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  1. Preface
  2. 1 Integration
    1. Introduction
    2. 1.1 Approximating Areas
    3. 1.2 The Definite Integral
    4. 1.3 The Fundamental Theorem of Calculus
    5. 1.4 Integration Formulas and the Net Change Theorem
    6. 1.5 Substitution
    7. 1.6 Integrals Involving Exponential and Logarithmic Functions
    8. 1.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  3. 2 Applications of Integration
    1. Introduction
    2. 2.1 Areas between Curves
    3. 2.2 Determining Volumes by Slicing
    4. 2.3 Volumes of Revolution: Cylindrical Shells
    5. 2.4 Arc Length of a Curve and Surface Area
    6. 2.5 Physical Applications
    7. 2.6 Moments and Centers of Mass
    8. 2.7 Integrals, Exponential Functions, and Logarithms
    9. 2.8 Exponential Growth and Decay
    10. 2.9 Calculus of the Hyperbolic Functions
    11. Key Terms
    12. Key Equations
    13. Key Concepts
    14. Chapter Review Exercises
  4. 3 Techniques of Integration
    1. Introduction
    2. 3.1 Integration by Parts
    3. 3.2 Trigonometric Integrals
    4. 3.3 Trigonometric Substitution
    5. 3.4 Partial Fractions
    6. 3.5 Other Strategies for Integration
    7. 3.6 Numerical Integration
    8. 3.7 Improper Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  5. 4 Introduction to Differential Equations
    1. Introduction
    2. 4.1 Basics of Differential Equations
    3. 4.2 Direction Fields and Numerical Methods
    4. 4.3 Separable Equations
    5. 4.4 The Logistic Equation
    6. 4.5 First-order Linear Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  6. 5 Sequences and Series
    1. Introduction
    2. 5.1 Sequences
    3. 5.2 Infinite Series
    4. 5.3 The Divergence and Integral Tests
    5. 5.4 Comparison Tests
    6. 5.5 Alternating Series
    7. 5.6 Ratio and Root Tests
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Chapter Review Exercises
  7. 6 Power Series
    1. Introduction
    2. 6.1 Power Series and Functions
    3. 6.2 Properties of Power Series
    4. 6.3 Taylor and Maclaurin Series
    5. 6.4 Working with Taylor Series
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  8. 7 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 7.1 Parametric Equations
    3. 7.2 Calculus of Parametric Curves
    4. 7.3 Polar Coordinates
    5. 7.4 Area and Arc Length in Polar Coordinates
    6. 7.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Checkpoint

4.2

55

4.3

y=2x2+3x+2y=2x2+3x+2

4.5

y=13x32x2+3x6ex+14y=13x32x2+3x6ex+14

4.6

v(t)=−9.8tv(t)=−9.8t

4.7


A graph of the direction field for the differential equation y’ = x ^ 2 – y ^ 2. Along y = x and y = -x, the lines are horizontal. On either side of y = x and y = -x, the lines slant and direct solutions along those two functions. The rest of the lines are vertical. The solution going through (-1, 2) is shown. It curves down from about (-2.75, 10), through (-1, 2) and about (0, 1.5), and then up along the diagonal to (10, 10).
4.8


A direction field with arrows pointing to the right at y = -4 and y = 4. The arrows point up for y > -4 and down for y < -4. Close to y = 4, the arrows are more horizontal, but the further away, the more vertical they become.


The equilibrium solutions are y=−2y=−2 and y=2.y=2. For this equation, y=−2y=−2 is an unstable equilibrium solution, and y=2y=2 is a semi-stable equilibrium solution.

4.9
nn xnxn yn=yn1+hf(xn1,yn1)yn=yn1+hf(xn1,yn1)
00 11 −2−2
11 1.11.1 y1=y0+hf(x0,y0)=−1.5y1=y0+hf(x0,y0)=−1.5
22 1.21.2 y2=y1+hf(x1,y1)=−1.1419y2=y1+hf(x1,y1)=−1.1419
33 1.31.3 y3=y2+hf(x2,y2)=−0.8387y3=y2+hf(x2,y2)=−0.8387
44 1.41.4 y4=y3+hf(x3,y3)=−0.5487y4=y3+hf(x3,y3)=−0.5487
55 1.51.5 y5=y4+hf(x4,y4)=−0.2442y5=y4+hf(x4,y4)=−0.2442
66 1.61.6 y6=y5+hf(x5,y5)=0.0993y6=y5+hf(x5,y5)=0.0993
77 1.71.7 y7=y6+hf(x6,y6)=0.5099y7=y6+hf(x6,y6)=0.5099
88 1.81.8 y8=y7+hf(x7,y7)=1.0272y8=y7+hf(x7,y7)=1.0272
99 1.91.9 y9=y8+hf(x8,y8)=1.7159y9=y8+hf(x8,y8)=1.7159
1010 22 y10=y9+hf(x9,y9)=2.6962y10=y9+hf(x9,y9)=2.6962
4.10

y=2+Cex2+3xy=2+Cex2+3x

4.11

y=4+14ex2+x17ex2+xy=4+14ex2+x17ex2+x

4.12

Initial value problem:

dudt=2.42u25,u(0)=3dudt=2.42u25,u(0)=3

Solution:u(t)=3027et/50Solution:u(t)=3027et/50

4.13
  1. Initial-value problem
    dTdt=k(T70),T(0)=450dTdt=k(T70),T(0)=450
  2. T(t)=70+380ektT(t)=70+380ekt
  3. Approximately 114114 minutes.
4.14
  1. dPdt=0.04(1P750),P(0)=200dPdt=0.04(1P750),P(0)=200


  2. A direction field with horizontal lines on the x axis and y = 15. The other lines are vertical, except for those curving into the x axis and y = 15. A solution is drawn that crosses the y axis at about (0, 4) and asymptotically approaches y = 15.
  3. P(t)=3000e.04t11+4e.04tP(t)=3000e.04t11+4e.04t

  4. After 1212 months, the population will be P(12)278P(12)278 rabbits.

4.15

y+15x+3y=10x20x+3;p(x)=15x+3y+15x+3y=10x20x+3;p(x)=15x+3 and q(x)=10x20x+3q(x)=10x20x+3

4.16

y=x3+x2+Cx2y=x3+x2+Cx2

4.17

y=−2x4+2e2xy=−2x4+2e2x

4.18
  1. dvdt=v9.8v(0)=0dvdt=v9.8v(0)=0
  2. v(t)=9.8(et1)v(t)=9.8(et1)
  3. limtv(t)=limt(9.8(et1))=−9.8m/s21.922mphlimtv(t)=limt(9.8(et1))=−9.8m/s21.922mph
4.19

Initial-value problem:

8q+10.02q=20sin5t,q(0)=48q+10.02q=20sin5t,q(0)=4

q(t)=10sin5t8cos5t+172e−6.25t41q(t)=10sin5t8cos5t+172e−6.25t41

Section 4.1 Exercises

1.

11

3.

33

5.

11

7.

11

19.

y=4+3x44y=4+3x44

21.

y=12ex2y=12ex2

23.

y=2e1/xy=2e1/x

25.

u=sin−1(e−1+t)u=sin−1(e−1+t)

27.

y=x+11x1y=x+11x1

29.

y=Cx+xlnxln(cosx)y=Cx+xlnxln(cosx)

31.

y=C+4xln(4)y=C+4xln(4)

33.

y=23t2+16(t2+16)+Cy=23t2+16(t2+16)+C

35.

x=2154+t(3t2+4t32)+Cx=2154+t(3t2+4t32)+C

37.

y=Cxy=Cx

39.

y=1t22,y=t221y=1t22,y=t221

41.

y=et,y=ety=et,y=et

43.

y=2(t2+5),t=35y=2(t2+5),t=35

45.

y=10e−2t,t=12ln(110)y=10e−2t,t=12ln(110)

47.

y=14(41e−4t),y=14(41e−4t), never

49.

Solution changes from increasing to decreasing at y(0)=0y(0)=0

51.

Solution changes from increasing to decreasing at y(0)=0y(0)=0

53.

v(t)=−32t+av(t)=−32t+a

55.

00 ft/s

57.

4.864.86 meters

59.

x=50t15π2cos(πt)+3π2,2x=50t15π2cos(πt)+3π2,2 hours 11 minute

61.

y=4e3ty=4e3t

63.

y=12t+t2y=12t+t2

65.

y=1k(ekt1)y=1k(ekt1) and y=xy=x

Section 4.2 Exercises

67.


A graph of the given direction field with a flat line drawn on the axis. The arrows point up for y < 0 and down for y > 0. The closer they are to the x axis, the more horizontal the arrows are, and the further away they are, the more vertical they become.
69.

y=0y=0 is a stable equilibrium

71.


A direction field with horizontal arrows at y = 0 and y = 2. The arrows point up for y > 2 and for y < 0. The arrows point down for 0 < y < 2. The closer the arrows are to these lines, the more horizontal they are, and the further away, the more vertical the arrows are. A solution is sketched that follows y = 2 in quadrant two, goes through (0, 1), and then follows the x axis.
73.

y=0y=0 is a stable equilibrium and y=2y=2 is unstable

75.


A direction field over the four quadrants. As t goes from 0 to infinity, the arrows become more and more vertical after being horizontal closer to x = 0.
77.


A direction field over [-2, 2] in the x and y axes. The arrows point slightly down and to the right over [-2, 0] and gradually become vertical over [0, 2].
79.


A direction field with horizontal arrows pointing to the right at y = 1 and y = -1. The arrows point up for y < -1 and y > 1. The arrows point down for -1 < y < 1. The closer the arrows are to these lines, the more horizontal they are, and the further away they are, the more vertical they are.
81.


A direction field with arrows pointing down and to the right for nearly all points in [-2, 2] on the x and y axes. Close to the origin, the arrows become more horizontal, point to the upper right, become more horizontal, and then point down to the right again.
83.


A direction field with horizontal arrows pointing to the right on the x axis and x = -3. Above the x axis and for x < -3, the arrows point down. For x > -3, the arrows point up. Below the x axis and for x < -3, the arrows point up. For x > -3, the arrows point down. The further away from the x axis and x = -3, the arrows become more vertical, and the closer they become, the more horizontal they become.
85.

E

87.

A

89.

B

91.

A

93.

C

95.

2.24,2.24, exact: 33

97.

7.739364,7.739364, exact: 5(e1)5(e1)

99.

−0.2535−0.2535 exact: 00

101.

1.345,1.345, exact: 1ln(2)1ln(2)

103.

−4,−4, exact: 1/21/2

105.


A direction field with horizontal arrows pointing to the right on the x axis and at y = 4. The arrows below the x axis and above y = 4 point down and to the right. The arrows between the x axis and y = 4 point up and to the right.
107.

y=2et2/2y=2et2/2

109.

22

111.

3.27563.2756

113.

2e2e

Step Size Error
h=1h=1 0.39350.3935
h=10h=10 0.061630.06163
h=100h=100 0.0066120.006612
h=1000h=1000 0.00066610.0006661
115.


A direction field with horizontal arrows pointing to the right on the x axis. Above the x axis, the arrows point down and to the right. Below the x axis, the arrows point up and to the right. The closer the arrows are to the x axis, the more horizontal the arrows are, and the further away they are from the x axis, the more vertical the arrows are.
117.

4.0741e−104.0741e−10

Section 4.3 Exercises

119.

y=et1y=et1

121.

y=1ety=1et

123.

y=Cxe−1/xy=Cxe−1/x

125.

y=1Cx2y=1Cx2

127.

y=2C+lnxy=2C+lnx

129.

y=Cex(x+1)+1y=Cex(x+1)+1

131.

y=sin(lnt+C)y=sin(lnt+C)

133.

y=ln(ex)y=ln(ex)

135.

y=12ex2y=12ex2

137.

y=tanh−1(x22)y=tanh−1(x22)

139.

x=sin(ttlnt)x=sin(ttlnt)

141.

y=ln(ln(5))ln(25x)y=ln(ln(5))ln(25x)

143.

y=Ce−2x+12y=Ce−2x+12

A direction field with horizontal arrows pointing to the right at y = 0.5. Above 0.5, the arrows slope down and to the right and are increasingly vertical the further they are from y = 0.5 Below0.5, the arrows slope up and to the right and are increasingly vertical the further they are from y = 0.5.
145.

y=12Cexy=12Cex

A direction field with arrows pointing to the right. They are horizontal at the y axis. The further the arrows are from the axis, the more vertical they become. They point up above the x axis and down below the x axis.
147.

y=Cexxxy=Cexxx

A direction field with arrows pointing to the right. The arrows are flat on y = 1. The further the arrows are from that, the steeper they become. They point up above that line and down below that line.
149.

y=rd(1edt)y=rd(1edt)

151.

y(t)=109ex/50y(t)=109ex/50

153.

134.3134.3 kilograms

155.

720720 seconds

157.

1212 hours 1414 minutes

159.

T(t)=20+50e−0.125tT(t)=20+50e−0.125t

161.

T(t)=20+38.5e−0.125tT(t)=20+38.5e−0.125t

163.

y=(c+ba)eaxbay=(c+ba)eaxba

165.

y(t)=cL+(IcL)ert/Ly(t)=cL+(IcL)ert/L

167.

y=40(1e−0.1t),40y=40(1e−0.1t),40 g/cm2

Section 4.4 Exercises

169.


A direction field with arrows pointing to the right. The arrows are horizontal along the x axis. The arrows point down above the x axis and below the x axis. The further away the arrows are from the x axis, the more vertical the lines become.


P=0P=0 semi-stable

171.

P=10e10xe10x+4P=10e10xe10x+4

173.

P(t)=10000e0.02t150+50e0.02tP(t)=10000e0.02t150+50e0.02t

175.

6969 hours 55 minutes

177.

77 years 22 months

179.


A direction field with arrows down for P < 1,000, pointing up for 1,000 < P < 8,500, and pointing down for P > 8,500. Right above P = 8,500, the arrows point down and to the right.
181.


A direction field with arrows pointing down and to the right. Around y = 4,000, the arrows are more horizontal. The further the arrows are from this line, the more vertical the arrows become.


P1P1 semi-stable

183.


A direction field with arrows pointing up for P < 10,000 and arrows pointing down for P > 10,000.


P2>0P2>0 stable

185.


A direction field with arrows pointing to the right at P = 0. Below 0, the arrows point down and to the right. Above 0, the arrows point down and to the right. The further away from 0, the more vertical the arrows become.


P1=0P1=0 is semi-stable

187.

y=−204×10−60.002e0.01ty=−204×10−60.002e0.01t

189.


A direction field with arrows pointing horizontally to the right along y = 2 and y = 10. For P < 2, the arrows point down and to the right. For 2 < P < 10, the arrows point up and to the right. For P > 10, the arrows point down and to the right. The further the arrows are from 2 and 10, the steeper they become, and the closer they are from 2 and 10, the more horizontal the arrows become.
191.

P(t)=850+500e0.009t85+5e0.009tP(t)=850+500e0.009t85+5e0.009t

193.

1313 years months

195.


A direction field with arrows pointing down and to the right for P < 0, up for 0 < P < 1,000, and down for P > 1,000. The further the arrows are from P = 0 and P = 1,000, the more vertical they become, and the closer they are, the more horizontal they are.
197.

31.46531.465 days

199.

September 20082008

201.

K+T2K+T2

203.

r=0.0405r=0.0405

205.

α=0.0081α=0.0081

207.

Logistic: 361,361, Threshold: 436,436, Gompertz: 309.309.

Section 4.5 Exercises

209.

Yes

211.

Yes

213.

yx3y=sinxyx3y=sinx

215.

y+(3x+2)xy=exy+(3x+2)xy=ex

217.

dydtyx(x+1)=0dydtyx(x+1)=0

219.

exex

221.

ln(coshx)ln(coshx)

223.

y=Ce3x23y=Ce3x23

225.

y=Cx3+6x2y=Cx3+6x2

227.

y=Cex2/23y=Cex2/23

229.

y=Ctan(x2)2x+4tan(x2)ln(sin(x2))y=Ctan(x2)2x+4tan(x2)ln(sin(x2))

231.

y=Cx3x2y=Cx3x2

233.

y=C(x+2)2+12y=C(x+2)2+12

235.

y=Cx+2sin(3t)y=Cx+2sin(3t)

237.

y=C(x+1)3x22x1y=C(x+1)3x22x1

239.

y=Cesinh−1x2y=Cesinh−1x2

241.

y=x+4ex1y=x+4ex1

243.

y=3x2(x21)y=3x2(x21)

245.

y=1etan−1xy=1etan−1x

247.

y=(x+2)ln(x+22)y=(x+2)ln(x+22)

249.

y=2e2x2x2x1y=2e2x2x2x1

251.

v(t)=gmk(1ekt/m)v(t)=gmk(1ekt/m)

253.

40.45140.451 seconds

255.

gmkgmk

257.

y=Cexa(x+1)y=Cexa(x+1)

259.

y=Cex2/2ay=Cex2/2a

261.

y=ektetk1y=ektetk1

Chapter Review Exercises

263.

F

265.

T

267.

y(x)=2xln(2)+xcos−1x1x2+Cy(x)=2xln(2)+xcos−1x1x2+C

269.

y(x)=ln(Ccosx)y(x)=ln(Ccosx)

271.

y(x)=eeC+xy(x)=eeC+x

273.

y(x)=4+32x2+2xsinxy(x)=4+32x2+2xsinx

275.

y(x)=21+3(x2+2sinx)y(x)=21+3(x2+2sinx)

277.

y(x)=−2x22x1323e3xy(x)=−2x22x1323e3x

279.


A direction field with arrows pointing up and to the right along a logarithmic curve that approaches negative infinity as x goes to zero and increases as x goes to infinity.


y(x)=Cex+lnxy(x)=Cex+lnx

281.

Euler: 0.6939,0.6939, exact solution: y(x)=3xe−2x2+ln(3)y(x)=3xe−2x2+ln(3)

283.

40494049 second

285.

x(t)=5000+2459493t2459e5/3t,t=307.8x(t)=5000+2459493t2459e5/3t,t=307.8 seconds

287.

T(t)=200(1et/1000)T(t)=200(1et/1000)

289.

P(t)=1600000e0.02t9840+160e0.02tP(t)=1600000e0.02t9840+160e0.02t

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