MATH 201 Final, Spring 2005

American University of Beirut MATH 201 Calculus and Analytic Geometry III Spring 2005 Final Exam TD FE eseeseeeeseeeeeeneereeseeecters Exercise 1 [9 points] Discuss whether the following series converges OF diverges: 1l+n? +00 ny a) S- vn n=l £2 1.3.5.7...(2n — 1) > (2n)! n=1 +00 n e c) >» gntnsl) n=0 +090 : i 1 Exercise 2 /10 points] Find the interval of convergence of the power series S Wavere| Ce 3)" n: 2 n=1 (be sure to check at the end points) points] Find an equation of the tangent plane and a parametric equation for Exercise 3 a) [5 24 — Se +1 at the point (2, 1,3). the normal line to the surface z = 3a b) [6 points] Suppose that 2 = e™ and that cg = 2u+vu,y = u/v. Use the chain rule to find 8z/du and Gz/dv. (give the expressions in terms in u and v) vy? a = can be extended by continuity at (0, 0). . ¢) [5 points] Prove or disprove: g{z,y) pay + J ustify. Exercise 4 /15 points] Find the absolute minimum and maximum values of f(z,y) = 8ay — 62 —3y+7 (0,0), (3,0), and (0, 5). on the closed triangular region R with vertices Exercise 5 /10 points} Evaluate the double integral dzdy ety? f ree r 0 40 egral in polar coordinates. by converting it to a double int 10 points] Let V be the volume of the tetrahedron bounded by the coordinages Exercise 6 /10 poin planes and the plane z = 4—4a—2y in the order: . - Express V as iterated triple integral in Cartesian Coordinates the region R cut from the cone z = Vze + 42 by the two Planes z = 1 and z= a) Sketch the region of integration, b) Express y 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(z,y,z) = 4yi + Qezj 4 3yk acting on an abject as it Moves along the line Segment from ( b) Evaluate the line integral gd (3. (2,1,1) to (2,2, 1). 2,0) (22 cos z ~ 2 de + (zo 2y)dy + (y ~z* sin z)dz.