MATH 201 Exam 2, Fall 2005

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2005-2006 quiz # 2 Name: ............................... ID #: ................................. 1. (25 points) Let f(x, y) = ln(4 − x2 − y2). a. what are the domain Df and the range R of f? b. what is the boundary of Df? is the domain closed, open, bounded? Justify. c. find the equation of the level curve that passes through the point ( √ 2, 1). d. what are the level curves of f? 2. (20 points) The Fourier series expansion of the function f(x) =    0 −π < x ≤ 0 x 0 ≤ x < π is π 4 + +∞ � n=1 1 π �(−1)n − 1 n2 � cos(nx) + +∞ � n=1 bn sin(nx) a. find bn b. use the series in part a) to find +∞ � k=0 1 (2k + 1)2 . 3. (30 points) Let (C) the curve of equation r = cos(2θ). a. show that the x-axis and the y-axis are axis of symmetry for (C). b. graph the curve (C). c. find the area inside one of the leaves of (C). 4. (10 points) Find lim (x,y)→(1,1) x2 − y2 √x − √y 5. (15 points) Use the two path test to show that f(x, y) = cos(x + y) − 1 x2 + y2 does not have a limit at (0, 0). 6. (bonus: 5 points) Let f(x, y) =    x2e− 1+y2 x2 if x ̸= 0 0 if x = 0 Find ∂f ∂x(0, 0).