Mathematics 241, section 1 – Multivariable Calculus Midterm Exam 2 November 1, 2013

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points.

I. All parts of this problem refer to the vector field

F(x, y) = (x^{2} − 2x, xy − y).

A. (10) Find all critical points of F(x, y).

B. (5) There are two vector fields plotted on the back of this sheet. Say which one shows F(x, y) and use that plot to classify each of the critical points as a source, sink, saddle, or center.

C. (20) Show that α(t) = (0, 4e^{−t}) and β(t) = (\frac{2}{1 + e^{2t}} , 0) are both flow lines of F. What are

lim_{t→∞} α(t) and lim_{t→∞} β(t)?

D. (5) Is there a scalar-valued function f(x, y) such that F(x, y) = ∇f(x, y)? Why or why not?

II. In the neighborhood of Eagle Pass, the landscape has elevation in feet above sea level given by f(x, y) = \frac{x^{2}}{4} − y^{2} + 1000.

A. (10) Sketch the contours of f(x, y) for c = 999, 1000, 1001 on the same set of axes.

B. (10) Compute the directional derivative D_{u}f(2, 1) for a general unit vector.

C. (5) In the direction of which unit vector u should you walk from the point with (x, y) = (2, 1) in order to decrease your elevation at the fastest rate?

III. All parts of this problem refer to the function

f(x, y) = \frac{x^{3} − 3xy^{2}}{x^{2} + y^{2}} if (x, y) ≠ (0, 0) and f(0, 0) = 0.

A. (15) Find the tangent plane to z = f(x, y) at (1, 1, f(1, 1)).

B. (10) Does \frac{∂f}{∂x}(0, 0) exist? If so, find it; if not say why not.

IV. (10) Can the curve α(t) = (2 cos(t), sin(t)) for t ∈ (0, ∞) be a flow line of the vector field ∇f(x, y) for a differentiable function f? Why or why not?

Extra Credit (10) Refer to the function in question III. Let m be arbitrary and compute lim_{t→0} f(t, mt) (the limit of the value of f along the line through the origin in the direction of the vector (1, m)). Is lim_{t→0}(f ∘ α)(t) = 0 for every differentiable curve α(t) with α(0) = (0, 0)? Explain.