Question 1. (a) Given the following information, use chain rule to find - and . u = xy +xz + yz, x= r, y = rcost , z = r sint (b) If f is differentiable function of the variable u , letu = mx - ky and prove that z = f(mx - ky) satisfies the equation k (2") + m (2") = 0, where k and m are constants. Question 2. Determine the gradient of the given function. f (x, y) = es + y* sin(ry) Question 3. For the given function, find the gradient of fat P, the rate of change of the function value in the direction of U at P. Also find the maximum rate of change of the function at the indicated point and the direction in which this maximum rate of change occurs. f (x, y, z) = 2x3 + xy2 + xz2 ; P = (1, 1, 1) ; vector U = V21 2VZ k Question 4. Let g(x, y) = x2+y2 (x, y) = (0,0) (x, y) = (0,0) Discuss the continuity of the above function. (c) Determine the domain of f, and sketch in R the domain of f. f(x, y) = In(x2 + y2 - 25) + In(4 - x2 - y2)