Consider the function z = f(x, y) = x²+9y² + 3.

(a) [2 marks] Find the domain and range of f. Briefly justify your answer.

(b) [2 points] Draw the contour lines k = f(x, y) for k = 3 and k = 12.

Show your work and indicate the value of k corresponding to each curve in your diagram. Do not forget to put the scales on the coordinate axes, and choose these scales so that the curves are clearly visible.

(c) [1 point] Draw the graph trace of f in the y = 0 plane.

(d) [2 points] Calculate the derivative of f at the point (x, y) = (2, 1) in the direction of vector v = 3, 4. Show off all your hard work!