Odd answers to appendix

Odd answers to appendix - Limits 439049092_11 ans.qxd A192...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Limits 439049092_11 ans.qxd A192 10/9/09 3:05 PM Page A192 Answers to Odd-Numbered Exercises and Tests Chapter 11 z 75. −3 −5 −4 −3 −2 y −2 2 (3, − 1, 0) 3 y 2 3 1 2 3 −3 x −2 x 1 −2 4 −4 (1, − 1, − 2) 1. zero 7. (a) (b) z 15. 5 (3, − 2, 5) 4 z 3 2 1 y 1 1 2 −3 3 x −2 −3 ( 1 −2 −2 1 1 ( 3 , 4, − 2 2 − 3 (− 2, 3, 1) −1 ge 2 4 2 2 3 y x 9. (a) 7, en 3, 4, 5 19. 8, 0, 0 21. Octant IV Octants I, II, III, and IV 25. Octants II, IV, VI, and VII 29. 29 units 31. 114 units 65 units 35. 12 units 110 units 2 39. 3 2 6 2 2 5 2 32 29 2 35 6, 6, 2 10; isosceles triangle 6, 6, 2 10; isosceles triangle 3 47. 0, 1, 7 49. 1, 1, 1 1, 2 2, 22 5 53. x 3 2 y 22 z 4 2 16 2 , 2, 6 x 5 2 y2 z 2 2 36 x 32 y 72 z 5 2 25 2 x3 y2 z 3 2 445 2 Center: 3, 0, 0 ; radius: 3 Center: 2, 1, 3 ; radius: 2 Center: 2, 0, 4 ; radius: 1 Center: 1, 1, 4 ; radius: 3 3 Center: 1, 1, 0 ; radius: 1 3 z z 73. 2 2 ga 3 z 11. (a) C 4 (1, 0, 0) x 10 (− 2, 3, 0) −2 2 x 6 8 2 3 −4 2 −3 2 2 1 1 3 3 3 4 −3 x −4 x z (c) z (d) 4 339 ,, 222 5 4 3 −4 2 −3 3 〈0, 0, 0〉 2 1 −4 −3 −2 y −4 −3 −2 2 3 2 1 3 4 3 z (b) 4 2 −3 y 1 1 2 3 5 2 5 y −5 −4 −3 −2 3 4 x −5 −4 2 −3 3 〈− 2, − 2, 1〉 1 −3 −2 −3 −4 −5 4 −4 3 y 3 −3 x z 13. (a) 2 −2 4 −2 y 1 1 2 −3 2 y 2 〈− 1, − 1, − 3〉 4 4 −2 4 1 1 2 3 y −4 −3 −2 y −4 −3 −2 1 2 2 4 〈2, 2, 6〉 3 of −2 5, 5 z 4 4 4 −6 11 7, 33 5 x −8 (c) (b) 6 ty er op (y − 3) + z = 5 (x − 1) 2 + z 2 = 36 (b) 3 11 5, 5 Pr 17. 23. 27. 33. 37. 41. 43. 45. 51. 55. 57. 59. 61. 63. 65. 67. 69. 71. 5. parallel Le 2 −3 5 (page 822) 3. component form 2, 3, 1 3 −4 y False. z is the directed distance from the xy-plane to P. 0; 0; 0 A point or a circle (where the sphere and the yz-plane meet) x 2, y2, z 2 2xm x1, 2ym y1, 2z m z1 Section 11.2 −5 5 ni −3 79. 81. 83. 85. 1 1 4 ng (2, 1, 3) 3 ar −4 2 −3 2 2 3 4 5 x 3 2 2052 4 z2 5 1. three-dimensional 3. octants x1 x2 y1 y2 z1 z2 5. 7. surface; space , , 2 2 2 9. A: 1, 4, 4 , B: 1, 3, 2 , C : 3, 0, 2 z z (− 4, 2, 2) 11. 13. 3 y2 6 (page 815) Section 11.1 77. x2 1 2 3 〈4, 4, − 2〉 x −2 −3 −4 1 2 3 Limits 439049092_11 ans.qxd 10/9/09 3:05 PM Page A193 A193 Answers to Odd-Numbered Exercises and Tests z (d) 3 4 2 3 −4 2 −3 1 1 1 2 3 2 3 −4 −3 −2 −2 4 5 6 −4 3 2 47. 3, 7, 6 6, 17. z 19. 9 2 23. 11 25. 74 27. 34 74 74 8i 3j k 8i 3j k (a) (b) 74 74 4 33. 0 35. About 124.45 37. About 109.92 Parallel 41. Neither 43. Orthogonal Orthogonal 47. Not collinear 49. Collinear 57 3 14 53. 6, , 55. ± 3, 1, 7 24 14 0, 2 2, 2 2 or 0, 2 2, 2 2 B: 226.52 N, C: 202.92 N, D: 157.91 N True 63. The angle between u and v is an obtuse angle. (page 829) Section 11.3 3. u 61. p 15 T 5.75 j −1 2 y 2 −2 x 2 y (b) −2 11. (a) (b) 13. (a) op 2 −2 (b) −1 −1 15. (a) (b) Pr 〈1, 0, 0〉 −1 2 1 2 y 17. (a) −2 x 11. 7i 11j 8k 13. 0 15. 21i 33j 24k 17. 7i 11j 8k 19. 0 21. 1, 2, 2 23. 0, 42, 0 25. 7i 13j 16k 27. 18i 6j 19 29. i 2j k 31. i 3j 3k 19 7602 2 33. 35. 71 i 44 j 25k ij 7602 2 37. 1 39. 30 3 41. 56 43. (a) AB 1, 2, 2 and is parallel to DC 1, 2, 2 . AD 3, 4, 4 and is parallel to BC 3, 4, 4 . \ \ (b) 40 45 13.41 15.32 17.24 (page 838) \ PQ 3. symmetric equations t z y x t, y 2t, z 3t (b) x 23 x 4 3t, y 1 8t, z 6t z x4y1 3 8 6 x 2 2t, y 3 3t, z 5 t x2y3 z5 2 3 x 2 t, y 4 t, z 2 5t y z2 x2 1 4 5 x 3 4 t, y 8 10 t, z 15 t x3y8 z 15 4 10 x 3 4t, y 1, z 2 3t No symmetric equations 1 1 3 t, y 2 5 t, z t x 2 2 2x 1 y 2 2z 1 6 5 2 z 19. 3 −3 2 −3 −2 −1 1 2 3 \ \ 11.49 35 ar (b) 9. (a) 1 er z 1 7. (a) ty 9. i −2 C −1 of (0, 0, − 1) −2 −1 (0, − 1, 0) 1 −1 x 1 −2 en −2 30 63. True. The cross product is not defined for two-dimensional vectors. 65. u v v u ; Answers will vary. 67. Yes. The area of the triangle is 1 u v . 69. Proof 2 5. (a) 2 2 −1 9.58 1. direction; z 1 25 7.66 Section 11.4 v sin 7. z −2 20 x 1 −2 (0, 2, 1) 1 2 3 y CHAPTER 11 1. cross product 5. k 51. \ Le 57. 59. 61. −4 1 2, ge 51. −3 ga 31. 39. 45. 4 \ \ 2 15. z 21. 21 3 −2 x 5, 5, − 5 2 \ \ y 1 1 4 −5 29. 2 3 −3 x \ 45. 〈0, 0, 0〉 4 \ ng y −3 −2 \ (b) Area is AB AD 6 10. (c) The dot product is not 0 and therefore the parallelogram is not a rectangle. (a) AB 5, 0, 2 and is parallel to CD 5, 0, 2 . AC 0, 3, 1 and is parallel to BD 0, 3, 1 . (b) Area is AB DB 286 . (c) The dot product is not 0 and therefore the parallelogram is not a rectangle. 3 13 1 49. 4290 2 2 6 53. 2 55. 2 57. 12 59. 84 p (a) T cos 40 2 (b) ni z (c) Limits 439049092_11 ans.qxd A194 21. 25. 29. 33. 37. 41. 10/9/09 3:06 PM Page A194 Answers to Odd-Numbered Exercises and Tests 23. 2x y 2z 10 0 20 27. 3x 9y 7z 0 x 2y z 2 0 31. y 5 0 6x 2y z 8 0 35. 7x y 11z 5 0 yz20 Orthogonal 39. Orthogonal 43. x 2 3t 45. x 5 2t x2 y3 y 3 2t y 3t z4t z4t z 4 3t 47. (a) 60.7 (b) x t 2, y 8t, z 7t 49. (a) 77.8 (b) x 6t 1, y t, z 7t 1 z z 51. 53. x 3. 9. 11. 15. 17. 19. 21. 23. 25. 5. 41 7. 10 5, 4, 0 29, 38, 67 2 2 2 29 38 67 13 13. 1, 2, 9 2 , 2, 5 x 22 y 32 z 52 1 2 2 x1 y5 z 2 2 36 2 2 2 y z 12 x Center: 0, 0, 4 ; radius: 4 Center: 5, 3, 2 ; radius: 2 z (a) (b) z (y − 3) 2 + z 2 = 16 x2 + z2 = 7 4 6 5 4 3 6 2 (0, 3, 0) 2 2 (4, 0, 0) 3 45 6 y 6 (6, 0, 0) 5 −1 4 x 3 4 (0, 2, 0) 3 4 6 y 27. (a) 1, 6, 4 31. 39. 45. y 3.90 3.81 2008 3.54 2009 3.42 2010 3.29 of 2007 C Model 2006 (b) The approximations are very similar to the actual values of z. (c) Answers will vary. op Year Pr 63. False. Lines that do not intersect and are not in the same plane may not be parallel. 65. Parallel. 10, 18, 20 is a scalar multiple of 15, 27, 30 . 67. (a) Sphere: x 4 2 y 12 z 12 4 (b) Two planes: 4x 3y z 10 ± 2 26 47. 51. 53. 55. 59. 61. 63. 67. 38 3 38 , , 38 19 (c) 1 1 (2, 0, 0) 3 2 −2 1 y 3 −2 1 −5 −4 y −2 (5, − 1, 2) 2 3 4 x 1 −2 −3 −4 −5 1 2 3 6 6 14 73. 6 7 77. Answers will vary. 71. 1 2 (0, 0, − 2) 3 −2 4 x (− 3, 3, 0) 2 −3 −1 1 −1 x 3 38 38 2 (0, 0, 2) 1 z 1. 38 (b) 185 10, 6, 7 2 185 6 185 7 185 (c) , , 37 185 185 33. 1 35. 90 37. Parallel 9 Orthogonal 41. Collinear 43. Not collinear A: 159.1 lb B: 115.6 lb C: 115.6 lb 49. 4 i 2 j 7k 10, 0, 10 11 3 11 11 i j k 11 11 11 22 7602 25 7602 71 7602 i j k 7602 3801 7602 57. 75 172 2 43 13.11 Area (a) x 1 4 t, y 3 3t, z 5 6 t x1y3z5 (b) 4 3 6 y z x (a) x 4 t, y 5t, z 2 t (b) 452 65. z 2 0 2x 12y 5z 0 z z 69. (0, − 3, 0) (page 842) Review Exercises (b) 1 ga 26 3 4 6 59. 6 ty 8 9 61. (a) 5 (0, 0, − 6) −6 −7 57. (0, 3, 0) 3 y en 4 5 29. (a) −2 er 6 x 6 (0, 3, 0) 5 Le −1 4 ge −2 −1 2 4 x 6 z (2, 0, 0) 2 4 y 2 x 55. (0, 3, 0) 2 −2 −1 −2 ng −2 −2 ni (0, 0, 2) 2 −4 −2 ar 4 3 2 −2 x 4 75. False. u (3, 0, 0) v v u y Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A195 A195 Answers to Odd-Numbered Exercises and Tests (page 844) Chapter Test 1. (page 847) Problem Solving z z 1. (a) 4 (b) Answers will vary. (c) a b 1 (d) Answers will vary. 3 3 2 2 1 −6 −5 −4 −3 −2 3 102 12 z 194 2 2 2 8 4 8 −4 400 6 y 0 − 300 \ ge ga en C \ \ er Pr 6 op 9 (0, 0, 9) (c) The distance between the two insects appears to lessen in the first 3 seconds but then begins to increase with time. (d) The insects get within 5 inches of each other. 32 17. (a) D (b) D 5 2 4 2 x 2 1 Chapter 12 y 4 6 4 10 3 z 6 − 10 (0, 0, − 5) 8 −6 −8 11 0 (2, 0, 0) 4 − 6 2 (0, − 10, 0) 5 \ −1 of \ ty \ (c) AB F 298.2 when 30 . (d) 51.34 (e) The zero is 141.34 ; the angle making AB parallel to F. 15. (a) d 70 when t 0. 20 (b) Section 12.1 1. limit 5. (a) (page 858) 3. oscillates − 10 (0, 3, 0) x y −4 −3 2 1 1 3 4 3 5 6 x 2(12 − x) (6, 0, 0) 17. 27x 4y 32z 8 4 14 18. 7 14 33 2(12 − x) 0 19. 88.5 (b) V lwh 2 12 x 2 12 4 x 12 x 2 x x CHAPTER 12 5. u 2, 6, 6 , v 12, 5, 5 6. (a) 84 (b) 0, 62, 62 19 3 19 3 19 7. (a) , , 19 19 19 6 194 5 194 5 194 (b) , , 97 194 194 8. 46.23 9. (a) x 8 2t, y 2 6t, z 5 6t x8y2z5 (b) 2 6 6 10. Neither 11. Orthogonal 12. Parallel 13. AB is parallel to CD 4, 8, 2 . AC is parallel to BD 1, 3, 3 . Area 2 230 14. 200 z 15. 16. 180 Le sphere ar −2 y ni \ 4 2 x 3 \ xz-trace 6 12 2 3. Answers will vary. 5. (a) Right triangle (b) Obtuse triangle (c) Obtuse triangle (d) Acute triangle 7. About 860.0 lb 9–11. Proofs 5 13. (a) AB j k, F 200 cos j sin k 4 (b) AB F 25 10 sin 8 cos 3. 7, 1, 2 19 z −4 v u1 x 2 −8 −3 3 −4 y −2 2 −3 2 −1 (2, 2, − 1) −2 4 5 − 10 −2 2 x 2. No. 76 4. x 7 2 1 1 1 −3 y ng (3, − 7, 2) Limits 439049092_12 ans.qxd A196 10/13/09 10:58 AM Page A196 Answers to Odd-Numbered Exercises and Tests (c) 15. x 3 3.5 3.9 4 V 972 1011.5 1023.5 1024 4.1 4.5 V 1023.5 1012.5 0.001 0.2236 1024 0.2247 0.2237 0.001 0.01 0.2236 0.2235 0.2225 5 ; 10 1200 3 − 0.8 12 1.999 fx 13.5 13.95 13.995 14 x 2.001 2.01 2.1 fx 14.005 14.05 14.5 fx x fx 1 2; 2.99 2.999 3 fx 0.1695 0.1669 0.1667 Error x 3.001 3.01 3.1 fx 0.1666 0.1664 0.1639 fx x fx 0.001 1.9999 1.999999 0.001 0.01 1.999999 1.9999 0.9 0.999 1 0.2506 0.2501 x 1.001 1.01 fx 0.2499 0.2494 fx fx 1; 0.5003 0.5025 3.9 0.5263 3 −3 0.1 0.01 0.9983 0 0.001 0.99998 0.9999998 0.001 0.01 0.1 0.9999998 0.99998 0.9983 2 −3 21. 3 x 0.1 0.01 0.001 0 fx 0.2439 0.1 0.01 0.001 Error x 0.001 0.01 0.1 fx 0.001 0.01 0.1 0; 4 3.99 Error 1.1 −3 x x Error 0.2564 −5 3.999 Error −2 0.99 3 0.4998 3 0 1.9867 fx 1 4; 4 0.1 Pr x 0.01 1.9867 2; No 13. en C 0.1 ty x of No er 11. 4.001 0.4975 −6 19. op 1 6; 4.01 0.4762 ga 2.9 x 4.1 ge 14; Yes 9. x 2 ni 1.99 ar 1.9 Le 17. x ng 0 7. Error 0.8 −3 0 0 0.1 fx 980 lim V 0.01 fx 5 x→4 0.1 x x (d) x 2 −3 3 −2 Error Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A197 A197 Answers to Odd-Numbered Exercises and Tests 23. x 0.1 0.01 0.001 0.9063 0.9901 0.9990 x 0.001 0.01 0.1 fx 1.0010 1.0101 45. (a) 12 (b) 9 (c) 1 (d) 3 2 47. (a) 8 (b) 3 (c) 3 (d) 681 8 9 49. 15 51. 7 53. 3 55. 10 35 7 57. 13 59. 1 61. 3 63. e 3 20.09 1.1070 fx 1; 0 Error 65. 0 69. True 6 71. (a) and (b) Answers will vary 73. (a) No. The function may approach different values from the right and left of 2. For example, 0, x < 2 fx 4, x 2 6, x > 2 implies f 2 4, but lim f x 4. 3 −3 3 −1 67. x 0.999 2.2314 2.0203 2.0020 Error x 1.001 1.01 1.1 fx 1.9980 1.9803 (b) No. The function may approach 4 as x approaches 2, but the function could be undefined at x 2. For example, in the function 4 sin x 2 , fx x2 the limit is 4 as x approaches 2, but f 2 is not defined. 2 75. lim tan x 0 1 1.8232 2; Le fx ni 0.99 ar 0.9 ng x→2 25. x→0 3 lim tan x x→ 1 4 − ge lim tan x does not exist. x→ ga 5 −1 8 4 7 x→5 9 No; No −9 9 x 2 4 6 8 Pr op er 5 29. 13 31. Limit does not exist. Answers will vary. 33. Limit does not exist. Answers will vary. 35. Limit does not exist. Answers will vary. 2 3 37. 39. −3 −3 (page 868) 1. dividing out technique 3. one-sided limit 5. (a) 1 (b) 3 (c) 5 7. (a) 2 (b) 0 (c) 0 g2 x 2x 1 g2 x xx 1 1 9. 12 11. 4 13. 4 15. 12 17. 80 19. 5 3 1 21. 23. 25. 1 27. 29. 1 10 6 4 ty −2 −3 Section 12.2 of 2 1 16 31. 37. 33. Does not exist 35. 0 39. 5 3 3 3 −2 −1 Limit does not exist. Answers will vary. 41. lim f x x→4 C 6 −2 6 en 27. −2 77. lim f x y 2 −1 43. 8 −3 Limit does not exist. Answers will vary. −6 Limit does not exist. Answers will vary. 3 −1 0 43. 4 −6 −3 6 2 −3 3 −2 −4 2.000 4 −1 2.000 41. 8 Limit exists. −2 3 −1 3 CHAPTER 12 −1 2 2 1.000 3 Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A198 A198 Answers to Odd-Numbered Exercises and Tests 45. 2 47. y 55. 1 y 57. 6 −3 − 1.5 3 4 1.5 3 2 2 −2 x −1 0.333 49. (a) 8 −3 10 −2 0.135 −2 2 The limit does not exist. lim x→1 y 59. 1 x2 y 0.999 0.9999 0.5263 0.5025 0.5003 0.50003 5 3 1 3 2 2 0.5 −4 lim f x −6 63. 5 −3 6 f (x) = x cos x 1.99 1.999 6.6923 74.1880 x 2.001 2.01 2.1 750.8125 75.8130 8.3171 fx 749.1875 y=x lim x cos x 67. 2 16.1 Pr 0.12481 fx x x → 16 0.12498 16.001 4 x 16.0001 0.124998 fx (c) lim 16.01 x −1 −1 1 3 4 5 −3 The limit does not exist. 65. 6 f (x) = ⏐x⏐sin x −9 9 9 y = −x y=x 0 −6 lim x sin x x→0 f (x) = x sin 1 x lim x sin −3 2 −2 x→0 1 x y = −x 0 0 3 ty er x op − 0.2 −6 x→0 of (c) The limit does not exist. − 53. (a) 100.1 22 Error en fx 2 C 1.9 ga −9 x 1 1 ge 6 4 −3 −3 5 x→2 (b) 3 ar 51. (a) 1 1 2 Le x→1 x x2 1 ni x −3 −2 −1 −1 (c) lim ng 0.99 1 2 1 61. 4 fx 3 −3 0.9 x 2 −2 −6 −2 (b) 1 −1 −4 2 −4 (b) x −1 y=x −2 y = −x 69. Limit (a) can be evaluated by direct substitution. (a) 0 (b) 1 1 1 71. 2 73. 75. 2x 3 77. x2 2x 79. 32 ft sec 83. (a) 140 2 81. Answers will vary. 0.1249998 x 16 0.125 0 10 80 (b) x Cx lim C x x → 5.5 5 5.3 5.4 5.5 5.6 5.7 6 105 110 110 110 110 110 110 110 Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A199 A199 Answers to Odd-Numbered Exercises and Tests 4 4.9 5 5.1 5.5 105 105 105 110 110 47. (a) (c) 6 100 110 1 4 2 −2 x −2 2 4 ga en 6 2 6 er 1 1 3 33. 6x 35. 0.125 0 2 1.5 1 0.5 0 0.5 1 1.5 2 fx 0.125 0.5 1.125 2 fx 0.5 1 1.5 2 53. 3 They appear to be the same. 2 −1 1.5 1 0.5 0 Pr 32 1 1.225 1.414 1.581 1.732 fx y 0.5 0.408 0.354 0.316 0.289 x 1 1.5 2 fx 2 0.5 1.871 2 2.121 2.236 fx x y 6 2 fx 2 x3 1 1 39. 41. x 62 2x 9 11 (b) y 4 x 5 45. (a) 1 (b) y (c) 0.267 0.25 0.236 0.224 3 5 2 4 (2, 3) 3 2 −3 1 x 2 −2 0.5 x op 31. −4 −3 −2 1.125 2 −2 ty −6 2x 43. (a) 4 (c) 0 of −2 −4 1 C (1, 2) x −2 0.5 CHAPTER 12 y 2 1 x 1 2 2 fx 6 −2 −6 27. 1.5 fx x −4 37. 2 y (1, − 1) 2 4 −6 Le −4 3 −4 ge −6 2 They appear to be the same. (1, 1) 8 2 x 1 −2 4 6 x −4 −1 6 4 (− 4, 1) −2 1 6 2 29. 0 −1 51. 4 2 4 2 −2 3 6 (3, 2) 15. x y 3 6 −2 (b) y 1 − 10 − 8 1. Calculus 3. secant line 5. 0 7. 1 9. 2 11. 2 13. 1 2 17. m 2x; (a) 0 (b) 2 1 1 1 19. m (b) ; (a) x 42 16 4 1 1 1 21. m (b) ; (a) 4 6 2x 1 y 23. 25. −4 49. (a) (c) 4 (page 878) x 5 4 y The limit does not exist. 85. True 87. (a) and (b) Graphs will vary. 89. Answers will vary. Section 12.3 1 4x (b) y ng Cx 4.5 ni x ar (c) 3 4 −2 −1 x 1 −1 −2 −3 2 (1, − 1) 3 55. y 57. y x1 6x ± 8 59. f x 2x 4; horizontal tangent at 2, 1 61. f x 9x 2 9; horizontal tangents at 1, 6 and 1, 63. 1, 1 , 0, 0 , 1, 1 55 65. ,3 , , 3 6 6 66 67. 0, 0 , 2, 4e 2 69. e 1, e 1 71. Answers will vary. 6 Limits 439049092_12 ans.qxd A200 10/13/09 10:58 AM Page A200 Answers to Odd-Numbered Exercises and Tests 73. (a) y 112.87x2 (b) 8000 256.45x (b) 380.3 2 −6 12 − 10 0 7 0 lim f x 37. (a) x 100 101 102 103 fx 0.7082 0.7454 0.7495 8000 104 x fx 7 0 The slopes are the same. (b) 75. True. The slope is dependent on x. 77. b 78. a 79. d 80. c 81. Answers will vary. Example: a sketch of any linear function with positive slope 83. Answers will vary. Example: a sketch of any quadratic function of the form y a x 1 2 k, where a > 0. 5 85. (a) 6. a 3 23. 1 2 7. d ni Le 53 39. 1, 3, 2, 17, 13 55 Limit: 0 43. 1, 4, 1, 16, 25 57 11 13 Limit does not exist. 47. 1, 1, 1, 1, 1 2 34 5 Limit: 0 49. lim an 3 2 n 100 101 102 103 an 2 1.55 1.505 1.5005 ty n 104 105 106 1.50005 1.500005 1.5000005 16 3 51. lim an er op 5 1 41. 3, 2, 3, 4, 11 579 1 Limit: 2 45. 2, 3, 4, 5, 6 Limit does not exist. an n→ 6 n Horizontal asymptote: y0 100 101 102 103 an −4 Pr 33. 16 6.16 5.4136 5.3413 n Horizontal asymptote: y 104 105 106 an 3 −6 3 4 4 −6 Horizontal asymptote: y 3 lim f x n→ of 1. limit; infinity 3. converge 5. c 8. b 9. 1 11. 1 13. 2 15. 17. Does not exist 19. 4 21. 1 3 25. 4 27. 5 2 29. 31. −6 −5 C (page 887) 8 6 x→ en (b) About 0.33, 0.33 (c) Slope at vertex is 0. (d) Slope of tangent line at vertex is 0. −4 1 −3 0.75 ga 5 −1 Section 12.4 0.7499995 ge −4 0.749995 106 ar 0 105 0.74995 ng (c) 0 x→ Slope when x 5 is about 1385.2. This represents $1385.2 million and the rate of change of revenue in 2005. 5.3341 5.33341 5.333341 1 6 53. (a) 1 (b) 2 −5 35. (a) x 100 101 102 103 fx 0.7321 0.0995 0.0100 0.0010 0 x fx 104 1.0 105 10 4 1.0 106 10 5 1.0 10 6 20 0 (c) Over a long period of time, the level of the oxygen in the pond returns to the normal level. 13.50 x 45,750 55. (a) C x x (b) $471; $59.25 Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A201 A201 Answers to Odd-Numbered Exercises and Tests (b) (c) $13.50. As the number of PDAs gets very large, the average cost approaches $13.50. 57. (a) 1100 The model is a good fit for the data. Sn Sn 100 50 x y1 x is undefined for x < 0. Answers will vary. C −3 3 1 x 1 1 10 1 100 of 102 1 1000 1 10,000 105 1 100,000 n2 9. 44,140 13. (a) S n n 0.2385 0.02338 0.002334 0.000233 102 103 104 0.66165 0.666167 0.666617 3n 6n2 1 ni 100 101 0 0.615 23. 1.2656 square units 4 8 20 50 18 21 22.8 23.52 4 8 20 50 3.52 2.85 2.48 2.34 29. 4 8 20 50 100 Approximate area 40 38 36.8 36.32 36.16 n 4 8 20 50 100 36 38 39.2 39.68 39.84 5. 420 36 31. 4 Approximate area 15.13 100 15.2528 n 1 14.8125 15.2932 40 20 50 Approximate area 7. 44,100 8 14.25 n 2 1 4 11. 5850 n2 2n 4n2 3. 3 Approximate area 33. (page 896) Pr 1. cn 104 1 does not exist. x Section 12.5 104 n op lim 103 ty 101 er 100 103 Approximate area − 1.5 x 102 n en 2.5 x→0 27. Diverges y2 101 Approximate area ga 1 2 3 4 5 6 7 8 9 10 1 100 ge 150 69. 3n 6n3 Le 200 −1 1.00015 (c) Limit: 2 3 21. 14.25 square units 25. n 250 Converges to 0 71. n 300 1 2 3 4 5 6 7 8 9 10 1.0015 (b) 350 x −1 1.0154 CHAPTER 12 10 9 8 7 6 5 4 3 2 1 1.185 (c) Limit: 0 4n2 19. (a) S n y 67. 6 ng n . x1 61. True. If the sequence converges, then the limit exists. 1 1 1 63. Let f x ,g x , and c 0. Now 2 increases without x2 x2 x bound as x → 0 and lim f x gx 0. x→0 104 (b) x2 y 103 (c) Limit: 1 14n2 17. (a) S n (b) $1598.39 (c) When t is slightly less than 29, a vertical asymptote is found. 65. 102 Sn 7 59. False. Graph y 101 ar 1 800 100 n 46 3 (b) 35. n 100 101 102 103 104 1 0.3025 0.25503 0.25050 0.25005 Sn (c) Limit: 15. (a) S n n 4 8 20 50 100 Approximate area 19 18.5 18.2 18.08 18.04 1 4 2n2 3n 2n2 7 37. 3 square units 41. 10 square units 3 45. 3 square unit 4 39. 2 square units 43. 17 square units 4 47. 51 square units 4 18 Limits 439049092_12 ans.qxd A202 10/13/09 10:58 AM Page A202 Answers to Odd-Numbered Exercises and Tests 49. (b) 500 x 0.1 0.01 0.001 fx 7.2E86 Error x − 100 4.85E8 0.001 0.01 0.1 0 1E–87 0 Error 2.1E–9 600 − 100 Area is 105,208.33 ft2 51. True 53. c fx 2.4153 acres Limit does not exist. (page 900) Review Exercises 1. 41. (a) x 2.9 2.99 2.999 3 fx 16.4 16.94 16.994 17 x 3.001 3.01 3.1 fx 17.006 17.06 17.6 3 −3 3 x 0.01 0.001 fx 1.0517 1.005 1.0005 x 0.001 0.01 0.9995 0.995 Error 0.9516 0 0.001 1.99947 1.999995 0.001 0.01 1.99947 Error 0.1 1.999995 fx 0.1 fx x 0 0.01 1.94709 fx 17; The limit can be reached. 0.1 0.1 ni x ar 3. 1 (b) 1.94709 Le lim 6x x→3 ng −1 2 1 1 4 (a) 4 −9 x 2.9 fx Pr op −4 0.1695 10 3 2.99 0.1669 3 (b) x 1.1 1.01 1.001 1.0001 0.5680 fx 0.5764 0.5773 0.5773 y 45. 3 . 3 y 47. 4 3 3 2 3.01 4 2 0.1664 1 1 3.1 0.1639 x 1 2 4 5 x −4 −3 −2 6 2 −2 4 −3 −4 3 −2 −3 0.1667 39. (a) −2 About 0.575 −2 −1 Error 5 About 0.577 Actual limit is er 3 1 6 (b) ga 4 (b) 7 (c) 20 (d) 5 17. 11 19. 77 21. 1 29. 1 31. 1 15 3 9. (a) 64 15. 0 1 27. 4 en 2 5 2e 1; The limit cannot be reached. C 5. 11. 23. 33. 37. e x 7. 2 3 13. 10 25. 35. 1 4 of 1 2 −1 x ty lim x→0 ge 2 43. (a) −4 6 Limit does not exist. −9 9 Limit does not exist. y 49. 7 6 −6 Limit does not exist. y 51. 6 4 5 2 4 x 6 8 −2 −4 −6 Limit does not exist. 10 3 2 1 −2 −1 −1 x 1 2 3 4 5 Limit does not exist. 6 Limits 439049092_12 ans.qxd 10/13/09 10:58 AM Page A203 A203 Answers to Odd-Numbered Exercises and Tests 55. 3 57. 2 61. 2x y 105. 50 square units 107. 15 square units 109. 6 square units 111. 4 square units 3 113. (a) y 3.376068 10 7 x 3 3.7529 0.17x 132 (b) 150 y 5 4 3 12 10 8 x −4 −3 −2 −1 4 23456 x −6 −4 −2 2 4 6 8 10 0 (c) About 87,695.0 ft2 (Answers will vary.) 115. False. The limit of the rational function as x approaches does not exist. 2 1. x 2 −2 6 8 4 (2, − 3) 3 81. ar ga 4x 2 3 lim 3 5 −2 −3 x 4 −5 −6 −4 −7 x2 1 3 4 2x x→ 2 3. The limit does not exist. y 4 3 2 1 x −1 of 1 1 2 3 4 6 7 −2 −3 −4 1 2 3 (0, − 1) 4 The limit does not exist. er −2 ty x −4 −3 −2 −1 −3 4. op Pr 3 0.99 0.8484 0.8348 x→0 −2 1 125 5. 0.02 2.9982 2.9996 20 50 Approximate area 6.375 5.74 5.4944 0 0.01 0.02 Error 2.9996 2.9982 0.01 7 lim x→0 0.8335 8 3 −1 fx 104 sin 3x x 2 x (c) Limit: 5 6 101. 27 6.75 square units 4 103. n 4 7.5 4 lim 83. 2 85. 1 87. 0 89. Limit does not exist. 91. 3 11 15 19 11 3 93. 4, 1, 10, 13, 16 95. 1, 1, 27, 64, 8 4 Limit: 3 Limit: 0 9 97. 1, 1, 11, 8, 25; Limit does not exist. 52 7 17 n 1 5n 4 99. (a) S n 6n2 (b) 101 102 103 100 n Sn ge 73. g x 4 s 52 4 2 1 2 −3 5 3 −1 −1 x −3 −2 −1 −2 1 en 79. 4; (a) 4 (b) 6 4 ; (a) 4 (b) m x 62 1 71. h x fx 0 2 1 77. g s ft 2t 5 1 gx 2x 432 (a) 0 (b) y 1 y (c) 1 −6 e 2x 1 x 2 6 −1 x fx 0.004 0.003 0.002 0.001 1.9920 1.9940 1.9960 1.9980 0 Error CHAPTER 12 75. −4 −3 2x C 69. Le −6 y 2. y 10 −4 67. (page 903) Chapter Test 4 ng 3 2 6 1000 0 1 4 y 65. m x2 (2, 2) −4 63. 4 2 −2 −3 −4 2 10 6 (2, 0) ni 53. 4 59. Limits 439049092_12 ans.qxd A204 10/13/09 10:58 AM Page A204 Answers to Odd-Numbered Exercises and Tests 6. (a) m 7. f x 5; 7 (b) m 6 x 2 8. f x 4x 4 6x 2 5 π 2 9. 6; 12 π 2 10. 1 10. 0 11. 3 x 32 The limit does not exist. 14. 0, 1, 0, 1, 0 0, 3, 14, 12, 36 4 19 17 53 2 1 Limit: 2 Limit: 0 25 16. 8 square units 17. 16 square units 2 square units 3 2 (a) y 8.786x 6.25x 0.4 (b) 81.6 ft sec 9. f x 1. Ellipse 5 4 1 3 x 1 2 3 5 (2, − 1) −2 1 −3 −2 −1 −4 x 1 −1 2 3 4 5 Limaçon with inner loop 12. 13. 0, 4, 0 14. 149 6, 1, 3 15. 3, 4, 5 ? 32 42 52 ? 9 16 25 25 25 16. 17. x 2 2 1, 2, 1 y 22 z 4 2 24 2 z 18. 19. u v 38 uv 18, 6, 14 −2 −5 2 1 4 37.98 5. ge 2 y y 14 ga y 12 10 8 6 4 2 10 8 en 4. (1, 2) 2 6 6 4 x −4 −2 2 4 6 8 10 2 4 6 2 8 10 12 14 ty The corresponding rectangular equation is y ex 2. π 2 op 3 7. 5 , 2, 4 8r cos 0 8. 9x2 20x 7 , 2, 4 4 3r sin 16y2 4 0 4 y xy-trace (x − 2) 2 + (y + 1) 2 = 4 x 0 6 0 or r 5 2 2 20. Neither 21. Orthogonal 22. Parallel 23. (a) x 2 7t, y 3 5t, z 25t z x2y3 (b) 7 5 25 24. x 25. 75x 50y 31z 1 2t y 2 4t zt 30 z 26. 27. 2.74 2 Pr 2 er (− 2, − 34π ( 1 2, −2 −2 of −2 2 yz-trace (0, − 1, 0) 4 x −6 −8 − 10 6. 8 9 10 Le −2 −1 −1 0 456 ni 2 ng y 3 x2 1 Dimpled limaçon π 2 11. 2. Circle y 3. Circle ( page 904) Cumulative Test for Chapters 10–12 ar 15. 18. 0 1 C 12. 13. 0 12345 5 8 cos 4 3 sin (0, − 4, 0) 2 y −6 2 4 8 x 4 (0, 0, − 2) 6 −4 (8, 0, 0) −6 28. 84.26 29. 32. 1 33. 1 9 8 36. m 1 x 3 2 1 4 1 30. 34. 2; 1 2 1 4 31. Limit does not exist. 1 35. m 2x; 4 1 37. m x 3 2 ; 16 Limits 439049092_AppA ans.qxd 10/9/09 3:07 PM Page A205 A205 Answers to Odd-Numbered Exercises and Tests 39. Limit does not exist. m 2x 1; 1 Limit does not exist. 41. 7 42. 3 43. 0 0 45. 42,875 46. 8190 47. 672,880 49. A 1.566 square units A 10.5 square units 3 5 51. 2 square units 52. 16 square units 4 square unit 3 1. rational 3. origin 5. composite 7. variables; constants 9. coefficient 11. (a) 5, 1, 2 (b) 0, 5, 1, 2 (c) 9, 5, 0, 1, 4, 2, 11 (d) 7, 2, 9, 5, 0, 1, 4, 2, 11 (e) 2 23 13. (a) 1 (b) 1 (c) 13, 1, 6 (d) 2.01, 13, 1, 6, 0.666 . . . (e) 0.010110111 . . . 15. (a) 6, 8 (b) 6, 8 (c) 6, 1, 8, 22 3 3 3 (d) 1, 6, 7.5, 1, 8, 22 (e) ,1 2 33 2 17. (a) − 2 − 1 0 1 2 3 4 x (c) g1, g4 y 3 2 1 x −1 1 (b) −1 −2 7 2 x −1 0 1 (c) f3 1 3 0 f1 1 1 (b) lim f x 4 1 x→1 lim f x x→1 0 fx lim 2 fx 1 ge 21. 0.123 ga −8 en C of ty er 9 Pr op (c) lim d m 3, lim d m 3. This indicates that the m→ m→ distance between the point and the line approaches 3 as the slope approaches positive or negative infinity. 13. The error was probably due to the calculator being in degree mode rather than radian mode. y 15. (a) (b) A 36 (c) Base 6, 12 height 9; 10 2 Area 3bh 36 6 4 −4 −2 −5 −4 8 2 4 6 3 4 5 6 7 <7 1 2 3 31. (a) x 5 denotes the set of all real numbers less than or equal to 5. x (b) (c) Unbounded 0 1 2 3 4 5 6 33. (a) x < 0 denotes the set of all real numbers less than 0. x (b) (c) Unbounded −2 −1 0 1 2 35. (a) 4, denotes the set of all real numbers greater than or equal to 4. x (b) (c) Unbounded 1 2 3 4 5 6 7 2 < x < 2 denotes the set of all real numbers greater than 2 and less than 2. x (b) (c) Bounded 37. (a) −2 −1 0 1 2 1 x < 0 denotes the set of all real numbers greater than or equal to 1 and less than 0. x (b) (c) Bounded 39. (a) −1 x x 2 x 0 > 3 2 3 2 25 36 5 6 27. 1 4> 2 −6 −6 2.5 < 2 23. x −7 x 0 41. (a) 2, 5 denotes the set of all real numbers greater than or equal to 2 and less than 5. x (b) (c) Bounded −2 −1 0 1 2 3 4 5 6 APPENDIX A 19. 0.625 25. 29. −4 1 − 5.2 lim g x 0. When x is close to 0, both parts of the function x→0 are close to 0. 9. y 1 3 x 3m 3 11. (a) d m m2 1 8 (b) x→0 x 0 −7 −6 −5 −4 −3 −2 −1 5. a 3, b 6 7. lim f x does not exist. No matter how close x is to 0, there x→0 are still an infinite number of rational and irrational numbers, so lim f x does not exist. −9 5 −5 (d) 2 x→ 1 2 4 −5 −4 −3 −2 −1 lim x→ 1 2 3 ar 4 2 Le 1 4 (a) f ng (b) g1, g3, g4 (page A11) Appendix A.1 (page 907) Problem Solving 1. (a) g1, g4 3. Appendix A ni 38. 40. 44. 48. 50. Limits 439049092_AppA ans.qxd A206 113. 121. 123. 127. 129. 131. 5 10 500 50,000 5,000,000 (b) The value of 5 n approaches infinity as n approaches 0. True. Because b < 0, a b subtracts a negative number from (or adds a positive number to) a positive number. The sum of two positive numbers is positive. 11 False. If a < b, then > , where a 0 and b 0. ab (a) No. If one variable is negative and the other is positive, the expressions are unequal. (b) No. u v u v The expressions are equal when u and v have the same sign. If u and v differ in sign, u v is less than u v. The only even prime number is 2, because its only factors are itself and 1. a if a < 0. Appendix A.2 ga ge 47. 51. 57. 61. 63. 65. 69. 73. 75. 77. ng 43. ni 39. exponent; base 3. square root 5. index; radicand like radicals 9. rationalizing 11. (a) 27 (b) 81 (a) 1 (b) 9 15. (a) 243 (b) 3 4 5 (a) 6 (b) 4 19. 1600 21. 2.125 25. 6 27. 54 29. 5 24 3 6 (a) 125z (b) 5x 33. (a) 24y 2 (b) 3x 2 2 7 4 x b5 (a) (b) x y 2 37. (a) 2 (b) 5 y a x 3 1 10 (a) 1 (b) 41. (a) 2x 3 (b) 4x 4 x b5 (a) 33n (b) 5 45. 1.02504 104 a 49. 5.73 107 mi2 1.25 10 4 5 3 53. 125,000 55. 0.002718 8.99 10 g cm 15,000,000 C 59. 0.00009 m (a) 6.8 105 (b) 6.0 104 (a) 954.448 (b) 3.077 1010 (a) 3 (b) 3 67. (a) 1 (b) 27 2 8 8 (a) 4 (b) 2 71. (a) 7.550 (b) 7.225 (a) 0.011 (b) 0.005 (a) 67,082.039 (b) 39.791 (a) 2 (b) 2 5 3 x 79. (a) 2 5 (b) 4 3 2 18 z (a) 6x 2x (b) z2 5x 3 (a) 2x 3 2x 2 (b) y2 (a) 34 2 (b) 22 2 87. (a) 2 x (b) 4 y (a) 13 x 1 (b) 18 5x 5 3> 5 3 3 14 2 95. 97. 5 > 32 22 3 2 2 2 101. 103. 2.51 2 2 35 3 ar 35. (page A24) Le 1. 7. 13. 17. 23. 31. 81. en 5n 125. 133. Yes. a C of ty 93. 95. 99. 101. 103. 105. 107. 109. 111. er 83. 85. 87. 89. 91. Page A206 Inequality Interval y0 0, 2<x 4 2, 4 10 t 22 10, 22 W > 65 65, 10 53. 5 55. 1 57. 1 59. 1 63. 5 3> 3 5 67. 51 69. 5 71. 128 2 2 2 75 75. y x5 3 6 57 236 179 mi $113,356 $112,700 $656 > $500 0.05 $112,700 $5635 Because the actual expense differs from the budget by more than $500, there is failure to meet the “budget variance test.” $37,335 $37,640 $305 < $500 0.05 $37,640 $1882 Because the difference between the actual expense and the budget is less than $500 and less than 5% of the budgeted amount, there is compliance with the “budget variance test.” $1453.2 billion; $107.4 billion $2025.5 billion; $236.3 billion $1880.3 billion; $412.7 billion 7x and 4 are the terms; 7 is the coefficient. 3 x2, 8x, and 11 are the terms; 3 and 8 are the coefficients. 4x 3, x 2, and 5 are the terms; 4 and 1 are the coefficients. 2 (a) 10 (b) 6 97. (a) 14 (b) 2 (a) Division by 0 is undefined. (b) 0 Commutative Property of Addition Multiplicative Inverse Property Distributive Property Multiplicative Identity Property Associative Property of Addition Distributive Property 5x 1 3 115. 117. 48 119. 2 8 12 (a) Negative (b) Negative (a) n 1 0.5 0.01 0.0001 0.000001 op 81. 3:07 PM Answers to Odd-Numbered Exercises and Tests Pr 43. 45. 47. 49. 51. 61. 65. 73. 77. 79. 10/9/09 83. 85. 89. 91. 93. 99. 107. 81 216 1 3 2 1 111. 113. 3, x > 0 x x 115. (a) (b) 3 x 1 2 3 105. 119. 4 2 121. (a) 109. 813 4 117. (a) 2 4 2 (b) 8 2x 1.57 sec h 0 1 2 3 4 5 6 t 0 2.93 5.48 7.67 9.53 11.08 12.32 h 7 8 9 10 11 12 t 13.29 14.00 14.50 14.80 14.93 14.96 (b) t → 8.64 3 14.96 123. True. When dividing variables, you subtract exponents. Limits 439049092_AppA ans.qxd 10/9/09 3:07 PM Page A207 A207 Answers to Odd-Numbered Exercises and Tests am an am n: am am m a0 1. am 127. No. A number is in scientific notation when there is only one nonzero digit to the left of the decimal point. 129. No. Rationalizing the denominator produces a number equivalent to the original fraction; squaring does not. 27. 81. 85. 89. 95. 101. 105. er op 35. 37. 41. 45. 51. 55. 61. 65. 69. 73. 75. 77. Pr 31. 33. ty of 29. 12 16 x 5 2x 1 83. 25 x2 9 25 87. 2.25x2 16 5.76x 14.4x 9 2 4 91. u 93. x y 2x 2x 16 97. 4 x 4 99. 2x x 2 x2 2 5 x 5 103. x 3 x 1 x 5 3x 8 1 107. 1 x x2 4x 10 8 2x 2 2 3 111. x 9 x 9 x 6x 3 115. 4x 1 4x 1 3 4y 3 4y 3 3 3 119. 3u 2v 3u 2v x 1x 3 123. 2t 1 2 125. 5y 1 2 x 22 22 2 129. x 3 131. 1 6x 1 2 3u 4v 9 2 135. y 4 y 2 4y 16 x 2x 2x 4 1 139. 2t 1 4t 2 2t 1 2 9x2 6x 4 27 3x 2 2 u 3v u 3uv 9v x y 2 x2 xy 4x y2 2y 4 147. s 3 s 2 x 2x 1 151. x 20 x 10 y 5y 4 155. 5x 1 x 5 3x 2 x 1 159. x 1 x 2 2 3z 2 3z 1 2 163. 3 x 2 x 3 2x 1 x 3 2 167. x 2 3x 4 3x 1 2x 1 171. 3x 1 5x 2 2x 1 3x 2 175. x2 x 1 6x 3 x 3 179. x 1 2 181. 1 2x 2 xx 4 x 4 1 185. 81 x 36 x 18 2x x 1 x 2 189. x x 4 x 2 1 3x 1 x 2 5 12 193. t 6 t 8 3 x 12 4x x 2x 4x 2x 4 199. 3 4x 23 60x 5 x 2 x 2 2x 4 5 1 x 2 3x 2 4x 3 x 2 2 x 1 3 7x 5 3 x2 1 4 x4 x2 1 4 3x 2 2 33x6 20x5 3 4x3 2x 1 3 2x2 2x 1 2x 5 3 5x 4 2 70x 107 8 213. 14, 14, 2, 2 5x 1 2 51, 51, 15, 15, 27, 27 Two possible answers: 2, 12 Two possible answers: 2, 4 (a) P 22 x 25,000 (b) $85,000 (a) V 4x3 88x2 468x (b) x (cm) 1 2 3 Le 211. 215. 217. 219. 221. 223. V (cm3) 225. 44 x 308 x 227. 384 x 1 x 1 x 1 x x 720 x x x 616 1 x 1 x 1 x x 1 x x 1 1 1 1 1 1 x x 1 APPENDIX A 25. ga 23. en 21. n; an; a 0 3. monomial; binomial; trinomial First terms; Outer terms; Inner terms; Last terms completely factored 9. d 10. e 11. b a 13. f 14. c 15. 2 x 3 4x2 3x 20 15x 4 1 (a) 1 x 5 14 x 2 (b) Degree: 5; Leading coefficient: 1 2 (c) Binomial (a) 3x 4 x 2 4 (b) Degree: 4; Leading coefficient: 3 (c) Trinomial (a) x6 3 (b) Degree: 6; Leading coefficient: 1 (c) Binomial (a) 3 (b) Degree: 0; Leading coefficient: 3 (c) Monomial (a) 4x 5 6x 4 1 (b) Degree: 5; Leading coefficient: 4 (c) Trinomial (a) 4 x 3y (b) Degree: 4; Leading coefficient: 4 (c) Monomial Polynomial: 3x3 2x 8 Not a polynomial because it includes a term with a negative exponent Polynomial: y 4 y 3 y 2 39. 5t3 5t 1 2x 10 43. 12z 8 8.3x 3 29.7x2 11 47. 15z 2 5z 49. 4x 4 4x 3x 3 6x 2 3x 53. 0.2x2 34x 4.5t3 15t 57. 6x 2 7x 5 59. x 2 100 x 2 7x 12 63. 4x 2 12x 9 x 2 4y 2 67. 8x 3 12x 2y 6xy 2 y 3 x 3 3x 2 3x 1 71. x 4 x 2 1 16x 6 24x 3 9 3x 4 x 3 12x 2 19x 5 m 2 n 2 6m 9 79. 4r 4 25 x2 2 xy y 2 6x 6y 9 C 1. 5. 7. 12. 17. 19. (page A35) 2 3 ge Appendix A.3 109. 113. 117. 121. 127. 133. 137. 141. 143. 145. 149. 153. 157. 161. 165. 169. 173. 177. 183. 187. 191. 195. 197. 201. 203. 205. 207. 209. ng 0, using the property ni 1, a ar 125. a0 Limits 439049092_AppA ans.qxd A208 10/9/09 3:07 PM Page A208 Answers to Odd-Numbered Exercises and Tests 1 1 x x 1 x 1 1 x x 1 x x 1 63. 1 x x x 67. 1 1 1 69. 1 1 231. 4 r 233. (a) V (b) V 1 71. hR hR rR rR 2 hR 2 R 2 235. 237. 239. 243. 245. 39. 43. 45. rh 85. 1 89. 93. 95. (page A45) 97. 99. 101. en domain 3. complex 5. equivalent All real numbers 9. All nonnegative real numbers All real numbers x such that x 3 All real numbers x such that x 1 All real numbers x such that x 3 All real numbers x such that x 7 All real numbers x such that x 5 2 All real numbers x such that x > 3 3x 3y 25. , x 0 27. 3x, x 0 ,x 0 2 y1 4y 1 31. 1, x 5 33. y 4, y ,y 4 2 5 2 xx 3 y4 37. ,x 2 ,y 3 x2 y6 x2 1 41. z 2 ,x 2 x2 When simplifying fractions, you can only divide out common factors, not terms. x 0 2 x 2x x x 3 3 1 1 2 3 4 5 6 1 2 3 Undef. 5 6 7 1 2 3 4 5 6 7 The expressions are equivalent except at x 3. 1 r1 47. , r 0 49. 51. ,x 1 ,r 4 5x 2 r t3 53. ,t 2 t 3t 2 x 6x 1 55. , x 6, 1 x2 75 55.9 48.3 14 16 18 20 22 T 45 43.3 42.3 41.3 41.1 40.9 40.7 41.7 40.6 (b) The model is approaching a T-value of 40. 103. False. In order for the simplified expression to be equivalent to the original expression, the domain of the simplified expression needs to be restricted. If n is even, x 1, 1. If n is odd, x 1. 105. Completely factor each polynomial in the numerator and in the denominator. Then conclude that there are no common factors. Appendix A.5 1. 7. 11. 19. 27. 1 T t C of ty er 35. R 81. op 29. r 2 r Pr 23. rR 75. False. 4x2 1 3x 1 12 x 3 4x2 3x True. a2 b2 a ba b 241. x 3 8x 2 2x 7 mn xn yn xn yn Answers will vary. Sample answer: x 2 3 Appendix A.4 1. 7. 11. 13. 15. 17. 19. 21. r r ge 1 ga 1 6x 13 x5 2 59. 61. x3 x1 x2 x2 3 2x2 3x 8 65. 2x 1 x 2 x 1x 2x 3 2x ,x 0 x2 1 The error is incorrect subtraction in the numerator. 1 73. x x 1 , x 1, 0 ,x 2 2 2x 1 x7 2 1 77. 79. 2 , x>0 2x x2 x 15 3x 1 2 x 3 2 x2 5 83. ,x 0 x 112 3 1 1 87. ,h 0 ,h 0 xx h x 4x h 4 1 1 91. ,t 0 x2 x t3 3 1 ,h 0 xh1 x1 x ,x 0 2 2x 1 1 x 120 (a) min (b) min (c) 2.4 min 50 50 50 288 MN P (a) 6.39% (b) ; 6.39% N MN 12P (a) 0 2 4 6 8 10 12 t ng 57. ni x x ar x x Le 229. 29. 35. 37. 39. 41. 47. 55. (page A60) equation 3. extraneous 5. Identity Conditional equation 9. Identity Conditional equation 13. 4 15. 9 17. 5 1 21. No solution 23. 96 25. 6 23 5 No solution. The x-terms sum to zero, but the constant terms do not. 10 31. 4 33. 0 No solution. The solution is extraneous. No solution. The solution is extraneous. No solution. The solution is extraneous. 0 43. 2x 2 8x 3 0 45. 3x 2 90x 10 0 49. 4, 2 51. 5, 7 53. 3, 1 0, 1 2 2 57. a 59. ± 7 61. ± 3 3 2, 6 Limits 439049092_AppA ans.qxd 10/9/09 3:07 PM Page A209 A209 Answers to Odd-Numbered Exercises and Tests 83. 1 ± 1 11 89 7 2 ± 3 3 −5 14 −4 −3 0 1 2 3 4 0 9 2 51. 3 4 53. 1 4 <x< 3 − 4 −1 4 2 −1 57. 0 5<x<5 −6 −4 −2 67. x 20 4 13.5 11 12 13 14 2, x > 2 x 0 1 2 3 3 4 6 3 2, 65. x 25 5, x 8 6 13.5 −3 −2 −1 x 26 15 4 x 10 x 3 −3 2 x 10 x 59. x < 61. No solution 63. 14 x 26 14 2 5 2 ge 0 0 10.5 1 −5 15 2 x x x 30 −2 −1 0 1 2 69. 4 < x < 5 11 11 x x − 15 − 10 − 5 0 29 2, 71. x − 5 10 3 4 5 6 15 11 2 x 29 2 − 11 2 x − 16 − 12 −8 73. −4 75. 10 10 − 10 10 10 − 10 x 2 79. 10 10 −10 10 x 24 −10 4 6 81. 83. 10 − 15 1 x 22 3 −4 8 3 4 3 15 2 −10 3 2 1 <x< 55. 10.5 −10 2 4 −9 2 5 x>2 1 3 x −1 − 10 1 2 1<x<3 −2 5 77. x 0 1 −6 −4 −2 ga 13 0 x −2 −1 en C of ty er 0 x 6 47. 2<x 49. op Pr 12 7 2 − 10 x 11 5 35. x > 2 12 10 4 4 −6 x 33. x 3 3 2 −1 6 2 − 10 5 −2 5 x 27 2, x 1 2 −5 (a) x (b) x 2 3 2 APPENDIX A 4 4 x (page A69) 3 1 43. x 45. x 2 7 93. x 2 0 4 2 solution set 3. negative 5. double (a) 0 x < 9 (b) Bounded (a) 1 x 5 (b) Bounded (a) x > 11 (b) Unbounded (a) x < 2 (b) Unbounded b 16. h 17. e 18. d f 20. a 21. g 22. c (a) Yes (b) No (c) Yes (d) No (a) Yes (b) No (c) No (d) Yes (a) Yes (b) Yes (c) Yes (d) No 31. x < 3 x<3 2 1 −1 41. x 6 97. 6 ± 11 99. 3.449, 1.449 2 101. 1.355, 14.071 103. 1.687, 0.488 105. 1 ± 2 1 107. 6, 12 109. ± 3 111. ± 1 113. 0, ± 5 2 115. ± 3 117. 6 119. 3, 1, 1 121. ± 1 123. ± 3, ± 1 125. 1, 2 127. 50 129. 26 13 131. No solution 133. 52 135. 6, 7 137. 10 139. 3 ± 5 5 141. 1 1 ± 31 143. 2, 3 145. 4, 5 147. 149. 3, 2 2 3 1 17 151. 3, 3 153. 3, 155. 61.2 in. 2 1849 3 157. About 1.12 in. 159. 43 cm; 800.6 cm2 4 161. (a) 1998 (b) 2011; Answers will vary. 163. False. See Example 14 on page A58. 165. Equivalent equations have the same solution set, and one is derived from the other by steps for generating equivalent equations. 2x 5, 2x 3 8 167. x2 3x 18 0 169. x 2 2 x 1 0 171. Sample answer: a 9, b 9 173. Sample answer: a 20, b 20 b 175. (a) x 0, (b) x 0, 1 a 1. 7. 9. 11. 13. 15. 19. 23. 25. 27. 29. x 3 x −2 95. 2 ± Appendix A.6 39. x < 5 2 7 5 4 3 91. 2 7 37. x x 7± 85. 3 89. 1±3 2 2 6, 11 6 15 ± 85 79. 10 ng 77. 4±2 5 87. 73. 5± 4 8 75. 2 ± 2 3 67. ni 71. 4, 14 ar 69. 2 1 81. , 2 2± 65. Le 63. 8, 16 Limits 439049092_AppA ans.qxd A210 10/9/09 Page A210 Answers to Odd-Numbered Exercises and Tests 85. 87. 6 −6 6 8 −5 10 −2 −2 ni ng (a) 2 x 4 (a) 1 x 5 (b) x 4 (b) x 1, x 7 91. 93. 5, 3, ,7 2 All real numbers within eight units of 10 99. x 7 101. x 12 < 10 x 3 3 105. 4.10 E 4.25 x 3 >4 109. 100 r 170 p 0.45 9.00 0.75x > 13.50; x > 6 115. x 36 r > 3.125% 160 x 280 (a) 5 (b) x 129 17. Factor within grouping symbols before applying exponent to each factor. x2 5x 1 2 x x 5 1 2 x1 2 x 5 1 2 19. To add fractions, first find a common denominator. 3 4 3y 4x x y xy 21. To add fractions, first find a common denominator. x y 3x 2y2 2y 3 6y 1 23. 5x 3 25. 2x 2 x 15 27. 3 29. 3y 10 1 36 9 , 31. 2 33. 35. 37. 3, 4 39. 1 5x 25 4 2x 2 5 41. 1 7x 43. 3x 1 45. 7 x 3 4 47. 2x5 3x 5 4 49. 3 x 1 4 x 4 7x 2x 1 3 1 x 4 2 51. 53. 4x 8 3 7x 5 3 13 3 x x 3 7x 2 4x 9 32 72 55. 1 2 5x 57. 2 x x x 33x 14 2 27x 24x 2 1 59. 61. 6x 1 4 x 3 2 3 x 2 74 4x 3 x 63. 65. 2 x 4 3x 1 4 3 3x 2 1 2 15x2 4x 45 67. 2 x2 5 1 2 69. (a) 0.50 1.0 1.5 2.0 x 75 Le ar 89. 95. 97. 103. 107. 111. 113. 117. 119. 3:07 PM 150 0 t 12 13 14 15 16 17 18 19 Appendix A.7 133. b ga of 131. False. c has to be greater than zero. 135. Sample answer: x > 5 (page A78) Pr op er ty 1. numerator 3. Change all signs when distributing the minus sign. 3y 4 2 x 3y 4 2x 5. Change all signs when distributing the minus sign. 4 4 16x 2x 1 14x 1 30z 2 7. z occurs twice as a factor. 5z 6z 9. The fraction as a whole is multiplied by a, not the numerator and denominator separately. x ax a y y 11. x 9 cannot be simplified. 13. Divide out common factors, not common terms. 2 x2 1 cannot be simplified. 5x 15. To get rid of negative exponents: 1 ab 1 ab . a 1 b 1 a 1 b 1 ab b a t 1.70 1.72 1.78 1.89 x 2.5 3.0 3.5 4.0 t en 17.5 C 13.7 ge 121. (a) 1.47 t 10.18 (Between 1991 and 2000) (b) t > 21.19 (2011) 123. 106.864 in.2 area 109.464 in.2 125. You might be undercharged or overcharged by $0.21. 127. 13.7 < t < 17.5 129. 20 h 80 2.02 2.18 2.36 2.57 (b) x 0.5 mi 3x x2 8x 20 x 4 x2 4 (c) 2 2 6x 4x 8x 20 71. You cannot move term-by-term from the denominator to the numerator. ...
View Full Document

Ask a homework question - tutors are online