1. Consider the function f defined by f(x, y) = x²y + y²x x² + y² 0 if (x, y) (0,0); " if (x, y)=(0,0). (a) Show that f is continuous at (0,0). (Hint: Evaluate the appro- priate limit by changing to polar coordinates.) (b) Using the principal definition of partial derivatives, show that (0,0) exists and equals 0. af მე (c) You may assume that af მყ (0, 0) exists and is equal to 0. Using the definition of differentiablity of a multivariable function, explain why f is not differentiable at (0,0). (Hint: Investigate the exis- tence or otherwise of the appropriate limit by changing to polar coordinates.) 2. Consider the region R in the xy-plane given by R={(x,y) R² x ≥ 0, y ≥ 0 and 1 ≤ x² + y² ≤ 2}. (a) Sketch the region R. (b) By changing to polar coordinates evaluate the double integral y sin (π (x² + y²)) √x² + y² dx dy. Recall that Cartesian and polar coordinates are related by (x, y) = (r cos 0, r sin 0), ✓ (1) where r = √√√x² + y², and 0 ≤ 0 < 27. You may assume that the Jacobian for the change of variables (1) is given by (x, y) ǝ (r, 0) = 1.