MATH 201 Final, Spring 2005

American University of Beirut MATH 201 Calculus and Analytic Geometry III Spring 2005 Final Exam Name: ............................... ID #: ................................. Exercise 1 [9 points] Discuss whether the following series converges or diverges: a) +∞ � n=1 n√n 1 + n2 b) +∞ � n=1 1.3.5.7...(2n − 1) (2n)! c) +∞ � n=0 en 3n(n+1) Exercise 2 [10 points] Find the interval of convergence of the power series +∞ � n=1 1 √ n2 + 1 (x − 1 2)n (be sure to check at the end points) Exercise 3 a) [5 points] Find an equation of the tangent plane and a parametric equation for the normal line to the surface z = 3x2y − 5x + 1 at the point (2, 1, 3). b) [6 points] Suppose that z = exy and that x = 2u + v, y = u/v. Use the chain rule to find ∂z/∂u and ∂z/∂v. (give the expressions in terms in u and v) c) [5 points] Prove or disprove: g(x, y) = x2y2 x4 + 3y4 can be extended by continuity at (0, 0). Justify. Exercise 4 [15 points] Find the absolute minimum and maximum values of f(x, y) = 3xy − 6x − 3y + 7 on the closed triangular region R with vertices (0, 0), (3, 0), and (0, 5). Exercise 5 [10 points] Evaluate the double integral � 2 0 � √ 1−(y−1)2 0 x x2 + y2 dxdy by converting it to a double integral in polar coordinates. Exercise 6 [10 points] Let V be the volume of the tetrahedron bounded by the coordinates planes and the plane z = 4−4x−2y. Express V as iterated triple integral in cartesian coordinates in the order: a) dxdzdy (do not evaluate the resulting integral). b) dzdydx, then evaluate the resulting integral. Exercise 7 [15 points] Let V be the volume of the region R cut from the cone z = � x2 + y2 by the two planes z = 1 and z = 2. a) Sketch the region of integration. b) Express V as iterated triple integral in cylindrical coordinates (do not evaluate the resulting integral). c) Express V as iterated triple integral in spherical coordinates, then evaluate the resulting integral. Exercise 8 [15 points] a) Find the work done by the force field defined by F(x, y, z) = 4yi + 2xzj + 3yk acting on an abject as it moves along the line segment from (1, 1, 1) to (2, 2, 1). b) Evaluate the line integral � (1,0,π) (3,−2,0) (2x cos z − x2)dx + (z − 2y)dy + (y − x2 sin z)dz.