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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

7.1


A curved line going from (−7, 8) through (−1, 0) and (2, −1) to (8, 3) with arrow going in that order. The point (−7, 8) is marked t = −3, the point (2, −1) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 − 1, and −3 ≤ t ≤ 2.
7.2

x=2+3y+1,x=2+3y+1, or y=−1+3x2.y=−1+3x2. This equation describes a portion of a rectangular hyperbola centered at (2,−1).(2,−1).

A curved line going from (3.5, 1) to (2.5, 5) with arrow going in that order. The point (3.5, 1) is marked t = 2 and the point (2.5, 5) is marked t = 6. On the graph there are also written three equations: x(t) = 2 + 3/t, y(t) = t − 1, and 2 ≤ t ≤ 6.
7.3

One possibility is x(t)=t,y(t)=t2+2t.x(t)=t,y(t)=t2+2t. Another possibility is x(t)=2t3,y(t)=(2t3)2+2(2t3)=4t28t+3.x(t)=2t3,y(t)=(2t3)2+2(2t3)=4t28t+3.

There are, in fact, an infinite number of possibilities.

7.4

x(t)=2t4x(t)=2t4 and y(t)=6t26,y(t)=6t26, so dydx=6t262t4=3t23t2.dydx=6t262t4=3t23t2.
This expression is undefined when t=2t=2 and equal to zero when t=±1.t=±1.

A curve going from (12, −4) through the origin and (−4, 0) to (−3, 36) with arrows in that order. The point (12, −4) is marked t = −2 and the point (−3, 36) is marked t = 3. On the graph there are also written three equations: x(t) = t2 – 4t, y(t) = 2t3 – 6t, and −2 ≤ t ≤ 3.
7.5

The equation of the tangent line is y=24x+100.y=24x+100.

7.6

d2ydx2=3t212t+32(t2)3.d2ydx2=3t212t+32(t2)3. Critical points (5,4),(−3,−4),and(−4,6).(5,4),(−3,−4),and(−4,6).

7.7

A=3πA=3π (Note that the integral formula actually yields a negative answer. This is due to the fact that x(t)x(t) is a decreasing function over the interval [0,2π];[0,2π]; that is, the curve is traced from right to left.)

7.8

s=2(103/223/2)57.589s=2(103/223/2)57.589

7.9

A=π(49413+128)1215A=π(49413+128)1215

7.10

(82,5π4)(82,5π4) and (−2,23)(−2,23)

7.12


The graph of r = 4 + 4 cosθ is given. It vaguely looks look a heart tipped on its side with a rounded bottom instead of a pointed one. Specifically, the graph starts at the origin, moves into the second quadrant and increases to a rounded circle-like figure. The graph is symmetric about the x axis, so it continues its rounded circle-like figure, goes into the third quadrant, and comes to a point at the origin.


The name of this shape is a cardioid, which we will study further later in this section.

7.13

y=x2,y=x2, which is the equation of a parabola opening upward.

7.14

Symmetric with respect to the polar axis.

A three-petaled rose is graphed with equation r = 2 cos(3θ). Each petal starts at the origin and reaches a maximum distance from the origin of 2.
7.15

A=3π/2A=3π/2

7.16

A=4π3+43A=4π3+43

7.17

s=3πs=3π

7.18

x=2(y+3)22x=2(y+3)22

A parabola is drawn with vertex at (−2, −3) and opening to the right with equation x = 2(y + 3)2 – 2. The focus is drawn at (0, −3). The directrix is drawn at x = −4.
7.19

(x+1)216+(y2)29=1(x+1)216+(y2)29=1

An ellipse is drawn with equation 9x2 + 16y2 + 18x – 64y − 71 = 0. It has center at (−1, 2), touches the x axis at (2, 0) and (−4, 0), and touches the y axis near (0, −1) and (0, 5).
7.20

(y+2)29(x1)24=1.(y+2)29(x1)24=1. This is a vertical hyperbola. Asymptotes y=−2±32(x1).y=−2±32(x1).

A hyperbola is drawn with equation 4y2 – 9x2 + 16x + 18y – 29 = 0. It has center at (1, −2), and the hyperbolas are open to the top and bottom.
7.21

e=ca=7471.229e=ca=7471.229

7.22

Here e=0.8e=0.8 and p=5.p=5. This conic section is an ellipse.

Graph of an ellipse with equation r = 4/(1 – 0.8 sinθ), center near (0, 11), major axis roughly 22, and minor axis roughly 12.
7.23

The conic is a hyperbola and the angle of rotation of the axes is θ=22.5°.θ=22.5°.

Section 7.1 Exercises

1.


A parabola open to the right with (−1, 0) being the point furthest the left with arrow going from the bottom through (−1, 0) and up.


orientation: bottom to top

3.


A straight line passing through (0, −3) and (6, 0) with arrow pointing up and to the right.


orientation: left to right

5.

y=x24+1y=x24+1

Half a parabola starting at the origin and passing through (2, 2) with arrow pointed up and to the right.
7.


A curve going through (1, 0) and (0, 3) with arrow pointing up and to the left.
9.


A graph with asymptotes at the x and y axes. There is a portion of the graph in the third quadrant with arrow pointing down and to the right. There is a portion of the graph in the first quadrant with arrow pointing down and to the right.
11.


An ellipse with minor axis vertical and of length 8 and major axis horizontal and of length 12 that is centered at the origin. The arrows go counterclockwise.
13.


An ellipse in the fourth quadrant with minor axis horizontal and of length 4 and major axis vertical and of length 6. The arrows go clockwise.
15.


A graph with asymptotes at y = x and y = −x. The first part of the graph occurs in the second and third quadrants with vertex at (−1, 0). The second part of the graph occurs in the first and fourth quadrants with vertex as (1, 0).


Asymptotes are y=xy=x and y=xy=x

17.


A curve starting slightly above the origin and increasing to the right with arrow pointing up and to the right.
19.


A curve with asymptote being the y axis. The curve starts in the fourth quadrant and increases rapidly through (1, 0) at which point is increases much more slowly.
21.

x=4y21;x=4y21; domain: x[1,).x[1,).

23.

x216+y29=1;x216+y29=1; domain x[−4,4].x[−4,4].

25.

y=3x+2;y=3x+2; domain: all real numbers.

27.

(x1)2+(y3)2=1;(x1)2+(y3)2=1; domain: x[0,2].x[0,2].

29.

y=x21;y=x21; domain: x[−1,1].x[−1,1].

31.

y2=1x2;y2=1x2; domain: x[2,)(,−2].x[2,)(,−2].

33.

y=lnx;y=lnx; domain: x(0,).x(0,).

35.

y=lnx;y=lnx; domain: x(0,).x(0,).

37.

x2+y2=4;x2+y2=4; domain: x[−2,2].x[−2,2].

39.

line

41.

parabola

43.

circle

45.

ellipse

47.

hyperbola

51.

The equations represent a cycloid.

A graph starting at (−6, 0) increasing rapidly to a sharp point at (−3, 2) and then decreasing rapidly to the origin. The graph is symmetric about the y axis, so the graph increases rapidly to (3, 2) before decreasing rapidly to (6, 0).
53.


A graph starting at roughly (−6, 0) increasing to a rounded point and then decreasing to roughly (0, −0.5). The graph is symmetric about the y axis, so the graph increases to a rounded point before decreasing to roughly (6, 0).
55.

22,092 meters at approximately 51 seconds.

57.


A graph with asymptotes roughly near y = x and y = −x. The first part of the graph is in the first and second quadrants with vertex near (0, 3). The second part of the graph is in the third and fourth quadrants with vertex near (0, −3).
59.


A graph starting at roughly (−6, −1) decreasing to a minimum in the third quadrant near (−1, −4.8) increasing through roughly (0, −4.7) and (3, 0) to a maximum near (1, 1.9) before decreasing through (0, 1.5) to the origin. The graph is symmetric about the y axis, so the graph increases through (0, 1.5) to a maximum in the second quadrant, decreases again through (0, −4.7), and then increases to (6, −1).
61.


A vaguely parabolic graph with vertex at the origin that is open to the right.

Section 7.2 Exercises

63.

0

65.

−35−35

67.

Slope=0;Slope=0; y=8.y=8.

69.

Slope is undefined; x=2.x=2.

71.

t=arctan(−2);t=arctan(−2); (45,−85).(45,−85).

73.

No points possible; undefined expression.

75.

y=(2e)x+3y=(2e)x+3

77.

y=–2x+3y=–2x+3

79.

π4,5π4,3π4,7π4π4,5π4,3π4,7π4

81.

dydx=tan(t)dydx=tan(t)

83.

dydx=34dydx=34 and d2ydx2=0,d2ydx2=0, so the curve is neither concave up nor concave down at t=3.t=3. Therefore the graph is linear and has a constant slope but no concavity.

85.

dydx=4,d2ydx2=−63;dydx=4,d2ydx2=−63; the curve is concave down at θ=π6.θ=π6.

87.

No horizontal tangents. Vertical tangents at (1,0),(−1,0).(1,0),(−1,0).

89.

sec3(πt)sec3(πt)

91.

Horizontal (0,−9);(0,−9); vertical (±2,−6).(±2,−6).

93.

1

95.

0

97.

4

99.

Concave up on t>0.t>0.

101.

e12 1 2 e12 1 2

103.

3π23π2

105.

6πa26πa2

107.

2πab2πab

109.

13(221)13(221)

111.

7.0757.075

113.

6a6a

115.

6262

119.

2π(24713+64)12152π(24713+64)1215

121.

59.101

123.

8π3(17171)8π3(17171)

Section 7.3 Exercises

125.


On the polar coordinate plane, a ray is drawn from the origin marking π/6 and a point is drawn when this line crosses the circle with radius 3.
127.


On the polar coordinate plane, a ray is drawn from the origin marking 7π/6 and a point is drawn when this line crosses the circle with radius 0, that is, it marks the origin.
129.


On the polar coordinate plane, a ray is drawn from the origin marking π/4 and a point is drawn when this line crosses the circle with radius 1.
131.


On the polar coordinate plane, a ray is drawn from the origin marking π/2 and a point is drawn when this line crosses the circle with radius 1.
133.

B(3,π3)B(−3,2π3)B(3,π3)B(−3,2π3)

135.

D(5,7π6)D(−5,π6)D(5,7π6)D(−5,π6)

137.

(5,−0.927)(−5,−0.927+π)(5,−0.927)(−5,−0.927+π)

139.

(10,−0.927)(−10,−0.927+π)(10,−0.927)(−10,−0.927+π)

141.

(23,−0.524)(−23,−0.524+π)(23,−0.524)(−23,−0.524+π)

143.

(3,−1)(3,−1)

145.

(32,−12)(32,−12)

147.

(0,0)(0,0)

149.

Symmetry with respect to the x-axis, y-axis, and origin.

151.

Symmetric with respect to x-axis only.

153.

Symmetry with respect to x-axis only.

155.

Line y=xy=x

157.

y=1y=1

159.

Hyperbola; polar form r2cos(2θ)=16r2cos(2θ)=16 or r2=16secθ.r2=16secθ.

A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.
161.

r=23cosθsinθr=23cosθsinθ

A straight line with slope 3 and y intercept −2.
163.

x2+y2=4yx2+y2=4y

A circle of radius 2 with center at (2, π/2).
165.

xtanx2+y2=yxtanx2+y2=y

A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.
167.


A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.


y-axis symmetry

169.


A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.


y-axis symmetry

171.


A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.


x- and y-axis symmetry and symmetry about the pole

173.


A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.


x-axis symmetry

175.


The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.


x- and y-axis symmetry and symmetry about the pole

177.


A spiral that starts at the origin crossing the line θ = π/2 between 3 and 4, θ = π between 6 and 7, θ = 3π/2 between 9 and 10, θ = 0 between 12 and 13, θ = π/2 between 15 and 16, and θ = π between 18 and 19.


no symmetry

179.


A line that crosses the y axis at roughly 3 and has slope roughly 3/2.


a line

181.


A geometric shape that resembles a butterfly with larger wings in the first and second quadrants, smaller wings in the third and fourth quadrants, a body along the θ = π/2 line and legs along the θ = 0 and π lines.
183.


A line with θ = 120°.
185.


A spiral that starts in the third quadrant.
187.

Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

Section 7.4 Exercises

189.

920πsin2θdθ920πsin2θdθ

191.

320π/2sin2(2θ)dθ320π/2sin2(2θ)dθ

193.

12π2π(1sinθ)2dθ12π2π(1sinθ)2dθ

195.

sin−1(2/3)π/2(23sinθ)2dθsin−1(2/3)π/2(23sinθ)2dθ

197.

0π(12cosθ)2dθ0π/3(12cosθ)2dθ0π(12cosθ)2dθ0π/3(12cosθ)2dθ

199.

40π/3dθ+16π/3π/2(cos2θ)dθ40π/3dθ+16π/3π/2(cos2θ)dθ

201.

9π9π

203.

9π49π4

205.

9π89π8

207.

18π273218π2732

209.

43(4π33)43(4π33)

211.

32(4π33)32(4π33)

213.

2π42π4

215.

02π(1+sinθ)2+cos2θdθ02π(1+sinθ)2+cos2θdθ

217.

201eθdθ201eθdθ

219.

103(e61)103(e61)

221.

32

223.

6.238

225.

2

227.

4.39

229.

A=π(22)2=π2and120π(1+2sinθcosθ)dθ=π2A=π(22)2=π2and120π(1+2sinθcosθ)dθ=π2

231.

C=2π(32)=3πand0π3dθ=3πC=2π(32)=3πand0π3dθ=3π

233.

C=2π(5)=10πand0π10dθ=10πC=2π(5)=10πand0π10dθ=10π

235.

dydx=f(θ)sinθ+f(θ)cosθf(θ)cosθf(θ)sinθdydx=f(θ)sinθ+f(θ)cosθf(θ)cosθf(θ)sinθ

237.

The slope is 13.13.

239.

The slope is 0.

241.

At (4,0),(4,0), the slope is undefined. At (−4,π2),(−4,π2), the slope is 0.

243.

The slope is undefined at θ=π4.θ=π4.

245.

Slope = −1.

247.

Slope is −2π.−2π.

249.

Calculator answer: −0.836.

251.

Horizontal tangent at (±2,π6),(±2,π6), (±2,π6).(±2,π6).

253.

Horizontal tangents at π2,7π6,11π6.π2,7π6,11π6. Vertical tangents at π6,5π6π6,5π6 and also at the pole (0,0).(0,0).

Section 7.5 Exercises

255.

y2=16xy2=16x

257.

x2=2yx2=2y

259.

x2=−4(y3)x2=−4(y3)

261.

(x+3)2=8(y3)(x+3)2=8(y3)

263.

x216+y212=1x216+y212=1

265.

x213+y24=1x213+y24=1

267.

(y1)216+(x+3)212=1(y1)216+(x+3)212=1

269.

x216+y212=1x216+y212=1

271.

x225y211=1x225y211=1

273.

x27y29=1x27y29=1

275.

(y+2)24(x+2)232=1(y+2)24(x+2)232=1

277.

x24y232=1x24y232=1

279.

e=1,e=1, parabola

281.

e=12,e=12, ellipse

283.

e=3,e=3, hyperbola

285.

r=45+cosθr=45+cosθ

287.

r=41+2sinθr=41+2sinθ

289.


Graph of a parabola open down with center at the origin.
291.


Graph of a parabola open to the left with center near the origin.
293.


Graph of an ellipse with center near (8, 0), major axis horizontal and roughly 18, and minor axis slightly more than 12.
295.


Graph of an circle with center near (0, −1.5) and radius near 2.5.
297.


Graph of a circle with center (0, −0.5) and radius 1.
299.


Graph of an ellipse with center the origin and with major axis vertical and of length 8 and minor axis of length 4.
301.


Graph of a hyperbola with center the origin and with the two halves open to the left and right. The vertices are on the x axis at ±2.
303.


Graph of a parabola with vertex the origin and open up.
305.


Graph of a parabola with vertex the origin and open to the right.
307.

Hyperbola

309.

Ellipse

311.

Ellipse

313.

At the point 2.25 feet above the vertex.

315.

0.5625 feet

317.

Length is 96 feet and height is approximately 26.53 feet.

319.

r=2.6161+0.995cosθr=2.6161+0.995cosθ

321.

r=5.1921+0.0484cosθr=5.1921+0.0484cosθ

Chapter Review Exercises

323.

True.

325.

False. Imagine y=t+1,y=t+1, x=t+1.x=t+1.

327.


Graph of a curve starting at (1, 0) and decreasing into the fourth quadrant.


y=1x3y=1x3

329.


Graph of an ellipse with center (0, 1), major axis horizontal and of length 8, and minor axis of length 2.


x216+(y1)2=1x216+(y1)2=1

331.


Graph of a five-petaled rose with initial petal at θ = 0.


Symmetric about polar axis

333.

r2=4sin2θcos2θr2=4sin2θcos2θ

335.


Graph of a peanut-shaped figure, with y intercepts at ±2 and x intercepts at ±4. The tangent line occurs in the second quadrant.


y=322+15(x+322)y=322+15(x+322)

337.

e22e22

339.

910910

341.

(y+5)2=−8x+32(y+5)2=−8x+32

343.

(y+1)216(x+2)29=1(y+1)216(x+2)29=1

345.

e=23,e=23, ellipse

Graph of an ellipse with center near (1.5, 0), major axis nearly 5 and horizontal, and minor axis nearly 4.
347.

y219.032+x219.632=1,y219.032+x219.632=1, e=0.2447e=0.2447

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