Mathematics 241 – Multivariable Calculus Exam 2 October 26, 2007

Information and Directions

Do all work in the blue exam booklet. Do not place anything you want considered for credit on this sheet. There are 100 possible points, plus the Extra Credit problem. Extra Credit points will be awarded as on the first exam – the Extra Credit problem will be counted only if it raises your average.

I. Let F (x, y) = (x − y^{2}, xy) and let g(t) = (sin(t), cos(t) + sin(t)) A) (10) Compute the derivative matrices D(F) and D(g). B) (10) Use the Chain Rule to compute D(F ∘ g)(π). C) (10) Compute the composition F ∘ g and find its derivative matrix directly to check

your answer from part B.

II. (20) Let z = f(x+y, x−y) have continuous partial derivatives with respect to u = x+y and v = x − y. Show that

\frac{∂z}{∂x} \frac{∂z}{∂y} = (\frac{∂z}{∂u})^{2} − (\frac{∂z}{∂v} )^{2} .

III. A mountain has the shape of the graph

z = \sqrt{4 − x^{2} − \frac{y^{2}}{4}} ,

where x, y are east-west and north-south map coordinates respectively, and z is the altitude. A) (10) If you are standing at the point on the mountain with (x, y, z) = (1, 2, \sqrt{2}), which

direction should you move (on the map) to increase your altitude the fastest? B) (10) At the same point as in part A, in which direction should you move (on the map)

to remain at a constant altitude?

IV. Let x(t) = (t^{2}/2, t, t) and y(t) = (t cos(t), t sin(t), t) be two curves in R^{3}. A) (10) Does the tangent line to x(t) at t = 2 cross the xy-plane? If so, where does this

occur? B) (10) Show that y(t) (for t > 0) is a flow line for the vector field

F (x, y, z) = (−y + \frac{x}{z}, x + \frac{y}{z} , 1) .

C) (10) Show that the segments of x(t) and y(t) for a ≤ t ≤ b have the same arclength for all a, b. (Hint: Think about how the arclength would be computed, but it’s not necessary to actually do the computation.)

Extra Credit (10) What is the curvature function for the curve x(t) from problem IV above?