,
,Student Solutions Manual
Tamas Wiandt
Rochester Institute of Technology
to accompany
CALCULUS
Early Transcendentals
Single Variable
Tenth Edition
Howard Anton
Drexel University
Irl C. Bivens
Davidson College
Stephen L. Davis
Davidson College
John Wiley& Sons, Inc.
,PUBLISHER Laurie Rosatone
ACQUISITIONS EDITOR David Dietz
PROJECT EDITOR Ellen Keohane
ASSISTANT CONTENT EDITOR Beth Pearson
EDITORIAL ASSISTANT Jacqueline Sinacori
CONTENT MANAGER Karoline Luciano
SENIOR PRODUCTION EDITOR Kerry Weinstein
SENIOR PHOTO EDITOR Sheena Goldstein
COVER DESIGNER Madelyn Lesure
COVER PHOTO © David Henderson/Getty Images
Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and
understanding for more than 200 years, helping people around the world meet their needs and
fulfill their aspirations. Our company is built on a foundation of principles that include
responsibility to the communities we serve and where we live and work. In 2008, we launched a
Corporate Citizenship Initiative, a global effort to address the environmental, social, economic,
and ethical challenges we face in our business. Among the issues we are addressing are carbon
impact, paper specifications and procurement, ethical conduct within our business and among
our vendors, and community and charitable support. For more information, please visit our
website: www.wiley.com/go/citizenship.
Copyright ¤ 2012 John Wiley & Sons, Inc. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted
under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-
copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923
(Web site: www.copyright.com). Requests to the Publisher for permission should be addressed to
the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-
5774, (201) 748-6011, fax (201) 748-6008, or online at: www.wiley.com/go/permissions.
ISBN 978-1-118-17381-7
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
, Contents
Chapter 0. Before Calculus ………..…………………………………………………………………………..……. 1
Chapter 1. Limits and Continuity ……………………………………………………………………………….. 21
Chapter 2. The Derivative ……………………………………………………………………………………..…….39
Chapter 3. Topics in Differentiation ……………………………..………………………………………..…….59
Chapter 4. The Derivative in Graphing and Applications ……………………………………..………. 81
Chapter 5. Integration …………………………………………………………………………………………..…… 127
Chapter 6. Applications of the Definite Integral in Geometry, Science, and Engineering… 159
Chapter 7. Principles of Integral Evaluation ……………………………………………………………….. 189
Chapter 8. Mathematical Modeling with Differential Equations …………………………………… 217
Chapter 9. Infinite Series ……………………………………………………………………………………..…….. 229
Chapter 10. Parametric and Polar Curves; Conic Sections ……………………………………….…….. 255
Appendix A. Graphing Functions Using Calculators and Computer Algebra Systems .………. 287
Appendix B. Trigonometry Review ……………………………………………………………………………….. 293
Appendix C. Solving Polynomial Equations …………………………………………………………………… 297
,
,Before Calculus
Exercise Set 0.1
1. (a) −2.9, −2.0, 2.35, 2.9 (b) None (c) y = 0 (d) −1.75 ≤ x ≤ 2.15, x = −3, x = 3
(e) ymax = 2.8 at x = −2.6; ymin = −2.2 at x = 1.2
3. (a) Yes (b) Yes (c) No (vertical line test fails) (d) No (vertical line test fails)
5. (a) 1999, $47,700 (b) 1993, $41,600
(c) The slope between 2000 and 2001 is steeper than the slope between 2001 and 2002, so the median income was
declining more rapidly during the first year of the 2-year period.
2 2 2 2
√
√ f2(0) = 3(0) − 2 = −2; f2 (2) = 3(2)2 − 2 = 10; f (−2) = 3(−2) − 2 = 10; f (3) = 3(3) − 2 = 25; f ( 2) =
7. (a)
3( 2) − 2 = 4; f (3t) = 3(3t) − 2 = 27t − 2.
√ √
(b) f (0) = 2(0) = 0; f (2) = 2(2) = 4; f (−2) = 2(−2) = −4; f (3) = 2(3) = 6; f ( 2) = 2 2; f (3t) = 1/(3t) for
t > 1 and f (3t) = 6t for t ≤ 1.
9. (a) Natural domain: x = 3. Range: y = 0. (b) Natural domain: x = 0. Range: {1, −1}.
√ √
(c) Natural domain: x ≤ − 3 or x ≥ 3. Range: y ≥ 0.
√
(d) x2 − 2x + 5 = (x − 1)2 + 4 ≥ 4. So G(x) is defined for all x, and is ≥ 4 = 2. Natural domain: all x. Range:
y ≥ 2.
(e) Natural domain: sin x = 1, so x = (2n+ 12 )π, n = 0, ±1, ±2, . . .. For such x, −1 ≤ sin x < 1, so 0 < 1−sin x ≤ 2,
1 1 1
and 1−sin x ≥ 2 . Range: y ≥ 2 .
2
−4
(f ) Division by 0 occurs for x = 2. For all other x, xx−2 = x + 2, which is nonnegative for x ≥ −2. Natural
√ √
domain: [−2, 2) ∪ (2, +∞). The range of x + 2 is [0, +∞). But we must exclude x = 2, for which x + 2 = 2.
Range: [0, 2) ∪ (2, +∞).
11. (a) The curve is broken whenever someone is born or someone dies.
(b) C decreases for eight hours, increases rapidly (but continuously), and then repeats.
h
t
13.
1
,2 Chapter 0
√
15. Yes. y = 25 − x2 .
√
2
17. Yes. y = √25 − x , −5 ≤ x ≤ 0
− 25 − x2 , 0<x≤5
19. False. E.g. the graph of x2 − 1 crosses the x-axis at x = 1 and x = −1.
21. False. The range also includes 0.
23. (a) x = 2, 4 (b) None (c) x ≤ 2; 4 ≤ x (d) ymin = −1; no maximum value.
25. The cosine of θ is (L − h)/L (side adjacent over hypotenuse), so h = L(1 − cos θ).
27. (a) If x< 0, then |x| = −x so f (x) = −x + 3x + 1 = 2x + 1. If x ≥ 0, then |x| = x so f (x) = x + 3x + 1 = 4x + 1;
2x + 1, x < 0
f (x) =
4x + 1, x ≥ 0
(b) If x < 0, then |x| = −x and |x − 1| = 1 − x so g(x) = −x + (1 − x) = 1 − 2x. If 0 ≤ x < 1, then |x| = x and
|x − 1| = 1 − x so g(x) = x + (1 − x) = 1. If x ≥ 1, then |x| = x and |x − 1| = x − 1 so g(x) = x + (x − 1) = 2x − 1;
⎧
⎨ 1 − 2x, x<0
g(x) = 1, 0≤x<1
⎩
2x − 1, x≥1
29. (a) V = (8 − 2x)(15 − 2x)x (b) 0 < x < 4
100
0 4
(c) 0 0 < V ≤ 91, approximately
(d) As x increases, V increases and then decreases; the maximum value occurs when x is about 1.7.
31. (a) The side adjacent to the building has length x, so L = x + 2y. (b) A = xy = 1000, so L = x + 2000/x.
120
20 80
(c) 0 < x ≤ 100 (d) 80 x ≈ 44.72 ft, y ≈ 22.36 ft
500 500 10
33. (a) V = 500 = πr2 h, so h = 2
. Then C = (0.02)(2)πr2 + (0.01)2πrh = 0.04πr2 + 0.02πr 2 = 0.04πr2 + ;
πr πr r
Cmin ≈ 4.39 cents at r ≈ 3.4 cm, h ≈ 13.7 cm.
10
(b) C = (0.02)(2)(2r)2 + (0.01)2πrh = 0.16r2 + . Since 0.04π < 0.16, the top and bottom now get more weight.
r
Since they cost more, we diminish their sizes in the solution, and the cans become taller.
(c) r ≈ 3.1 cm, h ≈ 16.0 cm, C ≈ 4.76 cents.
, Exercise Set 0.2 3
35. (i) x = 1, −2 causes division by zero. (ii) g(x) = x + 1, all x.
37. (a) 25◦ F (b) 13◦ F (c) 5◦ F
39. If v = 48 then −60 = WCT ≈ 1.4157T − 30.6763; thus T ≈ 15◦ F when WCT = −10.
Exercise Set 0.2
y y
1 2
x
1
–1 0 1 2
x
1. (a) –1 (b) 1 2 3
y y
1
2
x x
(c) –1 1 2 (d) –4 –2 2
y
y 1
1
x
x
1
–2 –1 1 2
–1 –1
3. (a) (b)
1 y y
1
x x
–1 1 2 3 –1 1 2 3
–1 –1
(c) (d)
5. Translate left 1 unit, stretch vertically by a factor of 2, reflect over x-axis, translate down 3 units.
y x
–6 –2 2 6
–20
–60
7. y = (x + 3)2 − 9; translate left 3 units and down 9 units.
,Student Solutions Manual
Tamas Wiandt
Rochester Institute of Technology
to accompany
CALCULUS
Early Transcendentals
Single Variable
Tenth Edition
Howard Anton
Drexel University
Irl C. Bivens
Davidson College
Stephen L. Davis
Davidson College
John Wiley& Sons, Inc.
,PUBLISHER Laurie Rosatone
ACQUISITIONS EDITOR David Dietz
PROJECT EDITOR Ellen Keohane
ASSISTANT CONTENT EDITOR Beth Pearson
EDITORIAL ASSISTANT Jacqueline Sinacori
CONTENT MANAGER Karoline Luciano
SENIOR PRODUCTION EDITOR Kerry Weinstein
SENIOR PHOTO EDITOR Sheena Goldstein
COVER DESIGNER Madelyn Lesure
COVER PHOTO © David Henderson/Getty Images
Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and
understanding for more than 200 years, helping people around the world meet their needs and
fulfill their aspirations. Our company is built on a foundation of principles that include
responsibility to the communities we serve and where we live and work. In 2008, we launched a
Corporate Citizenship Initiative, a global effort to address the environmental, social, economic,
and ethical challenges we face in our business. Among the issues we are addressing are carbon
impact, paper specifications and procurement, ethical conduct within our business and among
our vendors, and community and charitable support. For more information, please visit our
website: www.wiley.com/go/citizenship.
Copyright ¤ 2012 John Wiley & Sons, Inc. All rights reserved. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted in any form or by any means,
electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted
under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior
written permission of the Publisher, or authorization through payment of the appropriate per-
copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923
(Web site: www.copyright.com). Requests to the Publisher for permission should be addressed to
the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-
5774, (201) 748-6011, fax (201) 748-6008, or online at: www.wiley.com/go/permissions.
ISBN 978-1-118-17381-7
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
, Contents
Chapter 0. Before Calculus ………..…………………………………………………………………………..……. 1
Chapter 1. Limits and Continuity ……………………………………………………………………………….. 21
Chapter 2. The Derivative ……………………………………………………………………………………..…….39
Chapter 3. Topics in Differentiation ……………………………..………………………………………..…….59
Chapter 4. The Derivative in Graphing and Applications ……………………………………..………. 81
Chapter 5. Integration …………………………………………………………………………………………..…… 127
Chapter 6. Applications of the Definite Integral in Geometry, Science, and Engineering… 159
Chapter 7. Principles of Integral Evaluation ……………………………………………………………….. 189
Chapter 8. Mathematical Modeling with Differential Equations …………………………………… 217
Chapter 9. Infinite Series ……………………………………………………………………………………..…….. 229
Chapter 10. Parametric and Polar Curves; Conic Sections ……………………………………….…….. 255
Appendix A. Graphing Functions Using Calculators and Computer Algebra Systems .………. 287
Appendix B. Trigonometry Review ……………………………………………………………………………….. 293
Appendix C. Solving Polynomial Equations …………………………………………………………………… 297
,
,Before Calculus
Exercise Set 0.1
1. (a) −2.9, −2.0, 2.35, 2.9 (b) None (c) y = 0 (d) −1.75 ≤ x ≤ 2.15, x = −3, x = 3
(e) ymax = 2.8 at x = −2.6; ymin = −2.2 at x = 1.2
3. (a) Yes (b) Yes (c) No (vertical line test fails) (d) No (vertical line test fails)
5. (a) 1999, $47,700 (b) 1993, $41,600
(c) The slope between 2000 and 2001 is steeper than the slope between 2001 and 2002, so the median income was
declining more rapidly during the first year of the 2-year period.
2 2 2 2
√
√ f2(0) = 3(0) − 2 = −2; f2 (2) = 3(2)2 − 2 = 10; f (−2) = 3(−2) − 2 = 10; f (3) = 3(3) − 2 = 25; f ( 2) =
7. (a)
3( 2) − 2 = 4; f (3t) = 3(3t) − 2 = 27t − 2.
√ √
(b) f (0) = 2(0) = 0; f (2) = 2(2) = 4; f (−2) = 2(−2) = −4; f (3) = 2(3) = 6; f ( 2) = 2 2; f (3t) = 1/(3t) for
t > 1 and f (3t) = 6t for t ≤ 1.
9. (a) Natural domain: x = 3. Range: y = 0. (b) Natural domain: x = 0. Range: {1, −1}.
√ √
(c) Natural domain: x ≤ − 3 or x ≥ 3. Range: y ≥ 0.
√
(d) x2 − 2x + 5 = (x − 1)2 + 4 ≥ 4. So G(x) is defined for all x, and is ≥ 4 = 2. Natural domain: all x. Range:
y ≥ 2.
(e) Natural domain: sin x = 1, so x = (2n+ 12 )π, n = 0, ±1, ±2, . . .. For such x, −1 ≤ sin x < 1, so 0 < 1−sin x ≤ 2,
1 1 1
and 1−sin x ≥ 2 . Range: y ≥ 2 .
2
−4
(f ) Division by 0 occurs for x = 2. For all other x, xx−2 = x + 2, which is nonnegative for x ≥ −2. Natural
√ √
domain: [−2, 2) ∪ (2, +∞). The range of x + 2 is [0, +∞). But we must exclude x = 2, for which x + 2 = 2.
Range: [0, 2) ∪ (2, +∞).
11. (a) The curve is broken whenever someone is born or someone dies.
(b) C decreases for eight hours, increases rapidly (but continuously), and then repeats.
h
t
13.
1
,2 Chapter 0
√
15. Yes. y = 25 − x2 .
√
2
17. Yes. y = √25 − x , −5 ≤ x ≤ 0
− 25 − x2 , 0<x≤5
19. False. E.g. the graph of x2 − 1 crosses the x-axis at x = 1 and x = −1.
21. False. The range also includes 0.
23. (a) x = 2, 4 (b) None (c) x ≤ 2; 4 ≤ x (d) ymin = −1; no maximum value.
25. The cosine of θ is (L − h)/L (side adjacent over hypotenuse), so h = L(1 − cos θ).
27. (a) If x< 0, then |x| = −x so f (x) = −x + 3x + 1 = 2x + 1. If x ≥ 0, then |x| = x so f (x) = x + 3x + 1 = 4x + 1;
2x + 1, x < 0
f (x) =
4x + 1, x ≥ 0
(b) If x < 0, then |x| = −x and |x − 1| = 1 − x so g(x) = −x + (1 − x) = 1 − 2x. If 0 ≤ x < 1, then |x| = x and
|x − 1| = 1 − x so g(x) = x + (1 − x) = 1. If x ≥ 1, then |x| = x and |x − 1| = x − 1 so g(x) = x + (x − 1) = 2x − 1;
⎧
⎨ 1 − 2x, x<0
g(x) = 1, 0≤x<1
⎩
2x − 1, x≥1
29. (a) V = (8 − 2x)(15 − 2x)x (b) 0 < x < 4
100
0 4
(c) 0 0 < V ≤ 91, approximately
(d) As x increases, V increases and then decreases; the maximum value occurs when x is about 1.7.
31. (a) The side adjacent to the building has length x, so L = x + 2y. (b) A = xy = 1000, so L = x + 2000/x.
120
20 80
(c) 0 < x ≤ 100 (d) 80 x ≈ 44.72 ft, y ≈ 22.36 ft
500 500 10
33. (a) V = 500 = πr2 h, so h = 2
. Then C = (0.02)(2)πr2 + (0.01)2πrh = 0.04πr2 + 0.02πr 2 = 0.04πr2 + ;
πr πr r
Cmin ≈ 4.39 cents at r ≈ 3.4 cm, h ≈ 13.7 cm.
10
(b) C = (0.02)(2)(2r)2 + (0.01)2πrh = 0.16r2 + . Since 0.04π < 0.16, the top and bottom now get more weight.
r
Since they cost more, we diminish their sizes in the solution, and the cans become taller.
(c) r ≈ 3.1 cm, h ≈ 16.0 cm, C ≈ 4.76 cents.
, Exercise Set 0.2 3
35. (i) x = 1, −2 causes division by zero. (ii) g(x) = x + 1, all x.
37. (a) 25◦ F (b) 13◦ F (c) 5◦ F
39. If v = 48 then −60 = WCT ≈ 1.4157T − 30.6763; thus T ≈ 15◦ F when WCT = −10.
Exercise Set 0.2
y y
1 2
x
1
–1 0 1 2
x
1. (a) –1 (b) 1 2 3
y y
1
2
x x
(c) –1 1 2 (d) –4 –2 2
y
y 1
1
x
x
1
–2 –1 1 2
–1 –1
3. (a) (b)
1 y y
1
x x
–1 1 2 3 –1 1 2 3
–1 –1
(c) (d)
5. Translate left 1 unit, stretch vertically by a factor of 2, reflect over x-axis, translate down 3 units.
y x
–6 –2 2 6
–20
–60
7. y = (x + 3)2 − 9; translate left 3 units and down 9 units.
