Mathematics 241, section 1 – Multivariable Calculus Exam III – December 3, 2010

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points.

I. A) (10) Give a precise statement of the Second Derivative Test for local extrema of

functions of two variables. B) (20) Let f(x, y) = 3x^{4} + y^{3} − 4x^{3} − 3y. Find all critical points of f and apply the

Second Derivative Test to each.

II. A) (20) Find the maximum and minimum values of

f(x, y) = xy − y

on the closed and bounded set

S = {(x, y) ∈ R^{2} | x^{2} + y^{2} ≤ 4}.

B) (5) Give a sketch of the contours of f for the maximum and minimum values you found in part A, together with S.

III. All parts of this problem refer to the region

R = {(x, y) ∈ R^{2} | x^{2} + y^{2} ≤ 9, y ≥ 0, y ≤ 2x}.

and the solid S ⊂ R^{3} with “shadow” R in the xy-plane and 0 ≤ z ≤ x^{2}.

A) (5) Sketch the region R in the plane. B) (10) Set up iterated integral(s) of the form ∫ ∫ f(x, y) dy dx to compute the volume

of S. Do not compute the value. C) (10) Now set up iterated integral(s) of the form ∫ ∫ f(x, y) dx dy to compute the

volume of S. Do not compute the value. D) (10) Finally, set up a polar coordinate integral that will compute the volume of S. Do

not compute the value. E) (10) Finally, compute the volume of S using any one of your answers from parts B,C,D

(your choice). For full credit, you must show all work leading to your final answer.

Extra Credit (10) Evaluate the improper double integral

∫ ∫_{R^{2}} e^{−x^{2}−y^{2}} dA = ∫_{−∞}^{∞} ∫_{−∞}^{∞} e^{−x^{2}−y^{2}} dy dx.