MATH 201 Exam 2, Fall 2004

AMERICAN UNIVERSITY OF BEIRUT DEPARTMENT OF MATHEMATICS FALL SEMESTER 2004-05 MATH 201, QUIZ II Time: 90 minutes Answer the following questions: 1. Consider the polar curves r = 1 + 2 cos θ and r = 4 cosθ. (a) Sketch the graphs of the polar curves. (8 points) (b) Find their points of intersection. (8 points) (c) Find the area of the region that lies inside the curve r = 4 cosθ and outside the curve r = 1 + 2 cos θ. (9 points) 2. Let f(x) = x/2 if −π ≤ x ≤ π. (a) Show that the Fourier series of f is �∞ n=1(−1)n+1(sin nx)/n. (9 points) (b) For what values of x, −π ≤ x ≤ π, does the equality x 2 = ∞ � n=1 (−1)n+1 sin nx n hold? Justify your answer. (8 points) (c) Use (b) to conclude the value of π/4 as an alternating series. (8 points) 3. (a) By discussing the limit of the function f(x, y) = x4 + y4 (x2 + y2)3/2 ((x, y) ̸= (0, 0)) at (0, 0), decide whether f(0, 0) can be defined so that f becomes continuous at (0, 0). (8 points) (b) Find an equation for the tangent plane to the paraboloid z = 2x2 + 3y2 at the point P(1, −1, 5). (8 points) (c) Find and classify the critical points of the function f(x, y) = 2x3 −3x2 + y2 − 12x + 10. (9 points) 4. (a) The temperature T at (x, y, z) is given by T (x, y, z) = 4x2 − y2 + 16z2. Find the maximum rate of change of T at the point P(4, 2, −1) and the direction in which it is achieved. (9 points) (b) Find an equation for the normal line to the curve 2x3 + x2y = 1 − y3 at the point P(−1/2, 1). (8 points) (c) Sketch the level surface passing through the point P(1, 1, 1) of the func- tion f(x, y, z) = x2 + y2 − z2. (8 points) GOOD LUCK 1 www.amal-aub.org