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  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Key Terms
    13. Key Equations
    14. Key Concepts
    15. Chapter Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. Index

Checkpoint

6.1

1212 units2

6.2

310310 unit2

6.3

2+222+22 units2

6.4

5353 units2

6.5

5353 units2

6.7

π2π2

6.8

8π8π units3

6.9

21π21π units3

6.10

10π310π3 units3

6.11

60π60π units3

6.12

15π215π2 units3

6.13

8π8π units3

6.14

12π12π units3

6.15

11π611π6 units3

6.16

π6π6 units3

6.17

Use the method of washers; V=−11π[(2x2)2(x2)2]dxV=−11π[(2x2)2(x2)2]dx

6.18

16(551)1.69716(551)1.697

6.19

Arc Length3.8202Arc Length3.8202

6.20

Arc Length=3.15018Arc Length=3.15018

6.21

π6(5533)3.133π6(5533)3.133

6.22

12π12π

6.23

70/370/3

6.24

24π24π

6.25

88 ft-lb

6.26

Approximately 43,255.243,255.2 ft-lb

6.27

156,800156,800 N

6.28

Approximately 7,164,520,000 lb or 3,582,260 t

6.29

M=24,x=25mM=24,x=25m

6.30

(−1,−1)(−1,−1) m

6.31

The centroid of the region is (3/2,6/5).(3/2,6/5).

6.32

The centroid of the region is (1,13/5).(1,13/5).

6.33

The centroid of the region is (0,2/5).(0,2/5).

6.34

6π26π2 units3

6.35
  1. ddxln(2x2+x)=4x+12x2+xddxln(2x2+x)=4x+12x2+x
  2. ddx(ln(x3))2=6ln(x3)xddx(ln(x3))2=6ln(x3)x
6.36

x2x3+6dx=13ln|x3+6|+Cx2x3+6dx=13ln|x3+6|+C

6.37

4ln24ln2

6.38
  1. ddx(ex2e5x)=ex25x(2x5)ddx(ex2e5x)=ex25x(2x5)
  2. ddt(e2t)3=6e6tddt(e2t)3=6e6t
6.39

4e3xdx=43e−3x+C4e3xdx=43e−3x+C

6.40
  1. ddt4t4=4t4(ln4)(4t3)ddt4t4=4t4(ln4)(4t3)
  2. ddxlog3(x2+1)=x(ln3)(x2+1)ddxlog3(x2+1)=x(ln3)(x2+1)
6.41

x22x3dx=13ln22x3+Cx22x3dx=13ln22x3+C

6.42

There are 81,377,39681,377,396 bacteria in the population after 44 hours. The population reaches 100100 million bacteria after 244.12244.12 minutes.

6.43

At 5%5% interest, she must invest $223,130.16.$223,130.16. At 6%6% interest, she must invest $165,298.89.$165,298.89.

6.44

38.9038.90 months

6.45

The coffee is first cool enough to serve about 3.53.5 minutes after it is poured. The coffee is too cold to serve about 77 minutes after it is poured.

6.46

A total of 94.1394.13 g of carbon remains. The artifact is approximately 13,30013,300 years old.

6.47
  1. ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)ddx(tanh(x2+3x))=(sech2(x2+3x))(2x+3)
  2. ddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshxddx(1(sinhx)2)=ddx(sinhx)−2=−2(sinhx)−3coshx
6.48
  1. sinh3xcoshxdx=sinh4x4+Csinh3xcoshxdx=sinh4x4+C
  2. sech2(3x)dx=tanh(3x)3+Csech2(3x)dx=tanh(3x)3+C
6.49
  1. ddx(cosh−1(3x))=39x21ddx(cosh−1(3x))=39x21
  2. ddx(coth−1x)3=3(coth−1x)21x2ddx(coth−1x)3=3(coth−1x)21x2
6.50
  1. 1x24dx=cosh−1(x2)+C1x24dx=cosh−1(x2)+C
  2. 11e2xdx=sech−1(ex)+C11e2xdx=sech−1(ex)+C
6.51

52.95ft52.95ft

Section 6.1 Exercises

1.

323323

3.

13121312

5.

3636

7.


This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=-x^2+18x. The region between the graphs is shaded, bounded above by g(x) and below by f(x). It is in the first quadrant.


243 square units

9.


This figure is has two graphs. They are the functions y=cos(x) and y=cos^2(x). The graphs are periodic and resemble waves. There are four regions created by intersections of the curves. The areas are shaded.


4

11.


This figure is has two graphs. They are the functions f(x)=e^x and g(x)=e^-x. There are two shaded regions. In the second quadrant the region is bounded by x=-1, g(x) above and f(x) below. The second region is in the first quadrant and is bounded by f(x) above, g(x) below, and x=1.


2(e1)2e2(e1)2e

13.


This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=absolute value of x. There are two shaded regions. The first region is in the second quadrant and is between g(x) above and f(x) below. The second region is in the first quadrant and is bounded above by g(x) and below by f(x).


1313

15.


This figure is has three graphs. They are the functions f(x)=squareroot of x, y=12-x, and y=1. The region between the graphs is shaded, bounded above and to the left by f(x), above and to the right by the line y=12-x, and below by the line y=1. It is in the first quadrant.


343343

17.


This figure is has two graphs. They are the functions f(x)=x^3 and g(x)=x^2-2x. There are two shaded regions between the graphs. The first region is bounded to the left by the line x=-2, above by g(x) and below by f(x). The second region is bounded above by f(x), below by g(x) and to the right by the line x=2.


5252

19.


This figure is has two graphs. They are the functions f(x)=x^3+3x and g(x)=4x. There are two shaded regions between the graphs. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x), below by f(x).


1212

21.


This figure is has two graphs. They are the equations x=2y and x=y^3-y. The graphs intersect in the third quadrant and again in the first quadrant forming two closed regions in between them.


9292

23.


This figure is has two graphs. They are the equations x=y+2 and y^2=x. The graphs intersect, forming a region in between them


9292

25.


This figure is has two graphs. They are the equations x=cos(y) and x=sin(y). The graphs intersect, forming two regions bounded above by the line y=pi/2 and below by the line y=-pi/2.


332332

27.


This figure is has two graphs. They are the equations y=xe^x and y=e^x. The graphs intersect, forming a region in between them in the first quadrant.


e−2e−2

29.


This figure is has two graphs. They are the equations x=-y^2+1 and x=y^3+2y^2. The graphs intersect, forming two regions in between them.


274274

31.


This figure is has two graphs. They are the equations y=4-3x and y=1/x. The graphs intersect, having region between them shaded. The region is in the first quadrant.


43ln(3)43ln(3)

33.


This figure is has two graphs. They are the equations y=x^2-3x+2 and y=x^3-2x^2-x+2. The graphs intersect, having region between them shaded.


1212

35.


This figure is has two graphs. They are the equations 2y=x and y+y^3=x. The graphs intersect, forming two regions. The regions are shaded.


1212

37.


This figure is has two graphs. They are the equations y=arccos(x) and y=arcsin (x). The graphs intersect, forming two regions. The first region is bounded to the left by x=-1. The second region is bounded to the right by x=1. Both regions are shaded.


−2(2π)−2(2π)

39.

1.0671.067

41.

0.8520.852

43.

7.5237.523

45.

3π4123π412

47.

1.4291.429

49.

$33,333.33$33,333.33 total profit for 200200 cell phones sold

51.

3.2633.263 mi represents how far ahead the hare is from the tortoise

53.

3432434324

55.

4343

57.

π3225π3225

Section 6.2 Exercises

63.

8 units3

65.

32323232 units3

67.

7π12hr27π12hr2 units3

69.


This figure shows the x-axis and the y-axis with a line starting on the x-axis at (1,0) and ending on the y-axis at (0,1). Perpendicular to the xy-plane are 4 shaded semi-circles with their diameters beginning on the x-axis and ending on the line, decreasing in size away from the origin.


π24π24 units3

71.


This figure shows the x-axis and the y-axis in 3-dimensional perspective. On the graph above the x-axis is a parabola, which has its vertex at y=1 and x-intercepts at (-1,0) and (1,0). There are 3 square shaded regions perpendicular to the x y plane, which touch the parabola on either side, decreasing in size away from the origin.


22 units3

73.


This figure is a graph with the x and y axes diagonal to show 3-dimensional perspective. On the first quadrant of the graph are the curves y=x, a line, and y=x^2, a parabola. They intersect at the origin and at (1,1). Several semicircular-shaped shaded regions are perpendicular to the x y plane, which go from the parabola to the line and perpendicular to the line.


π240π240 units3

75.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^2, below by the x-axis, and to the right by the vertical line x=4.


4096π54096π5 units3

77.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=1, below by the curve y=x^4, and to the left by the y-axis.


8π98π9 units3

79.


This figure is a shaded region bounded above by the curve y=cos(x), below to the left by the y-axis and below to the right by y=sin(x). The shaded region is in the first quadrant.


π2π2 units3

81.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the line x + y=9, below by the x-axis, to the left by the y-axis, and to the left by the curve x^2-y^2=9.


207π207π units3

83.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=2x^3, below by the x-axis, and to the right by the line x=1.


4π54π5 units3

85.


This figure is a graph in the first quadrant. It is a quarter of a circle with center at the origin and radius of 2. It is shaded on the inside.


16π316π3 units3

87.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=pi/4, to the right by the curve x=sec(y), below by the x-axis, and to the left by the y-axis.


ππ units3

89.


This figure is a graph in the first quadrant. It is a shaded triangle bounded above by the line y=4-x, below by the line y=x, and to the left by the y-axis.


16π316π3 units3

91.


This figure is a graph above the x-axis. It is a shaded region bounded above by the line y=x+2, and below by the parabola y=x^2.


72π572π5 units3

93.


This figure is a shaded region bounded above by the curve y=4-x^2 and below by the line y=2-x.


108π5108π5 units3

95.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=squareroot(x), below by the curve y=x^2.


3π103π10 units3

97.


This figure is a shaded region bounded above by the curve y=squareroot(4-x^2) and, below by the curve y=squareroot(1+x^2).


26π26π units3

99.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the line y=x+2, below by the line y=2x-1, and to the left by the y-axis.


9π9π units3

101.


This figure is a graph in the first quadrant. It is a shaded region bounded above by the curve y=ln(2), below by the x-axis, to the left by the curve x=y^2, and to the right by the curve x=e^(2y).


π20(754ln5(2))π20(754ln5(2)) units3

103.

m2π3(b3a3)m2π3(b3a3) units3

105.

4a2bπ34a2bπ3 units3

107.

2π22π2 units3

109.

2ab2π32ab2π3 units3

111.

π12(r+h)2(6rh)π12(r+h)2(6rh) units3

113.

π3(h+R)(h2R)2π3(h+R)(h2R)2 units3

Section 6.3 Exercises

115.


This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.


54π54π units3

117.


This figure is a graph in the first quadrant. It is the line y=3x. Under the line and above the x-axis there is a shaded region. The region is bounded to the right at x=3.


81π81π units3

119.


This figure is a graph in the first quadrant. It is the increasing curve y=2x^3. Under the curve and above the x-axis there is a shaded region. The region is bounded to the right at x=2.


512π7512π7 units3

121.

2π2π units3

123.

2π32π3 units3

125.

2π2π units3

127.

4π54π5 units3

129.

64π364π3 units3

131.

32π532π5 units3

133.

π(e2)π(e2) units3

135.

28π328π3 units3

137.

−84π5−84π5 units3

139.

eππ2eππ2 units3

141.

64π564π5 units3

143.

28π1528π15 units3

145.

3π103π10 units3

147.

52π552π5 units3

149.

0.98760.9876 units3

151.


This figure is a graph. On the graph are two curves, y=cos(pi times x) and y=sin(pi times x). They are periodic curves resembling waves. The curves intersect in the first quadrant and also the fourth quadrant. The region between the two points of intersection is shaded.


3232 units3

153.


This figure is a graph in the first quadrant. It is the parabola y=x^2-2x. . Under the curve and above the x-axis there is a shaded region. The region begins at x=2 and is bounded to the right at x=4.


496π15496π15 units3

155.


This figure is a graph in the first quadrant. There are two curves on the graph. The first curve is y=3x^2-2 and the second curve is y=x. Between the curves there is a shaded region. The region begins at x=1 and is bounded to the right at x=2.


398π15398π15 units3

157.


This figure is a graph. There are two curves on the graph. The first curve is x=y^2-2y+1 and is a parabola opening to the right. The second curve is x=y^2 and is a parabola opening to the right. Between the curves there is a shaded region. The shaded region is bounded to the right at x=2.


15.907415.9074 units3

159.

13πr2h13πr2h units3

161.

πr2hπr2h units3

163.

πa2πa2 units3

Section 6.4 Exercises

165.

226226

167.

217217

169.

π6(171755)π6(171755)

171.

13138271313827

173.

4343

175.

2.00352.0035

177.

1233212332

179.

1010

181.

203203

183.

1675(2292298)1675(2292298)

185.

18(45+ln(9+45))18(45+ln(9+45))

187.

1.2011.201

189.

15.234115.2341

191.

49π349π3

193.

70π270π2

195.

8π8π

197.

120π26120π26

199.

π6(17171)π6(17171)

201.

92π92π

203.

1010π27(73731)1010π27(73731)

205.

25.64525.645

207.

2π2π

209.

10.501710.5017

211.

2323 ft

213.

22

215.

Answers may vary

217.

For more information, look up Gabriel’s Horn.

Section 6.5 Exercises

219.

150150 ft-lb

221.

200J200J

223.

11 J

225.

392392

227.

ln(243)ln(243)

229.

332π15332π15

231.

100π100π

233.

20π1520π15

235.

66 J

237.

55 cm

239.

3636 J

241.

18,75018,750 ft-lb

243.

323×109ft-lb323×109ft-lb

245.

8.65×105J8.65×105J

247.

a. 3,000,0003,000,000 lb, b. 749,000749,000 lb

249.

23.25π23.25π million ft-lb

251.

AρH22AρH22

253.

Answers may vary

Section 6.6 Exercises

255.

5454

257.

(23,23)(23,23)

259.

(74,32)(74,32)

261.

3L43L4

263.

π2π2

265.

e2+1e21e2+1e21

267.

π24ππ24π

269.

14(1+e2)14(1+e2)

271.

(a3,b3)(a3,b3)

273.

(0,π8)(0,π8)

275.

(0,3)(0,3)

277.

(0,4π)(0,4π)

279.

(58,13)(58,13)

281.

mπ3mπ3

283.

πa2bπa2b

285.

(43π,43π)(43π,43π)

287.

(12,25)(12,25)

289.

(0,289π)(0,289π)

291.

Center of mass: (a6,4a25),(a6,4a25), volume: 2πa492πa49

293.

Volume: 2π2a2(b+a)2π2a2(b+a)

Section 6.7 Exercises

295.

1x1x

297.

1x(lnx)21x(lnx)2

299.

ln(x+1)+Cln(x+1)+C

301.

ln(x)+1ln(x)+1

303.

cot(x)cot(x)

305.

7x7x

307.

csc(x)secxcsc(x)secx

309.

−2tanx−2tanx

311.

12ln(53)12ln(53)

313.

212ln(5)212ln(5)

315.

1ln(2)11ln(2)1

317.

12ln(2)12ln(2)

319.

13(lnx)313(lnx)3

321.

2x3x2+1x212x3x2+1x21

323.

x−2(1/x)(lnx1)x−2(1/x)(lnx1)

325.

exe1exe1

327.

11

329.

1x21x2

331.

πln(2)πln(2)

333.

1x1x

335.

e56units2e56units2

337.

ln(4)1units2ln(4)1units2

339.

2.86562.8656

341.

3.15023.1502

Section 6.8 Exercises

349.

True

351.

False; k=ln(2)tk=ln(2)t

353.

2020 hours

355.

No. The relic is approximately 871871 years old.

357.

71.9271.92 years

359.

55 days 66 hours 2727 minutes

361.

1212

363.

8.618%8.618%

365.

$6766.76$6766.76

367.

99 hours 1313 minutes

369.

239,179239,179 years

371.

P(t)=43e0.01604t.P(t)=43e0.01604t. The population is always increasing.

373.

The population reaches 1010 billion people in 2027.2027.

375.

P(t)=2.259e0.06407t.P(t)=2.259e0.06407t. The population is always increasing.

Section 6.9 Exercises

377.

exandexexandex

379.

Answers may vary

381.

Answers may vary

383.

Answers may vary

385.

3sinh(3x+1)3sinh(3x+1)

387.

tanh(x)sech(x)tanh(x)sech(x)

389.

4cosh(x)sinh(x)4cosh(x)sinh(x)

391.

xsech2(x2+1)x2+1xsech2(x2+1)x2+1

393.

6sinh5(x)cosh(x)6sinh5(x)cosh(x)

395.

12sinh(2x+1)+C12sinh(2x+1)+C

397.

12sinh2(x2)+C12sinh2(x2)+C

399.

13cosh3(x)+C13cosh3(x)+C

401.

ln(1+cosh(x))+Cln(1+cosh(x))+C

403.

cosh(x)+sinh(x)+Ccosh(x)+sinh(x)+C

405.

4116x24116x2

407.

sinh(x)cosh2(x)+1sinh(x)cosh2(x)+1

409.

csc(x)csc(x)

411.

1(x21)tanh−1(x)1(x21)tanh−1(x)

413.

1atanh−1(xa)+C1atanh−1(xa)+C

415.

x2+1+Cx2+1+C

417.

cosh−1(ex)+Ccosh−1(ex)+C

419.

Answers may vary

421.

37.3037.30

423.

y=1ccosh(cx)y=1ccosh(cx)

425.

−0.521095−0.521095

427.

1010

Chapter Review Exercises

435.

False

437.

False

439.

323323

441.

162π5162π5

443.

a. 4,4, b. 128π7,128π7, c. 64π564π5

445.

a. 1.949,1.949, b. 21.952,21.952, c. 17.09917.099

447.

a. 316,316, b. 452π15,452π15, c. 31π631π6

449.

245.282245.282

451.

Mass: 12,12, center of mass: (1835,911)(1835,911)

453.

17+18ln(33+817)17+18ln(33+817)

455.

Volume: 3π4,3π4, surface area: π(2sinh−1(1)+sinh−1(16)25716)π(2sinh−1(1)+sinh−1(16)25716)

457.

11:02 a.m.

459.

π(1+sinh(1)cosh(1))π(1+sinh(1)cosh(1))

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