as indicated next to the questions. You have until 5:30pm to work on these problems.

I. In this problem, P = (1, −1, 3), Q = (0, 3, 1), and R = (−4, 1, 2).

A) (10) Find the equation of the plane containing the points P, Q, R in R^{3}. B) (10) At what point does the line containing P, Q meet the xy-plane? C) (5) If v is the vector from P to Q and w is the vector from P to R, at what angle do

v, w meet?

II. Suppose you follow a flow line of the vector field ∇f for f(x, y) in the xy-plane. A) (5) As you traverse the flow line in the increasing t-direction, is the corresponding

path on the graph z = f(x, y) going uphill or downhill? Explain. B) (5) What does the vector field ∇f look like near a local maximum of f? Near a local

minimum of f? Make rough sketches by hand to illustrate.

III. All parts of this problem refer to f(x, y) = (x − 1)^{2} − y^{2}.

A) (5) Sketch the contours of f(x, y) for the values c = −1, 0, 1. B) (10) At the point (1, 2), in which direction is f increasing the fastest? Express your

answer as a unit direction vector.

IV. (20) After an ill-fated “three hour tour” goes awry, you are stranded on a island at the point with coordinates (1, 1). Fortunately, you have a radio transmitter with you. Unfortunately, it has a limited range – its signal can only be received at distances less than or equal to 2/3 from its position. You know that there is a Coast Guard patrol boat that makes a circuit of the path x^{2} + y^{2} = 4 every day, and they always carry a radio receiver and listen for transmissions. Will the patrol boat ever get within 2/3 of your position and receive your signal? (Note: Minimizing the distance from a point (a, b) is the same as minimizing the function f(x, y) = ∥(x, y) − (a, b)∥^{2}.)

V. Let f(x, y) = ye^{−x^{2}−2y^{2}}.

A) (10) Find the equation of the tangent plane to the graph z = f(x, y) at the point

(1, 1, e^{−3}). B) (10) Find all the critical points of f(x, y). C) (20) Use the Second Derivative Test to determine the type of each critical point you

found in part B.

VI. The region R in R^{2} is the set of points

R = {(x, y) ∈ R^{2} | x^{2} + y^{2} ≤ 25, y ≥ x, x ≥ 0, and y ≤ 4}

and let f(x, y) be some continuous function defined on R.

A) (5) Sketch R. B) (10) Set up the limits of integration of iterated integral(s) to compute ∫ ∫_{R} f(x, y) dA

integrating with respect to x first, then y. C) (10) Now reverse the order of the variables and set up iterated integral(s) to compute

the same integral, but integrating with respect to y first, then x.

VII. (20) The metal making up a thin plate with the shape of the region in R^{2} with x^{2} + y^{2} ≤ 4 has density δ(x, y) = 6 + y at all points. Determine the coordinates of its center of mass.

VIII. Consider the following triple integral:

∫_{−3}^{3} ∫_{−\sqrt{9−x^{2}}}^{√\overline{9−x^{2}}} ∫_{−\sqrt{9−x^{2}−y^{2}}}^{√\overline{9−x^{2}−y^{2}}} 1 + x dz dy dx

A) (5) Describe the solid over which you are integrating here. B) (20) Using any convenient coordinate system, evaluate this integral.

IX.

A) (10) State Green’s Theorem. B) (10) Let F(x, y) = (x − y^{2}, x^{2} + y). Verify that Green’s Theorem holds for the region

D = {(x, y) ∈ R^{2} : x^{2} + y^{2} ≤ 9} by computing both sides of the equation in the theorem and showing that they are equal.

Extra Credit

A function f(x, y) is said to be harmonic on an open set U in R^{2} if it satisfies the equation

f_{xx} + f_{yy} = 0

at all points in U. (Here f_{x}, f_{y}, f_{xx}, f_{yy} are the partial derivatives of the harmonic function f with respect to the indicated variables.) A) (5) How does a nondegenerate critical point of a harmonic function fit into our classi-

fication? Is it a local maximum, local minimum, or a saddle point? Explain how you can tell from the Second Derivative Test.

B) (5) If f is harmonic, what is true about the total flux of the vector field

F(x, y) = ∇f(x, y)

across any simple closed curve in U? Explain.

Have a peaceful and joyous holiday season!