MATH 201 Final, Fall 2010

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2010-2011 Final Exam Exercise 1 Determine if the following series converges or diverges. Justify your answers. a) ∞ � n=0 e−n2 b) ∞ � n=1 � 1 − 1 n �n c) ∞ � n=2 5 ln2 n Exercise 2 Give the parametric equations of the line tangent to the curve of intersection of the surfaces xyz = 1 and x2 + 2y2 + 3z2 = 6 at the point P(1, 1, 1) Exercise 3 a) Find the absolute maximum and minimum values of the function f(x, y) = 2x2 − xy + y2 − 7x on the region R = {(x, y) ; x ≥ 0 , y ≤ 3 , 2x − y ≤ 3} (group your points and their values in a table) b) Use Lagrange multipliers to find the point on the plane 2x + 3y + 4z = 12 at which the function f(x, y, z) = 4x2 + y2 + 5z2 has its least value. Exercise 4 Change the order of integration, then evaluate the integral � 2 0 � 1 y/2 ex2 dx dy Exercise 5 Let V be the the volume of the region D enclosed by the cylinder x2 + y2 = 4 and the planes z = 0 and y + z = 4 . a) Express, but do not evaluate, V as a triple integral in cartesian coordinates in the order dzdydx. b) Express V as a triple integral in cylindrical coordinates, then evaluate the resulting integral. Exercise 6 Convert to spherical coordinates the integral: � 1 0 � √ 1−x2 − √ 1−x2 � 1 √ x2+y2 dz dy dx then evaluate the resulting integral. Exercise 7 Evaluate � 1 0 � 1−x 0 √x + y (y − 2x)2 dy dx by applying the transformation u = x + y and v = y − 2x −→ Exercise 8 a) Find the work done by the force F = xyi+(y −x)j over the straight line from (1, 1) to (2, 3). b) Find the counterclockwise circulation of the field F = xyi + (x − y)j around the curve C in the first quadrant, bounded by the curve y = x2 and y = x. i) by direct calculation ii) by Green’s theorem