MATH 201 Final, Fall 2006

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2007 Final Exam Name: ............................... ID #: ................................. circle your section please: 1 (Th 2-2:50) 2 (F 2-2:50) 3 (F 4-4:50) 4 (F 9-9:50) Exercise 1 (12 points) Determine if the following series converge or diverge. Justify your answers a. +∞ � n=0 23n 1 + 32n b. +∞ � n=1 n! en2 c. +∞ � n=1 lnn n n2 Exercise 2 (8 points) Assuming that the following equation 2xy + ex+y − 2 = 0 defines y as a differentiable function of x, find the value of dy dx at the point P(0, ln 2) Exercise 3 (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 4 (15 points) Express the volume V under the parabolic cylinder z = x2 above the region enclosed by the parabola y = 6 − x2 and the line y = x in the xy-plane. a. by using triple integral in the order dzdydx b. by using triple integral in the order dzdxdy c. Evaluate V Exercise 5 (15 points) Convert � 2π 0 � √ 2 0 � √ 4−r2 r 3rdzdrdθ to (a) rectangular coordinates with the order dzdxdy and (b) spherical coordinates. Then (c) evaluate one of the integrals. Exercise 6 (10 points) Evaluate � 2 0 � 0 −√ 1−(y−1)2 xy2dxdy by converting it to polar coordi- nates Exercise 7 (15 points) Find the absolute minimum and maximum values of f(x, y) = 3xy − 6x − 3y + 7 on the closed triangular region R whose vertices are (0, 0), (3, 0) and (0, 5). Exercise 8 (20 points) a. Find the work done by the force F = xyi+(y −x)j along the straight line from (1, 1) to (2, 3) b. Green’s Theorem: The counter clockwise circulation of a field F = Mi + Nj around a simple closed curve C equals the double integral of (curl F).k over the region R enclosed by C: � C Mdx + Ndy = � � R �∂N ∂x − ∂M ∂y � dA (1) Evaluate � C y2dx + 3xydy Where C is the counter clockwise-oriented boundary of the upper-half unit disk. i. by using the line integral in the left side of equation (1) ii. ((bonus: 5 points)) by using the double integral in the right side of equation (1) good luck