SalasSV_11_08_ex_ans

# SalasSV_11_08_ex_ans - A-80 ANSWERS TO ODD-NUMBERED...

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Unformatted text preview: A-80 ANSWERS TO ODD-NUMBERED EXERCISES (k + 1)k 2 (n + k β 1)! n x + Β·Β·Β· + x + Β·Β·Β· 2! n!(k β 1)! β SECTION 11.8 1. 1 + 2x + 3x2 + Β· Β· Β· + nxnβ1 + Β· Β· Β· 5. ln (1 β x2 ) = βx2 β β 3. 1 + kx + 14 16 1 2n+2 x β x β Β·Β·Β· β x β Β·Β·Β· 2 3 n+1 β 7. 1 + x2 + 2 x4 + 3 β 17 6 x 45 + Β·Β·Β· β 9. β72. ( β 1)k 3k +3 x k! 11. k =0 ( β 1)k 4k +2 x (2k + 1)! 13. k =0 3k 3k x k! 27. 15. 2 k =0 β x2k +1 β 17. k =0 (k ! + 1) k x k! β 19. k =1 k ( β 1)k +1 3k +1 x k 21. k =0 23. 1 2 25. β 1 2 k =1 (β1) k2 k β1 x k , β1 β€ x β€ 1 29. k =0 (β1) x 2 k +1 (2k + 1)2 41. e x β k =0 3 31. 0. 804 β€ I β€ 0. 808 43. 3x2 e x 3 33. 0. 600 β€ I β€ 0. 603 35. 0. 294 β€ I β€ 0. 304 β 37. 0. 9461 1 39. 0. 4485 1 0 β k =0 45. (a) k =0 1 k +1 x k! (b) 0 xex dx = 1 = 1 k +1 x dx = k! (k ) 1 1 =+ k !(k + 2) 2 β k =1 1 k !(k + 2) 47. Let f (x) be the sum of these series; ak and bk are both f (0)/k !. f (2k β1) (0) = 0 for all k . (2k β 1)! 49. (a) If f is even, then f (2k β1) is odd for k = 1, 2, . . . This implies that f (2k β1) (0) = 0, and so a2k β1 = (b) If f is odd, then f (2k ) is odd for k = 1, 2, . . . , which implies a2k = 0 for all k . 51. f (x) = x β 23 4 8 x + x5 β x7 + Β· Β· Β· = 3! 5! 7! 3 16 β k =0 β ( β 1)k 2k 2k +1 1 x ; β sin (x 2) (2k + 1)! 2 55. 0. 2640 β€ I β€ 0. 2643; I = 1 β 2/e βΌ 0. 2642411 = 53. 0. 0352 β€ I β€ 0. 0359; I = β 3 8 ln 1. 5 βΌ 0. 0354505 = SECTION 11.9 1. 1 + 1 x β 1 x2 + 2 8 9. 8 + 3x + 15. 2. 0799 32 x 16 13 x 16 β 54 x 128 3. 1 + 1 x2 β 1 x4 2 8 β 5. 1 β 1 x + 3 x2 β 2 8 β1/2 k β 53 x 16 + 35 4 x 128 7. 1 β 1 x β 4 R=1 32 x 32 β 73 x 128 β 77 4 x 2048 β 13 x 128 + 3 x4 4096 11. (a) k =0 (β1)k x 2k (b) k =0 (β1)k β1/2 k 1 x2k +1 , 2k + 1 13. 9. 8995 17. 0. 4925 19. 0. 3349 21. 0. 4815 CHAPTER 12 SECTION 12.1 1. z 3. z 5. z = β2 7. y = 1 9. x = 3 A(0,β2,5) A(2,0,0) B (0,0,β 4) x y B (4,1,0) y length AB: 2 β5 midpoint: (1,0,β2) x length AB: 5 β2 midpoint: (2, β 1 , 5 ) 22 11. x2 + ( y β 2)2 + (z + 1)2 = 9 13. (x β 2)2 + ( y β 4)2 + (z + 4)2 = 36 19. center (β2, 4, 1), radius 4 31. (2, β5, 5) 33. (β2, 1, β3) 15. (x β 3)2 + ( y β 2)2 + (z β 2)2 = 13 21. center: (3, β5, 1); radius: 6 23. (2, 3, β5) 25. (β2, 3, 5) 17. (x β 2)2 + ( y β 3)2 + (z + 4)2 = 25 27. (β2, 3, β5) 29. (β2, β3, β5) 35. (x β 3)2 + ( y β 3)2 + (z β 3)2 = 9, (x β 7)2 + ( y β 7)2 + (z β 7)2 = 49 37. not a sphere; the equation is equivalent to (x β 2)2 + (y + 2)2 + (z + 3)2 = β3 ...
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