MATH 201 Final, Fall 2007

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2007-2008 Final Exam Name: ............................... ID #: ................................. Exercise 1 (12 points) Discuss the convergence of the following series: a) +∞ � n=0 1 √ n! b) +∞ � n=0 n + 10 n ln3 n c) +∞ � n=0 (−1)n2 1 n2 + √n Exercise 2 (10 points) If w = f(x, y) is differentiable and x = r + s, y = r − s, show that ∂w ∂r × ∂w ∂s = �∂f ∂x �2 − �∂f ∂y �2 Exercise 3 (15 points) Find the absolute extrema of f(x, y) = 5 + 4x − 2x2 + 3y − y2 on the region R bounded by the lines y = 2, y = x, and y = −x. Exercise 4 (8 points) Use Lagrange multipliers to find the maximum and minimum of f(x, y) = 4xy subject to x2 + y2 = 8. Exercise 5 (15 points) Evaluate the integral � 2 0 � 4−x2 0 � x 0 sin(2z) 4 − z dydzdx Exercise 6 (10 points) Evaluate the integral � � D cos(x2 + y2)dA, where D = {(x, y) ∈ R2; 1 ≤ x2 + y2 ≤ 2, y ≥ 0} Exercise 7 (20 points) Let V be the volume of the region D that is bounded form below by the plane z = 0, from above by the sphere x2 + y2 + z2 = 4 and on the sides by the cylinder x2 + y2 = 1 a. express V as an iterated triple integral cartesian coordinates in the order dzdydx b. express V as an iterated triple integral cartesian coordinates in the order dydzdx c. express V as an iterated triple integral spherical coordinates d. express V as an iterated triple integral cylindrical coordinates, then evaluate the resulting integral. Exercise 8 (10 points) Set up an integral in rectangular coordinates equivalent to the integral � π/2 0 � √ 3 1 � √ 4−r2 1 r3(sin θ cos θ)z2dzdrdθ (do not evaluate the integral) good luck