MATH 201 Exam 2, Fall 2008

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2008-2009 quiz # 2 1. (15 points) Find the domain and the range of the function f(x, y) = √xy. Determine the boundary of the domain, is the domain open or closed, also determine if the domain is bounded or unbounded. Find the equation of the level curve passing through (1, 1). Deter- mine all the level curves of f. 2. a. (8 points) Give the first four terms of the Maclaurin series of f(x) = √ 1 − x2, then deduce f(4)(0). b. (10 points) Find the values of a for which lim x→0 �sin x x3 − 1 x2 + a � = 0 3. a. (10 points) Let w = xe2y cos(z). Use the chain rule to find the value of dw/dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln(t + 2), z = t. b. (8 points) Find the equation of the tangent plane to the surface z = y − x2 at (1, 2, 1). 4. (16 points) Find the absolute minimum and maximum of the function f(x, y) = x2−xy+y2+1 on the closed triangular plate in the first quadrant bounded by the lines x = 0, y = 4, y = x. 5. a. (8 points) Find the direction of maximum increase of f(x, y) = x2 − 3xy + 4y2 at P(1, 2). Is there is a direction u in which the rate of change of f at P(1, 2) equals 14? Justify your answer. b. (10 points) Find the points on the surface (y + z)2 + (z − x)2 = 16 where the normal line is parallel to the yz-plane. 6. a. (8 points) Show that the limit of f(x, y) = 3x2y x2 + 4y2 exists at (0, 0). b. (7 points) Show that f(x, y) = sin(x − y) |x| + |y| has no limit at (0, 0).