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# Calculus: One and Several Variables

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-01 JWDD027-Salas-v1 November 25, 2006 15:52 SECTION 1.2 1 CHAPTER 1 SECTION 1.2 1. rational 2. rational 3. rational 4. irrational 5. rational 6. irrational 7. rational 8. rational 9. rational 10. rational 11. 3 4 = 0 . 75 12. 0 . 33 < 1 3 13. 2 > 1 . 414 14. 4 = 16 15. 2 7 < 0 . 285714 16. π < 22 7 17. | 6 | = 6 18. | − 4 | = 4 19. | − 3 7 | = 10 20. | − 5 | − | 8 | = 3 21. | − 5 | + | − 8 | = 13 22. | 2 π | = π 2 23. | 5 5 | = 5 5 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. bounded, lower bound 0, upper bound 4 42. bounded above by 0 43. not bounded 44. bounded above by 4 45. not bounded 46. bounded; lower bound 0, upper bound 1 47. bounded above, upper bound 2 48. 2 < 3 π < 2 π < π 3 < 3 π 49. x 0 = 2 , x 1 = 2 . 75 , x 2 = 2 . 58264 , x 3 = 2 . 57133 , x 4 = 2 . 57128 , x 5 = 2 . 57128; bounded; lower bound 2, upper bound 3 (the smallest upper bound = 2 . 57128 · · · ); x n = 2 . 5712815907 (10 decimal places)

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P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-01 JWDD027-Salas-v1 November 25, 2006 15:52 2 SECTION 1.2 50. x n 2 . 970 ... ; bounded 51. x 2 10 x + 25 = ( x 5) 2 52. 9( x 2 3 )( x + 2 3 ) 53. 8 x 6 + 64 = 8( x 2 + 2)( x 4 2 x 2 + 4) 54. 27( x 2 3 )( x 2 + 2 3 x + 4 9 ) 55. 4 x 2 + 12 x + 9 = (2 x + 3) 2 56. 4( x 2 + 1 2 ) 2 57. x 2 x 2 = ( x 2)( x + 1) = 0; x = 2 , 1 58. 3 , 3 59. x 2 6 x + 9 = ( x 3) 2 ; x = 3 60. 1 2 , 3 61. x 2 2 x + 2 = 0; no real zeros 62. 4 63. no real zeros 64. no real zeros 65. 5! = 120 66. 5! 8! = 1 8 · 7 · 6 = 1 336 67. 8! 3!5! = 8 · 7 · 6 3 · 2 · 1 = 56 68. 9! 3!6! = 9 · 8 · 7 3 · 2 · 1 = 84 69. 7! 0!7! = 7! 1 · 7! = 1 70. p 1 q 1 + p 2 q 2 = p 1 q 2 + p 2 q 1 q 1 q 2 , p 1 q 2 + p 2 q 1 and q 1 q 2 are integers, and q 1 q 2 = 0 71. Let r be a rational number and s an irrational number. Suppose r + s is rational. Then ( r + s ) r = s is rational, a contradiction. 72. p 1 q 1 p 2 q 2 = p 1 p 2 q 1 q 2 , p 1 p 2 and q 1 q 2 are integers, and q 1 q 2 = 0 73. The product of a rational and an irrational number may either be rational or irrational; 0 · 2 = 0 is rational, 1 · 2 = 2 is irrational. 74. 2 + 3 2 = 4 2 irrational; π + (1 π ) = 1, rational. ( 2)( 3) = 6 irrational; ( 2)(3 2) = 6, rational. 75. Suppose that 2 = p/q where p and q are integers and q = 0. Assume that p and q have no common factors (other than ± 1). Then p 2 = 2 q 2 and p 2 is even. This implies that p = 2 r is even. Therefore 2 q 2 = 4 r 2 which implies that q 2 is even, and hence q is even. It now follows that p and q are both even, contradicting the assumption that p and q have no common factors. 76. Assume 3 = p q , where p and q have no common factors. Then 3 = p 2 q 2 , so p 2 = 3 q 2 . Thus p 2 is divisible by 3, and therefore p is divisible by 3, say p = 3 a . Then 9 a 2 = 3 q 2 , so 3 a 2 = q 2 , where q must also be divisible by 3, contracting our assumption.
P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-01 JWDD027-Salas-v1 November 25, 2006 15:52 SECTION 1.3 3 77. Let x be the length of a rectangle that has perimeter P . Then the width y of the rectangle is given by y = (1 / 2) P x and the area is A = x 1 2 P x = P 4 2 x P 4 2 .

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