INSTRUCTOR S OLUTIONS MANUAL SINGLE VARIABLE

[Pages:134]Single Variable Calculus Early Transcendentals 2nd Edition Briggs Solutions Manual Full Download:

INSTRUCTOR'S SOLUTIONS MANUAL

SINGLE VARIABLE

MARK WOODARD

Furman University

CALCULUS

EARLY TRANSCENDENTALS

SECOND EDITION

William Briggs

University of Colorado at Denver

Lyle Cochran

Whitworth University

Bernard Gillett

University of Colorado at Boulder

with the assistance of

Eric Schulz

Walla Walla Community College

Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto

Delhi Mexico City S?o Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

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The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs.

Reproduced by Pearson from electronic files supplied by the author.

Copyright ? 2015, 2011 Pearson Education, Inc. Publishing as Pearson, 75 Arlington Street, Boston, MA 02116.

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, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.

ISBN-13: 978-0-321-95422-0 ISBN-10: 0-321-95422-X

1 2 3 4 5 6 ?? 17 16 15 14 13



Contents

1 Functions

5

1.1 Review of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Representing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Inverse, Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.4 Trigonometric Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter One Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 Limits

67

2.1 The Idea of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.2 Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3 Techniques of Computing Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.4 Infinite Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

2.5 Limits at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.7 Precise Definitions of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter Two Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3 Derivatives

135

3.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.2 Working with Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

3.3 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.4 The Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.5 Derivatives of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.6 Derivatives as Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

3.7 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

3.8 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

3.9 Derivatives of Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 220

3.10 Derivatives of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 229

3.11 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Chapter Three Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

4 Applications of the Derivative

255

4.1 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

4.2 What Derivatives Tell Us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

4.3 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

4.4 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

4.5 Linear Approximation and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

4.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

4.7 L'Ho^pital's Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

4.8 Newton's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

4.9 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Chapter Four Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396

1

2

Contents

5 Integration

411

5.1 Approximating Areas under Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

5.2 Definite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

5.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446

5.4 Working with Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

Chapter Five Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

6 Applications of Integration

495

6.1 Velocity and Net Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

6.2 Regions Between Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

6.3 Volume by Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

6.4 Volume by Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

6.5 Length of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

6.6 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

6.7 Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

6.8 Logarithmic and Exponential Functions Revisited . . . . . . . . . . . . . . . . . . . . . . . . 562

6.9 Exponential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

6.10 Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Chapter Six Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583

7 Integration Techniques

597

7.1 Basic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

7.2 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

7.3 Trigonometric Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

7.4 Trigonometric Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

7.5 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

7.6 Other Integration Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

7.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669

7.8 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677

7.9 Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

Chapter Seven Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

8 Sequences and Infinite Series

715

8.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

8.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722

8.3 Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735

8.4 The Divergence and Integral Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746

8.5 The Ratio, Root, and Comparison Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755

8.6 Alternating Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Chapter Eight Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767

9 Power Series

775

9.1 Approximating Functions With Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 775

9.2 Properties of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794

9.3 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

9.4 Working with Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812

Chapter Nine Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823

10 Parametric and Polar Curves

831

10.1 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

10.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851

10.3 Calculus in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 871

10.4 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883

Chapter Ten Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903

Copyright c 2015 Pearson Education, Inc.

Contents

3

Appendix A

921

Copyright c 2015 Pearson Education, Inc.

4

Contents

Copyright c 2015 Pearson Education, Inc.

Chapter 1

Functions

1.1 Review of Functions

1.1.1 A function is a rule which assigns each domain element to a unique range element. The independent variable is associated with the domain, while the dependent variable is associated with the range.

1.1.2 The independent variable belongs to the domain, while the dependent variable belongs to the range.

1.1.3 The vertical line test is used to determine whether a given graph represents a function. (Specifically, it tests whether the variable associated with the vertical axis is a function of the variable associated with the horizontal axis.) If every vertical line which intersects the graph does so in exactly one point, then the given graph represents a function. If any vertical line x = a intersects the curve in more than one point, then there is more than one range value for the domain value x = a, so the given curve does not represent a function.

1.1.4

f (2) =

1 23 +1

=

1 9

.

f (y2) =

1 (y 2 )3 +1

=

1 y 6 +1

.

1.1.5 Item i. is true while item ii. isn't necessarily true. In the definition of function, item i. is stipulated.

However, item ii. need not be true ? for example, the function f (x) = x2 has two different domain values

associated with the one range value 4, because f (2) = f (-2) = 4.

1.1.6 (g f

(f g)(x) = f )(x) = g(f (x))

(g(x)) = = g( x)

f (x3 - 2) = x3/2 -

= 2.

x3

-

2

(f f )(x) = f (f (x)) = f ( x) = x = 4 x.

(g g)(x) = g(g(x)) = g(x3 - 2) = (x3 - 2)3 - 2 = x9 - 6x6 + 12x3 - 10

1.1.7 f (g(2)) = f (-2) = f (2) = 2. The fact that f (-2) = f (2) follows from the fact that f is an even function.

g(f (-2)) = g(f (2)) = g(2) = -2.

1.1.8 The domain of f g is the subset of the domain of g whose range is in the domain of f . Thus, we need to look for elements x in the domain of g so that g(x) is in the domain of f .

1.1.9

When f is an even function, we have f (-x) = f (x) for all x in the domain of f , which ensures that the graph of the function is symmetric about the y-axis.

y 6

5

4

3

2

1

2

1

x

1

2

5

6

1.1.10

When f is an odd function, we have f (-x) = -f (x) for all x in the domain of f , which ensures that the graph of the function is symmetric about the origin.

Chapter 1. Functions

y

5

2

1

5

x

1

2

1.1.11 Graph A does not represent a function, while graph B does. Note that graph A fails the vertical line test, while graph B passes it.

1.1.12 Graph A does not represent a function, while graph B does. Note that graph A fails the vertical line test, while graph B passes it.

1.1.13 The domain of this function is the set of a real numbers. The range is [-10, ).

f 15

10

5

2

1

5

x

1

2

10

1.1.14 The domain of this function is (-, -2)(-2, 3) (3, ). The range is the set of all real numbers.

g 3

2

1

4

2

1

y

2

4

6

2

3

The domain of this function is [-2, 2]. The range 1.1.15 is [0, 2].

f 4

2

4

2

2

x

2

4

4

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