微积分_英文版_原书第9版_Varberg D

手助 CHAPTER 0 xuebazhushou.com 0.1 Concepts Review 3. If not Q then not P. 1. rational numbers 2. dense 霸学 4. theorems Problem Set 0.1 Preliminaries 8. − 1 2 ⎡ ⎢ 3 5 ⎣ − 1 1 1 2 3 5 − ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ = − ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ 3 − − 1 3 1 5 2 ⎡ ⎛ ⎜ ⎢ 2 15 15 5 ⎝ ⎣ 1 2 1 ⎤ ⎡ ⎢ ⎥ 3 5 15 ⎣ ⎦ 1 ⎞ ⎟ 9 ⎠ = − − = − = − − 1 2 1 2 ⎡ ⎛ ⎜ ⎢ 2 15 3 5 ⎝ ⎣ 1 1 6 ⎞ ⎛ ⎜ ⎟ 3 15 15 ⎝ ⎠ − ⎞ ⎟ ⎠ ⎤ ⎥ ⎦ = − = − 1 5 ⎛ ⎜ 3 15 ⎝ 9. 14 21 ⎛ ⎜ ⎜ ⎝ 2 − 5 1 3 2 ⎞ ⎟ ⎟ ⎠ 2 = = 14 2 14 21 3 14 3 21 7 ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ 2 ⎞ ⎟ ⎠ 2 = 14 6 21 14 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = 2 9 3 49 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ = 6 49 1. 4 2(8 11) 6 − − + = − − + = + + = 4 2( 3) 6 4 6 6 16 2. 3 2 4 7 12 ⎡ ⎣ − − ( ) ⎤ ⎦ = = [ 3 2 4( 5) − − ] [ 3 2 20 = + 3. –4[5(–3 12 – 4) 2(13 – 7)] –4[25 12] + = + = = –4[5(5) 2(6)] –4(37) –148 + + = 4. + ] [ 2 5 1(7 12 16) 4 + − + − ] [ ( 5 2 5 1(3) 4 3 4 = + = − − + ( ) 5 2 5 1 7 + = + = = + 2 5. 5 1 7 13 – = 65 7 91 91 – = 58 91 手助霸 = 66 ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ 2 7 1 10. ] 3(22) xuebazhushou.com 11. 12. + 2 ) 学 − 5 − 1 7 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ – 11 12 7 21 11 12 7 21 + 1 2 1 2 − + + − 3 4 3 4 2 ⎛ ⎜ 7 ⎝ 7 ⎛ ⎜ 7 ⎝ − − 35 7 1 7 ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ = ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ 33 7 6 ⎞ ⎟ 7 ⎠ − ⎛ ⎜ ⎝ = − 33 6 = − 11 2 – 11 4 7 7 11 4 7 7 + = 7 7 15 7 = 7 15 = 4 8 4 8 − + + − 6 8 6 8 7 8 7 8 = 5 8 3 8 = 5 3 = = 7 8 7 8 6. 3 − 4 7 + 3 1 21 6 − = + 3 3 − 42 42 3 1 21 6 − + 6 42 − 7 42 = − 43 42 = − 7. 1 1 1 ⎛ ⎜ 3 2 4 ⎝ ⎡ ⎢ ⎣ – 1 3 ⎞ ⎟ ⎠ + 1 6 ⎤ ⎥ ⎦ = = = = 1 ⎤ ⎥ 6 ⎦ 1 ⎤ ⎥ 6 ⎦ + + ⎤ ⎥ ⎦ – 1 1 3 – 4 ⎡ ⎛ ⎜ ⎢ 12 3 2 ⎝ ⎣ 1 1 1 ⎡ ⎛ ⎜ ⎢ 3 2 12 ⎝ ⎣ 1 1 ⎡ ⎢ 24 3 ⎣ 1 3 ⎞ ⎛ ⎜ ⎟ 3 24 ⎝ ⎠ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ 4 24 1 24 + = – Instructor’s Resource Manual 13. 1– 1 + 1 1 2 14. 2 + 3 1 + 5 2 15. ( 5 + 3 = 1– 1 3 2 = 1– 2 3 = 3 3 – 2 3 = 1 3 手助霸 2 2 2 3 − = + = + + = 2 2 6 7 = + = 5 2 14 7 ) 3 7 2 20 7 xuebazhushou.com 6 7 ) ( 5 5 – 3 Section 0.1 1 5 – 3 – 2 = = )( ) ( = 3 2 2 学学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

16. ( 3 手助霸 2 2 x x x 3 5 2 5 2 3 5 2 4 + = − − − − ) ) ) (3 x x 4)( 3 3 3 x 1) + = = x 4 − + 4 − − )( ( + = − ( ( 5 2 15 3 8 2 15 = − xuebazhushou.com x 3) 3)(2 − − x x 6 6 − + − x 9 12 + − + –15 – 9 x 3 –18 – 9 x x (2 2 x 4 x 4 1) + = = (3 – 9)(2 = = = 6 6 x x (2 3) − 9 x x x x 2 2 2 2 学 17. 18. 19. 2 ) 27. 28. 20. (4 x − 11)(3 x − 7) 12 12 = = x x 2 2 − − x 28 x 61 − + 33 77 x + 77 2 + + + 12 + 4 x 2 x 2 + x 4( 2) + x x ( 2) + x 8 2 + + 2) + 10) 2) x 2 12 x x 2) ( + x 12 4 + x x ( x 2(3 + x x ( + = 2) + x 2 x x ( + x 20 6 + x x ( 2) + y 2 2 y − + 1 y 9 + − 2(3 2 2 y (3 1) − y 2(3 1) + y y 1)(3 2(3 + y 2 2 6 + + y y 2(3 1)(3 + y 2(4 1) + y y 1)(3 + 2(3 − y − − 1) 1) 1) y 1)(3 y y + − 1) y 2 y 1)(3 + y 2 8 + y y 1)(3 + y 4 1 + y 1)(3 + − + 2(3 y = = 2(3 (3 y 1) − − 1) 1) x = = = 6 = = = = 21. 2 t (3 t 9 = t 9 = t 1) − + 4 3 t 3 − t 6 − 4 3 2 + + 2 2 = t 3 t 7 t (3 2 − − 2 t t 1)(3 − + 3 2 t t t 3 + − + t 1 2 + t − + 2 t 3 1) t − + 1 22. t (2 + 3) 3 = = = = t (2 t (4 3 t 8 t 8 3 t 3)(2 + 2 t 12 + 2 t 12 + t 36 + 2 t 3)(2 + t 9)(2 + 2 t 24 + t 54 + 3) + 3) + t 36 + 27 + 23. 2 – 4 x x – 2 = x x ( – 2)( – 2 x + 2) = + , 2 x 24. 2 x 6 x − − x 3 − ( x = 2) − ( 3)( x − x + 3) = + x ⋅ = 0 0 0 手助霸 + + 27 t 18 c. 29. a. xuebazhushou.com 30. If 0 0 x ≠ 3 0 17 e. x ≠ 2 50 , 学 0= 0 = 2 a= , then 0 b. 0 0 d. 3 0 is undefined. is undefined. f. 017 1= 0 a = ⋅ , but this is meaningless because a could be any real number. No single value satisfies 0 0 a= . t 25. 2 – 4 – 21 t + 3 t t ( + = 3)( – 7) t + t 3 = t – 7 , t ≠ − 3 26. x 2 3 − − 2 x 2 2 x x 2 + x = = x x 2 (1 ) − 2 x x x 2 ( − + x x 1) 2 ( − − x x x ( 1)( − − 2 − 1 x 1) 1) = − 2 Section 0.1 31. .083 12 1.000 96 40 36 4 手助霸 xuebazhushou.com Instructor’s Resource Manual 学学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

35. 3.6 3 11.0 9 20 18 2 36. .846153 13 11.000000 10 4 60 52 80 78 20 13 70 65 50 39 11 手助霸 xuebazhushou.com 38. 学 1000 999 0.2941176470588235 37. x = 0.123123123... x x x = = = x = 123.123123... 0.123123... 123 123 999 41 333 = x = 1000 10 990 … = = = 0.217171717 x x x 217.171717... 2.171717... 215 215 990 43 198 x = = 学手助霸 32. .285714 7 2.000000 xuebazhushou.com 1 4 60 56 40 35 50 49 10 7 30 28 2 33. .142857 21 3.000000 2 1 90 84 60 42 180 168 120 105 150 147 3 34. .294117... 17 5.000000... → 3 4 160 153 70 68 20 17 30 17 130 119 11 40. 99 x = 100 99 39. x = 2.56565656... 100 x x x = = = x = 手助霸 … 256.565656... 2.565656... 254 254 99 xuebazhushou.com 392.929292... 3.929292... 389 389 99 Section 0.1 3 学 3.929292 x = x = x = Instructor’s Resource Manual x = 学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

41. x = 0.199999... 手助霸 x = = 1 5 x x x = = = 100 10 90 19.99999... 1.99999... 18 18 90 xuebazhushou.com 39.99999... 3.99999... 36 36 90 0.399999 x = x = x = x = 100 10 90 … 2 5 = = x 学 42. 43. Those rational numbers that can be expressed by a terminating decimal followed by zeros. 52. ( 2 − 3 )4 ≈ 0.0102051443 53. 4 1.123 – 1.09 3 ≈ 0.00028307388 54. ( 3.1415 ) 1/ 2 − ≈ 0.5641979034 55. 8.9 2 π + 1 – 3 π ≈ 0.000691744752 56. 4 (6 2 π π− 2) ≈ 3.661591807 44. 2 = If then m 1 5 1 , ⎛ ⎞ p = ⎜ ⎟ q ⎝ ⎠ n m 5 , ⋅ n 1 2 so we only need to look at 1 . q p q q = 1 q of any number of terminating decimals is also a terminating decimal, so (0.5) and (0.2) and hence their product, 1 , q is a terminating (0.5) (0.2) m , ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ = m n n . ⋅ has a terminating decimal decimal. Thus p q expansion. 助霸 a Note that The product a 10 −= n xuebazhushou.com Given a choose 2 on. 59. r = 学 1 12 π ≈ 0.0000010819... 45. Answers will vary. Possible answer: 0.000001, a < . Let n b . Since n S = > b } . Let ab 57. Let a and b be real numbers with be a natural number that satisfies nkk −/0 S − /)1 1 − n Thus, k r such that n ≤ k 0 n k n . Then 1 n b ≤ 1 n <−10 < −10 n n r . Otherwise, choose −> but . If > − = b a b k b 0 k 手 b , then choose k 20 −= n . < − < . b r 1 n b< , choose r so that r 3,r r so that < 1 r 2 a < a < r < 3 b r < . Then 1 b < , and so 58. Answers will vary. Possible answer: ≈ 120 in 3 4000 mi 5280 × = 21,120,000 ft ft mi equator = ≈ 2 (21,120,000) = rπ π 2 132,700,874 ft 46. Smallest positive integer: 1; There is no smallest positive rational or irrational number. 47. Answers will vary. Possible answer: 3.14159101001... 48. There is no real number between 0.9999… (repeating 9's) and 1. 0.9999… and 1 represent the same real number. = 61. V 49. Irrational 50. Answers will vary. Possible answers: , π π − and − 2 and 2 51. ( 3 1) + 3 ≈ 20.39230485 4 Section 0.1 60. Answers will vary. Possible answer: day year beats min 735,840,000 beats min hr hr day 365 60 70 24 × × × = π ⋅ ⎛ ⎜ ⎝ = π 2 r h 16 12 2 ⎞ ⎟ ⎠ 3 93,807,453.98 in. ≈ × 20 yr 手助霸 ⋅ 2 (270 12) xuebazhushou.com 651,441 board ft Instructor’s Resource Manual ≈ 学 × volume of one board foot (in inches): 3 1 12 12 144 in. × number of board feet: 93,807,453.98 = 144 学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

(8) (270) 3 54.3 ft. ≈ If I stay home from work today then it rains. If I do not stay home from work, then it does not rain. If the candidate will be hired then she meets all the qualifications. If the candidate will not be hired then she does not meet all the qualifications. 62. 手助霸 2 2 − = π b. V π 63. a. (8.004) (270) xuebazhushou.com 64. a. b. 学 If I pass the course, then I got an A on the final exam. If I did not pass the course, thn I did not get an A on the final exam. If I take off next week, then I finished my research paper. If I do not take off next week, then I did not finish my research paper. b. Every circle has area less than or equal to 9π. The original statement is true. c. Some real number is less than or equal to its square. The negation is true. 71. a. True; If x is positive, then 2x is positive. b. False; Take x < . 0 x = − . Then 2 2 x > but 0 c. False; Take x = 1 2 . Then 2 x 1 <= 4 x d. True; Let x be any number. Take y x= 2 1 + . Then y x> 2 . e. True; Let y be any positive number. Take x = y 2 . Then 0 < < . x y x ( + − x ) ( < + + − 1 x x ) < : 0 1 72. a. True; 65. a. b. 66. a. 2 2 If a triangle is a right triangle, then a If a triangle is not a right triangle, then 2 a 2. 2. + = b b + ≠ c c 2 If the measure of angle ABC is greater than 0o and less than 90o, it is acute. If the measure of angle ABC is less than 0o or greater than 90o, then it is not acute. If angle ABC is an acute angle, then its measure is 45o. If angle ABC is not an acute angle, then its measure is not 45o. 手助霸 xuebazhushou.com 73. a. If 2 a then b≥ 2 学 b. 2 If 2 b< a b≥ a . then a b< . 67. a. The statement, converse, and contrapositive are all true. b. The statement, converse, and contrapositive are all true. 68. a. The statement and contrapositive are true. The converse is false. b. b. False; There are infinitely many prime numbers. c. True; Let x be any number. Take y = 1 1 + . Then x y > . 1 x d. True; 1/ n can be made arbitrarily close e. True; 1/ 2n can be made arbitrarily close to 0. to 0. If n is odd, then there is an integer k such that n = = k= n + Then 1. 2 2 k k 4 1) (2 + = + 2 k k 2 ) 1 2(2 + + + 4 1 k 2 2 b. The statement, converse, and contrapositive are all false. 69. a. Some isosceles triangles are not equilateral. The negation is true. b. All real numbers are integers. The original statement is true. c. Some natural number is larger than its square. The original statement is true. 70. a. Some natural number is not rational. The original statement is true. Instructor’s Resource Manual 74. 手助 2 2 k k 4 . = = = 2(2 k (2 ) k= 2 . Parts (a) and (b) prove that n is odd if and only if 2n is odd. Prove the contrapositive. Suppose n is even. Then there is an integer k such that Then 2 2 n n ) Thus 2n is even. xuebazhushou.com 4 31 2 2 31 or 2 Section 0.1 5 = ⋅ = ⋅ 31 2 ⋅ ⋅ ⋅ 学霸 75. a. 243 3 3 3 3 3 ⋅ = ⋅ ⋅ b. 124 学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

76. For example, let 6 4 手助霸 ⋅ ⋅ ⋅ ⋅ 2 2 2 c d b = ⋅ ⋅ 3; c. 5100 then = ⋅ = ⋅ = ⋅ 2 2550 = ⋅ 2 2 3 425 ⋅ ⋅ 2 2 2 3 5 5 17 or 2 ⋅ 2 2 1275 ⋅ 2 2 3 5 85 ⋅ = ⋅ ⋅ 3 5 17 ⋅ ⋅ ⋅ A b c = ⋅ 2 A , so the square of the number is the product of primes which occur an even number of times. xuebazhushou.com factors of p2 must occur an even number of times, 2q2 would not be valid and 2 Since the prime p 2 q p q ; 2 ; 2 d = = = q p 2 = 2 2 2 ; ⋅ 学 77. p q must be irrational. 78. 3 = p q ; 3 = 2 p 2 q 2 q ; 3 = 2 p ; Since the prime factors of 2p must occur an even number of times, 23q would not be valid and must be irrational. 82. a. –2 b. –2 c. x = 2.4444...; 10 x x x = = = 9 24.4444... 2.4444... 22 22 9 x = d. 1 e. n = 1: x = 0, n = 2: x = 5 4 x = n = 4: The upper bound is 3 . 2 n = 3: x = 2– , 3 3 , 2 f. 2 83. a. Answers will vary. Possible answer: An 2 = example x x is { : Here the least upper bound is 5, which is real but irrational. a rational number}. 5, < x 手助霸 3 S = p q xuebazhushou.com 3. (b) and (c) 2. b > 0; b < 0 1. [ 1,5);( − ≤ ≤ 1 5 4. − x 学 b. True 0.2 Concepts Review −∞ − , 2] 79. Let a, b, p, and q be natural numbers, so a b aq bp are rational. a b and p q sum is the quotient of natural numbers, so it is also rational. This + bq p q + = 80. Assume a is irrational, p q ≠ is rational, and 0 ⋅ a r s p q = is rational. Then q r ⋅ p s ⋅ rational, which is a contradiction. = a is 81. a. – 9 = –3; rational b. 0.375 = 3 8 ; rational c. (3 2)(5 2) = 15 4 = 30; rational d. + 2 3) (1 irrational = + 1 2 3 3 + = + 4 2 3; 6 Section 0.2 Problem Set 0.2 1. a. b. c. d. 手助霸 xuebazhushou.com Instructor’s Resource Manual 学学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

9. 10. 4 –6 < < –3 1– 6 x –4 2 3 > ≥ x x ≤ 3 ≤ 1 2 1 2 – ; – , 2 3 ⎡ ⎢⎣ ⎞ ⎟ ⎠ x 4 5 3 < − x 3 1 < − < − 1 3 > > − x < 2 2 3 7 ; ⎛ −⎜ ⎝ 2 1 , 3 3 ⎞ ⎟ ⎠ 11. x2 + 2x – 12 < 0; 手助霸 f. e. xuebazhushou.com x 2 x ;( 2, 5 − − ∞ 7 − < 2 − < −∞ − , 2] (2,7) 2. a. c. d. [ 1,3] 3. − − x ( ) 学 b. [ 3,4) 4. 3 x − < < 5 1 x 4 − ( x ; 1, 6 ∞ ) x 7 – 2 9 2 ≤ ≤ –5 5. + 3 5 – ; – , 2 ⎡ ⎢⎣ ⎞ ∞⎟ ⎠ x ≥ x x 5 2 6. 5 x − > > x 3 6 4 − x ;( 1 −∞ ,1) 7. x 2 5 4 3 + < − < x 6 3 3 < − < x 1;( 2, 1) 2 − < < − − 8. x x 3 4 − < 6 4 < 3 2 9 11 − < 20 < 35; ⎛ < < ⎜ 2 ⎝ x ,5 ⎞ ⎟ ⎠ x = –2 ± 2 (2) – 4(1)(–12) 2(1) –2 = ± 2 52 –1 = ± ( x – –1 13 + 13 –1– 13, –1 ) + x ⎤ ⎡ ⎦ ⎣ 13 ( – –1– 13 ) ) ⎤ ⎦ < 0; 手助霸 ⎡ ⎣ ( xuebazhushou.com 12. 学 2 5 x x 0 6 − − > x x 0; 6) 1)( ( > − + −∞ − ∪ ∞ ( (6, , 1) ) 13. 2x2 + 5x – 3 > 0; (2x – 1)(x + 3) > 0; ( −∞ − ∪ , 3) 1 ⎛ , ∞⎜ 2 ⎝ ⎞ ⎟ ⎠ 14. 24 x − 5 (4 x + 3)( x 手助霸 x 0 6 < − 0; ⎞ ⎟ ⎠ 2) ,2 3 4 − < ⎛ −⎜ ⎝ xuebazhushou.com Section 0.2 [–4, 3) 7 学 0; 15. x x 4 + – 3 ≤ Instructor’s Resource Manual 学霸助手[xuebazhushou.com]-课后答案|期末试卷|复习提纲

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