Solution Manual for Calculus with Applications, 12th Edition

Type:

Solution Manual

Resource:

Calculus with Applications

Edition:

12th Edition

Author(s):

Margaret L. Lial Nathan P. Ritchery Raymond N. Greenwell

Complete Instructor Answers Answers to selected writing exercises are provided.

Answers to Prerequisite Skills Diagnostic Test 1. 20% 2. 51 / 35 3. x + y = 75 4. s Ú 4p 5. -20 / 3 (Sec. R.4) 6. -11 / 5 (Sec. R.4) 7. 1-2, 54 (Sec. R.5) 8. x … -3 (Sec. R.5) 9. y Ú -17 / 2 (Sec. R.5) 10. p 7 3 / 2 (Sec. R.5) 11. -y 2 + 4y - 6 (Sec. R.1) 12. x 3 - x 2 + x + 3 (Sec. R.1) 13. a2 - 4ab + 4b2 (Sec. R.1) 14. 3pq11 + 2p + 3q2 (Sec. R.2) 15. 13x + 521x - 22 (Sec. R.2) 16. 1a - 62/ 1a + 22 (Sec. R.3) 17. 1x 2 + 5x - 22/ 3x1x - 121x + 124 (Sec. R.3) 18. 1-2 ± 172/ 3 (Sec. R.4) 19. 1-∞, -32 ∪ 31, ∞2 (Sec. R.5) 20. x 6y / 4 (Sec. R.6) 21. 2 / 1p2q2 (Sec. R.6) 22. 1m - k2/ 1km2 (Sec. R.6) 23. 1x 2 + 12-1/213x 2 + x + 52 (Sec. R.6) 24. 4b2 (Sec. R.7) 25. 14 + 2102/ 3 (Sec. R.7) 26. y - 5 (Sec. R.7)

Chapter R Algebra Reference For exercises . . .

1–8 9, 10 11–16 Refer to example . . . 2 3 4

Exercises R.1 (page R-5–R-6)

17–24 5

1. -x 2 + x + 9 2. -6y 2 + 3y + 10 3. 13 / 22z 2 + 15 / 62z + 2 4. 11 / 22t 2 + 11 / 22t + 2 / 3 5. -16q2 + 4q + 6 6. 9r 2 - 4r + 19 7. -0.327x 2 - 2.805x - 1.458 8. 0.8r 2 + 3.6r - 1.5 9. -18m3 - 27m2 + 9m 10. -12x 4 + 30x 2 + 36x 11. 9t 2 + 9ty - 10y 2 12. 18k 2 - 7kq - q2 13. 4 - 9x 2 14. 36m2 - 25 15. 16 / 252y 2 + 111 / 402yz + 11 / 162z 2 16. 115 / 162r 2 - 17 / 122rs - 12 / 92s 2 17. 27p3 - 1 18. 15p3 + 13p2 - 10p - 8 19. 8m3 + 1 20. 12k 4 + 21k 3 - 5k 2 + 3k + 2 21. 3x 2 + xy + 2xz - 2y 2 - 3yz - z 2 22. 2r 2 + 2rs - 5rt - 4s 2 + 8st - 3t 2 23. x 3 + 6x 2 + 11x + 6 24. x 3 - 2x 2 - 5x + 6 25. x 2 + 4x + 4 26. 4a2 - 16ab + 16b2 27. x 3 - 6x 2y + 12xy 2 - 8y 3 28. 27x 3 + 27x 2y + 9xy 2 + y 3

25–28 6

For exercises . . .

1–4 5–15 16–20 21–32 Refer to example . . . 1 2,3 3, 2nd CAUTION 4

Exercises R.2 (page R-9)

1. 7a21a + 22 2. 3y1y 2 + 8y + 32 3. 13p2q1p2q - 3p + 2q2 4. 10m216m2 - 12mn + 5n22 5. 1m + 221m - 72 6. 1x + 521x - 12 7. 1z + 421z + 52 8. 1b - 721b - 12 9. 1a - 5b21a - b2 10. 1s - 5t21s + 7t2 11. 1y - 7z21y + 3z2 12. 13x + 721x - 12 13. 13a + 721a + 12 14. 15y + 2213y - 12 15. 17m + 2n213m + n2 16. 61a - 1021a + 22 17. 3m1m + 321m + 12 18. 212a + 321a + 12 19. 2a214a - b213a + 2b2 20. 12x 21x - y212x + 5y2 21. 1x + 821x - 82 22. 13m + 5213m - 52 23. 101x + 421x - 42 24. Prime 25. 1z + 7y22 26. 1s - 5t22 27. 13p - 422 28. 1a - 621a2 + 6a + 362 29. 13r - 4s219r 2 + 12rs + 16s 22 30. 31m + 521m2 - 5m + 252 31. 1x - y21x + y21x 2 + y 22 32. 12a - 3b212a + 3b214a2 + 9b22

For exercises . . .

1–12 13–38 Refer to example . . . 1 2

Exercises R.3 (page R-12)

1. v / 7 2. 5p / 2 3. 8 / 9 4. 2 / 1t + 22 5. x - 2 6. 41y + 22 7. 1m - 22/ 1m + 32 8. 1r + 22/ 1r + 42 9. 31x - 12/ 1x - 22 10. 1z - 32/ 1z + 22 11. 1m2 + 42/ 4 12. 12y + 12/ 1y + 12 13. 3k / 5 14. 25p2 / 9 15. 9 / 15c2 16. 2 17. 1 / 4 18. 3 / 10 19. 21a + 42/ 1a - 32 20. 2 / 1r + 22 21. 1k - 22/ 1k + 32 22. 1m + 62/ 1m + 32 23. 1m - 32/ 12m - 32 24. 212n - 12/ 13n - 52 25. 1 26. 16 + p2/ 12p2 27. 112 - 15y2/ 110y2 28. 137 / 130m2 29. 13m - 22/ 3m1m - 124 30. 1r - 62/ 3r12r + 324 31. 14 / 331a - 124 32. 23 / 3201k - 224 33. 17x + 12/ 31x - 221x + 321x + 124 34. 1y 2 + 12/ 31y + 321y + 121y - 124 35. k1k - 132/ 312k - 121k + 221k - 324 36. m13m - 192/ 313m - 221m + 321m - 424 37. 14a + 12/ 3a1a + 224 38. 15x 2 + 4x - 42/ 3x1x - 121x + 124

For exercises . . .

1–8 9–26 27–37 Refer to example . . . 2 3–5 6,7

Exercises R.4 (page R-17)

1. -12 2. 3 / 4 3. 12 4. -3 / 8 5. -7 / 8 6. -6 / 11 7. 4 8. -10 / 19 9. -3, -2 10. -1, 3 11. 7 12. -2, 5 / 2 13. -1 / 4, 2 / 3 14. 2, 5 15. -3, 3 16. -4, 1 / 2 17. 0, 4 18. 15 + 2132/ 6 ≈ 1.434, 15 - 2132/ 6 ≈ 0.232 19. 12 + 2102/ 2 ≈ 2.581, 12 - 2102/ 2 ≈ -0.581 20. 1-1 + 252/ 2 ≈ 0.618, 1-1 - 252/ 2 ≈ -1.618 21. 5 + 25 ≈ 7.236, 5 - 25 ≈ 2.764 22. 14 + 262/ 5 ≈ 1.290, 14 - 262/ 5 ≈ 0.310 23. 1, 5 / 2 24. No real number solutions 25. 1-1 + 2732/ 6 ≈ 1.257, 1-1 - 2732/ 6 ≈ -1.591 26. -1, 0 27. 3 28. 12 29. -59 / 6 30. 6 31. 3 32. -5 / 2 33. 2 / 3 34. 1 35. 2 36. No solution 37. No solution For exercises . . .

1–14 Refer to example . . . Figure 1, Example 2

Exercises R.5 (page R-22) 1. 1-∞, 42 2. 3-3, ∞2

0 –3

0

15–26 2

27–38 3

39–42 4

43–54 5–7

4

3. 31, 22

0

1

2

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A-9

A-10

Complete Instructor Answers

4. 3-2, 34

–2

0

5. 1-∞, -92

3

–9

6. 36, ∞2

0

0

6

7. -7 … x … -3 8. 4 … x 6 10 9. x … -1 10. x 7 3 11. -2 … x 6 6 12. 0 6 x 6 8 13. x … -4 or x Ú 4 14. x 6 0 or x Ú 3 15. 1-∞, 24 17. 13, ∞2

0

20. 11 / 3, ∞2

0 1

1

–5

0

26. 1-∞, 50 / 94

21. 1-4, 62

–3 2

34. 3-1 / 2, 2 / 54

–6 –4

0

36. 1-∞, -22 ∪ 15 / 3, ∞2 38. 1-∞, 02 ∪ 116, ∞2

–2 0

0

0

1

0 0

5 3

0

4

7 3

0

4

0

– 17

3 1

2

31. 1-∞, -42 ∪ 14, ∞2

–4

–1 0

5

–1

01

37. 1-∞, -34 ∪ 33, ∞2

41. 1-∞, 02 ∪ 11, 62

0

4

3

–3

39. 3-2, 04 ∪ 32, ∞2

16

1

5

7

35. 1-∞, -12 ∪ 11 / 3, ∞2

0

01

25. 3-17 / 7, ∞2

2

–5

1

22. 37 / 3, 44

6

33. 1-∞, -14 ∪ 35, ∞2

–4 –3 –1

–1

0

19. 11 / 5, ∞2

2

2 5

40. 1-∞, -44 ∪ 3-3, 04 42. 1-1, 02 ∪ 14, ∞2

01 5

2

0

29. 11, 22

0 1

0

–1

–4

27. 1-5, 32

50 9

30. 1-∞, -42 ∪ 11 / 2, ∞2 32. 3-3 / 2, 54

0

24. 3-1, 24

3 0

28. 1-∞, -64 ∪ 31, ∞2

16. 1-∞, 12

2

18. 1-∞, 14

3

3

23. 3-5, 32

0

–2 0

0 0

3

2

1

6

43. 1-5, 34 44. 1-∞, -12 ∪ 11, ∞2 45. 1-∞, -22

46. 1-2, 3 / 22 47. 3-8, 52 48. 1-∞, -3 / 22 ∪ 3-13 / 9, ∞2 49. 32, 32 50. 1-∞, -12 51. 1-2, 04 ∪ 13, ∞2 52. 1-4, -22 ∪ 10, 22 53. 11, 3 / 24 54. 1-∞, -22 ∪ 1-2, 22 ∪ 34, ∞2

Exercises R.6 (page R-26)

For exercises . . .

1–8 Refer to example . . . 1

9–26 27–36 2 3,4

37–50 5

51–56 6

For exercises . . .

1–22 23–26 Refer to example . . . 1,2 3

27–40 4

41–44 5

1. 1 / 64 2. 1 / 81 3. 1 4. 1 5. -1 / 9 6. 1 / 9 7. 36 8. 27 / 64 9. 1 / 64 10. 85 11. 1 / 10 8 12. 7 13. x 2 14. 1 15. 8k 3 16. 1 / 13z 72 17. x 5 / 13y 32 18. m3 / 54 19. a3b6 20. 49 / 1c 6d 42 21. 1a + b2/ 1ab2 22. 11 - ab22/ b2 23. 21m - n2/ 3mn1m + n224 24. 13n2 + 4m2/ 1mn22 25. xy / 1y - x2 26. y 4 / 1xy - 122 27. 11 28. 3 29. 4 30. -25 31. 1 / 2 32. 4 / 3 33. 1 / 16 34. 1 / 5 35. 4 / 3 36. 1000 / 1331 37. 9 38. 3 39. 64 40. 1 41. x 4 / y 4 42. b / a3 43. r 44. 123 / y 8 45. 3k 3/2 / 8 46. 1 / 12p22 47. a2/3b2 48. y 2 / 1x 1/6z 5/42 49. h1/3t 1/5 / k 2/5 50. m3p / n 51. 3x1x 2 + 3x221x 2 - 52 52. 6x1x 3 + 721-2x 3 - 5x + 72 53. 5x1x 2 - 12-1/21x 2 + 12 54. 316x + 22-1/2127x + 52 55. 12x + 521x 2 - 42-1/214x 2 + 5x - 82 56. 14x 2 + 1212x - 12-1/2136x 2 - 16x + 12

Exercises R.7 (page R-30)

1. 5 2. 6 3. -5 4. 5 22 5. 20 25 6. 4y 2 22y 7. 9 8. 8 3 3 3 9. 7 22 10. 9 23 11. 9 27 12. -2 27 13. 5 2 2 14. 3 2 5 15. xyz 2 22x 16. 4r 3s 4t 6 210rs 17. 4xy 2z 3 2 2y 2 6 5 2 2 4 3 3 2 3 2 4 2 2 4 18. x yz 2y z 19. ab 2ab1b - 2a + b 2 20. p 2pq1pq - q + p 2 21. 2a 22. b 2b 23. 4 - x 24. 3y + 5 25. Cannot be simplified 26. Cannot be simplified 27. 5 27 / 7 28. 210 / 2 29. - 23 / 2 30. 22 31. -311 + 222 32. -512 + 262/ 2 33. 312 - 222 34. 15 - 2102/ 3 35. 1 2r + 232/ 1r - 32 36. 51 2m + 252/ 1m - 52 37. 2y + 25 38. 1z + 25z - 2z - 252/ 1z - 52 39. -2x - 2 2x1x + 12 - 1 40. 3p2 + p + 2 2p1p2 - 12 - 14/ 1-p2 + p + 12 41. -1 / 3211 - 2224 42. 1 / 13 + 232 43. -1 / 32x - 2 2x1x + 12 + 14 44. 2 / 3p + 2p1p - 224

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Complete Instructor Answers

A-11

Chapter 1 Linear Functions For exercises . . .

5–8 9–12 17,33,34 18,35–38 19–21 Refer to example . . . 1 3 8 9 4

Exercises 1.1 (page 13–17)

22,31 7

23–28 5

29,30 2

32 6

49–64 11,12

W1. -3 W2. y = -2x - 13 W3. y = 12 / 52x + 19 / 30 W4. y = 12 / 32x - 7 / 3 1. False 2. True 3. False 4. False 5. 3 / 5 6. -7 / 4 7. Not defined 8. 0 9. 1 10. 3 11. 5 / 9 12. -4 / 7 13. Not defined 14. 0 15. 0 16. 0 17. 2 18. -1 / 4 19. y = -2x + 5 20. y = -x + 6 21. y = -7 22. x = -8 23. y = - 11 / 32x + 10 / 3 24. y = -x + 7 25. y = 6x - 7 / 2 26. y = 121 / 322x + 33 / 16 27. x = -8 28. y = 3 29. x + 2y = -6 30. 2x - y = -4 31. x = -6 32. y = 7 33. 3x + 2y = 0 34. 2x - y = 9 35. x - y = 7 36. 3x + 2y = 6 37. 5x - y = -4 38. 3x + 6y = -2 39. No 40. (a) k = -1 / 2 (b) k = -7 / 2 43. (a) 44. (f) 45. -4 46. 1 / 2 48. (a) y = - 1b / a2x + b (b) a and b y y 49.

50.

51.

52. y = 4x + 5

5

6

y = –6x + 12

1

53.

y

54.

0

x

0

–1

3x – y = –9

9

55.

0

0

56.

y

3

y

x

2

5y + 6x = 11 4 x

0

x

3

x

6

3y – 7x = –21 –7

57.

58.

59.

y

y

60.

x

0 x=4 0

4

x

–8

61.

62.

63.

y

64.

y+8=0

y 3

y = –5x 1 x

0

5 x

0 3x – 5y = 0

–5

Tuition and fees

65. (a) 12,000; y = 12,000x + 3000 (b) 8 years 1 month 66. (a) 0.9; each additional cupcake costs $0.90 (b) y = 0.9x + 36 (c) $198 67. (a) Yes (b) y = 598t + 25,522; the slope indicates that the annual cost of tuition and fees at y 40,000

30,000 25,000

68. (a)

private four-year colleges is increasing by about $598 per year. (c) The year 2035 is too far in the future to rely on this equation to predict costs; too many other factors may influence these costs by then.

35,000

8

10

12 14 16 18 Years after 2000

20

The number of subscribers appears to be increasing at a nearly linear rate. (b) y = 15.729t + 138.67 (c) y = 15.688t + 139.42 (e) 390.33 million; 390.43 million

y 400 Subscribers

t

300 200 100 0

0

5

10 15 Years after 2000

20

t

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65–79 10,13

A-12

Complete Instructor Answers

69. (a) y = 4.317t + 87.049 (b) 173.4, which is slightly more than the actual CPI (c) It is increasing at a rate of approximately 4.3 per year. 70. (a) y = 0.53t - 0.043 (b) About 10.2 yr 71. (a) u = 0.851220 - x2 = 187 - 0.85x, l = 0.71220 - x2 = 154 - 0.7x (b) 140 to 170 beats per minute (c) 126 to 153 beats per minute (d) The women are 16 and 52. Their pulse is 143 beats per minute. 72. Approximately 4.3 m / sec 73. About 86 yr 74. (a) y = -1.93t + 267 (b) 2026 75. (a) y = 13,104.18t - 406,022 (b) About 1,166,480 76. (a) y = 0.12t + 25.12 (b) y = 0.14t + 22.54 (c) Women (d) 2038 (e) 30.6 77. (a) There appears to be a linear relationship. (b) y = 76.9x (c) About 780 megaparsecs (about 1.5 * 10 22 mi) (d) About 12.4 billion yr 78. (a) T = 0.03t + 15 (b) About 2103 (c) T = 0.02t + 15; about 2170 79. (a) yo = 4.625t - 19.25 (b) y = 4.75t - 41.5 (c) The percent of Americans who had listened to online radio in the previous month increased by 4.625% per year, while the percent of U.S. cellphone users who had ever listened to online radio in a car using a phone increased by 4.75% per year.

Exercises 1.2 (page 24–27) W1. 60 W2.

p 8 6 4 2

For exercises . . .

9–18 23–26 Refer to example . . . 1 4

27–30,41,42,49,50 31–36 5,6 2,3

37–40,43–48 51–53 7 8

y = 7 – 2.5x

0

2

4 q

1. True 2. False 3. True 4. True 5. False 6. True 7. True 8. True 9. -3 10. -13 11. 22 12. 12 13. 0 14. 2 15. -4 16. -9 / 2 17. 7 - 5t 18. 2k 2 - 3 23. If R1x2 is the cost of renting a snowboard for x hours, then R1x2 = 2.25x + 10. 24. If C1x2 is the cost of downloading x songs, then C1x2 = 0.99x + 10. 25. If C1x2 is the cost of parking a car for x hours, then C1x2 = 0.75x + 2. 26. If R1x2 is the cost of renting a car for x miles, then R1x2 = 44 + 0.28x.

27. C1x2 = 30x + 100 28. C1x2 = 45x + 35 29. C1x2 = 75x + 550 30. C1x2 = 120x + 12,500 31. (a) $16 (b) $11 (c) $6 (d) 640 watches (e) 480 watches (f) 320 watches (g) p (h) 0 watches (i) About 1333 watches (j) About 2667 watches (k) p 16 16 (l) 800 watches, $6 p = 16 – 1.25q 14 12 10 8 6 4 2

0

5 4 3 2 1

33. (a)

5 p 125 100

10

15

20

25

30

0

(b) 100 tubs, $40 34. (a) 4 p = 120 2 q 5

50

100

p = 0.75q

150

2 4 6 8 10 12 14 q

p = 6 – 0.25q p = 0.35q (10, 3.5)

5

10

15

20

25

30

q

(b) About 1120 lb; about $0.96 p = –2q + 3.2 p = 1.4q – 0.6

2

2 q 5

1

25 0

p 3

p=

(100, 40)

p 7 6 5 4 3 2 1

q

75 50

(8, 6)

0

2 4 6 8 10 12 14 q

32. (a) $6 (b) $5 (c) $3.90 (d) 600 quarts (e) 1100 quarts (f) 1440 quarts (g) p (h) 0 quarts (i) 800 quarts (j) 1800 quarts (k) 7 (l) 1000 quarts; $3.50 p = 6 – 0.25q 6

0

p = 16 – 1.25q

14 12 10 8 6 4 2

200

q

0

1

q

35. D1q2 = 6.9 - 0.4q 36. D1q2 = 9 - 0.35q 37. (a) 2 units (b) $980 (c) 52 units 38. (a) 3 units (b) $3211 (c) 13 units 39. (a) C1x2 = 3.50x + 90 (b) 17 shirts (c) 108 shirts 40. (a) C1x2 = 2.15x + 525 (b) 188 (c) 545 books 41. (a) C1x2 = 0.097x + 1.32 (b) $1.32 (c) $98.32 (d) $98.417 (or $98.42) (e) 9.7. (f) 9.7., the cost of producing one additional cup of coffee would be 9.7.. 42. (a) C1x2 = 500,000 + 4.75x (b) $500,000 (c) $975,000 (d) $4.75; each additional item costs $4.75 to produce. 43. Break-even quantity is 45 units; don’t produce; P1x2 = 20x - 900 44. Break-even quantity is about 41 units; produce; P1x2 = 145x - 6000 45. Break-even quantity is -50 units; impossible to make a profit when C1x2 7 R1x2 for all positive x; P1x2 = -10x - 500 (always a loss) 46. Break-even quantity is -50 units; impossible to make a profit when C1x2 7 R1x2 for all positive x; P1x2 = -100x - 5000 (always a loss). 47. 5 48. 26

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Complete Instructor Answers

A-13

49. (a) About 23 acorns per square meter (b) 34; the number of deer tick larvae per 400 squares meters in the spring will increase by 34 for each additional acorn per square meter in the preceding fall. 50. (a) ft1t2 = 2.8t + 35 (b) fd1t2 = -2.2t + 100 (c) 2003 51. (a) 14.4°C (b) -28.9°C (c) 122°F 52. (a) 98.6°F (b) 97.7°F to 99.5°F 53. -40° 54. (a) C1x2 = 3.308 * 10 9 + 7.17x (b) $4.168 billion (c) 166 million MWh

Exercises 1.3 (page 33–40) For exercises . . .

7(c),14(a),15(a), 16(a),17(a),18(a), 20(c),21(a)(c), 22(a)(c), 23(b),24(c), 25(a),26(a),27(c), 29(a),30(b),31(a), 32(b),33(a)(b), 34(a)(b),35(c) 1

7(b),8,11(a), 14(d),15(d), 16(e),17(e),18(e), 20(b),21(b),22(b), 23(d), 24(b),25(d), 26(c),27(b),29(d), 30(c),32(a),33(d), 34(d),35(d),36

Refer to example . . . 4

1. True 2. False 3. False 4. True 7. (a) y (b) 0.993 (c) Y = 0.555x - 0.5

y 6

6

7(d),14(b), 15(b),16(c), 17(c),18(c), 25(b),26(b), 27(d), 29(b)(c), 31(b)(c), 32(c)

9(a)(b), 10(a)(b), 12(a), 19(b)(d), 28(a)(b)

9(c),10(c), 14(c),15(c), 20(d) 16(d),17(d), 18(d),19(f), 25(c),33(c), 34(c)

2

1,4

5

3

(d) 5.6

5

5

4

4

3

3

2

2

1

1 0

2

4

6

8

0

10 x

2

4

6

8

10 x

8. (c) 9. (a) Y = 0.9783x + 0.0652; 0.9783 (b) Y = 1.5; 0 (c) The point (9, 9) is an outlier that has a strong effect on the least y squares line and the correlation coefficient.

10. (a) Y = -2.887x + 8.969; -0.8880 (b) Y = x; 1 10 (c) y The point 19, -202 is an outlier 8 10 that has a strong effect on the 6 5 least squares line and the 4 0 2 4 6 8 10 x –5 2 correlation coefficient.

0

11. (a) 0.7746 (b)

2

4

8

–10 –15 –20 –25

10 x

12. (a) Y = 2; 0 (b)

y 2

14. (a) Y = 0.06815x + 1.844 (b) $3.548 trillion (c) 2061 (d) 0.7907; fairly strong positive correlation

y 5 4 3

1

0

6

2

2

4

6

1 0

x

1

2

3

4

5

x

15. (a) Y = -0.2333x + 8.872 (b) About 2900 (c) 2030 (d) -0.9991; strong negative correlation 16. (a) Y = 0.815x + 19.4 (b) By about 0.815 per year (c) 40.6 (d) 2032 (e) 0.9953; strong positive correlation 17. (a) Y = -3.21x + 98.7 (b) By about 3.21 percent per year (c) 21.7% (d) 2025 (e) -0.9896; strong negative correlation 18. (a) Y = 176.84x + 812.28 (b) By about $176.84 billion per year (c) About $6117 billion (d) 2035 (e) 0.9952; strong positive correlation. 19. (a) 70,000 Yes (b) Y = 615.16x + 26,104; 0.9516; strong positive correlation (c) By about $615 per year (d) Y = 1071.91x + 48,383; 0.9568; strong positive correlation (e) By about $1072 per year (f) 2088; 2030 20. (a)

0 20,000 300

0

20

No (b) -0.2050; there is a negative correlation between the price and the distance. (c) Philadelphia; 0.0006244 (d) Y = 3.964 * 10 -5x + 191.16; 3.964 * 10 -5 dollars (much less than a penny)

3000 0

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A-14

Complete Instructor Answers

21. (a) Y = 0.0721x + 1.2648 (b) 0.4298 (c) y

12 10 8 6 4 2 0

Y 5 0.0721x 1 1.2648 8 10 12 14 16 18 20 x y Length (cm)

23. (a)

Y 5 20.7467x 1 17.264

8 10 12 14 16 18 20 x

Yes (b) Y = 1.585x - 0.487

8 6 4 2

8 6 4

Y 5 1.585x 2 0.487

2

0

24. (a)

(c) No; it gives negative values for small widths. (d) 0.999

y Length (cm)

3.5 3 2.5 2 1.5 1 0.5 0

22. (a) Y = -0.7467x + 17.264 (b) -0.9900 (c) y

2 4 6 Width (cm)

x

0

(b) 0.959, yes (c)

90

2 4 6 Width (cm)

x

25. (a) Y = 0.212x - 0.309 (b) 15.2 chirps per second (c) 86.4°F (d) 0.835

90

Y 5 3.98x 1 22.7 0

0

16

16 10

10

Income (in thousands)

26. (a) Y = -0.06857x + 17.53 (b) 15.1 (c) -0.7286; fairly strong negative correlation y 27. (a) Yes (b) 0.9998; yes (c) Y = 0.4411x + 13.335 (d) About $30,979 30 25 20 15 10 5 0 0

5 10 15 20 25 Years (since 1990)

30

x

28. (a) Y = -0.08915x + 74.28; r = -0.1035. The taller the student, the shorter is the ideal partner’s height. (b) Females: Y = 0.6674x + 27.89; r = 0.9459; males: Y = 0.4348x + 34.04; r = 0.7049 (c) 80 29. (a) Y = -0.0067x + 14.75 (b) 12 (c) 11 (d) -0.13 (e) There is no linear relationship.

55

80 60

30. (a)

3 Seconds

Seconds

2 1 0 1

2

3

4

5

x

Y = 0.366x + 0.803; the line seems to fit the data. (c) r = 0.995 indicates a good fit, which confirms the conclusion in part (b).

y

(b)

y 3

2 1 0 1

2

3

4

5

x

Feet

Feet

31. (a) Y = 14.9x + 2820 (b) 5060, compared to actual 5000; 6990, compared to actual 7000; 9080, compared to actual 9000 (c) 6250 BTUs; 6500 BTU air conditioner 32. (a) -0.995; yes (b) Y = -0.0769x + 5.91 (c) 2.07 points 33. (a) Y = -0.1178x + 113.10 (b) Y = -0.3441x + 147.89 (c) x ≈ 154; the women will catch up to the men in the year 2054. (d) rmen = -0.9608; rwomen = -0.9208; both sets of data points closely fit a line with negative slope. (e) 150 34. (a) Y = -0.009498x + 10.79 (b) Y = -0.01488x + 12.27 (c) x ≈ 275; the women will catch up to the men in the year 2175. (d) rmen = -0.8834; rwomen = -0.8915; both sets of data points closely fit a line with negative slope. (e) 12.5 0 100

Y 5 –0.01488x 1 12.27

120

Y 5 –0.009498x 1 10.79 0

120 9

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Complete Instructor Answers

35. (a) 3.816 miles per hour (b)

Yes (c) Y = 3.714x + 3.809 (d) 0.9969; yes (e) 3.714 miles per hour

110

0

A-15

36. (a) 0.5259 (b) 0.2926 (c) -0.0956

28 0

For exercises . . .

1–6,15–47,57(a)(b), 63(a)(b) Refer to section . . . 1

Chapter 1 Review Exercises (page 42–46)

7–10,13, 48–56,59,62 2

11,12,14,57(c)(f),58, 60,61,63(c)(e),64,65 3

1. False 2. False 3. True 4. False 5. True 6. False 7. True 8. False 9. False 10. False 11. False 12. True 14. ©x, ©y, ©xy, ©x 2, ©y 2, and n. 15. 1 16. 2 17. -2 / 11 18. Undefined 19. -4 / 3 20. 4 21. 0 22. 0 23. 5 24. 1 / 5 25. y = 12 / 32x - 13 / 3 26. y = - 11 / 42x + 2 27. y = -x - 3 28. y = - 17 / 52x - 1 / 5 29. y = -10 30. y = 5 31. 2x - 7 = 10 32. 5x - 8y = -40 33. x = -1 34. x = 7 35. y = -5 36. x = -3 y 37.

38.

39.

40. y 6

3

4x + 6y = 12

y = 6 – 2x

3

x

x 3

41.

42.

y

43.

y

44.

x–3=0 1

3 0

y x + 3y = 0

y=1

1 x

x

–3

x

45. (a) y = 24.4t + 100 (b) Imports from China are increasing by about $24.4 billion per year. (c) $588 billion (d) 2027 46. (a) y = 5.78t + 16 (b) Exports to China are increasing by about $5.78 billion per year. (c) $132 billion (d) 2041 47. (a) y = 1187t - 76,887 (b) The median income for all U.S. households is increasing by about $1187 per year. (c) $71,488 (d) 2033 48. (a) 2.5 lb; 5 lb (b) 3.5 lb; 4 lb (c) 5.5 lb; 2 lb (d) p (e) $25 per pound (f) 3.75 lb 40 35 30 25 20 15 10 5 0

S(q) 5 4q + 10 D(q) 5 40 – 4q

0

2

4

6

8

10 12

q

49. (a) p = S1q2 = 0.5q + 10 (b) p = D1q2 = -0.5q + 72.50 (c) $41.25, 62.5 dietary supplement pills 50. C1x2 = 30x + 60 51. C1x2 = 180x + 2000 52. C1x2 = 30x + 85 53. C1x2 = 46x + 120 54. (a) 5 cartons (b) $2000 55. (a) 40 pounds (b) $280 56. (a) E1x2 = 352 + 42x (where x is in thousands) (b) R1x2 = 130x (where x is in thousands) (c) More than 4000 chips 57. (a) y = 778t + 21,245 (b) y = 721.75t + 22,258 (c) y = 744.11t + 21,938 (d) 36,000 (f) 0.9888 Y 5 744.11t 1 21,938 Y 5 721.75t 1 22,258 Y 5 778t 1 21,245 0 30,000

20

58. (a) Y = 13.981x - 288.43 (b) $1459.20 (c) 1400 Yes (d) 0.9928; strong positive correlation 59. (a) b1t2 = -0.725t + 64.5; p1t2 = -0.075t + 47.8; 1t2 = 0.556t + 54.2 (b) Beef is decreasing by about 0.725 lb / year; c Y 5 13.981x 2 288.43 pork is decreasing by about 0.075 lb / yr; chicken is increasing by about 0.556 lb / yr (c) 2009 (d) Beef: 50.0 lb; pork: 46.3 lb; chicken: 65.3 lb 30 120 0

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Yes (c) Y = 0.01315t + 35.92 (d) 81.87 61. (a) Y = 0.9724x + 31.43 (b) About 216 (c) 0.93 62. (a) m1t2 = 0.2t + 29.6 (b) The percent of never-married males is increasing by about 0.2 percent per year. (c) ƒ1t2 = 0.24t + 22.8 (d) The percent of never-married females is increasing by about 2000 4000 50 0.24 percent per year. (e) Males: 35.6%; females: 30.0% 100 63. (a) y = -0.9364t + 129.0 (b) y = -0.5t + 92.3 (c) Y = -0.9256t + 124.5

60. (a) 0.8988; yes (b)

85

Y 5 20.9364t 1 129.0 Y 5 20.5t 1 92.3 Y 5 20.9256t 1 124.5 30

90 40

(e) -0.9614 64. (a) 0.6683; yes, but the fit is not very good. (b)

(c) Y = 4.159t + 112.78 (d) $4159 65. (a) 0.4529 (b) 0.3955 (c) -0.4768

190

0

20 90

Extended Application: Predicting Life Expectancy (page 46) 1. Y = 0.134t - 188.63 2. 66.09 years 3. The poor prediction isn’t surprising, since we were extrapolating far beyond the range of the original data.

5.

1 x y Predicted value Residual 74.7 77.4 78.8 78.9 79.3 80.1 81.0 81.1

75.48 76.82 78.16 78.83 79.50 80.17 80.84 81.51

-0.78 0.58 0.64 0.07 -0.20 -0.07 0.16 -0.41

Residual

1970 1980 1990 1995 2000 2005 2010 2015

1970

1990

2010

y

21 Year

6. You’ll get 0 slope and 0 intercept, because you’ve already subtracted out the linear component of the data.

7. A cubic would fit the data: Y = 0.0001052x 3 - 0.6303x 2 + 12.5936x - 838,720.8.

Chapter 2 Nonlinear Functions Exercises 2.1 (page 58–62)

For exercises . . .

5–12

13–20

21–36

37–44

Refer to example . . .

2,3

4(b)

4(a),(c)–(e)

3(d)

45–60, 82,83 5

61–66

67–74

75,76,81

77–80

84,85

6

7

1

8

9

W1. 3 / 2, -3 / 2 W2. 1 / 2, -5 W3. 3-4, 44 W4. 1-∞, -44 ∪ 37, ∞2 1. False 2. False 3. False 4. True 5. Not a function 6. Function 7. Function 8. Not a function 9. Function 10. Function 11. Not a function 12. Not a function 13. 1-2, -12, 1-1, 12, 10, 32, 11, 52, 12, 72, 13, 92; 14. 1-2, 152, 1-1, 122, 10, 92, 11, 62, 15. 1-2, 3 / 22, 1-1, 22, 10, 5 / 22, 12, 32, 13, 02; range: 50, 3, 6, 9, 12, 156 11, 32, 12, 7 / 22, 13, 42; range: 5-1, 1, 3, 5, 7, 96 y y range: 53 / 2, 2, 5 / 2, 3, 7 / 2, 46 (–2, 15)

(3, 9)

9

y

14

(3, 4)

4 3 (–2, –1)

3 –2, –

(

6 3

x

2 0

(3, 0) 2

x

Copyright © 2022 Pearson Education, Inc.

2

) 2

x

Complete Instructor Answers

16. 1-2, -112, 1-1, -52, 10,12, 11, 72, 12, 132, 13, 192; range: 5-11, -5, 1, 7, 13, 196

17. 1-2, 02, 1-1, -12, 10, 02, 11, 32, 12, 82, 13, 152; range: 5-1, 0, 3, 8, 156 y

(3, 19)

y

(3, 15)

y 20

18. 1-2, 02, 1-1, -32,10, -42, 11, -32, 12, 02, 13, 52; range: 5-4, -3, 0, 56 (3, 5)

10

(–2, 0)

2 0

0

x

1

2

x

(–2, 0) 0

x

2

(–2, –11)

19. 1-2, 42, 1-1, 12, 10, 02, 11, 12, 12, 42, 13, 92; range: 50, 1, 4, 96

20. 1-2, -162, 1-1, -42, 10, 02, 11, -42, 12, -162, 13, -362; range: 5-36, -16, -4, 06

y

y (3, 9)

(–2, 4)

5

(–2, –16)

3

x

3

x

–20

–35

(3, –36)

21. 1-∞, ∞2 22. 1-∞, ∞2 23. 1-∞, ∞2 24. 1-∞, ∞2 25. 3-2, 24 26. 1-∞, ∞2 27. 33, ∞2 28. 3-5 / 3, ∞2 29. 1-∞, -12 ∪ 1-1, 12 ∪ 11, ∞2 30. 1-∞, -62 ∪ 1-6, 62 ∪ 16, ∞2 31. 1-∞, -42 ∪ 14, ∞2 32. 1-∞, ∞2 33. 1-∞, -14 ∪ 35, ∞2 34. 1-∞, -2 / 54 ∪ 31 / 3, ∞2 35. 1-∞, -12 ∪ 11 / 3, ∞2 36. 1-∞, 32 37. Domain: 3-5, 42; range: 3-2, 64 38. Domain: 3-5, ∞2; range: 30, ∞2 39. Domain: 1-∞, ∞2; range: 1-∞, 124 40. Domain: 1-∞, ∞2; range: 1-∞, ∞2 41. Domain: 3-2, 44; range: 30, 44 (a) 0 (b) 4 (c) 3 (d) -1.5, 1.5, 2.5 42. Domain: 3-2, 44; range: 30, 54 (a) 5 (b) 0 (c) 1 (d) -0.2, 0.5, 1.2, 2.8 43. Domain: 1-2, 44; range: 3-3, 24 (a) -3 (b) -2 (c) -1 (d) 2.5 44. Domain: 3-2, 44; range: {3} (a) 3 (b) 3 (c) 3 (d) Nowhere 45. (a) 33 (b) 15 / 4 (c) 3a2 - 4a + 1 (d) 12 / m2 - 8 / m + 1 or 112 - 8m + m22/ m2 (e) 0, 4 / 3 46. (a) 0 (b) -45 / 4 (c) 1a + 321a - 42 (d) 212 + 3m211 - 2m2/ m2 (e) 11 ± 2532/ 2 47. (a) 7 (b) 0 (c) 12a + 12/ 1a - 42 if a Z 4, 7 if a = 4 (d) 14 + m2/ 12 - 4m2 if m Z 1 / 2, 7 if m = 1 / 2 (e) -5 48. (a) 0 (b) 10 (c) 1a - 42/ 12a + 12 if a Z -1 / 2, 10 if a = -1 / 2 (d) 12 - 4m2/ 14 + m2 if m Z -4, 10 if m = -4 (e) -5 49. 6t 2 + 12t + 4 50. 6r 2 - 24r + 22 51. r 2 + 2rh + h2 - 2r - 2h + 5 52. z 2 - 2zp + p2 - 2z + 2p + 5 53. 9 / q2 - 6 / q + 5 or 19 - 6q + 5q22/ q2 54. 25 / z 2 + 10 / z + 5 or 125 + 10z + 5z 22/ z 2 55. (a) 2x + 2h + 1 (b) 2h (c) 2 56. (a) x 2 + 2xh + h2 - 3 (b) 2xh + h2 (c) 2x + h 57. (a) 2x 2 + 4xh + 2h2 - 4x - 4h - 5 (b) 4xh + 2h2 - 4h (c) 4x + 2h - 4 58. (a) -4x 2 - 8hx - 4h2 + 3x + 3h + 2 (b) -8hx - 4h2 + 3h (c) -8x - 4h + 3 59. (a) 1 / 1x + h2 (b) -h / 3x1x + h24 (c) -1 / 3x1x + h24 60. (a) -1 / 1x 2 + 2xh + h22 (b) 12xh + h22/ 3x 21x 2 + 2xh + h224 (c) 12x + h2/ 3x 21x 2 + 2xh + h224 61. Function 62. Function 63. Not a Function 64. Not a function 65. Function 66. Not a function 67. Odd 68. Odd 69. Even 70. Even 71. Even 72. Odd 73. Odd 74. Neither 75. (a) No (b) Year, t (c) Average price of silver, S1t2 (d) [2000, 2019] (e) 34, 354 (f) $15 (g) 2011 76. (a) Year; energy consumption (b) [2005, 2040]; [17, 55] (c) No (d) About 100 quadrillion Btu, or 100 * 1015) Btu; about 155 quadrillion Btu, or 155 * 1015 Btu; about 35 quadrillion Btu, or 35 * 1015 Btu (e) About 2028 (f) About 2009 77. (a) $36 (b) $36 (c) $64 (d) $120 (e) $120 (f) $148 (g) $148 (i) x, the number of full and partial days (j) S, the cost of renting a saw 79. (a) $93,300. Attorneys can receive a maximum of $93,300 on a 78. (a) $98 (b) $98 (c) $98 (d) $152 (e) $206 f(x) jury award of $250,000. (f) (500,000, 169,950) 160,000 (b) $124,950. Attorneys can receive a maximum of $124,950 on a jury 120,000 award of $350,000.

(300,000, 109,950) 80,000 (c) $181,950. Attorneys can receive a maximum of $181,950 on a jury (150,000, 60,000) 40,000 award of $550,000. x (d) f(x) 0 200,000

600,000

(500,000, 169,950)

160,000

(g) Yes (h) No (i) x, the number of full and partial days (j) C, the cost of renting the car

120,000 80,000 40,000 0

(300,000, 109,950) (150,000, 60,000)

200,000

x 600,000

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Complete Instructor Answers

80. (a) $407.50. A tax of $407.50 was due on an income of $10,000. (b) $499.75. A tax of $499.75 was due on an income of $12,000. (c) $841.40. A tax of $841.40 was due on an income of $18,000. (d) $2852.38. A tax of $2852.38 was due on an income of $50,000. (e) 30, ∞2, 30, ∞2

(f)

f(x) 6000 5000

(80,650, 4793)

4000 3000

(21,400, 1042)

2000 1000 0

(13,900, 599.5) (11,700, 484) (8500, 340) x 40,000 80,000

81. (a) About 140 m (b) About 250 m 82. (a) (i) 66 kcal / day (ii) 222 kcal / day (b) z = g1z2 = 0.454z (c) y = 10.9z 0.753 83. (a) (i) 3.6 kcal / km (ii) 61 kcal / km (b) x = g1z2 = 1000z (c) y = 4.4z 0.88 84. (a) A = 13000 - w2w 85. (a) P1w2 = 1000 / w + 2w (b) 0 … w … 3000 (b) 10, ∞2 (c) (c) A 5 (3000 2 w)w P 5 (1000/w) 1 2w 300

2,250,000

0

0

3000

0

Exercises 2.2 (page 72–76)

100

0

For exercises . . .

7–12

13–16

17–28

29–42

43–50

Refer to example . . .

1–3

4

4–6

10

Before Example 10

53–57,61,62, 64–67,70–76 8

58–60

63,68,69

7

9

W1. 12x - 92 W2. 1x + 5 / 82 W3. 2, -5 / 3 W4. 1-5 ± 2572/ 4 1. True 2. True 3. True 4. False 7. D 8. F 9. A 10. D 11. C 12. E 13. y = 31x + 3 / 222 - 7 / 4, 1-3 / 2, -7 / 42 14. y = 41x - 5 / 222 - 32; 15 / 2, -322 15. y = -21x - 222 - 1; 12, -12 16. y = -51x + 4 / 522 + 31 / 5; 1-4 / 5, 31 / 52 17. Vertex is 1-5 / 2, -1 / 42; axis is x = -5 / 2; 18. Vertex is 1-2, -92; axis is x = -2; 19. Vertex is 1-3, 22; axis is x-intercepts are -3 and -2; y-intercept is 6. x-intercepts are -5 and 1; y-intercept is -5. x = -3; x-intercepts are -4 and -2; y-intercept is -16. y 2

2

10 8 6 4 2

y = x 2 + 5x + 6

6 4

0 –6

2

x

–4 –2 –2 0 –4

x

y = x 2 + 4x – 5

20. Vertex is 1-1, 72; axis is x = -1; x-intercepts are -1 ± 221 / 3 ≈ 0.53 and -2.53; y-intercept is 4.

21. Vertex is 1-2, -162; axis is x = -2; x-intercepts are -2 ± 2 22 ≈ 0.83 and -4.83; y-intercept is -8.

y

y

y 4

8 y = –3x 2 – 6x + 4 –4

–2

0

2

–6 –4

0

f(x) = –x2 + 6x – 6

2

x

–2

–8

4x

22. Vertex is 13, 32; axis is x = 3; x-intercepts are 3 ± 23 ≈ 4.73 and 1.27; y-intercept is -6.

2

4

6

8 x

–4 –6

f(x) = 2x 2 + 8x – 8

23. Vertex is 11, 32; axis is x = 1; no x-intercepts; y-intercept is 5.

24. Vertex is 1-6, 62; axis is x = -6; no x-intercepts; y-intercept is 24. y

25. Vertex is 14, 112; axis is x = 4; x-intercepts are 4 ± 222 / 2 ≈ 6.35 and 1.65; y-intercept is -21. y 20

25

f (x) = –2x 2 + 16x – 21

10 f(x) = 12 x 2 + 6x + 24 –15

–10

0 –5

0

5x

–10 –20

Copyright © 2022 Pearson Education, Inc.

5

10 x

Complete Instructor Answers

26. Vertex is 11 / 3, -25 / 62; axis is x = 1 / 3; x-intercepts are -4 / 3 and 2; y-intercept is -4.

27. Vertex is 14, -52; axis is x = 4; x-intercepts are 4 ± 215 ≈ 7.87 and 0.13; y-intercept is 1 / 3.

y 15

28. Vertex is 1-1, -32; axis is x = -1; no x-intercepts; y-intercept is -7 / 2. y

y

2 –4

10

–5

5 –4

–2

0

0

4

6

8

f(x) = 13 x 2 – 83 x + 13

f(x) = 32 x 2 – x – 4

30. A

2

x

–4

4 x

2

–5

29. D 35.

31. C

32. B

y

33. E

1 2 – x – –7 f(x) = – –x 2 2

34. F

36.

y

37.

38.

(1, 4)

(–3, 2)

(5, 0)

(–5, 0) x

x (3, –2)

(–1, –4)

39. y 5

x

4

y 5

40.

43.

4

8x

6

6

8 10 x

44.

–5 –10

45.

y

x

48.

46.

y

x

y

(2, –2)

f(x) = –√2 – x – 2

(–2, –3) –5

y

47.

x

–8 –6 –4 –2 –2 0

(2, 2) 2

42.

f(x) = √x + 2 – 3

f(x) = √x – 2 + 2

0

y 5

41.

y

x

49.

y

x

50.

y

y

x

x x

(b) -r

51. (a) r 53. (a)

x

(c) -r

52.

y 20

(c) -b y

y

55. (a)

10

30 20

20 x

0

–30 –20 –10

(a) -b (b) b 54. (a)

–10

0

x –10

–20

0

(b) 1 batch (b) 1 batch (c) $16,000 56. (a) y

(c) $12,500

60 40

57. Maximum revenue is $9225; 35 seats are unsold.

58. (a) R1x2 = 20,000 + 200x - 4x 2 (b)

20 2

4

6

(b) 2.5 batches

x

(c) $31,250

(d) $5000

(d) $4000

80

(b) 2 batches (d) $1000

(d) $3125

10

8 10 x

59. (a) 960 + 30x (b) 80 - x (c) R1x2 = 76,800 + 1440x - 30x 2 (d) 24 (e) $94,080 60. (a) 80 - 4x (b) 100 + 5x (c) R1x2 = 8000 - 20x 2 (d) Now (e) $80.00 per tree

(c) $81,000 (c) 25

(d) $22,500

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Complete Instructor Answers

Revenue (in thousands of dollars)

61. (a) R1x2 = x1500 - x2 = 500x - x 2 R(x) (b) 72 60 48 36 24 12

62. (a) $15,000 (c)

(b) $40,000

(250, 62,500)

0

(c) $250

100 200 300 Demand

x

(d) $62,500 (c) ƒ1t2 = -2.67t 2 + 32.1t + 926

63. (a) Quadratic 1100

1100

(d) No

(e) 49 thousand short tons, 60 thousand short tons

f(t) 5 –2.71(t – 6)2 + 1027

1100

f(t) 5 –2.67t 2 1 32.1t 1 926

f(t) 5 –2.67t 2 1 32.1t 1 926 0

20 t

0

0

20 0

20 0

(b) ƒ1t2 = -2.711t - 622 + 1027

64. (a) 2.024 mm; 6.104 mm (b) 8.48 mm; 39.4 weeks after conception 65. (a) 87 yr (b) 98 yr 66. About 2005; 2002–2005; 2005–2016 67. (a) 28.5 weeks (b) 0.81 (c) 0 weeks or 57 weeks of gestation; no (c) ƒ1t2 = 0.01375t 2 - 0.4309t + 65.92

80

68. (a)

(e) No

f(t) 5 0.01375t2 – 0.4309t + 65.92

80

80 f(t) 5 0.009723(t – 10)2 1 62.04

f(t) 5 0.01375t2 – 0.4309t + 65.92 0

50 50

0

(b) Quadratic

50

0

50

(d) ƒ1t2 = 0.0097231t - 102 + 62.04 28

69. (a)

(f) 74.38; 73.95

f(t) 5 0.00111t2 2 0.0712t 1 20.8

(c)

f(t) 5 0.00111t2 2 0.0712t 1 20.8

(e) No

28

40

50 50

2

28

120

f(t) 5 0.00223(t 2 60)2 1 20.3

18 40

(b) Quadratic

120

40

18

(d) ƒ1t2 = 0.002231t - 6022 + 20.3 70. 49 yr; 3.98

120 18

(f) 29.2; 29.7

71. (a) 16 ft (b) 2 sec 72. (a) 61.70 ft (b) 43.08 mph 73. 95 ft by 190 ft

75. y = - 11 / 152x 2; 10 23 m ≈ 17.32 m

74. 9025 ft2

76. y = 14 / 272x 2; 6 23 ft ≈ 10.39 ft 12

y

(9, 12)

4 –9

Exercises 2.3 (page 84–89)

For exercises . . .

Refer to example . . .

7–10 1

(0, 0)

11–19,25–30 4

9

x

20–24,31–46,63,64 7

W1. The graph of ƒ1x2 shifted 2 units to the left and 3 units down.

W2. The graph of ƒ1x2 reflected across the x-axis and shifted 3 units to the right.

W3. The graph of ƒ1x2 reflected across the y-axis and shifted 2 units up.

W4. The graph of ƒ1x2 reflected across the y-axis and shifted 2 units to the right.

1. True 2. False 3. False 4. False

Copyright © 2022 Pearson Education, Inc.

50-53,56–58 8

59,67,68 5

60–62,65 2,3

Complete Instructor Answers

y

7.

y

8. f(x) = (x +1)3 – 2

5

–1 0

1

y

f(x) = – (x – 1)4 + 2

(–2, 0)

(1 + 4√2, 0)

2

x

0

–6

y

10.

f(x) = – (x +3)4 + 1

–2

0

4

(1 – 4√2, 0)

6 x

–5

5x

3

9.

(–4, 0)

–3

f(x) = (x – 2)3 + 3

1

x

–1 0

–3

3

(–1 + 3√2, 0)

1

(2 – 3√3, 0)

11. D 12. C 13. E 14. B 15. I 16. F 17. G 18. H 19. A 20. B 21. D 22. A 23. E 24. C 25. 4, 6, etc. 1true degree = 42; + 26. 5, 7, etc. 1true degree = 52; + 27. 5, 7, etc. 1true degree = 52; + 28. 6, 8, etc. 1true degree = 62; - 29. 7, 9, etc. 1true degree = 72; - 30. 7, 9, etc. 1true degree = 72; + 31. Horizontal asymptote: y = 0; vertical asymptote: x = -2; no x-intercept; y-intercept = -2

32. Horizontal asymptote: y = 0; vertical asymptote: x = -3; no x-intercept; y- intercept = -1 / 3

33. Horizontal asymptote: y = 0; vertical asymptote: x = -3 / 2; no x-intercept; y-intercept = 2 / 3

y y = –4 x+2

4

y

y

2 y = 3 + 2x

2

4

–2

0

x x=2

34. Horizontal asymptote: y = 0; vertical asymptote: x = 5 / 3; no x-intercept; y-intercept = 8 / 5

0

x

1

y=

8 5 – 3x

x= 5 3

0

x

3

–2 x=–3 2

35. Horizontal asymptote: y = 2; vertical asymptote: x = 3; x@intercept = 0; y@intercept = 0 y

36. Horizontal asymptote: y = -2; vertical asymptote: x = 3 / 2; x-intercept = 0; y-intercept = 0

y

y = 2x x–3

y=2

39. Horizontal asymptote: y = -1 / 2; vertical asymptote: x = -5; x@intercept = 3 / 2; y@intercept = 3 / 20

40. Horizontal asymptote: y = -3 / 4; vertical asymptote: x = -3; x-intercept = 2; y-intercept = 1 / 2

y=1 x = –1

41. Horizontal asymptote: y = -1 / 3; vertical asymptote: x = -2; x@intercept = -4; y@intercept = -2 / 3

2 –2

y 10 y = –2x + 5 x+3 2x

y=–1 3

–2 –x–4 y = 3x + 6

y = 3 – 2x 4x + 20

43. No asymptotes; hole at x = -4; x@intercept = -3; y - intercept = 3

44. No asymptotes; hole at x = 3; no x-intercept; y-intercept = 3 y

y

3

3 –3

0

x 2 y = x + 7x + 12 x+4

y = 9 – 6x + x 3–x

2

3

x

x

4

42. Horizontal asymptote: y = -2; vertical asymptote: x = -3; x-intercept = 5 / 2; y-intercept = 5 / 3

2

0 2x

0

–4

x = –2 y

y = –1 2

0

y= x–4 x+1

x

4 x+1 y= x–4

y x = –5

y y=1

x

3

38. Horizontal asymptote: y = 1; vertical asymptote: x = -1; x-intercept = 4; y-intercept = -4

x=4

1 0

x=3

2

37. Horizontal asymptote: y = 1; vertical asymptote: x = 4; x@intercept = -1; y@intercept = -1 / 4

x = –3

8

x

y = –2

45. One possible answer is y = 2x / 1x - 12.

46. One possible answer is y = 3 / 1x + 22 47. (a) 0 (b) 2, -3 (d) 1x + 121x - 121x + 22 (e) 31x + 121x - 121x + 22 (f) 1x - a2 48. (a) Two; one at x = -1.4 and one at x = 1.4 (b) Three; one at x = -1.414, one at x = 1.414, and one at x = 1.442 49. (a) Two; one at x = -1.4 and one at x = 1.4 (b) Three; one at x = -1.414, one at x = 1.414, and one at x = 1.442

Copyright © 2022 Pearson Education, Inc.

A-21

A-22

Complete Instructor Answers

50. (a) C1x2 = 1200 + 0.85x2/ x (b) $0.85 (c) y

51. (a) C1x2 = 190 + 0.68x2/ x (b) $0.68 (c) y

50

50

40 30 20 10 0

40 30 20 10 0

0 10 20 30 40 50

x

52. (a) $20; $15; $8.57; $6.32; $5 (c) 10, ∞2; it is not reasonable to discuss the average cost per unit of zero units. (c) Vertical asymptote at x = -20, a horizontal asymptote at y = 0 (the x-axis), and y-intercept = 30 y (d) x = –20

0 10 20 30 40 50

– C(x) = 600 x + 20

x 30

–30

53. (a) $440; $419; $383; $326; $284; $251 (b) Vertical asymptote at x = -475; horizontal asymptote at y = 0 (c) y = 463.2 y (d)

55. f11x2 = x1100 - x2/ 25, f21x2 = x1100 - x2/ 10, ƒ1x2 = x 21100 - x22 / 250

54. (a) $54 billion (b) $504 billion (c) $750 billion (d) $1104 billion (e)

y

25,000

C(x) = 220,000 x + 475

x = –475

5000 10

200

56. (a)

y5 30

x

200

x

58. (a) $0; $6250; $23,600; $48,800; $88,200; $214,500; $325,000 (b)

57. (a) $6700; $15,600; $26,800; $60,300; $127,300; $328,300; $663,300 (b) No (c) y

300x 2 3x 2 5x 1 100

0

100

2 2 f(x) = x (100 – x) 250

Cost (in thousands of dollars)

0

x

0

100 0

(b) 29.0%; $25.2 million

20

(c)

59. (a) 1100

y = 6.7x 100 – x x = 100

7 2 0

25 50 75 100 x Percent removed

y 5 –0.1743t2 1 19.95t 1 336.0 1100

(d) y = -0.01792t 3 + 1.652t 2 - 26.35t + 504.8 (e) y 5 –0.01792t3 1 1.652t2 2 26.35t 1 504.8 1100

0

70 0

(b) y = -0.1743t 2 + 19.95t + 336.0

0

70 0

0

70 0

60. (a)

g(t) 5 20.006t 4 1 0.140t 3 2 0.053t 2 1 1.79t

61. (a)

A(x) 5 0.003631x 3 2 0.03746x 2 1 0.1012x 1 0.009

20

0

6 0

62. (a)

0.1

0

60,000

5

0 30,000

0

(b) Close to 2 hours (c) About 1.1 to 2.7 hours

A(t) 5 0.1953t4 2 13.70t3 1 282.9t2 2 836.7t 1 35,842

(b) 2002

Copyright © 2022 Pearson Education, Inc.

(c) 2008

20

Complete Instructor Answers

(d) The population of the next generation, ƒ1x2, gets smaller when the current generation, x, is larger.

63. (a) 30, ∞2

(b)

(c)

64. (a) 30, ∞2

(b) 6y

65. (a) 220 g; 602.5 g; 1220 g (b) c 6 19.68 (c) 1500 c3 m(c) 5

2 0

1200 f(x) = 5x x+2

2

12

A-23

100

2

66. (a) k = 337 (b) 8.32 (using k = 337)

c

x

(d) Maximum growth rate 20

50 0

(d) 41.9 cm (b) y = -9.5037t 2 + 12,771t + 383,087

67. (a) 1,000,000

(c) y = -30.486t 3 + 2224.2t 2 - 28,111t + 472,959 y 5 –30.486t 3 1 2224.2t2 2 28,111t 1 472,959

y 5 –9.5037t 2 1 12,771t 1 383,087

1,000,000

1,000,000

0

50 0 0

(d) 1,057,000; 583,000 (e) cubic 68. (a) 1.80; 0.812; 0.366 (b) L 5 1.80T

0

50

50 0

0

(c) 2.48 sec (d) The period increases by a factor of 22. (e) L = 0.822T 2, which is very close to the function found in part (b).

L 5 0.812T 2 4

0

2.5 0

L 5 0.366T 3

Exercises 2.4 (page 97–101)

For exercises . . .

Refer to example . . .

7–15,33–36 1,2

17–32 3

41–50 4,5

52,53,55,56 6

51,54,57–60 7

W1. 2-6x W2. 335x - 15 W3. 53x + 1 W4. x a - b 1. False 2. True 3. False 4. True 5. E 6. D 7. C 8. F 9. F 10. D 11. A 12. B 13. C 15. 2, 4, 8, 16, 32, c, 1024; 1.125899907 * 10 15 16. 71,079,539.57 mi 17. 5 18. 3 19. -4 20. -5 21. -3 22. -6 23. 21 / 4 24. -1 25. -12 / 5 26. 3, -3 27. 2, -2 28. -4, 4 29. 0, -1 30. 4 / 3, -1 31. 0, 1 / 2 32. -3, -2 y y y y 33.

34.

35.

36.

40

2

30

–2

–1

20 10 –1 –10

40.

3

0.1

y=2 1 2 x y = 5ex + 2

f(x) 5 (1 1 1/x)x

50 0

1

2 x y = –3

10

y= 2 –2

8 –1

2

y = –3e–2x + 2

–12 –18

1

x

y = –2ex – 3

–20

4 –2 y=–1 –4

2 4 6 y = 4e2x/2 – 1

x

41. (a) $2166.53 (b) $2189.94 (c) $2201.90 (d) $2209.97 (e) $2214.03 42. (a) $6824.40 (b) $6936.02 (c) $6993.62 (d) $7032.72 (e) $7052.48 43. She should choose the 5.9% investment, which would yield $23.74 additional interest.

44. (a) 4.06% (b) 4.02% 45. (a) $10.94 (b) $11.27 (c) $11.62 46. (a) $26,413.52 (b) $32,913.27 (c) $43,331.33 47. 5.22% 48. (a) 10.16% (b) 10.08% 49. (a) $209,162 (b) $56.29 50. a

e ≈ 2.71828 Copyright © 2022 Pearson Education, Inc.

A-24

Complete Instructor Answers

(d)

51. (a) 22,000

52. (a) y1t2 = 10,000122t/5 (b) y1t2 = 10,00011.1492t (c) 10 hours (d) 279,000 53. (a) The function gives a population of about 3729 million, which is close to the actual population. (b) 7592 million (c) 8892 million

2029

y 5 42,858 45,000

0

f(t) 5 103(1.0699)t

80 0

0 100 Exponentially 0 (b) ƒ1t2 = 10311.06992t (c) 6.99% 54. (a) y Yes (b) ƒ1t2 = 0.88111.0322t (c) 1.76, 3.64. These are fairly close to the values in the data.

10 (d) Y = 0.72311.0342t. This is close to the function found in part (b).

8 6 4 2 0

0

20

40

60

55. (a) 59.51 million

80 t

(b) 19.09 million (c) 1.51%, 1.59%; Asian

(e) Hispanic: 2060;

Asian: 2057;

h(t) 5 46.82(1.0151)t

y 5 36.64

140

(d) 43.25 million Black: 2123 y 5 86.00

40

120

y 5 114.94 0

80

0

80 0

0

56. (a) 0.196 g

0

150 0

a(t) 5 14.83(1.0159)t

b(t) 5 0.3985t 1 36.87

(b) 25.4 days

57. (a) P = 101310.9998662t; P = -0.0748x + 1013; P = 1 / 12.79 * 10 -7x + 9.87 * 10 -42 (b) P 5 20.0748x 1 1013

58. (a) y = 807,591t - 3,230,066; y = 15,237.6t 2 - 243,501; y = 105.99211.297072t (b) y 5 807,591t 2 3,230,066

1100

40,000,000 P 5 1013(0.999866)t y 5 15,237.6t2 2 243,501

0

10,000

0

200 P5

50

y 5 105.992(1.29707)t

0

1 2.79 3 1027x 1 9.87 3 1024

y = 105.99211.297072t is the best fit. (c) 173,086,000 (d) Y = 53.717911.312612t

P = 101310.9998662t is the best fit. (c) 829 millibars, 232 millibars (d) P = 103810.999986612x.

This is very close to the function found in part (b).

59. (a) C = 49,941.25t - 302,386.5; 60. (a) C = 1783.62t 2 + 18,664.4; C = 49,330.311.148532t (b) C 5 49,941.25t 2 302,386.5

(b) ƒ1t2 = 316.1911.00448632t

450

(c) 0.45%

600,000 0

C 5 1783.62t 2 1 18,664.4

y 5 632.38

(d)

C 5 49,330.3(1.14853)t 0 150,000

60 300

20

Quadratic (c) C = 1402.75t 2 + 11,578.1t - 60,590; C = 51,815.611.149062t (d) 646,497; 662,551; 685,201; 665,787; 726,059

2115

700

f(t) 5 316.19(1.0044863)t

0

200 300

(e) ƒ1t2 = 311.5611.00452672t; this is close to the function found in part (b).

61. y = 1001102x

Exercises 2.5 (page 110–114) For exercises . . .

5–16

17–28

31–40

Refer to example . . .

1

2

3

41–44, 94,96, 98,100 4

45–60

61–68

69–72

5

6

7

73–74

79–81,83, 84,93, 94(d) first CAUTION 8

Copyright © 2022 Pearson Education, Inc.

79(d), 80(c),82

87–89

9

10

A-25

Complete Instructor Answers

W1. 3, -2 W2. 20 / 3 W3. -2 / 3 W4. 4 1. True 2. False 3. False 4. False 5. log5125 = 3 6. log7 49 = 2 7. log381 = 4 8. log2128 = 7 9. log3 11 / 92 = -2 10. log5/4116 / 252 = -2 11. 25 = 32 12. 34 = 81 13. e -1 = 1 / e 14. 2-3 = 81 15. 10 5 = 100,000 16. 10 -3 = 0.001 17. 2 18. 2 19. 3 20. 3 21. -4 22. -4 23. -2 / 3 24. -1 / 12 25. 1 26. 3 27. 5 / 3 28. 0 29. log34 31. log53 + log5k 32. log9 4 + log9 m 33. 1 + log3p - log35 - log3k 34. log715 + log7 p - 1 - log7 y 35. ln 3 + 11 / 22 ln 5 - 11 / 32 ln 6 36. ln 9 + 11 / 32 ln 5 - 11 / 42ln 3 37. 5a 38. a + 2c 39. 2c + 3a + 1 40. 2c + 2 41. 2.113 42. 2.152 43. -0.281 44. -2.059 45. x = 1 / 6 46. m = 3 / 2 47. z = 4 / 3 48. y = 16 49. r = 25 50. x = 3 51. x = 1 52. x = 1 53. No solution 54. x = 0 55. x = 3 56. x = 1, 2 57. x = 5 58. x = 1e 2 - 42/ 5 ≈ 0.6778 59. x = 1 / 23e ≈ 0.3502 60. x = 1 / 1e - 12 ≈ 0.5820 61. x = 1ln62/ 1ln22 ≈ 2.5850 62. x = 1ln 122/ 1ln 52 ≈ 1.5440 63. k = 1 + ln 6 ≈ 2.7918 64. y = 1ln 152/ 2 ≈ 1.3540 65. x = 1ln32/ 1ln15 / 322 ≈ 2.1507 66. x = 1ln 122/ 1ln 32 ≈ 2.2619 67. x = 1ln1.252/ 1ln1.22 ≈ 1.224 68. x = 1ln 0.752/ 1ln 11.01 / 1.0522 ≈ 7.4069 2 69. e 1ln1021x + 12 70. e 1ln 102x 71. 20.09x 72. 0.0183x 73. x 6 5 74. x 6 -3 or x 7 3 79. (a) 23.4 yr (b) 11.9 yr (c) 9.0 yr (d) 23.3 yr; 12 yr; 9 yr 80. (a) 11 yr (b) 17 yr (c) 11 yr 81. 5.19% 82.

r

0.001

0.02

0.05

0.08

0.12

ln 2 / In11 + r2

693.5

35

14.2

9.01

6.12

70 / 1100r2

700

35

14

8.75

5.83

70 / 1100r2

720

36

14.4

9

6

For 0.001 … r 6 0.05, the rule of 70 is more accurate. For 0.05 … r … 0.12, the rule of 72 is more accurate. At r = 0.05, the two are equally accurate.

83. 2030 84. 2043 85. (b) 22: 0.0161; 35: 0.0214; 45: 0.0255; 55: 0.0298; 65: 0.0317; 75: 0.0317 86. -0.6 87. (a) About 0.693 (b) ln 2 (c) Yes 88. 1.101 89. (a) About 1.099 (b) About 1.386 91. About every 7 hr, T = 13 ln52/ ln2 92. (a) 1.3 cm3 / g / hr / , 0.41 cm3 / g / hr (c) y = 2.3x -0.25 (d) 0.73 cm3 / g / hr 93. (a) 2060 (b) 2057 94. (a) 84.6 (b) 51.9 (c) 55.6 (d) $35,600 (e) $410 (f) $16,310 95. s / n = 2C/B - 1 96. (a) 21 (b) 70 (c) 91 (d) 120 (e) 140 (f) 240,000 microbars; 179,000,000 microbars 97. No; 1 / 10 98. (a) 6 (b) 8 (c) 12,600,000I0 (d) 7,940,000I0 (e) 1.6 times (f) The 2017 earthquake had an energy about twice that of the 2018 earthquake. 99. (a) 1000 times greater (b) 1,000,000 times greater 100. (a) 69 (b) 81

Exercises 2.6 (page 120–123)

For exercises . . .

10,32(b), 33(b) 3

Refer to example . . .

11–14, 19–21, 25(c) 4

15–18, 22–24 6

23(d), 24(d) 7

27,28 8

29–32, 34,35,46 1

25(d),(e), 26,33,45 5

36–44 2

W1. ln152/ 2 ≈ 0.8047 W2. ln132 - 2 ≈ -0.9014 W3. ln14.52/ 3 ≈ 0.5014 1. True 2. True 3. True 4. True 6. Initial quantity; rate of growth or decay 8. The time period for the quantity to decay to one-half of the initial amount 11. 4.06% 12. 6.17% 13. 8.33% 14. 5.13% 15. $6209.93 16. $42,769.89 17. $6283.17 18. $17,302.93 19. 9.20% 20. 6.35% 21. 6.17% 22. $15,431.86 23. (a) $257,107.67 (b) $49,892.33 (c) $68,189.54 (d) 14.28% 24. (a) $29,119.63 (b) $10,880.37 (c) $12,527.12 (d) 13.86% 25. (a) The 8% investment compounded quarterly (b) $759.26 (c) 8.24% and 8.06% (d) 3.71 yr (e) 3.75 yr 26. (a) 12.75 yr (b) 12.52 yr 27. (a) 200 (b) About 1 / 2 year (c) No (d) Yes; 1000 28. (a) $1,000,000 (b) About 2 years (a) $5,000,000 29. (a) P1t2 = 0.01428e 0.006540t (b) 14,374 (c) No; it is too small. Exponential growth does not accurately describe population growth for the world over a long period of time. 30. (a) y = y0e 0.05776t (b) y = y02t/12 (c) 1,048,576; 1,073,741,824 31. (a) y = 20,000122t/17 (b) y = 20,00011.04162t (c) y = 20,000e 0.0408t (d) About 125,000 (e) About 56 minutes (e) 670 thousand, 824 thousand; neither 32. (a) y = 1797e 0.3947t (b) y = 117611.54782t (c) y 5 1797e0.3947t 30,000

y 5 1176(1.5478)t

0

8 0

33. (a) 17.9 days (b) January 17 35. (a) 2, 5, 24, and 125 (b) 0.061 (c) 0.24 (d) No, the values of k are different. (e) Between 3 and 4 36. About 4100 years old 37. About 5.29 years 38. About 10.8 years 39. (a) 0.0325 g (b) 76.6 years 40. (a) 3.8 g (b) About 8600 years 41. (a) y = 500e -0.0863t (b) y = 5001386 / 5002t/3 (c) 8.0 days 42. (a) y = 25.0e -0.00497t (b) y = 2510.782t/50 (c) 139 days 43. (a) 19.5 watts (b) 173 days 44. 0.5% 45. (a) 67% (b) 37% (c) 23 days (d) 46 days 46. (a) y = 10e 0.0095t (b) 42.7°C 47. 18.02° 48. About 30 minutes 49. 1 hour

Chapter 2 Review Exercises (page 126–131) For exercises . . .

1,2,6,20,21,35–42, 87,110,115,122

3,4,31–34,105,109

Refer to section . . .

3

2

5,9,43–46,51–54,83, 88–93,101,108,111, 112,114,116–118,120 4

7,10–17,22,29,30, 47–50,55–82,84, 94–95,121 5

Copyright © 2022 Pearson Education, Inc.

8,23–28,86

18,96–100,102–104

1

6

A-26

Complete Instructor Answers

1. True 2. False 3. True 4. True 5. False 6. False 7. True 8. False 9. False 10. False 11. False 12. False 13. False 14. True 15. False 16. False 17. True 18. True 24. 1-3, -3 / 102, 1-2, -2 / 52, 1-1, -1 / 22, 23. 1-3, 142, 1-2, 52, 1-1, 02, 25. (a) 17 (b) 4 (c) 5k 2 - 3 10, 02, 11, 1 / 22, 12, 2 / 52, 13, 3 / 102; range: 10, -12, 11, 22, 12, 92, 13, 202; (d) -9m2 + 12m + 1 5-1 / 2, -2 / 5, -3 / 10, 0, 3 / 10, 2 / 5, 1 / 26 range: 5-1, 0, 2, 5, 9, 14, 206 (e) 5x 2 + 10xh + 5h2 - 3 y y (f) -x 2 - 2xh - h2 + 4x + 4h + 1 1 (g) 10x + 5h (h) -2x - h + 4 15 0.5 26. (a) 23 (b) 19 (c) 18m2 + 5 (d) 3k 2 - 4k - 1 10 x –1 (e) 2x 2 + 4xh + 2h2 + 5 5 (f) 3x 2 + 6xh + 3h2 + 4x + 4h - 1 –1 (g) 4x + 2h (h) 6x + 3h + 4 3x 27. 1-∞, 02 ∪ 10, ∞2 y 31.

28. 32, ∞2 32.

29. 1-7, ∞2

30. 1-∞, -42 ∪ 14, ∞2 y 33.

2 y = –x + 4x + 2

4

4 0

0

36.

0

y

2

x

4

–4 –2 0

0

–1 –2 –3 x = – 1– 3

3

39.

43.

–2

4 f (x) =

–8 y

44.

8 4

y 3

48.

y = log 2 (x – 1)

0

3

–3 x=1

6

0

y = 4x

x

2 3x – 6

2x – 3

50 25

y=3 4

) )

y = 1– 5

x

100 75

y = 4–x + 3

47.

y

45.

125

–4

46.

0

8 x

y

3 x

2

y

2

–8

f(x)

y = 4– 3

3– 3

x=2

f(x) = 8– x

f(x) = 4x – 2 3x + 1

1

40.

f(x) 8

x

x

42.

f (x) = x

–4

y = –(x + 2) 3 – 2 –3

x

2

y = 3x 2 – 9x + 2

y

38. y = –(x – 1) 4 + 4

4

0

0

–2

37.

f(x) 3 2

x

4

x

2

y = 2x 2 + 3x – 1

41.

y

35.

5

2 –2

y

34.

0

49.

–1

x

2

0 x = –3

1

2 x

50.

y

9x

0

2

x

y = –ln(x + 3)

Cost (in thousands)

Cost (in dollars)

51. -5 52. 1/2 53. -6 54. 1/4 55. log3243 = 5 56. log5 25 = 1 / 2 57. ln 2.22554 ≈ 0.8 58. log 12 ≈ 1.07918 59. 25 = 32 60. 91/2 = 3 61. e 4.41763 ≈ 82.9 62. 10 0.50651 ≈ 3.21 63. 4 64. 4/5 65. 3 / 2 66. 3/2 67. log5 121k 42 68. log31y / 42 69. log3 1y 4 / x 22 70. log41r 42 71. p ≈ 1.581 72. z ≈ 4.183 73. m ≈ -1.807 74. k ≈ -0.884 75. x ≈ -3.305 76. z ≈ 1.213 77. m ≈ 2.156 78. p ≈ 1.830 or -6.830 79. k = 2 80. x = 119 81. p = 3 / 4 82. m = 14 83. (a) 1-∞, ∞2 (b) 10, ∞2 (c) 1 (d) y = 0 is a horizontal asymptote. (e) greater than 1 (f) between 0 and 1 84. (a) 10, ∞2 (b) 1-∞, ∞2 (c) 1 (d) x = 0 is a vertical asymptote (e) greater than 1 (f) between 0 and 1 86. (a) $100 (b) $100 (c) $100 (d) $160 (e) $220 y 87. (a) $28,000 (b) $7000 88. $4173.68 89. $921.95 90. $13,701.92 (f) 340 (c) $63,000 91. $15,510.79 92. $2574.01 93. $17,901.90 280 y (d) (e) No 94.

12 years; 19 years 95. 70 quarters or 17.5 years; 111 220 y = 7x 100 – x 160 quarters or 27.75 years 96. 7.19% 97. 6.17% 98. 5.13% 100 99. $1494.52 100. $6245.97 101. $17,339.86 20 0 102. 3.7 years 103. About 9.59% 104. $20,891.12 1 2 3 4 5 x Days 105. (a) n = 1500 - 10p (b) R = p11500 - 10p2 (g) The number of days, or x 0 x (c) 50 … p … 150 (d) R = 11500n - n22/ 10 50 100 Percent removed (h) The cost, or C(x) Copyright © 2022 Pearson Education, Inc.

(e) 0 … n … 1000 (f) $75 (g) 750 (h) $56,250 y (i) R = p(1500 – 10p)

107. (a)

106. (a)

30000

0

20

x

60

(j) The revenue starts at $50,000 when the price is $50, rises to a maximum of $56,250 when the price is $75, and falls to 0 when the price is $150.

108. (a) ƒ1t2 = 29.611.03762t (b) ƒ1t2 = 32.211.03962t (c) f(t) 5 32.2(1.0396)t

(e)

f(t) 5 0.0209t2 1 2.84t 1 21.1 700

700 f(t) 5 32.2(1.0396)t 0 f(t) 5 29.6(1.0376)t 0

5

C(x) = x 2 + 4x + 7 3 5 x Hundreds of nails

60

109. The third day; 104.2°F 110. (a) 1/8. The fraction of radiation let in is 1 over the SPF rating. (b) y 80 60

0

40

f(t) 5 20.00119t3 1 0.124t2 1 0.624t 1 27.3

20

60

0

0

The two graphs are somewhat close to each other. (d) ƒ1t2 = 0.0209t 2 + 2.84t + 21.1; ƒ1t2 = -0.00119t 3 + 0.124t 2 + 0.624t + 27.3 (c) 111. (a) 60

y 30

(b) 2x + 5 (c) A1x2 = x + 4 + 7 / x (d) 1 - 7 / 3x1x + 124

(b) 2 / 31x + 221x + 124 (c) 15x + 32/ 3x1x + 124 (d) 1-5x - 62/ 3x1x + 121x + 224

10000

Production cost (in hundreds of dollars)

Complete Instructor Answers

0.2 0.4 0.6 0.8 x

(c) UVB = 1 - 1 / SPF (d) 12.5% (e) 3.3% (f) It decreases to 0. y 5 0.0635t 3 2 0.670t2 1 4.34t 1 34.6 60

y 5 0.0919t 2 1 2.05t 1 35.7

112. (a) [0, 36) (b) Decreasing (c) 1.82411 2 0.0125995x 1 0.00013401x 2 y1 5 10 100

0

y 5 2.78t + 34.8;)t

10 30

(b) y = 2.78t + 34.8; y = 0.0919t 2 + 2.05t + 35.7; y = 0.0635t 3 - 0.670t 2 + 4.34t + 34.6; y = 35.511.06282t

y 5 35.5(1.0628)t 0

10 30

0

(d) 68.2, 73.5, 100.0, 73.8

36 0

y2 5 101.72858 2 0.0139928x 1 0.00017646x

2

(d) 10.7 breaths per minute 113. 187.9 cm; 345 kg 114. (a) y = 26,00012t/32 (b) 29 days 115. (a) 26.6 billion; this is more than three times the estimate of 7.795 billion. (b) 37.3 billion; 96.3 billion 116. About 7.7 m 117. 0.25; 0.69 minute 118. (a) When it is first injected; 0.08 g (b) Never (c) It approaches c / a = 0.0769 g 119. (a) S = 21.35 + 104.6 ln A 120. (a) y = 100,000e -0.05t (b) 7.1 years 121. (a) 0 yr (b) 1.85 * 10 9 yr 0.3040 (b) S = 85.49 A (c) As r increases, t increases, but at a slower and slower rate. As r decreases, t decreases at a (c) faster and faster rate.

S 5 21.35 1 104.6 ln A 1200

S 5 85.49A0.3040 0

2000 0

(d) 742.2, 694.7 122. (a) P = 5.48D; P = 1.00D1.5; P = 0.182D2 (b) P 5 5.48D P 5 1.00D1.5 (c) 248.3 yr

(d) P = 1.00D1.5, the same as the function found in part (b).

175

0

32 0

P 5 0.182D 2

P = 1.00D1.5 is the best fit.

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A-28

Complete Instructor Answers

Extended Application: Power Functions (page 131) 1. (a)

Yes (b) Y = 5.065 - 0.3289X

5

(c)

It is decreasing; the exponent is negative. As the price increases the demand decreases.

130

5

0

4

0

0

0

4

25 30

0

(d) y = 158.4x -0.3289 2. (a) 5

Yes

(b) Y = 0.7617X + 1.2874

(c) y = 3.623x 0.7617

5

2

(d) An approximate value for the power is 0.76.

100

5 3

2

5 3

0

100 0

Chapter 3 The Derivative Exercises 3.1 (page 150–154)

For exercises . . .

7,13(b), 14(b), 15(a), 65–66, 92–94

Refer to example . . . 5

8,13(a), 15(a), 16(a), 63–64

9,11(b), 10,77 12(a), 23–26

3

2

4

11(a), 17,18 12(b), 14(a), 15(b), 16(b), 21,22 1 11

27–34

35–38

39–46, 49,50

47,48

51–61, 69–74, 75,76, 78, 85–90, 81–84 95-105

6

7

8

9

12

W1. 12x + 3214x + 52 W2. 13x - 4214x + 32 W3. 13x + 72/ 1x + 22 W4. 12x - 52/ 1x - 32 1. True 2. False 3. True 4. False 5. False 6. True 7. (c) 8. (a) 9. (b) 10. (c) 11. (a) 3 (b) 1 12. (a) 4 (b) 4 13. (a) 0 (b) Does not exist 14. (a) 2 (b) Does not exist 15. (a) (i) -1 (ii) -1 / 2 (iii) Does not exist (iv) Does not exist (b) (i) -1 / 2 (ii) -1 / 2 (iii) -1 / 2 (iv) -1 / 2 16. (a) (i) 1 (ii) 1 (iii) 1 (iv) 2 (b) (i) 0 (ii) 0 (iii) 0 (iv) 0 17. 3 18. ∞ (does not exist) 21. 4 22. 5 23. 10 24. -4 25. Does not exist 26. Does not exist 27. 1 / 3 28. 243 29. 3 30. 2 31. 512 32. 100 33. 2 / 3 34. -137 / 8 35. 11 36. -6 37. -1 38. 3 39. 6 40. -4 41. 3 / 2 42. 6 / 5 43. -5 44. 7 45. -1 / 9 46. 1 / 4 47. 1 / 10 48. 1 / 12 49. 2x 50. 3x 2 51. ∞ (does not exist) 52. -∞ (does not exist) 53. 3 / 7 54. 2 55. 3 / 2 56. 1 / 3 57. 0 58. 0 59. ∞ (does not exist) 60. ∞ (does not exist) 61. -∞ (does not exist) 62. -∞ (does not exist) 63. 1 64. -1 65. (a) 2 (b) Does not exist 66. (a) Does not exist (b) 7 67. -5; 7 68. -3; 9 / 2 69. 6 70. -4 71. 1.5 72. 1.2 73. (a) Does not exist (b) x = -2 (c) If x = a is a vertical asymptote for the graph of ƒ1x2, then xlim ƒ1x2 does not exist. 74. (a) Does not exist (b) x = 4 Sa (c) If x = a is a vertical asymptote for the graph of ƒ1x2, then xlim ƒ1x2 does not exist. 75. (a) 0 (b) y = 0 76. (a) 0 Sa (b) 0 (c) 0 77. (a) -∞ (does not exist) (b) x = 0 78. (a) 0 (b) 0 (c) 0 81. 5 82. 8.25 83. 0.3333 or 1 / 3 84. 2.4 85. (a) 1.5 86. (a) -1.5 87. (a) -2 88. (a) 2 89. (a) 8 90. (a) 8 92. (a) 3 million gallons (b) Does not exist (c) 2 million gallons (d) 16 months 93. (a) 8.25 cents per dollar (b) 7.5 cents per dollar (c) 7.25 cents per dollar (d) Does not exist (e) 7.25 cents per dollar 94. (a) 49 cents (b) 50 cents (c) Does not exist (d) 50 cents 95. $6; the average cost approaches $6 as the number of flashdrives becomes very large. 96. 0.00003964; the average cost approaches $0.00003964 per mile as the number of miles becomes very large. 97. 63 items; the number of items a new employee produces gets closer and closer to 63 as the number of days of training increases. 98. (a) 20 units (b) About 38 units (c) About 59 units (d) 60 units; the number of items a new worker should be able to assemble gets closer and closer to 60 units as the number of months increases. 99. (a) About 18,880 units (b) About 28,167 units (c) About 89,116 units (d) 90,000 units; the number of units sold approaches 90,000 as the number of months becomes large.

100. R / i 101. R / 1i - g2 102. (a) About 65 teeth (b) About 72 teeth 103. (a) 36.2 cm; the depth of the sediment layer deposited below the bottom of the lake in 1970 is 36.2 cm. (b) 155 cm; as we go back in time, the depth of the sediment approaches 155 cm. 104. 0; the concentration of drug in the bloodstream approaches 0 as the number of hours after injection increases. 105. (a) 0.572 (b) 0.526 (c) 0.503 (d) 0.5; the numbers in (a), (b), and (c) give the probability that the legislator will vote yes on the second, fourth, and eighth votes. In (d), as the number of roll calls increases, the probability of a yes vote approaches 0.5 but is never less than 0.5.

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10

Complete Instructor Answers

Exercises 3.2 (page 161–164)

For exercises . . .

5–10,38,39 15–26,43–45 27–30 Refer to example . . . 1 3 2

31–40 4

W1. 3 W2. 10 W3. 4 W4. Does not exist W5. 25 1. True 2. False 3. True 4. False 5. a = -1: (a) ƒ1-12 does not exist. (b) 1 / 2 (c) 1 / 2 (d) 1 / 2 (e) ƒ1-12 does not exist. 6. a = -1: (a) 2 (b) 2 (c) 4 (d) Limit does not exist. (e) Limit does not exist.

7. a = 1 (a) 2 (b) -2 (c) -2 (d) -2 (e) ƒ112 does not equal the limit. 8. a = -2: (a) 1 (b) -1 (c) -1 (d) -1 (e) ƒ1-22 does not equal the limit. 9. a = -5: (a) ƒ1-52 does not exist. (b) ∞ (does not exist) (c) -∞ (does not exist) (d) Limit does not exist. (e) ƒ1-52 does not exist and the limit does not exist; a = 0: (a) ƒ102 does not exist. (b) 0 (c) 0 (d) 0 (e) ƒ102 does not exist. 10. a = 0: (a) ƒ102 does not exist. (b) -∞ (does not exist) (c) -∞ (does not exist) (d) -∞ (does not exist) (e) ƒ102 does not exist and the limit does not exist; a = 2: (a) ƒ122does not exist. (b) -2 (c) -2 (d) -2 (e) ƒ122 does not exist. 11. a = -1: removable 12. a = -1: nonremovable 13. a = -5: nonremovable; a = 0: removable 14. a = 0: nonremovable; a = 2: removable 15. a = 0, limit does not exist; a = 2, limit does not exist. 16. a = -1 / 2, limit does not exist; a = -2, limit does not exist.

17. a = 2, limit is 4. 18. a = -5, limit is -10. 19. Nowhere 20. Nowhere 21. a = -2, limit does not exist.

22. a = 5, limit does not exist. 23. a 6 1, limit does not exist. 24. a = 0, limit does not exist 25. a = 0, -∞ (limit does not exist); a = 1, ∞ (limit does not exist) 26. a = -2, -∞ (limit does not exist); a = 3, ∞ (limit does not exist) 27. 1-∞, ∞2 28. 1-∞, -42 ∪ 1-4, ∞2 29. 35 / 2, ∞2 30. 1- ∞, 14 y 31. (a) (b) 5; does not exist; discontinuous (c) 6; 6; continuous (d) 7; 7; continuous 8

4

0

–2

2

4

6 x

32. (a)

(b) 0; 0; continuous (c) 0; 0; continuous

(b) -1 (c) 11, 3

y 15

33. (a)

(d) 0; does not exist; discontinuous

34. (a)

(b) 5

(c) 0, 5

10 5 5 x

0

–5

–5 y 10

35. (a)

y 10

(b) None 36. (a)

0

–5 –3

0

(c) -10, 2 37. 2 / 3 38. 16

39. 4

5

5

–6

(b) 1

3

6x

x

–10

–5

Weight (lbs.)

40. -4 43. (a) Discontinuous at x = 1.2 44. (a) Approximately -0.926 45. (a) 47. (a) $500 (b) $1500 (c) $1000 (d) Does not exist (e) Discontinuous at x = 10; a change in shifts (f) 15 48. (a) $620 (b) $700 (c) $730 (d) $1300 (e) $1350 (f) Discontinuous at x = 150 and x = 400 49. (a) $100 (b) $150 (c) $125 (d) Discontinuous at x = 100 50. (a) $36 (b) $36 (c) $30 (d) About $25.71 (e) $27 (f) $36 (g) $30 (h) Discontinuous at 5, 6, 7, 8, 9, 10, 11 51. (a) $1.40 (b) $1.60 (c) Does not exist (d) $1.40 (e) $2.60 (f) $2.60 (g) $2.60 (h) $2.60 (i) 1, 2, 3, . . . , 12 y 53. (a) About 687 g (b) No (c) 3000 52. (a) (b) Discontinuous at t = 40 140

(40, 140)

130 120 0

1 20 40 60 Time (weeks)

80

t

Copyright © 2022 Pearson Education, Inc.

0

56

A-29

47–53 5

A-30

Complete Instructor Answers

Exercises 3.3 (page 174–178)

For exercises . . .

5–12 13–22,37,40,42,48 23–26,32,33,41 29–31 34–36,38,39,43–45 46,47 Refer to example . . . 1 3 6 4 and 5 2 7

W1. 2h2 + 11h + 18 W2. 2h + 11 W3. 2 / 15 + h2 W4. -2 / 3515 + h24 1. True 2. True 3. True 4. False 5. 6 6. -32 7. -15 8. 20 9. 1 / 3 10. 1 11. 11 - e -22/ 2 ≈ 0.4323 12. 1ln 22/ 2 ≈ 0.3466 13. 17 14. 7 15. 18 16. 28 17. 5 18. 50 19. 2 20. -16 21. 2 22. 0 23. 6.773 24. 56.66 25. 1.121 26. 2.449 28. The function is increasing.

29. (a) $700 per item (b) $500 per item (c) $300 per item (d) $1100 per item 30. (a) $5998 per item (b) $6000; this is approximately the revenue generated by the 1000th unit produced. (c) $5998 (d) The answers to parts (a) and (c) are the same and are approximately equal to the answer to part (b). 31. (a) -25 boxes per dollar (b) -20 boxes per dollar (c) -30 boxes per dollar (d) Demand is decreasing. Yes, a higher price usually reduces demand. 32. (a) $31.85 per year (b) $36.93 per year (c) $34.27 per year 33. (a) $32.37 per year (b) $37.60 per year (c) $34.86 per year 34. (a) 0.56% per year (b) -0.78, per year (c) -0.66, per year 35. (a) 42 cents per gallon per year (b) -14.1 cents per gallon per year (c) -1.7 cents per gallon per year 36. (a) $13.8 billion per year (b) -$30.16 billion per year 37. (a) Increase about 334,000 people per year (b) Increase about 321,000 people per year 38. (a) 2010: increase 55 million per year; 2030: increase 65 million per year; 2050: increase 75 million per year (b) 2010: increase 7.5 million per year; 2030: increase 7.5 million per year; 2050: increase 10 million per year 39. (a) Increase 151 cases per year (b) Decrease 182.3 cases per year (c) Increase 135.4 cases per year 40. (a) 0.288 mm per week (b) 0.348 mm per week (c) L(t) 5 20.01t 2 1 0.788t 2 7.048 41. (a) F(t) 5 210.28 1 175.9te2t/1.3 100

9

0 0

50

6

220

0

(b) 81.51 kilojoules per hour per hour (c) 18.81 kilojoules per hour per hour (d) 1.3 hours 42. (a) 0.18 kg per day (b) 0.09 kg per day (c) 43. (a) -15,540 immigrants per year (b) 15,560 immigrants per year M(t) 5 27.5 1 0.3t 2 0.001t 2 65

5

125 25

(c) 10 immigrants per year (d) They are equal; no (e) About 1,167,000 immigrants 44. (a) -0.425 percent per year; -0.075 percent per year; -0.233 percent per year (b) 0.075 percent per year; 0.725 percent per year; 0.067 percent per year (c) 0.15 percent per year; -0.25 percent per year; -0.011 percent per year 45. (a) 10°F per 1000 ft (b) 7.5°F per 1000 ft (c) -6.7°F per 1000 ft (d) 0°F per 1000 ft (e) 5000 ft; 1000 ft; if 7000 ft is changed to 10,000 ft, the lowest temperature would be at 10,000 ft. (f) About 8500 ft 46. (a) 5 ft / sec (b) 2 ft / sec (c) 3 ft / sec (d) 5 ft / sec (e) (i) 2.5 ft / sec (ii) 2.5 ft / sec (f) (i) 4 ft / sec (ii) 4 ft / sec 47. (a) 50 mph (b) 44 mph 48. (a) 15 ft / sec (b) 14 ft / sec (c) 13 ft / sec

Exercises 3.4 (page 192–197)

For exercises . . .

5–10,60, 11–14 61,64–66 Refer to example . . . 3 4,5

15,16

17,18

19,20

21–26

27–34, 35–38 46–49,59 2 11

W1. 6x + 3h - 2 7 8 6 1 and 10 W2. -3 / 31x + h - 221x - 224 W3. y = - 15 / 62x + 7 / 3 W4. y = - 13 / 82x + 17 / 4 1. True 2. True 3. False 4. False 5. 2 6. -1 7. 1 / 4 8. -4 / 5 9. 0 10. Undefined 11. ƒ′1x2 = 3; 3; 3; 3 12. ƒ′1x2 - 2; -2; -2; -2 13. ƒ′1x2 = -8x + 9; 25; 9; -15 14. ƒ′1x2 = 12x - 5; -29; -5; 31 15. ƒ′1x2 = -12 / x 2; -3; does not exist; -4 / 3 16. ƒ′1x2 = -3 / x 2; -3 / 4; does not exist; -1 / 3 17. ƒ′1x2 = 1 / 12 2x2; does not exist; does not exist; 1 / 12 232 18. ƒ′1x2 = -3 / 12 2x2; does not exist; does not exist; -3 / 12 232 19. ƒ′1x2 = 6x 2; 24; 0; 54 20. ƒ′1x2 = 12x 2; 4; 0; 108 21. (a) y = 10x - 15 (b) y = 8x - 9 22. (a) y = -2x + 3 (b) y = 2x + 7 23. (a) y = - 11 / 22x + 7 / 2 (b) y = - 15 / 42x + 5 24. (a) y = 11 / 42x - 7 / 4 (b) y = 13 / 42x - 9 / 4 25. (a) y = 14 / 72x + 48 / 7 (b) y = 12 / 32x + 6 26. (a) y = 11 / 112x + 30 / 11 (b) y = 11 / 102x + 5 / 2 27. -5; -117; 35 28. 20; 188; -40 29. 7.3891; 8,886,112; 0.0498 30. 0.5; 0.0625; -0.3333 31. 0.5; 0.0078; 0.2222 32. -1.5; -0.0234; -0.6667 33. 0.3536; 0.125; does not exist 34. -1.0607; -0.375 does not exist 35. x = 0 36. x = -6 37. x = -3; x = -1; x = 0; x = 2; x = 3; x = 5 38. x = -5; x = -3; x = 0; x = 2; x = 4 39. (a) Does not exist 40. At x = -2 41. (a) 0 (b) 1 (c) -1 (d) Does not exist (e) m 42. The tangent line is horizontal at x = a. 43. (a) 1a, 02 and 1b, c2 (b) 10, b2 (c) x = 0 and x = b 44. (a) Distance (b) Velocity 45. (a) Distance (b) Velocity 46. (a) and (b) 6.773 47. (a) and (b) 56.66 48. (a) and (b) 0.1085 49. (a) and (b) -0.0158 50. The derivative of ƒ1x2 = ln x is ƒ′1x2 = 1 / x. The derivative of ƒ1x2 = e x is ƒ′1x2 = e x. The derivative of ƒ1x2 = x 3 is ƒ′1x2 = 3x 2. 52. (a) -13; -13.4; -13 (b) -13; -13.04; -13 (c) 2 / 9 ≈ 0.222222; 0.215054; 0.222469 (d) 2 / 9 ≈ 0.222222; 0.221484; 0.222225 (e) 1 / 12 232 ≈ 0.288675; 0.286309; 0.288715 Copyright © 2022 Pearson Education, Inc.

53–58, 62,63 9

Complete Instructor Answers

A-31

(f) 1 / 12 232 ≈ 0.288675; 0.288435; 0.288676 53. (a) D′1p2 = -4p - 4 (b) -44; demand is decreasing at a rate of about 44 items for each increase in price of $1. 54. (a) P′182 = 0; no, do not increase expenditures since the marginal profit is $0. (b) P′162 = 8; yes, increase expenditures since the marginal profit is $8000. (c) P′1122 = -16; no, do not increase expenditures since the marginal profit is -$16,000. (d) P′1202 = -48; no, do not increase expenditures since the marginal profit is -$48,000.

55. (a) R′1x2 = 20 - x / 250 (b) $16 per table (c) $15.998 (or $16) (d) The marginal revenue gives a good approximation of the actual revenue from the sale of the 1001st table. 56. (a) C′1x2 = -600 / 1x + 2022 (b) -0.12; the average cost is decreasing by $0.12 per unit of yogurt. (c) -0.04; the average cost is decreasing by $0.04 per unit of yogurt.

57. (a) ƒ′1x2 = 1908 / 1106 - x22 (b) 1.985; the cost is increasing by $1985 for every additional percent of pollution removed. (c) 7.453; the cost is increasing by $7453 for every additional percent of pollution removed. (d) 15.769; the cost is increasing by $15,769 for every additional percent of pollution removed. 58. (a) C′1x2 = -0.0075x + 1.5, for 0 … x … 180 (b) $0.75 (c) $0.74625 (or $0.75) (d) The marginal cost gives a good approximation of the actual cost of the 101st taco.

59. Answers are in trillion of dollars. (a) 2.54; 2.03; 1.02 (b) 0.061; -0.019; -0.079; -0.120; -0.133 60. (a) 0; the power expenditure is not changing. (b) 0.54; the power expended is increasing 0.54 unit per unit increase in speed. (c) The power level first decreases to Vmp, then increases at greater rates. (d) Vmr is the speed which produces the smallest slope of the line.

61. At (2, 4000): 1000; the population is increasing at a rate of 1000 shellfish per time unit. At (10, 10,300): 570; the population is increasing more slowly at 570 shellfish per time unit. At (13, 11,250): 200; the population is increasing at a much slower rate of 200 shellfish per time unit. 62. (a) 57; the rate of change of intake of food 5 minutes into a meal is 57 grams per minute. (c) 0 … t … 24 63. (a) 1690 m per sec (b) 4.84 days per m per sec; an increase in velocity from 1700 m per sec to 1701 m per sec indicates an approximate increase in the age of the cheese of 4.84 days. 64. (a) About 270; the temperature was increasing at a rate of about 270° per hour at 9:00 a.m. (b) About -150°; the temperature was decreasing at a rate of about 150° per hour at 11:30 a.m. (c) About 0; the temperature staying constant at 12:30 p.m. (d) About 11:15 a.m. 65. (a) 16-ounce bat: 0 mph per oz; 25-ounce bat: about -0.6 mph per oz (b) 16 oz 66. (a) 40-oz bat: 0 mph per oz; 30-oz bat: about 0.5 mph per oz (b) 40 oz

Exercises 3.5 (page 202–206) W1. 2 W2. -5 1. True 2. True 3. False 4. True f'(x) (c) 1-∞, 12, 12, ∞2 (d) 11, 22 (e)

For exercises . . .

Refer to example . . .

7. (a) x = 2 (b) x = 1

11–14 5

15–32 1,2,3, and 4

(b) x = -2, 2 (c) 1-∞, -22, 12, ∞2

8. (a) Nowhere

2 –3

3

x

–2

(d) 1-2, 22 (e)

(c) 1-1, 02, 11, ∞2 (d) 1-∞, -12, 10, 12

9. (a) x = -1, 0, 1 (b) Nowhere

f'(x) 2 –3

3

x

–2

(e)

10. (a) x = 0 (b) x = -2, 2 (c) 1-2, 02, 12, ∞2 (d) 1-∞, -22, 10, 22 (e)

f'(x) 4

f'(x) 4 –4

–4

x

4

4

–4

11. ƒ:Y2; ƒ′:Y1

–4

12. ƒ:Y1; ƒ′:Y2

13. ƒ:Y1; ƒ′:Y2

14. ƒ:Y2; ƒ′:Y1

f'(x)

15.

16.

2 –4

17.

18.

f'(x)

19.

2

–4

2

4

x

–4

2

–2

22.

2 –2

x

23.

f'(x)

24.

2 4

–2

4

–2 f'(x)

–4

x

20.

f '(x)

2

21.

4

–2

x

0

x

–4

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f'(x) –1

0 –2

2

x

x

A-32

(b) 16; the tipping point occurs 16 months after the price began rising. G′1162 is undefined.

0

2 4 6

–0.2 –0.3 –0.4 –0.5

20 t

14

(b)

2.0 1.0 6 10

–1.0

20

–2.0 –3.0

Rate of change of BMI

Rate of Change of Consumption

26. (a)

Rate of change of BMI

25. (a)

Complete Instructor Answers

27. (a) 1.0

1000 6 10

–1.0

20

800 600 400

Age (years)

Age (years)

200 0

28. (a)

(b) y′1t2 approaches 0 as time goes to infinity. The change in the population size approaches zero, so the population approaches a constant value.

y9(t) 1200

2 4 6 8 10 12 1416 18 20 t

(b) At Vmp the tangent line has slope 0, so P′1Vmp2 = 0. Vmp represents the minimum amount of power expended by flight muscles for a bird to achieve.

P'(V) 50 2

8 12 16 V

11

12

–50 –100

–1

13 14 15 16 17 Skeletal age (years)

18

19

20

30. (a)

4

–2 –3 –4

2

0

600 400 200 0

3/7/19 3/12/19 3/17/19 3/22/19 3/27/19 3/31/19

–200

Rate of change of transformed scores

32. (a)

800 Rate of change of discharge

3

1

About 9 cm; about 2.6 cm less per year 31.

(b) Growth is the slowest when the boy is close to 18 years of age since the derivative is the smallest there.

5

kg/yr.

Rate of change of growth

29.

Day

Chapter 3 Review Exercises (page 207–211)

12 16 8 Age (years)

(b) Around a student test score of 50.

0.6 0.5 0.4 0.3 0.2 0.1 0

–400

4

20

40 60 80 100 Student scores

For exercises . . .

1–3,17–34,45–46, 4–6,15,35–44, 7–9,47–50, 10–12,16,51–58, 61,66 63,74 63–65 62,68–69,73 Refer to section . . . 1 2 3 4

1. True 2. True 3. True 4. False 5. True 6. False 7. False 8. True 9. True 10. True 11. False 12. False 17. (a) 4 (b) 4 (c) 4 (d) 4 18. (a) -2 (b) 2 (c) Does not exist (d) -2 19. (a) ∞ (does not exist) (b) -∞ (does not exist) (c) Does not exist (d) Does not exist 20. (a) 1 (b) 1 (c) 1 (d) Does not exist 21. ∞ (does not exist) 22. -3 23. 19 / 9 24. Does not exist 25. 8 26. 7 27. -13 28. 16 29. 1 / 6 30. 1 / 8 31. 2 / 5 32. 0 33. 3 / 8 34. -6 35. Discontinuous at x2 and x4 36. Discontinuous at x1 and x4 37. 0, does not exist, does not exist; -1 / 3, does not exist, does not exist 38. -3, does not exist, does not exist; 1, does not exist, does not exist 39. -5, does not exist, does not exist 40. -3, does not exist, does not exist 41. Continuous everywhere 42. Continuous everywhere f '(x)

43. (a)

(b) 1

(c) 0, 2

44. (a)

(b) 2

(c) 0, 1

45. (a) and (b) 2

2 –4

–2

–2

2

4

x

46. (a) and (b) -13 47. 126; 18 48. -68, -12 49. 9 / 77; 18 / 49 50. -5 / 4, -5 51. (a) y = 13x - 17 (b) y = 7x - 5 52. (a) y = - 12 / 32x + 7 / 3 (b) y = -4x + 4 53. (a) y = -x + 9 (b) y = -3x + 15 54. (a) y = 12 / 52x + 2 (b) y = 11 / 22x + 3 / 2 55. ƒ′1x2 = 8x + 3 56. ƒ′1x2 = 10x - 6 57. (a) and (b) 1.332 58. (a) and (b) 1.121

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59–60,67, 70–72,74 5

Complete Instructor Answers

f'(x)

59.

60.

–4

2

–2

4 x

–2

(b) $187.50

(c) $189

(d)

C(x) 200 Cost (in dollars)

63. (a) $150

62. (a) R′1x2 = 16 - 6x (b) R′1102 = -44; An increase of $100 spend on advertising when advertising expenditures are $1000 will result in the revenue decreasing by $44. (e) Discontinuous at x = $125 (f) $1.50 (g) $1.50 (125, 187.50) (h) $1.35 (i) 1.5; when 100 lb are purchased, an (125, 168.75) additional pound will cost about $1.50 more. (j) 1.35; when 140 lb are purchased, an additional pound will cost about $1.35 more.

61. (e)

2

100

0

50

100 150 200 Weight (in pounds)

x

64. (a) 340 cents ($3.40) (b) 327.5 cents ($3.28) (c) 317.5 cents ($3.18) (d) 315 cents ($3.15) (e) 1015 cents ($10.15) (f) 1515 cents ($15.15) (g) 30, ∞2 (h) No, marginal profit will not decrease (i) P1x2 = 15 + 25x (j) P′1x2 = 25 (k) No, the profit per pound never changes, no matter how many pounds are sold. 65. (b) x = 7.5 (c) The marginal cost equals the average cost at the point where the average cost is smallest. 66. (a) $4395 (b) $4350 (c) Does not exist. (e) x = 29,300 (d) (k) 0.15 for 0 … x … 29,300 (f) A1x2 = e 0.27 - 3561 / x for x 7 29,300 (g) 0.15 (h) 0.14846 (i) Does not exist (j) 0.27

Rate of change of unemployment rate

67.

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 –0.5 –1.0 –1.5 –2.0

2008 2000

2004

68. (a) 0.13; the number of people aged 65 and over with Alzheimer’s disease is increasing at a rate of about 0.13 million per year. (b) 0.34; the number of people aged 65 and over with Alzheimer’s disease is increasing at a rate of about 0.34 million per year. (c) 0.16

2016 2012

69. (a)

about 8.2; about -0.5

100

Rate (m/min)

80 60 40

2 1

–1

6 10

–2

20

Rate of change of Bm2

Rate of change of BMI

(b) 30.8, 5.24 (c) 3 weeks; 500 cases (d) V′1t2 = -2t + 6 (e) 0 (ii) 85 meters per minute (b) 71. (a) (b)

(f) +; -

70. (a) (i) 90 meters per minute

2 1 –1

6 10

20

–2

Age (years)

Age (years)

20 17:35

17:37 17:39 17:41 Time (hours:minutes)

72. The remaining growth is about 14 cm and the rate of change is about -2.75 cm per year Rate of Change of Growth

Skeletal Age in Years

x

73. (a) Nowhere (b) 50, 130, 230, 770 (c)

2 4 6 8 10 12 14 16 –1 –2 –3 –4

y

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A-34

Complete Instructor Answers

Extended Application: A Model for Drugs Administered Intravenously (page 214) 1. (a) A1t2 = 500e -0.077t (b) -35.68 mg / hr; -17.84 mg / hr 3. 19.25 mg/hr 4. t 0 0.5 1 1.5 2

2. (a) A1t2 = 152211 - e -0.23t2 (b) 252.9 mg/hr 2.5

3

3.5

4

4.5

A(t)

500

481.116

462.945

445.46

428.636

412.447

396.87

381.881

367.458

353.579

t

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

A(t)

340.225

327.376

315.011

303.114

291.666

280.65

270.05

259.851

250.037

240.593

t

10

10.5

11

11.5

12

12.5

13

13.5

14

14.5

A(t)

231.507

222.763

214.35

206.254

198.464

190.968

183.756

176.816

170.138

163.712

t

15

15.5

16

16.5

17

17.5

18

18.5

19

19.5

A(t)

157.529

151.579

145.854

140.346

135.045

129.945

125.037

120.314

115.77

111.398

t

20

20.5

21

21.5

22

22.5

23

23.5

24

A(t)

107.191

103.142

99.247

95.498

91.891

88.421

85.081

81.868

78.776

5. t A(t)

0

0.5

1

1.5

2

2.5

3

3.5

0

165.341

312.72

444.089

561.186

665.563

758.601

841.532

5

5.5

6

6.5

7

7.5

t

4

4.5

A(t)

915.454

981.345

t

8

8.5

A(t)

1280.28

6. t

1040.079 1092.432 1139.097 1180.694 1217.771 1250.821 9

9.5

10

1306.539 1329.945 1350.809 1369.406

0

0.5

1

1.5

2

2.5

3

3.5

A(t)

250

250

250

250

250

250

250

250

t

4

4.5

5

5.5

6

6.5

7

7.5

250

250

250

A(t)

250

250

250

250

250

t

8

8.5

9

9.5

10

A(t)

250

250

250

250

250

The results show that the chosen infusion rate has the desired effect of keeping the drug at a constract level of 250 mg in the bloodstream.

Chapter 4 Calculating the Derivative Exercises 4.1 (page 228–233)

For exercises . . .

Refer to example . . .

7–28,33–36, 50,51 1,2,3,5

37–49,52

57,58

59–61

6

7

8,9

63,64,66–68, 70,74–83 10

65,69,71,73 4

W1. y = -2x + 11 W2. y = 12 / 52x + 26 / 5 1. True 2. True 3. False 4. False 5. True 6. True 7. dy / dx = 36x 2 - 16x + 7 8. dy / dx = 24x 2 - 10x - 1 / 12 9. dy / dx = 12x 3 - 18x 2 + 11 / 42x 10. dy / dx = 20x 3 + 27x 2 + 24x - 7 11. ƒ′1x2 = 21x 2.5 - 5x -0.5 or 21x 2.5 - 5 / x 0.5 12. ƒ′1x2 = -3x 0.5 + 6x -0.5 or -3x 0.5 + 6 / x 0.5 13. dy / dx = 4x -1/2 + 19 / 22x -1/4 or 4 / x 1/2 + 9 / 12x 1/42 14. dy / dx = -50x -1/2 - 122 / 32x -1/3 or - 50 / x 1/2 - 22 / 13x 1/32 15. dy / dx = -30x -4 - 20x -5 - 8 or -30 / x 4 - 20 / x 5 - 8 16. dy / dx = -25x -6 + 12x -3 - 13x -2 or -25 / x 6 + 12 / x 3 - 13 / x 2 17. f′1t2 = -7t -2 + 15t -4 or -7 / t 2 + 15 / t 4 18. ƒ′1t2 = -14t -2 - 48t -5 or -14 / t 2 - 48 / t 5 19. dy / dx = -24x -5 + 21x -4 - 3x -2 or -24 / x 5 + 21 / x 4 - 3 / x 2 20. dy / dx = -18x -7 - 5x -6 + 14x -3 or -18 / x 7 - 5 / x 6 + 14 / x 3 21. p′1x2 = 5x -3/2 - 12x -5/2 or 5 / x 3/2 - 12 / x 5/2 22. h′1x2 = 1-1 / 22x -3/2 + 21x -5/2 or -1 / 12x 3/22 + 21 / x 5/2 23. dy / dx = 1-3 / 22x -5/4 or -3 / 12x 5/42 24. dy / dx = 12 / 32x -4/3 or 2 / 13x 4/32 25. ƒ′1x2 = 2x - 5x -2 or 2x - 5 / x 2 26. g′1x2 = 15 / 22x 3/2 - 2x -1/2 or 15 / 22x 3/2 - 2 / x 1/2 or 15 / 22x 3/2 - 2 / 2x 27. g′1x2 = 256x 3 - 192x 2 + 32x 28. h′1x2 = 6x 5 - 12x 3 + 6x 29. (b) 30. (a) 32. (a), (d) 33. - 19 / 22x -3/2 - 3x -5/2 or -9 / 12x 3/22 - 3 / x 5/2 34. -2x -5/4 + 19 / 22x -5/2 or -2 / x 5/4 + 9 / 12x 5/22 35. -25 / 3 36. -39 37. -28; y = -28x + 34 38. -31; y = -31x + 24 39. 25 / 6; y = 125 / 62x - 33 / 2 40. -1 / 16; y = - 11 / 162x + 1 / 4 41. 14 / 9, 20 / 92 42. 1-2, -202, 1-4, -62 43. -5, 2 44. -3, -7 45. 14 ± 2372/ 3 46. 15 ± 272/ 3 47. 1-1 / 2, -19 / 22 48. 14, -592, 1-1, 62 49. 1-2, -242 50. 42 51. 7 52. (a) 2 (b) 1 / 2 (c) y = 11 / 22x + 3 / 2 56. (a) 57.18 (b) -0.88, 0.88 57. (a) C′1x2 = -0.08x + 80

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Complete Instructor Answers

(b) $72; after 100 devices have been produced, the cost to produce one more device is approximately $72. (c) $71.96; the approximate cost of $72 is close to the actual cost. 58. (a) R′1x2 = 48 - 0.6x (b) $36; after 20 book bags have been sold, the revenue from the sale of one more bag is approximately $36. (c) $35.70; the approximate revenue of $36 is close to the actual revenue.

59. (a) R1q2 = 1000 / q + 1000q; R′1q2 = -1000q-2 + 1000 or -1000 / q2 + 1000 (b) $990; after 10 systems have been sold, the revenue from the sale of one more system is approximately $990. (c) P′1q2 = 840 - 0.8q - 1000 / q2 (d) $822; after 10 systems have been sold, the profit from the sale of one more system is approximately $822. 60. (a) R1q2 = 50q - q2 / 100; R′1q2 = 50 - q / 50 (b) $40; after 500 units have been sold, the revenue increases by $40 per unit. (c) $30; after 1000 units have been sold, the revenue increases by $30 per unit. (d) P′1q2 = 70 - 0.08q (e) $30; after 500 units have been sold, the profit increases by $30 per unit. (f) 875 units 61. (a) C′1x2 = 2 (b) R′1x2 = 6 - x / 500 (c) P′1x2 = 4 - x / 500 (d) x = 2000 (e) $4000 63. (a) 31.4.; 48.4. (b) 0.897., postage is increasing by about 0.897.; per year; 0.799., postage is increasing by about 0.799. per year. (c) C1t2 = 0.000247t 3 - 0.0203t 2 + 1.35t + 6.06; 0.798. per year; 1.02. per year 64. M′1t2 = 0.0966t 2 + 3.42t + 18.2 (a) $37.7 billion per year (b) $62.1 billion per year (c) $91.2 billion per year (d) $118.1 billion per year 65. (a) 0.4824 (b) 2.216 66. (a) 450 points (b) 325 points (c) -4; after 10 units of insulin are injected, the blood sugar level is decreasing at a rate of 4 points per unit of insulin. (d) -10; after 25 units of insulin are injected, the blood sugar level is decreasing at a rate of 10 points per unit of insulin. 67. (a) 1232.62 cm3 (b) 948.08 cm3 / year; at 3 years, the horn volume is increasing by 948.08 cm3 per year. 68. (a) 220 g (b) 86 / 3 g / cm; when the circumference of the brain is 30 cm, it is increasing by 86 / 3 g with every centimeter the circumference increases. 69. dv / dl ≈ 5.00l 0.86 70. (a) 18 … x … 44 (b) l′1x2 = 0.2356 - 0.005348x (c) 0.1019 cm / week; at 25 weeks, the left ventricular length of a human fetus is increasing by 0.1019 cm / week. 71. (a) 3 minutes, 58.1 seconds (b) 0.118 sec / m; at 100 meters, the fastest possible time increases by 0.118 second for each additional meter. (c) Yes 72. R = 12 / 32R0 73. (a) 27.5 (b) 23 pounds (c) -175,750 / h3 (d) -0.64; for a 125-lb female with a height of 65 in., the BMI decreases by 0.64 for each additional inch of height. (f) Bm = wm / h2m 74. (a) v1t2 = 22t + 4 (b) 4; 114; 224 75. (a) v1t2 = 36t - 13 (b) -13; 167; 347 76. (a) v1t2 = 12t 2 + 16t + 1 (b) 1; 381; 1361 77. (a) v1t2 = -9t 2 + 8t - 10 (b) -10; -195; -830 78. (a) -32 ft / sec; -64 ft / sec (b) 3 seconds (c) -96 ft / sec 79. (a) 0 ft / sec; -32 ft / sec (b) 2 seconds (c) 64 ft 80. (a) 1.3275 g / cm3 (b) -0.43 g / cm3 per percent; when the level of the Dead Sea decreases to 50% of the current level, the density of the brine is decreasing at the rate of 0.43 g / cm3 per percent. 81. 340 cycles / sec; -680 cycles / sec / m 82. (a) (i) About 22.3 percent; about 2.64 percent per point on the PSAT/NMSQT. (ii) About 95.8 percent; about 0.386 percent per point on the PSAT/NMSQT. 83. (a) 35; 36 (b) dy1 / dx = 4.13; dy2 / dx ≈ 4.32; these values are fairly close. As a dog’s actual age increases from 5 to 6 years, the dog’s human age increases by four years. (c) y = 4x + 16

Exercises 4.2 (page 239–241)

For exercises . . .

5–14,33,37, 38,45,54,56

15–30,34–36, 39–41,46,53, 55,57–64 3

31,32

49–52

W1. ƒ′1x2 = 12x 3 + 12x 2 Refer to example . . .

1,2 4 5 W2. ƒ′1x2 = -6 / x 4 + 3 / 2x -1/3 -3/2 2 W3. ƒ′1x2 = 6x - 2x 1. True 2. False 3. False 4. True 5. dy / dx = 18x - 6x + 4 6. dy / dx = 60x 2 + 30x - 4 7. dy / dx = 8x - 20 8. dy / dx = 98x - 84 9. k′1t2 = 4t 3 - 4t 10. g′1t2 = 36t 3 + 24t 11. dy / dx = 13 / 22x 1/2 + 11 / 22x -1/2 + 2 or 3x 1/2 / 2 + 1 / 12x 1/22 + 2 12. dy / dx = 3x 1/2 - 13 / 22x -1/2 - 2 or 3x 1/2 - 3 / 12x 1/22 - 2 13. p′1y2 = -8y -5 + 15y -6 + 30y -7 14. q′1x2 = -9x -4 + 12x -5 - 24x -7 + 28x -8 15. ƒ′1x2 = 57 / 13x + 1022 16. ƒ′1x2 = 101 / 17x + 322 17. dy / dt = -17 / 14 + t22 18. dy / dt = 2 / 11 - t22 19. dy / dx = 1x 2 - 2x - 12/ 1x - 122 20. dy / dx = 1x 2 + 6x - 122/ 1x + 322 21. ƒ′1t2 = 2t / 1t 2 + 322 22. dy / dx = 1-32x 2 + 10x - 402/ 14x 2 - 522 23. g′1x2 = 14x 2 + 2x - 122/ 1x 2 + 322 24. k′1x2 = 1-7x 2 - 142/ 1x 2 - 222 25. p′1t2 = 3- 2t / 2 - 1 / 12 2t24/ 1t - 122 or 1-t - 12/ 32 2t1t - 1224 26. r′1t2 = 3- 2t + 3 / 12 2t24/ 12t + 322 or 1-2t + 32/ 32 2t12t + 3224 27. dy / dx = 15x 1/2 - 6x -1/22/ 2x or 15x - 62/ 12x 2x2 28. dy / dx = 14x 1/2 + 3x -1/22/ 2x or 14x + 32/ 12x 2x2 29. h′1z2 = 1-z 4.4 + 11z 1.22/ 1z 3.2 + 522 30. g′1y2 = 1-1.1y 2.9 - 2.5y 1.5 + 2.8y 0.42/ 1y 2.5 + 222 31. ƒ′1x2 = 160x 3 + 57x 2 - 24x + 132/ 15x + 422 32. g′1x2 = 1120x 3 - 186x 2 - 56x - 1412/ 16x - 722 33. 77 34. -13 / 16 35. In the first step, the numerator should be 1x 2 - 122 - 12x + 5212x2. 36. In the first step, the denominator, x 6, was omitted. 37. y = 36x - 56 38. y = 14x - 30 39. y = -0.75x + 1.25 40. y = 6x + 8 41. (a) ƒ′1x2 = 17x 3 - 42/ x 5/3 (b) ƒ′1x2 = 7x 4/3 - 4x -5/3 42. The result is the same as applying the rule for differentiating a constant times a function. 45. 0, -1.307, and 1.307 46. -0.828, 4.828 47. (a) 2cx + bc (b) 3cx 2 + 2x1d + bc2 + bd 49. (a) 350; the total cost to produce 50 T-shirts is $350. (b) 2; after 50 T-shirts have been produced, the total cost is increasing by $2 per shirt. (c) 7; when 50 T-shirts are produced, the average cost is $7 per shirt. (d) -0.1; after 50 T-shirts have been produced, the average cost is decreasing by $0.10 per shirt. 50. (a) 5000; the total profit for 100 watches is $5000. (b) 98; after 100 watches have been sold, the total profit is increasing by $98 per watch. (c) 50; when 100 watches have been sold, the average profit is $50 per watch. (d) 0.48; after 100 watches have been sold, the average profit is increasing by $0.48 per watch.

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A-36

Complete Instructor Answers

51. (a) C1x2 = 13x + 22/ 1x 2 + 4x2 (b) C′1x2 = 1-3x 2 - 4x - 82/ 1x 2 + 4x22 (c) 0.2286 hundreds; at 10 units, the average cost is $22.86 per unit. -0.0178 hundreds; at 10 units, the average cost is decreasing by $1.78 per unit. (d) 0.1292 hundreds; at 20 units, the average cost is $12.92 per unit. -0.0056 hundreds; at 20 units, the average cost is decreasing by $0.56 per unit.

52. (a) P1x2 = 15x - 62/ 12x 2 + 3x2 (b) P′1x2 = 1-10x 2 + 24x + 182/ 12x 2 + 3x22 (c) 0.224 tens; at 8 books, the average profit is $2.24 per book. -0.019 tens; at 8 books, the average profit is decreasing by $0.19 per book. (d) 0.139 tens; at 15 books, the average profit is $1.39 per book. -0.008 tens; at 15 books, the average profit is decreasing by $0.08 per book. 53. (a) M′1d2 = 2000d / 13d 2 + 1022 (b) 8.3; the new employee can assemble about 8.3 additional bicycles per day after 2 days of training. 1.4; the new employee can assemble about 1.4 additional bicycles per day after 5 days of training.

56. Revenue is increasing by $700 per month. 57. Increasing at a rate of $0.03552 per gallon per month 58. (a) s′1x2 = m / 1m + nx22 (b) The rate of contraction is about 0.000391 mm per ml. 59. (a) ƒ′1x2 = AK / 1A + x22 (b) K / 14A2 60. (a) dW / dH = 1H 2 - 1.86H - 17.73512/ 1H - 0.9322 (b) 5.24 m (c) Crows apply optimal foraging techniques. 61. (a) 8.57 min (b) 16.36 min (c) 6.12 min2 / kcal; 2.48 min2 / kcal 62. (a) -100 facts per hour (b) -0.01 facts per hour 63. (a) 0.1173 vehicles (b) 2.625 vehicles 64. (a) The 2019 Houston Astros winning percentage was 0.660. W = 0.674 (b) -0.000687 (c) 0.673; W = 0.673 (d) W = 0.659; dW / dH = -0.000636

Exercises 4.3 (page 250–252)

For exercises . . .

5–12

Refer to example . . .

1

11–18, 19–26 27–34, 59,65 49,50, 60,63, 69,71–73 2 3 5,6,7

35–40, 68,70

41–46, 62

64,65

61

66,67

51–56

8

9

10

11

4

6 (in 1st section of chapter)

W1. ƒ′1x2 = 12x - x 42/ 1x 3 + 122 W2. ƒ′1x2 = 13 - 21x 42/ 3 2x1x 4 + 1224 W3. ƒ′1x2 = 11 + 4x -1/32/ 331x 1/3 + 1224 1. True 2. False 3. False 4. False 5. 1767 6. 6919 7. 131 8. 1083 9. 320k 2 + 224k + 39 10. 1000z 2 - 80z + 3 11. 16x + 552/ 8; 13x + 1642/ 4 12. 1-8x - 1152/ 5; 1-8x + 292/ 5 13. 1 / x 2; 1 / x 2 2 14. 2 / 12 - x24; 12x 4 - 22/ x 4 15. 28x 2 - 4; 8x + 10 16. 36x + 72 - 22 2x + 2; 2 29x 2 - 11x + 2 17. e 2x - 2 + 1; e x 18. ln14x 2 - 4x + 52; 2 ln1x 2 + 42 - 1 19. If ƒ1x2 = x 3/5 and g1x2 = 5 - x 2, then y = ƒ 3 g1x24. 20. If ƒ1x2 = x 2/3 and g1x2 = 3x 2 - 7, then y = ƒ 3 g1x24. 21. If ƒ1x2 = - 2x and g1x2 = 13 + 7x, then y = ƒ 3 g1x24. 22. If ƒ1x2 = 2x and g1x2 = 9 - 4x, then y = ƒ 3 g1x24. 23. If ƒ1x2 = e x and g1x2 = 5x - 3, then y = ƒ 3 g1x24. 24. If ƒ1x2 = e x and g1x2 = x 2, then y = ƒ 3 g1x24. 25. If ƒ1x2 = ln x and g1x2 = 4x + 7, then y = ƒ 3 g1x24. 26. If ƒ1x2 = ln x and g1x2 = 1 + x 3, then y = ƒ 3 g1x24.

27. dy / dx = 418x 4 - 5x 2 + 123132x 3 - 10x2 28. dy / dx = 512x 3 + 9x2416x 2 + 92 29. k′1x2 = 288x112x 2 + 52-7 30. ƒ′1x2 = 336x 313x 4 + 22-5 31. s′1t2 = 11215 / 22t 213t 3 - 821/2 32. s′1t2 = 144t 312t 4 + 521/2 33. g′1t2 = -63t 2 / 12 27t 3 - 12 34. ƒ′1t2 = 32t / 24t 2 + 7 35. m′1t2 = -615t 4 - 123185t 4 - 12 36. r′1t2 = 412t 5 + 323142t 5 + 32 37. dy / dx = 3x 213x 4 + 123119x 4 + 64x + 12 38. dy / dx = x1x 2 - 123111x 3 - 3x + 162 39. q′1y2 = 2y1y 2 + 121/419y 2 + 42 40. p′1z2 = 16z + 121/3114z + 12 41. dy / dx = 60x 2 / 12x 3 + 123 42. dy / dz = -30x / 13x 2 - 426 43. r′1t2 = 215t - 623115t 2 + 18t + 402/ 13t 2 + 422 44. p′1t2 = 212t + 32214t 2 - 12t - 32/ 14t 2 - 122 45. dy / dx = 1-18x 2 + 2x + 12/ 12x - 126 46. dy / dx = -2115x 4 + 66x 3 - 2x - 42/ 13x 3 + 225 49. (a) -2 (b) -24 / 7 50. (a) -18 / 7 (b) -5 51. y = 13 / 52x + 16 / 5 52. y = x + 3 53. y = x 54. y = 40x - 72 55. 1, 3 56. -2 27 / 7; 2 27 / 7 59. D1c2 = 1-c 2 + 10c + 12,4752/ 25 60. (a) 11,213.96; when 100 sets are sold, the total revenue is $11,213.96. (b) 148.78, when 100 sets are sold, the total revenue is increasing by about $148.78. (c) 187.29; when 200 sets are sold, the total revenue is increasing by about $187.29. (d) R1x2 = 241x 2 + x22/3 / x; R′1x2 = 81x - 12/ 3x1x 2 + x21/34 (e) 112.14; when 100 sets are sold, the average revenue is $112.14 per set. (f) 0.37; when 100 sets are sold, the average revenue is increasing by about $0.37 per set. (g) 0.23; when 200 sets are sold, the average revenue is increasing by about $0.23 per set. 61. (a) 101.22; at 6%, the balance is increasing by about $101.22 per percentage point. (b) 111.86; at 8%, the balance is increasing by about $111.86 per percentage point. (c) 117.59; at 9%, the balance is increasing by about $117.59 per percentage point.

62. dq / dp = -30 / 1p2 + 123/2 63. (a) -$10,500; at 2 years, the value of the truck is decreasing by $10,500 per year. (b) -$4570.64; at 4 years, the value of the truck is decreasing by $4570.64 per year. 64. R1q2 = 130,000q - 2q32/ 3 (b) P1q2 = 8000q - 2q3 / 3 - 3500 (c) P′1q2 = 8000 - 2q2 (d) -$7000 65. P 3 ƒ1a24 = 18a2 + 24a + 9 66. (a) A3r1t24 = pt 4; this function represents the area of the oil slick as a function of time t after the beginning of the leak. (b) 4,000,000p; at 100 minutes, the area of the spill is changing at the rate of 4,000,000p ft2 / min. 67. (a) A3r1t24 = A1t2 = 4pt 2; this function gives the area of the pollution in terms of the time since the pollutants were first emitted. (b) 32p; at 12 p.m. the area of pollution is changing at the rate of 32p mi2 per hour. 68. (a) 6; at 0 hours, the number of bacteria is increasing by 6,000,000 per hour. (b) 9.75; at 7/5 hours, the number of bacteria is increasing by 9,750,000 per hour. (c) 19.71; at 8 hours, the number of bacteria is increasing by about 19,710,000 per hour. 69. (a) -0.5; at 0 hours, the calcium level in the bloodstream is decreasing by 0.5 mg / cm3 per day. (b) -1 / 54 ≈ -0.02; after 4 days, the calcium level in the bloodstream is decreasing by about 0.2 mg / cm3 per day.

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A-37

(c) -1 / 128 ≈ -0.008; after 7.5 days, the calcium level in the bloodstream is decreasing by about 0.008 mg / cm3 per day. (d) Always negative; the rate of change of the calcium level is always negative (the calcium level is decreasing).

70. (a) R′1Q2 = -Q / 361C - Q / 321/24 + 1C - Q / 321/2 (b) 2.83 (c) Increasing 71. (a) 34 minutes (b) - 1108 / 172p; at 17 minutes, the volume of the jawbreaker is decreasing by 1108 / 172 ≈ 19.96 mm3 per minute. - 172 / 172p; at 17 minutes, the surface area of the jawbreaker is decreasing by 172 / 172p ≈ 13.31 mm2 per minute. 72. (a) 8x 7 (b) x 8; 8x 7 73. (a) 12x 11 (b) x 12; 12x 11

Exercises 4.4 (page 258–262)

For exercises . . .

5–8

9–20,33–38, 41,42

Refer to example . . .

1

2

21–32,39,40,51, 52–55,59, 57,61, 60,64–66, 62,69, 56,58,63,67, 70,72,76, 71 73–75,77 78,79 4,5 3 7

68

43–46

6

6 (in first section of chapter)

W1. ƒ′1x2 = 3x 5 / 2x 6 + 5 W2. ƒ′1x2 = 1014x 2 + 3x + 229 18x + 32 W3. ƒ′1x2 = 10x1x 2 - 123/2 / 1x 2 + 121/2 1. True 2. False 3. True 4. False 5. ƒ′1x2 = 1.2e x 6. g′1t2 = -7.3e x 7. ƒ′1t2 = 1ln 1002100 t 8. g′1x2 = 1ln p2px 9. dy / dx = 4e 4x 10. dy / dx = -2e -2x 11. dy / dx = -24e 3x 12. dy / dx = 6e 5x 13. dy / dx = -32e 2x + 1 14. dy / dx = 1.2e -0.3x 2 2 2 3 2 15. dy / dx = 2xe x 16. dy / dx = -2xe -x 17. dy / dx = 12xe 2x 18. dy / dx = -60x 2e 4x 19. dy / dx = 16xe 2x - 4 3x2 + 5 x x x 2 -2x -2x -2x 20. dy / dx = -18xe 21. dy / dx = xe + e = e 1x + 12 22. dy / dx = -2x e + 2xe = 2xe 11 - x2 23. dy / dx = 21x + 3212x + 72e 4x 24. dy / dx = 1-15x 3 + 9x 2 + 20x - 42e -5x 25. dy / dx = 12xe x - x 2e x2/ e 2x = x12 - x2/ e x 26. dy / dx = e x12x - 12/ 12x + 122 27. dy / dx = 3x1e x - e -x2 - 1e x + e -x24/ x 2 28. dy / dx = 1xe x + xe -x - e x + e -x2/ x 2 = 3e x1x - 12 + e -x1x + 124/ x 2 29. dp / dt = 8000e -0.2t / 19 + 4e -0.2t22 2 2 2 2 30. dp / dt = 1250e -0.5t / 112 + 5e -0.5t22 31. ƒ′1z2 = 412z + e -z 211 - ze -z 2 32. ƒ′1t2 = 31e t + 5t2212e t + t2 3x + 1 -5x + 2 x2 + 2 3x2 - 4 2ln 10 33. dy / dx = 31ln 727 34. dy / dx = -51ln 424 35. dy / dx = 6x1ln 424 36. dy / dx = -6x110 37. ds / dt = 1ln 3232t / 2t 38. ds / dt = 15 ln 2222t - 2 / 32 2t - 244 39. dy / dt = 1-te 3t + e 3t + te t + e t - 4e 2t2/ 1e 2t + 122 40. dy / dx = 1t 2e 2t + 2t 3e 2t + 2te 5t - t 2e 5t2/ 1t + e 3t22 41. ƒ′1x2 = 19x + 42e x23x + 2 / 32 23x + 24 2 3 42. ƒ′1x2 = x14 - x 32e x /1x + 22 / 1x 3 + 222 43. y = 3e 2x - 2e 2 44. y = 8e 2x - 12e 2 45. y = 4ex + 3e 46. y = 1e 3 / 272x 48.

y5

e x10.0001 2 e x 2 0.0001 16

The graphs coincide, providing graphical evidence that Dx 3e x4 = e x.

21

4 21

50. (a) 0.6931; this value agrees with the value given by the formula, ln 2 ≈ 0.693147. (b) 1.099; this value agrees with the value given by the formula, ln 3 ≈ 1.098612. (c) a = 2.718; this value is approximately equal to e = 2.71828 c.

51. (a) $3.81; at 0 units, the cost is increasing by $3.81 per meal kit. (b) $0.20; at 20 units, the cost is increasing by $0.20 per meal kit. (c) C′1x2 approaches zero. 52. (a) 20 thousand; after 1 year, the sales are increasing by about 20,000 computers per year. (b) 6 thousand; after 5 years, the sales are increasing by about 6,000 computers per year. (c) As time goes on, the rate of change of sales is decreasing. (d) No; although the rate of change of sales never equals zero, it gets closer and closer to zero as t increases. 53. (a) $304.52; after 1 year, the compound amount is increasing by about $304.52 per year. (b) $383.26; after 5 years, the compound amount is increasing by about $383.26 per year. (c) $510.93; after 10 years, the compound amount is increasing by about $510.93 per year. 54. (a) $31.86; at 2%, the compound amount is increasing by about $31.86 per percent. (b) $34.86; at 5%, the compound amount is increasing by about $34.86 per percent. (c) $40.50; at 10%, the compound amount is increasing by about $40.50 per percent.

S 55. (a) (b) 48,843; they anticipate sales of about $48,843 when $2500 is spent on local advertising. 6.694; S(A) 5 c 2 de2uA c when $2500 is spent on local advertising, sales are expected to increase by about $6.94 per dollar of advertising. (c) 55,464; they anticipate sales of about $55,464 when $4000 is spent on local c–d advertising. 2.722; when $4000 is spent on local advertising, sales are expected to increase by about A $2.72 per dollar of advertising.

56. (a) 3.07; after 2 months, the percent of the public who are aware of the product is increasing by about 3.07% per month. (b) -1.93; after 4 months, the percent of the public who are aware of the product is decreasing by about 1.93% per month. (c) Public awareness increased at first but then decreased. 57. (a) G1t2 = 315 / 11 + 156.5e -0.2961t2 (b) 8.6 million users; 2.48 million users per year (c) 110.8 million users; 21.27 million users per year (d) 287.6 million users; 7.42 million users per year (e) It increases for a while and then gradually decreases to 0. 58. (b) 59. (a) 22,000,000 people

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(b) 350,000 people/year 60. (a) 129.8; in 2020, the world population is growing by about 129,800,000 people per year. (b) 140.5; in 2025, the world population is growing by about 140,500,000 people per year. 61. (a) G1t2 = 31.4 / 11 + 12.5e -0.393t2 (b) 3.33 g; 1.17 g / day (c) 11.4 g; 2.85 g / day (d) 25.2 g; 1.95 g / day (e) It increases for a while and then gradually decreases to 0. 62. (a) G1t2 = 787 / 11 + 60.8e -0.0473t2 (b) 13 cranes (c) 18 cranes; 0.815 cranes per year (d) 104 cranes; 4.26 cranes per year (e) 395 cranes; 9.3 cranes per year (f) It increases for a while but will eventually decrease to 0.

63. (a) 3.857 cm3 (b) 0.973 cm (c) 18 years (d) 1100 cm3 (e) 0.282; at 240 months old, the tumor is increasing in volume at the instantaneous rate of 0.282 cm3 / month. 64. (a) 0.026%; 0.286%; 3.130% (b) 0.0025% per year; 0.0274% per year; 0.300% per year (c) The percentage of people in each age group that die in a given year is increasing. The formula implies that everyone will be dead by age 112. 65. (a) 2.7 cases per 1000 people; 0.084, the number of cases increases for 35-year-olds by about 0.084 cases per 1000 people for each additional year. (b) 5.0 cases per 1000 people; 0.16, the number of cases increases for 55-year-olds by about 0.16 cases per 1000 people for each additional year. (c) 9.4 cases per 1000 people; 0.29, the number of cases increases for 75-year-olds by about 0.29 cases per 1000 people for each additional year. 66. (a) 3 kg (b) 3.1 kg (c) 124 days (d) 2.75 grams per day (f) (e) 2e20.022(t256) M(t) 5 3102e

3200

Growth is initially rapid, then tapers off.

0

300 0

Day

Weight

Rate

50

991

24.88

100

2122

17.73

150

2734

7.60

200

2974

2.75

250

3059

0.94

300

3088

0.32

67. (a) 509.7 kg, 498.4 kg; no (b) 1239 days, 1095 days (c) 0.22 kg / day, 0.22 kg / day (e) The graphs of the rates of change of the two functions are also are (d) The growth patterns of the two functions are very similar. very similar.

1

525

0

2500

0

0

2500

0

68. 0.840 69. (a) G1t2 = 1 / 11 + 270e -3.5t2 (b) 0.109, 0.341 per century (c) 0.802, 0.555 per century (d) 0.993, 0.0256 per century (e) It increases for a while and then gradually decreases to 0. 70. (a) 36.8 (b) 0.00454 (c) 0 (d) H′1N2 is always positive since powers of e are always positive. This means that repetition always makes a habit stronger.

71. (a) G1t2 = 6.8 / 11 + 3.242e -0.2992t2 (b) 3.435 million students; 0.509 million students/yr (c) 6.068 million students; 0.195 million students/yr (d) 6.700 million students; 0.029 million students/yr (e) The rate increases at first and then decreases toward 0. 72. (a) -44.9 grams per year (b) -24.1 grams per year (c) -6.98 grams per year (d) The rate of change approaches 0. (e) No 73. (a) lc = 1V / R2e -t/RC (b) 1.35 * 10 -7 amps 74. (a) H1T2 = T + 2.4385e 0.02086T (b) 92.94°F (c) 1.2699; the rate of change of the heat index when the temperature is 80°F is 1.2699°F per degree increase in the temperature.

75. (a) 29.24°, 0.3361 (b) 35.18°, 0.6918 (c) 44.46°, 1.205 76. 3.90; the temperature is decreasing at 3.90 degrees per hour.

77. (a) 625.14 ft (b) 0; yes (c) -1.476 ft per ft from center 78. 4.319 deaths per 100 students; 0.2907 deaths per 100 students per year 79. (a) 218.2 seconds (b) -0.01675; the record is decreasing by about 0.01675 second per year at the end of 2020. (c) 218 seconds. If the estimate is correct, then this is the least amount of time that it will ever take for a human to run a mile.

Exercises 4.5 (page 267–269)

For exercises . . .

5–8,11–14, 35,36–42, 73,75 1,2

9,10,15–34, 43–48,71,72, 77, 80 3

49–56

75,76, 78,79

57,58

W1. ƒ′1x2 = 2xe x Refer to example . . .

4 5 6 (in first W2. ƒ′1x2 = 2x12x + 12e 4x section of 2x 3x 3x 2 W3. ƒ′1x2 = e 12 - e 2/ 1e + 12 chapter) 1. True 2. False 3. True 4. True 5. dy / dx = 1 / x 6. dy / dx = 1 / x 7. dy / dx = -3 / 18 - 3x2 or 3 / 13x - 82 8. dy / dx = 3x 2 / 11 + x 32 9. dy / dx = 18x - 92/ 14x 2 - 9x2 10. dy / dx = 1-12x 2 + 12/ 1-4x 3 + x2 11. dy / dx = 1 / 321x + 524 12. dy / dx = 1 / 12x + 12 13. dy / dx = 312x 2 + 52/ 3x1x 2 + 524 14. dy / dx = 3115x 2 - 22/ 3215x 3 - 2x24 15. dy / dx = -15x / 13x + 22 - 5 ln 13x + 22 16. dy / dx = 213x + 72/ 12x - 12 + 3 ln12x - 12 17. ds / dt = t + 2t ln t = t11 + 2 ln t 2 18. dy / dx = -2x 2 / 12 - x 22 + ln 2 - x 2 19. dy / dx = 32x - 41x + 32ln1x + 324/ 3x 31x + 324 20. dv / du = 11 - 3 ln u2/ u4 2

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21. dy / dx = 14x + 7 - 4x ln x2/ 3x14x + 7224 22. dy / dx = -213x - 1 - 3x ln x2/ 3x13x - 1224 23. dy / dx = 16x ln x - 3x2/ 1ln x22 24. dy / dx = 13x 3 ln x - x 3 + 12/ 32x1ln x224 25. dy / dx = 41ln x + 1 23 / 1x + 12 2 2 26. dy / dx = 1 / 321x - 32 2ln x - 3 4 27. dy / dx = 1 / 1x ln x2 28. dy / dx = 1ln 42/ x 29. dy / dx = e x / x + 2xe x ln x 2x - 1 2x - 1 x x 2 30. dy / dx = 2e ln12x - 12 + 2e / 12x - 12 31. dy / dx = 1xe ln x - e 2/ 3x1ln x2 4 32. p′1y2 = 11 - y ln y2/ 1yey2 2z 2 2z 33. g′1z2 = 31e + ln z2 12e + 1 / z2 34. s′1t2 = 11 - te -t2/ 32t 2e -t + ln 2t4 35. dy / dx = 1 / 1x ln 102 36. dy / dx = 4 / 31ln 10214x - 324 37. dy / dx = -1 / 31ln 10211 - x24 or 1 / 31ln 1021x - 124 38. dy / dx = 1 / 1x ln 102 39. dy / dx = 5 / 312 ln 5215x + 224 40. dy / dx = 2 / 31ln 7214x - 324 41. dy / dx = 31x + 12/ 31ln 321x 2 + 2x24 42. dy / dx = 514x - 12/ 312 ln 2212x 2 - x24 43. dw / dp = 1ln 222p / 31ln 8212p - 124 or 2p / 3312p - 124 44. dz / dy = 10 y / 31ln 102y4 + 1log y21ln 10210 y 45. ƒ′1x2 = e 2x51 / 32 2x1 2x + 524 + ln1 2x + 52/ 32 2x46 46. ƒ′1x2 = e 2x1 2x + 22/ 321xe 2x + 224 47. ƒ′1t2 = 31t 2 + 12ln1t 2 + 12 - t 2 + 2t + 14 /51t 2 + 123ln1t 2 + 12 + 1426 48. ƒ′1t2 = 316t 2 + 3t 1/22ln12t 3/2 + 12 - 6t 24/ 512t 3/2 + 123ln12t 3/2 + 12426 49. ƒ′1x2 = 5 / 3215x - 324 50. g′1x2 = 3x / 13x 2 - 72 51. h′1x2 = 313x 2 + 5x2/ 12x 3 + 5x 22 52. ƒ′1t2 = 516t + 72/ 3313t 2 + 7t - 424 53. ƒ′1x2 = 2 / x + e x / 1e x + 12 = 12e x + 2 + xe x2/ 3x1e x + 124 54. g′1x2 = 5 / x + 20x / 1x 2 + 52 = 125x 2 + 252/ 3x1x 2 + 524 55. h′1t2 = 2 / 1t + 42 - 1 / t = 1t - 42/ 3t1t + 424 56. r′1x2 = 2x / 1x 2 - 42 - 3 / x = 1-x 2 + 122/ 3x1x 2 - 424 57. y = 12 / 32x - 2 / 3 + ln 3 58. y = x / 2 + 1 / 2 + ln 16 62. ln ux 1 0.0001u 2 lnuxu y5

2

0.0001 3

The graphs coincide, providing graphical evidence that Dx 3ln x4 = 1 / x.

23

3

23

68. h′1x2 = x x11 + ln x2 69. h1x2 = 1x 2 + 125x 310x 2 / 1x 2 + 12 + 5 ln1x 2 + 124 70. (c) Continuity 71. (a) dR / dq = 100 + 501ln q - 12/ 1ln q22 (b) $112.48 (c) To decide whether it is reasonable to sell additional items 72. (a) C′1q2 = 100 (b) P1q2 = 50q / ln q - 100 (c) $12.48 (d) To decide whether it is profitable to make and sell additional items 73. (a) -$0.19396; when 10 reeds are produced, the average cost is decreasing by about $0.19 per reed. (b) -$0.06099; when 20 reeds are produced, the average cost is decreasing by about $0.06 per reed.

74. (a) R′1x2 = 60 / 12x + 12 (b) P1x2 = 30 ln12x + 12 - x / 2 (c) 0 (d) When 60 items are manufactured, there is no -e2.54197 - 0.2167t profit from selling additional items. 75. (a) N1t2 = 1000e 9.8901e (b) 1,307,416 bacteria per hour; the number of bacteria is increasing at a rate of 1,307,416 per hour, 20 hours after the experiment began. (e) 9.8901; 1000e 9.8901 ≈ 19,734,033 (d) 20,000,000 (c) 12 76. (a) About 2590 cm2 (b) 0.46 g / cm2; when the infant has a mass of 4000 g, the BSA is increasing by 0.46 cm2 for each additional gram of mass.

0

35 0

35 0

(c) A(w) 5 4.688w0.816820.0154 log w 6000

2000

0

77. (b) (i) 3.343 imagoes per mated female per day (ii) 1.466 imagoes per mated female per day (c) (i) -0.172 imagoes per mated female per day per fly per bottle (ii) -0.0511 imagoes per mated female per day per fly per bottle

10,000 0

78. (a) About 4 kJ / day (b) 1.3 * 10 -5; when a fawn is 25 kg in size, the rate of change of the energy expenditure of the fawn is about 1.3 * 10 -5 kJ / day per gram.

79. 26.9 ants/day; after 2 days, the ant population is growing by 26.9 ants per day. 13.1 ants/ F(x) 5 0.774 1 0.727 log x (c) 5 day; after 8 days, the ant population is growing by 13.1 ants per day.

80. (a) 21.28; in 1970, the percent of persons 65 years and over with family income below the poverty level is 21.28%. -1.158; in 1970, that percentage is decreasing by about 1.158 percentage per year. (b) 10.91; in 1995, the percent of persons 65 years and over with family income below the poverty level is 10.91%. -0.193; in 1995, that percentage is decreas5000 30,000 3 ing by about 0.193 percentage per year. (c) 7.40; in 2020, the percent of persons 65 years and over with family income below the poverty level is 7.4%. -0.105; in 2020, that percentage is decreasing by about 0.105 percentage per year. 81. (a) 1.567 * 10 11 kWh

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A-40

Complete Instructor Answers

(b) 63.4 months (c) 4.14 * 10 -26 (d) dM/dE decreases and approaches zero. 82. (a) 817; for a street 30 ft wide, the maximum traffic flow is 817 vehicles per hour. -41.2; the maximum rate of traffic flow is decreasing by 41.2 vehicles/hr per foot. (b) 522, for a street that is 40 ft wide, the maximum traffic flow is 522 vehicles per hour. -20.9; the maximum rate of traffic flow is decreasing by 20.9 vehicles/hr per foot.

Chapter 4 Review Exercises (page 271–275)

For exercises . . .

1,2,4,11–16, 53,54,75, 81–83

3,17–20, 55,56,67, 68,76,77,90

5,21–30,51, 52,57,58, 69–72,78,93

Refer to section . . .

1

2

3

6–7,31–36, 43–46,59, 60,73,79, 84–89,91,92 4

8–10,37–42, 47–50,61, 62,74,80 5

1. False 2. True 3. False 4. True 5. False 6. False 7. False 8. True 9. True 10. False 11. dy / dx = 15x 2 - 14x - 9 12. dy / dx = 21x 2 - 8x - 5 13. dy / dx = 24x 5/3 14. dy / dx = 12x -4 or 12 / x 4 15. ƒ′1x2 = -12x -5 + 3x -1/2 or -12 / x 5 + 3 / x 1/2 16. ƒ′1x2 = -19x -2 - 4x -1/2 or -19 / x 2 - 4 / x 1/2 17. k′1x2 = 21 / 14x + 722 18. r′1x2 = -8 / 12x + 122 19. dy / dx = 1x 2 - 2x2/ 1x - 122 20. dy / dx = 14x 3 + 7x 2 - 20x2/ 1x + 222 21. ƒ′1x2 = 24x13x 2 - 223 22. k′1x2 = 90x 215x 3 - 125 23. dy / dt = 7t 612t 7 - 52-1/2 or 7t 6 / 12t 7 - 521/2 24. dy / dt = -48t 318t 4 - 12-1/2 or -48t 3 / 18t 4 - 121/2 25. dy / dx = 312x + 12218x + 12 26. dy / dx = 4x13x - 224121x - 42 27. r′1t2 = 1-15t 2 + 52t - 72/ 13t + 124 28. s′1t2 = 1-4t 3 - 9t 2 + 24t + 62/ 14t - 325 29. p′1t2 = t1t 2 + 123/217t 2 + 22 3 2 30. g′1t2 = t 21t 4 + 525/2117t 4 + 152 31. dy / dx = -12e 2x 32. dy / dx = 4e 0.5x 33. dy / dx = -6x 2e -2x 34. dy / dx = -8xe x 35. dy / dx = 10xe 2x + 5e 2x = 5e 2x12x + 12 36. dy / dx = 21x 2e -3x - 14xe -3x = 7xe -3x13x - 22 37. dy / dx = 2x / 12 + x 22 38. dy / dx = 5 / 15x + 32 39. dy / dx = 1x - 3 - x ln 3x 2/ 3x1x - 3224 40. dy / dx = 321x + 32 - 12x - 12ln 2x - 1 4/ 312x - 121x + 3224 41. dy / dx = 3e x1x + 121x 2 - 12ln1x 2 - 12 - 2x 2e x4/ 51x 2 - 123ln1x 2 - 12426 42. dy / dx = e 2x 32x1ln x21x 2 + 1 + x2 - 1x 2 + 124/ 3x1ln x224 43. ds / dt = 21t 2 + e t212t + e t2 44. dq / dp = 8e 2p + 11e 2p + 1 - 223 2 45. dy / dx = -6x1ln 10210 -x 46. dy / dx = 51ln 2222x / x 1/2 47. g′1z2 = 13z 2 + 12/ 31ln 221z 3 + z + 124 z 48. h′1z2 = e / 31ln 10211 + e z24 49. ƒ′1x2 = 1x + 12e 3x / 1xe x + 12 + 2e 2x ln1xe x + 12 50. ƒ′1x2 = e 2x31 2x + 12ln1 2x + 12 - 14/ 52 2x1 2x + 123ln1 2x + 12426 51. (a) -3 / 2 (b) -24 / 11 52. (a) -36 / 13 (b) -21 / 10 53. -2; y = -2x - 4 54. -2; y = -2x + 9 55. -3 / 4; y = - 13 / 42x - 9 / 4 56. -5 / 9; y = 1-5 / 92x + 16 / 9 57. 3 / 4; y = 13 / 42x + 7 / 4 58. -4 / 5; y = - 14 / 52x - 13 / 5 59. 1; y = x + 1 60. 2e; y = 2ex - e 61. 1; y = x - 1 62. 2; y = 2x - e 63. No points if k 7 0; exactly one point if k = 0 or if k 6 -1 / 2; exactly two points if -1 / 2 … k 6 0. 65. 5%; 5.06% 67. abn2x n - 1 / 1n - x2n + 1 68. ake kx/1k - 12 / 3b1k - 124 69. C′1x2 = 1-x - 22/ 32x 21x + 121/24 70. C′1x2 = - 13x + 42/ 32x 213x + 221/24 71. C′1x2 = 1x 2 + 32215x 2 - 32/ x 2 72. C′1x2 = 14x + 323112x - 32/ x 2 73. C′1x2 = 3e -x1x + 12 - 104/ x 2 74. C′1x2 = 3x - 1x + 52ln1x + 524/ 3x 21x + 524 75. (a) 22; when $9000 is spent on research, sales will increase by $22 million when $1000 more is spent on research. (b) 19.5; when $16,000 is spent on research, sales will increase by $19.5 million when $1000 more is spent on research. (c) 18; when $25,000 is spent on research, sales will increase by $18 million when $1000 more is spent on research. (d) As more is spent on research, the increase in sales is decreasing. 76. (a) 0.4938; profit will increase by about $49.38 from selling the fifth unit. (b) 0.4992; profit will increase by about $49.92 from selling the thirteenth unit. (c) 0.4997; profit will increase by about $49.97 from selling the twenty-first unit. (d) As the number sold increases, the marginal profit increases. (e) 0.0123; when 4 units are sold, the average profit is increasing by $1.23 per unit. 77. (a) -2.201; when $900 is spent on training, costs will decrease by $2201 for the next $100 spent on training. (b) -0.564; when $1900 is spent on training, costs will decrease by $564 for the next $100 spent on training. (c) Decreasing 78. 215.15; when the interest rate is 5%, the balance increases by approximately $215.15 for every 1% increase in the interest rate 79. 218.65; when the interest rate is 5%, the balance increases by approximately $218.65 for every 1% increase in the interest rate. 80. -2.77; when the interest rate is 5%, the doubling time decreases by approximately 2.77 years for every 1% increase in the interest rate. 81. (a) -6.51 billion / yr; the volume of mail is decreasing by about 6,510,000,000 pieces per year. (b) -3.97 billion / yr; the volume of mail is decreasing by about 3,970,000,000 pieces per year. 82. (a) y = 0.0001365t 3 - 0.02294t 2 + 0.5698t + 52.31 (b) -0.68; in 1965, the percent of men aged 65 and older in the workforce was decreasing by about 0.68% per year. 0.71; in 2015, the percent of men aged 65 and older in the workforce was increasing by about 0.71% per year. 83. (a) y = 1.135 * 10 -6t 3 + 0.003147t 2 - 0.2019t + 4.099 (b) 0.15; in 1955, the number of dollars required to equal $1 in 1913 is increasing by about $0.15 per year. 0.57; in 2015, the number of dollars required to equal $1 in 1913 is increasing by about $0.57 per year. 84. 50,000; the population is growing at a rate of 50,000 per year. 85. (a) G1t2 = 30,000 / 11 + 14e -0.15t2 (b) 4483; after 6 years, the population is 4483. 572; after 6 years, the population is growing by 572 per year. 86. (a) 28.1 cm (b) 4.34 cm / yr (c) 205 grams (d) 21.2 grams / cm (e) 92.0 grams / year 87. (a) 3493.76 grams (b) 3583 grams (c) 84 days (d) 1.76 g / day

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(e)

(f)

20.020(t266)

M(t) 5 3583e2e 3600

Growth is initially rapid, then tapers off.

0

300 0

Day

Weight

Rate

50

904

24.90

100

2159

21.87

150

2974

11.08

200

3346

4.59

250

3494

1.76

300

3550

0.66

A-41

88. (a) 1834; after 2 hours, the number of bacteria is increasing by 1834 bacteria per hour. (b) 2421; after 4 hours, the number of bacteria is increasing by 2421 bacteria per hour. (c) 5570; after 10 hours, the number of bacteria is increasing by 5570 bacteria per hour. 89. (a) 27,286 megawatts / yr (b) 54,531 megawatts / yr (c) 108,982 megawatts / yr 90. (a) -5 ft / sec (b) -1.7 ft / sec 91. 0.390; the production of corn is increasing at a rate of 0.390 billion bushels a year in 2020.

92. (a) 169 words; actual number is 167 words. (b) 165 words (c) -36; in 2050, the number of words in use will decrease by 36 words per millennium. 93. (a) -0.4677 fatalities per 1000 licensed drivers per 100 million miles per year; at the age of 20, each extra year results in a decrease of 0.4677 fatalities per 1000 licensed drivers per 100 million miles. (b) 0.003672 fatalities per 1000 licensed drivers per 100 million miles per year; at the age of 60, each extra year results in an increase of 0.003672 fatalities per 1000 licensed drivers per 100 million miles.

Chapter 5 Graphs and the Derivative Exercises 5.1 (page 288–291)

5–12,51,52, 17–24,29,30, 25–28 58,68,69 37–42,45–47, 57,59,62–67 Refer to example . . . 1 2 4 For exercises . . .

31,32,43, 33–36 44,48

53,60,61 54–56

W1. -3 / 2, 5 / 4 W2. -1 / 3, 0, 15 / 4 W3. -3, 3 W4. -2, 0, 2 3 5 7 W5. ƒ′1x2 = -x1x 3 + 12x - 102/ 1x 3 + 522 3 W6. ƒ′1x2 = -x / 23 - x 2 W7. ƒ′1x2 = x115x 3 + 22e 5x W8. ƒ′1x2 = 3x 1/2 / 1x 3/2 + 52 1. False 2. True 3. False 4. False 5. (a) 11, ∞2 (b) 1-∞, 12 6. (a) 1-∞, 42 (b) 14, ∞2 7. (a) 1-∞, -22 (b) 1-2, ∞2 8. (a) 13, ∞2 (b) 1-∞, 32 9. (a) 1-∞, -42, 1-2, ∞2 (b) 1-4, -22 10. (a) 11, 52 (b) 1-∞, 12, 15, ∞2 11. (a) 1-7, -42, 1-2, ∞2 (b) 1-∞, -72, 1-4, -22 12. (a) 1-3, 02, 13, ∞2 (b) 1-∞, -32, 10, 32 13. (a) 1-∞, -12, 13, ∞2 (b) 1-1, 32 14. (a) 13, 52 (b) 1-∞, 32, 15, ∞2 15. (a) 1-∞, -82, 1-6, -2.52, 1-1.5, ∞2 (b) 1-8, -62, 1-2.5, -1.52 16. (a) 1-∞, ∞2 (b) Nowhere 17. (a) 17 / 12 (b) 1-∞, 17 / 122 (c) 117 / 12, ∞2 18. (a) -1 / 2 (b) 1-∞, -1 / 22 (c) 1-1 / 2, ∞2 19. (a) -3, 4 (b) 1-∞, -32, 14, ∞2 (c) 1-3, 42 20. (a) -1, 2 (b) 1-∞, -12, 12, ∞2 (c) 1-1, 22 21. (a) -3 / 2, 4 (b) 1-∞, -3 / 22, 14, ∞2 (c) 1-3 / 2, 42 22. (a) -1, 5 / 2 (b) 1-∞, -12, 15 / 2, ∞2 (c) 1-1, 5 / 22 23. (a) -2, -1, 0 (b) 1-2, -12, 10, ∞2 (c) 1-∞, -22, 1-1, 02 24. (a) -3, 0, 1 (b) 1-3, 02, 11. ∞2 (c) 1-∞, -32, 10, 12 25. (a) None (b) Nowhere (c) 1-∞, ∞2 26. (a) None (b) 1-∞, ∞2 (c) Nowhere 27. (a) None (b) Nowhere (c) 1-∞, -12, 1-1, ∞2 28. (a) None (b) Nowhere (c) 1-∞, 42, 14, ∞2 29. (a) 0 (b) 10, ∞2 (c) 1-∞, 02 30. (a) 0 (b) 1-∞, 02 (c) 10, ∞2 31. (a) 0 (b) 10, ∞2 (c) 1-∞, 02 32. (a) -1 (b) 1-1, ∞2 (c) 1-∞, -12 33. (a) -1 (b) 1-∞, ∞2 (c) Nowhere 34. (a) 2 (b) Nowhere (c) 1-∞, ∞2 35. (a) 0 (b) Nowhere (c) 1-∞, ∞2 36. (a) 1 (b) 1-∞, ∞2 (c) Nowhere 37. (a) 7 (b) 17, ∞2 (c) 13, 72 38. (a) 0 (b) 10, ∞2 (c) 1-∞, 02 39. (a) 1 / 3 (b) 1-∞, 1 / 32 (c) 11 / 3, ∞2 40. (a) 1 / 2, 1 (b) 1-∞, 1 / 22, 11, ∞2 (c) 11 / 2, 12 41. (a) 0, 2 / ln 2 (b) 10, 2 / ln 22 (c) 1-∞, 02, 12 / ln 2, ∞2 42. (a) -1 / 22 ln 2, 1 / 22 ln 2 (b) 1-1 / 22 ln 2, 1 / 22 ln 22 (c) 1-∞, -1 / 22 ln 22, 11 / 22 ln 2, ∞2 43. (a) 0, 2 / 5 (b) 10, 2 / 52 (c) 1-∞, 02, 12 / 5, ∞2 44. (a) -1 / 4, 0 (b) 1-1 / 4, ∞2 (c) 1-∞, -1 / 42 45. Vertex: 1-b / 12a2, 14ac - b22/ 14a22; increasing on 1-b / 12a2, ∞2; decreasing on 1-∞, -b / 12a22 46. Vertex: 1-b / 12a2, 14ac - b22/ 14a22; increasing on 1-∞, -b / 12a22; decreasing on 1-b / 12a2, ∞2 47. Increasing on 1-∞, ∞2; decreasing nowhere; tangent line is horizontal nowhere 48. Increasing on 10, ∞2; decreasing nowhere; tangent line is horizontal nowhere 49. (a) 0.5 (b) 0.5 (c) The answers are the same. (d) 1.25 (e) 1.25 (f) The answers are the same. 51. (a) About 1567, ∞2 (b) About 10, 5672 52. (a) About (0, 558) (b) About 1558, ∞2 53. (a) Nowhere; as the mortgage rates increase, the number of housing starts never increases. (b) 10, ∞2; as the mortgage rates increase, the number of housing starts always decreases.

54. (a) Nowhere, as the quantity manufactured increases, the total cost never decreases. (b) 10, ∞2; as the quantity manufactured increases, the total cost always increases. 55. 10, 22002; the profit is increasing for the first 2200 games sold, then the profit starts to decrease. 56. (a) 10, 32; profit is increasing when production is between 0 and 300 phones per month. (b) 13, 3.92; profit is decreasing when production is between 300 and 390 phones per month. 57. (a) 10, 7.42; Social Security assets are increasing from 2000 to about the middle of 2007. (b) (7.4, 50); Social Security assets are decreasing from about the middle of 2007 to 2050.

58. Unemployment rate is increasing on (2000, 2003) and (2007, 2010), decreasing on (2003, 2007) and (2010, 2019), and constant nowhere. 59. (a) (0, 1.85); the alcohol concentration is increasing for about 1.85 hours after consuming alcohol.

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6

A-42

Complete Instructor Answers

(b) (1.85, 5); after about 1.85 hours, the alcohol concentration decreases. 60. (a) (0, 3); the concentration of the drug is increasing for the first 3 hours. (b) 13, ∞2; after 3 hours, the concentration of the drug is decreasing. 61. (a) (0, 1); the concentration of the drug is increasing for the first hour. (b) 11, ∞2; after 1 hour, the concentration of the drug is decreasing. 62. (55, 130); the function is increasing on the entire interval and is decreasing nowhere. 63. (a) F′1t2 = 175.9e -t/1.311 - 0.769t2 (b) 10, 1.32; the thermic effect of food increases for the first 1.3 hours after eating a meal. 11.3, ∞2; 1.3 hours after eating a meal, the thermic effect of food decreases. 64. 10, ∞2; the weight of the Holstein cow is increasing every day.

65. (a) dR / dN = -N0 / 32N 3/2 22pD1N0 - N24 (b) Decreases (c) It approaches -∞; it approaches -∞. 66. About day 9 67. Increasing on 1-∞, 02; decreasing on 10, ∞2 68. (a) (1945, 1965), (1970, 1973) (b) (1987, 2019) 69. (a) 12500, 57502 (b) 15750, 60002 (c) 12800, 48002 (d) 12500, 28002 and 14800, 60002 70. (a) Negative (b) mpg/lb, or miles per gallon per pound

Exercises 5.2 (page 301–303)

For exercises . . .

5–12,45–47, 60 Refer to example . . . 2

17–26, 43,44 3

27–32

33–42,62–64, 51–54, 66 56–59 5 6

W1. Increasing on 1-∞, -22 and 13, ∞2; 4 decreasing on 1-2, 32 W2. Increasing on 1-3 / 2, 02 and 14, ∞2; decreasing on 1-∞, -3 / 22 and 10, 42 1. False 2. True 3. False 4. False 5. Relative minimum of -4 at x = 1 6. Relative maximum of 1 at x = 4 7. Relative maximum of 3 at x = -2 8. Relative minimum of -4 at x = 3 9. Relative maximum of 3 at x = -4; relative minimum of 1 at x = -2 10. Relative minimum of -6 at x = 1; relative maximum of 2 at x = 5 11. Relative maximum of 3 at x = -4; relative minimum of -2 at x = -7 and x = -2 12. Relative maximum of 4 at x = 0; relative minimum of 0 at x = -3 and x = 3 13. Relative maximum at x = -1; relative minimum at x = 3 14. Relative minimum at x = 3; relative maximum at x = 5 15. Relative maxima at x = -8 and x = -2.5; relative minima at x = -6 and x = -1.5 16. No relative extrema 17. Relative minimum of 8 at x = 5 18. Relative minimum of -11 at x = -4 19. Relative maximum of -8 at x = -3; relative minimum of -12 at x = -1 20. Relative maximum of 82 at x = -4; relative minimum of -26 at x = 2 21. No relative extrema 22. No relative extrema 23. Relative maximum of 827 / 96 at x = -1 / 4; relative minimum of -377 / 6 at x = -5 24. Relative maximum of -13 / 6 at x = 1; relative minimum of -59 / 8 at x = -3 / 2 25. Relative maximum of -4 at x = 0; relative minimum of -85 at x = 3 and x = -3 26. Relative maximum of 9 at x = 0; relative minimum of -7 at x = -2 amd x = 2 27. Relative maximum of 3 at x = -8 / 3 28. Relative minimum of 1 at x = 5 / 9 29. Relative maximum of 1 at x = -1; relative minimum of 0 at x = 0 30. Relative maximum of 0 at x = 0; relative minimum of -9 # 22/3 ≈ -14.287 at x = 2 31. No relative extrema 32. No relative extrema 33. No relative extrema 3 3 34. Relative minimum of 3 2 2 / 2 ≈ 1.890 at x = 2 4 / 2 35. Relative maximum of 0 at x = 1; relative minimum of 8 at x = 5 36. Relative maximum of -20 at x = -7; relative minimum of 0 at x = 3 37. Relative maximum of 4 / e 2 - 3 ≈ -2.46 at x = -2; relative minimum of -3 at x = 0 38. Relative minimum of -3 / e + 2 ≈ 0.90 at x = -1 39. No relative extrema 40. Relative minimum of 5.445 at x = 2e ≈ 1.65 41. Relative minimum of e ln 2 at x = 1 / ln 2 42. Relative minimum of 3ln1ln 82 + 14/ ln 8 at x = ln1ln 82/ ln 8 43. (3, 13) 44. 1-b / 12a2, 14ac - b22/ 14a22 45. Relative maximum of 6.211 at x = 0.085; relative minimum of -57.607 at x = 2.161 46. Relative maximum of 12.821 at x = 0.183; relative minimum of -143.572 at x = -2.703 47. Relative minimum at x = 5 48. (a) When graphing g1x2 in a standard window, no graph seems y 5 2zx 1 1z 1 4zx 2 5z 2 20 30 to appear. (b) Minimum of -10 36 at x = -0.001 and x = 0.001 49. Relative maximum at x = -1; relative minimum at x = 3 50. Relative minima at x = 1 and x = 4; relative maximum at x = 2 51. (a) 13 (b) $44 (c) $258 52. (a) 1625 (b) $57.50 210 10 (c) $47,812.50 53. (a) 100 (b) $14.72 (c) $635.76

55,61, 65,67 1

215

54. (a) 120 (b) $25.66 (c) $550.84 55. Relative maximum of 20,620 megawatts at midnight 1t = 02; relative minimum of 20,000 megawatts at 1:59 a.m.; relative maximum of 29,930 megawatts at 4:17 p.m.; relative minimum of 18,130 megawatts at midnight 1t = 242 56. (a) 6 units (b) About $5784 57. q = 10; p ≈ $73.58 58. Maximum revenue occurs when the price is about $4600 and 25 servers are sold. 59. 120 units 60. Relative minimum of 4.0% in 2000; relative maximum of 6% in 2003; relative minimum of 4.6% in 2006–2007; relative maximum of 9.6 in 2010; relative minimum of 3.7% in 2019 61. 5:04 p.m.; 6:56 a.m. 62. (a) The maximum daily consumption is 9.54 kg at 9.68 weeks. (b) The maximum daily consumption is a1b / c2be -b kg and it occurs at b / c weeks. 63. 4.96 years; 458.22 kg 64. 1.3 hours 65. Degree of discrepancy of 10 66. 10 minutes 67. (a) 28 ft (b) 2.57 sec

Exercises 5.3 (page 314–318)

For exercises . . .

7–34, 84,85 Refer to example . . . 2,3

35–58,64,67–70,86, 67–74,91, 80–83,89, 101,102,104 87,88,90,95–100 92,103 93,94 5,6 7 8 4

W1. Relative maximum of 196 at x = -4; relative minimum of -304 at x = 6 W2. Relative maximum of 10 at x = 0; relative minimum of -38 at x = -2 and of -2038 at x = 8 1. False 3. True 4. True 5. False 6. False 7. ƒ″1x2 = 30x - 14; -14; 46 8. ƒ″1x2 = 24x + 10; 10; 58 Copyright © 2022 Pearson Education, Inc.

2. True

Complete Instructor Answers

9. ƒ″1x2 = 48x 2 - 18x - 4; -4; 152 10. ƒ″1x2 - 12x 2 + 42x - 1; -1; 35 11. ƒ″1x2 = 6; 6; 6 12. ƒ″1x2 = 16; 16; 16 13. ƒ″1x2 = 2 / 11 + x23; 2; 2 / 27 14. ƒ″1x2 = -2x1x 2 + 32/ 11 - x 223; 0; 28 / 27 15. ƒ″1x2 = 4 / 1x 2 + 423/2; 1 / 2; 1 / 14 222 16. ƒ″1x2 = 18 / 12x 2 + 923/2; 2 / 3; 18 / 117 2172 17. ƒ″1x2 = -6x -5/4 or -6 / x 5/4; ƒ″102 does not exist; -3 / 21/4 2 2 18. ƒ″1x2 = 14 / 32x -5/3 or 4 / 13x 5/32; ƒ″102 does not exist; 21/3 / 3 19. ƒ″1x2 = 20x 2e -x - 10e -x ; -10; 70e -4 ≈ 1.282 2 x2 x2 x2 2 4 3 20. ƒ″1x2 = 2x e + e or e 12x + 12; 1; 9e ≈ 491.4 21. ƒ″1x2 = 1-3 + 2 ln x2/ 14x 2; ƒ″102 does not exist; -0.050 22. ƒ″1x2 = 12 - x2/ x 3; ƒ″102 does not exist; 0 23. f‴1x2 = 168x + 36; f 1421x2 = 168 24. f‴1x2 = -48x + 42; f 142 = -48 25. ƒ‴1x2 = 300x 2 - 72x + 12; f 1421x2 = 600x - 72 26. f‴1x2 = 120x 2 + 72x - 30; f 142 = 240x + 72 27. ƒ‴1x2 = 181x + 22-4 or 18 / 1x + 224; f 142 = -721x + 22-5 or -72 / 1x + 225 28. f‴1x2 - 6x -4 or -6 / x 4; f 142 = 24x -5 or 24 / x 5 29. f‴1x2 = -361x - 22-4 or -36 / 1x - 224; f 1421x2 = 1441x - 22-5 or 144 / 1x - 225 30. f‴1x2 = 2412x + 12-4 or 24 / 12x + 124; f 1421x2 = -19212x + 12-5 or -192 / 12x + 125 31. (a) n! (b) 0 32. (a) ƒ′1x2 = x -1 = 1 / x; ƒ″1x2 = -x -2 = -1 / x 2; f‴1x2 = 2x -3 = 2 / x 3; f 1421x2 = -6x -4 = -6 / x 4; f 1521x2 = 24x -5 = 24 / x 5 (b) f 1n21x2 = 1-12n - 11n - 12! / x n 33. ƒ″1x2 = e x; f‴1x2 = e x; f 1n21x2 = e x 34. ƒ″1x2 = ax1ln a22; f‴1x2 = ax1ln a23; f 1n21x2 = ax1ln a2n 35. Concave upward on 12, ∞2; concave downward on 1-∞, 22; inflection point at (2, 3) 36. Concave upward on 1-∞, 32; concave downward on 13, ∞2; inflection point at (3, 7) 37. Concave upward on 1-∞, -12 and 18, ∞2; concave downward on 1-1, 82; inflection points at 1-1, 72 and (8, 6) 38. Concave upward on 1-2, 62; concave downward on 1-∞, -22 and 16, ∞2; inflection points at 1-2, -42 and 16, -12 39. Concave upward on 12, ∞2; concave downward on 1-∞, 22; no inflection points 40. Concave upward on 1-∞, 02; concave downward on 10, ∞2; no inflection points 41. Always concave upward; no inflection points 42. Always concave downward; no inflection points 43. Concave upward on 1-∞, 3 / 22; concave downward on 13 / 2, ∞2; inflection point at 13 / 2, 525 / 22 44. Concave upward on 1-∞, -42; concave downward on 1-4, ∞2; inflection point at 1-4, 542 45. Always concave downward; no inflection points 46. Always concave upward; no inflection points 47. Concave upward on 15, ∞2; concave downward on 1-∞, 52; no inflection points 48. Concave upward on 1-∞, -12; concave downward on 1-1, ∞2; no inflection points 49. Concave upward on 1-10 / 3, ∞2; concave downward on 1-∞, -10 / 32; inflection point at 1-10 / 3, -250 / 272 50. Concave upward on 1-∞, 22; concave downward on 12, ∞2; inflection point at 12, -22 51. Never concave upward; always concave downward; no inflection points 52. Concave upward on 1-∞, - 22 / 22 and 1 22 / 2, ∞2; concave downward on 1- 22 / 2, 22 / 22; inflection points at 1- 22 / 2, 2 / 2e2 and 1 22 / 2, 2 / 2e2 53. Concave upward on 1-∞, 02 and 11, ∞2; concave downward on (0, 1); inflection points at (0, 0) and 11, -32 54. Concave upward on 1-8, ∞2; concave downward on 1-∞, -82; inflection point at 1-8, -7862 55. Concave upward on 1-1, 12; concave downward on 1-∞, -12 and 11, ∞2; inflection points at 1-1, ln 22 and 11, ln 22 56. Concave upward on 1-∞, -32 and 11, ∞2; concave downward at 1-3, -12 and 1-1, 12; inflection points at 1-3, 9 + 8 ln 22 and 11, 1 + 8 ln 22 57. Concave upward on 1-∞, -e -3/22 and 1e -3/2, ∞2; concave downward on 1-e -3/2, 02 and 10, e -3/22; inflection points at 1-e -3/2, -3e -3 / 12 ln 1022 and 1e -3/2, -3e -3 / 12 ln 1022 58. Concave upward on 1-∞, -1 / 22 ln 52 and 11 / 22 ln 5, ∞2; concave downward on 1-1 / 22 ln 5, 1 / 22 ln 52; inflection points at 1-1 / 22 ln 5, e -1/22 and 11 / 22 ln 5, e -1/22 59. Concave upward on 1-∞, 02 and 14, ∞2; concave downward on 10, 42; inflection points at 0 and 4 60. Concave upward on (0, 5); concave downward on 1-∞, 02 and 15, ∞2; inflection points at 0 and 5 61. Concave upward on 1-7, 32 and 112, ∞2; concave downward on 1-∞, -72 and 13, 122; inflection points at -7, 3, and 12 62. Concave upward on 1-∞, -52 and (1, 9); concave downward on 1-5, 12 and 19, ∞2; inflection points at -5, 1, and 9 63. Choose ƒ1x2 = x k where 1 6 k 6 2. For example, if ƒ1x2 = x 4/3 then ƒ has a relative minimum at x = 0. If ƒ1x2 = x 5/3 then ƒ has an inflection point at x = 0.

64. (a) (c) ƒ″102 = 0, but g″102 is undefined. (d) No 65. (a) Close to 0 (b) Close to 1 g(x) 5 x 5/3 66. Approaches 0; approaches ∞ 67. Relative maximum at x = -5 68. Relative minimum f(x) 5 x7/3 2 at x = 6 69. Relative maximum at x = 0; relative minimum at x = 2 / 3 70. Relative maximum at x = 0; relative minium at x = 4 / 3 71. Relative minimum at x = -3 72. No relative extrema 73. Relative maximum at x = -4 / 7; relative minimum at x = 0 22 2 74. Relative minimum at x = -5 / 8 75. (a) Relative minimum at about -0.4 and 4.0; relative maximum at about 2.4 (b) Increasing on about 1-0.4, 2.42 and about 14.0, ∞2; decreasing on 22 about 1-∞, -0.42 and 12.4, 4.02 (c) Inflection points at about 0.7 and about 3.3 (d) Concave upward on about 1-∞, 0.72 and 13.3, ∞2; concave downward on about 10.7, 3.32 76. (a) Relative minimum at 1; relative maximum at 0.6 (b) Increasing on 1-∞, 0.62 and 11, ∞2; decreasing on 10.6, 12 (c) Inflection points at 0, about 0.36, and about 0.84 (d) Concave upward on about 10, 0.362 and 10.84, ∞2; concave downward on 1-∞, 02 and about 10.36, 0.842 77. (a) Relative maximum at 1; relative minimum at -1 (b) Increasing on 1-1, 12; decreasing on 1-∞, -12 and 11, ∞2 (c) Inflection points at about -1.7, 0, and about 1.7 (d) Concave upward on about 1-1.7, 02 and 11.7, ∞2; concave downward on about 1-∞, -1.72 and 10, 1.72 78. (a) Relative minimum at about 0.5671 (b) Increasing on about 10.5671, ∞2; decreasing on about 10, 0.56712 (c) Inflection point at about 0.2315 (d) Concave upward on about 10.2315, ∞2; concave downward on about 10, 0.23152 79. One example is ƒ1x2 = 2x. 80. 114, 26,6882 81. 122, 6517.92 82. 11.11, 13.52 83. 12.06, 20.72 84. For U1M2 = 2M, risk aversion is I1M2 = 1 / 12M2. For U1M2 = M 2/3, risk aversion is I1M2 = 1 / 13M2. The utility function U1M2 = 2M indicates a greater aversion to risk. 86. Around the middle of 2045 87. (a) R′1x2 = C1xke -kx + 1 - e -kx2, all x in 30, 14 (b) R″1x2 = Cke -kx12 - kx2, 0 … x 6 2 / k 89. (a) Initial population (b) Inflection point Copyright © 2022 Pearson Education, Inc.

A-43

A-44

Complete Instructor Answers

(c) Maximum carrying capacity 90. c1t2 is increasing and concave downward; c′1t2 7 0 and c″1t2 6 0 91. (a) After 2 hours (b) 3 / 4% 92. (a) After 3 hours (b) 2 / 9% 93. 16.427, 15.72 94. 186.8, 3932 95. Inflection point at t = 1ln c2/ k ≈ 2.96 years; this signifies the time when the rate of growth begins to slow down, since L changes from concave up to concave down at this inflection point. 96. Inflection point at t = 1ln 27.32/ 0.011 ≈ 301 days; this signifies the time when the rate of growth begins to slow down, since L changes from concave up to concave down at this inflection point.

97. Always concave down 98. 132.01, 26.412; this signifies the time when the rate of increase in the number of teeth begins to slow down. 99. 1 / 26pD; 110 100. ƒ1t2 is decreasing and concave up; ƒ′1t2 6 0 but ƒ″1t2 7 0. 101. (a) -96 ft / sec (b) -160 ft / sec (c) -256 ft / sec (d) -32 ft / sec2 102. (a) 190 ft (b) At 5 seconds velocity is 19 ft / sec; at 10 seconds velocity is 34 ft / sec. (c) v1t2 7 0 for all t Ú 0 (d) At 5 seconds acceleration is 3 ft / sec2; at 10 seconds acceleration is 3 ft / sec2. (e) As t increases, the velocity increases, but the acceleration is constant. 103. v1t2 = 256 - 32t; a1t2 = -32; maximum height is 1024 ft; it hits the ground 16 seconds after being shot. 104. t = 6

Exercises 5.4 (page 327–328) W1. 1-1, 02, 12, -272 W2. 13, -140.72, 14, -160.72 1. False 2. False 3. False 4. False 5. 0 f(x) 7. (a) Increasing on 1-6, 32; (c) 250 (3, 179) decreasing on 1-∞, -62 and 13, ∞2; relative maximum of 179 x –6 3 at x = 3 and relative minimum of (–1.5, –185.5) –250 -550 at x = -6 (b) Concave upward on 1-∞, -1.52; concave –500 (– 6, –550) downward on 1-1.5, ∞2; inflection f(x) = –2x 3 – 9x 2 + 108x – 10 point at 1-1.5, -185.52 9. (a) Increasing nowhere; decreasing on 1-∞, ∞2; no relative extrema (b) Concave upward on 1-∞, 2 / 32; concave downward on 12 / 3, ∞2; inflection point at 12 / 3, -17 / 192

f(x)

(c)

4 2 –1

0 –2

(

1

2 , – 17 3 9

x

)

For exercises . . .

7–14,33–34 Refer to example . . . 1

8. (a) Increasing on 1-∞, -12 and 16, ∞2; decreasing on 1-1, 62; relative maximum of 8.5 at x = -1 and relative minimum of -163 at x = 6 (b) Concave upward on 15 / 2, ∞2; concave downward on 1-∞, 5 / 22; inflection point at 15 / 2, -77.252

(c)

10. (a) Increasing on 1-∞, ∞2; decreasing nowhere; no relative extrema (b) Concave upward on 12, ∞2; concave downward on 1-∞, 22; inflection point at 12, -32

(c)

f(x) = –3x3 + 6x2 – 4x – 1

f(x) (2, –16) (0, 0)

–16 –32

(4, 0) x

3 (3, –27)

f(x) = x 4

– 4x 3

(

17–22 3

)

5 , –77.25 2

– 11

11. (a) Increasing on f(x) (c) 1-2 23, 02 and 12 23, ∞2; decreasing on 1-∞, -2 232 and (0, 80) 10, 2 232; relative maximum of 200 (2√5, 0) (–2√5, 0) 80 at x = 0 and relative (–2, 0) (2, 0) minimum of -64 at x = -2 23 x (–2√3, –64) (2√3, –64) and x = 2 23 f(x) = x 4 – 24x 2 + 80 (b) Concave upward on 1-∞, -22 and 12, ∞2; concave downward on 1-2, 22; inflection points at 1-2, 02 and 12, 02 13. (a) Increasing on 13, ∞2; (c) decreasing on 1-∞, 32; relative minimum of -27 at x = 3 (b) Concave upward on 1-∞, 02 and 12, ∞2; concave downward on 10, 22; inflection points at (0, 0) and 12, -162

15,16 2

f(x) 12. (a) Increasing on (c) (–√3, 9) 10 (√3, 9) 1-∞, - 232 and 10, 232; (–1, 5) (1, 5) decreasing on 1- 23, 02 and 1 23, ∞2; relative maximum of (–√6, 0) (√6, 0) 9 at x = - 23 and x = 23 and x –1 (0, 0) relative minimum of 0 at x = 0 –5 (b) Concave upward on 1-1, 12; f (x) = –x 4 + 6x 2 concave downward on 1-∞, -12 and 11, ∞2; inflection points at 1-1, 52 and 11, 52

14. (a) Increasing on 1-∞, -32 and 13, ∞2; decreasing 1-3, 32; relative maximum of 162 at x = -3 and relative minimum of -162 at x = 3 (b) Concave upward on 1-3 / 22, 02 and 13 / 22, ∞2; concave downward on 1-∞, -3 / 222 and 10, 3 / 222; inflection points at (0, 0), 1-3 / 22, 100.232 and 13 / 22, -100.232

Copyright © 2022 Pearson Education, Inc.

(c)

25–32 4

Complete Instructor Answers

15. (a) Increasing on 1-∞, - 252 and 1 25, ∞2; decreasing on 1- 25, 02 and 10, 252; relative maximum of -4 25 at x = - 25 and relative minimum of 4 25 at x = 25 (b) Concave upward on 10, ∞2; concave downward on 1-∞, 02; no inflection points

(c)

17. (a) Increasing nowhere; decreasing on 1-∞, -22 and 1-2, ∞2; no relative extrema (b) Concave upward on 1-2, ∞2; concave downward on 1-∞, -22; no inflection points

(c)

16. (a) Increasing on 1-∞, 02 and (c) 11 / 2, ∞2; decreasing on 10, 1 / 22; relative minimum of 12 at x = 1 / 2 (b) Concave upward at 1-∞, 02 and 10, ∞2; concave downward nowhere; no inflection points

f(x) (√5, 4√5)

10 0

x

1 (–√5, –4√5)

f(x) 1 3 ,0 2√2

–1

18. (a) Increasing nowhere; decreasing on 1-∞, 22 and 12, ∞2; no relative extrema (b) Concave upward on 12, ∞2; concave downward on 1-∞, 22; no inflection points

(4, 0) x

0 1

f(x)

(c)

10 5 (0, 0) –2

6

3 –5

1

x

(–2, –1)

(c)

23. (a) Increasing on 1-∞, -32 and 1-3, 02; decreasing on 10, 32 and 13, ∞2; relative maximum of -1 / 9 at x = 0 (b) Concave upward on 1-∞, -32 and 13, ∞2; concave downward on 1-3, 32; no inflection points

(c)

f(x) 1 (1, 1/2) (√3, √3/4) (0, 0)

8

–1 f(x) =

x x2 + 1

f(x) 2

( ) 0, – 19

0.8

4

–0.8

x

–2 f(x) = 2 1 x –9

25. (a) Increasing on 1-∞, -1 / e2 and 11 / e, ∞2; decreasing on 1-1 / e, 02 and 10, 1 / e2; relative maximum of 1 / e at x = -1 / e and relative minimum of -1 / e at x = 1 / e (b) Concave upward on 10, ∞2; concave downward on 1-∞, 02; no inflection points

x

(–√3, –√3/4) (–1, –1/2)

–5

y

(c)

)– 1e , 1e ) –3

4

0 –4

f(x)

(3, ) 1 2

0

1 x 2 + 4x + 3

–8

x

–10

(c) 20. (a) Increasing on 13, 72 and 17, ∞2; decreasing on 1-∞, -12 and 1-1, 32; relative minimum of 1 / 2 at x = 3 (b) Concave upward on 1-1, 72; concave downward on 1-∞, -12 and 17, ∞2; no inflection points

f(x)

21. (a) Increasing on 1-1, 12; decreasing on 1-∞, -12 and 11, ∞2; relative maximum of 1 / 2 at x = 1 and relative minimum of -1 / 2 at x = -1 (b) Concave upward on 1- 23, 02 and 1 23, ∞2; concave downward on 1-∞, - 232 and 10, 232; inflection points at 1- 23, - 23 / 42, 10, 02, and 1 23, 23 / 42

x

f (x) = 3x x–2

(c)

f(x) =

1

–32

f(x) = –x + 4 x+2

19. (a) Increasing on 1-∞, -32 and 1-3, -22; decreasing on 1-2, -12 and 1-1, ∞2; relative maximum of -1 at x = -2 (b) Concave upward on 1-∞, -32 and 1-1, ∞2; concave downward on 1-3, -12; no inflection points

0

–16

f(x) = 16x + 12 x

f(x)

–5

1 2

(– )

f (x) = 2x + 10 x

2

( , 12)

4

) 1e , – 1e)

f (x) = x 1n x

x

22. (a) Increasing on 1-∞, 02; decreasing on 10, ∞2; relative maximum of 1 / 4 at x = 0 (b) Concave upward on 1-∞, -2 / 232 and 12 / 23, ∞2; concave downward on 1-2 / 23, 2 / 232; inflection points at 1-2 / 23, 3 / 162 and 12 / 23, 3 / 162

(c)

24. (a) Increasing on 1-∞, -22, 1-2, 22, and 12, ∞2; decreasing nowhere; no relative extrema (b) Concave upward on 1-∞, -22 and 10, 22; concave downward on 1-2, 02 and 12, ∞2; inflection point at (0, 0)

(c)

26. (a) Increasing on 1-∞, 02 and 11, ∞2; decreasing on 10, 12; relative minimum of 1 at x = 1 (b) Concave upward on 1-∞, 02 and 10, ∞2; concave downward nowhere; no inflection point

(c)

Copyright © 2022 Pearson Education, Inc.

x

1

5

f (x) =

–8 x 2 – 6x – 7

(0, ) , ( ) ( , ) f(x)

1 4

0.24

–2

2

3

2 1 x2 + 4

4

3

√3 16

√3 16

0.08 –4 –2 0 f(x) =

x

f(x) 2 (0, 0) –3

x 1

–4 f(x) = –2x x2 – 4

A-45

A-46

Complete Instructor Answers

(c) 27. (a) Increasing on 10, e2; decreasing on 1e, ∞2; relative maximum of 1 / e at x = e (b) Concave upward on 1e 1.5, ∞2; concave downward on 10, e 1.52; inflection point at 1e 1.5, 1.5 / e 1.52 ≈ 14.48, 0.332

29. (a) Increasing on 1-∞, 12; (c) decreasing on 11, ∞2; relative maximum of 1 / e at x = 1 (b) Concave upward on 12, ∞2; concave downward on 1-∞, 22; inflection point at 12, 2 / e 22

y

35. 7, 11, 13, 15, 19

(4.48, 0.33)

.3

) ) 1 e, e

0

y 1

e

5

(1, e –1) (2, 2e –2) 1

x

2 3

–2

y

(c)

1 –1

y 3

(c)

2 1 (–0.2, 0.410) –4 –3 –2 –1 0 –1

36. 8, 10, 14, 16, 20

x

9

f(x) = xe –x

–1

31. (a) Increasing on 1-∞, 22; decreasing on 12, ∞2; relative maximum of 1 / e 2 at x = 2 (b) Concave upward on 13, ∞2; concave downward on 1-∞, 32; inflection point at 13, 2 / e 32

33. (a) Increasing on 10, 2 / 52; decreasing on 1-∞, 02 and 12 / 5, ∞2; relative maximum of about 0.326 at x = 0.4 and relative minimum of 0 at x = 0 (b) Concave upward on 1-∞, -1 / 52; concave downward on 1-1 / 5, ∞2; inflection point at 1-0.2, 0.4102

1n x f(x) = x

.5

(2, e –2)

(3, 2e –3)

1

3

2

f(x) = x 2/3 – x5/3 (0.4, 0.326) 1 2 3 4 x

x

32. (a) Increasing on 10, ∞2; decreasing on 1-∞, 02; relative minimum of 2 at x = 0 (b) Concave upward on 1-∞, ∞2; concave downward nowhere; no inflection point

34. (a) Increasing on 1-1 / 4, ∞2; decreasing on 1-∞, -1 / 42; relative minimum of about -0.472 at x = -0.25 (b) Concave upward on 1-∞, 02 and 11 / 2, ∞2; concave downward on 10, 1 / 22; inflection points at 10, 02 and about 10.5, 1.1912

37. 21, 23, 27, 29, 31

In Exercises 39–43, other answers are possible. f(x) 40.

39.

38. 22, 24, 28 f(x)

41.

4

2

–4 2

42.

4

x

f(x)

43.

0 –2

4

x

f(x) 5

3 –2

3

x 1

(c)

30. (a) Increasing on (0, 2); decreasing (c) on 1-∞, 02 and 12, ∞2; relative maximum of 4 / e 2 at x = 2 and relative minimum of 0 at x = 0 (b) Concave upward on about 1-∞, 0.62 and 13.4, ∞2; concave downward on about 10.6, 3.42; inflection points at about 10.6, 0.192 and 13.4, 0.382

f(x) = (x – 1)e –x

–1

28. (a) Increasing on 1-∞, - 2e2 and 10, 2e2; decreasing on 1- 2e, 02 and 1 2e, ∞2; relative maxima of 1 / e at x = - 2e and x = 2e (b) Concave upward on 1-∞, -e 5/62 and 1e 5/6, ∞2; concave downward on 1-e 5/6, 02 and 10, e 5/62; inflection points at 1-e 5/6, 5 / 13e 5/322 and 1e 5/6, 5 / 13e 5/322

x

Copyright © 2022 Pearson Education, Inc.

(c)

(c)

Complete Instructor Answers

Chapter 5 Review Exercises (page 330–333)

For exercises . . .

1,2,12,13, 3,4,14,15, 17–24,70 25–32,63, 64,74,75(a), 76(b),(c) Refer to section . . . 1 2

5–9,16,33–38, 61,62,65–68, 73,75(b),76(a)

1. True 2. False 3. False 4. True 5. False 6. True 7. False 8. False 9. False 10. False 3 11. True 12. False 17. Increasing on 1-9 / 2, ∞2; decreasing on 1-∞, -9 / 22 18. Increasing on 1-∞, 7 / 42; decreasing on 17 / 4, ∞2 19. Increasing on 1-5 / 3, 32; decreasing on 1-∞, -5 / 32 and 13, ∞2 20. Increasing on 1-∞, -22 and 12 / 3, ∞2; decreasing on 1-2, 2 / 32 21. Never decreasing; increasing on 1-∞, 32 and 13, ∞2 22. Never increasing; decreasing on 1-∞, -7 / 22 and 1-7 / 2, ∞2 23. Decreasing on 1-∞, -12 and 10, 12; increasing on 1-1, 02 and 11, ∞2 24. Increasing on 1-∞, -1 / 42; decreasing on 1-1 / 4, ∞2 25. Relative maximum of -4 at x = 2 26. Relative minimum of -5 at x = 3 27. Relative minimum of -7 at x = 2 28. Relative maximum of -14 / 3 at x = 1 / 3 29. Relative maximum of 101 at x = -3; relative minimum of -24 at x = 2 30. Relative maximum of 25 at x = -2; relative minimum of -2 at x = 1 31. Relative maximum of 0.206 at x = -0.618; relative minimum of 13.203 at x = 1.618 32. Relative of about 0.83 at 2e / 3 ≈ 0.55 33. ƒ″1x2 = 36x 2 - 10; 26; 314 34. ƒ″1x2 = 54x + 2x -3 or 54x + 2 / x 3; 56; -4376 / 27 35. ƒ″1x2 = 18013x - 62-3 or 180 / 13x - 623; -20 / 3; -4 / 75 36. ƒ″1x2 = 11214x + 52-3 or 112 / 14x + 523; 112 / 729; -16 / 49 37. ƒ″1t2 = 1t 2 + 12-3/2 or 1 / 1t 2 + 123/2; 1 / 23/2 ≈ 0.354; 1 / 10 3/2 ≈ 0.032 38. ƒ″1t2 = 515 - t 22-3/2 or 5 / 15 - t 223/2; 5 / 8; does not exist 39. (a) Increasing on 1-1 / 2, 1 / 32; decreasing on 1-∞, -1 / 22 and 11 / 3, ∞2; relative maximum of -2.80 at x = 1 / 3 and relative minimum of -3.375 at x = -1 / 2 (b) Concave upward on 1-∞, -1 / 122; concave downward on 1-1 / 12, ∞2; inflection point at 1-1 / 12, -3.092 41. (a) Increasing on 1-1, 02 and 12, ∞2; decreasing on 1-∞, -12 and 10, 22; relative maximum of 1 at x = 0 and relative minima of -29 / 3 at x = 2 and of -2 / 3 at x = -1 (b) Concave upward on 1-∞, 11 - 272/ 32 and 111 + 272/ 3, ∞2; concave downward on 111 - 272/ 3, 11 + 272/ 32; inflection points at 111 + 272/ 3, -5.122 and 111 - 272/ 3, 0.112

f(x) 6

(c) –2

(

2

)

– 1 , –3.375 2 – 1 , –3.09 12

(

f(x) 12

–6

–1, – 2 3

( √

)

)

1– 7 , 0.11 3

)

)

0

3x

1+ 7 , –5.12 –10 3 f(x) = x

(c) 43. (a) Increasing on 1-∞, -1 / 22 and 1-1 / 2, ∞2; decreasing nowhere; no relative extrema (b) Concave upward on 1-∞, -1 / 22; –3 concave downward on 1-1 / 2, ∞2; no inflection points

(

(0, 1)

–2

(√

)

x 1 , –2.80 3 (0, –3)

f(x) = –2x3 – 1 x2 + x – 3 2

(c)

(

0

4–

(

)

2, – 29 3

4 x3 – 4x 2 + 1 3

f(x)

10

(1, 0) (0, –1)

f(x) =

3

x

40. (a) Increasing on 1-5 / 2, 32; (c) decreasing on 1-∞, -5 / 22 and 13, ∞2; relative maximum of 56 at x = 3 and relative minimum of -54.91 at x = -5 / 2 (b) Concave upward on 1-∞, 1 / 42; concave downward on 11 / 4, ∞2; inflection point at 11 / 4, 0.542 42. (a) Increasing on 1-1 / 2, 52; decreasing on 1-∞, -1 / 22 and 15, ∞2; relative maximum of 55.17 at x = 5 and relative minimum of -0.29 at x = -1 / 2 (b) Concave upward on 1-∞, 9 / 42; concave downward on 19 / 4, ∞2; inflection point at 19 / 4, 27.442

(c)

44. (a) Increasing on 1-∞, -32 and 1-3, ∞2; decreasing nowhere, no relative extrema (b) Concave upward on 1-∞, -32; concave downward on 1-3, ∞2; no inflection points

–10 x–1 2x + 1

Copyright © 2022 Pearson Education, Inc.

(c)

(

)

– 5 , –54.9 2

A-47

10,11,39–60, 63(e),69–71

4

A-48

Complete Instructor Answers

45. (a) Increasing on 1-2 / 3, 1 / 22; decreasing on 1-∞, -2 / 32 and 11 / 2, ∞2; relative maximum of 6.25 at x = 1 / 2 and relative minimum of 3.07 at x = -2 / 3 (b) Concave upward on 1-∞, -1 / 122; concave downward on 1-1 / 12, ∞2; inflection point at 1-1 / 12, 4.662

(c)

47. (a) Increasing on 10, ∞2; decreasing on 1-∞, 02; relative minimum of 0 at x = 0 (b) Concave upward on 1-∞, ∞2; concave downward nowhere; no inflection points

(c)

(

f(x) 10

(

–

2 , 3.07 3

)

–3

–2

–

5 2.5

–1

)

1 , 4.66 12 1 , 6.25 2 (0, 5)

(

1

–5

)

3 x

2

–10 f(x) = –4x3 – x2 + 4x + 5

48. (a) Increasing on 1-∞, 9 / 22; decreasing on 19 / 2, ∞2; relative maximum of 136.7 at x = 9 / 2 (b) Concave upward on (0, 3); concave downward on 1-∞, 02 and 13, ∞2; inflection points at 10, 02 and 13, 812

f(x) 20 10 2 x

–2 0 (0, 0) f(x) = x4 + 2 x2

49. (a) Increasing on 1-∞, -22 and (c) 12, ∞2; decreasing on 1-2, 02 and (0, 2); relative maximum of -4 at x = -2 and relative minimum of 4 at x = 2 (b) Concave upward on 10, ∞2; concave downward on 1-∞, 02; no inflection point

f(x) 6

(2, 4)

4

–4 –2 0

x

2 4 –4

(–2, –4)

51. (a) Increasing on 1-∞, 32 and (c) 13, ∞2; decreasing nowhere; no relative extrema (b) Concave upward on 1-∞, 32; concave downward on 13, ∞2; no inflection point

f(x) =

x2 + 4 x

f(x)

(0, 0) x

3

2

46. (a) Increasing on 1-∞, -22 and 11 / 3, ∞2; decreasing on 1-2, 1 / 32; relative maximum of 3 at x = -2 and relative minimum of -3.35 at x = 1 / 3 (b) Concave upward on 1-5 / 6, ∞2; concave downward on 1-∞, -5 / 62; inflection point at 1-5 / 6, -0.182

(c)

(c)

50. (a) Increasing on 1-∞, -2 222 and 12 22, ∞2; decreasing on 1-2 22, 02 and 10, 2 222; relative maximum of -4 22 at x = -2 22 and relative minimum of 4 22 at x = 2 22 (b) Concave upward on 10, ∞2; concave downward on 1-∞, 02; no inflection point

(c)

52. (a) Increasing nowhere; (c) decreasing on 1-∞, -1 / 22 and 1-1 / 2, ∞2; no relative extrema (b) Concave upward on 1-1 / 2, ∞2; concave downward on 1-∞, -1 / 22; no inflection point

f(x) = 2x 3–x

53. (a) Increasing on 1-1 / 2, ∞2; decreasing on 1-∞, -1 / 22; relative minimum of -1 / 12e2 at x = -1 / 2 (b) Concave upward on 1-1, ∞2; concave downward on 1-∞, -12; inflection point at 1-1, -e -22

f(x) 2

(c)

)– 21 , –12e) 1 –2

(0, 0) 1 x

(–1, –e –2) –1

f(x) = xe2x

55. (a) Increasing on 10, ∞2; decreasing on 1-∞, 02; relative minimum of ln 4 at x = 0 (b) Concave upward on 1-2, 22; concave downward on 1-∞, -22 and 12, ∞2; inflection points at 1-2, ln 82 and 12, ln 82

(c)

f(x) 4 3 (–2, ln 8) 2 1 –6 –4 –2 0

(2, ln 8) (0, ln 4)

2

4

6

x

54. (a) Increasing on 1-∞, -12 and (c) 10, ∞2; decreasing on 1-1, 02; relative maximum of e -2 at x = -1 and relative minimum of 0 at x = 0 (b) Concave upward on 1-∞, -1 - 22 / 22 and 1-1 + 22 / 2, ∞2; concave downward on 1-1 - 22 / 2, -1 + 22 / 22; inflection points at 1-1 - 22 / 2, 0.095882 and 1-1 + 22 / 2, 0.047752 56. (a) Increasing on 1e -1/2, ∞2; decreasing on 10, e -1/22; relative minimum of -1 / 12e2 at x = e -1/2 (b) Concave upward on 1e -3/2, ∞2; concave downward on 10, e -3/22; inflection point at 1e -3/2, -3 / 12e 322

f (x) = ln(x 2 + 4)

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(c)

f(x)

(

)

–1 – √2, 0.09588 2 1 (–1, e –2)

(

)

–1 + √2, 0.04775 2

(0, 0) 1 x

f(x) = x 2e2x

f(x)

1

0 –1

(e–3/2, –3/(2e3))

(1, 0) (e–1/2, –1/(2e)) f(x) = x 2ln x

2

x

Complete Instructor Answers

57. (a) Increasing on 1-1, ∞2; decreasing on 1-∞, -12; relative minimum of -3 at x = -1 (b) Concave upward on 1-∞, 02 and 12, ∞2; concave downward on 10, 22; inflection points at 10, 02 and 12, 61221/32

(c)

f(x) (2, 6(2)1/3) 2 (0, 0) 2

x

(–1, –3) f(x) = 4x

1/3

+x

4/3

58. (a) Increasing on 1-∞, -22 and 10, ∞2; decreasing on 1-2, 02; relative maximum of, 31222/3 at x = -2 and relative minimum of 0 at x = 0 (b) Concave upward on 11, ∞2; concave downward on 1-∞, 02 and 10, 12; inflection point at (1, 6)

A-49

f(x)

(c)

(–2, 3(2)2/3)

(1, 6)

6

(–5, 0) –4

1

(0, 0)

–2 f(x) = 5x

x

2/3 + x 5/3

In Exercises 59 and 60, other answers are possible. y 59.

60.

6

–2 4

–6 –3

6

x

–6

61. (a) Both are negative. 62. (a) P′1t2 = 0; P″1t2 6 0 63. (a) P1q2 = -q3 + 7q2 + 49q (b) 7 brushes (c) $229 (d) $343 (e) q = 7 / 3; between 2 and 3 brushes 64. (a) Increasing on (10, 12.02) and (17.20, 19); that is, increasing from 2010 to 2012, and from early 2017 to 2019 (b) Decreasing on (12.02, 17.20); that is, decreasing from 2012 to early 2017 (c) Relative maximum of about $3.70 in 2012 and a relative minimum of about $2.39 in early 2017 65. ƒ1t2 is increasing and concave downward. ƒ′1t2 is positive and decreasing. ƒ″1t2 is negative. 67. (a) The first derivative has many critical numbers. (b) The curve is always decreasing except at frequent inflection points. 68. (a) Metabolic rate and life span are increasing functions of mass. Heartbeat is a decreasing function of mass. Metabolic rate and life span have graphs that are concave downward.

Heartbeat has a graph that is concave upward.

69.

70. f(v) 71.

0.1

f(v) = v(0.25 – v) (v – 1)

1

v

–0.1

72. (a) 1486 ml / m2; for males with 1.88 m2 of surface area, the red cell volume increases approximately 1486 ml for each additional square meter of surface area. (b) S ≈ 1.57 m2; about 2593 ml (Hurley) and about 2484 ml (Pearson et al.) (c) 1578 ml / m2; for males with 1.57 m2 of surface area, the red cell volume increases approximately 1578 ml for each additional square meter of surface area. 73. 7.6108 yr; the age at which the rate of learning to pass the test begins to slow down.

74. (a) At 1965, 1973, 1976, 1983, 1986, and 1988 (b) Concave upward; this means that the stockpile was increasing at an increasingly rapid rate. 75. (a) v1t2 = 512 - 32t; a1t2 = -32 (b) 4096 ft (c) 32 seconds; -512 ft / sec 76. (b) 30, 444, 356, 654, 375, 844, 397, 1004 (c) 150, 1.52, 160, 3.02, 170, 3.52, 180, 4.02, 190, 4.52

Extended Application: A Drug Concentration Model for Orally Administered Medications (page 333–335) 1. The maximum steady-state concentration of 776.2 mcg/mL occurs at 1.28 hours. The minimum steady-state concentration of 185 mcg/mL occurs at 0 and 12 hours (immediately after a dose and immediately before the next dose). 2. The maximum steady-state concentration of 2325.6 mcg/mL occurs at 1.28 hours. The minimum steady-state concentration of 555 mcg/mL occurs at 0 and 12 hours. 3. The dose should be between 216 mg and 258 mg.

Chapter 6 Applications of the Derivative Exercises 6.1 (page 342–346)

For exercises . . .

5–10,35–42 Refer to example . . . 2

11,12,15–34,43 45–50,56–66 51–55 1 3 On Graphical Optimization

W1. -4 / 3, 5 / 2 W2. 0, 5 / 3 1. True 2. True 3. False 4. True 5. Absolute maximum at x3; no absolute minimum 6. Absolute minimum at x1; no absolute maximum 7. No absolute extrema 8. No absolute extrema 9. Absolute minimum at x1; no absolute maximum 10. Absolute maximum at x1; no absolute minimum 11. Absolute maximum at x1; absolute minimum at x2 12. Absolute maximum at x2; absolute minimum at x1

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A-50

Complete Instructor Answers

15. Absolute maximum of 12 at x = 5; absolute minimum of -8 at x = 0 and x = 3 16. Absolute maximum of 33 at x = -2; absolute minimum of -75 at x = 4 17. Absolute maximum of 17.5 at x = -3; absolute minimum of -7 / 6 ≈ -1.17 at x = 1 18. Absolute maximum of 31 / 3 ≈ 10.33 at x = -2; absolute minimum of -25 / 3 ≈ -8.33 at x = 2 19. Absolute maximum of 1 at x = 0; absolute minimum of -80 at x = -3 and x = 3 20. Absolute maximum of 137 at x = 6; absolute minimum of -263 at x = -4 and x = 4 21. Absolute maximum of 1 / 3 at x = 0; absolute minimum of -1 / 3 at x = 3 22. Absolute maximum of 7 at x = 6; absolute minimum of 3 at x = 4 23. Absolute maximum of 1 22 - 12/ 2 ≈ 0.21 at x = 1 + 22 ≈ 2.4; absolute minimum of 0 at x = 1 24. Absolute maximum of 22 / 4 ≈ 0.35 at x = 22; absolute minimum of 0 at x = 0 25. Absolute maximum of 51/3 ≈ 1.710 at x = 3; absolute minimum of 1-421/3 ≈ -1.587 at x = 0 26. Absolute maximum of 482/3 ≈ 13.208 at x = 8; absolute minimum of 0 at x = -4 and x = 4 27. Absolute maximum of 7 at x = 1; absolute minimum of 0 at x = 0 28. Absolute maximum of 4 at x = -8 and x = 1; absolute minimum of 0 at x = 0 29. Absolute maximum of about 4.910 at x = 4; absolute minimum of about -1.545 at x = 2 30. Absolute maximum of about 0.1839 at x = e 1/2; absolute minimum of 0 at x = 1 31. Absolute maximum of about 19.09 at x = -1; absolute minimum of about 0.6995 at x = 1ln 32/ 3 32. Absolute maximum of about 2.165 at x = 4; absolute minimum of about 1.472 at x = 2 33. Absolute maximum of 1.356 at about x = 0.6085; absolute minimum of 0.5 at x = -1 34. Absolute maximum of 0.5 at x = 0; absolute minimum of -8.10 at about x = -2.35 35. Absolute minimum of 7 at x = 2; no absolute maximum 36. Absolute maximum of 6 at x = 3; no absolute minimum 37. Absolute maximum of 137 at x = 3; no absolute minimum 38. Absolute minimum of 1 at x = 0 and x = 2; no absolute maximum 39. Absolute maximum of 0.1 at x = 4; absolute minimum of -0.5 at x = -2 40. Absolute maximum of 0.5 at x = 1; absolute minimum of -0.5 at x = -1 41. Absolute maximum of 1 / 13e2 ≈ 0.1226 at x = e 1/3; no absolute minimum 42. Absolute minimum of -e -1 ≈ -0.3679 at x = e -1; no absolute maximum 43. (a) Absolute minimum of -5 at x = -1; absolute maximum of 0 at x = 0 (b) Absolute maximum of about -0.76 at x = 2; absolute minimum of -1 at x = 1 44. (a) 45. (a) Relative maxima of 5546 in 2010 and 4185 in 2016; relative minima of 3879 in 2014 and 2975 in 2018 (b) Absolute maximum of 5546 in 2010 and absolute minimum of 2975 in 2018 46. (a) Relative maxima of 74 in 2010 and in 2012, 71 in 2014, and 65 in 2016; relative minima of 60 in 2011, 47 in 2013, 53 in 2015, and 56 in 2017 and 2018 (b) Absolute maximum of 74 in 2010 and 2012; absolute minimum of 47 in 2013 47. The maximum profit is $700,000 when 1,000,000 tires are sold. 48. Maximum weekly profit is $20,000 when 100 units per week are made 49. (a) 112 (b) 162 50. (a) 341 (b) 859.4 51. About 11.5 units 52. 20 units 53. 100 units 54. 280 units 56. (b) 1 / 16 (1 manager per 16 workers) 57. 6 mo; 6% 58. 12°C 59. About 7.2 mm 60. About 21.92°C 61. Absolute maximum of 25 mpg at 45 mph; absolute minimum of 16.1 mpg at 65 mph 62. Absolute maximum of 21.3 mpg at 43.7 mph; absolute minimum of 17.3 mpg at 60 mph 63. The piece formed into a circle should have length 12p / 14 + p2ft, or about 5.28 ft. 64. The piece formed into a circle should have length 12 ft.

66. (b) p = 1 / 2

Exercises 6.2 (page 352–357)

For exercises . . .

5–12,19–22,31, 13–18,23–28, 29,30,33–36, 37,38,47, 42–46, 39,40,41,52 32,55–56 51 48,53,54 49,50 Refer to example . . . 1 3 4 2 5

W1. 0, -4 / 5, 7 / 2 W2. 1-1 ± 2132/ 2 1. True 2. False 3. False 4. False 5. (a) y = 180 - x (b) P = x1180 - x2 (c) 30, 1804 (d) dP / dx = 180 - 2x; x = 90 (e) P102 = 0; P11802 = 0; P1902 = 8100 (f) 8100; x = 90 and y = 90 6. (a) y = 140 - x (b) P = 2x 2 - 280x + 19,600 (c) 30, 1404 (d) dP / dx = 4x - 280; x = 70 (e) P102 = 19,600; P1702 = 9800; P11402 = 19,600 (f) 9800; x = 70 and y = 70 7. (a) y = 90 - x (b) P = x 2190 - x2 (c) 30, 904 (d) dP / dx = 180x - 3x 2; x = 0, x = 60 (e) P102 = 0, P1602 = 108,000, P1902 = 0 (f) 108,000; x = 60 and y = 30 8. (a) y = 105 - x (b) P = 11,025x - 210x 2 + x 3 (c) 30, 1054 (d) dP / dx = 11,025 - 420x + 3x 2; x = 35, x = 105 (e) P102 = 0; P1352 = 171,500; P11052 = 0 (f) 171,500; x = 35 and y = 70 9. C1x2 = x 2 / 2 + 2x - 3 + 35 / x; x ≈ 2.722 10. C1x2 = 10x -1 + 20x -1/2 + 16x 1/2; x ≈ 2.110 11. (a) R1x2 = 160,000x - 100x 2 (b) x = 800; 800,000 candy bars (c) $640,000 12. (a) R1x2 = 12,000x - 125x 2 (b) x = 48; 48,000 compact discs (c) $288,000 13. (a) 1400 - 2x (b) A1x2 = 1400x - 2x 2 (c) 350 m (d) 245,000 m2 14. 75 m by 75 m; 5625 m2 15. 405,000 m2 16. 240,000 m2 17. $960 18. $1600 19. (a) 125 passengers (b) $156,250 20. (a) 65 seats (b) $422.50 21. In 10 days; $960 22. Maximum revenue is $40,612.50; minimum revenue is $0; fire the assistant. 23. 4 in. by 4 in. by 2 in.

24. 3 ft by 6 ft by 2 ft 25. 2 / 3 ft (or 8 in.) 26. (a) Both are 64 in.2. (b) Both are 100 / 9 ft2. (c) It appears that the area of the base and the total area of the walls for the box with maximum volume are equal. (This conjecture is true.)

27. 20 cm by 20 cm by 40 cm; $7200 28. Width is 2 26 + 2 ≈ 6.9 in.; length is 3 26 + 3 ≈ 10.3 in.

30. Radius ≈ 1.08 ft; height ≈ 4.34 ft; cost ≈ $44.11 31. 12.5 ft per side; $28,906.25 32. 233 m by 934 m; 467 lots; $50,814,737 33. Radius ≈ 5.206 cm; height ≈ 11.75 cm 34. Radius ≈ 5.454 cm; height ≈ 10.70 cm 35. Radius ≈ 5.242 cm; height ≈ 11.58 cm 36. Radius = 10 cm; height = 10 cm 37. 1 mile from point A 38. Point A 39. (a) 15 days (b) 16.875% 40. (a) 8 days (b) 29.43% 41. (a) 12 days (b) 50 per mL (c) 1 day (d) 81.365 per mL

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Complete Instructor Answers

A-51

42. 12.98 thousand 43. 5.071 thousand 45. 49.37 46. 237.10 47. Point P is 4 / 3 mi from Point A. 48. Point P is at point L. 49. (a) Replace a with e r and b with r / P. (b) Shepherd: ƒ′1S2 = a3a + 11 - c21S / b2c4/ 31 + 1S / b2c42; Ricker: ƒ′1S2 = ae -bS11 - bS2; Beverton-Holt: ƒ′1S2 = a / 31 + 1S / b242 (c) Shepherd: a; Ricker: a; Beverton-Holt: a; the constant a represents the slope of the graph of ƒ1S2 at S = 0. (d) 194,000 tons (e) 256,000 tons 50. S = b1 2a / r - 12 51. radius = 2.2 mm, height = 4.4 mm 53. He should travel 156 - 2 2212/ 7 ≈ 6.7 mi along the river. 54. He should travel 0 mi along the river. 55. 36 in. by 18 in. by 18 in. 56. (b) radius = 10 / p m (c) Arc length = 10 m; height = 10 m; they are equal. (d) radius = 10 / p m; arc length = 10 m

Exercises 6.3 (page 365–367)

For exercises . . .

5,6,13,14, 15,16 17–20 Refer to example . . . 1 2

12,27,28,29(a)–(c) 11,21–26,29(d), 30(a)–(c),31–34,36,37 30(d),35,38 3,4 5

W1. ƒ′1x2 = -k / x 2 W2. ƒ′1x2 = -2a / x 3 1. True 2. True 3. True 4. False 7. (c) 9. E is negative. 10. The demand function has a horizontal tangent line at the value of p where E = 0. 12. (a) E = k 13. (a) 10,000 lamps (b) 10 batches per year 14. (a) 310 cases (b) 45 batches per year 15. (a) 4899 copies (b) About 20.4 times per year 16. (a) 95 bottles of wine (b) About 9.5 times per year 18. 913 books 20. 10 production runs per year 21. (a) E = p / 1200 - p2 (b) 25 22. (a) E = p / 1500 - p2 (b) 12,5000 23. (a) E = 2p2 / 17500 - p22 (b) 25,000 24. (a) E = 2p2 / 14800 - p22 (b) 32,000 25. (a) E = 5 / q (b) 5 26. (a) E = 1 / 110 - ln p2 (b) 1 27. (a) E = 0.5; inelastic; a percentage increase in price will result is a smaller percentage change in demand. (b) E = 8; elastic; a percentage increase in price will result in a greater percentage decrease in demand.

28. (a) E = 2; elastic; a percentage increase in price will result in a greater percentage decrease in demand. (b) E = 0.5; inelastic; a percentage increase in price will result is a smaller percentage change in demand.

29. (a) E = 50p2 / 1-25p2 + 63,0752 (b) E = 0.66; inelastic; yes (c) E = 1.89; elastic; no (d) p = $29; maximum revenue is $1,219,450. 30. (a) E = 2p2 / 1400 - p22 (b) E = 0.13; inelastic; yes (c) E = 2.57; elastic; no (d) p = $11.55; maximum revenue is $307,920. 31. E = 0.06; the demand is inelastic (for any price); a 1% increase in the price of crude oil will result in only a 0.06% decrease in the demand for crude oil. 32. E = 0.13; the demand is inelastic (for any price); a 1% increase in the price of rice will result in only a 0.13% decrease in the demand for rice. 33. E = 2.826; the demand is elastic (for any price); a 1% increase in the price for the software will result is a 2.826% decrease in the demand for the software. 34. E ≈ 0.5314; the demand is inelastic (for any price); a 1% increase in the price of beef will result in about a 0.5314% decrease in the demand for beef. 35. (a) E = 0.071 (b) Inelastic; yes (c) $1255 36. (a) E = 10.604q2 - 20.16q + 263.0672/ 3q120.16 - 1.208q24 (b) E ≈ 1.51 (c) Elastic (d) E approaches infinity. 38. (a) The line intercepts the p-axis at 1-b / m, 02 and the q-axis at 10, b2. (b) 1-b / 12m2, b / 22

Exercises 6.4 (page 373–375)

For exercises . . .

5–21,42, 21–41 47–50 Refer to example . . . 1,2 3

43–46

W1. ƒ′1x2 = ln1x 2 + 12 + 2x 2 / 1x 2 + 12 W2. ƒ′1x2 = e x 13x 3 - 22/ x 3 5 1. False 2. True 3. False 4. True 5. dy / dx = -6x / 15y2 6. dy / dx = 7x / 14y2 7. dy / dx = 18x - 5y2/ 15x - 3y2 8. dy / dx = 17x - 2y2/ 12x + 5y2 9. dy / dx = 15x 2 / 16y + 42 10. dy / dx = 9x 2 / 116y + 102 11. dy / dx = -3x12 + y22 / 2 12. dy / dx = 5 / 32y15 - x224 13. dy / dx = 2y / 3 2x15 2y - 224 14. dy / dx = 2 2y / 3 2x19y + 424 15. dy / dx = 14x 3y 3 + 6x 1/22/ 19y 1/2 - 3x 4y 22 16. dy / dx = 14xy 4/3 + 12/ 118x 2/3y 5 - 4x 2y 1/32 2 2 17. dy / dx = 15 - 2xye x y2/ 1x 2e x y - 42 18. dy / dx = 13x 2 - 2xe y2/ 1x 2e y + 12 19. dy / dx = y12xy 3 - 12/ 11 - 3x 2y 32 20. dy / dx = 13x 3/2y 5/2 - 2y2/ 3x12 ln x - 5x 3/2y 3/224 21. y = 13 / 42x + 25 / 4 22. y = 14 / 32x - 50 / 3 23. y = x + 2 24. y = - 11 / 42x + 3 25. y = x / 64 + 7 / 4 26. y = - 11 / 42x + 3 27. y = 111 / 122x - 5 / 6 28. y = x 29. y = - 137 / 112x + 59 / 11 30. y = 7x - 5 31. y = 15 / 22x - 1 / 2 32. y = - 14 / 92x + 35 / 9 33. y = 1 34. y = 14x - 12 35. y = -2x + 7 36. y = - 1675 / 42x + 1377 37. y = -x + 2 38. y = - 12 / 112x + 15 / 11 39. y = 18 / 92x + 10 / 9 40. y = 111 / 122x - 5 / 6 41. (a) y = - 13 / 42x + 25 / 2; y = 13 / 42x - 25 / 2 y (b) 42. dy / dx = 13x 2 + a2/ 12y2 43. d 2x / dy 2 = -80 / 19y 22 44. d 2x / dy 2 = 1 / x 3/2 10 45. d 2x / dy 2 = 8x / y 5 46. d 2x / dy 2 = -21x 2 / y 7 47. (a) du / dv = -2u1/2 / 12v + 121/2 (6, 8) 5 (b) dv / du = - 12v + 121/2 / 12u1/22 (c) They are reciprocals.

2 2 2 2 48. (a) du / dv = 11 + e u - v2/ 12ue u - v2 (b) dv / du = 12ue u - v2/ 11 + e u - v2 –10 –5 0 5 10 x (c) They are reciprocals. 49. dy / dx = -x / y; there is no function y = ƒ1x2 that satisfies –5 (6, –8) x 2 + y 2 + 1 = 0. 50. dw / dx = 1 / 12e2 3

–10

51. (a) $0.94; at 5 units, the approximate increase in cost of an additional unit is $0.94. (b) $0; at 5 units, the approximate change in revenue for a unit increase in sales is $0. 52. (a) dq / dp = -2p / q; this is the rate of change of demand with respect to price. (b) dp / dq = -q / 12p2; this is the rate of change of price with respect to demand. 53. (a) E = 0.44; the demand is inelastic. (b) E = 0.44 54. (a) E = 0.678; the demand is inelastic. (b) E = 0.678 55. (a) and (b) R′1w2 = -29.0716w-1.43 56. dy / dx = -y / 1ax2 57. a = 1 / 13 232 60. ds / dt = 1-s + 6 2st2/ 18s 2st + t2 61. ds / dt = 14s - 6t 2 + 52/ 13s 2 - 4t2

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51–61 4

A-52

Complete Instructor Answers

Exercises 6.5 (page 380–382)

For exercises . . .

5–12,20–26 Refer to example . . . 1

13–18 5

19 6

27,28,33,35,36,37 29 3 2

W1. dy / dx = - 13x 2y + y 42/ 1x 3 + 4xy 32 W2. dy / dx = 13y 3 - 2x2/ 12y - 9xy 22 1. True 2. False 3. False 4. True 5. dy / dt = -64 6. dy / dt = -1 / 2 7. dy / dt = -9 / 7 8. dy / dt = -3 / 14 9. dy / dt = 1 / 5 10. dy / dt = 555 / 71 11. dy / dt = -3 / 2 12. dy / dt = -5 13. Cost is increasing at a rate of $384 per month. 14. Revenue is increasing at a rate of $135 per month. 15. (a) Revenue is increasing at a rate of $180 per day. (b) Cost is increasing at a rate of $50 per day. (c) Profit is increasing at a rate of $130 per day.

16. (a) Revenue is decreasing at a rate of $168 per day. (b) Cost is increasing at a rate of $60 per day. (c) Profit is decreasing at a rate of $228 per day. 17. Demand is decreasing at a rate of approximately 98 units per unit time. 18. Revenue is increasing at a rate of $1650 per day. 19. The blood velocity is increasing by 0.067 mm per minute. 21. The total mass is increasing by about 1.9849 g per day. 22. The rate of change of the energy expenditure is about -0.0067 cal / g / hr2.

23. (a) dr / dt = 105.15m-0.251dm / dt2 (b) The rate of change of the average daily metabolic rate is about 52.89 kcal per day2.

24. The rate of change of the energy expenditure is about -0.0521 kcal / kg / km / day. 25. The crime rate is rising at the rate of 25.6 crimes per month. 26. The number of words memorized is increasing by 0.008 words per minute. 27. The ladder is sliding down the building at the rate of 24 / 5 = 4.8 ft per minute. 28. (a) The distance between the two cars is increasing at the rate of 50 mph. (b) The distance between the two cars is increasing at a rate of about 47.15 mph. 29. The area is increasing at the rate of 16p ft2 per minute. 30. The volume of the snowball is changing at the rate of -16p in3 per hour. 31. The side of the ice cube is decreasing at the rate of 2 / 27 cm per minute. 32. The volume of the sand pile is increasing at the rate of 54p in3 per minute.

33. (a) The length of the shadow is increasing at the rate of 10 / 3 ft per second. (b) The tip of the shadow is moving at the rate of 25 / 3 ft per second. 34. The height of the water is increasing at a rate of 1 / 16 ft per minute. 35. The boat is approaching the dock at the rate of 22 ≈ 1.41 ft per second. 36. She must let out the string at a rate of 25 23 ≈ 43.3 ft per minute. 37. The runner’s distance from home plate is increasing at the rate of about 12.5 ft per second.

Exercises 6.6 (page 388–390)

For exercises . . .

5–12 13–20 Refer to example . . . 1 3

21,22,25–39 2

23,24 4

W1. ƒ′1x2 = 2x 3 / 2x 4 + 2 W2. ƒ′1x2 = 2e 2x / 1e 2x + 12 1. True 2. False 3. False 4. True 5. 1.9 6. 21 7. 0.1 8. -4.8 9. 0.060 10. 0.037 11. -0.023 12. -0.010 13. 12.0417; 12.0416; 0.0001 14. 4.8; 4.7958; 0.0042 15. 0.995; 0.9950; 0 16. 4.1275; 4.1255; 0.0020 17. 1.01; 1.0101; 0.0001 18. 0.998; 0.9980; 0 19. 0.05; 0.0488; 0.0012 20. -0.02; -0.0202; 0.0002 21. (a) Demand decreases by about 4.4 thousand lb. (b) Demand decreases by about 52.2 thousand lb. 22. (a) Average cost increases by about $2.18. (b) Average cost increases by about $4.50. 23. Revenue increases by about $60.

24. Profit decreases by about $15. 25. About 9600 in3 of coating should be ordered. 26. About 21,608p in.3 of material would be needed. 27. (a) Alcohol concentration increases by about 0.007435. (b) Alcohol concentration decreases by about 0.005105.

28. (a) Drug concentration increases by 0.2. (b) Drug concentration increases by about 0.037. 29. (a) Population increases by about 0.347 million. (b) Population decreases by about 0.022 million. 30. Area decreases by about 0.34p mm2.

31. Volume increases by about 1568p mm3. 32. Area increases by about 0.48p mi2. 33. Area increases by about 80p mm2.

34. (a) (0, 94) (b) About 0.47 years 35. (a) The pig will gain about 9.3 kg. (b) The pig will gain about 9.5 kg.

36. Volume increases by 12.8p cm3. 37. Volume decreases by 7.2p cm3. 38. Side increases by 0.0037 mm.

39. Volume increases by 0.472 cm3. 40. ±0.0138 in2 41. 0.00125 cm 42. ±1.224 in2; 0.01643 43. ±1.273 in3; 0.001549 44. 0.004 ft 45. ±0.116 in3

Chapter 6 Review Exercises (page 392–394)

For exercises . . .

1–3,11–18 4,5,49–54 6,19–30,43, 44,56 Refer to section . . . 1 3 4

7,8,31–38,55, 9,10,39–42, 59–61 62,63 5 6

40–45 5

45–48,57,58, 64–68 2

1. False 2. True 3. False 4. True 5. True 6. True 7. True 8. True 9. True 10. True 11. Absolute maximum of 33 at x = 4; absolute minimum of 1 at x = 0 and x = 6 12. Absolute maximum of -3 at x = 0; absolute minimum of -16 at x = -1 13. Absolute maximum of 39 at x = -3; absolute minimum of -319 / 27 at x = 5 / 3 14. Absolute maximum of 29 at x = -3; absolute minimum of -3 at x = -1 and x = 1 17. (a) Absolute maximum is 0.37; absolute minimum is 0. (b) Absolute maximum is 0.35; absolute minimum is 0.13.

18. (a) Absolute maximum is 13.65; absolute minimum is 7.39. (b) Absolute maximum is 44.83; absolute minimum is 7.39.

21. dy / dx = 12x - 9x 2y 42/ 18y + 12x 3y 32 22. dy / dx = 1-2xy 3 - 4y2/ 13x 2y 2 + 4x2 23. dy / dx = 6 2y - 1 / 3x 1/311 - 2y - 124 24. dy / dx = 9 2y / 32 2x11 - 12y 5/224 25. dy / dx = - 130 + 50x2/ 3 26. dy / dx = 12y - 2y 1/22/ 14y 1/2 + 9 - x2 27. dy / dx = 12xy 4 + 2y 3 - y2/ 1x - 6x 2y 3 - 6xy 22 28. dy / dx = 11 - 2x 2 - 2xy2/ 13xy 2 + 3y 3 - 12 29. y = 1-16 / 232x + 94 / 23 30. y = 11 / 122x + 5 / 4 33. 272 34. 10 / 3 35. -2 36. -2 / 9 37. -8e 3 38. -6e / 1e + 122 41. 0.00204 42. 0.1 43. (a) 12, -52 and 12, 42 (b) 12, -52 is a relative minimum; 12, 42 is a relative maximum. (c) No 45. (a) 600 boxes (b) $720 46. 2 m by 4 m by 4 m 47. 3 in. 48. Radius = 1.684 in., height = 4.490 in. 49. 1789 batteries 50. 775 cases 51. 80 lots 52. 127 batches 53. 0.47; inelastic 54. The demand is elastic when k 7 1 and inelastic when k 6 1. 55. Area increases by 56p ft2 per minute.

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30–32,34 4

Complete Instructor Answers

57. (a)

300

(b) About the 15th day 58. (a)

7.5

5

(b) Maximum number of polygons is about 237; minimum number is about 44.

0

51

A-53

95 0

0

59. 8 / 3 ft per min 60. About 0.0248 ft per min 61. 21 / 16 = 1.3125 ft per min 62. About 4.021 in3 63. ±0.736 in.2 64. About 43.1 in. 65. 1.25 + 2 ln 1.5 66. 225 m by 450 m 67. 10 ft; 18.67 sec 68. 0 ft; 15.72 sec

Extended Application: A Total Cost Model for a Training Program (page 395) 1. Z′1m2 = -C1 / m2 + DC3 / 2 2. m = 22C1 / 1DC32 3. m ≈ 3.33 4. m + = 4 and m- = 3 5. Z1m +2 = $11,400; Z1m-2 = $11,300 6. The optimal time interval is 3 months. There should be 9 trainees per batch.

7. Demand D for trainees per month varies from Marginal cost per trainee per month C2 varies from $50 to $250, with other values as in Exercise 3: 1 to 10, with other values as in Exercise 3: 2-104 Total cost

Total cost

1.5-104 1-104 5000

1-104

0 0

50

0

5 10 Trainee demand

Fixed cost C1 of training a batch varies from $5000 to $20,000, with other values as in Exercise 3:

250

2-104 Total cost

Total cost

150 200 Marginal cost

Salary for jobless trainee C3 varies from $100 to $1800, with other values as in Exercise 3:

2-104

1-104

0

100

0

1.104

2.104

3.10

4

Fixed cost

4

1-10

0

0

500

1000 1500 2000 Jobless salary

Chapter 7 Integration Exercises 7.1 (page 406–409)

For exercises . . .

9–28, 31,32

29,30,33–36, 39,40,45,46, 64,67,70 5,6

37,38, 41–44, 65 4

47,48

49–56, 61–63, 69 7

57–60

W1. ƒ′1x2 = 20x 3 + 3 / 2x Refer to example . . . 2,3 11 8 W2. ƒ′1x2 = 21e 3x - 20x 3/2 1. False 2. False 3. False 4. True 5. They differ only by a constant. 9. 6k + C 10. 9y + C 11. z 2 + 3z + C 12. 3x 2 / 2 - 5x + C 13. 2t 3 - 4t 2 + 7t + C 14. 5x 3 / 3 - 3x 2 + 3x + C 15. z 4 + z 3 + z 2 - 6z + C 16. 4y 4 + 3y 3 - 3y 2 + 3y + C 17. 10z 3/2 / 3 + 2 2z + C 18. 4t 5/4 / 5 + p1/4t + C 19. 5x 4 / 4 - 20x 2 + C 20. x 7 / 7 + x 4 + x 3 + C 21. 8v 3/2 / 3 - 6v 5/2 / 5 + C 22. 6x 5/2 + 4x 3/2 / 3 + C 23. 4u5/2 - 4u7/2 + C 24. 16t 7/2 + 4t 9/2 + C 25. -7 / z + C 26. -2 / x 2 + C 27. -p3 / 12y 22 - 2 2py + C 28. 2u3/2 / 3 - 1 / u + C 29. 6t -1.5 - 2 ln t + C 30. -4x -2.5 + 4 ln x + C 31. -1 / 13x2 + C 32. -2x 3 / 9 + C 33. -15e -0.2x + C 34. -20e 0.2v + C 35. -3 ln x - 10e -0.4x + e 0.1x + C 36. 9 ln x + 15e -0.4x / 2 + C 37. 11 / 42ln t + t 3 / 6 + C 38. 2y 1/2 / 3 - y 2 / 4 + C 39. e 2u / 2 + 2u2 + C 40. 1v 3 - e 3v2/ 3 + C 41. x 3 / 3 + x 2 + x + C 42. 4y 3 / 3 - 2y 2 + y + C 43. 6x 7/6 / 7 + 3x 2/3 / 2 + C 44. 3z 2/3 / 2 - 2z + C 45. 10 x / 1ln 102 + C 46. 32x / 321ln 324 + C 47. ƒ1x2 = 3x 5/3 / 5 48. ƒ1x2 = 2x 3 - 2x 2 + 3x + 1 49. C1x2 = 2x 2 - 5x + 8 50. C1x2 = 0.2x 3 / 3 + 5x 2 / 2 + 10 51. C1x2 = 3e 0.01x + 5 52. C1x2 = 2x 3/2 / 3 + 7 / 3 53. C1x2 = 3x 5/3 / 5 + 2x + 114 / 5 54. C1x2 = x 2 / 2 - 1 / x + 4 55. C1x2 = 5x 2 / 2 - ln x - 153.50 56. C1x2 = 1.2x + 8 57. p = 175 - 0.01x - 0.01x 2 58. p = 50 - 3x 2/3 59. p = 500 - 0.1 2x 60. p = 600 + 25,00011 - e 0.0002x2/ x 61. (a) p1t2 = 0.00528t 2 + 1.94t + 9.131 (b) Approximately 60,931 design patent applications 62. P1x2 = 2x 3/2 / 3 + x / 2 - 1 63. (a) P1x2 = 25x 4 / 2 + 10x 3 - 40 (b) $240 64. (a) ƒ1t2 = -e -0.01t + k (b) 0.095 units 65. a ln x - bx + C 66. (a) c′1t2 = -kA1c0 - C2e -kAt/V / V 67. (a) N1t2 = 155.3e 0.3219t + 144.7 (b) 7537 cells 68. V1t2 = 1kP0 / m2e -mt + V0 - kP0 / m 69. (a) B1t2 = 0.00979t 3 - 0.101t 2 + 8.95t + 840

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9,10

A-54

Complete Instructor Answers

(b) About 2,656,000 bachelor’s degrees 70. (a) D1t2 = 1597e 0.03537t + 40.50 (b) About 3907 dental degrees 71. v1t2 = 5t 3 / 3 + 4t + 6 72. s1t2 = 3t 3 - 2t 3/2 + 7 73. s1t2 = -16t 2 + 6400; 20 sec 74. s1t2 = 3t 3 + 4t 2 - 2t + 14 75. s1t2 = 2t 5/2 + 3e -t + 1 77. 160 ft / sec; 12 ft 78. (a) v1t2 = -32t + v0; s1t2 = -16t 2 + v0t (b) v0 = 224 (c) 3136 feet

Exercises 7.2 (page 416–417)

For exercises . . .

9,10,35, 11–14, 39,40 27,28 Refer to example . . . 1 3

15,16,33, 17–22,37, 23–26, 35,36 41,42 38 2 5 4

29–34

W1. 2x 5 + 4x 3/2 + C W2. 5 ln x - 2 / x 2 + C 6 W3. e 6x / 6 + C 1. True 2. True 3. False 4. False 5. Let u = 3x 2 - 5, then du = 6x dx. 6. Let u = 1 - x, then du = -dx. 7. Let u = 2x 3 + 1, then du = 6x 2 dx.

8. Let u = x 4, then du = 4x 3dx. 9. 212x + 325 / 5 + C 10. - 1-4t + 124 / 16 + C 11. -1 / 3212m + 1224 + C 12. 213u - 521/2 + C 13. -1 / 331x 2 + 2x - 4234 + C 14. -2 / 12x 3 + 721/2 + C 15. 14z 2 - 523/2 / 12 + C 3 2 2 3 16. 15r 2 + 223/2 / 15 + C 17. e 2x / 2 + C 18. -e -r / 2 + C 19. e 2t - t / 2 + C 20. e x - 3x / 3 + C 21. -e 1/z + C 2y 2 2 4 2 22. e + C 23. 3ln1t + 224/ 2 + C 24. -2 ln1x + 32 + C 25. 3ln1x + 4x + 724/ 4 + C 26. 3ln t 3 + 6t + 3 4/ 3 + C 27. -1 / 321x 2 + x224 + C 28. 12y 3 + 3y 2 + 121/3 / 2 + C 29. 1p + 127 / 7 - 1p + 126 / 6 + C 30. 818 - r25/2 / 5 - 6418 - r23/2 / 3 + C 31. 21u - 123/2 / 3 + 21u - 121/2 + C 32. -1 / 321x + 5244 + 2 / 1x + 525 + C 33. 1x 2 + 12x23/2 / 3 + C 34. 1x 2 - 6x23/2 / 3 + C 35. 11 + 3 ln x23 / 9 + C 36. 12 / 3212 + ln x23/2 + C 37. 11 / 22ln1e 2x + 52 + C 38. ln ln x + C 39. 1ln 1021log x22 / 2 + C 40. 1ln 223log215x + 1243 / 15 + C 2 41. 83x + 1 / 16 ln 82 + C 42. 2110252x + 2 / 15 ln 102 + C 45. (a) R1x2 = 61x 2 + 27,00021/3 - 180 (b) 150 devices 46. (a) D1t2 = 301t 2 + 12t23/2 + 900 (b) 7 years 47. (a) C1x2 = 6 ln15x 2 + e2 + 4 (b) Yes 2 48. (a) P1t2 = -e -t / 2 + 0.01 (b) The profit approaches $10,000.

49. (a) ƒ1t2 = 4.0674 * 10 -431t - 197022.3 / 2.3 + 19701t - 197021.3 / 1.34 + 61.298 (b) About 162,000 vehicles 50. (a) ƒ1t2 = 0.00148331t - 198022.75 / 2.75 + 19801t - 198021.75 / 1.754 + 262.951 (b) 1,344,000,000 outpatient visits 2 51. (a) P1t2 = 3250 - 1250e -t /5 (b) 3043 52. (a) N1t2 = 50 ln1t 2 + 22 + 2.343 (b) 307 people

Exercises 7.3 (page 425–429) 1. False 6.

2. False

45–52 7

For exercises . . .

9–18,21–26,30,31 19,20,28–29,32–36,39–41 37,38,42–44 Refer to example . . . 1,2 3 4

3. True 4. True

n 2 a i = 11x i + 32∆x; 4 n; any value x in the ith interval

/

5 /2

8

7. (a) 88 (b) 3 12x + 52 dx 8. (a) 25 / 12 (b) 3 11 / x2 dx 1 /2

0

9. (a) 21 (b) 23 (c) 22 (d) 22 10. (a) 14.5 (b) 17.5 (c) 16 (d) 16 11. (a) 10 (b) 10 (c) 10 (d) 11 12. (a) 30 (b) 54 (c) 42 (d) 41 13. (a) 8.22 (b) 15.48 (c) 11.85 (d) 10.96 14. (a) 27.19 (b) 80.79 (c) 53.99 (d) 47.43 15. (a) 6.70 (b) 3.15 (c) 4.93 (d) 4.17 16. (a) 1.283 (b) 0.95 (c) 1.117 (d) 1.090 17. (a) 4 (b) 4 18. (a) 12.5 (b) 12.5 19. (a) 4 (b) 5 20. (a) 8 + 4p (b) p 21. 4p 22. 9p / 2 23. 24 24. 14 25. (b) 0.385 (c) 0.33835 (d) 0.334334 (e) 0.333333 or 1 / 3 26. (b) 0.3025 (c) 0.255025 (d) 0.251001 (e) 0.25 or 1 / 4 28. Left: 153.0 million kilowatts; right: 154.6 million kilowatts; average: 153.8 million kilowatts 29. Left: 15,501 trillion BTUs; right: 23,499 trillion BTUs; average: 19,500 trillion BTUs 30. Left: $6000; right: $3500; average: $4750 31. Left: $235; right: $185; average: $210 32. Left: 35.7 liters; right: 35.9 liters; average: 35.8 liters 33. Left: 472,000 cases; right: 613,000 cases; average: 542,750 cases 34. Left: 22,908 fatal collisions; right: 24,462 fatal collisions; average 23,685 fatal collisions 35. About 1300 ft; yes 36. About 1900 ft; yes 37. Left: 4161 ft; right: 4607 ft; average: 4384 ft 38. Left: 4280 ft; right: 4737 ft; average: 4509 ft 39. (a) About 680 BTU per ft2 (b) About 320 BTU per ft2 40. (a) About 360 BTU per ft2 (b) About 170 BTUs per ft2 41. (a) 9 ft (b) 2 sec (c) 4.6 ft (d) Between 3 and 3.5 sec (e) About 1.9 sec (f) Car B 42. Left: 38 ft; right: 56 ft 43. Left: 22.5 ft; right: 18 ft 44. (a) 0.0486 mi (b) 0.0690 mi (c) 0.0622 mi

Exercises 7.4 (page 436–439)

For exercises . . .

5–10,13–16, 11,12,17,18,25, 19–24,27,28 26,31–34,58 Refer to example . . . 1,2,3 4

35–48,51 59–65, 67–76 5,6 7

W1. 21x 2 + 423/2 + C W2. 2 ln1x 2 + 22 + C W3. 5e x + C 1. True 2. False 3. False 4. True 5. -18 6. 5 22 ≈ 7.071 7. -3 / 2 8. 12 9. 28 / 3 10. 35 / 6 11. 13 12. 56 / 3 13. -16 / 3 14. -656 / 15 15. 76 16. -3038 / 15 17. 4 / 5 18. -1 / 3 19. 108 / 25 20. 5 / 81 21. 20e 0.3 - 20e 0.2 + 3 ln 2 - 3 ln 3 ≈ 1.353 22. 2 ln 2 + 10e -0.3 - 10e -0.6 ≈ 3.306 23. e 8 / 4 - e 4 / 4 - 1 / 6 ≈ 731.4 24. 15 / 64 - e 4 / 4 + e 2 / 4 ≈ -11.57 25. 91 / 3 26. 9,150,624 / 3 ≈ 3,050,208 27. 447 / 7 ≈ 63.86 28. 9 / 2 29. 1ln 222 / 2 ≈ 0.2402 30. 21ln 323/2 / 3 ≈ 0.7677 31. 49 32. 3 ln11 + ln 22 ≈ 1.580 33. 1 / 8 - 1 / 3213 + e 224 ≈ 0.07687 34. 21 + e 2 - 22 ≈ 1.482 35. 10 36. 26 37. 76 38. 54 39. 41 / 2 40. 52 41. e 2 - 3 + 1 / e ≈ 4.757 42. e - 1 + e -2 ≈ 1.854 43. e - 2 + 1 / e ≈ 1.086 44. e - 2 + 1 / e ≈ 1.086 45. 23 / 3 3

c

46. 8 / 3 47. e 2 - 2e + 1 ≈ 2.952 48. 1

b

c

49. 3 ƒ1x2 dx = 3 ƒ1x2 dx + 3 ƒ1x2 dx 50. (a) Yes a

a

b

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(b) Yes

51. -8

Complete Instructor Answers

A-55

55. -12 56. (a) 2.92530 (b) 14.98998 57. (a) ƒ1x2 = x 5 / 5 - 1 / 5 (c) ƒ′112 ≈ 2.746, and g112 = e ≈ 2.718 58. (a) 0 59. (a) 19000 / 821174/3 - 24/32 ≈ $46,341 (b) 19000 / 821264/3 - 174/32 ≈ $37,477 (c) It is slowly increasing without bound. 60. (a) 75 hours (b) 100 hours 61. No 62. (a) About 414 barrels (b) About 191 barrels (c) The number of barrels of oil leaking per day is decreasing to 0. 63. (a) 0.8778 ft (b) 0.6972 ft 64. About 760.3 ft 60

65. (a) 18.12

(b) 8.847

66. F1t2 = k1bT - 12/ ln b 67. (b) 3 n1x2 dx (c) 21513/2 - 263/22/ 15 ≈ 30.89 million 0

68. About 5.486 mg 69. (a) Q1R2 = pkR4 / 2 (b) 0.04k mm per min 70. About 33.8 cm 71. (b) About 505,000 kJ / W0.67 72. About 178 g 73. (a) About 308 million; the total population aged 0 to 90 (b) About 57 million 10

74. About 14,500,000 families 75. (a) c′1t2 = 1.2e 0.04t

(b) 3 1.2e 0.04t dt

(c) 30e 0.4 - 30 ≈ 14.75 billion

0

(d) About 12.8 yr (e) About 14.4 yr

76. 6.64 billion barrels

Exercises 7.5 (page 448–450)

For exercises . . .

5,6 7–10,15–22, 11–14,23,25,26, 31–34, 24,27,28 29,30,45,47 43,44 Refer to example . . . 1 3 2 4

35–41

W1. 1839 / 28 ≈ 65.68 W2. 1e 6 - 12/ 2 ≈ 201.2 5 1. False 2. True 3. True 4. True 5. 21 6. 53 / 4 7. 20 8. 22 / 3 9. 23 / 3 10. 26 11. 2197 / 6 ≈ 366.2 12. 343 / 6 ≈ 57.17 13. 4 / 3 14. 1 / 12 15. 2 ln 2 - ln 6 + 3 / 2 ≈ 1.095 16. ln 31 23 + 122 / 54 + 1 / 2 + 23 ≈ 2.633 17. 6 ln13 / 22 - 6 + 2e -1 + 2e ≈ 2.605 18. e 2 + e -2 + e + e -1 - 4 ≈ 6.611 19. 1e -2 + e 42/ 2 - 2 ≈ 25.37 20. 2 ln 2 - ln 3 + 1 / 4 ≈ 0.5377 21. 1 / 2 22. 1 / 2 23. 1 / 20 24. 1 / 6 25. 3124/32/ 2 - 3127/32/ 7 ≈ 1.620 26. 4 / 15 27. 1e 9 + e 6 + 12/ 3 ≈ 2836 28. e 4 + e 3 - 4e 2 + e + 1 ≈ 48.85 29. x ≈ -1.9241, x ≈ -0.4164; 0.6650 30. x ≈ 1.4027, x ≈ 3.4482; 3.3829 31. (a) 8 yr (b) About $148 (c) About $771 32. (a) 5 yr (b) $43.33 million 33. (a) 39 days (b) About $3369.18 (c) About $484.02 (d) About $2885.16 34. (a) 10 yr (b) About $834,000 35. About $12,931.66 36. About $1999.54 p 37. $54 38. $81 39. (a) (b) (15, 375) (c) $4500 (d) $3375 40. (a)

45,46 6

1000 D(q) = 900 – 20q – q2 800 600 400

S(q) = q2 + 10 q

200 0

5 10 15 20 25 q

(b) (9, 100) (c) About $1402.59 (d) $567 41. (a) 12 (b) $5616, $1116 (c) $1872, $1503 (d) $387 42. (a) About $290 thousand; the total income, from 2000 to 2018, of a family with an income in the lowest fifth of the U.S. population was about $290,000. (b) About $6330 thousand; the total income, from 2000 to 2018, of a family with an income in the top 5% of the U.S. population was about $6,329,000. (c) About 6040 thousand; the difference in total income of the top 5% and the bottom fifth was approximately $6,040,000 over this period. 43. (a) About 71.25 gal (b) About 25 hr (c) About 105 gal (d) About 47.91 hr 44. (a) About 23.04 gal (b) About 44.63 hr (c) About 102.88 gal (d) About 73.47 hr 45. (a) 0.019; the lower 10% of the income producers earn 1.9% of the total income of the population. (b) 0.184; the lower 40% of the income producers earn 18.4% of the total income of the population. (c) 0.30 46. (a) 0.269; the lower 60% of the Hispanic income producers earn about 26.9% of the total income (b) 0.46 47. 4 / 3

Exercises 7.6 (page 456–458)

For exercises . . .

5(a)–16(a),19, 5(b)–16(b),20–22,25, 23,24,29–32, 27(a)–28(a),33 26,27(b)–28(b),33,37 34–36 Refer to example . . . 1 2 3

1. False 2. True 3. False 4. False 5. (a) 12.25 (b) 12 (c) 12 6. (a) 7.5 (b) 22 / 3 ≈ 7.333 (c) 22 / 3 ≈ 7.333 7. (a) 3.35 (b) 3.3 (c) 3 ln 3 ≈ 3.296 8. (a) 3.997 (b) 3.909 (c) 3 ln111 / 32 ≈ 3.898 9. (a) 11.34 (b) 10.5 (c) 10.5 10. (a) 46.03 (b) 43.5 (c) 43.5 11. (a) 0.9436 (b) 0.8374 (c) 4 / 5 = 0.8 12. (a) 0.0973 (b) 0.0940 (c) 3 / 32 ≈ 0.09375 13. (a) 1.236 (b) 1.265 (c) 2 - 2e -1 ≈ 1.264 14. (a) 32.30 (b) 31.40 (c) 1333/2 - 12/ 6 ≈ 31.43 15. (a) 5.991 (b) 6.167 (c) 2p ≈ 6.283; Simpson’s rule 16. (a) 9.186 (b) 9.330 (c) 3p ≈ 9.4248; Simpson’s rule 17. Case (b) is true. 18. (a) Case (b) is true. (b) Case (a) is true. (c) Case (c) is true. 19. (a) 0.2 (b) 0.220703; 0.205200; 0.201302; 0.200325; 0.020703; 0.005200; 0.001302; 0.000325 (c) p = 2 (d) The error is multiplied by 1 / 4. 20. (a) 0.2 (b) 0.2005208; 0.2000326; 0.2000020; 0.2000001; 0.0005208; 0.0000326; 0.0000020; 0.0000001 (c) p = 4 (d) The error is multiplied by 1 / 16. 21. M = 0.7355; S = 0.8048 22. M = 0.09198; S = 0.09377 23. (a) 19,500 trillion BTUs (b) 19,322 trillion BTUs 24. (a) $57.75 thousand; total cost for given period is about $57,750. (b) $57.63 thousand; total cost for given period is about $57,530. 25. $3979 hundreds; the total revenue between the 12th and 36th months is about $397,900.

26. 1401 kg; the total amount of milk consumed by a calf from 7 to 182 days is about 1401 kg. 27. (a) 1.831 (b) 1.758 28. (a) 2.612 ft (b) 2.595 ft 29. About 30 mcg1h2/ ml; using Formulation A, this represents the total amount of drug available to the patient for each ml of blood. 30. About 33 mcg1h2/ ml; using Formulation B, this represents the total amount of drug

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A-56

Complete Instructor Answers

available to the patient for each ml of blood. 31. About 9 mcg1h2/ ml; for Formulation A, this represents the total effective amount of the drug available to the patient for each ml of blood. 32. About 4.4 mcg1h2/ ml; for Formulation B, this represents the total effective amount of the drug available to the patient for each ml of blood. 33. (a) y = b01t / 72b1e -b2t/7 (b) About 1212 kg; about 1231 kg (c) About 1224 kg; about 1250 kg 34. About 539,800 cases 35. (a) 71.5 (b) 69.0 36. (a) 128 (b) 128 37. (a) 0.6827 (b) 0.9545 (c) 0.9973

Chapter 7 Review Exercises (page 460–464)

For exercises . . .

1–4, 19–30, 79,80,84 Refer to section . . . 1

6–8,15, 41–43, 81,92,96 3

5,16,17, 31–40, 87,94 2

9,10,45–62, 82,83, 89–91,95 4

11,12, 63–66, 85,86 5

1. True 2. False 3. False 4. True 5. True 6. False 7. False 8. True 9. True 10. False 11. True 12. False 13. False 14. True 19. x 2 + 3x + C 20. 5x 2 / 2 - x + C 21. x 3 / 3 - 3x 2 / 2 + 2x + C 22. 6x - x 3 / 3 + C 23. 2t 3/2 + C 24. t 3/2 / 3 + C 25. 2x 3/2 / 3 + 9x 1/3 + C 26. 6x 7/3 / 7 + 2x 1/2 + C 27. 2y -2 + C 28. -5 / 13y 32 + C 29. -3e 2x / 2 + C 30. -5e -x + C 2 2 31. e 3x / 6 + C 32. e x + C 33. 13 ln u2 - 1 2/ 2 + C 34. 1ln 2 - u2 2/ 2 + C 35. -1 / 391x 3 + 5234 + C 4 2 36. 1x 2 - 5x25 / 5 + C 37. -e -3x / 12 + C 38. e 3x + 4 / 6 + C 39. 13 ln z + 225 / 15 + C 40. 215 ln z + 523/2 / 15 + C

13,14,18, 44,67–77, 93 6

T

41. 20

42. (a) 0

(b) 4.5

43. 24

44. 28

45. (a) s1T2 - s102 (b) 3 v1t2 dt = s1T2 - s102 is equivalent to the 0

Fundamental Theorem with a = 0 and b = T because s1t2 is an antiderivative of v1t2. 47. 12 48. 965 / 6 ≈ 160.83 49. 3 ln 5 + 12 / 25 ≈ 5.308 50. 2 ln 3 + 2 / 3 ≈ 2.864 51. 19 / 15 52. 1254/3 - 12/ 12 ≈ 6.008 53. 311 - e -42/ 2 ≈ 1.473 54. 251e 2 - e 0.42/ 4 ≈ 36.86 55. p / 32 56. 2p 57. 25p / 4 58. 9p 59. 13 / 3 60. 5504 / 7 61. 1e 4 - 12/ 2 ≈ 26.80 62. 5 - e -4 ≈ 4.982 63. 64 / 3 64. 1 / 6 65. 149 / 3 66. 32 67. 0.5833; 0.6035 68. 10.46; 10.20 69. 4.187; 4.155 70. 0.4688; 0.4908 71. 0.6011 72. 10.28 73. 4.156 74. 0.4937 75. (a) 4 / 3 (b) 1.146 (c) 1.252 76. (a) 8 - ln 5 ≈ 6.391 (b) 6.317 (c) 6.378 77. (a) 0 (b) 0 78. (c) 79. C1x2 = 12x - 123/2 + 145 80. C1x2 = 4 ln12x + 12 + 18 81. $8420 billion 82. About 26.3 years 83. $38,000 84. (a) ƒ1t2 = 0.0672t 2 - 0.9852t + 102.1 (b) 107.6 85. (a) $916.67 (b) $666.67 86. The company should use the machinery for 2.5 years. The net savings are about $99,000. 87. (a) 27.7345 billion barrels (b) 27.7345 billion barrels (d) y = 0.2585x - 0.6511; 27.87 billion barrels 88. (c) 89. 782 spiders 90. Approximately 142 people 91. (a) 0.2784 (b) 0.2784 92. (a) Left: 4710 pM; right: 4480 pM; average: 4595 pM (b) Left: 3016 pM; right: 2590 pM; average 2803 pM (c) The area under the curve is about 64% more for the fasting sheep. 93. (a) About 8208 kg (b) About 8430 kg (c) About 8558 kg 94. (a) $2,009,588 (b) $2,009,588 (d) y = 4431x + 196,190; $2,030,000 95. s1t2 = t 3 / 3 - t 2 + 8

Extended Application: Estimating Depletion Dates for Minerals (page 466) 1. About 177 yr 2. About 75.8 yr 3. About 45.4 yr 4. About 90.5 yr T 6. (a) 10 16,900e 3t/1t + 1002dt (b) About 77 years (in 2047)

5. (b) k1t2 = 3 / 1t + 1002

Chapter 8 Further Techniques and Applications of Integration Exercises 8.1 (page 475–477)

For exercises . . .

Refer to example . . .

5–8,40,45,46,48,49,51,52 1,3

9,10,39,43,44,47,50 2 5

11–14,24–26 4

17–23 1–3

W1. 3 / x W2. x 6 / 2 + 1 / x 2 + C W3. 3x 5/3 / 5 + 2x 1/2 + C W4. e 2x / 2 - 5 / e x + C W5. e x / 5 + C W6. 38 + e 3 ≈ 58.09 1. True 2. True 3. False 4. True 5. xe x - e x + C 6. 1x + 52 e x + C 7. 1-x / 2 + 23 / 162e -8x + C 8. - 11 / 2216x + 32 e -2x - 13 / 22e -2x + C 9. 1x 2 ln x2/ 2 - x 2 / 4 + C 10. 1x 4 ln x2/ 4 - x 4 / 16 + C 11. -5e -1 + 3 ≈ 1.161 12. 1e -3 + 22/ 3 ≈ 0.6833 13. 26 ln 3 - 8 ≈ 20.56 14. ln 20 - 1 ≈ 1.996 15. e 4 + e 2 ≈ 61.99 16. 1e 2 + 52/ 4 ≈ 3.097 17. x 2e 2x / 2 - xe 2x / 2 + e 2x / 4 + C 18. 11 / 62 ln 2x 3 + 1 + C 19. 12 / 32x 21x + 423/2 - 18 / 152x1x + 425/2 + 116 / 10521x + 427/2 + C or 12 / 721x + 427/2 - 116 / 521x + 425/2 + 132 / 321x + 423/2 + C 2 20. 1x 2 - x2 ln 3x - x 2 / 2 + x + C 21. 14x 2 + 10x2 ln 5x - 2x 2 - 10x + C 22. e x / 4 + C 23. 1-e 2 / 4213e 2 + 12 ≈ -42.80 3 24. 11 / 621ln 32 ≈ 0.1831 25. 2 23 - 10 / 3 ≈ 0.1308 26. 243 / 8 - 22 / 4 ≈ 29.43 27. 16 ln x + 2x 2 + 16 + C 28. ln 1x - 52/ 1x + 52 + C 29. - 13 / 112 ln 111 + 2121 - x 22/ x + C 30. - 12 / 152ln x / 13x - 52 + C 31. -1 / 14x + 62 - 11 / 62ln x / 14x + 62 + C 32. 1x / 22 2x 2 + 15 + 115 / 22ln x + 2x 2 + 15 + C 33. The product rule 35. -18 36. 101 37. 15 38. -35 41. (a) 12 / 32x1x + 123/2 - 14 / 1521x + 125/2 + C (b) 12 / 521x + 125/2 - 12 / 321x + 123/2 + C 42. The integration constant is missing. 43. 1169 / 22 ln 13 - 42 ≈ $174.74 44. e 4 + 2 ln 2 - 10 45. 300 - 700 / e 46. 25e - 25 47. 72 ln 12 + 12 - 80 ln 8 48. 3.431 49. 15e 6 + 3 ≈ 6054

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A-57

50. 18 ln 9 - 116 / 32 ln 4 - 76 / 9 ≈ 23.71 sq cm 51. About 219 kJ 52. (a) 1 - 1 / k + 11 / k2e -k; k = 1 / 12: 12e -1/12 - 11 ≈ 0.0405; k = 1 / 24: 24e -1/24 - 23 ≈ 0.0205; k = 1 / 48: 48e -1/48 - 47 ≈ 0.0103 (b) e -6k / 15k2 + 11 - 115k22e -k; k = 1 / 12: 112 / 52e -1/2 - 17 / 52e -1/12 ≈ 0.1676; k = 1 / 24: 124 / 52e -1/4 - 119 / 52e -1/24 ≈ 0.0933; k = 1 / 48: 148 / 52e -1/8 - 143 / 52e -1/48 ≈ 0.0493 For exercises . . .

Refer to example . . .

Exercises 8.2 (page 482–484)

5–27,36,37,46,49 1–3

28–35,38–43,45,47,48 4

44 Derivation of volume formula

W1. 218 / 3 W2. 243 / 5 W3. 14 / 3 W4. 14 W5. 1e 2 - 12/ 2 ≈ 3.1945 W6. 19 ln 32/ 2 - 2 ≈ 2.9438 1. True 2. True 3. False 4. False 5. 9p 6. 24p 7. 364p / 3 8. 72p 9. 386p / 27 10. 1685p / 12 11. 15p / 2 12. 14p 13. 18p 14. 12p 15. p1e 4 - 12/ 2 ≈ 84.19 16. 2p1e 2 - e -42 ≈ 46.31 17. 4p ln 3 ≈ 13.81 18. 4p ln 4 ≈ 17.42 19. 3124p / 5 20. 256p / 5 21. 16p / 15 22. 64p 22 / 15 23. 4p / 3 24. 288p 25. 4pr 3 / 3 26. 4ab2p / 3 27. pr 2h 28. -11 29. 13 / 3 ≈ 4.333 30. 31 / 9 ≈ 3.444 31. 38 / 15 ≈ 2.533 32. e - 1 ≈ 1.718 33. e - 1 ≈ 1.718 34. 1e 2 + 12/ 341e - 124 ≈ 1.221 35. 15e 4 - 12/ 8 ≈ 34.00 36. 0.5048 37. 3.758 38. $38.83 39. $42.49 40. 200 cases 41. 200 cases 42. (a) 38.6% (b) 54.3% 43. (a) 2.27 billion (b) 12.27 billion R

44. (a) 2pk 3 r1R2 - r 22 dr

(b) pkR4 / 2 45. (a) 110e -0.1 - 120e -0.2 ≈ 1.284 (b) 210e -1.1 - 220e -1.2 ≈ 3.640

0

(c) 330e -2.3 - 340e -2.4 ≈ 2.241 46. (a) (i) 151.5 cubic cm (ii) 2.238 cubic cm (iii) 8.586 cubic cm (iv) 0.5236 cubic cm (v) 28.51 cubic cm 47. (a) 916 ln 6 - 52 ≈ 51.76 (b) 5110 ln 10 - 92 ≈ 70.13 (c) 3131 ln 31 - 302/ 2 ≈ 114.7 48. (a) 40 words/minute (b) 100 words per minute; 4 minutes (c) 83.75 words/minute 49. 1.083 * 10 21 m3

Exercises 8.3 (page 491–492)

For exercises . . .

Refer to example . . .

5(a)–12(a),20 2

15(b)–12(b),19,21,22 3,4

13–18,23,24 5

W1. 30,000 e 0.01t + C W2. 553.51 W3. -10,000 te -0.04t - 250.000 e -0.04t + C W4. 10,951.94 1. True 2. False 3. False 4. True 5. (a) $6883.39 (b) $15,319.26 6. (a) $2065.02 (b) $4595.78 7. (a) $3441.69 (b) $7659.63 8. (a) $13,766.78 (b) $30,638.52 9. (a) $3147.75 (b) $7005.46 10. (a) $6911.51 (b) $15,381.86 11. (a) $32,968.35 (b) $73,372.42 12. (a) $6321.21 (b) $14,068.10 13. (a) $746.91 (b) $1662.27 14. (a) $1493.81 (b) $3324.54 15. (a) $688.64 (b) $1532.59 16. (a) $3443.19 (b) $7662.96 17. (a) $11,351.78 (b) $25,263.84 18. (a) $31,965.78 (b) $71,141.15 19. $63,748.43 20. (a) $45,231.83 (b) $41,469.08 (c) $38,121.66 21. $28,513.76; $54,075.81; $25,562.05 22. $4122.44; $4741.93; $619.49 23. $4560.94 24. $3227.34

Exercises 8.4 (page 496–498)

For exercises . . .

Refer to example . . .

5–30,35–40,43 1

31–34,41,42 2

48–54,57 4

55,56,58–62 3

W1. 0 W2. 0 W3. 0 W4. ∞ 1. False 2. False 3. True 4. False 5. 1 / 3 6. 1 / 32 7. Divergent 8. Divergent 9. -1 10. 1 / 64 11. 10,000 12. Divergent 13. 1 / 10 14. -1 / 18 15. 3 / 5 16. -1 / 6 17. 1 18. 1 19. 1000 20. 1 / 12 21. Divergent 22. Divergent 23. 1 24. 1 / 8 25. Divergent 26. Divergent 27. Divergent 28. 1 / ln 2 29. Divergent 30. -25 31. 0 32. 2 33. Divergent 34. Divergent 35. Ce -ka / k 36. C / 31k - 12ak - 14 37. Divergent 38. Divergent 39. 1 40. 3 / 2 41. 0 42. 0 45. (a) 2.808, 3.724, 4.417, 6.720, 9.022 (b) Divergent (c) 0.8770, 0.9070, 0.9170, 0.9260, 0.9269 (d) Convergent 46. (a) 0.7468, 0.8862, 0.8862, 0.8862 (b) Convergent; 0.8862 47. (a) 9.9995, 49.9875, 99.9500, 995.0166 (b) Divergent (c) 100,000 48. $3,750,000 49. $20,000,000 50. (a) $144,000 (b) $72,000 51. $30,000 52. $200,000 53. $30,000 54. Be -kB / 11 - e -kB2 55. 4478 million lb 56. 4 57. K1a + br2/ r 2 58. Na / 3b1b + k24 59. 33,544 60. About 833.3 61. 1250 62. (a) 1 (b) 1 (c) 2 (d) 6

Chapter 8 Review Exercises (page 500–502)

For exercises . . .

Refer to section . . .

1–4,14–27,61 1

5–7,28–37,59,60,65 2

8,9,46–57 3

10,38–45,58,62–64 4

1. False 2. True 3. False 4. True 5. False 6. False 7. True 8. True 9. True 10. False 14. - 12x / 5218 - x25/2 - 14 / 35218 - x27/2 + C 15. 6x1x - 221/2 - 41x - 223/2 + C 16. xe x - e x + C 17. - 1x + 22e -3x - 11 / 32e -3x + C 18. 11 / 4214x + 521ln 4x + 5 - 12 + C 19. 1x 2 / 2 - x2 ln x - x 2 / 4 + x + C 20. 1-1 / 182 ln 25 - 9x 2 + C 21. 11 / 82 216 + 8x 2 + C 22. 13e 4 + 12/ 16 ≈ 10.30 23. 10e 1/2 - 16 ≈ 0.4872 24. 17 / 421e 2 - 12 ≈ 11.18 25. 234 / 7 ≈ 33.43 26. 5 27. -19 28. 125p / 9 ≈ 43.63 29. 81p / 2 ≈ 127.2 30. p1e 4 - e -22/ 2 ≈ 85.55 31. p ln 3 ≈ 3.451 32. 406p / 15 ≈ 85.03 33. 64p / 5 ≈ 40.21 34. 7pr 2h / 12 36. 13 / 6 37. 2,391,484 / 3 38. Divergent 39. 1 / 5 40. 1 / 3 41. 6 / e ≈ 2.207 42. Divergent 43. Divergent 44. 5 45. 3 47. 16,250 / 3 ≈ $5416.67 48. $28,513.76 49. $174,701.45 50. $713.72 51. $15.58 52. $5830.98

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A-58

Complete Instructor Answers

53. $5354.97 54. $390.61 55. $30,035.17 56. $5715.89 57. $176,919.15 58. $555,555.56 59. 24% 60. 926 thousand short tons 61. 0.4798 62. 5000 gallons 63. 2825 million tons 64. 145 billion tons 65. (a) 158.3° (b) 125° (c) 133.3°

Extended Application: Estimated Learning Curves with Manufacturing with Integrals (page 502) 1. C(280) ≈ C(1) * 280 - 0.152 ≈ 271,000 * 280 - 0.152 ≈ $115,000 4. (a) No (b) About -3.7%

2. No

3. Choose a = 1, b = 1ln r2/ 1ln 22.

Chapter 9 Multivariable Calculus Exercises 9.1 (page 513–517)

y

W1.

For exercises . . .

5–10,33,34, 11–18 42–52,55 Refer to example . . . 1–3 4–6 y

W2.

7,8

material after Example 8

8

W4. -4 W5. 18a2 + 9a - 6 W6. 4x + 2h + 3 y=2

2 x

3

37–40

x=4

2 0

27–32

y

W3.

2x + 3y = 6

19–22

0

x

4

x

0

1. False 2. True 3. True 4. False 5. (a) 12 (b) -6 (c) 10 (d) -19 6. (a) 84 (b) -11 (c) 43 (d) 16 7. (a) 243 (b) 6 (c) 219 (d) 211 8. (a) 10 (b) 2905 / 2 (c) 10 210 (d) - 225.9 9. (a) e (b) e 2 (c) 2 (d) 3 10. (a) e (b) 2 (c) 3e 5 (d) -e 3 11.

12.

z 9

13.

z 15

14.

z

3 9

15 y

y

x+y+z=9

9

15

x

x z

15.

6

x + y + z = 15

16.

y

4

2x + 3y + 4z = 12

x z

17.

z

18.

4 4 4

y

y 5

x+y=4 x

x

19.

y z=4

x=5

y 12 10 8 6 4 2 0

20.

y

21.

8 z=4

6 z=2

z=0

z=2

4 2

1 2 3 4 5 6 7 8

x 2

z=0

y 2.5 2 1.5 1 0.5 0

4 x

x

z=0 z=2 z=4

1 2 3 4 5 6 7 8 x

22.

y 12 10 8 6 4 2 0

z=4 z=2 z=0

1 2 3 4 5 6 7 8 x

z=4

27. (c) 28. (f) 29. (e) 30. (a) 31. (b) 32. (d) 33. (a) 8x + 4h (b) -4y - 2h (c) 8x (d) -4y 34. (a) 15x 2 + 15xh + 5h2 (b) 6y + 3h (c) 15x 2 (d) 6y 35. (a) 3e 2 (b) 3e 2 36. ƒ1x, y2 = 4 + 2x + 3y 37. (a) 1987 (rounded) (b) 595 (rounded) (c) 359,768 (rounded) 38. y = 500 4 / 11.01x 3/424 ≈ 6 # 10 10 / x 3

1 3 1010

39. y = 500 5/2 / x 3/2 ≈ 5,590,170 / x 3/2

y

y 2000

8 3 109 6 3 109

1500

4 3 109

1000

2 3 109

500 0

2

4

6

8 10

x

Copyright © 2022 Pearson Education, Inc.

0

500 1000 1500 2000 x

Complete Instructor Answers

A-59

40. z is multiplied by 20.75; z is multiplied by 20.25; z is doubled. 41. C1x, y, z2 = 250x + 150y + 75z 42. 1.14; the IRA account grows faster. 43. 1.416; the IRA account grows faster. 44. (a) 183 W (b) 135 W 45. (a) 1.89 m2 (b) 1.62 m2 (c) 1.78 m2 46. (a) 1.5 m per sec; 5.5 m per sec (b) 1 m per sec 47. (a) 8.7% (b) 48% (c) Multiple solutions: W = 19.75, R = 0, A = 0 or W = 10, R = 10, A = 4.59 (d) Wetland percentage 48. (a) About 211 (b) Average temperature 49. (a) 397 accidents 50. (a) (i) 2.91 (ii) 2.53 (iii) 2.35 (b) It decreases. (c)

(d)

H

R

0.0383T + 0.0224H = 2.968

120 80

300

40

200 T

0

20 40 60 80 100 T

Exercises 9.2 (page 524–529)

100

6 4 0

100 200 H

300

51. (a) T = 242.257C 0.18 / F 3 (b) 58.82; a tethered sow spends nearly 59% of the time doing repetitive behavior when she is fed 2 kg of food a day and neighboring sows spend 40% of the time doing repetitive behavior. 52. (a) 20.0 yr (b) 363 yr 53. ƒ1L, W, H2 = L + 2H + 2W 54. G1L, W, H2 = 2LW + 2WH + 2LH ft2 55. (a) 4.69 in. by 3.75 in. (b) 4.04 in. by 3.75 in.

For exercises . . .

4,6,37–40,47,48 7–24,57 25–36 Refer to example . . . 3 1–3 6,7

41–46 8

49–51,59–75 52–56,58 4 5

W1. ƒ′1x2 = 6x 2 + 7 W2. ƒ′1x2 = 1 / 12 2x2 - 10 / x 3 + 12e 2x W3. ƒ′1x2 = e x + xe x W4. ƒ′1x2 = 16x 5 - 60x 32/ 1x 2 - 522 2 2 W5. ƒ′1x2 = 1 / 22x + 8 W6. ƒ′1x2 = 271e x + 5x2212xe x + 52 W7. ƒ′1x2 = 6x 2 / 12x 3 + 52 W8. ƒ′1x2 = ln x 2 + 21x + 12/ x 1. True 2. True 3. True 4. True 5. (a) 12x - 4y (b) -4x + 18y (c) 12 (d) -40 6. (a) 8 + 12xy (b) 6x 2 + 4y (c) 54 (d) 32 7. ƒx1x, y2 = -4y; ƒy1x, y2 = -4x + 18y 2; 4; 178 8. ƒx1x, y2 = 18xy 2; ƒy1x, y2 = 18x 2y - 8y; 36; 840 9. ƒx1x, y2 = 10xy 3; ƒy1x, y2 = 15x 2y 2; -20; 2160 10. ƒx1x, y2 = -12x 3y 3; ƒy1x, y2 = -9x 4y 2; 96; -20,736 11. ƒx1x, y2 = e x + y; ƒy1x, y2 = e x + y; e 1 or e; e -1 or 1 / e 12. ƒx1x, y2 = 12e 3x + 2y; ƒy1x, y2 = 8e 3x + 2y; 12e 4; 8e -6 13. ƒx1x, y2 = -24e 4x - 3y; ƒy1x, y2 = 18e 4x - 3y; -24e 11; 18e -25 14. ƒx1x, y2 = 56e 7x - y; ƒy1x, y2 = -8e 7x - y; 56e 15; -8e -31 15. ƒx1x, y2 = 1-x 4 - 2xy 2 - 3x 2y 32/ 1x 3 - y 222; ƒy1x, y2 = 13x 3y 2 - y 4 + 2x 2y2/ 1x 3 - y 222; -8 / 49; -1713 / 5329 16. ƒx1x, y2 = 6xy 5 / 1x 2 + y 222; ƒy1x, y2 = 19x 4y 2 + 3x 2y 42/ 1x 2 + y 222; -12 / 25; 24,624 / 625 17. ƒx1x, y2 = 15x 2y 2 / 11 + 5x 3y 22; ƒy1x, y2 = 10x 3y / 11 + 5x 3y 22; 60 / 41; 1920 / 2879 18. ƒx1x, y2 = 18x 2 - 2y 22/ 12x 3 - xy 22; ƒy1x, y2 = -2y / 12x 2 - y 22; 15 / 7; -6 / 23 2 2 19. ƒx1x, y2 = e x y12x 2y + 12; ƒy1x, y2 = x 3e x y; -7e -4; -64e 48 20. ƒx1x, y2 = y 2e x + 3y; ƒy1x, y2 = ye x + 3y 13y + 22; e -1; 33e 5 3 4 4 21. ƒx1x, y2 = 14x + 3y2/ 321x + 3xy + y + 1021/24; ƒy1x, y2 = 13x + 4y 32/ 321x 4 + 3xy + y 4 + 1021/24; 29 / 12 2212; 48 / 2311 22. ƒx1x, y2 = 114x + 18y 22/ 3317x 2 + 18xy 2 + y 322/34; ƒy1x, y2 = 136xy + 3y 22/ 3317x 2 + 18xy 2 + y 322/34; 46 / 3316322/34; -135 / 150922/3 23. ƒx1x, y2 = 36xy1e xy + 22 - 3x 2y 2e xy4/ 1e xy + 222; ƒy1x, y2 = 33x 21e xy + 22 - 3x 3ye xy4/ 1e xy + 222; -241e -2 + 12/ 1e -2 + 222; 2 2 1624e -12 + 962/ 1e -12 + 222 24. ƒx1x, y2 = 7e x + 2y1e x + y 2 + 22 + 2xe x 17e x + 2y + 42; 2 ƒy1x, y2 = 14e x + 2y1e x + y 2 + 22 + 2y17e x + 2y + 42; 51e 4 + 21; 14e 21e 16 + 112 + 617e 2 + 42 25. ƒxx1x, y2 = 8y 2 - 32; ƒyy1x, y2 = 8x 2; ƒxy1x, y2 = ƒyx1x, y2 = 16xy 26. gxx1x, y2 = 60x 2y 2; gyy1x, y2 = 10x 4 + 72y; gxy1x, y2 = gyx1x, y2 = 40x 3y 27. Rxx1x, y2 = 8 + 24y 2; Ryy1x, y2 = -30xy + 24x 2; Rxy1x, y2 = Ryx1x, y2 = -15y 2 + 48xy 28. h xx1x, y2 = 10y; h yy1x, y2 = 24x; h xy1x, y2 = h yx1x, y2 = 10x + 24y 29. rxx1x, y2 = 12y / 1x + y23; ryy1x, y2 = -12x / 1x + y23; rxy1x, y2 = ryx1x, y2 = 16y - 6x2/ 1x + y23 30. k xx1x, y2 = 84y / 12x + 3y23; k yy1x, y2 = -126x / 12x + 3y23; k xy1x, y2 = k yx1x, y2 = 163y - 42x2/ 12x + 3y23 31. z xx = 9ye x; z yy = 0; z xy = z yx = 9e x 32. z xx = 0; z yy = -6xe y; z xy = z yx = -6e y 33. rxx = -1 / 1x + y22; ryy = -1 / 1x + y22; rxy = ryx = -1 / 1x + y22 34. k xx = -2515x - 7y2-2 or -25 / 15x - 7y22; k yy = -4915x - 7y2-2 or -49 / 15x - 7y22; k xy = k yx = 3515x - 7y2-2 or 35 / 15x - 7y22 35. z xx = 1 / x; z yy = -x / y 2; z xy = z yx = 1 / y 36. z xx = -31y + 12/ x 2; z yy = -1 / y 2 + 1 / y; z xy = z yx = 3 / x 37. x = -4, y = 2 38. x = -13 / 3, y = 14 / 3 39. x = 0, y = 0; or x = 3, y = 3 40. x = 0, y = 0; or x = 4, y = -18 41. ƒx1x, y, z2 = 4x 3; ƒy1x, y, z2 = 2z 2; ƒz1x, y, z2 = 4yz + 4z 3; ƒyz1x, y, z2 = 4z 42. ƒx1x, y, z2 = 18x 2 - 2xy 2; ƒy1x, y, z2 = -2x 2y + 5y 4; ƒz1x, y, z2 = 0; ƒyz1x, y, z2 = 0 43. ƒx1x, y, z2 = 6 / 14z + 52; ƒy1x, y, z2 = -5 / 14z + 52; ƒz1x, y, z2 = -416x - 5y2/ 14z + 522; ƒyz1x, y, z2 = 20 / 14z + 522 44. ƒx1x, y, z2 = 14x + y2/ 1yz - 22; ƒy1x, y, z2 = 1-2x - 2x 2z2/ 1yz - 222; ƒz1x, y, z2 = - 12x 2y + xy 22/ 1yz - 222; ƒyz1x, y, z2 = 14x 2 + 4xy + 2x 2yz2/ 1yz - 223 45. ƒx1x, y, z2 = 12x - 5z 22/ 1x 2 - 5xz 2 + y 42; ƒy1x, y, z2 = 4y 3 / 1x 2 - 5xz 2 + y 42; ƒz1x, y, z2 = -10xz / 1x 2 - 5xz 2 + y 42; ƒyz1x, y, z2 = 40xy 3z / 1x 2 - 5xz 2 + y 422 46. ƒx1x, y, z2 = 18y - 3x 22/ 18xy + 5yz - x 32; ƒy1x, y, z2 = 18x + 5z2/ 18xy + 5yz - x 32; ƒz1x, y, z2 = 5y / 18xy + 5yz - x 32; ƒyz1x, y, z2 = -5x 3 / 18xy + 5yz - x 322 47. (a) 6.773 (b) 3.386 48. (a) 938.9 (b) 1077 49. (a) 80 (b) 150 (c) 80 (d) 440 (e) Mxx1x, y2 = 90 (f) Myy1x, y2 = 80 (g) Mxy1x, y2 = -20 50. (a) R would increase by $70. (b) R would increase by $54. (c) Rxx1x, y2 = 10, Ryy1x, y2 = 18, Rxy1x, y2 = -4 51. (a) 150 (b) 200 52. (a) 50.57 hundred units (b) ƒx116,812 = 1.053 hundred units and is the rate at which production is changing when labor changes by 1 unit (from 16 to 17) and capital remains constant; ƒy116,812 = 0.4162 hundred units and is the rate at which production is changing when capital changes by 1 unit (from 81 to 82) and labor remains constant. (c) Production would increase by approximately 105 units.

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A-60

Complete Instructor Answers

53. 0.75Cx -0.25y 0.25; 0.25Cx 0.75y -0.75 54. 0.65Cx -0.35y 0.35; 0.35Cx 0.65y -0.65 55. (a) 1.5625, which is the approximate change in production (in thousands of units) for a 1-unit change in labor; 1.6, which is the approximate change in production (in thousands of units) for a 1-unit change in capital. (b) Increase of approximately 1563 batteries (c) Increasing capital 56. (a) 800 (b) 12.5; when the quantity produced is held constant at 40,000, and the storage cost is held constant at $4, the number of gallons produced when the setup cost is $32 goes up by approximately 12.5 gallons for each $1 increase in the setup cost. (c) -100; when the quantity produced is held constant at 40,000, and the fixed setup cost is held constant at $32, the number of gallons produced when the storage cost is $4 goes down by approximately 100 gallons for each $1 increase in the storage cost.

57. (c) ƒx1x, y2 = y1-x 2 - x - xy - 12e -x / 1x + y22, ƒy1x, y2 = x11 + x2e -x / 1x + y22 59. (a) 1279 kcal per hr (b) 2.906 kcal per hr per g; the instantaneous rate of change of energy usage for a 300-kg animal traveling at 10 km per hr is about 2.9 kcal per hr per g. 60. (a) 5.87 W (b) -5.97 W 61. (a) 0.0142 m2 (b) 0.00442 m2 62. (a) 5.714 (b) -0.1633 (c) 0.0286 (d) 0.1633 (e) Changing a by 1 unit produces the greatest decrease in the liters of blood pumped, while changing v by 1 unit produces the same amount of increase in the liters of blood pumped. 63. (a) 4.125 lb (b) 0ƒ / 0n = n / 4; the rate of change of weight loss per unit change in workouts (c) An additional loss of 3 / 4 lb 64. (a) 36.4 (b) 703 / h2; -1406w / h3 (c) Bm = wm / h 2m 65. (a) 0.0783 (b) 0.0906 per m (c) -0.0212 per m (d) Increase in waist 66. (a) Decrease of 0.036 yr; decrease of 0.035 yr (b) Increase of 0.015 yr; increase of 0.015 yr 67. (a) 12ax - 3x 22t 2e -t (b) x 21a - x212t - t 22e -t (c) 12a - 6x2t 2e -t (d) 12ax - 3x 2212t - t 22e -t (e) 0R / 0x gives the rate of change of the reaction per unit of change in the amount of drug administered. 0R / 0t gives the rate of change of the reaction for a 1-hour change in the time after the drug is administered. 68. (a) 1.75 atmospheres (b) 0.0227 atm / ft; 0.0347 atm / min (c) 6.172 minutes 69. (a) -24.9°F (b) 15 mph (c) WV120,102 = -1.114; while holding the temperature fixed at 10°F, the wind chill decreases approximately 1.1°F when the wind velocity increases by 1 mph; WT120,102 = 1.429; while holding the wind velocity fixed at 20 mph, the wind chill increases approximately 1.429°F if the actual temperature increases from 10°F to 11°F. (d) Sample table 70. (a) 90 (b) 109 (c) 86 (d) 1 (e) 0.4 (f) 4.2 (g) 1 71. -10 ml per year, 100 ml per in. 72. (a) 52.39% (b) 41.83% (c) 2.683%; 0.05% 73. (a) Fm = gR2 / r 2; T\V 5 10 15 20 the rate of change in force per unit change in mass while the distance is held constant; 30 27 16 9 4 Fr = -2mgR2 / r 3; the rate of change in force per unit change in distance while the mass is 20 16 3 -5 -11 held constant 74. (a) 164,456 m / sec (b) 0.238 m / sec per m / sec (c) c 75. (a) 1055 10 6 -9 -18 -25 (b) Ts13, 0.52 = 127.4 msec per ft. If the distance to move an object increases from 3 ft to 4 ft, while keeping w fixed at 0.5, the approximate increase in movement time is 127.4 msec.

0 -5 -21 -32 -39 Tw13, 0.52 = -764.6 msec per ft. If the width of the target area increases by 1 ft, while keeping s fixed at 3 ft, the approximate decrease in movement time is 764.6 msec.

Exercises 9.3 (page 536–539)

For exercises . . .

5–22,25–32 Refer to example . . . 1–3

W1. Relative maximum at 1-2, 332; relative minimum at 14, -752 W2. Relative maximum at 10, 12; relative minima at 1-2, -152 and 12, -152 1. False 2. True 3. False 4. True 5. Saddle point at 1-1, 22 6. Saddle point at 1-2, 5 / 32 7. Relative minimum at 1-3, -32 8. Relative minimum at 14, -22 9. Relative maximum at 1-2, -22 10. Relative minimum at 15, -52 11. Relative minimum at 115, -82 12. Relative maximum at 1-8, -232 13. Relative maximum at 12 / 3, 4 / 32 14. Saddle point at 110, -32 15. Saddle point at 12, -22 16. Relative minimum at 12, -12 17. Saddle point at 10, 02; relative minimum at 127, 92 18. Saddle point at 10, 02; relative minimum at 1126, 26462 19. Saddle point at 10, 02; relative maximum at 19 / 2, 3 / 22 20. Saddle point at 10, 02; relative minimum at (98, 14) 21. Saddle point at 10, -12 22. No extrema and no saddle points 25. Relative maximum of 9 / 8 at 1-1, 12; saddle point at 10, 02; (a) 26. Relative maximum of 17 / 16 at (0, 1); relative minimum of -15 / 16 at 10, -12; saddle points at 1 26 / 2, 02 and 1- 26 / 2, 02; (d) 27. Relative minima of -33 / 16 at 10, 12 and at 10, -12; saddle point at 10, 02; (b) 28. Relative maximum of 17 / 16 at (1, 1); relative maximum of 17 / 16 at 11, -12; (c) 29. Relative maxima of 17 / 16 at 11, 02 and 1-1, 02; relative minima of -15 / 16 at 10, 12 and 10, -12; saddle points at 10, 02, 1-1, 12, 11, -12, 11, 12, and 1-1, -12; (e) 30. Saddle point at (0, 0); relative maximum of 17 / 16 at (1, 1) and at 1-1, -12; (f) 34. (e) 35. (a) all values of k (b) k Ú 0 38. Maximum profit is $351,600 when x = 12 and y = 30. 39. Minimum cost of $59 when x = 4 and y = 5 40. 38 units of electrical tape and 52 units of packing tape should be produced to yield a minimum cost of $1832.

41. Sell 12 spas and 7 solar heaters for a maximum revenue of $166,600. 42. 6 tons of steel and 3 tons of aluminum produce a maximum profit of $216,000. 43. $2000 on quality control and $1000 on consulting, for a minimum time of 8 hours 44. (a) 0.5148; 0.4064; the jury is less likely to make the correct decision in the second situation. (b) If r = s = 1 then P1a, 1, 12 = 1 (c) P1a, 1, 12 = 1 is a maximum value. 45. (a) r = 3.98375, s = 0.272021, y = 53.717911.312612t (b) Same as (a) (c) Same as (a) 46. (a) 436.16 kJ / mol (b) 137.66 kJ / mol (c) Saddle point at 11.75, 133°C2

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38–44,46 4

Exercises 9.4 (page 545–547)

Complete Instructor Answers

A-61

For exercises . . .

5–12,15,16,19,25 13,14,17,18,41–46 27–30 Refer to example . . . 1 3 material after Example 3

31–40 2

W1. ƒx1x, y2 = 12x 2y 2 - 24x 3; ƒy1x, y2 = 8x 3y - 15y 4 W2. ƒx1x, y, z2 = 2xy 3z 5 - 8xy 5 + 3z 6; ƒy1x, y, z2 = 3x 2y 2z 5 - 20x 2y 4 - 16y 7z 7; ƒz1x, y, z2 = 5x 2y 3z 4 + 18xz 5 - 14y 8z 6 1. True 2. True 3. False 4. True 5. ƒ18, 82 = 256 6. ƒ110, 102 = 204 7. ƒ15, 52 = 125 8. ƒ10, -92 = 0 9. ƒ15, 32 = 28 10. ƒ117 / 2, 42 = 797 / 4 = 244.75 11. ƒ120, 22 = 360 12. ƒ19, 72 = 528 13. ƒ13 / 2, 3 / 2, 32 = 81 / 4 = 20.25 14. ƒ14, 4, 22 = 48 15. x = 8, y = 16 16. x = 32, y = 16 17. 30, 30, 30 18. 80, 80, and 80 19. Minimum value of 128 at 12, 52, maximum value of 160 at 1-2, 72 25. (a) Minimum value of -5 at 13, -22 (d) 13, -22 is a saddle point. 27. Purchase 20 units of x and 20 units of y for a maximum utility of 8000. 28. Purchase 16 units of x and 48 units of y for a maximum utility of 28,311,552.

29. Purchase 20 units of x and 5 units of y for a maximum utility of 4,000,000. 30. Purchase 6 units of x and 8 units of y for a maximum utility of 884,736. 31. 60 feet by 60 feet 32. 40 feet by 48 feet 33. 10 large kits and no small kits 34. x = 2, y = 4 35. 167 units of labor and 178 units of capital 36. 189 units of labor and 35 units of capital 37. 125 m by 125 m 38. 45,000 m2 39. Radius = 5 in.; height = 10 in. 40. Radius ≈ 1.585 in.; height ≈ 3.169 in.

41. 12.91 m by 12.91 m by 6.455 m 42. 5.698 in. by 5.698 in. by 5.698 in. 43. 5 m by 5 m by 5 m 44. 4 ft by 4 ft for the base; 2 ft for the height 45. (b) 4 yd by 1 yd by 1 / 2 yd 46. (a) F1r, s, t, l2 = rs11 - t2 + 11 - r2st + r11 - s2t + rst - l1r + s + t - a2 (b) r = s = t = 0.25 (c) r = s = t = 1.0

Exercises 9.5 (page 551–553)

For exercises . . .

5–10,22–24,27–33 11–18 Refer to example . . . 1 2

19–21,25,26,34–37 38,39 3 4

W1. 2.7 W2. -0.016 W3. 4.9 W4. 1.01 1. True 2. False 3. True 4. True 5. 0.12 6. 0.49 7. 0.0311 8. –0.00769 9. -0.335 10. 0.0730 11. 10.022; 10.0221; 0.0001 12. 13.04; 13.0401; 0.0001 13. 2.0067; 2.0080; 0.0013 14. 1.975; 1.9748; 0.0002 15. 1.07; 1.0720; 0.0020 16. 0.94; 0.9416; 0.0016 17. -0.02; -0.0200; 0 18. 0.04; 0.0402; 0.0002 19. 20.73 cm3 20. 18.4 cm3 21. 86.4 in3 22. Decrease $130 23. 0.07694 unit 24. 0.348 unit 25. 6.65 cm3 26. 6.258 cm3 27. 2.98 liters 28. 16.0 W 29. Increase of 3.51 yr; increase of 3.51 yr 30. (a) 2052.5 cm3 (b) 2163.51 cm3; 2164.37 cm3 31. (a) 87% (b) 75% (d) 89%; 87% 32. (a) 0.2649 (b) Actual 0.2817; approximation 0.2816 33. (a) 0.06569 sec; 0.0379 sec; in a close race, this could certainly affect the outcome. (b) 0.001993 sec; this is the approximate change in time when the temperature decreases 5 degrees and the swimmer stands 0.5 m farther away from the starter in the y direction while the starter’s position is held fixed.

34. 7.305 cm3 35. 26.945 cm2 36. 2a + b, the same as for a cylinder 37. 3% 38. 960 39. 8

Exercises 9.6 (page 562–564)

For exercises . . .

5–14 15–24 Refer to example . . . 1 2,3

25–32,65–68,70–75 33–42,69 43–50 4 5,6 7

51–60 8

W1. -4 W2. 38 / 3 W3. ln 8 ≈ 2.0794 W4. 1e 6 - 12/ 6 ≈ 67.0715 W5. 1e 8 - e2/ 3 ≈ 992.7466 W6. 98 / 3 1. False 2. True 3. False 4. False 5. 630y 6. 11x / 4 7. 12x / 9231x 2 + 1523/2 - 1x 2 + 1223/24 8. 11 / 323136 + 3y23/2 - 19 + 3y23/24 9. 6 + 10y 10. 255 / 12 2x2 2 2 11. 11 / 221e 12 + 3y - e 4 + 3y2 12. 11 / 321e 2x + 3 - e 2x-32 13. 11 / 221e 4x + 9 - e 4x2 14. 1y / 421e 20 + y - e 4 + y 2 15. 945 16. 99 / 8 17. 12 / 4521395/2 - 125/2 - 75332 18. 12 / 4521245/2 - 215/2 - 155/2 + 125/22 19. 21 20. 255 21. 1ln 322 22. 2 ln 5 23. 8 ln 2 + 4 24. 9 / 5 + 17 / 22 ln 2 25. 171 26. 88 27. 14 / 152133 - 25/2 - 35/22 28. 12 / 4521145/2 - 65/2 - 85/22 29. -3 ln13 / 42 or 3 ln14 / 32 30. 11 / 1521316 - 130 252 31. 11 / 221e 7 - e 6 - e 3 + e 22 32. 11 / 621e 14 - e 7 - e 10 + e 32 33. 96 34. 351 / 2 35. 40 / 3 36. 72 37. 12 / 152125/2 - 22 38. 11 / 1521175/2 - 10252 39. 11 / 42 ln117 / 82 40. e 2 - 2e + 1 41. e 2 - 3 42. e - 2 43. 97,632 / 105 44. 44 45. 128 / 9 46. 12 / 152185/2 - 62 - 25/22 47. ln 16 or 4 ln 2 48. 4 ln 4 - 2 49. 64 / 3 50. e 2 / 5 - e 3 / 3 + 2 / 15 51. 34 52. 568 53. 10 / 3 54. 1 - ln 2 55. 71e - 12/ 3 56. 4 / 5 57. 16 / 3 58. 7 / 18 59. 4 ln 2 - 2 60. 13e 4 - 72/ 8 61. 1 62. e - 1 65. 49 66. 13 / 3 67. 1e 6 + e -10 - e -4 - 12/ 60 68. 1e 7 - e 6 - e 5 + e 42/ 2 69. 9 in.3 70. $3518 71. 14,753 units 72. $933.33 73. $34,833 74. 78.4 hr 75. $32,000

Chapter 9 Review Exercises (page 568–571)

For exercises . . .

61,62 9

1–5,17–26,87 6,13,14,27–44, 7,8,45–53,90, 9,54–57, 10–12,67–86, 15,59–66,88, 89,101–103 105,106 91,92 98,99,102 93–96,100 Refer to section . . . 1 2 3 4 6 5

1. True 2. True 3. True 4. True 5. False 6. False 7. False 8. True 9. False 18. 23; 594 19. -5 / 9; -4 / 3 20. - 25 / 3; 25 / 3

10. False 11. True

12. False

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17. -19; -255

A-62

21.

Complete Instructor Answers

z

22.

23.

z

24.

4

1 y

3

x

25.

x + 2y + 6z = 6

6

z

26.

4 x

y=4

y

y

3 4x + 3z = 12 x

27. (a) 9x 2 + 8xy (b) -12 (c) 16 28. (a) 4xy1x - y 222 (b) -1 / 2 (c) 0 29. ƒx1x, y2 = 12xy 3; ƒy1x, y2 = 18x 2y 2 - 4 30. ƒx1x, y2 = 20x 3y 3 - 30x 4y; ƒy1x, y2 = 15x 4y 2 - 6x 5 31. ƒx1x, y2 = 4x / 14x 2 + y 221/2; ƒy1x, y2 = y / 14x 2 + y 221/2 32. ƒx1x, y2 = 1-6x 2 + 2y 2 - 30xy 22/ 13x 2 + y 222; ƒy1x, y2 = 130x 2y - 4xy2/ 13x 2 + y 222

33. ƒx1x, y2 = 3x 2e 3y; ƒy1x, y2 = 3x 3e 3y 34. ƒx1x, y2 = 1y - 222e x + 2y; ƒy1x, y2 = 21y - 221y - 12e x + 2y 35. ƒx1x, y2 = 4x / 12x 2 + y 22; ƒy1x, y2 = 2y / 12x 2 + y 22 36. ƒx1x, y2 = -2xy 3 / 12 - x 2y 32; ƒy1x, y2 = -3x 2y 2 / 12 - x 2y 32 37. ƒxx1x, y2 = 30xy; ƒxy1x, y2 = 15x 2 - 12y 38. ƒxx1x, y2 = -6y 3 + 6xy; ƒxy1x, y2 = -18xy 2 + 3x 2 39. ƒxx1x, y2 = 12y / 12x - y23; ƒxy1x, y2 = 1-6x - 3y2/ 12x - y23 40. ƒxx1x, y2 = 213 + y2/ 1x - 123; 2 ƒxy1x, y2 = -1 / 1x - 122 41. ƒxx1x, y2 = 8e 2y; ƒxy1x, y2 = 16xe 2y 42. ƒxx1x, y2 = 2ye x 12x 2 + 12; x2 2 2 2 2 2 2 ƒxy1x, y2 = 2xe 43. ƒxx1x, y2 = 1-2x y - 4y2/ 12 - x y2 ; ƒxy1x, y2 = -4x / 12 - x y2 44. ƒxx1x, y2 = -9y 4 / 11 + 3xy 222; ƒxy1x, y2 = 6y / 11 + 3xy 222 45. Saddle point at 10, 22 46. Relative minimum at 1-9 / 2, 42 47. Relative minimum at 12, 12 48. Relative maximum at 1-3 / 4, -9 / 322; saddle point at (0, 0) 49. Saddle point at 13, 12 50. Relative maximum at 111 / 4, -22 51. Saddle point at 1-1 / 3, 11 / 62; relative minimum at 11, 1 / 22 52. Relative maximum at 1-8, -232 54. Minimum of 0 at (0, 4); maximum of 256 / 27 at 18 / 3, 4 / 32 55. Minimum of 18 at 1-3, 32 56. x = 160 / 3, y = 80 / 3 57. x = 25, y = 50 58. No 59. 1.22 60. -0.0168 61. 13.0846; 13.0848; 0.0002 62. 2.095; 2.0972; 0.0022 63. 8y - 6 64. 1e 15 + 5y - e 3 + 5y2/ 3 65. 13 / 2231100 + 2y 221/2 - 12y 221/24 66. 12 / 32317x + 29721/2 - 17x + 1121/24 67. 1232 / 9 68. 395 69. 12 / 1352314225/2 - 12425/2 - 13925/2 + 12125/24 70. 1e 3 + e -8 - e -4 - e -12/ 14 71. 2 ln 2 or ln 4 72. ln 2 73. 110 74. 12 / 1521115/2 - 85/2 - 75/2 + 322 75. 14 / 1521782 - 85/22 76. 1e 2 - 2e + 12/ 2 77. 105 / 2 78. 308 / 3 79. 1 / 2 80. 52 / 5 81. 1 / 48 82. 1 / 12 83. ln 2 84. 1615 25 - 12/ 3 85. 3 86. 26 / 105 87. (a) $1325 + 2102 ≈ $328.16 (b) $1800 + 2152 ≈ $803.87 (c) $12000 + 2202 ≈ $2004.47 88. (a) $26 (b) $2572 89. (a) 0.7y 0.3 / x 0.3 (b) 0.3x 0.7 / y 0.7 90. (a) Relative minimum at x = 11, y = 12 (b) $431 91. Purchase 10 units of x and 15 units of y for a maximum utility of 33,750. 92. Purchase 3 units of x and 2 units of y for a maximum utility of 972. 93. Decrease by $243.82 94. 7.92 cm3 95. 4.19 ft3 96. 15.6 cm3 97. (a) $200 spent on fertilizer and $80 spent on seed will produce a maximum profit of $266 per acre. (b) Same as (a) (c) Same as (a) 98. 13,493 units 99. $19,685 100. 1.341 cm3 101. (a) 49.68 liters (b) -0.09, the approximate change in total body water if age is increased by 1 yr and mass and height are held constant is -0.09 liter; 0.34, the approximate change in total body water if mass is increased by 1 kg and age and height are held constant is 0.34 liter; 0.25, the approximate change in total body water if height is increased by 1 cm and age and mass are held constant is 0.25 liter. 102. (a) About 33.98 cm (b) About 0.02723 cm per g; about 0.2821 cm / year; the approximate change in the length of a trout if its mass increases from 450 to 451 g while age is held constant at 7 years is 0.027 cm; the approximate change in the length of a trout if its age increases from 7 to 8 years while mass is held constant at 450 g is 0.28 cm.

103. (a) 50; in 1900, 50% of those born 60 years earlier are still alive. (b) 75; in 2000, 75% of those born 70 years earlier are still alive. (c) -1.25; in 1900, the percent of those born 60 years earlier who are still alive was dropping at a rate of 1.25 percent per additional year of life. (d) -2; in 2000, the percent of those born 70 years earlier who are still alive was dropping at a rate of 2 percent per additional year of life. 104. (a) 2.828 ft2 (b) An increase of 0.6187 ft2 105. 5 in. by 5 in. by 5 in.

106. 20,000 ft2 with dimensions 100 ft by 200 ft

Extended Application: Using Multivariable Fitting to Create a Response Surface Design (page 574) 1. 10 2. Close to orange = 56, banana = 48 3. x ≈ 55.254, y ≈ 48.589 4. D 7 0, Gxx 6 0 6. Either 400 minutes at about 130°C or 20 minutes at about 160°C 7. The allowable regions in the three contour plots do not overlap. 8. A processing temperature between 145°C and 155°C with treatment times from 50 to 70 seconds

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Complete Instructor Answers

A-63

Chapter 10 Differential Equations Exercises 10.1 (page 585–590)

For exercises . . .

5,6

7,8,21–24

Refer to example . . .

1

3

9–20,25–34,39, 40,44,45,57,59 4,5

35–38 8

43,47,48, 62,64 5

46,53,54, 58,69 7

49–52,59, 60,65–68 6

W1. 11 / 22e 2x + 2x 2 + 12 / 32x 3/2 + ln x + C W2. 11 / 62e 2x + C W3. 2 ln1x 2 + 12 + C W4. 13 / 52xe 5x - 13 / 252e 5x + C W5. - 11 / 32x 2e -3x - 12 / 92xe -3x - 12 / 272e -3x + C 1. False 2. True 3. False 4. False 5. y = -2x 2 + 2x 3 + C 6. y = -4e -3x / 3 + C 7. y = x 4 / 2 + C 8. y = x 3 / 3 - 2x / 3 + C 9. y 2 = 2x 3 / 3 + C 10. y 2 = 2x 3 / 3 - x 2 + C 2 2 3 2 11. y = Ke x 12. y = Ke x /3 13. y = Ke x - x 14. 2y 3 - 3y 2 = 3x 2 + C 15. y = Cx 16. y = Ke -1/x 2 17. ln1y 2 + 62 = x + C 18. e -y /3 = -2x + C 19. y = -1 / 1e 2x / 2 + C2 20. y = ln1e x + C2 21. y = x 2 - x 3 + 5 22. y = x 4 - x 3 + x 2 / 2 - 1 / 2 23. y = -2xe -x - 2e -x + 44 24. y = 1x / 32e 3x - e 3x / 9 + 1 25. y 2 = x 4 / 2 + 25 -1/ 2 2 26. y = e -2x 27. y = e x + 3x 28. y 2 - y = x 3 / 3 + 5x + 110 29. y 2 / 2 - 3y = x 2 + x - 4 30. y = -e 1-1/x2 + 1 31. y = -3 / 13 ln x -42 32. y = -9 / 16x 3/2 - 492 33. y = 1e x - 1 - 32/ 1e x - 1 - 22 34. y = -ln 310 - 1x + 223 / 34 35. y = -1, 1: unstable; y = 0: stable 36. y = -2, 2: stable; y = -1: unstable 37. y = 0: stable; y = 3: unstable 38. y = e: unstable; y = 5: stable 43. (a) dy / dt = -0.15y (b) y = Me -0.15t (c) About 9.2 yr 44. q = 2-4p2 + C (b) y = 32.35011.21432t (c) y = 6252 / 11 + 106.97e -0.1952t2 45. q = C / p2 46. (a) 5000 (d) 6252 3

y 5 32.350(1.2143)t

y 5 32.350(1.2143)t

5000

0

y 5 6252/(1 1 106.97e20.1952t )

5000

30 0 0

30 0

0

30 0

47. (d) 48. 4.4 cc 49. (a) I = 2.4 - 1.4e -0.088W (b) I approaches 2.4. 50. 987 fish 51. (a) dw / dt = k1C - 17.5w2; the calorie intake per day is constant. (b) lb/calorie (c) dw / dt = 1C - 17.5w2/ 3500 (d) w = C / 17.5 - e -0.005Me -0.005t / 17.5 (e) w = C / 17.5 + 1w0 - C / 17.52e -0.005t 52. (a) w = 143 + 37e -0.005t (b) The asymptote is w = 143; 143 will never be attained. (b) y = 25,538 / 11 + 110.28e -0.01819t2 (c) 584 days 53. (a) 15,000 w 143 37e 0.005t 200

0 0 120

(c)

y

25,538 1 110.28e

300 0

300

(d) 25,538

(b) y = 1383 / 11 + 1.867e -0.05615t2

54. (a) 2000

0.01819t

y = 1383 (1 + 1.867e–0.05615t ) 2000

15,000

0

110 0

0

0

300

(c) 1383 million (d)

110 0

0 2000

y = 1959 (1 + 4.800e–0.03415t )

y = 1959 / 11 + 4.800e -0.03415t2; 1959 million

2000

0

110 0

0

110 0

55. y = 45.2e 0.01499t 56. y = 14.2e 0.01588t 57. P = 1 / 11 + 2pDR22 58. (a) y = 11.74 / 31 + 11.423 * 10 222e -0.02554t4 (b) The function seems to fit the data from 1927 on very well. For the year 1804, the function does not fit the data very well. y 11.74/[1 (1.423 1022)e 0.02554t] 12

(c) y = 9.803 / 31 + 12.612 * 10 292e -0.03391t4

(d)

y 9.803/[1 (2.612 1029)e 0.03391t] 12

Yes 1800

2100

1800

0

2100 0

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A-64

Complete Instructor Answers

59. (b) q = 1 / 11 + kp2 (c) q = 1 / 11 + p2 (d) 83% 60. (a) q = C / pk; as p approaches 0, q becomes infinitely large. (b) q = 1P / 1p + P22k (c) q = 10.1 / 1p + 0.1220.289 (d) 73% 61. About 10 62. (a) k ≈ 0.8 (b) 11 (c) 55 (d) About 3000 63. 7:22:55 a.m. 64. (a) dy / dt = -0.03y (b) y = Me -0.03t (c) y = 75e -0.03t (d) 56 g 66. The temperature approaches TM, the temperature of the surrounding medium. 67. (a) 83.8°F (b) 4.5 hours (c) 24.3 hours 68. (a) T = 88.6e -0.24t + 10 (b) (c) Just after death—the graph shows that the most rapid decrease T 88.6e 0.24t 10 occurs in the first few hours. (d) About 43.9°F 100 (e) About 4.5 hours

0

30

0

(b) y = 250,724 / 11 + 9.9928e -0.23631t2

69. (a) 225000

(c)

y 5 250,724/(1 1 9.9928e20.23631t)

(d) 250,724

225000

5 100000

20 5 100000

20

For exercises . . .

Refer to example . . .

Exercises 10.2 (page 595–597)

5–25,31–36 2–4

W1. 4 ln x + C W2. -10e -0.1t + C W3. 11 / 22e 3x + C W4. 11 / 32e t - 3t + C 1. True 2. True 3. False 4. True 2 2 5. y = 2 + Ce -3x 6. y = 12 / 5 + Ce -5x 7. y = 2 + Ce -x 8. y = 1 + Ce -2x 9. y = x ln x + Cx 10. y = x / 2 - 1 / 4 + Ce -2x 2 2 11. y = -1 / 2 + Ce x /2 12. y = -1 / 6 + Ce -x 13. y = x 2 / 4 + 2x + C / x 2 14. y = x 2 - 4x ln x + Cx 15. y = -x 3 / 2 + Cx 2 16. y = -x 2 - x / 2 - 1 / 4 + Ce 2x 17. y = 2e x + 48e -x 18. y = e 5x + 24e -4x 19. y = -2 + 22e x - 1 20. y = 2 / 3 + 22x 3 / 3 2 3 3 21. y = x 2 / 7 + 2560 / 17x 52 22. y = -5 / 4 + 145 / 42e x - 1 23. y = 13 + 197e 4 - x2/ x 24. y = x 2e -x + 1000e -x 25. (a) dA / dt = 0.05A - 50 (b) A1t2 = 1000 + 1000e 0.05t (c) $2051.27; $2284.03; $2648.72 (d) 2.56%; 2.84%; 3.24% (e) $1000 26. A1t2 = 1000 + 1000e 0.05t 27. (a) y = c / 1p + Kce -cx2 (b) y = cy0 / 3py0 + 1c - py02e -cx4 (c) c / p 28. G = a / K + Ce -Kt 31. (a) y = a / 1a + b2 + 3y0 - a / 1a + b24e -1a + b2t (b) a / 1a + b2 32. y = -0.98e -t + 10,000.98e -0.02t 33. y = 1.02e t + 9999e 0.02t 1rounded2 34. y = -50t - 2500 + 12,500.98e 0.02t 35. y = 50t + 2500 + 7500e 0.02t 36. T = Ce -kt + TM 2

3

Exercises 10.3 (page 602–604)

For exercises . . .

Refer to example . . .

1. True 2. False 3. True 4. True 5. 8.273 6. 2.123 7. 4.315 8. 0.151 9. 1.491 10. 0.452 11. 6.191 12. 2.783 13. -0.540; -0.520 14. 3.900; 4.000 15. 4.010; 4.016 16. 3.053; 3.009 17. 3.806; 4.759 18. 1.056; 1.075 19. 3.112; 3.271 20. 3.099; 3.118 21. 73.505; 74.691 22. 21.446; 23.146 23. 3.186, -0.085, 2.6% 24. 3.108, -0.01, 0.3% 25. xi yi f 1 xi 2 yi − f 1 xi 2 ∣ 1 yi − f 1 xi 2 2 /f 1 xi 2 ∣ : 100

26.

0

0

0

0

0

0.2

0

0.08772053

-0.08772053

100

0.4

0.11696071

0.22104189

-0.10408118

47.09

0.6

0.26432197

0.37954470

-0.11522273

30.36

0.8

0.43300850

0.55699066

-0.12398216

22.26

1.0

0.61867206

0.75000000

-0.13132794

17.51

xi

yi

0

1

f 1 xi 2

yi − f 1 xi 2

1

0

∣ 1 yi − f 1 xi 2 2 /f 1 xi 2 ∣ : 100 0

0.2

1.2

1.2214028

-0.0214028

1.75

0.4

1.44

1.4918247

-0.0518247

3.47

0.6

1.728

1.8221188

-0.0941188

5.17

0.8

2.0736

2.2255409

-0.1519409

6.83

1.0

2.48832

2.7182818

-0.2299618

8.46

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5–42 1,2

Complete Instructor Answers

27.

28.

29.

xi

yi

f 1 xi 2

yi − f 1 xi 2

∣ 1 yi − f 1 xi 2 2 /f 1 xi 2 ∣ : 100

0

0

0

0

0

0.2

0.8

0.725077

0.07492

10.33

0.4

1.44

1.3187198

0.12128

9.20

0.6

1.952

1.8047535

0.14725

8.16

0.8

2.3616

2.2026841

0.15892

7.21

1.0

2.68928 2.5284822

0.16080

6.36

xi

yi

f 1 xi 2

yi − f 1 xi 2

∣ 1 yi − f 1 xi 2 2 /f 1 xi 2 ∣ : 100

0

1

1

0

0

0.2

1

0.9803947

0.01961

2.00

0.4

0.96

0.9260719

0.03393

3.66

0.6

0.8864

0.8488382

0.03756

4.43

0.8

0.793664

0.7636462

0.03002

3.93

1.0

0.699692

0.6839397

0.01575

2.30

y

30.

0.8 0.4 0.2

0.2 0.4 0.6 0.8 1.0

x

2.5 2.0 1.5 1.0 0.5 0 0.2 0.4 0.6 0.8 1.0

y 3.0 2.5 2.0 1.5 1.0 0.5

f(x)

0.6

0

31.

y

f(x)

A-65

0 x

32. f(x)

0.2 0.4 0.6 0.8 1.0 x

y 1.2 1.0 0.8 0.6 0.4 0.2

f(x)

0.2 0.4 0.6 0.8 1.0 x

0

33. (a) 4.109 (b) y = 1 / 11 - x2; y approaches ∞. 34. (a) dy / dt = 0.25y - 0.002y 2 (b) About 48 firms 35. (a) dy / dt = 0.01y 1500 - y2 = 5y - 0.01y 2 (b) About 484 thousand 36. About 20 species 37. About 75 insects 38. About 7000 whales 39. About 8.07 kg 40. (a) t (b) 1 pi i 41. About 157 people 0 0.1 42. 79.5° 5 0.1675 10

0.297678

15

0.536660

20

0.846644

25

0.963605

30

0.980024

Exercises 10.4 (page 610–612)

For exercises . . .

Refer to example . . .

5–9 1

10–12 2

13–21 3

3 3 W1. y = 2 x + 8 W2. y = e -1/x - 1 W3. y = x - 1 + 4e -x W4. y = -5 / 2 + 3e x 1. True 2. True 3. True 4. False 5. $50,216.53 6. 6.2 years 7. About 6.9 years 8. $142,914.80 9. (a) dA / dt = 0.06A - 1200 (b) $6470.04 (c) 8.51 years 11. (a) 2y - 3 ln y - 4 ln x + 2x = 4 (b) x = 2, y = 3 / 2, or x = 0, y = 0 12. (a) 4y - 2x - 4 ln y + 3 ln x = 3 ln 5 - 6 (b) x = 3 / 2, y = 1 or x = 0, y = 0 (c) Both populations increase. The x population increases while the y population decreases. 13. (a) y = 24,995,000 / 14999 + e 0.25t2 (b) 3672 (c) 91 (d) 34th day 14. (b) y (c) (4.6 days, 50 people) (d) This is the point where the number of infected people equals the 100 number of uninfected people. (e) For y it is 100 people. For N - y it is 0 people.

Infected 80 individuals 15. (a) y = 20,000 / 11 + 199e -0.14t2 or 20,000e 0.14t / 1e 0.14t + 1992 (b) About 38 days 60 -0.02t 40 16. (a) y ≈ 982,000e -9.89e (b) In about 31 days 17. (a) y = 0.005 + 0.015e -1.010t Uninfected -1.1t 20 individuals (b) Y = 0.00727e + 0.00273 18. y = 1500 / 1497e -1.456t + 32; about 449 x 0 19. (a) y = 45 / 11 + 14e -0.54t2 (b) About 6 days 20. y = 100 / 11 + 19e -1.558t2; about 85 2 4 6 8 10 2

Days

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22–28 4

A-66

Complete Instructor Answers

-0.1t

21. (a) y = 347e -4.24e (b) About 5.5 days 22. (a) y = 200 - 180e -0.02t (b) It increases, approaching 200 lb.

23. (a) y = 321t + 10023 - 1,800,0004/ 1t + 10022 (b) About 250 lb of salt (c) Increases 24. (a) y = 21100 - t2 - 0.0181100 - t22 (b) About 51 lb (c) It increases at first but then decreases. 25. (a) y = 20e -0.02t (b) About 6 lb of salt (c) Decreases 26. (a) y = 500 / 1t + 1002 (b) About 3.8 g 27. (a) y = 30.251t + 10022 - 20004/ 1t + 1002 (b) About 17.1 g 28. 34.7 minutes

Chapter 10 Review Exercises (page 613–616)

For exercises . . .

Refer to section . . .

1–3,6,7,13,14,25–32, 5,15–24 4,8,9,33–36, 43–46 37–42,47,48,54–56, 59,60,65–69,71–74 1 1, 2 2

10,11,49–53

12,57,58, 61–64,70

3

4

1. True 2. False 3. True 4. False 5. False 6. True 7. False 8. True 9. False 10. False 11. True 12. True 17. Neither 18. Separable 19. Separable 20. Linear 21. Both 22. Both 23. Linear 24. Neither 25. y = x 3 + 3x 2 + C 26. y = x 4 + x 6 + C 27. y = 2e 2x + C 28. y = 11 / 32 ln 3x + 2 + C 29. y 2 = 3x 2 + 2x + C -x 30. y 2 / 2 - y = e x + x 2 / 2 + C 31. y = 1Cx 2 - 12/ 2 32. y = 3 + Me e 33. y = x - 1 + Ce -x 34. y = 1ln x + C2/ x 3 35. y = 1x 2 + C2/ ln x 36. y = e 2x / 12x2 - e 2x / 14x 22 + C / x 2 37. y = x 3 / 3 - 3x 2 + 3 38. y = -5e -x - 5x + 22 2 3/2 39. y = -ln35 - 1x + 224 / 44 40. y = 5e -x + 3x 41. y 2 + 6y = 2x - 2x 2 + 352 42. y = 2.054e 12/32x 3 2 3 43. y = -x 2e -x / 2 - xe -x / 2 + e -x / 4 + 41.75e x 44. y = 1 / 3 + 15 / 32e -x 45. y = 3 / 2 + 27e x / 2 46. y = 1e 2x + 5e 22/ 16x 42 47. y = 2; stable; y = 0; unstable 48. y = 1: stable; y = -1 and y = 8: unstable 50. 2.138 51. 2.608 52. 1.215; 1.223; -0.008 y 53. x 55. (a) $10,099 (b) $71,196 56. (a) 100 items (b) 10 days yi i 5 57. (a) dA / dt = 0.05A - 20,000 (b) $235,127.87 58. About 27.7 years 0 0 4 59. (a) About 40 (b) About 1.44 * 10 10 hours 60. 219 3 0.2 0.6 61. 0.2 ln y - 0.5y + 0.3 ln x - 0.4x = C; x = 3 / 4 units, y = 2 / 5 units 2 0.4 1.355 1 62. 17.3 min 63. It is not possible (t is negative). 64. (a) y = 489,300 / 1699 + e 1.140t2 x (b) 135 people (c) 5.7 wk 66. (a) 185 million (b) 207 million (c) 326 million 0 0.2 0.4 0.6 0.8 1.0 0.6 2.188 67. (a) N = 326, b = 7.20; k = 0.248 (b) y ≈ 266 million, which is less than the 0.8 3.084 table value of 308.7 million. (c) About 287 million for 2030, about 301 million for 2050 1.0 4.035 68. (a)

300

Yes

(b) y = 487 / 11 + 58.1e -0.208t2 (c) The equation seems to fit the data well. (d) 487 million y 487/(1 58.1e 0.208t ) 300

0

22 0

0

22 0

69. (a) All three models: increasing for all t (b) Exponential: concave upward for all t; limited growth: concave downward for all t; logistic: concave upward for t 6 1ln b2/ k, concave downward for t 7 1ln b2/ k 70. (a) y = 200 / 11 + 19e -0.4646t2 (b) About 70 people 71. (a) and (b) x = 1 / k + Ce -kt (c) 1 / k 72. 213° 73. 3 hr 74. (a) v = 1G / K21e 2GKt - 12/ 1e 2GKt + 12 (b) G / K (c) v = 881e 0.727t - 12/ 1e 0.727t + 12

Extended Application: Pollution of a Lake (page 617–618) 1. 1.8 yr; 3.6 yr; 12.0 yr 2. 21.3 yr; 42.7 yr; 142 yr 3. 5.4 yr; 10.8 yr; 36.0 yr 4. (b) Yes; it becomes 1. (c) About 0.0325; 3.25% 5. C1023e 1k - p2t - 14 / 1k - p2 6. (a) About 67 years, compared with 21 years assuming pollution-free inflow. (b) The ratio becomes 1, which means the pollution level is constant.

Chapter 11 Probability and Calculus Exercises 11.1 (page 627–631)

For exercises . . .

W1. 304 W2. 4 ln 2 - 7 / 4 ≈ 1.023 W3. 4e 5 - 4 ≈ 589.7 1. False 2. False 3. True 4. False 5. True 6. True 7. Yes

Refer to example . . . 1

7–16

17–24, 35–40 2

25–30,41(c),(d), 42(c),(d),54(d),(e), 55(d),(e) 5

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41(a),(b),42(a),(b),45–48, 49,57 50–53,54(a),(b),(c), 55(a),(b),(c),56 3 4

Complete Instructor Answers

3

3

A-67

1

2

8. Yes 9. Yes 10. Yes 11. No; 3 4x 3 dx Z 1 12. No; 3 1x 3 / 812 dx Z 1 13. No; 3 1x 2 / 162 dx Z 1 14. No; 3 2x 2 dx Z 1 0

-1

-2

0

15. No; ƒ1x2 6 0 for at least one x-value in 3-1, 14. 16. No; ƒ1x2 6 0 for at least one x-value in 30, 44. 17. k = 3 / 14 18. k = 5 / 422 19. k = 3 / 125 20. k = 1 / 3 21. k = 2 / 9 22. k = 2 / 5 23. k = 1 / 156 24. k = 1 / 60 25. F1x2 = 1x 2 - x - 22/ 18, 2 … x … 5 26. F1x2 = 1x 2 - x - 62/ 6, 3 … x … 4 27. F1x2 = 1x 3 - 12/ 63, 1 … x … 4 28. F1x2 = 1x 3 - 272/ 98, 3 … x … 5 29. F1x2 = 1x 3/2 - 12/ 7, 1 … x … 4 30. F1x2 = 1x 5/2 - 322/ 211, 4 … x … 9 31. 1 35. (a) 0.4226 (b) 0.2071 (c) 0.4082 36. (a) 0.6321 (b) 0.2325 (c) 0.8647 37. (a) 0.3935 (b) 0.3834 (c) 0.3679 38. (a) 0.0476 (b) 0.1524 (c) 0.8 39. (a) 0.3333 (b) 0.6667 (c) 0.6829 40. (a) 0.4444 (b) 0.5556 (c) 0.9653 41. (a) 0.9975 (b) 0.0024 (c) F1t2 = 1 - e -t/2, t Ú 0 (d) 0.9502 42. (a) 0.5730 (b) 0.2197 (c) 0.6250 (d) F1t2 = 1t + 6 2t - 162/ 11, 4 … t … 9 (e) 0.8155 43. (c) 44. (b) 45. (a) 0.2679 (b) 0.4142 (c) 0.3178 46. (a) 0.5372 (b) 0.2314 47. (a) 0.8131 (b) 0.4901 48. (a) 0.5024 (b) 0.6673 60 ;yes 49. (a) 60 polynomial function (b) N1t2 = -0.00008621t 4 + 0.01632t 3 1.128t 2 + 32.66t - 280.8

0

0

60

60 0

0

(c) S1t2 = 1-0.00008621t 4 + 0.01632t 3 - 1.128t 2 + 32.66t - 280.82/ 1995 (Your answer may differ due to rounding.) (d) 0.3131; 0.2844; 0.4354 50. (a) 0.5714 (b) 0.2381 51. (a) 0.2 (b) 0.6 (c) 0.6 52. (a) 0.3163 (b) 0.3679 53. (a) 0.1640 (b) 0.1353 54. (a) 0.3676 (b) 0.4081 (c) 0.2601 (d) F1t2 = 1.7395 - 7.6034t -0.532, 16 … t … 80 (e) 0.2343 55. (a) 0.2829 (b) 0.4853 (c) 0.2409 (d) F1t2 = 1.883810.5982 - e -0.03211t2, 16 … t … 84 (e) 0.1671 56. (a) 0.875 (b) 0.0290 (c) 0.0370 57. (a) 6000 polynomial function (b) T1t2 = -2.067t 3 + 78.97t 2 - 6000 ; yes 704.6t + 4633;

0

24

0

0

24 0

(c) S1t2 = 1-2.067t 3 + 78.97t 2 - 704.6t + 46332/ 100,716 (Your answer may differ due to rounding.) (d) 0.0800; 0.0785

Exercises 11.2 (page 638–641)

41,48 15(d),(e)–18(d),(e), 19(a)–24(a), 5–12, 15(a)(b)(c)–18(a)(b)(c), 19(b)–26(b),28–30, 28(d),30(d), 35–39,42,43,45–47 35(d),37(d), 27 40,45(d),46(d) Refer to example . . . 1,2 3 5 4

For exercises . . .

W1. 3 / 4 W2. 14 22 - 12/ 31 ≈ 0.1502 1. True 2. False 3. False 4. True 5. m = 5; Var1X2 ≈ 1.33; s ≈ 1.15 6. m = 5; Var1X2 ≈ 8.33; s ≈ 2.89 7. m = 14 / 3 ≈ 4.67; Var1X2 ≈ 0.89; s ≈ 0.94 8. m = 1 / 3 ≈ 0.33; Var1X2 ≈ 0.06; s ≈ 0.24 9. m = 17 / 6 ≈ 2.83; Var1X2 ≈ 0.57; s ≈ 0.76 10. m = 141 / 22 ≈ 6.41; Var1X2 ≈ 2.09; s ≈ 1.45 11. m = 4 / 3 ≈ 1.33; Var1X2 ≈ 0.22; s ≈ 0.47 12. m = 1.5; Var1X2 = 0.75; s ≈ 0.87 15. (a) 5.40 (b) 5.55 (c) 2.36 (d) 0.5352 (e) 0.6043 16. (a) 3.2 (b) 5.76 (c) 2.4 (d) 0.4571 (e) 0.5729 17. (a) 1.6 (b) 0.11 (c) 0.33 (d) 0.5904 (e) 0.6967 18. (a) 0.75 (b) 0.24 (c) 0.49 (d) 0.4639 (e) 0.6137 19. (a) 5 (b) 0 20. (a) 5 (b) 0 21. (a) 4.828 (b) 0.0556 22. (a) 0.2929 (b) 0.0556 23. (a) 1.189 (b) 0.1836 24. (a) 1.260 (b) 0.2037 25. 3.2; does not exist; does not exist 26. 10 / 9 ≈ 1.111; 0.1940; 0.4405 27. (d) 28. (a) About 310.3 hr (b) About 267.0 hr (c) 0.2059 (d) About 240.3 hr 29. (a) About 6.409 yr (b) About 1.447 yr (c) 0.4910 30. (a) 2 mo (b) 2 mo (c) 0.6321 (d) 1.386 mo 31. (c) 32. (d) 33. (c) 34. (c) 35. (a) 6.342 seconds (b) 5.135 sec (c) 0.7518 (d) 4.472 sec 36. (a) 6.37 days (b) 4.08 days (c) 0.3903 37. (a) 2.333 cm (b) 0.8692 cm (c) 0 (d) 2.25 cm 38. 2.990 m 39. 111 40. 3.778 min 41. About 39.05 years; about 13.76 years 42. 960 days; 960 days 43. (a) 1.806 cm (b) 1.265 cm (c) 0.1886 45. (a) About 35.56 years (b) About 16.98 years (c) 0.1330 (d) 30.35 years 46. (a) About 38.51 years (b) About 17.56 years (c) 0.1656 (d) About 34.26 years 47. 1.5 minutes 48. About 1:22 p.m.

Exercises 11.3 (page 651–654)

For exercises . . .

5,6,33,39, 7–10,34,35,40–42,44, 43,52 45,47–49,453,54–57 Refer to example . . . 1 2

11–18,36–38, 50 46,51,52,58 3 4

(Normal probabilities were generated using a graphing calculator. If you use the normal curve table, your answers may be slightly different.)

W1. 1 / 4 W2. 2 / 27 1. True 2. True 3. True 4. False 5. (a) 3.7 cm (b) 0.4041 cm (c) 0.2886 6. (a) $287.50 (b) $7.22 (c) 0.2888 7. (a) 0.25 years (b) 0.25 years (c) 0.2325 8. (a) 20 yr (b) 20 yr (c) 0.2325 9. (a) 3 days (b) 3 days (c) 0.2325 10. (a) 10 m (b) 10 m (c) 0.2325 11. 49.98% 12. 45.35% Copyright © 2022 Pearson Education, Inc.

A-68

Complete Instructor Answers

13. 8.01% 14. 46.75% 15. -1.28 16. -2.05 17. 0.92 18. 0.77 22. m = 1a + b2/ 2 23. m = 1-ln 0.52/ a or 1ln 22/ a 27. (a) 1.00000 (b) 1.99999 (c) 7.999998 29. (a) m ≈ 0 (b) s = 0.9999999251 ≈ 1 30. (a) m ≈ 0.3583 (b) s ≈ 0.2432 31. F1x2 = 1x - a2/ 1b - a2, a … x … b 32. F1x2 = 1 - e -ax, x Ú 0 33. (a) $47,500 (b) 0.4667 34. (a) ƒ1x2 = 0.2e -0.2x for 30, ∞2 (b) 0.3691 35. (a) ƒ1x2 = 0.235e -0.235x for 30, ∞2 (b) 0.0954 36. (a) 0.2335 (b) 0.4279 37. (a) 0.1587 (b) 0.7699 38. $45.29 to $63.51 39. (c) 40. (c) 41. (d) 42. (d) 43. (a) 28 days (b) 0.375 44. (a) 0.5 (b) 0.4866 45. (a) 1 hour (b) 0.3935 46. 3.065 ft to 3.335 ft 47. (a) 58 minutes (b) 0.0907 48. (a) 0.4210 (b) 0.4994 49. (a) 0.2518 (b) 0.2500 50. (a) 0.5776 (b) 0 51. 60.29 mph 52. 35.18 mph 53. (a) 4.37 millennia; 4.37 millennia (b) 0.6325 54. (a) 38 in. (b) 0.1667 55. (a) 0.2865 (b) 0.2212 56. (a) 609.5 days; 609.5 days (b) 0.5494 57. (a) 0.5457 (b) 0.0039 58. 0.5857

Chapter 11 Review Exercises (page 656–659) 1. True 2. True 6. True 7. True

For exercises . . .

3. True 4. False 5. False 8. True 9. False 10. False b

1–4,11–22,41, 46(a)–47(a), 60(a)–(d) Refer to section . . . 1

7–10,33–40,43,48–51, 5,6,23–31,42, 46(b),(c)–47(b),(c), 53,54,56,57–59,62 52,55,60(e),(f),61 2 3

13. (i) ƒ1x2 Ú 0 for all x in 3a, b4; (ii) 3 ƒ1x2 dx = 1 14. 0 15. Not a probability density function a 16. Probability density function 17. Probability density function 18. Probability density function 19. k = 1 / 21 20. k = 3 / 28 21. (a) 1 / 5 = 0.2 (b) 9 / 20 = 0.45 (c) 0.54 22. (a) 0.8284 (b) 0.5359 (c) 0.3643 24. (b) 25. (a) 4 (b) 0.5 (c) 0.7071 (d) 4.121 (e) F1x2 = 1x - 222 / 9, 2 … x … 5 26. (a) 6.5 (b) 2.083 (c) 1.443 (d) 6.5 (e) F1x2 = 1x - 42/ 5, 4 … 9 27. (a) 1.25 (b) 0.1042 (c) 0.3227 (d) 1.149 (e) F1x2 = 1 - 1 / x 5, x Ú 1 28. (a) 4.6 (b) 5.493 (c) 2.334 (d) 4.406 (e) F1x2 = 1x + 6 2x - 72/ 20, 1 … x … 9 29. (a) 0.5833 (b) 0.2444 (c) 0.4821 (d) 0.6123 30. m = 0.5970; 0.0180 31. (a) 100 (b) 100 (c) 0.8647 32. (a) 13.6 (b) 6.7 (c) 0.5840 33. 33.36% 34. 5.26% 35. 34.31% 36. 81.75% 37. 11.51% 38. 99.38% 39. z = -0.05 40. z = -0.81 41. (a) Uniform (b) Domain: 310, 304, range: 50.056 42. (a) Exponential with a = 1 (b) Domain: 30, ∞2; range: 10, 14 (c) f(x) (d) m = 1; s = 1 (c) (d) m = 20; s ≈ 5.77 (e) 0.8647 (e) 0.577 11. probabilities

f(x) = e 2x for [0, `)

1

0.5

0

43. (a) Normal (c)

(b) Domain: 1-∞, ∞2, range: 10, 1 / 2p4 (d) m = 0; s = 1 / 22 (e) 0.6827

0

1.5

2

x

(b) 0.4928

56. 0.1377

57. 0.2266

polynomial function (b) N1t2 = -0.002155t 4 + 0.4934t 3 - 38.5057t 2 + 1090.5874t - 4695.1476 6000 The function models the data well.

6000

0

1

44. (b) 0.4422 45. (b) 0.6819 (c) 0.9716 (d) 1; yes 46. (a) 0.4063 (b) $8.56 (c) $0.29 47. (a) 0.9107 (b) 13.57 years (c) 6.68 years 48. (a) 6 outlets (b) 6 outlets (c) 0.3679 49. (a) ƒ1x2 = e -x/8 / 8; 30, ∞2 (b) 8 repairs (c) 8 repairs (d) 0.2488 50. 0.1912 51. (d) 52. (a) 2.3765 g (b) 1.534 g (c) 0.8506

53. 0.6321 54. (a) 16 in. (b) 0.3571 55. (a) 40.07°C 58. (a) 0.4138 (b) 0.4447 59. (a) 0.2921 (b) 0.1826 60. (a)

0.5

90 0

0

90

(c) S1t2 = 1-0.002155t 4 + 0.4934t 3 - 38.5057t 2 + 1090.5874t - 4695.14762/ 196,875 (d) Estimates: 0.3971, 0.1320, 0.0265; Actual: 0.3692, 0.1741, 0.0166 (e) 31.80 years (f) 16.36 years 61. 3650.1 days; 3650.1 days 62. 0.2209

Extended Application: Exponential Waiting Times (page 659–661) 1. Since P1X = c2 = 0, it does not matter whether we use strict or inclusive inequalities. 2. 2 times 3. e -2 ≈ 0.135 4. Exponential: 0.039; uniform: 0.1 5. Exponential: 0.148 of arrivals; uniform: all arrivals 6. The trick used at the HotBits site is the following: they measure times between successive pairs of decay events. If the first wait is shorter than the second, it omits a 0, and if the first wait is longer than the second, it omits a 1.

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Complete Instructor Answers

Chapter 12 Sequences and Series Exercises 12.1 (page 667–668)

For exercises . . .

5–11 11–26

27–34

35–44

43,44,45(a)-(c), 45(d),46,47, 48,49–51 52,53 3,4 7

1. False 2. True 3. False 4. True 5. 2, 6, 18, 54 Refer to example . . . 1 2 5 6 6. 4, 8, 16, 32, 64 7. 1 / 2, 2, 8, 32 8. 2 / 3, 4, 24 9. 3 / 2, 3, 6, 12, 24 10. 27, 9, 3, 1 11. a5 = 324; an = 4132n - 1 12. a5 = 2048; an = 8142n - 1 13. a5 = -1875; an = -31-52n - 1 14. a5 = -64; an = -41-22n - 1 15. a5 = 3 / 2; an = 2411 / 22n-1 or an = 24 / 2n - 1 16. a5 = 2 / 9; an = 1811 / 32n - 1 or an = 18 / 3n - 1 17. a5 = -256; an = - 1-42n - 1 18. a5 = -243; an = 1-32n 19. r = 4; an = 4n 20. r = 2; an = 6122n - 1 21. r = 2; an = 13 / 42122n - 1 22. Not geometric 23. Not geometric 24. Not geometric 25. r = -2 / 3; an = 1-5 / 821-2 / 32n - 1 26. r = -1 / 3; an = 17 / 421-1 / 321n - 12 27. 93 28. 1705 29. 33 / 4 30. 1111 / 72 31. 33 32. -1705 33. 464.4 34. -6.221 35. 2040 36. 4372 37. 262,143 / 2 38. 3069 / 4 39. 183 / 4 40. -63 / 2 41. 511 / 4 42. 2059 / 9 43. (a) $3932 (b) $2013 44. $23,530 45. (a) $30,000; $31,500; $33,075 (b) r = 1.05; an = 30,00011.052n - 1 (c) $46,539.85; $201,142.53 (d) $3,623,993.23 46. $14,398,193 47. $230 or $1,073,741,824; $231 - $1 or $2,147,483,647 48. About 41% 49. 3.125 * 10 13 50. About 95 times 51. (a) 32.77 in. (b) 9.007 * 10 12 in., or about 142 million miles 52. (a) 32, 16, 8, 4, 2, 1 or 25, 24, 23, 22, 21, 20 (b) 63 (c) 2n - 1, 2n - 2, g, 22, 21, 20; Sn = 2n - 1 53. (a) 27, 9, 3, 1 or 33, 32, 31, 30 (b) 40 (c) 3n - 1, 3n - 2, g, 32, 31, 30; Sn = 13n - 12/ 2 (d) Sn = 1t n - 12/ 1t - 12

Exercises 12.2 (page 677–679)

For exercises . . .

1–6,31, 37–40 Refer to example . . . 1

7–10,32 11–14,33–36, 15–24, 41–43 44–47 2 3,4 5

25–30,48, 49–52 53,54 6 8

(Note: Answers in this section may differ by a few cents, depending on how calculators are used.)

1. True 2. True 3. False 4. False 5. $1509.35 6. $22,538.71 7. $278,150.87 8. $4,654,133.66 9. $833,008.00 10. $597,173.82 11. $205,785.64 12. $128,254.48 13. $142,836.33 14. $288,307.46 15. $526.95 16. $3416.70 17. $952.33 18. $120.67 19. $39,434.37 20. $8339.50 21. $17,585.54 22. $20,975.21 23. $1,367,773.96 24. $440,559.38 25. $111,183.87 26. $103,796.58 27. $97,122.49 28. $85,594.79 29. $476.90 30. $160.08 31. $11,942.55 32. $5570.59 33. $1673.21 34. $320.10 35. (a) $132,318.77 (b) $121,909.27 (c) $10,409.50 36. (a) $9600 (b) $12,282.85 (c) $2682.25 37. (a) $491.54 (b) $533.42 38. $1063.69 39. $1398.12 40. (a) $4765.40 (b) $4482.90 41. $112,796.87 42. $152,667.08 43. $209,348.00 44. $290,335.90 46. (a) $120 (b) $681.83 45. (a) $1200 (b) $3511.58 (c) Payment Amount of (c) Payment Amount of Interest Interest Total in Number

Deposit

Earned

Total

Number

Deposit

Earned

Account

1

$3511.58

$0

$3511.58

1

$681.83

$0

$681.83

2

$3511.58

$105.35

$7128.51

2

$681.83

$54.55

$1418.21

3

$3511.58

$213.86

$10,853.95

3

$681.83

$113.46

$2213.50

4

$3511.58

$325.62

$14,691.15

4

$681.83

$177.08

$3072.41

5

$3511.58

$440.73

$18,643.46

5

$681.80

$245.79

$4000.00

6

$3511.58

$559.30

$22,714.34

7

$3511.58

$681.43

$26,907.35

8

$3511.58

$807.22

$31,226.15

47. (a) $32.49 (b) $195.52; $10.97 48. $6699.00 49. $12,493.78 50. $573,496.06 51. (a) $623,110.52 (b) $456,427.28 (c) $563,757.78 (d) $392,903.18 52. $547.47; $4278.56 53. $1885.00; $229,612.44 54. $1267.07; $162,628.91 55. $2583.01; $336,107.59 56. $1461.16; $180,417.11 57. (a) $4025.90 (b) $2981.93 58. (a) $17,584.58 (b) $13,278.95

9

$3511.58

$936.78

$35,674.51

10

$3511.58

$1070.24

$40,256.33

11

$3511.58

$1207.69

$44,975.60

12

$3511.58

$1349.27

$49,836.45

13

$3511.58

$1495.09

$54,843.12

14

$3511.59

$1645.29

$60,000.00

Amount of Payment

Interest for Period

Portion to Principal

Principal at End of Period

0

—

—

—

$4000

1

$1207.68

$320.00

$887.68

$3112.32

2

$1207.68

$248.99

$958.69

$2153.63

3

$1207.68

$172.29

$1035.39

$1118.24

4

$1207.70

$89.46

$1118.24

$0

59. Payment Number

Copyright © 2022 Pearson Education, Inc.

55–58 7

A-70

Complete Instructor Answers

60. Payment Number

Amount of Payment

Interest for Period

Portion to Principal

Principal at End of Period

0

—

—

—

$72,000.00

1

$10,017.22

$3420.00

$6597.22

$65,402.78

2

$10,017.22

$3106.63

$6910.59

$58,492.19

3

$10,017.22

$2778.38

$7238.84

$51,253.35

4

$10,017.22

$2434.53

$7582.69

$43,670.66

5

$10,017.22

$2074.36

$7942.86

$35,727.80

6

$10,017.22

$1697.07

$8320.15

$27,407.65

7

$10,017.22

$1301.86

$8715.36

$18,692.29

8

$10,017.22

$887.88

$9129.34

$9562.95

9

$10,017.19

$454.24

$9562.95

$0

Amount of Payment

Interest for Period

Portion to Principal

Principal at End of Period

0

—

—

—

—

1

$183.93

$62.86

$121.07

$7062.93

2

$183.93

$61.80

$122.13

$6940.80

3

$183.93

$60.73

$123.20

$6817.60

4

$183.93

$59.65

$124.28

$6693.32

5

$183.93

$58.57

$125.36

$6567.96

6

$183.93

$57.47

$126.46

$6441.50

Amount of Payment

Interest for Period

Portion to Principal

Principal at End of Period

0

—

—

—

$20,000.00

1

$2465.82

$800.00

$1665.82

$18,334.18

2

$2465.82

$733.37

$1732.45

$16,601.73

3

$2465.82

$664.07

$1801.75

$14,799.98

4

$2465.82

$592.00

$1873.82

$12,926.16

5

$2465.82

$517.05

$1948.77

$10,977.39

6

$2465.82

$439.10

$2026.72

$8950.67

7

$2465.82

$358.03

$2107.79

$6842.88

8

$2465.82

$273.72

$2192.10

$4650.78

9

$2465.82

$186.03

$2279.79

$2370.99

10

$2465.83

$94.84

$2370.99

$0

61. Payment Number

62. Payment Number

Exercises 12.3 (page 687–688)

For exercises . . .

5–24,39–43,44–48,49,50 25–38,44–48 Refer to example . . . 1,2,4 3,5

W1. ƒ′1x2 = 1 / 22x + 5, ƒ″1x2 = -1 / 12x + 523/2, ƒ‴1x2 = 3 / 12x + 525/2 W2. ƒ′1x2 = -1 / 1x + 222, ƒ″1x2 = 2 / 1x + 223, ƒ‴1x2 = -6 / 1x + 224 W3. ƒ′1x2 = 3e 3x, ƒ″1x2 = 9e 3x, ƒ‴1x2 = 27e 3x W4. ƒ′1x2 = 2 / 11 + 2x2, ƒ″1x2 = -4 / 11 + 2x22, ƒ‴1x2 = 16 / 11 + 2x23 1. False 2. False 3. True 4. False 5. 1 - 2x + 2x 2 - 14 / 32x 3 + 12 / 32x 4 6. 1 + 3x + 19 / 22x 2 + 19 / 22x 3 + 127 / 82x 4 7. e + ex + ex 2 / 2 + ex 3 / 6 + ex 4 / 24 8. 1 - x + x 2 / 2 - x 3 / 6 + x 4 / 24 9. 3 + x / 6 - x 2 / 216 + x 3 / 3888 - 15 / 279,9362x 4 10. 4 + x / 8 - x 2 / 512 + x 3 / 16,384 - 15 / 2,097,1522x 4 11. -1 + x / 3 + x 2 / 9 + 15 / 812x 3 + 110 / 2432x 4 12. 2 + x / 12 - x 2 / 288 + 15 / 20,7362x 3 - 15 / 248,8322x 4 13. 1 + x / 4 - 13 / 322x 2 + 17 / 1282x 3 - 177 / 20482x 4 14. 2 + x / 32 - 13 / 40962x 2 + 17 / 262,1442x 3 - 177 / 67,108,8642x 4 15. -x - x 2 / 2 - x 3 / 3 - x 4 / 4 16. 2x - 2x 2 + 18 / 32x 3 - 4x 4 17. 2x 2 - 2x 4 18. -x 3 19. x - x 2 + x 3 / 2 - x 4 / 6 20. x 2 + x 3 + x 4 / 2 21. 27 - 19 / 22x + x 2 / 8 + x 3 / 432 + x 4 / 10,368 22. 1 + 13 / 22x + 13 / 82x 2 - x 3 / 16 + 13 / 1282x 4 23. 1 - x + x 2 - x 3 + x 4 24. -1 - x - x 2 - x 3 - x 4 25. 0.9608 26. 1.0618 27. 2.7732 28. 0.9324 29. 2.9866

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Complete Instructor Answers

A-71

30. 4.0373 31. -1.0164 32. 1.9925 33. 1.0147 34. 1.9962 35. -0.0305 36. 0.0583 37. 0.0080 38. -0.0080 39. P31x2 = 3 + 6x + 6x 2 + 4x 3 40. P41x2 = 1 + x + x 2 + x 3 + x 4; Pn1x2 = 1 + x + g + x n - 1 + x n 41. (b) 4.04167; actual value is 4.04124. 42. (b) 1320; 1357 43. (a) 1 + lN + l2N 2 / 2 (b) N = 22k / l 44. $399; $399 45. $4623; $4623 46. $0.91; $0.90 47. $718; $718 48. 0.15 ml; 0.2 ml

Exercises 12.4 (page 693–695)

For exercises . . .

5–18,25,26,28,31–38 19–24 Refer to example . . . 2 1

29,30 4

35–42 3

9–26,28,41–43 33–38 3 4

39,40 5

1. False 2. True 3. True 4. False 5. Converges to 40 6. Converges to 5 7. Diverges 8. Diverges 9. Converges to 81 / 2 10. Converges to 256 / 3 11. Converges to 1000 / 9 12. Converges to 88 13. Converges to 5 / 2 14. Converges to 8 / 5 15. Converges to 1 / 5 16. Converges to 101 17. Converges to e 2 / 1e + 12 18. Diverges 19. S1 = 1; S2 = 3 / 2; S3 = 11 / 6; S4 = 25 / 12; S5 = 137 / 60 20. S1 = 1 / 2; S2 = 5 / 6; S3 = 13 / 12; S4 = 77 / 60; S5 = 29 / 20 21. S1 = 1 / 7; S2 = 16 / 63; S3 = 239 / 693; S4 = 3800 / 9009; S5 = 22,003 / 45,045 22. S1 = 1 / 2; S2 = 7 / 10; S3 = 33 / 40; S4 = 403 / 440; S5 = 3041 / 3080 23. S1 = 1 / 6; S2 = 1 / 4; S3 = 3 / 10; S4 = 1 / 3; S5 = 5 / 14 24. S1 = 1 / 12; S2 = 37 / 300; S3 = 103 / 700; S4 = 1027 / 6300; S5 = 24,169 / 138,600 25. 2 / 9 26. 2 / 11 27. (a) First 3.12; second 2.90: Viete’s formula (b) 38 28. (a) 1111 units (b) 1111 units 29. (a) $2000 (b) 10 30. (a) $1333.33 (b) 33.33% 32. (d) 33. (a) 34. 70 meters 35. 1600 rotations 36. 200 centimeters 37. 12 meters 38. 4 23 / 3 square meters 39. (a) 10 p.m. (b) 10 p.m. 40. (a) 10 / 9 sec (b) 10 / 9 sec 41. 15 miles 42. (c) All x in 10, 12

Exercises 12.5 (page 702–704)

For exercises . . .

5–9,27 Refer to example . . . 2

1. False 2. True 3. True 4. True 5. 6 + 6x + 6x 2 + 6x 3 + g + 6x n + g; 1-1, 12 6. -3 - 3x - 3x 2 - 3x 3 - g - 3x n - g; 1-1, 12 7. x 2 + x 3 + x 4 / 2! + x 3 / 3! + g + x n + 2 / n! + g; 1-∞, ∞2 8. x 5 + x 6 + x 7 / 2! + x 8 / 3! + g + x n + 5 / n! + g; 1-∞, ∞2 9. 5 / 2 + 15 / 222x + 15 / 232x 2 + 15 / 242x 3 + g + 15 / 2n + 12x n + g; 1-2, 22 10. -3 / 4 - 13 / 162x - 13 / 642x 2 - 13 / 2562x 3 - g - 13 / 4n + 12x n - g; 1-4, 42 11. 8x - 8 # 3x 2 + 8 # 32x 3 - 8 # 33x 4 + g + 1-12n # 8 # 3nx n + 1 + g; 1-1 / 3, 1 / 32 12. 7x - 14x 2 + 28x 3 - 56x 4 + g + 1-12n # 7 # 2nx n + 1 + g; 1-1 / 2, 1 / 22 13. x 2 / 4 + x 3 / 42 + x 4 / 43 + x 5 / 44 + g + x n + 2 / 4n + 1 + g; 1-4, 42 14. 9x 4 + 9x 5 + 9x 6 + 9x 7 + g + 9x n + 4 + g; 1-1, 12 15. 4x - 142 / 22x 2 + 143 / 32x 3 - 144 / 42x 4 + g + 1-12n4n + 1x n + 1 / 1n + 12 + g; 1-1 / 4, 1 / 44 16. -x / 2 - x 2 / 8 - x 3 / 24 - x 4 / 64 - g - x n + 1 / 31n + 122n + 14 - g; 3-2,22 17. 1 + 4x 2 + 142 / 2!2x 4 + 143 / 3!2x 6 + g + 14n / n!2x 2n + g; 1-∞, ∞2 18. 1 - 3x 2 + 19 / 22x 4 - 19 / 22x 6 + g + 1-12n13n / n!2x 2n + g; 1-∞, ∞2 19. x 3 - x 4 + x 5 / 2! - x 6 / 3! + g + 1-12nx n + 3/n! + g; 1-∞, ∞2 20. x 4 + 2x 5 + 2x 6 + 14 / 32x 7 + g + 12n / n!2x n + 4 + g; 1-∞, ∞2 21. 2 - 2x 2 + 2x 4 - 2x 6 + g + 1-12n2x 2n + g; 1-1, 12 22. 2 - 12 / 32x 2 + 12 / 92x 4 - 12 / 272x 6 + g + 1-12n12 / 3n2x 2n + g; 1- 23, 232 23. 1 + x 2 / 2! + x 4 / 4! + x 6 / 6! + g + x 2n / 12n2! + g; 1-∞, ∞2 24. x + x 3 / 6 + x 5 / 120 + x 7 / 5040 + g + x 2n + 1 / 12n + 12! + g; 1-∞, ∞2 4 4 25. 2x 4 - 122 / 22x 8 + 123 / 32x 12 - 124 / 42x 16 + g + 1-12n2n + 1x 4n + 4 / 1n + 12 + g; 3-1 / 2 2, 1 / 2 24 2 4 6 8 n + 1 2n + 2 1 2 1 2 1 2 1 2 1 26. -5x - 25 / 2 x - 125 / 3 x - 625 / 4 x - g - 5 x / n + 1 - g; -1 / 25, 1 / 252 27. 1 + 2x + 2x 2 + 2x 3 + g + 2x n + g 28. 2x + 12 / 32x 3 + 12 / 52x 5 + 12 / 72x 7 + g + 2x 2n + 1 / 12n + 12 + g 33. 0.3461 34. 0.5168 35. 0.1729 36. 1.9972 37. 0.1554 38. 0.2257 39. About 14.94 years; about 14.74 years; a difference of 0.2 year, or about 10 weeks 40. About 11.9 years; about 12 years, a difference of 0.1 year, or about 5 weeks 41. (b) l (c) 0.2254 42. (b) 1 / p (c) 3.514 (d) 0.6338 43. (a) 6 (b) 0.5787

Exercises 12.6 (page 708–709)

For exercises . . .

5–21,31–34,36–40 21–30 Refer to example . . . 1 2

W1. ƒ′1x2 = 2x -2/3 + 3 / x W2. ƒ′1x2 = 1-3x 2 + 2x2e -3x 1. True 2. False 3. False 4. True 5. 1.13 6. 3.58 7. 3.06 8. 2.70 9. 2.24 10. 1.60 11. -1.13; 2.37 12. -2.36; 1.50 13. -0.58 14. 2.14 15. 0.44 16. 0.47 17. 1.25 18. -0.79 19. 1.56 20. 1.81 21. 1.414 22. 1.732 23. 3.317 24. 3.873 25. 15.811 26. 17.321 27. 2.080 28. 2.466 29. 4.642 30. 4.946 31. Relative maximum at x = -1.65; relative minimum at x = 3.65 32. Relative maximum at x = -6.32; relative minimum at x = 0.32 33. Relative maximum at x = 1.19; relative minima at x = -0.71 and x = 1.77 34. Relative minimum at x = 0.75 36. 138.06 units 37. 4.80 years 38. (a) ƒ′1i2 = 3-1 + ni11 + i2-n - 1 + 11 + i2-n4/ i 2 (b) ƒ1i2/ ƒ′1i2 = 3Mi - Mi11 + i2-n - Pi 24/ 5M3-1 + ni11 + i2-n - 1 + 11 + i2-n46 (c) 0.013712295 (d) 0.1383273 39. i 2 = 0.02075485; i 3 = 0.02075742 40. i 2 = 0.01036283; i 3 = 0.01036541

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A-72

Complete Instructor Answers

Exercises 12.7 (page 715–716)

For exercises . . .

5–7,9,10,12–14,17–28,31, 8,11 15,16,33, 29,30 32,35,36,47–50,53,55 34,51 Refer to example . . . 1,3 2 5 4

37–40, 54 6

W1. ƒ′1x2 = 13x + 22/ 23x 2 + 4x + 5 W2. ƒ′1x2 = 10 3ln15x + 324/ 15x + 32 1. True 2. False 3. False 4. True 5. 4 6. 22 / 3 7. 0 8. Does not exist 9. 1 10. 1 11. Does not exist 12. -2 13. 1 14. 1 / 4 15. Does not exist 16. Does not exist 17. 1 / 12 222 or 22 / 4 18. 1 / 6 19. 1 / 4 20. 1 / 6 21. 1 / 12 22. 1 / 27 23. 53 24. -112 25. 0 26. -1 27. 1 / 8 28. 1 / 12 29. 1 / 9 30. 25 / 3 31. 1 32. -1 / 23 or - 23 / 3 33. Does not exist 34. Does not exist 35. 5 36. 7 37. 0 38. ∞ (does not exist) 39. 0 40. 0 41. ∞ (does not exist) 42. ∞ (does not exist) 43. 1 / 5 44. 1 / 3 45. 0 46. ∞ (does not exist) 47. 1 / 2 48. 2 49. 1 / 2 50. 2 51. lim 1x 2 + 32 Z 0, so l’Hospital’s rule does not apply. 52. 16a / 9 55. (b) -s′112

41–46 7

xS0

Chapter 12 Review Exercises (page 718–720)

For exercises . . .

1,13–16,75, 85,86 Refer to section . . . 1

2,3, 76–82 2

4–6, 17–32 3

7,8, 33–40 4

9,10,41–50, 83,84 5

11,67–74 12,51–66 6

1. True 2. False 3. True 4. True 5. False 6. True 7. False 8. True 9. True 10. False 11. False 12. False 13. a4 = -40; an = 51-22n - 1; S5 = 55 14. a4 = 16; an = 12811 / 22n - 1; S5 = 248 15. a4 = 1; an = 2711 / 32n - 1; S5 = 121 / 3 16. a4 = -250; an = 21-52n - 1; S5 = 1042 17. e 2 - e 2x + 1e 2 / 2!2x 2 - 1e 2 / 3!2x 3 + 1e 2 / 4!2x 4 18. 5 + 10x + 10x 2 + 120 / 32x 3 + 110 / 32x 4 19. 1 + x / 2 - x 2 / 8 + x 3 / 16 - 15 / 1282x 4 20. 3 + x / 27 - x 2 / 2187 + 15 / 531, 4412x 3 - 110 / 43,046,7212x 4 21. ln 2 - x / 2 - x 2 / 8 - x 3 / 24 - x 4 / 64 22. ln 3 + 12 / 32x - 12 / 92x 2 + 18 / 812x 3 - 14 / 812x 4 23. 1 + 12 / 32x - x 2 / 9 + 14 / 812x 3 - 17 / 2432x 4 24. 8 + 3x + 13 / 162x 2 - 11 / 1282x 3 + 13 / 40962x 4 25. 6.8895 26. 5.2041 27. 1.0149 28. 2.9978 29. 0.7178 30. 1.1184 31. 0.9459 32. 8.0601 33. Converges to 27 / 5 34. Converges to 20 / 3 35. Diverges 36. Diverges 37. Converges to 1 / 3 38. Converges to 432 / 11 39. S1 = 1; S2 = 4 / 3; S3 = 23 / 15; S4 = 176 / 105; S5 = 563 / 315 40. S1 = 1 / 12; S2 = 2 / 15; S3 = 1 / 6; S4 = 4 / 21; S5 = 5 / 24 41. 4 / 3 + 14 / 322x + 14 / 332x 2 + 14 / 342x 3 + g + 14 / 3n + 12x n + g; 1-3, 32 42. 2x - 6x 2 + 18x 3 - 54x 4 + g + 1-12n # 2 # 3nx n + 1 + g; 1-1 / 3, 1 / 32 43. x 2 - x 3 + x 4 - x 5 + g + 1-12nx n + 2 + g;1-1, 12 44. 13 / 22x 3 + 13 / 42x 4 + 13 / 82x 5 + 13 / 162x 6 + g + 13 / 2n + 12x n + 3 + g; 1-2, 22 45. -2x - 122 / 22x 2 - 123 / 32x 3 - 124 / 42x 4 - g - 2n + 1x n + 1 / 1n + 12 - g; 3-1 / 2, 1 / 22 46. x / 3 - x 2 / 18 + x 3 / 81 - x 4 / 324 + g + 1-12nx n + 1 / 33n + 11n + 124 + g; 1-3, 34 47. 1 - 2x 2 + 122 / 2!2x 4 - 123 / 3!2x 6 + g + 1-12n12n / n!2x 2n + g; 1-∞, ∞2 48. 1 - 5x + 125 / 22x 2 - 1125 / 62x 3 + g + 1-12n15n / n!2x n + g; 1-∞, ∞2 49. 2x 3 - 2 # 3x 4 + 12 # 32 / 2!2x 5 - 12 # 33 / 3!2x 6 + g + 1-12n12 # 3n / n!2x n + 3 + g; 1-∞, ∞2 50. x 6 - x 7 + x 8 / 2 - x 9 / 6 + g + 1-12nx n + 6 / n! + g; 1-∞, ∞2 51. 7 / 4 52. 2 53. Does not exist 54. 3 55. 5 / 7 56. 1 / 12 252 or 25 / 10 57. -1 / 2 58. 1 / 8 59. Does not exist 60. 1 / 12 252 or 25 / 10 61. 0 62. ∞ (Does not exist) 63. 9 / 2 64. -1 / 3 65. -8 66. Does not exist 67. 4.73 68. 2.33 69. 2.65 70. 3.87 71. 6.132 72. 7.190 73. 4.558 74. 3.183 75. $11,495,247 76. $542.64 77. $27,320.71 78. $18,156.82 79. $3322.43 80. $1302.44 81. $1184.01 82. $1355.55 83. About 21.67 years; about 21.54 years; differ by 0.13 year, or about 7 weeks 84. About 8.04 years; about 8 years; differ by about 0.04 year, or about 2 weeks 85. 64,000 bacteria 86. xn = 22,70010.922n; about 14,961 crimes

Extended Application: Living Assistance and Subsidized Housing (page 720) 1. R ≈ 21.28; rent is 300 + R = 321.28, or $321.28. S ≈ 6.38; stipend is 1000 + 100 + S = 1106.38, or $1106.38. 2. Stipend: 1000 - 50 - 5010.3210.22 - 50 310.3210.2242 - g; final stipend is $946.81.

Rent: 300 - 10 - 1010.062 - 1010.0622 - g; final rent is $289.36. 3. Eastville (measured from Eastville toward Westville): 611 + 1 / 4 + 1 / 16 + g2; Eastville will locate the fire station 8 miles from Eastville. Westville (measured from Westville toward Eastville): 311 + 1 / 4 + 1 / 16 + g2; Westville will locate the fire station 4 miles from Westville. 4. Yes, the chosen location for the pool is halfway between the two towns. 5. Three terms give the value to the nearest dollar; five terms give the value to the nearest penny. 6. About 50 terms must be summed to get within 0.01 of ln 2.

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7

Complete Instructor Answers

A-73

59–66,81,86,88, 67–78, 89,93,94,96 85,87 6 8

83,84, 90–92 9

Chapter 13 The Trigonometric Functions Exercises 13.1 (page 735–739)

For exercises . . .

4–20 21–24, 97,98 Refer to example . . . 1 2,3

25–29

29–52, 79,80 4,5

53–58, 99–101 7

1. False 2. True 3. True 5 4. True 5. p / 3 6. p / 2 7. 5p / 6 8. 3p / 4 9. 3p / 2 10. 16p / 9 11. 11p / 4 12. 17p / 6 13. 225° 14. 120° 15. -390° 16. -45° 17. 288° 18. 100° 19. 105° 20. 900° Note: In Exercises 21–28 we give the answers in the following order: sine, cosine, tangent, cotangent, secant, and cosecant.

21. 4 / 5; -3 / 5; -4 / 3; -3 / 4; -5 / 3; 5 / 4 22. -5 / 13; -12 / 13; 5 / 12; 12 / 5; -13 / 12; -13 / 5 23. -24 / 25; 7 / 25; -24 / 7; -7 / 24; 25 / 7; -25 / 24 24. 3 / 5; 4 / 5; 3 / 4; 4 / 3; 5 / 4; 5 / 3 25. + + + + + + 26. + - - - - + 27. - - + + - - 28. - + - - + - 29. 23 / 3; 23; 2 30. 22 / 2; 22 / 2; 22; 22 31. 23 / 2; 23 / 3; 2 23 / 3 32. -1 / 2; - 23 / 2; -2 33. -1; -1 34. -1 / 2; - 23; -2 23 / 3 35. - 23 / 2; -2 23 / 3 36. 23; 23 / 3 37. 23 / 2 38. 23 / 2 39. 1 40. 23 / 3 41. 2 42. -1 43. -1 44. -1 45. - 22 / 2 46. Undefined 47. - 22 48. -1 49. 1 50. 23 / 3 51. 1 / 2 52. 23 / 2 53. p / 3, 5p / 3 54. 7p / 6, 11p / 6 55. 3p / 4, 7p / 4 56. p / 3, 4p / 3 57. 5p / 6, 7p / 6 58. p / 4, 7p / 4 59. 0.6293 60. 0.3907 61. -1.5399 62. 1.3764 63. 0.3558 64. 1.6004 65. 0.3292 66. 0.9994 67. a = 1; T = 2p / 3 68. a = 1 / 2; T = 1 / 2 69. a = 2; T = 8 70. a = 3; T = 1 / 440 y y 71.

72.

73.

74.

2 –2p –p

–2

y

75.

–2p

p 2p x y = 2 cos x

y

x

y = – 1 cos x 2

y

77.

78. y = –3 tan x

3

–4p

0 –2

(

2p 4p

)

y = 4 sin 1 x + p + 2 2

x – 2p 3

10

2p

1/2 –1/2

76.

82,95

2p x 3

0 –1

(

–p – p2

p 2

p

x

)

y = 2 cos 3x – p + 1 4

79. (a) All are 60° (b) 30°, 60°, 90° (c) 1, 23, 2 80. (a) 22 (b) 45°, 45°, 90° 81. (a) 1000 snowblowers (b) 750 snowblowers (c) 500 snowblowers (d) 0 snowblowers (e) 500 snowblowers (f) 82. (a) 550 ; yes (b) C1t2 = 80.63 sin10.8998t + 1.1542 + 416.1 (Answers using a different calculator may be slightly different.)

C(t) 5 80.63 sin (0.8998t 1 1.154) 1 416.1 550 0

13 0

0

13 0

(c) About 7 months (d) About 362.5 trillion BTUs (Answers using a different calculator may be slightly different.)

83. (a) 29.54; there is a lunar cycle every 29.54 days. (b) Feb. 15; 1.8% (c) 98.75% 84. (a) y = 2 sin12pt / 0.3502 (b) 0.0875 second (c) -1.95° 85. (a) T 5 37.29 1 0.46 cos(2p(t 2 16.37)/24) No, the two functions never cross. 87. P(t) 5 7[1 2 cos(2pt)](t 1 10) 1 100e0.2t 38 600 (b) About 2:55 p.m. (c) About 4:22 p.m.

86. (a) 100 (b) 258 (c) 122 (d) 296 0

30

0

36

6 0

T 5 36.91 1 0.32 cos(2p(t 2 14.92)/24)

88. 2.2 * 10 8 m per second 89. 2.1 * 10 8 m per second 90. 240°

91. 120°

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A-74

Complete Instructor Answers

92. (a)

(b) 0, 1 / 880, 1 / 440 (c) T = 1 / 440; frequency: 440 cycles per second 93. (a) 34°F (b) 58°F (c) 80°F (d) 90°F (e) 39°F 94. (a) About 8°F (b) About 37°F (c) About 45°F (d) About 1°F (e) 62°F and -12°F (f) 365

P(t) 5 0.002sin (880pt) 0.004

0

0.003

20.004

95. (a)

; yes (b) s1t2 = 94.1528 sin10.0166t - 1.21622 (Answers using a different calculator may be slightly different.) (c) 5:27 p.m.; 7:53 p.m.; 7:22 p.m. (d) 81st and 294th days 96. 2.2 m; 7.6 m 97. 60.2 m 98. 206 ft 99. 0.28° 100. (1.309, 0.951), (0.5, 1.539), 1-0.309, 0.9512 101. 34.6, 6.34

450

0 250

365

Exercises 13.2 (page 748–752)

For exercises . . .

5–30 31–36 Refer to example . . . 2–6 7

37–39 2–4

40 41–44 1 8

61–75 9

W1. dy / dx = ln 2x + 1 W2. dy / dx = e x1x - 22/ x 3 2 W3. dy / dx = 1011 + 2x24 W4. dy / dx = 110x - 32e 5x - 3x W5. dy / dx = 2x / 1x 2 + 12 1. False 2. True 3. True 4. True 5. dy / dx = 4 cos 8x 6. dy / dx = 2 sin 2x 7. dy / dx = 108 sec219x + 12 8. dy / dx = 56x sin 17x 2 - 422 9. dy / dx = -4 cos3x sin x 10. dy / dx = -45 sin4 x cos x 11. dy / dx = 8 tan7x sec2x 12. dy / dx = -15 cot4x csc2x 13. dy / dx = -12x cos 2x - 6 sin 2x 14. dy / dx = 8x sec 4x tan 4x + 2 sec 4x 15. dy / dx = - 1x csc x cot x + csc x2/ x 2 16. dy / dx = 31x - 12sec2x - tan x4/ 1x - 122 17. dy / dx = 4e 4x cos e 4x 18. dy / dx = -8e 2x sin 4e 2x 19. dy / dx = 1-sin x2e cos x 20. dy / dx = 12 csc2x2e cot x 21. dy / dx = 14 / x2cos1ln 3x 42 22. dy / dx = 1-3 / x2 sin1ln 2x 3 2 23. dy / dx = 12x cos x 22/ sin x 2 or 2x cot x 2 24. dy / dx = 2 sec2x / tan x 25. dy / dx = 16 cos x2/ 13 - 2 sin x22 26. dy / dx = -151sin x2/ 15 - cos x22 27. dy / dx = 2sin 3x1sin 3x cos x - 3 sin x cos 3x2/ 12 2sin x 1sin2 3x22 28. dy / dx = 1-4 cos x sin 4x + cos 4x sin x2/ 12 cos3/2x cos1/2 4x2 29. dy / dx = 13 / 42sec21x / 42 - 8 csc2 2x + 5 csc x cot x - 2e -2x 30. dy / dx = 81sin 3x + cot1x 322713 cos 3x - 3x 2 csc21x 322 31. 1 32. 22 / 2 33. 1 / 2 34. 22 / 2 35. 1 36. -2 37. -csc2 x 41. (a) np / 2, where n is an odd integer (b) 11n - 1 / 22p, 1n + 1 / 22p2 where n is an even integer (c) 11n + 1 / 22p, 1n + 3 / 22p2, where n is an even integer 42. (a) None (b) 1np / 2, 1n + 22p / 22 where n is an odd integer (c) None 43. Relative maximum of 1 at x = c, -7 / 2, -3 / 2, 1 / 2, 5 / 2, c; relative minimum of -1 at x = g, -5 / 2, -1 / 2, 3 / 2, 7 / 2, c 44. Relative maximum of 1 at x = c, -2p, -p, 0, p, 2p, c; relative minimum of -1 at x = g, -3p / 2, -p / 2, p / 2, 3p / 2, c 45. ƒ″1x2 = -9x 4 cos1x 32 - 6x sin1x 32; 0; -144 cos 8 - 12 sin 8 46. ƒ″1x2 = 6 sin2 x cos x - 3 cos3 x; -3; 6 sin2 2 cos 2 - 3 cos3 2 47. ƒ′1x2 = cos x; ƒ″1x2 = -sin x; ƒ″1x2 = -cos x; ƒ1421x2 = sin x; ƒ14n21x2 = sin x 48. ƒ′1x2 = -sin x; ƒ″1x2 = -cos x; ƒ‴1x2 = sin x; ƒ1421x2 = cos x; ƒ14n21x2 = cos x 49. Concave upward on c ∪ 1-3p / 2, -p2 ∪ 1-p / 2, 02 ∪ 1p / 2, p2 ∪ c; concave downward on c ∪ 1-p, -p / 22 ∪ 10, p / 22 ∪ 1p, 3p / 22 ∪ c; inflection points at 1np / 2, 02, where n is an integer 50. Concave upward on c ∪ 1-1, -1 / 22 ∪ 10, 1 / 22 ∪ 11, 3 / 22 ∪ c; concave downward on c ∪ 1-1 / 2, 02 ∪ 11 / 2, 12 ∪ 13 / 2, 22 ∪ c; inflection points at 1n, 02, where n is an integer f(x) f(x) f(x) = x + sin x 51.

52. f(x) = x + cos x 8 6 4 (0, 1) 2

–10

–8 –6 –4 p

p

_– 2 , – 2 +

8

(2p, 2p)

–4 –6 –8 –10

+

6

8

_

_p2 , p2 + 2

3p , 3p 2 2

4

4 (0, 0) 10 x

–8 –4 (–p, –p)

(p, p) 4

8 x

–4 –8

53. Absolute maximum of p / 2 at x = p; absolute minimum of p / 6 - 23 / 2 at x = p / 3 54. Absolute maximum of 0 at x = 0; absolute minimum of 1 - p / 2 at x = p / 4 55. dy / dx = sec1xy2/ x - y / x 56. dy / dx = -cos2y 57. dy/dt = 18 - 4p2/ 3p14 - p24 58. dy / dt = 6 59. 0.03; 0.0300; 0 60. 1; 1.0000; 0 61. (a) R′1t2 = -240p sin 2pt (b) -$120p per year (c) $0 per year (d) $120p per year 62. (a) C′1t2 = 75.55 cos10.8998t + 1.1542 (b) -54.9; in March, the amount of electricity consumed is decreasing by about 54.9 trillion BTUs per month. (c) 69.9; in June, the amount of electricity consumed is increasing by about 69.9 trillion BTUs per month.

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76,77 10

Complete Instructor Answers

63. (a)

y5 0.4

(b) v = dy / dt = 1-3p2 / 82 sin 33p1t - 1 / 324; a = d 2y / dt 2 = 1-9p3 / 82 cos 33p1t - 1 / 324 (d) At t = 1 second, the force is clockwise and the arm makes an angle p / 8 radians forward from the vertical. The arm is moving clockwise. At t = 4 / 3 seconds, the force is counterclockwise and the arm makes an angle of -p / 8 radians from the vertical. The arm is moving counterclockwise.

At t = 5 / 3 seconds, the answer corresponds to t = 1 second. So the arm is moving clockwise and makes an angle of p / 8 from the vertical.

p 1 cos[3p_t 2 3 +] 8

4 3

0

A-75

20.4 y

64. (a)

1_ 5 _ _1 5

1

2

3

y = 1_ sin [p(t – 1)] 5

65. (a)

t

(b) v = dy / dt = 1p / 52 cos 3p1t - 124; a = d 2y / dt 2 = 1-p2 / 52 sin 3p1t - 124 (d) At t = 1.5, acceleration is negative, arm is moving clockwise and is at an angle of 1 / 5 radian from vertical; at t = 2.5, acceleration is positive, arm is moving counterclockwise and is at an angle of -1 / 5 radian from vertical; at t = 3.5, acceleration is negative, arm is moving clockwise and is at an angle of 1 / 5 radian from vertical.

L(t) 5 0.022t 2 1 0.55t 1 316 1 3.5sin (2pt) 360

0

30

310

(b) C1252 = 370 parts per million; C135.52 = 401.71 parts per million; C150.22 ≈ 468.05 parts per million (c) 19.18 parts per million per year; the level of carbon dioxide will be increasing at the beginning of 2010 at 19.18 parts per million.

380

66. (a)

(b) L1252 = 343.5 parts per million; L135.52 = 363.25 parts per million; L150.22 = 402.38 parts per million (c) 9.55 parts per million per year; the level of carbon dioxide was increasing at the beginning of 2010 at 9.55 parts per million.

0 320

25

67. (a) About 1488 (b) About 5381 (c) 2000 (d) About 2916 (e) (f) Maximum is 7389 when t = p / 2 + 2pn, where n is any f(t) 5 1000 e2 sin t integer; minimum is 135 when t = 3p / 2 + 2pn.

9000 68. (a) 10.17 in. (b) ds / du = -2.625 sin u11 + cos u / 215 + cos2 u2 (c) 4.944 radians 0

11

0

69. (a)

0.004

0

0.01

(b) The pressure is decreasing at a rate of 1.05 lb per ft2 per sec when t = 0.002.

70. (a) About 340 cm (b) About 242 cm (c) About 445 cm 71. (a) 13.55 ft (c) 52.39 ft (d) dx / da = 1V 2 / 162cos12a2 and x is maximized when a = p / 4. (e) 242 ft 72. (a) ds / dt = -2.625 sin u11 + cos u / 215 + cos2u2 du / dt (b) 21.6 mph 73. (a) 1 (b) - 22 / 2 ≈ -0.7071 (c) 2 (d) -2 (e) -3 22 / 2 ≈ -2.1213 (f) 2

20.004

74. (a) 200p m / min (b) 400p m / min 75. (a) 5 / p rev per minute

Exercises 13.3 (page 758–759)

(b) 5 / 12p2 rev per minute

76. 14.38 ft

77. 20.81 ft

For exercises . . .

5–34 35–40,43–49,51,52,54–56 50,53 Refer to example . . . 1–3 4 5

W1. e x / 2 + C W2. ln1x 2 + 62/ 2 + C W3. xe x + C W4. 4x 2 ln x - 2x 2 + C 1. False 2. False 3. True 4. True 5. 11 / 32 sin 3x + C 6. 1-1 / 52 cos 5x + C 7. 3 sin x + 4 cos x + C 8. -9 cos x + 8 sin x + C 9. 1-cos x 22/ 2 + C 10. sin x 2 + C 11. -tan 3x + C 12. 11 / 42 cot 8x + C 13. 11 / 82 sin8x + C 14. 11 / 52 sin5x + C 15. -2 1cos x23/2 + C 16. 2 sin1/2x + C 17. -ln 1 + cos x + C 18. -ln 1 - sin x + C 19. 11 / 42sin x 8 + C 20. 1-1 / 52 cos1x + 225 + C 21. -3 ln cos1x / 32 + C 22. 1-8 / 32 ln sin1-3x / 82 + C 23. 11 / 62 ln sin x 6 + C 24. -2 ln cos1x / 422 + C 25. -cos e x + C 26. ln cos e -x + C 27. -csc e x + C 28. 11 / 52 sec x 5 + C 29. 1-6 / 52x sin 5x - 16 / 252cos 5x + C 30. 1-7 / 52 x cos 5x + 17 / 252 sin 5x + C 31. -4x cos x + 4 sin x + C 32. -11 x sin x - 11 cos x + C 33. 1-3 / 42x 2 sin 8x - 13 / 162x cos 8x + 13 / 1282 sin 8x + C 34. -20x 2 cos1x / 22 + 80x sin1x / 22 + 160 cos1x / 22 + C 35. 1 - 22 / 2 36. 1 37. -ln1 23 / 22 38. 11 / 22 ln 2 39. 23 / 2 - 1 40. 1 23 - 222/ 2 41. 0.5 42. 0.5 43. 3 44. 13 / 22 ln 2 ≈ 1.040 45. 22 - 1 46. 1 / 2 47. 22 / 2 - 1 + 1ln 22/ 2 48. 4 23 - 4 - p / 3 49. 6000 2

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50. (a)

160

; yes (b) C1t2 = 75.02 sin10.3860t + 2.0042 + 107.0 (Answers using a different calculator may be slightly different.)

C(t) 5 75.02 sin (0.3860t 1 2.004) 1 107.0 160

0

13

0

0

13

0

(c) 77.3 trillion BTUs; 70 trillion BTUs (d) Decreasing 26.6 trillion BTUs per month (e) 1020 trillion BTUs; 1002 trillion BTUs (f) 12 months; 16.3 months 51. 60,000 53. 4430 hours; this result is relatively close to the actual value. 54. sin k (“sink”) 55. tan k (“tank”) 56. cos t (“cost”)

Chapter 13 Review Exercises (page 762–765)

For exercises . . .

1–5,11–40, 6–8,41–72,93(c),94, 9,10,73–93, 95(a)–(h),96 95(i)–(o),97–100 101 Refer to section . . . 1 2 3

1. False 2. True 3. False 4. False 5. True 6. False 7. True 8. False 9. False 10. False 14. Any integer multiple of π/6 or π/4 15. p / 2 16. 8p / 9 17. 5p / 4 18. 3p / 2 19. 900° 20. 135° 21. 81° 22. 54° 23. 23 / 2 24. - 23 25. 22 / 2 26. -2 23 / 3 27. 2 23 / 3 28. - 23 / 3 29. 1 / 2 30. 1 / 2 31. 2 32. 2 23 / 3 33. -2.1445 34. -0.8387 35. 0.7058 36. 0.6811 y 37. y 38.

39.

40. y y = 4 cos x

2 0 –2

p

2 3

1 2p x

–p 2

2p x

p x 2

0 –1

–2 3

y = – 2 sin x 3

y = 1 tan x 2

41. -1, 1 42. x = p / 6 or x = 5p / 6 43. dy / dx = 10 sec2 5x 44. dy / dx = -28 cos 7x 45. dy / dx = 6x csc216 - 3x 22 46. dy / dx = 8x sec2 14x 2 + 32 47. dy / dx = 64x sin3 14x 22 cos14x 22 48. dy / dx = -10 sin x cos4 x 49. dy / dx = -2x sin 11 + x 22 50. dy / dx = -2x 3 csc21x 4 / 22 51. dy / dx = e -2x1cos x - 2 sin x2 52. dy / dx = -x 2 csc x cot x + 2x csc x 53. dy / dx = 1-2 cos x sin x + cos2x sin x2/ 11 - cos x22 54. dy / dx = 2 cos x / 1sin x + 122 55. dy / dx = 1sec2x + x sec2x - tan x2/ 11 + x22 56. dy / dx = - 31 + 16 - x2 tan x4/ sec x 57. dy / dx = 1cos x2/ 1sin x2 or cot x 58. dy / dx = -tan x 59. Never increasing; decreasing on 1p / 4 + np / 2, 3p / 4 + np / 22, where n is an integer 60. Increasing on 1p / 8 + np / 2, 3p / 8 + np / 22, where n is an integer; decreasing on 1-p / 8 + np / 2, p / 8 + np / 22, where n is an integer 61. Relative maxima of 2 at x = 0, ± 2, ± 4, c; relative minima of -2 at x = ±1, ±3, c 62. Relative maxima of 5 at x = c, -p / 6, 3p / 6, 7p / 6, c; relative minima of -5 at x = c, p / 6, 5p / 6, 9p / 6, c 63. ƒ″1x2 = 98 sec2 7x tan 7x; ƒ″112 = 98 sec2 7 tan 7; ƒ″1-32 = 98 sec21-212 tan1-212 or -98 sec2 21 tan 21 64. ƒ″1x2 = -9x cos 3x - 6 sin 3x; ƒ″112 = -9 cos 3 - 6 sin 3; ƒ″1-32 = 27 cos 1-92 - 6 sin 1-92 or 27 cos 9 + 6 sin 9 y f(x) = x – cos x f(x) 65.

66.

67. Absolute maximum of p - 1 at p; absolute 8 8 minimum of 1 at 0 68. Absolute maximum of 5p , 5p 3p 3p 6 , _2 2 + (2p, 2p) 6 _2 2 + f(x) = x – sin x 4 1 + 22 at 3p / 4; absolute minimum of 0 at 0 4 (p, p) p p 2 _– 2 , – 2 + 2 69. dy / dx = 31 - sin 1x + y24/ 32y + sin1x + y24 _p2 , p2 + x –3p –2p –p–2 p 2p 3p x 70. dy / dx = 3y sec21xy2 + 3x 24/ 31 - x sec21xy24 –3p –2p –p p 2p 3p (–p, –p) –4 3p, – 3p –4 – 71. dy / dt = -1 / 2 72. dy / dt = 6 73. 11 / 52 sin 5x + C (–2p, –2p) _ 2 2+ –6 –6 1 74.

-1 / 22 cos 2x + C 75. 11 / 52 tan 5x + C –8 (–3p, –3p) –8 –10 _– 5p2 , – 5p2 + 76. 1-1 / 72 ln cos 7x + C 77. -4 cot x + C 78. 8 tan x + C 79. 15 / 42sec 2x 2 + C 80. 1-1 / 122 cos 4x 3 + C 81. 1-1 / 92 cos9 x + C 82. 1-2 / 321cos x23/2 + C 83. 11 / 242 ln sin 8x 3 + C 84. 1-1 / 222 ln cos 11x 2 + C 85. 31cos x2-1/3 + C 86. 11 / 102 tan2 5x + C 87. 1 88. 1 / 2 89. 20p 90. p - 3 / 2 91. - 1x + 22cos x + sin x + C 92. 1x 2 / 2 - 1 / 42 sin 2x + 1x / 22 cos 2x + C 93. (a) 55,000 ; yes (b) C1t2 = 35,072 sin10.32954t + 2.31602 + 35,706 (Answers using a different calculator may be slightly different.)

C(t) 5 35,072 sin (0.32954t 1 2.3160) 1 35,706 55,000 0

0

12

0

0

12

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Complete Instructor Answers

(c) 10,888; the residential consumption of natural gas in Pennsylvania is increasing by 10,888 million cubic feet per month. (d) 249,882 million cubic feet; 238,391 million cubic feet (e) 19.07 months; 12 months 94. 105;75 y 95. (a) sin u = s / L 2 (b) L 2 = s / sin u (c) cot u = 1L 0 - L 12/ s (d) L 1 = L 0 - s cot u P(t) = 90 + 15 sin 144pt (e) R1 = k # L 1 / r 41 (f) R2 = k # L 2 / r 42 (g) R = k1L 1 / r 41 + L 2 / r 422 105 (h) R = k1L 0 - s cot u2/ r 41 + ks / 1r 42 sin u2 (i) dR / du = ks csc2u / r 41 - ks cos u / 1r 42 sin2 u2 90 (j) 0 = 1ks csc2u2/ r 41 - 1ks cos u2/ 1r 42 sin2 u2 (k) k / r 41 - k cos u / r 42 = 0 (l) cos u = r 42 / r 41 (n) cos u ≈ 0.0039; u ≈ 90° (o) 84° to the nearest degree 75 0

1 ___ 144

1 __ 72

t

70

96. (a)

0

30

12

(b) y = 12.5631 sin10.548067t - 2.353262 + 50.2793 (Answers using a different calculator may be slightly different.) (c) y 5 12.5631 sin (0.548067t 2 2.35326) 1 50.2793 (d) About 11.5 months; 12 months 97. (a) Yes (b) 0.18 … a … 0.41 in radians or 70 10.3 … a … 23.5 in degrees (c) 0.995 feet / degree; the distance the tennis ball travels will increase by approximately 1 foot by increasing the angle of the tennis racket by one degree.

0

12

30

98. (a)

p y 512.5 sin _ (t 1 1.2)+ 1 14.7 6 28

0

(b) 27.13 MCF; 3.28 MCF (c) -2.66 MCF / month; in July, the monthly gas usage is decreasing by 2.66 MCF per month. (d) 176.4 MCF 100. u = p / 4 or 45° 101. (c) ln sec x + tan x + C and -ln sec x - tan x + C (d) 8.63 in. (e) 5.15 in.

12 0

Extended Application: The Shortest Time and the Cheapest Path (pages 767–768) 3. 6.2

5. About 4.68

6. Below the horizon

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Solution Manual for Calculus with Applications, 12th Edition.

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