Mathematics 241 – Multivariable Calculus Quiz 5 – October 22, 2010

Your Name:

A) Let

f(x, y) = \{ \begin{matrix}\frac{xy}{x^{2}+y^{2}} + x & if (x, y) ≠ (0, 0) \\ 0 & if (x, y) = (0, 0). \end{matrix}

(1) (5) Find \frac{∂f}{∂x}(x, y) if (x, y) ≠ (0, 0).

(2) (5) Does \frac{∂f}{∂x}(0, 0) exist? If so, what is its value; if not, why not?

B) (10) Sketch the set S = {(x, y) ∈ R^{2} | |x| < 1, |y| ≤ 2} Is S open, closed, or neither? Explain.