ANSWERS TO ODD NUMBERED PROBLEMS

Section 1.1

1 (a) (IV)

(b) (II)

(c) (III)

3 Increasing

5 Decreasing

7 Argentina produced 14 million metric tons of wheat in 2010

9 (a) 8; 7

(b) 10

11 f(5) = 25

13 f(5) = 2

15 (a) 2

(b) 11

(c) 3 and −3

(d) No

17 f(0) meters

19 f(0) = f(1) + 0.001

23 Greatest number of species at intermediate number of snails; yes

25 (a) $1000

(b) About $2200

(c) About 20 years

27

29

35

37

Section 1.2

1 y = (1/2)x + 2

3 y = (1/2)x + 2

5 Slope:−12/7

Vertical intercept: 2/7

7 Slope: 2

Vertical intercept: −2/3

9 (a) l₁

(b) l₃

(c) l₂

(d) l₄

11 (a) P = 30,700 + 850t

(b) 39,200 people

(c) In 2026

13 (a) 300 miles

(b) 50 mph

(c) D = 300 + 50t

15 (a) y = −2x + 27

(b) s = 2t + 32

17 (a) 1.8 billion dollars/year

(b) 19.1 billion dollars

(c) 37.1 billion dollars

(d) 2013

19 (a) About 480 million tons

(b) About 10 million tons/year

(c) M = 480 + 10t

21 (a) P = 1241 + 26.9t

(b) 26.9 million tons per year

(c) 1241 million tons in 1975

(d) 2317 million tons

(e) In the year 2021

23 (b) P = 100 − 0.5d

(c) −0.5%/ft

(d) 100%; 200 ft

25 (a) N = 36.67 − 0.2424l

(b) −0.2424 species per degree latitude 36.67 species at the equator

(c) number of species

27 (c)

29 Female 6 years older than male

31 Yes

33 (a) 3.5% annual increase in production when unemployment rate constant

(b) Decreases 2.5%

(c) Increases 1.75%

Section 1.3

1 Concave down

3 Concave up

5 Decreasing

Concave up

7 (a) D to E, H to I

(b) A to B, E to F

(c) C to D, G to H

(d) B to C, F to G

9 −3

11 (a) −$134 million dollars

(b) −$67 million dollars per year

(c) Yes, 2009-2010

13 (a) 155,000 people/year

(b) 0.07, 0.08, 0.41, 0.06

(c) 155,000 people/year

15 1 meter/sec

17 (a) 0.077 billion people per year

0.980 million cars per year

207.7 million subscribers per year

(b) (i) Population

(ii) Cell phone subscribers

19 2 meters/sec

21 (a) Approximately −11.86 million pounds/year

(b) Yes, between 2003 and 2004, and any year between 2005 and 2009

23 1490 thousand people/year

912.9 thousand people/year

1879 thousand people/year

25 (a) −$35 billion dollars

(b) −$7 billion dollars per year

(c) Yes; 2006–2007, 2007–2008

27 (a) Negative

(b) −0.087 mg/hour

29 15.468, 57.654, 135.899, 146.353, 158.549 people/min

31 (a) About −11 cm/sec

(b) About −5.5 (cm/sec)/kg

33 (a) (i) f(1985) = 13

(ii) f(1990) = 99

(b) (f(1990)–f(1985))/(1990–1985) = 17.2 billionaires/yr

(c) f(t) = 17.2t − 34,129

35 (a) Concave up; no

(b) About 2.6 m/sec

37 Decreasing, concave down

39 Increases by 25%

41 Decreases by 83.3%

43 Change in 1931

45 The small class

47 (a) 100%

(b) −50%

(c) 0.1%

49 Increase by 2.2%

51 (a)

(b) Yes, last two quarters

53 (a) 2005–2007

(b) 2004–2007

55 (a) 25%

(b) 12%

(c) 0.48

Section 1.4

3 (a) About $75; $7.50 per unit

(b) About $150

5 Fixed cost: $5.7 million Variable cost: $2000 per unit

7 (a) Price $12, sell 60

(b) Decreasing

9 5500: Quantity demanded at price 0

100: Drop in quantity demanded if price increases $1

11 C(q) = 500 + 6q, R(q) = 12q, π(q) = 6q − 500

13 C(q) = 5000 + 15q, R(q) = 60q, π(q) = 45q − 5000

15 (a) C(q) = 5000 + 30q

R(q) = 50q

(b) $30/unit, $50/unit

(c)

(d) 250 chairs and $12,500

17 (a) First price list:

C₁(q) = 100 + 0.03q dollars

Second price list:

C₂(q) = 200 + 0.02q dollars

(b) First price list

(c) 10,000

19 (a) When there are more than 1000 customers

(b)

21 (a) C(q) = 650,000 + 20q

R(q) = 70q

π(q) = 50q − 650,000

(b) $20/pair, $70/pair, $50/pair

(c) More than 13,000 pairs

23 (a) V(t) = −2000t + 50,000

(c) (0 years, $50,000) and (25 years, $0)

25 (a) Roughly 360 scoops

(b) Roughly 120 scoops

27 (a) First: demand curve;

Second: supply curve

(b) Roughly 14

(c) Roughly 24

(d) Lower

(e) Any price less than or equal to $143

(f) Any price greater than or equal to $110

29 (a) C = 5q + 7000

R = 12q

(b) q = 1520, π(12) = $3640

(c) C = 17,000 − 200p

R = 2000p − 40p²

π(p) = −40p² + 2200p − 17,000

(d) At $27.50 per shirt the profit is $13,250

31 (a) q = 820 − 20p

(b) p = 41 − 0.05q

33

35 (a) 25,000r + 100m = 500,000

(b) m = 5000 − 250r

(c)

37 (a)

(b) Equilibrium price will increase; equilibrium quantity will decrease

(c) Equilibrium price and quantity will decrease

39 q = 90 − 5p

41 Pre-tax: p = $190, q = 70 units

Post-tax: p = $194, q = 68 units

43 (a) Demand:q = 100 − 2.1p

Supply:q = 3p − 50

(b) p = $29.41

q = 38.23 units

(c) Producer pays $0.59

Consumer pays $0.88

Total tax $1.47

(d) $56.20

Section 1.5

1 (a) (i), 12%

(b) (ii), 1000

(c) Yes, (iv)

3 (a) II

(b) I

(c) III

(d) V

5 y = 30(0.94)^t

7 (a) P = 1000 + 50t

(b) P = 1000(1.05)^t

9 (a) Q = 30 − 2t

(b) Q = 30(0.88)^t

11 (a) A = 50(0.94)^t

(b) 11.33 mg

(c) A(mg)

(d) About 37 hours

13 35.7%

15 (a)

(b)

(c)

(d)

17 g(t) = 5.50(0.8)^t

19 Table D

21 (a) a = 0.9 and P₀ = 27.435

(b) Initial quantity 27.435, decaying 10% per unit time

23 (a) a = 1.265 and P₀ = 454.05

(b) Initial quantity 454.05, growing 26.5% per unit time

25 1.7%

27 (a) y = 2700 + 486t

486 zebra mussels per year

(b) y = 2700(1.18)^t

18% per year

29 (a) h(x) = 31 − 3x

(b) g(x) = 36(1.5)^x

31 (a) Exponential

(b) P = 161.0(1.058)^t

(c) 5.8% per year

33 (a) $3486.78

(b) Approx 11 years (or 21 years from initial investment)

35 (a) w = 1108.773(1.12557)^t

(b) 12.557% per year

(c) No; recent growth rate higher

37 (a) 261 million gallons, 358 million gallons

(b)

39 (a) 16 trillion BTUs, 32 trillion BTUs

(c) 2007, 13 trillion BTUs

41 (a) Increased: 2006, 2008; decreased: none

(b) Yes

Section 1.6

1 t = (ln 10)/(ln 2) ≈ 3.3219

3 t = (ln 2)/(ln 1.02) ≈ 35.003

5 t = ln 10 ≈ 2.3026

7 t = (ln 5)/(ln 3) ≈ 1.465

9 t = (ln 100)/3 ≈ 1.535

11 t = 30.54

13 t = (ln B − ln P)/r

15 t = (ln 7 − ln 5)/(ln 2 − ln 3) ≈ −0.8298

17 5; 7%

19 15; −6% (continuous)

21 P = 15(1.2840)^t; growth

23 P = P₀ (1.2214)^t; growth

25 P = 15e^0.4055t; growth

27 P = 174e^−0.1054t

29 (a) k = 0.168 and P₀ = 84.575

(b) Initial quantity 84.575, growing 16.8% per unit time

31 (a) 6%

(b) P = 100(1.0618)^t, 6.18%

33 8.33%

35 (a) D

(b) C

(c) B

37 (a) P = 5.4(1.034)^t

(b) P = 5.4e^0.0334t

(c) Annual = 3.4%

Continuous = 3.3%

39 P = 7e^0.01143t

41 (a) S = 6.1e^0.042t

(b) 7.21 billion dollars

(c) Midway through 2017

43 (a) P = 50,000e^0.045t

(b) 78,416

(c) 15.403 years

45 27 meters

47 2023

Section 1.7

1 A: continuous

B: annual

$20

3 (a) $1534.69

(b) $1552.71

5 $10,976.23

7 About 11.6 years

9 Just over 6 hours

11 347 days

13 (a) A = 100e^−0.17t

(b) t ≈ 4 hours

(c) t = 4.077 hours

15 (a) 47.6%

(b) 23.7%

17 8.45%

19 (a) P(t) = (0.975)^t

(c) About 27 years

(d) About 8%

21 (a) About 4 years

(b) About 4 years

23 About 173 hours

25 96.336 years

27 (a) 2023

(b) 338.65 million people

29 It is a fake

31 (a) 0.00664

(b) t = 2.167; March 2, 2013

33 $12,712.49

35 $6549.85

37 (a) Choice 1

(b) Yes. Above 25%

39 (a) 8.75 years

(b) About 9.01 years

41 (a) Option 1

(b) $2102.54, $2051.27, $2000

(c) $2000, $1951.23, $1902.46

43 (a) Option 1

(b) Option 1: $10.929 million;

Option 2: $10.530 million

45 Buy

Section 1.8

1 (a) 15x + 9

(b) 15x − 1

(c) 25x − 6

3 (a) 3e^2x

(b) e^6x

(c) 9x

5 (a) h² + 6h + 11

(b) 11

(c) h² + 6h

7 (a) 4

(b) 2

(c) (x + 1)²

(d) x² + 1

(e) t²(t + 1)

9 (a) e

(b) e²

(c) e^x2

(d) e^2x

(e) e^tt²

11 (a) 5

(b) 2

(c) 3

(d) 4

(e) 5

(f) 2

13(a)

(b)

(c)

(d)

(e)

(f)

15 (a) y = 2^u, u = 3x − 1

(b)

(c) w = 2 ln u, u = 3r + 4

17 2zh + h²

19 4hz

21 6

23 3

25 4

27 1.1

29 About 0

31 2(y − 1)³ − (y − 1)²

33 About 18

35 Cannot be done

37

39

41

43 (a)

(b)

(c)

(d)

(e)

(f)

45 (a)

(b)

(c)

(d)

(e)

(f)

47

49

51

53 (a) y = 2x² + 1

(b) y = 2(x² + 1)

(c) No

55 g(2r) ft³

57 f⁻¹(g⁻¹(10,000)) min

Section 1.9

1 y = (1/5)x

3 y = 8x⁻¹

5 Not a power function.

7 y = 9x¹⁰

9 Not a power function

11 y = 125x³

13 S = kh²

15 v = d/t

17 (a) y = (x − 2)³ + 1

(b) y = −(x + 3)² − 2

19 N = kA^1/4, with k > 0, Increasing, concave down

21 (a) C = kW^0.75

(b)

(c) Horse: 5,716 calories

Rabbit: 218 calories

(d) Mouse

23 N = k/L²; small

25 (a) T = kB^1/4

(b) k = 17.4

(c) 50.3 seconds

27 (a) N = kP^0.77

(b) A has 5.888 times more than B

(c) Town

29 (a) q = −8p + 700

(b) R = −8p² + 700p

(c) At roughly $44 per unit, revenue ≈ $15,300

Section 1.10

1 Amplitude = 3; Period = 2π

3 Amplitude = 3; Period = π

5 Amplitude = 1; Period = π

7 (b) Max: 2nd quarter; Min: 4th quarter

(c) Period = 4 quarters or 1 year; amplitude = 5 million barrels

9 (a) 5

(b) 8

(c) f(x) = 5 cos((π/4)x)

11 Period = 25, amplitude = 0.45 f(15) = 2.0, f(75) = 1.4, f(135) = 2.3

13 Yes; About 28 days; day 12; days 17–21

15 (a)

(b) 9 species, 12 months

(c) N = 19 + 9 cos(πt/6)

17 x = 50 − 40 sin(2t)

19 f(x) = 2 sin(x/4)

21 f(x) = (sin x) + 2

23 f(x) = sin (2(π/5)x)

25 f(x) = 2 cos(5x)

27 f(x) = 3 sin(πx/9)

29 Depth = 7 + 1.5 sin(πt/3)

31 (a) Period is 12; amplitude is 4

(b) g(34) = 11; g(60) = 14

Chapter 1 Review

1 Pop 12 million in 2005

5

7 (a) (I)

(b) (IV)

(c) (II) and (III)

(d) Bread in (III) heats up faster

9 y = 8/3 − x/3

11 x = −1

13 y = 14x − 45

15 (a) S = 113 − 0.94t

(b) During 2038

17 8 mm/sec

19 4 mm/sec

21 −1/2 mm/sec

23 (b) 200 bushels

(c) 80 lbs

(d) About 0 ≤ Y ≤ 550

(e) Decreasing

(f) Concave down

25 (a) k(t)

(b) h(t)

(c) g(t)

27 (b) Lung cancer: 1.1 (answers may vary)

(c) Stomach cancer: −0.5 (answers may vary)

29 (a) (i) Positive

(ii) Positive

(iii) Negative

(iv) Positive

(b) (i) 0 ≤ t ≤ 5

(ii) 0 ≤ t ≤ 20

(c) About 25 m³/week

31 More than 875 students

33 Graph (a): supply

Graph (b): demand

35 y = (−3/7)x + 3

37 y = e^0.4621x or y = (1.5874)^x

39 y = 3e^0.2197t

or y = 3(1.2457)^t

41 f(x) is neither,

g(x) = 30.8 − 3.2x is linear,

h(x) = 15,000(0.6)^x is exponential

43 P = 40,000,000(1.2)^t/10

57,600,000 in 2020

Doubling time = 38.02 years

45 (ln(2/5))/(ln 1.04) = −23.4

47 ln(0.4)/3 = −0.305

49 (a) 15%

(b) P = 10(1.162)^t

(c) 16.2%

(d) Graphs are the same since functions are equal

51 (a) 15,678.7 years

(b) 5728.489, or about 5730 years

53 7.925 hours

55 3.68e^0.2899t

28.99%

57 Yes

59 (a) ln(2x + 3)

(b) 2 ln x + 3

(c) 4x + 9

61 (a) 10x² + 3

(b) 20x² + 60x + 45

(c) 4x + 9

63

65

67 (a)

(b)

(c)

(d)

69 (a)

(b)

(c)

(d)

71

73

75 y = −(x + 3)² + 2

77 No

79 2π/3; 7

81 2; 0.1

83 (a) Period about 12 months;

Amplitude about 4500 cases

(b) About 2000 cases and 2000 cases

85 f(x) = 2 sin(x/4) + 2

Ch. 1 Understanding

1 False

3 True

5 True

7 True

9 False

11 True

13 True

15 True

17 False

19 False

21 False

23 False

25 True

27 False

29 True

31 True

33 True

35 False

37 True

39 False

41 False

43 True

45 False

47 True

49 False

51 True

53 True

55 False

57 True

59 True

61 True

63 True

65 True

67 False

69 False

71 True

73 True

75 False

77 True

79 True

81 False

83 True

85 False

87 False

89 False

91 False

93 True

95 True

97 True

99 True

101 False

103 False

105 True

Section 2.1

1 (a) Negative

(b) f′(1960)

3 (a) 2.5 ft/sec

(b) 6.5 ft/sec

(c) 4.5 ft/sec

5 (a) Between x = 0 and x = 3

(b) At x = 1

(c) Thousands of dollars/kilogram

7 (a) 0.080 billion people/yr

(b) About 0.088 billion people/yr

9 g′(1) ≈ 5.549

11 Positive: A and D

Negative: C and F

Most positive: A

Most negative: F

13 f′(1) ≈ 3.3; greater

15 (a) 0.401 percent/year

(b) 0.15–0.38 percent/year

(c) Between 0.10 and 0.36 percent/year

17 (4, 25); (4.2, 25.3); (3.9, 24.85)

19

21 f′(1) ≈ 1.0005;

f′(2) ≈ 1.6934;

f is concave up between 1 and 2

23 (a) Negative or zero;

Positive;

Positive

(b) (i) −0.2 hr/yr, 0 hr/yr

(ii) $0.27/yr, $0.47/yr

(iii) $8.90/yr, $16.18/yr

Section 2.2

1

3

5

7 About 1.0, 0.3, −0.5, −1

9(a) x₃

(b) x₄

(c) x₅

(d) x3

11 R′(0) ≈ 9.531

13

15

17

19 IV

21 VI

23 5.2

25

27 (a) f′(1) ≈ 0.95

f′(2) ≈ 0.49

f′(3) ≈ 0.33

f′(4) ≈ 0.25

f′(5) ≈ 0.20

(b) 1/x

29 (a) (III)

(b) (III)

(c) (III), (IV)

(d) (II), (IV)

Section 2.3

1 dD/dt; feet per minute

3 dN/dD; gallons per mile

5 (a) ml; minutes

(b) ml; minutes/ml

7 (a) 12 pounds, 5 dollars

(b) Positive

(c) 12 pounds, 0.4 dollars/pound, extra pound costs about 40 cents

9 (a) Liters per centimeter

(b) About 0.042 liters per centimeter

(c) Cannot expand much more

11 (a) Negative

(b) °F/min

13 (a) °C/km

(b) Air temp decreases about 6.5° for a 1 km increase in altitude

15 (a) Negative until 2008, then positive

(b) Between 2008 and 2009

(c) About 160 metric tons/year

(d) About 2610 metric tons,

About 3410 metric tons

17 (a) Costs $1300 for 200 gallons

(b) Costs about $6 for 201^st gallon

19 (a) Positive

(b) Child weighs 45 pounds at 8 years

(c) lbs/year

(d) At 8 years of age, child weighs about 4 pounds more after the next year

(e) Decrease

21 (a) Shells thinner with more PCB

(b) 200 ppm PCB corresponds 0.28 mm shell 1 ppm increase PCB corresponds 0.0005 mm decrease shell

23 f(21) ≈ 65

f(19) ≈ 71

f(25) ≈ 53

25 (a) g′(36)

(b) Grows more rapidly at week 36 than at week 20

27 (a) about 0.09 kg/week

(b) about 0.2 kg/week

(c) about 0.08 kg/week

29 (a) Dose for 140 lbs is 120 mg

Dose increases by 3mg/lb

(b) About 135 mg

31 (a) 20 minutes,

0.36 mg,

−0.002 mg/minute

(b) f(21) ≈ 0.358

f(30) ≈ 0.34

33 (a) About −4 (cm/sec)/kg

(b) About −0.20cm/sec

(c) v′(2) ≈ 4

35 Dollars/percent; positive

37 Used at constant rate for 4 weeks

39 Switch from fat to protein

41 (a), (b)

43 (b) f′(100) = 2: Yes

f′(100) = 0.5: No

45 (a) f(t) volume of ice sheet t years after 2011

(b) −224 < f′(0) < −82

47 f(4) = 200 million users;

f′(4) ≈ 12.5 million users/month;

Increasing at about 6.25%/month

49 0.20

51 (a) Weighs 5.67 kg at 2 1/2 months

(b) Weight increasing 13% per month at 2 1/2 months

53 (a) About 6.1% per month

(b) About 5.9% per month

(c) About 5.9% per month

Section 2.4

1 (a) Negative

(b) Negative

3 (a)

(b)

(c)

(d)

5 f′(x) = 0

f″(x) = 0

7 f′(x) < 0

f″(x) > 0

9 f′(x) < 0

f″(x) < 0

11 Derivative:

Pos. about −2.3 < t < −0.5

Neg. about −0.5 < t < 4

Second derivative:

Pos. about 0.5 < t < 4

Neg. about −2.3 < t < 0.5

13 s′(t): positive

s″(t): positive or zero

15 A positive second derivative indicates a successful campaign

A negative second derivative indicates an unsuccessful campaign

17 (a) Positive; negative

(b) Neither; positive

(c) Number of cars increasing at about 600,000 million cars per year in 2005

19

21 (b)

23 22 only possible value

25 (a)

(b) dN/dt is positive.

d²N/dt² is negative.

27 (a) dP/dt > 0, d²P/dt² > 0

(b) dP/dt < 0, d²P/dt² > 0

(but dP/dt is close to zero)

29 (a) a′(t) > 0, m′(t) > 0

a″(t) > 0, m″(t) > 0

(b) 2 < a′(t) < 4; 2 < m′(t) < 10

(c) (i) Between 200 mm and 400 mm

(ii) 100 years

31 (a)

(b) Positive 0 < t < 8; negative 8 < t < 18

Section 2.5

1 About $3 per item

3 About $16.67 (answers may vary)

5 About $0.42 (answers may vary)

7 C′(2000) ≈ $0.37/ton

The marginal cost is smallest on the interval

2500 ≤ q ≤ 3000.

9 (a) About $2408

(b) About $2192

11 (a) About $4348

(b) $11 profit

(c) No, company will lose money

13 (a) $1.8 million

(b) About $28,000 increase

(c) About $35,000 decrease

(d) About $4000 increase

About $5000 decrease

15 (a) Fixed costs

(b) Decreases slowly, then increases

17

Chapter 2 Review

1 (a) (i) 6.3 m/sec

(ii) 6.03 m/sec

(iii) 6.003 m/sec

(b) 6 m/sec

3 (a) Negative

(b) f′(1) = −3

5 (a) Positive at C and G

Negative at A and E. Zero at B, D, and F

(b) Largest at G

Most negative at A

7 (a) (i) −1.00801 m/sec

(ii) −0.8504 m/sec

(iii) −0.834 m/sec

(b) −0.83 m/sec

9 f′(2) ≈ 40.268

11

13

15

17

19

21 (a) About 3

(b) Positive: 0 < x < 4

Negative: 4 < x < 12

23 kilograms/meter

25 (a) Positive

(b) °F/min

27 about 6 cm/yr

29 Wind stronger at 15.1 km than at 15 km

31 About 1.389 billion people in 2020;

growing at about 5.5 million people per year

33 (a) f′(t) > 0: depth increasing

f′(t) < 0: depth decreasing

(b) Depth increasing 20 cm/min

(c) 12 meters/hr

35 B

39 (a) Negative

(b) Degrees/min

41 Dollars/year; negative

43 (a)

(b)

(c)

(d)

45 (a) Minutes/kilometer

(b) Minutes/kilometer²

47 (a) x₄, x₅

(b) x₃, x₄

(c) x₃, x₄

(d) x₂, x₃

(e) x₁, x₂, x₅

(f) x₁, x₄, x₅

49 (a) t₃, t₄, t₅

(b) t₂, t₃

(c) t₁, t₂, t₅

(d) t₁, t₄, t₅

(e) t₃, t₄

51 (a)

(b) Student C's

Ch. 2 Understanding

1 True

3 True

5 False

7 False

9 False

11 True

13 False

15 True

17 False

19 True

21 True

23 True

25 True

27 True

29 True

31 False

33 False

35 True

37 True

39 True

41 True

43 True

45 False

47 True

49 False

51 True

53 True

55 True

Theory: Limits, Derivatives

1

3 (a) 0/0, undefined

(b) 6.487, 5.127, 5.013, 5.001

(c) ≈ 5

5 (a) 0/0, undefined

(b) 0.953, 0.995, 0.9995, 0.99995

(c) ≈ 1

7 1.6

9 1.9

11 0

13 No; yes

15 No; yes

17 (a) Yes

(b) Yes

(c) No

(d) No

19 2

21 2

23 Yes

25 Yes

27 No

29 Not continuous

31 Continuous

33 Not continuous

Section 3.1

1 3

3 −12x⁻¹³

5 24t²

7 5

9 3q²

11 18x² + 8x − 2

13 24t² − 8t + 12

15 −12x³ − 12x² − 6

17 6.1z^−7.1

19 (1/2)x^−1/2

21 −(3/2)x^−5/2

23 2t

25 2z − z⁻²

27 −3t⁻² − 8t⁻³

29 (1/2)θ^−1/2 + θ⁻²

31 2ax + b

33 2at − 2b/t³

35 (8πrb)/3

37 a/c

39 (a) P′(1) : Positive;

P′(3): Zero;

P′(4): Negative

(b) P′(1) = 4, P′(3) = 0, P′(4) = −2

41 (a) 2t − 4

(b) f′(1) = −2, f′(2) = 0

43 Height = 625 cm,

Changing (eroding) at −30 cm/year

45 69.6% per year

47 4800 mussels after 4 months, increasing by 2400 mussels per month

49 f′(t) = 6t² − 8t + 3

f″(t) = 12t − 8

51 y = −4 − x

53 y = −2t + 16

55 (a) C(w) = 42w^0.75;

C′(w) = 31.5w^−0.25

(b) (i) 236.2 calories a day;

17.7 calories/pound

(ii) 1328 calories a day;

10 calories/pound

(iii) 7,469 calories a day;

5.6 calories/pound

57 (a) dA/dr = 2πr

(b) Circumference of a circle

61 (a) R(p) = 300p − 3p²

(b) $240 per dollar price increase when price is $10

(c) Positive for p < 50, negative for p > 50

63 (a) 770 bushels per acre

(b) 40 bushels per acre per pound of fertilizer

(c) Use more fertilizer

65 (a) dC/dq = 0.24q² + 75

(b) C(50) = $14,750;

C′(50) = $675 per item

67 (a) R(q) = bq + mq²

(b) R′(q) = b + 2mq

Section 3.2

1 9t² + 2e^t

3 3x² + 3^x ln 3

5 5 · 5^t ln 5 + 6 · 6^t ln 6

7 4(ln 10)10^x − 3x²

9 5(ln 2)(2^x) − 5

11 0.7e^0.7t

13 −0.2e^−0.2t

15 24e^0.12t

17 12.41(ln 0.94)(0.94)^t

19 Ae^t

21 (ln 10)10^x − 10/x²

23 −1/p

25 2q − 2/q

27 Ae^t + B/t

29 15/(15t + 12)

31 −7

33 −4/t

35 f′(−1) ≈ −0.736

f′(0) = −2

f′(1) ≈ −5.437

37 y = −2t + 1

39 (a) 13,394 fish

(b) 8037 fish/month

41 f(2) = 6065, f′(2) = −1516

43 f(5) = $563.30;

f′(5) = $70 per week;

Relative rate = 12.4% per week

45 −544 people/year

47 c = −1/ln 2

49 Cs(50) ≈ 1365, C′(50) ≈ 18.27

51 About $2864; $0.60

53 (a) P = 1.166(1.015)^t

(b)

55 g(x) = x²/2 + x + 1

Section 3.3

1 99(x + 1)⁹⁸

3 200t(t² + 1)⁹⁹

5 15(5r − 6)²

7 −6x + 6e^3x

9 30e^5x − 2xe^−x2

11 −6te^−3t2

13 5/(5t + 1)

15 2t/(t² + 1)

17 e^x/(e^x + 1)

19 1/(x ln x)

21 3/(3t + 2)

23 5 + 1/(x + 2)

25 0.5/(x(1 + ln x)^0.5)

27

29 49.7% per year

31 1.5

33 2/t

35 (a) P(1 + r/100)^t ln(1 + r/100)

(b) Pt(1 + r/100)^t−1/100

37 v(t) = 10e^t/2

39 Approx 0.8

41 Approx −0.4

43 1/2

45 −1

47 0.5

49 (a) g′(1) = 3/4

(b) h′(1) = 3/2

Section 3.4

1 5x⁴ + 10x

3 −2te^−2t + e^−2t

5 2t(3t + 1)³ + 9t²(3t + 1)²

7 ln x + 1

9 (t³ − 4t² − 14t + 1)e^t

11 −3qe^−q + 3e^−q

13 1 − 3/x²

15 e^−t2 (1 − 2t²)

17 2p/(2p + 1) + ln(2p + 1)

19 2we^w2 (5w² + 8)

21 9(te^3t + e^5t)⁸(e^3t + 3te^3t + 5e^5t)

23 −2/(1 + t)²

25 (15 + 10y + y²)/(5 + y)²

27 (ak − bc)/(cx + k)²

29 ae^−bx − abxe^−bx

31 (1 − 2α)e^−2αe^αe−2α

33 f′(x) = 12x + 1 and

f″(x) = 12

35 y = 0

37 (a) f′(15) > 0, f′(45) < 0

(b) f(30) ≈ 181 mg/ml, f′(30) ≈ −1.2 mg/ml/min

39 Revenue R(10) ≈ 22,466.

R′(10) ≈ $449/dollar.

41 1/t

43 (fg)′/(fg) = (f′/f) + (g′/g)

47 (a) 0.555 = 55.5%

(b) Initially not in search area; unrealistic

(c) I(1 − I)/(1 − (1 − I)E)²; positive; O increases

Section 3.5

1 −sin t

3 A cost t

5 5 cos x − 5

7 5 cos(5t)

9 −10 sin(5t)

11 AB cos(Bt)

13 12 cos(2t) − 4 sin(4t)

15 2 sin(3x) + 6xcos(3x)

17 ((2te^t2 + 1) sin(2t) − (e^t2 + t)2 cos(2t))/sin²(2t)

19 (θ cos θ − sin θ)/θ²

21 y = −x + π

23 At x = 0:

y = x, sin(π6) ≈ 0.524

At x = π/3:

,

sin(π/6) ≈ 0.604

25 (a) 5 sec

(b) A′(1) = 239 cc³/second

27 (a) v(t) = 2π cos(2πt)

(b)

29 (a) D(40) = 9.4 hours daylight; D′(40) = 0.05 hours/day

(b) D(172) = 16 hours daylight; D′(172) = 0 hours/day

Chapter 3 Review

1 24t³

3 2e^2t

5 0.08e^0.08q

7 e^3x(1 + 3x)

9 6(1 + 3t)e^{(1 + 3t)2}

11 5 + 3.6e^1.2t

13 90(5x − 1)²

15 10e^x(1 + e^x)⁹

17 x(2 ln x + 1)

19 8t + 7 cos t

21 (−e^−t − 1)/(e^−t − t)

23 (50x − 25x²)/(e^x)

25 2x cos x − x² sin x

27 e

29 (e^x − 2 − e^−x)/(1 − e^−x)²

31 xe^x + e^x

33 −3x² sin(x³)

35 (t² + 6t + 13)/(t + 3)²

37 ((ln 3)3^x)/3 − (33x^−3/2)/2

39 −7ke^−kt

41 f′(0) = 0, f′(1) = 2,

f′(2) = 4, f′(−1) = −2

43 y = 2x − 1

45 81.6 m/yr

47 (−1, 7) and (7, −209)

49 Q(t) = (1 + ln 2)t − 2 ≈ 1.69t − 2

51 (a) 177°F

(b) 74°F; room temperature

(c) C(5) = 161°F;

C′(5) = −2.88°F/minute

(d) Less

53 (a) −0.000121e^−0.000121t

(b)

55 Increasing for a > 1,

decreasing for a < 1

57 (a) Y(0) = 105°

(b) 350°

(c) After about 42 minutes

(d) About 1.67 degrees/minute

59 y = x and y = ln(x + 1) look the same (like the line y = x);

y =

the line x = 0;

y = x², y = x³ + x², y = x³, and

y = ln(x² + 1) all look like the line y = 0

61 (a) 2

(b) 15

(c) 11

(d) −1/4

63 (a) CD − D²

(b) D < C

65 Approx 0.6

67 Approx 1.6

69 Approx −8.8

71 For x < 0 or 2 < x < 3

75 y = 2x and y = −6x

77 (a) 0.04 gal/mile; 0.06 gal/mile

(b) 25 mpg; 16.67 mpg

(d) 1.4 gallons

(e) 2.8 gal/hour; 1.8 gal/hour

79 331.3 + 0.606T m/sec

Ch. 3 Understanding

1 True

3 False

5 False

7 True

9 True

11 False

13 True

15 True

17 False

19 False

21 False

23 True

25 True

27 False

29 True

31 False

33 True

35 False

37 True

39 False

41 True

43 True

45 False

47 False

49 True

Practice: Differentiation

1 2t + 4t³

3 15x² + 14x − ³

5 −2/x³ + 5/

7 10e^2x − 2 · 3^x(ln 3)

9 2pe^p2 + 10p

11

13 16/(2t + 1)

15 2^x(ln 2) + 2x

17 6(2q + 1)²

19 bke^kt

21 2x ln (2x + 1) + 2x²/(2x + 1)

23 10 cos (2x)

25 15 cos (5t)

27 2e^x + 3 cos x

29 17 + 12x^−1/2

31 20x³ − 2/x³

33 1, x ≠ −1

35 −3 cos(2 − 3x)

37 6/(5r + 2)²

39 ae^ax/(e^ax + b)

41 5(w⁴ − 2w)⁴(4w³ − 2)

43 (cos x − sin x)/(sin x + cos x)

45 6/w⁴ + 3/(2 )

47 (2t − ct²)e^−ct

49 (cos θ)e^{sin θ}

51 3x²/a + 2ax/b − c

53 2r(r + 1)/(2r + 1)²

55 2e^t + 2te^t + 1/(2t^3/2)

57 x² ln x

59 6x (x² + 5)² (3x³ − 2)(6x³ + 15x − 2)

61 (2abr − ar⁴)/(b + r³)²

63 20w/(a² − w²)³

Section 4.1

1 One

3 Four

5 (a)

(b)

7 After 18 hours

9 Local max: (−1.4, 6.7)

Local min: (1.4, −4.7)

11 Local min: (2.3, −13.0)

13 Alternately incr/decr

15 Critical points: x = −1, 1

x = −1 local maximum; x = 1 local minimum

17 Critical points:

x = 0 and x = ±2

Extrema:

f(0) local minimum

f(−2) and f(2) are not

local extrema.

19 Critical points:

x = ±1

Extrema:

f(−1) local minimum

f(1) local maximum

21 x = 0: not max/min

x = 3/7: local max

x = 1: local min

23

25 (a) Increasing for all x

(b) No maxima or minima

27 (a) Increasing: −1 < x < 0 and x > 1

Decreasing: x < −1 and 0 < x < 1

(b) Local max: f(0)

Local min: f(−1) and f(1)

29 (a) Increasing weeks 0–2 and 6–10,

decreasing weeks 3–5

(b) Local max week 2–3 and week 10,

local min week 0 and week 5–6

31 Increasing: 0 < t < 2, t > 4

Decreasing: 2 < t < 4

Local max: t = 2, t = 5

Local min: t = 0, t = 4

33 a = 4, b = 1

35 b = 2, a = 5/(2 − 2 ln 2) ≈ 8.147

37 (a) x = 0, x = a²/4

(b)

39 (a)

(b) Critical point moves right

(c) x = a

41 (a)

(b) Nonzero critical point moves down to the left

(c) x = 0, 2/a

Section 4.2

1 Two

3 Three

5

7

9 (a) 6 pm

(b) Noon; another between noon and 6 pm

11 x = −1, 1/2

13 Critical points: x = −1,

local max; x = 1, local min

Inflection point: x = 0

15 Critical points: x = −2,

local max; x = 1, local min

Inflection point: x = −1/2

17 Critical points:

x = 0 (neither) and x = 1 (local max)

Inflection points: x = 0 and x = 2/3

19 Critical points: x = 0 (not an extrema) and x = 3 (local minimum)

Inflection points: x = 0 and x = 2

21 Critical points:

x = −1 (local maximum)

x = 0 (not an extrema)

x = 1 (local minimum)

Inflection points:

x = 0 and x = ±1/

23 (a) Critical points: x = 0, x = x = −

Inflection points: x = ,

x = −

(b) a = 4, b = 21

(c)

25

27

29

31 (a) cm/week

(b) Growing at 1.6 cm/wk in week 24

33 (a) Week 14

(b) Point of fastest growth

35 (a)

(b)

37(a) Concavity changes at y₁ and y₃

(b) f(t) grows most quickly where vase is skinniest and most slowly where vase is widest. Ratio is about 16 : 1

39 y = (2 + 2e)/(1 + e^1–t)

Section 4.3

1

3 (a) (IV)

(b) (I)

(c) (III)

(d) (II)

5

7

9 (a) Local minima at x = −1,

x ≈ 0.91, x = 4

Local maxima at x ≈ −0.46, x ≈ 3.73

f(0.91) is global minimum

f(3.73) is global maximum

(b) Local minima at x = −3, x ≈ 0.91

Local maxima at x ≈ −0.46, x = 2

f(−3) is global minimum

f(2) is global maximum

11

13

15 (a) t ≈ 50 days

(b) Throughout interval

t ≈ 50 days

17 (a) f′(x) = 6x² − 18x + 12,

f″(x) = 12x − 18.

(b) x = 1, 2

(c) x = 3/2

(d) Local minimum: x = −0.5, 2

Local maximum: x = 1, 3

Global minimum: x = −0.5

Global maximum: x = 3

19 (a) f′(x) = 1 + cos x,

f″(x) = −sin x.

(b) x = π

(c) x = 0, π, 2π

(d) Global, local minimum: x = 0

Global, local maximum: x = 2π

(e)

21 x = −b/2a,

Max if a < 0, min if a > 0

23 2500

25 1536

27 Global max = −1 at x = 2

No global min

29 Global max = 1/e at t = 1

No global min

31 Global max = 1/2 at t = 1

Global min = −1/2 at t = −1

33 w = −(5p)/(6q)

35 40 feet by 80 feet

39 306 children

41 (a) H = 1/b, S = ae^−1/b

(b) a: Increases

b: Decreases

43 (a) 10

(b) 9

45 (a) 120 mm Hg, 80 mm Hg

(b) 0.8 sec

47 (a) 0 ≤ y ≤ a

(b) y = 0

Section 4.4

1 5.5 < q < 12.5 psositive; 0 < q < 5.5 and q > 12.5 negative;

Maximum at q ≈ 9.5

3 Global maximum of $6875 at q = 75

5 (a)

7 (a) $9

(b) −$3

(c) C′(78) = R′(78)

9 (a) Increase production

(b) q = 8000

11 (a) Increase

(b) Decrease

(c) Decrease

13 q = 4000

15 Above 2000

17 $0.20/item

19 (a) $10; $30,000; $50,000

(b) R(q) = 70q − 0.02q²

(c) 1750

(d) $35

(e) $61,250

21 $14

23 (a) 10,000 + 2q

(b) q = 37,820 − 5544p

(c) π = −0.00018q² + 4.822q − 10,000

(d) 13,394 items, $22,294

25 $8378.54

27 (a) Ordering: a/q

Storage: bq

(b)

29 (a) q/r months

(b) (ra/q) + rb dollars

(c) C = (ra/q) + rb + kq/2 dollars

(d)

31 L = [βpcK^α/w]^1/(1–β)

Section 4.5

1 (a) No

(b) Yes

3 (a) (i) About $8 per unit

(ii) About $4 per unit

(b) About 30 units

5 MC = $20; a(q) = $25

7 (a) q = 6

(b) q = 6

9 (a) C(q) = 0.01q³ − 0.6q² + 13q

(b) $1

(c) q = 30, a(30) = 4

(d) Marginal cost is 4

11

(b) (i) N′(x) = 20

(ii) N(x)/x = (100/x) + 20

15 (b) q = [Fa/(K(1 − a))]^a

Section 4.6

1 (a) 1.5% decrease

(b) 1.5% increase

5 Elastic

7 Elastic

9 E = 2/3; inelastic

11 (a) E ≈ 0.470, inelastic

(c) About P = 1.25 and 1.50

15 $12.91

27 1% more time gives 1.3% more sales

Section 4.7

1 (a) 40 billion

(c) 2020; 2095

3

(b) 60,000

5 (a) and (b)

(c) P′ maximum

7 (a) About 0.252

(c) 1975

(d) 1975

11 (a) 5000

(b) 499

(c) P(t) = 5000/(1 + 499e^−1.78t)

(d) About 3.5 weeks; 2500

13 (b) f′(10) < 11

f′(20) < 11

15 (a) C: largest

B: smallest

(b) A: largest

B: smallest

(c) C: safest

17 10 mg to 18 mg

Section 4.8

1 (a)

(b) 5 hrs, 22.8 ng/ml

(c) 1 to 14.4 hrs

(d) At least 20.8 hrs

3 (a) After about 6 hours

(b) After about 5 hours

9 a = 49.3; b = 0.769

13 (b) Min effective ≈ 0.2 Max safe ≈ 1.0

(c) Min effective ≈ 1.2 Max safe ≈ 2.0

Chapter 4 Review

3 Critical points: x = 2, x = 4

Inflection point: x = 3

5 Critical points: x = 0, x = −12

Inflection points: x = −9

7 Critical point: x = −1/3

Inflection point: x = −2/3

9 (a) f(1) local minimum; f(0.1), f(2) local maxima

(b) f(0.1) global maximum f(1) global minimum

11 Local maximum, minimum, or neither Price relatively constant around July 1

13

15 A: local max

B: local min

C: neither

17 (a) 5x⁴ + 1; positive

(b) One

19 Domain: All real numbers except x = b; Critical points: x = 0, x = 2b

21 (a)

(b) x₁, x₃, and x₅

(c) x₂ and x₄

23 A has a = 1, B has a = 2, C has a = 5

25

27 (a) Yes, at 2000 rabbits

(b) 1787, 1000 rabbits

(c) 1787, 1000 rabbits

29 Intercepts: (0, 0), (1.77, 0), (2.51, 0)

Critical Points: (0, 0), (1.25, 1), (2.17, −1), (2.80, 1)

Inflection Points: (0.81, 0.61), (1.81, −0.13), (2.52, 0.07)

31 (a)

(b) −2 < a < 2

33 (a) D = C

(b) D = C/2

35 (a) V = Ax/4 − x³/2

(b)

(c) (A/6)^3/2

37 Max area = 1/(2e³) at x = 1

39 1/

41 (a) (9/4, ± /4)

(b) (3, 0)

43 (a) q = 400

(b) $5 per unit

(c) $700.

45 q = 1000 or q = 4500

47 (a) No

(b) Yes

49 (a) $0

(b) $96.56

(c) Raise the price by $5

51 (a) π(q) max when

R(q) > C(q) and R and Q are farthest apart

(b) C′(q₀) = R′(q₀) = p

53

55 C′(2)

57 (C(75) − C(50))/25

59 C(3)/3

61 E = 0.05, demand is inelastic

63 50 m by 50 m

65 Max: 0.552; Min: 0.358

67 (a) 1/(2e)

(b) (ln 2) + 1

69 (a) x = μ

(b) Yes; x = μ ± σ

75

(b) 7 worms

(c) increases

Ch. 4 Understanding

1 False

3 False

5 False

7 False

9 False

11 True

13 True

15 False

17 True

19 False

21 True

23 False

25 False

27 False

29 True

31 True

33 False

35 False

37 False

39 True

41 True

43 False

45 False

47 True

49 False

51 False

53 False

55 False

57 False

59 True

61 True

63 False

65 True

67 True

69 True

71 False

73 False

75 False

77 True

79 False

Section 5.1

1 (a) 160 miles

3 (a) Right sum

(b) Lower estimate

(c) 5

(d) Δt = 3

(e) Lower estimate ≈ 160.5

5 352.5 feet

7 Between 140 and 150 meters

9 669.5 bn barrels

11 250 meters

13 (a) About 420 kg

(b) 336 and 504 kg

15 220 fish

17 (a) Car A

(b) Car A

(c) Car B

19 (b) $6151

21 (a) 3400 million people

(b) 3530 million people; differs by 130 mn

23 No

25 About 0.009 miles or 48 feet

27 (a) 663 ft

(b) Upper estimate

(c) 489 ft

(d) Lower estimate

(e) 576 ft

29 (a) 72; 328

(b) 120; 248

(c) 148; 212

31 (a) 4; 0, 4, 8, 12, 16; 25, 23, 22, 20, 17

(b) 360; 328

(c) 8; 0, 8, 16; 25, 22, 17

(d) 376; 312

Section 5.2

1 (a) Right

(b) Upper

(c) 3

(d) 2

3 27

5 (a) 0.808

(b) 0.681

7 17,000, n = 4, Δx = 10

9 16.1, n = 5, Δt = 0.2

11 About 60

13 About 20

15 8.5

17 13

19 2.350

21 About 0.865

23 6.111

25 0

27 (a) 3.6

(b) 3.4641

29 (a) 2

(b) 2.05045

31 (a) 78

(b) 46; underestimate

(c) 118; overestimate

Section 5.3

1 84

3 7.667

5 (3x − x²) dx = 4.5

7(a) Negative

(b) Positive

(c) Negative

(d) Positive

9 (a) About 16.5

(b) About −3.5

11 (a) About −0.25

(b) About 0

(c) About 0.5

13 (a) 13

(b) −2

(c) 11

(d) 15

15 II

17 III

19 (a) −2

(b) −A/2

21 (a) 1

(b) 2π

(c) 2π − 1/2

(d) π − 3/2

23 1.977

25 7.799

27 21.527

29 2.828

31 −0.136

33 (a) 16.25

(b) 15.75

(c) No

Section 5.4

1 Change in position; meters

3 Change in world pop; bn people

5 Total amount = .

7 13.295 billion tons

9 (a) 1770 million metric tons; 2830 million metric tons

(b) 46,000 million metric tons

11 15 cm to the left

13 25 cm to the right

15 3.406 ft

17 2627 acres

19 (a)

(b) 100

(c) 108.33

21 1417 antibodies

23 t = 1

About 16.667 miles

25 (a) (i) I(t) dt; Total income from 2000 to 2015

(ii) E(t) dt; Total expenditures from 2000 to 2015

(iii) I(t) − E(t) dt; Change in value from 2000 to 2015

(b) About 2500 billion dollars

27 I(t) − E(t) dt

29 6 months: A more

First year: B more

Same: roughly 9 months

A roughly 170 sales

B roughly 250 sales

31 (a) Boys: black curve; girls: colored curve

(b) About 43 cm

(c) Boys: about 23 cm; girls: about 18 cm

(d) About 13 cm taller

33 (a) About 750 liters

(b)

(c) About 150 liters

35 Product B has a greater peak concentration

Product A peaks sooner

Product B has a greater overall bioavailability

Product A should be used

37

39 About $13,800

41 (a) 49(1 − (0.8187)^t) dt (meters)

(b) T ≈ 107 seconds

Section 5.5

1 45.8°C.

3 4,250,000 riyals

5 (a) About $10,550

(b) About $150

(c) About 10

7 (a)

(b) $22,775

(c) C′(150) = 18.5

(d) C(151) ≈ $22,793.50

9 Fixed cost is $500

Variable cost is $866.67

Total cost is $1,366.67

11 (a)

(b) $12,000

(c) Marginal revenue is $80/unit

Total revenue is $12,080

13 $16,000, $56,000

17

19 0.5; cost of preparing is $0.5 million

21 r(t) dt < r(t) dt

23 r(t) dt < 64 million

Section 5.6

1 (a) 20

(b) 10/3

3 2

5 (a) 0.79

7 8

9 About 17

11 (a) 527.25

(b)

13 (a) $26,667 per year

(b) Less 25–65; more 65–85

15 (a) 120 mm Hg

(b) 80 mm Hg

(c) 100 mm Hg

(d) Less

17 (a) 0.375 thousand/hour

(b) 1.75 thousands

19 (a) 22°C

(b) 183°C

(c) Smaller

21 (a) E(t) = 1.4e^0.07t

(b) ≈ 219 million megawatt-hours

(c) 1972

(d) Graph E(t) and estimate t such that E(t) = 219

23 (a) < (c) < (b) < (d)

Chapter 5 Review

1 (a) 408

(b) 390

3 1692.5

5 (a) Right sum

(b) Upper estimate

(c) 4

(d) Δt = 2

(e) Upper estimate ≈ 122.6

7 Dollars

9 (a) About 3.6 acres

(b) About 6.9 acres

(c) About 5.25 acres

11 About 7

13 28.5

15 1.15

17 −0.083

19 About 1.772

21 0.0833

23 About 0.1667

25 (a) 430 ft

(b) (ii)

27 (a)

(b) 3 sec, 144 feet

(c) 80 feet

29 (a) Species B for both

(b) Species A

31 (a) Concave up

(b) About 3.1 kg

33 Negative

35 Approximately zero

37 (a) −4

(b) 0

(c) 8

41 (a) f(t) dt

(b) 657.11 billion barrels

(c) Lower estimate of 3 years' consumption

43 About $14,667

45 (a)

(b) 7 years

(c) 69.3 cubic yards

47 (a) About times 17, 23, 27 seconds

(b) Right: t = 10 seconds

Left: t = 40 seconds

(c) Right: t = 17 seconds

Left: t = 40 seconds

(d) t = 10 to 17 seconds, 20 to 23 seconds, and 24 to 27 seconds

(e) At t = 0 and about t = 35

49 F(0) = 0, F(0.5) = 1.958,

F(1) = 3.667,

F(1.5) = 4.875,

F(2) = 5.333,

F(2.5) = 4.792

51 Maximum at x = 2 and F(2) = 5.333.

53 2202.55

55 (a) 100 cases; 5 cases

(b) 32 cases

57 About 8.5

59 (a) III

(b) I

(c) II and IV

Ch. 5 Understanding

1 True

3 False

5 True

7 False

9 True

11 False

13 False

15 True

17 False

19 True

21 False

23 False

25 True

27 False

29 True

31 True

33 True

35 True

37 False

39 True

41 False

43 False

45 False

47 False

49 True

51 False

53 True

55 False

57 False

59 True

Theory: Second Fund. Thm.

1 x³

3 xe^x

5 (a) 0

(b) F increases

(c) F(1) ≈ 1.4, F(2) ≈ 4.3, F(3) ≈ 10.1

9 9

11 8c

Section 6.1

1

3

5 F(0) = 0

F(1) = 1

F(2) = 1.5

F(3) = 1

F(4) = 0

F(5) = −1

F(6) = −1.5

7 3.5, 2, 1.5, 2, 2.5

9

11

13

15

17 Largest: F(2)

Smallest: F(4)

None negative

19 (a) Increasing for x < −1, x > 1,

Decreasing for −1 < x < 1,

Local maximum at x = −1,

Local minimum at x = 1

(b)

21 Critical points: (0, 5), (2, 21), (4, 13), (5, 15)

23 Min: (1.5, −20), max: (4.67, 5)

25 (a) x = 1, x = 3

(b) Local min at x = 1, local max at x = 3

(c)

27 f(1) < f(0)

29

31

Section 6.2

1 Yes

3 No

5 No

7 Family functions

9 Family functions

11 Family functions

13 Family functions

15 x³/3

17 5x

19 (t⁸/8) + (t⁴/4)

21 3x⁴/2 + 4x

23 10x + 2x⁴

25 F(x) = (x²/2) + (x⁶/6) − (x⁻⁴/4) + C

27 (x³/3) − 3x² + 17x

29 t⁴/4 − t³/6 − t²/2

31 −1/(2z²)

33 F(x) = x⁷/7 + x⁻⁵/35 + C

35 2z^5/2/5

37 sin t

39 G(θ) = −cos θ − 2 sin θ + C

41 g(x)

43 g(x)

45 F(x) = 2x + 2x² + (5/3)x³ (only possibility)

47 F(x) = (2/3)x^3/2 (only possibility)

49 e^x − 1 (only possibility)

51 3x³ + C

53 p + ln |p| + C

55 x³/3 − 1/x + C

57 5e^z + C

59 t⁴/4 + 2t³ + C

61 x³/3 + 2x² − 5x + C

63 x²/2 + 2x^1/2 + C

65 e^x + 5x + C

67 x⁶/6 − 3x⁴ + C

69 −cos t + C

71 25e^4x + C

73 2 ln |x| − π cos x + C

75 3 sin x + 7 cos x + C

77 10x − 4 cos(2x) + C

79 −6 cos(2x) + 3 sin(5x) + C

81 (2/5)x^5/2

83 (e^2x + e^−2x)/2

85 (a) 20q − 2q²

(b) p = 20 − 2q

87 F(x) = 3x² − 5x + 5

89 F(x) = −4 cos(2x) + 9

91 C(x) =

5x² + 4000x + 1,000,000

Section 6.3

1 48

3 81/4

5 22

7 2

9 125

11 75/4

13 8/15

15 609/4 − 39π ≈ 29.728

17 sin 1 − sin(−1) = 2 sin 1

19 20(e^0.15 − 1)

21 1/2

23 (300)^1/3 = 6.694

25 (a) 462e^0.019tdt

(b) About 2423 quadrillion BTUs

27 (a) First case: 19,923

Second case: 1.99 billion

(b) In both cases, 6.47 yrs

(c) 3.5 yrs

29 (a) −(1/b) + 1

(b) Converges to 1

31 (1/x²) dx

33 (a) 1000te^−0.5tdt

(b)

Section 6.4

1 (a) p* = $30, q* = 6000

(b) Consumer surplus = $210,000; Producer surplus ≈ $70,000

3 250

5 200

7 (a) $400

(b) $266.7

9

11 (a) About $2250, $2625, $4875

(b) Consumer surplus: less Producer surplus: greater Total gains: less

13 (a) Less

(b) Cannot tell

(c) Less

15 (a) No, yes

(b) Yes, no

Section 6.5

3 P = $21,105

F = $44,680

5 (a) $16,198.31

(b) $13,994.35

(c) Lump sum; Better to get money earlier

7 (a) $84,160.82

(b) $30,000

(c) $54,160.82

9 (a) (i) $18,846.59

(ii) $16,484.00

(b) (i) $21,249.47

(ii) $24,591.24

11 2,936,142.74 euros

13 (a) $5820 per year

(b) $36,787.94

15 No; present value = $306,279

17 (a) $33.58 billion

(b) $35.66 billion

19 (a) 80.1 and 102.1 billion dollars

(b) 268.6 and 342.2 billion dollars

21 (a) 10.6 years

(b) 624.9 million dollars

Section 6.6

1 (a) 2x cos(x² + 1);

3x² cos(x³ + 1)

(b) (i) sin(x² + 1) + C

(ii) sin(x³ + 1) + C

(c) (i) −cos(x² + 1) + C

(ii) −cos(x³ + 1) + C

3 Yes

5 Yes

7

9 (x² + 1)^3/2 + C

11 e^5t+2 + C

13 −500e^−0.2t + C

15 −cos(x²) + C

17 (1/6)(x² + 3)³ + C

19 ln(x⁴ + 1) + C

21 (1/148)(2t − 7)⁷⁴ + C

23 (1/5)y⁵ + (1/2)y⁴ + (1/3)y³ + C

25 (1/7) sin⁷ θ + C

27 cos(3 − t) + C

29 (1/3)e^x3+1 + C

31 −cos(4x²) + C

33 e^3x–4 + C

35 e^3x2 + C

37 ln(e^x + e^−x) + C

39 (1/2) ln(y² + 4) + C

41 ln |e^t + t| + C

43 2 sin + C

45 ln(e^t + 1) + C

47 +C

49 (a) (ln 2)/2

(b) (ln 2)/2

51 ln 10

53 2e(e − 1)

55 1

57 1 − e⁻¹

59 1/40

61 ln 3

63 (a) 10

(b) 5

65 (a) x⁴ + 2x² + C

(b) (x² + 1)² + C

(c) Both correct but differ by a constant

Section 6.7

1

3 (1/2)y² ln y − (1/4)y² + C

5 (1/6)q⁶ ln 5q − (1/36)q⁶ + C

7 (x⁴/4) ln x − (x⁴/16) + C

9 −2y(5 − y)^1/2 − (4/3)(5 − y)^3/2 + C

11 −2t(5 − t)^1/2 − (4/3)(5 − t)^3/2 − 14(5 − t)^1/2 + C

13 −t cost + sin t + C

15 0.386, 2 ln 2 − 1

17 5 ln 5 − 4 ≈ 4.047

19 (9/2) ln 3 − 2 ≈ 2.944

21 t(ln t)² − 2t ln t + 2t + C

23 (a) Substitution

(b) Substitution

(c) Substitution

(d) Substitution

(e) Parts

(f) Parts

25 2 ln 2 − 1

27 (a) −(1/a)Te^−aT + (1/a²)(1 − e^−aT)

(b) lim_T→∞ E = 1/a²

Chapter 6 Review

1 1, 0, −1/2, 0, 1

3

5 (a) Increasing for x < −2, x > 2

Decreasing for −2 < x < 2

Local maximum at x = −2

Local minimum at x = 2

(b)

7

9 t³ + (7t²/2) + t

11 2x³ − 4x² + 3x

13 x³ + 5x

15 P(y) = ln |y| + y²/2 + y + C

17 (2x + 1)⁴/8 + C

19 2t² + 7t + C

21 (x⁴/4) − (x²/2) + C

23 −5/t − 3/t² + C

25 2x² + 2e^x + C

27 2x³ + C

29 x + ln |x| + C

31 1/2

33 1 − cos 1 ≈ 0.460

35 (ln 2)/2 ≈ 0.35

37 (6x² + 1) dx = 18

39 b = 7

41 1/30

43 (a) 18, 61.2, 198

(b)

(c) Does not converge

45 (a) No

(b) (60/50^t) dt

(c) Yes; 15.34 miles

47 (a) p^* = $11.43, q^* = 322 units

(b) Consumer surplus = $2513.52;

Producer surplus = $1033.62

49 $85,750,000

51 $4026.35

53 $1,147.75

55 33 billion m³/year

57 F(x) = −7x²/2 (only possibility)

59 0.5 sin(t²) + C

61 ln(x² + 1) + C

63 (x² + 9)⁷ + C

65 −e^−x + C

67 −2

69 Approximately 77

71 −y cos y + sin y + C

73 2x ln x − 2x + C

Ch. 6 Understanding

1 True

3 True

5 True

7 True

9 False

11 True

13 False

15 True

17 False

19 True

21 False

23 True

25 False

27 False

29 False

31 True

33 False

35 True

37 False

39 True

41 True

43 False

45 False

47 True

49 False

51 False

53 False

55 False

57 True

59 True

61 False

63 False

65 False

67 True

69 False

Chap. 6: Practice

1 (1/18)(y² + 5)⁹ + C

3 u⁵/5 + 5u + C

5 −e^−3t/3 + C

7 ax³/3 + bx + C

9 (x⁴/4) + 2x² + 8x + C

11 q²/2 − 1/(2q²) + C

13 q³/3 + 5q²/2 + 2q + C

15 4 ln |x| − 5/x + C

17 −3 cos θ + C

19 p³/3 + 5 ln |p| + C

21 −5 cos x + 3 sin x + C

23 Aq²/2 + Bq + C

25 πhr³/3 + C

27 5p³q⁴ + C

29 x³ + 3e^2x + C

31 p⁴/4 + ln |p| + C

33 ln |y + 2| + C

35 x³/3 + 8x + e^x + C

37 a ln |x| − b/x + C

39 e^2t + 5t + C

41 P₀e^kt/k + C

43 −(A/B) cos(Bt) + C

45 ln(2 + e^x) + C

47 (1/2)x² ln x − (1/4)x² + C

Section 7.1

1 (a) 0.25

(b) 0.7

(c) 0.15

3 (a) 0.4375

(b) 0.49

(c) 0.2475

5 10–12: About 27%

< 8: About 12%

> 12: About 45%

12–13 days

7 For small Δx around 70, fraction of families with incomes in that interval about 0.05Δx

9 0.04

11 0.008

13 (a) 0.19

(b) Tenth; both same

(c) 0.02, 0.38, 0.21

15

17

Section 7.2

1 (a)

(b) fraction of patients

3 (a) Cumulative distribution increasing

(b) Vertical 0.2,

horizontal 2

5 pdf; 1/4

7 cdf; 1

9 cdf; 1/3

11

13 (a) About 2/3

(b) About 1/3

(c) Possibly many work just to pass

(d) fraction of students

15 (a) Cumulative distribution

(b)

(c) More than 50%: 1%

Between 20% and 50%: 49%

Most likely: C ≈ 28%

17 (a)

(b) About 25%

19 Fraction dead at time t

21 (a) −e⁻² + 1 ≈ 0.865

(b) −(ln 0.05)/2 ≈ 1.5 km

Section 7.3

1 About 5.35 tons

3 Mean 2/3; Median 2 − = 0.586

5 2.48 weeks

7 (a) 0.684 :1

(b) 1.6 hours

(c) 1.682 hours

9 (a) 0.2685

(b) 0.067

11 (a) 16.7%; 12.9%

(b) About $39,900

(c) False

Chapter 7 Review

1 1/15

3 1/5

5

7 (a)-(II), (b)-(I), (c)-(III)

9

11 (a) Twelfth

(b) 1/4

(c) 7/16

13 About 30%

15

17 Density function

19 14.6 days

21 False

23 True

25 True

Ch. 7 Understanding

1 False

3 True

5 False

7 True

9 True

11 True

13 False

15 False

17 False

19 False

21 True

23 False

25 True

27 True

29 True

Section 8.1

3 (a) Decreasing

(b) Decreasing

5 Increasing function

9 Decreasing function of p

Increasing function of a

11 (a) 81°F

(b) 30%

13 Incr of A and r Decr of t

15

Section 8.2

1 Decreasing function of x

Increasing function of y

3 Answers in °C:

(a)

(b)

(c)

(d)

5 Contours evenly spaced parallel lines

7 Contours evenly spaced parallel lines

9 Contours evenly spaced parallel lines

11

13

15

17

19 (a) π = 3q₁ + 12_q2 −4 (thousand dollars)

21 Contours evenly spaced

23 (a) False

(b) True

(c) False

(d) True

25 (a) About 0°F

(b) About −16°F

(c) About 23 mph

(d) About 25°F

27 (a) A

(b) B

(c) A

29

31 (a) 4 hours

(b) 40%

(c) Contours approx horizontal

(d) Increasing

(e) Increasing

33 (a) is (II)

(b) is (I)

35 (a) s(x, y) = xy = 16

(b) 4

(c) 6

Section 8.3

1 (a) Positive

(b) Negative

(c) Positive

(d) Zero

3

5 (a) f_c is negative

f_t is positive

7 ∂Q/∂b < 0

∂Q/∂c > 0

9 z_x(1, 0) ≈ 2

z_x(0, 1) ≈ 0

z_y(0, 1) ≈ 10

11 (a) Negative

13 2.9; 0.02

15 ∂f/∂P₁ < 0

∂f/∂P₂ > 0

19 2777

21 14 %

23 (a)

(b)

(c)

(d)

25 (a) −2 Grapes/Cherry

(b) No change in happiness when replacing 2 grapes with one cherry

Section 8.4

1 f_x = 2x + 2y

f_y = 2x + 3y²

3 200xy; 100x²

5 2xe^y

7 15a²t²

9 f_x = 20xe^3y,

f_y = 30x²e^3y.

11 (1/2)v²

13 (a + b)/2

15 15; 5; 30

17 (a) 3.3, 2.5

(b) 4.1, 2.1

(c) 4, 2

19 80; 30; 313 tons

2.9 tons per worker

2.6 tons per $25,000

21 f_xx = 2y, f_xy = 2x, f_yy = 0, f_yx = 2x

23 f_xx = 2, f_xy = 2, f_yy = 2, f_yx = 2

25 f_xx = 2y², f_xy = 4xy, f_yy = 2x², f_yx = 4xy

27 B_xx = 0, B_tt = 20xe^−2t, B_xt = B_tx = −10e^−2t

29 f_rr = 100t²e^rt, f_tt = 100r²e^rt, f_tr = f_rt = 100(rt + 1)e^rt

31 V_rr = 2πh, V_hh = 0, V_rh = V_hr = 2πr

33 f(x, y) = x⁴y² − 3xy⁴ + C

35 (a) (c + 1)/(c + r)

(b) Positive

37 (a) (r − 1)B/(c + r)²

(b) Negative

Section 8.5

1 f(2, 10) ≈ 0.5 local and global min

f(6, 4) ≈ 9.5 local max

f(6.5, 16) ≈ 10 local and global max

f(9, 10) ≈ 4 local min

3 Max: 11 at (5.1, 4.9)

Min: −1 at (1, 3.9)

5 Max: 1 at (π/2, 0); (π/2, 2π)

Min: −1 at (π/2, π)

7 Saddle pt: (−3, 6)

9 Saddle pt: (4, 2)

11 Local max: (−1, 0)

Saddle pts: (1, 0), (−1, 4)

Local min: (1, 4)

13 Local min: (2, 2)

15 Local max: (1, 5)

17 A = 10, B = 4, C = −2

19 (b) p₁ = p₂ = 25

Max revenue is 4375

21 q₁ = 300, q₂ = 225.

Section 8.6

1 Min = −, max =

3 f(10, 25) = 250

5 f(12, 4) = 240

7 Min = , no max

9 Min = 11.25; no max

11 (a) Point F

(b) Point D

13 (a) Q =

(b) 10x₁ + 25x₂ = 50,000

15 (b) L = 40, K = 30

17 (a) Reduce K by 1/2 unit, increase L by 1 unit.

19 (a) P(x, y); C(x, y) = 50,000

(b) C(x, y); P(x, y) = 2000

21 (a) P(K, L)

(b) C(K, L) = 600,000

(c) Tons/dollar

(d) Extra dollar produces approximately extra 3.17 tons

23 (a) Quantity of fuel, x₁ + x₂

(b) Terminal velocity (as function of x₁ and x₂) = v₀

(c) Liters per meter/sec

(d) 51 meters/sec requires about 8 more liters than 50 meters/sec

25 1820.04; about 209

Chapter 8 Review

1 Decreasing function of x

Increasing function of y

3 Lines with slope 3/5, evenly spaced

5

9 x-axis: apple juice

y-axis: orange juice

13 (a) About $122

(b) About $350

15(A) I

(B) II

(C) III

17 (a) Payment $376.59/mo at 1% for 24 mos

(b) 4.7¢ extras/mo for $1 increase

(c) Approx $44.83 increase for 1% interest increase

19 f_T(5, 20) ≈ 1.2°F/°F

21 ∂q/∂I > 0

∂q/∂p₁ < 0

∂q/∂p₂ > 0

25 f_x = 2x + y, f_y = 2y + x

27

29

31 $102,116

Additional $201.20 per unit

Additional $101.50 per unit

33

(b)

(c) The “wave” at a sports arena

35 Local min: (1, 0)

Saddle pt: (−1, 0)

37 (a) 517p₁ − + 770p₂ − + 2.2p₁p₂

(b) p₁ = 110, p₂ = 115

39 43

41 (a) C = 127x₁ + 92x₂

(b)

(c) 1 unit decrease in production gives approx $219 reduction in cost

43 (a) 2599

(b) 129; $4,712,958

(c) $8572.54 per car

45 (a) 475 units

(b) 505 units

Ch. 8 Understanding

1 False

3 False

5 True

7 True

9 False

11 True

13 True

15 True

17 True

19 True

21 True

23 True

25 False

27 True

29 True

31 True

33 True

35 False

37 False

39 True

41 False

43 True

45 False

47 True

49 True

51 False

53 False

55 True

57 True

59 False

Theory: Least Squares

1 y = 2/3 − (1/2)x

3 y = 2/3 − (1/2)x

5 y = x − 1/3

7 (a) (i) Power function

(ii) Linear function

(b) ln N = 1.20 + 0.32 ln A Agrees with biological rule

Section 9.1

1 (a) (III)

(b) (IV)

(c) (I)

(d) (II)

3 dP/dt = kP, k > 0

5 dQ/dt = kQ, k < 0

7 dP/dt = −0.08P − 30

9 dA/dt = −1

11 (a) dA/dt = −0.17A

(b) −17 mg

13 (a) Increasing, decreasing

(b) W = 4

15 dN/dt = B + kN

Section 9.2

1 (a) Yes

(b) No

3 y = t² + C

5 F

7 E

9 A

11 D

13 4, 4, 4, 4

15 12, 18, 27, 40.5

17 74, 78.8, 84.56 million

19 k = −0.03 and C is any number, or C = 0 and k is any number

21 k = 5

Section 9.3

1 Other answers possible

3

5(a)

(b) y = −x − 1

7 (a) III

(b) VI

(c) V

(d) I

(e) IV

(f) II

9 III: dP/dt = 3P(1 − P)

11 For starting points y > 0: y → ∞ as x → ∞ For starting points y = 0: y = 0 for all x For starting points y < 0: y → −∞ as x → ∞

13 As x → ∞, y → ∞

15 y → 4 as x → ∞

Section 9.4

1 P = 20e^0.02t

3 y = 5.6e^−0.14x

5 p = 164.87e^−0.1q

7 (a) dB/dt = 0.015B

(b) B = 5000e^0.015t

(c) $5809.17

9 (a) dB/dt = 0.10B + 1000

(b) B = 10,000e^0.1t − 10,000

11 Michigan: 72 years

Ontario: 18 years

13 (a) dQ/dt = −0.5365Q Q = Q₀e^−0.5365t

(b) 4 mg

15 (a) 69,300 barrels/year

(b) 25.9 years

17 (b) 2070

Section 9.5

1 y = 200 − 150e^0.5t

3 H = 75 − 75e^3t

5 B = 25 − 5e^4t

7 B = 25 + 75e^2–2t

11 dB/dt = 0.08B − 5000, B = 62,500 − 12,500e^0.08t, Yes, in 20.1 years

13 (a) dB/dt = 0.07B − 1000

(b) B ≈ $14,285.71

(c) B = 14,285.71 − (4285.71)e^0.07t

(d) B(5) ≈ $8204

(e) Balance is $0 in long run

15 (b) dQ/dt = 43.2 − 0.082Q

(c) Q = 526.8 − 526.8e^−0.082t

Q → 526.8 mg as t → ∞

17 (a) dy/dt = −k(y − a)

(b) y = (1 − a)e^−kt + a

(c) a: fraction remembered in the long run k: relative rate material is forgotten

19 (a) y = 3 and y = −2

(b) y = 3 is unstable y = −2 is stable

21 (a) H = 200 − 180e^−kt

(b) k ≈ 0.027 (if t is in minutes)

23 (a) dT/dt = −k(T − 68)

(b) T = 68 + 22.3e^−0.06t; 3:45 am.

Section 9.6

1 (a) x → ∞ exponentially

y → 0 exponentially

(b) y is helped by the presence of x

3 (a) x population grows exponentially

y population grows exponentially

(b) Competitor relationship

5 dx/dt = −x − xy,

dy/dt = −y − xy

7 dx/dt = −x + xy,

dy/dt = y

11 Symmetric about the line r = w;

solutions closed curves

13 Robins:

Max ≈ 2500

Min ≈ 500

When robins are at a max, the worm population is about 1 million

17 (a) dw/dt = 0

dr/dt = 1.2

(b) w ≈ 2.2, r ≈ 1.1

(c) At t = 0.2:

w ≈ 2.2, r ≈ 1.3

At t = 0.3

w ≈ 2.1, r ≈ 1.4

19 (a)

(b) Down and left

(c) r = 3.3, w = 1

(d) w = 3.3, r = 1

21 x and y increase, about same rate

23 x decreases quickly while y increases more slowly

25 (a) dQ₁/dt = A − k₁Q₁ + k₂Q₂

(b) dQ₂/dt = −k₃Q₂ + k₁Q₁ − k₂Q₂

Section 9.7

5 (a) I₀ = 1, S₀ = 349

(b) Increases; spreads

7 About 300 boys;

t ≈ 6 days

9 5

11 (a) b/a

Chapter 9 Review

1 (a) (III)

(b) (V)

(c) (I)

(d) (II)

(e) (IV)

3 Yes

5 (a) I is y′ = 1 + y;

II is y′ = 1 + x

(b)

(c) I: y′ = −1, unstable;

II: None

7 III: y′ = (1 + y)(2 − y)

9 (a) dB/dt = 0.07B

(b) B = B₀e^0.07t

(c) B = 5000e^0.07t

(d) B(10) ≈ $10,068.76

11 P = (1/2)t² + C

13 y = (5/2)t² + C

15 A = Ce^−0.07t

17 P = Ce^−2t + 5

19 y = Ce^0.2x + 40

21 dS/dt = −k(S − 65), k < 0 S = 65 − 25e^−kt

23 (a) k ≈ 0.000121

(b) 779.4 years

25 (a)

(b) dQ/dt = −0.0187Q

(c) 3 days

27 (b) dQ/dt = −0.347Q + 2.5

(c) Q = 7.2 mg

29 (a) dW/dt = (1/3500)(I − 20W)

(b) W = I/20, stable

(c) W = I/20 + (W₀ − I/20) e^−(1/175)t

(d)

31 (a) y = 1, y = 8, y = 16

(d) Stable: y = 1

Unstable: y = 8, y = 16

Ch. 9 Understanding

1 True

3 False

5 True

7 True

9 True

11 True

13 False

15 False

17 False

19 True

21 False

23 True

25 True

27 True

29 False

31 False

33 False

35 True

37 False

39 False

41 True

43 False

45 False

47 False

49 True

51 False

53 True

55 False

57 True

59 True

61 False

63 False

65 False

67 False

69 True

Theory: Separation of Vars

1 P = e^−2t

3

5 u = 1/(1 − (1/2)t)

7 R = 1 − 0.9e^1–y

9 z = −ln(1 − t²/2)

11 y = −2/(t² + 2t − 4)

13 (a) Yes (b) No (c) Yes

(d) No (e) Yes (f) Yes

(g) No (h) Yes (i) No

(j) Yes (k) Yes (l) No

15 Q = b − Ae^−t

17 R = −(b/a) + Ae^at

19 y = −1/(k(t + t³/3) + C)

21 (a)

(b)

(c) y(x) = Ae^x2/2

Section 10.1

1 21

3 Yes, a = 5, ratio = −2

5 Yes, a = 2, ratio = 1/2

7 Yes, a = 1, ratio = 2z

9 Yes, a = y², ratio = y

11 555.10

13 96.154

15 333.33

17 3(2¹¹ − 1)/2¹⁰

19 400

21 −4/3

23 30.51, 37.75,

39.47, 39.87, Yes

25 (a) $3007.51, $2507.51

(b) $6033.11, $5533.11

27 1.375 mg

29 (a) 40 mg

(b) 0.57 mg/kg, Yes

(c) (i) Greater than 100 kg

(ii) Less than 13.3 kg

Section 10.2

1 Balance = $24,297.37, $20,000 from deposits, $4297.37 from interest

3 (a) $427,767.74

(b) $1,419,863.01

5 $2.02 million

7 (a) $153,237.99

(b) $343,333.33

9 (a) $1.27

(b) $163.83

(c) $20,971.51

(d) $2,684,354.55

11 (a) 5000 units

(b) S₅ = 3362, S₁₀ = 4463,

S₁₅ = 4824, S₂₀ = 4942

13 $1081.11

15 (a) $1000

(b) When the interest rate is 5%, the present value equals the principal

(c) The principal

(d) Because the present value is more than the principal

17 (a) $400 billion

(b) $900 billion

19 (a) $1250

(b) 12.50

Section 10.3

1 (a) 98 mg

(b) 121.5 mg

(c) 125 mg

3 (a) Oil supply never runs out

(b) Oil supply exhausted in 39.1 years

5 (a) 400 mg

(b) 400(0.30) = 120 mg

7 (a) 0.7937

(b) 194.27 mg

(c) 242.37 mg

9 1604.0 micrograms

1596.0 micrograms

11 24.5 years

13 (a) 60 years

(b) 27.7 years

15 About 34 years

17 Lasts forever

Chapter 10 Review

1 2046

3 Does not exist

5 200

7 1.9961

9 (a) (i) $16.43 million

(ii) $24.01 million

(b) $16.87 million

11 $27,979.34

13 $400 million

15 (a) N(k/(1 − k))

(b) 5.667N

17 (a) ≈ 7%

(b) Q_n = 50(1 − (0.07)^{n + 1})/(1 − 0.07)

(c) P_n = 0.07(50)(1–(0.07)ⁿ)/(1–0.07)

Ch. 10 Understanding

1 True

3 False

5 True

7 False

9 True

11 True

13 True

15 False

17 True

19 True

21 False

23 False

25 False

27 False

29 False

Appendix A

1 (a)

(b) Increasing at a rate of $734 billion/year

(c) $43.9 trillion, $54.9 trillion, More confidence in the 2005 prediction

3 (a)

(c) For 1985: 4891

For 2020: 15,197

More confidence in 1985

5 (a) S = 0.08v + 1.77

(b)

(c) At v = 18 ft/sec, S = 3.21

At v = 10 ft/sec, S = 2.57

v = 18 better

7 (a) 0.0026 = 0.26%

(b) For 1900, 272.27

For 1980, 335.1

9 (a)

(b) Exponential

(c) D = 5309(1.048)^t, answers may vary

(d) About 4.8%

(e) No

11 (a) r = 1

(b) r = 0.7

(c) r = 0

(d) r = −0.98

(e) r = −0.25

(f) r = −0.5

13 (a) Exponential

(b) 2.6(1.0165)^t; answers may vary

(c) 1.65%

(d) At year 2020, 8.175 billion

At year 2050, 13.357 billion

2020 prediction more accurate

15

(b) Exponential

(c) C = 15.9·(1.725)^t, answers may vary

(d) About 73%

(e) No

17 (a) Negative

(b) f(t) = −0.03t² + 1.01t + 13.82

19 (a) Exponential

(b) S = 29.96(1.30)^t, answers may vary Increasing 30%/yr

(c) 21,141 megawatts

(d)

21 (a) Linear

(b) C = 320 + 1.5t

Increasing at 1.5 ppm/yr

(c) 387.5 ppm

(d)

Appendix B

1 27%

3 (a) $160,356.77

(b) $165,510.22

(c) $165,891.05

(d) $165,989.48

(e) $166,005.85

5 (a) 1.0408107

1.0408108

1.0408108

4% compounded continuously ≈ 4.08108%

(b) e^0.04 ≈ 1.048108

7 4.88%

9 (a) (i) 5.126978 . . .%

(ii) 5.127096 . . .%

(iii) 5.127108 . . .%

(b) 5.127%

(c) e^0.05 = 1.05127109 . . .

11 (a) = V, (b) = III, (c) = IV,

(d) = I, (e) = II

13 (a) 13,900 cruzados

(b) 24.52%