MATH 201 Exam 2, Fall 2008

Quiz 2-Math 201 • Write your name and your I.D. on the booklet • The duration of the test is one hour • Calculators are allowed 1. (20 points) a) Find the first four terms of the Taylor series of e−x and cos(5x) centered at a = 0. b) Compute the limit lim x→0 e−x − 1 + x cos(5x) − 1 c) Find an approximation of � 0.1 0 cos(5x)−1 x dx with an error less than 10−4.(Use the alternating series error estimation theorem) 2. (10 points) Define f(x) = 1 − x if 0 ≤ x ≤ π and 0 if π < x ≤ 2π. In the Fourier series f(x) = a0/2 + �∞ n=1 an cos nx + �∞ n=1 bn sin nx, find a3. 3. (25 points)Given f(x, y) = 5xy +�∞ n=0 (−1)nx2n+1y2n+1 (2n+1)! .(Hint: Write the series as a function of x and y) a) Find the equation of the tangent plane to z = f(x, y) at the point P(π, 1, 5π). b) Find the directional derivative of f at the point P1(π, 3) in the direction of v(3, 1). c) Find the standard linear approximation of f(x, y) at the point (π, 3)then use to estimate f(π, 2.99). Approximate the error. 4. (10 points) Show that lim(x,y)→(0,0) x4 x4+y2 does not exist while lim(x,y)→(0,0) x4 x2+y2 = 0. 5. (20 points)Suppose that over a certain region of space the electric potential V is given by V (x, y, z) = 5x2 − 3xy + xyz 1 a) Find the rate of change of the potential at P(3, 4, 5) in the direction of the vector v = i + j − k. b) In what direction does V change most rapidly at P. c) What is the maximum rate of change at P. 6. (15 points)The length l, width w and height h of a box change with time. At a certain instant the dimensions are l = 1m and w = h = 2m, and l and w are are increasing at a rate of 2m/s while h is decreasing at a rate of 3m/s. At that instant, find the rates at which the volume and the surface area are changing. (The box is closed from above.) 2