MATH 201 Exam 2, Fall 2006

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2006-2007 quiz # 2 Name: ............................... ID #: ................................. 1. (10 points) Find the value of a for which the limit lim x→0 √1 + x − 1 − ax x2 is finite and evaluate this limit. 2. (20 points) Let f(x, y) = ln � 1 e − x2 − y2 � a. find the domain Df and the range R of f b. what’s the boundary of Df, is the domain bounded? Justify. c. is the domain closed or open? Justify. d. find the equation of the level curve that passes through the point (0, 1). e. sketch the level curves of f? 3. (20 points) Find the area that lies inside the cardioid r = 2 + 2 cos θ and outside the circle r = 2 (sketch the two curves) 4. (20 points) The Fourier series expansion of the function f(x) =    1 0 ≤ x ≤ π 2 π < x ≤ 2π is a0 + +∞ � n=1 bn sin(nx) a. find a0, and bn b. use the series in part a) to show that +∞ � k=0 (−1)k 2k + 1 = π 4 . 5. (7 points) Find lim (x,y)→(0,0) x2y2 � x2 + y2 6. (13 points) Use the two paths test to show that the function f(x, y) = ln(1 + xy) x2 + y2 does not have a limit at (0, 0)