114Final-2012A

(a 4 5(1 e 2(b 1 5 e 3 e 2(c 4 5 e 3 e 4(d 1 5 e 4

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Unformatted text preview: (A) 4 5 (1- e 2 ) (B) 1 5 ( e 3- e 2 ) (C) 4 5 ( e 3- e 4 ) (D) 1 5 ( e 4- 1) (E) 3 5 (1- e 2 ) (F) 6 5 ( e 3- e 2 ) (G) 3 5 ( e 3- e 4 ) (H) 6 5 ( e 4- 1) Answer to 11: 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 . (A) (0 , , 1) (B) (0 , , 1 / 2) (C) (0 , , 2 / 3) (D) (0 , , 3 / 4) (E) (0 , ,π/ 2) (F) (0 , ,π/ 3) (G) (0 , , 2 π/ 3) (H) (0 , , 3 π/ 4) 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: 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: (A) V =- i (B) V = i + 2 k (C) V =- i- k (D) V = 1 √ 5 ( i + 2 k ) (E) V =- 1 √ 5 ( i + 2 k ) (F) V = 1 √ 2 ( i + k ) (G) V =- 1 √ 2 ( i + k ) (H) V = 1 √ 2 ( i- k ) Answer to 14: 16 (15) The function f ( x,y ) = xy 2 + x 2 y + 12 x has: (A) No critical points at all. (B) Exactly one saddle point and no other critical points. (C) Exactly one local maximum and no other critical points. (D) Exactly one local minimum and no other critical points. (E) One local maximum and one saddle point. (F) One local maximum and one local minimum. (G) One local minimum and one saddle point. (H) Two saddle points. Answer to 15: 17 (16) If three resistors having resistances x , y and z are connected in parallel, then the resistance R of the entire circuit satisfies the equation 1 R = 1 x + 1 y + 1 z . Use differentials to estimate R to three decimal places if x = 2 . 012, y = 4 . 008 and z = 3 . 992. (Note, 1 = 1 2 + 1 4 + 1 4 .) (A) 0 . 998 (B) 0 . 999 (C) 1 . 000 (D) 1 . 001 (E) 1 . 002 (F) 1 . 003 (G) 1 . 004 (H) 1 . 005 Answer to 16: 18 Math 114: Final Exam Answer Record Answer Record Instructions: (1) Record your answers on this sheet. (2) Tear off this sheet and take it with you after the exam. (3) Check your answers against the key to be posted on your courses web page. Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16...
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(A 4 5(1 e 2(B 1 5 e 3 e 2(C 4 5 e 3 e 4(D 1 5 e 4 1(E 3...

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