Mathematics 241 – Multivariable Calculus Final Examination – December 11, 2007

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 20 extra credit points, distributed as indicated next to the questions.

Notation: In the following, for simplicity, the subscript notation is used for partial derivatives: f_{x} = \frac{∂f}{∂x}, f_{y} = \frac{∂f}{∂y} , f_{xx} = \frac{∂^{2}f}{∂x^{2}} , and so forth.

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

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 displacement vector from P to Q and w is the displacement vector from

P to R, at what angle do v, w meet?

II.

A) (7.5) Let F : R_{u,v}^{2} → R_{x,y}^{2} be F (u, v) = (x(u, v), y(u, v)) = (u^{2} − v^{2}, 2uv) and let g : R_{x,y}^{2} → R be g(x, y) = sin(x) cos(y). Find the derivative matrix D(g ∘ F) by direct substitution and differentiation.

B) (7.5) Compute D(g ∘ F) for the functions in part A using the Chain Rule and show

you get the same result as in part A. C) (10) Now let z = g(x, y) for a general function g – not specifically g(x, y) as above in

part A, and x = u^{2} − v^{2}, y = 2uv as in the definition of F above. Show that

\frac{1}{4(u^{2} + v^{2})} ((z_{u})^{2} + (z_{v})^{2}) = (z_{x})^{2} + (z_{y})^{2}.

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

A) (5) Sketch the level curves of f(x, y) for the values c = 1, 4, 9. B) (10) At the point (1, 2), in which direction is f increasing the fastest? Express your

answer as a unit direction vector. C) (15) Find the points on the curve g(x, y) = \frac{x^{2}}{4} + y^{2} = 1 at which f(x, y) takes its

largest and smallest values. What is true about the vectors ∇f and ∇g at your points?

IV. Let f(x, y) = xe^{−2x^{2}−y^{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 (Hessian criterion) to determine the type of each

critical point you found in part B.

V. A region R in R^{2} is the set of points satisfying x^{2} + y^{2} ≥ 1, y ≥ x, x ≥ 0, and y ≤ 4.

A) (5) Sketch the region 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.

VI. (20) The metal making up a solid slug having the shape of the region in R^{3} with x^{2} + y^{2} ≤ 4 and −1 ≤ z ≤ 1 has density δ(x, y, z) = 5 + x at all points. Determine the coordinates of its center of mass.

VII.

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}. C) (5) We can think of a vector field on R^{2} as a vector field on R^{3} by making the last

component equal to zero at all points. Compute curl(F) for F = (M(x, y), N(x, y), 0) and relate your result to the statement of Green’s Theorem.

VIII. 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. 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 Hessian criterion.

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

F(x, y) = (−f_{y}, f_{x})

around any simple closed curve in U?

Extra Credit (20)

Suppose you follow a flow line of the vector field ∇f for f(x, y) in the xy-plane. 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. What does the vector field ∇f look like near a local maxmimum of f? near a local minimum of f?

Have a peaceful and joyous holiday season!