Student solutions manual [for] Calculus of a single variable by Ron Larson, the Pennsylvania State University, the Behrend College ; Bruce Edwards, University of Florida

574 Pages·2014·71.322 MB·English

by  Edwards, Bruce H.;Larson, Ron

Student Solutions Manual

Calculus of a Single Variable

TENTH EDITION

Ron Larson
The Pennsylvania State University,
The Behrend College

Bruce Edwards
University of Florida

Prepared by

Bruce Edwards
University of Florida

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial
review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to
remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous
editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by
ISBN#, author, title, or keyword for materials in your areas of interest.
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
© 2014 Brooks/Cole, Cengage Learning ISBN-13: 978-1-285-08571-5
ISBN-10: 1-285-08571-X
ALL RIGHTS RESERVED. No part of this work covered by the
copyright herein may be reproduced, transmitted, stored, or Brooks/Cole
used in any form or by any means graphic, electronic, or 20 Channel Center Street
mechanical, including but not limited to photocopying, Boston, MA 02210
recording, scanning, digitizing, taping, Web distribution, USA
information networks, or information storage and retrieval
systems, except as permitted under Section 107 or 108 of the Cengage Learning is a leading provider of customized
1976 United States Copyright Act, without the prior written learning solutions with office locations around the globe,
permission of the publisher. including Singapore, the United Kingdom, Australia,
Mexico, Brazil, and Japan. Locate your local office at:
www.cengage.com/global

For product information and technology assistance, contact us at Cengage Learning products are represented in
Cengage Learning Customer & Sales Support, Canada by Nelson Education, Ltd.
1-800-354-9706
To learn more about Brooks/Cole, visit
For permission to use material from this text or product, submit www.cengage.com/brookscole
all requests online at www.cengage.com/permissions
Further permissions questions can be emailed to Purchase any of our products at your local college
[email protected]
store or at our preferred online store

www.cengagebrain.com

Printed in the United States of America
1 2 3 4 5 6 7 17 16 15 14 13

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
C O N T E N T S

Chapter P Preparation for Calculus......................................................................................................1

Chapter 1 Limits and Their Properties...............................................................................................27

Chapter 2 Differentiation....................................................................................................................57

Chapter 3 Applications of Differentiation........................................................................................110

Chapter 4 Integration........................................................................................................................187

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions....................................231

Chapter 6 Differential Equations......................................................................................................290

Chapter 7 Applications of Integration..............................................................................................321

Chapter 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals.................................371

Chapter 9 Infinite Series...................................................................................................................449

Chapter 10 Conics, Parametric Equations, and Polar Coordinates.....................................................514

iii
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
C H A P T E R P
Preparation for Calculus
Section P.1 Graphs and Models.................................................................................2
Section P.2 Linear Models and Rates of Change......................................................6
Section P.3 Functions and Their Graphs.................................................................12
Section P.4 Fitting Models to Data..........................................................................18
Review Exercises..........................................................................................................20
Problem Solving...........................................................................................................23

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
C H A P T E R P
Preparation for Calculus
Section P.1 Graphs and Models
1. y = −3x + 3 9. y = x + 2
2
x-intercept: (2, 0)
x −5 −4 −3 −2 −1 0 1
y-intercept: (0, 3)
y 3 2 1 0 1 2 3
Matches graph (b).
2. y = 9 − x2 y
6
x-intercepts: (−3, 0), (3, 0)
4
(−5, 3)
y-intercept: (0, 3) (1, 3)
(−4, 2) 2 (0, 2)
Matches graph (d).
(−3, 1) (−1, 1)
x
−6 −4 (−2, 0) 2
3. y = 3 − x2 −2
( ) ( )
x-intercepts: 3, 0, − 3, 0
11. y = x − 6
y-intercept: (0, 3)

x 0 1 4 9 16
Matches graph (a).
y −6 −5 −4 −3 −2
4. y = x3 − x
x-intercepts: (0, 0), (−1, 0), (1, 0) y
2
y-intercept: (0, 0)
x
Matches graph (c). −4 4 8 12 16
−2 (9, −3)
(16, −2)
5. y = 1x + 2 −4 (4, −4)
2 (1, −5)
−6 (0, −6)

x −4 −2 0 2 4 −8
y 0 1 2 3 4 3
13. y =
x
y

6 x −3 −2 −1 0 1 2 3
(4, 4)
4 (2, 3)
y −1 −3 −3 Undef. 3 3 1
(0, 2) 2 2
(−2, 1)
x
−4 −2 2 4 y
(−4, 0)
−2 (1, 3)
3
2 (2, 32(
7. y = 4 − x2 (3, 1)
(−3, −1) 1
x
x −3 −2 0 2 3 −3 −2 −1 1 2 3
−1
y −5 0 4 0 −5 −2 (−2, − 32(
(−1, −3)
y
6
(0, 4)
2
(−2, 0) (2, 0)
x
−6 −4 4 6
−2
(−3, −5) −4 (3, −5)
−6
2
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
Section P.1 Graphs and Models 3
15. y = 5 − x 27. Symmetric with respect to the y-axis because
y = (−x)2 − 6 = x2 − 6.
5
(−4.00, 3)
(2, 1.73) 29. Symmetric with respect to the x-axis because
−6 6 (−y)2 = y2 = x3 − 8x.
−3 31. Symmetric with respect to the origin because
(a) (2, y) = (2,1.73) (y = 5 − 2 = 3 ≈ 1.73) (−x)(−y) = xy = 4.
(b) (x, 3) = (−4, 3) (3 = 5 − (−4)) 33. y = 4 − x + 3
No symmetry with respect to either axis or the origin.
17. y = 2x − 5
35. Symmetric with respect to the origin because
y-intercept: y = 2(0) − 5 = −5; (0, −5)
−x
−y =
x-intercept: 0 = 2x − 5 (−x)2 +1
5 = 2x x
y = .
x = 5; (5, 0) x2 +1
2 2
37. y = x3 + x is symmetric with respect to the y-axis
19. y = x2 + x − 2
y-intercept: y = 02 + 0 − 2 because y = (−x)3 + (−x) = −(x3 + x) = x3 + x.
y = −2; (0, −2)
39. y = 2 − 3x
x-intercepts: 0 = x2 + x − 2 y = 2 − 3(0) = 2, y-intercept
0 = (x + 2)(x −1)
0 = 2 − 3(x) ⇒ 3x = 2 ⇒ x = 2, x-intercept
x = −2,1;(−2, 0), (1, 0) 3
Intercepts: (0, 2), (2, 0) y
3
21. y = x 16 − x2
Symmetry: none
y-intercept: y = 0 16 − 02 = 0; (0, 0) 2 (0, 2)
1
x-intercepts: 0 = x 16 − x2 (2, 0(
3
x
0 = x (4 − x)(4 + x) −1 2 3
−1
x = 0, 4, −4;(0, 0), (4, 0), (−4, 0)
41. y = 9 − x2
23. y = 2 − x y = 9 − (0)2 = 9, y-intercept
5x +1
0 = 9 − x2 ⇒ x2 = 9 ⇒ x = ±3, x-intercepts
2 − 0
y-intercept: y = 5(0) +1 = 2 ; (0,2) Intercepts: (0, 9), (3, 0), (−3, 0) y
10 (0, 9)
2 − x y = 9 − (−x)2 = 9 − x2
x-intercept: 0 =
5x +1 6
Symmetry: y-axis
4
0 = 2 − x
2
(−3, 0) (3, 0)
x = 4 ; (4,0) x
−6 −4 −2 2 4 6
−2
25. x2y − x2 + 4y = 0
y-intercept: 02(y) − 02 + 4y = 0
y = 0; (0, 0)
x-intercept: x2(0) − x2 + 4(0) = 0
x = 0; (0, 0)
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.
4 Chapter P Preparation for Calculus
43. y = x3 + 2 51. y = 6 − x
y = 03 + 2 = 2, y-intercept y = 6 − 0 = 6, y-intercept

0 = x3 + 2 ⇒ x3 = −2 ⇒ x = −3 2, x-intercept 6 − x = 0
Intercepts: (−3 2, 0), (0, 2) 6 = x
y
x = ±6, x-intercepts
Symmetry: none 5
4 Intercepts: (0, 6), (−6, 0), (6, 0)
3
(0, 2) y = 6 − −x = 6 − x
(− 3 2, 0) 1
x Symmetry: y-axis
−3 −2 1 2 3
−1 y
8
6 (0, 6)
45. y = x x + 5
4
y = 0 0 + 5 = 0, y-intercept (−6, 0) 2 (6, 0)
x
−8 −4−2 2 4 6 8
−2
x x + 5 = 0 ⇒ x = 0,−5, x-intercepts
−4
Intercepts: (0, 0), (−5, 0) y −6
−8
3
Symmetry: none
2 53. y2 − x = 9
(−5, 0) (0, 0)
x y2 = x + 9
−4 −3 −2 −1 1 2
y = ± x + 9
−3
−4 y = ± 0 + 9 = ± 9 = ±3, y-intercepts
± x + 9 = 0

47. x = y3 x + 9 = 0
x = −9, x-intercept
y3 = 0 ⇒ y = 0, y-intercept

x = 0,x-intercept Intercepts: (0, 3), (0, −3), (−9, 0)
y
Intercept: (0, 0) 4 (−y)2 − x = 9 ⇒ y2 − x = 9
3
−x = (−y)3 ⇒ −x = −y3 2
Symmetry: x-axis
(0, 0)
Symmetry: origin −4−3−2−1 1 2 3 4 x y
−2 6
−3 4
−4 (−9, 0) 2 (0, 3)
x
8 −10 −6 −4 −2 2
49. y = −2 (0, −3)
x
−4
8 −6
y = ⇒ Undefined ⇒ no y-intercept
0

8
= 0 ⇒ Nosolution ⇒ no x-intercept
x
Intercepts: none y
8
8 8
−y = ⇒ y = 6
−x x 4
2
Symmetry: origin x
−2 2 4 6 8

Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.