Mathematics 241, section 1 – Multivariable Calculus Make-up Midterm Exam 2 November 8, 2010

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} − x, y).

A. (5) Find all critical points of F(x, y).

B. (10) 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. (10) Is α(t) = (\frac{2}{1 − 2e^{2t}} , \frac{−2}{1 − 3e^{−2t}} ) a flow line of F? Why or why not?

D. (10) Show that any f(x, y) = \frac{x^{3}}{3} − \frac{x^{2}}{2} + \frac{y^{2}}{2} + c satisfies ∇f = F. What are the critical points of such an f? Classify each of them as a local maximum, local minimum, or saddle point. (Don’t “start over” here; use previous parts of this question as appropriate.)

II. All parts of this problem refer to the function

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

A. (10) Compute \frac{∂f}{∂x}(x, y) and \frac{∂f}{∂y} (x, y) for (x, y) ≠ (0, 0).

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

C. (10) 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)). Does this prove anything about the existence of the limit lim_{(x,y)→(0,0)} f(x, y)? Explain your answer using the definition of the statement lim_{(x,y)→(0,0)} f(x, y) = L.

D. (10 Extra Credit) Is f(x, y) differentiable at (0, 0)? Explain your answer.

III. All parts of this problem refer to f(x, y) = x^{2} + 9y^{2}.

A. (15) Sketch the contours of f(x, y) for c = 0, 1, 4 on the same set of axes.

B. (10) Consider

S = {(x, y) ∈ ℝ^{2} | 1 < f(x, y) ≤ 4}.

Is S open, closed, or neither? Why? (An intuitive explanation referring to the definition of open or closed is OK here.)

C. (10) For which unit vectors u is the directional derivative D_{u}f(1, 1) = 0?

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