MTH 206 Final, Spring 2013

206-Final-S’12 1 | P a g e LEBANESE AMERICAN UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND MATHEMATICS MTH 206 - CALCULUS 3 FINAL EXAM – SPRING 2012 Instructors (circle one): Dr. May Hamdan Dr. Leila Issa Duration: 120 minutes Name: ID#: • This exam consists of 12 pages and 11 problems. • Answer the questions below on the space provided. You can use the back pages for scratch or for more space for your answers. Please specify. • Make sure you justify all your answers. Question Number Grade 1. 8% 2. 8% 3. 8% 4. 8% 5. 8% 6. 8% 7. 12% 8. 9% 9. 8% 10. 8% 11. 15% TOTAL Problem 1: (8%) If f(x, y, z) is a differentiable function, x = r − s, y = s − t and z = t − r, show that fr + fs + ft = 0 2 Problem 2: (8%) Find a direction in which the function f(x, y) = xe−4y2 incurs no change at the point P(1, 0). 3 Problem 3: (8%) Find the points on the surface (y + z)2 + (z − x)2 = 16 where the normal line is paral- lel to the y − z plane. 4 Problem 4: (8%) Let f(x, y) = e10x−5y. Using linearization, approximate f(0.05, −0.03). 5 Problem 5: (8%) If the sum of squares of two real numbers is 10, find the maximum of their product. 6 Problem 6: (8%) Find the volume of the region bounded above by the sphere x2 + y2 + z2 = 2 and below by the paraboloid z = x2 + y2. 7 Problem 7: (12%) At a point P, the velocity and acceleration of a particle moving in the plane are ⃗v = 3⃗i + 4⃗j and ⃗a = 5⃗i + 15⃗j. Find the curvature of the particle’s path at the point P. 8 Problem 8: (9%) Name formulas and theorems that connect the following. Make sure you mention the rele- vant conditions in each case (you may need more than one step to go from one integral to another): (a) Line integral to surface integral (b) Line integral to triple integral (c) Surface integral to double integral 9 Problem 9: (8%) Use the most efficient method to compute � C ⃗F · d⃗r, where ⃗F = 1 y ⃗i + (1 z − x y2)⃗j − y z2⃗k, and C is the circle in the y − z plane centered at (0, 2, 1) with radius 1. 10 Problem 10: (8%) Use the most efficient method to find the outward flux of the vector field ⃗F = x⃗i + y⃗j + z⃗k across the boundary of the cube bounded by x = ±1, y = ±1 and z = ±1. 11 Problem 11: (15%) (a) Let f(x) be the periodic function of period T = 2 defined as f(x) = |x| for − 1 ≤ x ≤ 1 Find the Fourier series of the function f(x) (you may want to use integration by parts in the process). (b) Let g(x) be the periodic function of period T = 2 defined as g(x) = � x, 0 ≤ x < 1, 2 + (x − 1)2, 1 ≤ x < 2. To what value does the Fourier series of g(x) converge at the point x = 1? Justify. (c) Write in your own words why are Fourier series useful. 12