MATH
114Final-2012A

A particle in space accelerates according to a t 2 i

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(9) A particle in space accelerates according to ~a ( t ) = 2 i + ( t 2 - 1) j + k with initial velocity ~v 0 = 3 i + 4 j and initial position ~ r 0 ( t ) = i + 5 k . Which of the following is its position at time t = 2? (Recall that i = (1 , 0 , 0), j = (0 , 1 , 0) and k = (0 , 0 , 1).) (A) 4 i + 4 j + 5 k (B) 2 i + 3 j + k (C) 11 i + 7 1 3 j + 7 k (D) 3 i + 9 j + 6 k (E) i - 2 j + 5 k (F) i - 2 3 j + k (G) 2 i - 3 j + k (H) None of the above. Answer to 9:

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11 (10) Evaluate ZZ R e - y 2 dA , where R is the triangular region with vertices (0 , 0), (0 , 1) and (1 , 1). Answer to 10
12 (11) Evaluate ZZZ D e x - y + z dV , where D is the parallelepiped bounded by the planes: x - y + z = 2 , x - y + z = 3 , x + 2 y = - 2 , x + 2 y = 1 , x - z = 0 , x - z = 2 . Answer to 11:

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13 (12) A solid cylinder of height 1 and radius 1 is formed by all the points ( x, y, z ) such that x 2 + y 2 1 and 0 z 1. Find the center of mass of the cylinder if its density at ( x, y, z ) is given by ρ ( x, y, z ) = z . Answer to 12:
14 (13) Calculate the work done by the force field F = h y 2 , x i on a particle which moves along the parabola y = x 2 from the point (0 , 0) to the point (1 , 1). (A) 2 / 5 (B) 1 / 3 (C) 7 / 15 (D) 3 / 5 (E) 2 / 3 (F) 11 / 15 (G) 4 / 5 (H) 13 / 15 Answer to 13:

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15 (14) Suppose f ( x, y, z ) = xe 2 yz . A unit length vector V in the direction of fastest de- crease of f at the point (1,1,0) is: Answer to 14:
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