Problem

Suppose that

$$f(x,y)= \begin{cases} x+2y, & \text{if }2x+y\ge 0,\\ 3x+y, & \text{if }2x+y<0. \end{cases}$$

(a) (5 pts) Determine whether $f$ is continuous at $(1,-1)$. Justify your answer.

(b) (5 pts) Find $f_x(0,1)$.

(c) (5 pts) Let

$$S=\{(x,y):0\le x\le 1\text{ and }2\le y\le 3\}.$$

Find $\int_S f(x,y)\,d(x,y)$.

(d) (5 pts) Let

$$S=\{(x,y):-3\le 2x+y\le 3\text{ and }-3\le x-y\le 0\}.$$

Find $\int_S f(x,y)\,d(x,y)$.