MATH 201 Final, Fall 2005

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2005-2006 Final Exam Name: ............................... ID #: ................................. good luck Exercise 1 (10 points) Given the surface z = x2 − 4xy + y3 + 4y − 2 containing the point P(1, −1, −2) a. Find an equation of the tangent plane to the surface at P. b. Find an equation of the normal line to the surface at P. Exercise 2 (15 points) a. Which of the following series converges and which diverges? justify. i. +∞ � n=0 en 1 + e2n ii. +∞ � n=1 1 − cos n nln n iii. +∞ � n=1 n 10 + n2 b. Find the radius of convergence of the series +∞ � n=0 x2n 2n , then find its sum. Exercise 3 (10 points) a. 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 b. Prove or disprove: The function f(x, y) = x2y 2x4 + 3y2 can be extended by continuity at (0, 0). Exercise 4 (10 points) Find the absolute minimum and maximum values of the function f(x, y) = 4x − 8xy + 2y + 1 on the triangular plate whose vertices are (0, 0), (0, 1) and (1, 0). Exercise 5 (10 points) Evaluate the integral � 4 0 � 2 √x sin (y3) dydx. Exercise 6 (10 points) Let I = � � � G xyz dV where G is the solid in the first octant that is bounded by the parabolic cylinder z = 2 − x2 and the planes z = 0, y = x, and y = 0. a. Express I as an iterated triple integral in the order dzdydx, then evaluate the resulting integral. b. Express I as an iterated triple integral in the order dxdzdy (do not evaluate the integral). Exercise 7 (15 points) Rewrite the following triple integral J = � 2 0 � √ 4−y2 0 � √ 8−x2−y2 √ x2+y2 z2 dzdxdy a. in the order dxdzdy (do not evaluate the integral). b. in cylindrical coordinates (do not evaluate the integral). c. in spherical coordinates, then evaluate the resulting integral. Exercise 8 (20 points) a. Evaluate � (1,π/2,e) (0,0,1) (ln z + ex sin y)dx + ex cos ydy + (x/z − z)dz b. The outward flux of a field F = Mi + Nj across a simple closed curve C equals the double integral of divF over the region R enclosed by C: � C Mdy − Ndx = � � R �∂M ∂x + ∂N ∂y � dA (1) Find the outward flux of the field F = 2xyi + x2j across the curve C in the first quadrant, bounded by the parabola y = x2 and the line y = 1. i. by using the line integral in the left side of equation (1) ii. by using the double integral in the right side of equation (1)